Null-controllability of cascade reaction-diffusion systems with odd coupling terms

Kévin Le Balc’h Sorbonne Université, CNRS, Inria, Laboratoire Jacques-Louis Lions, Paris, France Takéo Takahashi Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France

Abstract

In this paper, we consider a nonlinear system of two parabolic equations, with a distributed control in the first equation and an odd coupling term in the second one. We prove that the nonlinear system is small-time locally null-controllable. The main difficulty is that the linearized system is not null-controllable. To overcome this obstacle, we extend in a nonlinear setting the strategy introduced in [LB19] that consists in constructing odd controls for the linear heat equation. The proof relies on three main steps. First, we obtain from the classical $\displaystyle L^{2}$ parabolic Carleman estimate, conjugated with maximal regularity results, a weighted $\displaystyle L^{p}$ observability inequality for the nonhomogeneous heat equation. Secondly, we perform a duality argument, close to the well-known Hilbert Uniqueness Method in a reflexive Banach setting, to prove that the heat equation perturbed by a source term is null-controllable thanks to odd controls. Finally, the nonlinearity is handled with a Schauder fixed-point argument.

Keywords: Null-controllability, parabolic system, nonlinear coupling, Carleman estimate

2020 Mathematics Subject Classification. 35K45, 35K58, 93B05, 93C10

1 Introduction

Let $\displaystyle T>0$ be a positive time, $\displaystyle d\in\mathbb{N}^{*}$ , $\displaystyle\Omega$ be a bounded, connected, open subset of $\displaystyle\mathbb{R}^{d}$ of class $\displaystyle C^{2}$ corresponding to the spatial domain and $\displaystyle\omega$ be a nonempty open subset such that $\displaystyle\overline{\omega}\subset\Omega$ . In what follows, we use the notation $\displaystyle 1_{\omega}$ for the characteristic function of $\displaystyle\omega$ .

The null-controllability of the heat equation described below was first obtained by Fattorini and Russell [FR71] for $\displaystyle d=1$ and by Lebeau, Robbiano [LR95] and Fursikov, Imanuvilov [FI96] for $\displaystyle d\geqslant 1$ . More precisely for any $\displaystyle y_{0}\in L^{2}(\Omega)$ , there exists $\displaystyle h\in L^{2}((0,T)\times\omega)$ such that the solution $\displaystyle y$ of the system

\left\{\begin{array}[]{ll}\partial_{t}y-\Delta y=h1_{\omega}&\mathrm{in}\ (0,T)\times\Omega,\\ y=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ y(0,\cdot)=y_{0}&\mathrm{in}\ \Omega,\end{array}\right.

(1.1)

satisfies $\displaystyle y(T,\cdot)=0$ . These results were then extended to a large number of other parabolic systems, linear or nonlinear. For instance, the null-controllability of linear coupled parabolic systems has been a challenging issue for the control community in the last two decades. In that direction, we can quote, among the large literature devoted to this problem, [AKBDGB09], where Ammar-Khodja, Benabadallah, Dupaix, Gonzalez-Burgos exhibit sharp conditions for the null-controllability of systems of the form

\left\{\begin{array}[]{ll}\partial_{t}Y-D\Delta Y=AY+Bh1_{\omega}&\mathrm{in}\ (0,T)\times\Omega,\\ Y=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ Y(0,\cdot)=Y_{0}&\mathrm{in}\ \Omega.\end{array}\right.

(1.2)

Here, at time $\displaystyle t\in(0,T]$ , $\displaystyle Y(t,.):\Omega\rightarrow\mathbb{R}^{n}$ is the state, $\displaystyle h=h(t,.):\Omega\rightarrow\mathbb{R}^{m}$ is the control, $\displaystyle D:=\operatorname{diag}(d_{1},\dots,d_{n})$ with $\displaystyle d_{i}\in(0,+\infty)$ is the diffusion matrix, $\displaystyle A\in\mathbb{R}^{n\times n}$ is the coupling matrix and $\displaystyle B\in\mathbb{R}^{n\times m}$ represents the distribution of controls. One objective is to reduce the number of controls $\displaystyle m$ (and in particular to have $\displaystyle m<n$ ) by using the coupling matrices $\displaystyle A$ and $\displaystyle B$ . Let us also quote the survey [AKBGBdT11] for other results and open problems in that direction.

In this article, we consider the following controlled semi-linear reaction-diffusion system

\left\{\begin{array}[]{ll}\partial_{t}y_{1}-d_{1}\Delta y_{1}=a_{11}y_{1}^{N_{1}}+h1_{\omega}&\mathrm{in}\ (0,T)\times\Omega,\\ \partial_{t}y_{2}-d_{2}\Delta y_{2}=a_{21}y_{1}^{N_{2}}+a_{22}y_{2}^{N_{3}}&\mathrm{in}\ (0,T)\times\Omega,\\ y_{1}=y_{2}=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ y_{1}(0,\cdot)=y_{1,0},\quad y_{2}(0,\cdot)=y_{2,0}&\mathrm{in}\ \Omega,\end{array}\right.

(1.3)

where $\displaystyle d_{1},d_{2}\in(0,+\infty)$ , $\displaystyle N_{1},N_{2},N_{3}\in\mathbb{N}^{*}$ and $\displaystyle a_{ij}\in\mathbb{R}$ . In (1.3), at time $\displaystyle t\in[0,T]$ , $\displaystyle(y_{1},y_{2})(t,\cdot):\Omega\to\mathbb{R}^{2}$ is the state while $\displaystyle h(t,\cdot):\omega\to\mathbb{R}$ is the control. We are interested in the null-controllability of (1.3), that is find a control $\displaystyle h=h(t,x)$ , supported in $\displaystyle(0,T)\times\omega$ , that steers the state $\displaystyle(y_{1},y_{2})$ to zero at time $\displaystyle T$ , i.e. $\displaystyle(y_{1},y_{2})(T,\cdot)=0$ . Note that (1.3) is a so-called “cascade system” because the first equation is decoupled from the second equation. For such a system, the basic idea is to use the nonlinear coupling term $\displaystyle a_{21}y_{1}^{N_{2}}$ , as an indirect control term, that acts on the second component $\displaystyle y_{2}$ .

1.1 Main results

Our control results on (1.3) are written in the framework of weak solutions. More precisely, we define the Banach space

\mathcal{W}:=L^{2}(0,T;H_{0}^{1}(\Omega))\cap H^{1}(0,T;H^{-1}(\Omega))\cap L^{\infty}((0,T)\times\Omega),

(1.4)

and we consider solutions of (1.3) such that $\displaystyle y_{1},y_{2}\in\mathcal{W}$ . The precise definition of the weak solutions of (1.3) is given in 2.7 and a corresponding well-posedness result is stated in 2.8 for controls $\displaystyle h\in L^{p}((0,T)\times\omega)$ with

p\in\left(\frac{d+2}{2},\infty\right],\quad\text{and}\quad p\geqslant 2\ \text{if}\ d=1.

(1.5)

Our first main result can be stated as follows.

Theorem 1.1.

Let $\displaystyle p$ satisfies (1.5). Assume

a_{2,1}\neq 0,\quad N_{2}\ \text{is odd}.

(1.6)

Then there exists $\displaystyle\delta>0$ such that for any initial data satisfying

\left\|y_{1,0}\right\|_{L^{\infty}(\Omega)}+\left\|y_{2,0}\right\|_{L^{\infty}(\Omega)}\leqslant\delta,

(1.7)

there exists a control $\displaystyle h\in L^{p}((0,T)\times\omega)$ satisfying

\left\|h\right\|_{L^{p}((0,T)\times\omega)}\lesssim\delta,

(1.8)

such that the solution $\displaystyle(y_{1},y_{2})\in\mathcal{W}\times\mathcal{W}$ of (1.3) satisfies

(y_{1},y_{2})(T,\cdot)=0.

(1.9)

Here and in all that follows, we use the notation $\displaystyle X\lesssim Y$ if there exists a constant $\displaystyle C>0$ such that we have the inequality $\displaystyle X\leqslant CY$ . In the whole paper, we use $\displaystyle C$ as a generic positive constant that does not depend on the other terms of the inequality. The value of the constant $\displaystyle C$ may change from one appearance to another. Our constants may depend on the geometry ( $\displaystyle\Omega$ , $\displaystyle\omega$ ), on the time $\displaystyle T$ and on the dimension $\displaystyle d$ . If we want to emphasize the dependence on a quantity $\displaystyle k$ , we write $\displaystyle X\lesssim_{k}Y$ .

As we will see, the smallness conditions on the initial data i.e. (1.7) and on the control i.e. (1.8) are sufficient conditions to guarantee the well-posedness of the system (1.3), see 2.8 below.

Before continuing, let us make some comments related to 1.1.

•

The sufficient condition (1.6) ensuring the local null-controllability of (1.3) is actually necessary. Indeed, if $\displaystyle a_{21}=0$ then the second equation of (1.3) is decoupled from the first equation so $\displaystyle y_{2}$ cannot be driven to $\displaystyle 0$ at time $\displaystyle T$ . Moreover, if $\displaystyle N_{2}$ is even, the strong maximum principle shows that we can not control $\displaystyle y_{2}$ : assume for instance that $\displaystyle a_{21}\geqslant 0$ , then

$\partial_{t}y_{2}-d_{2}\Delta y_{2}-a_{22}y_{2}^{N_{3}}=a_{21}y_{1}^{N_{2}}\geqslant 0\ \text{in}\ (0,T)\times\Omega$ (1.10)

and thus $\displaystyle\widetilde{y}_{2}(t,x):=y_{2}(t,x)e^{-\lambda t}$ , with $\displaystyle\lambda\geqslant|a_{22}|\left\|y_{2}\right\|_{L^{\infty}((0,T)\times\Omega)}^{N_{3}-1}$ satisfies

$\displaystyle\partial_{t}\widetilde{y}_{2}-d_{2}\Delta\widetilde{y}_{2}+c\widetilde{y}_{2}\geqslant 0\ \text{in}\ (0,T)\times\Omega$

with $\displaystyle c\geqslant 0$ and we can apply the standard strong maximum principle (see, for instance, [Eva10, Theorem 12, p.397]): if $\displaystyle y_{2,0}\geqslant 0$ and $\displaystyle y_{2,0}\neq 0$ then for all $\displaystyle t\in(0,T]$ , $\displaystyle y_{2}(t,\cdot)>0$ in $\displaystyle\Omega$ .
•

The linear case

$\displaystyle N_{1}=N_{2}=N_{3}=1,$

is already treated in [dT00] by de Teresa. To obtain such a result, the author shows a Carleman estimate and deduce from it an observability inequality for the adjoint system.
•

For the semi-linear case, the main idea is to linearize the system in order to use the previous result. However, if $\displaystyle N_{2}\geqslant 2$ , in the linearized system around the trajectory $\displaystyle((\overline{y_{1}},\overline{y_{2}}),\overline{h})=((0,0),0)$ , we can see that the second equation is decoupled from the first one and thus can not be controlled; the linearized system is thus not null-controllable.
•

To overcome this difficulty, Coron, Guerrero, Rosier [CGR10] use the return method in the case

$\displaystyle N_{2}=3,\quad N_{3}=1.$

More precisely, they construct a reference trajectory $\displaystyle((\overline{y_{1}},\overline{y_{2}}),\overline{h})$ of (1.3) starting from $\displaystyle(\overline{y_{1}},\overline{y_{2}})(0,\cdot)=0$ , reaching $\displaystyle(\overline{y_{1}},\overline{y_{2}})(T,\cdot)=0$ and satisfying $\displaystyle|\overline{y_{1}}|\geqslant\varepsilon>0$ in $\displaystyle(t_{1},t_{2})\times\omega$ . Then they linearize (1.3) around the reference trajectory and obtain for the second equation

$\partial_{t}y_{2}-d_{2}\Delta y_{2}=3a_{21}\overline{y_{1}}^{2}y_{1}+a_{22}y_{2}\ \text{in}\ (0,T)\times\Omega.$ (1.11)

They can then use [dT00] to obtain that the null-controllability of the linearized system and then the local null-controllability of the nonlinear system (1.3) by a fixed-point argument.
•

In [LB19], the first author employs a new direct strategy in order to deal with the case

$\displaystyle N_{2}\ \text{odd and}\ N_{3}=1,$

that we adapt here to prove 1.1 for the more general case

$\displaystyle N_{2}\ \text{odd and}\ N_{3}\geqslant 1.$

We describe below the idea of the proof.

Strategy of the proof. We proceed in two steps: in the first step, we control the first equation of (1.3) in the time interval $\displaystyle(0,T/2)$ . Using the small-time local null-controllability of the semi-linear heat equation, there exists a control $\displaystyle h$ such that $\displaystyle y_{1}(T/2,\cdot)=0$ . Using the smallness assumptions, we can ensure that the second equation of (1.3) admits a solution on $\displaystyle(0,T/2)$ . In the second step, we control this second equation thanks to a fictitious odd control $\displaystyle H$ . More precisely, we can consider the control problem

\left\{\begin{array}[]{rl}\partial_{t}y_{2}-\Delta y_{2}=H\chi_{\omega}+y_{2}^{N_{3}}&\mathrm{in}\ (T/2,T)\times\Omega,\\ y_{2}=0&\mathrm{on}\ (T/2,T)\times\partial\Omega,\end{array}\right.

(1.12)

where $\displaystyle\chi_{\omega}=\widetilde{\chi}_{\omega}^{N_{2}}$ and where $\displaystyle\widetilde{\chi}_{\omega}\in C^{\infty}(\Omega)$ has a compact support in $\displaystyle\omega$ , $\displaystyle\widetilde{\chi}_{\omega}\not\equiv 0$ . We then need a control $\displaystyle H$ such that $\displaystyle y_{2}(T,\cdot)=0$ , satisfying $\displaystyle H(T/2,\cdot)=H(T,\cdot)=0$ and such that $\displaystyle H^{1/N_{2}}$ is regular. Such a control is given by our second main result (1.2) stated below. We can then set in $\displaystyle(T/2,T)$

\displaystyle y_{1}:=\left(H\chi_{\omega}\right)^{1/N_{2}},\quad h:=\partial_{t}y_{1}-c_{1}\Delta y_{1}-y_{1}^{N_{1}}.

By construction, $\displaystyle((y_{1},y_{2}),h)$ is a trajectory of (1.3) satisfying (1.9).

To simplify the work and without loss of generality, we assume in what follows that

\displaystyle d_{1}=d_{2}=1,\quad a_{11}=a_{21}=a_{22}=1,\quad N_{1},N_{2},N_{3}\geqslant 2.

The proof of 1.1 crucially relies on the construction of odd controls for the semi-linear heat equation that we present now. For $\displaystyle N\geqslant 2$ , we thus consider the system

\left\{\begin{array}[]{ll}\partial_{t}y-\Delta y=h\chi_{\omega}+y^{N}&\mathrm{in}\ (0,T)\times\Omega,\\ y=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ y(0,\cdot)=y_{0}&\mathrm{in}\ \Omega.\end{array}\right.

(1.13)

The definition of the weak solutions for the above system and a corresponding well-posedness result are given in 2.4 and 2.5. Our second main result states as follows.

Theorem 1.2.

Assume that $\displaystyle N\geqslant 2$ , $\displaystyle n\in\mathbb{N}$ , and $\displaystyle p\geqslant 1$ . There exists $\displaystyle\delta>0$ such that for every initial data $\displaystyle y_{0}\in L^{\infty}(\Omega)$ such that

\left\|y_{0}\right\|_{L^{\infty}(\Omega)}\leqslant\delta,

(1.14)

there exists a control $\displaystyle h\in L^{\infty}((0,T)\times\omega)$ satisfying

\left\|h\right\|_{L^{\infty}((0,T)\times\omega)}\lesssim\left\|y_{0}\right\|_{L^{\infty}(\Omega)},

(1.15)

h^{1/(2n+1)}\in L^{p}(0,T;W^{2,p}(\Omega))\cap W^{1,p}(0,T;L^{p}(\Omega)),\quad h^{1/(2n+1)}(0,\cdot)=h^{1/(2n+1)}(T,\cdot)=0,

(1.16)

and such that the solution $\displaystyle y\in\mathcal{W}$ of (1.13) satisfies

\left\|y\right\|_{\mathcal{W}}\lesssim\left\|y_{0}\right\|_{L^{\infty}(\Omega)},

(1.17)

and

y(T,\cdot)=0.

(1.18)

As for 1.1, the smallness conditions (1.14) and (1.15) are sufficient to guarantee the well-posedness of the semi-linear heat equation (1.13), see 2.5 below.

Before continuing, let us make some comments related to 1.2.

•

The crucial property in 1.2 is the odd behavior of the control, stated in (1.16). Actually, the small-time local null-controllability of (1.13) with controls in $\displaystyle L^{\infty}((0,T)\times\omega)$ is a consequence of [AT02, Lemma 6].
•

For $\displaystyle N=1$ , that is the linear case, the result of 1.2 is still true and has already been established by the first author, see [LB19, Proposition 3.7]. One can even obtain a (small-time) global null-controllability result with odd controls due to the linear setting. Note that here, we extend the result of [LB19] in the case of a linear heat equation with a source term, see Section 3.3.

Strategy of the proof. First, we use a classical Carleman estimate for the nonhomogeneous heat equation to obtain a weighted $\displaystyle L^{2}$ observability inequality stated in 3.3. From this result and after that, we need to take care about the weights appearing in the norm of the adjoint system they have to be “comparable”. We then deduce from this result a weighted $\displaystyle L^{p}$ observability inequality, see 3.4 below with an arbitrary large $\displaystyle p$ . As a consequence, a null-controllability result is obtained for the heat equation with a source term and with odd controls. Let us remark that taking $\displaystyle p$ large enough allows us to do only one bootstrap argument for getting the desired odd behavior for the control, see 3.6 below. This is different from [LB19, Theorem 4.4 and Proposition 4.9] where two such arguments are used for obtaining the null-controllability of the heat equation with odd controls. Another bootstrap argument is then required in order to deal with the nonlinearity in the fixed-point argument, see 3.8 below. Finally, a Schauder fixed-point argument, see Section 3.5, is performed to obtain 1.2. We can remark that here due to our method for constructing the control, in this fixed point argument, the corresponding nonlinear mapping is $\displaystyle\alpha$ -Hölder continuous with $\displaystyle\alpha<1$ . In particular, a Banach fixed point argument does not seem to apply.

1.2 Outline of the paper

The outline of the paper is as follows. In Section 2, we recall some standard facts about well-posedness, regularity results for linear and nonlinear heat equations in various functional settings. We notably prove that (1.13) and (1.3) are locally well-posed, see 2.5 and 2.8 below. Section 3 and Section 4 are devoted to the proofs of the main results, i.e. 1.1 and 1.2.

2 Well-posedness and regularity results for the heat equation

In this section, we give the notion of solutions that we consider in what follows. Then we recall standard well-posedness results for both linear and semi-linear heat equations in various functional settings we will use in what follows.

2.1 Functional spaces

In this article, we use in a crucial way a $\displaystyle L^{p}$ framework with $\displaystyle p\in(1,\infty)$ . First, we introduce the standard notation for the dual exponent $\displaystyle p^{\prime}\in(1,\infty)$ of $\displaystyle p$ defined by the relation

\displaystyle\frac{1}{p}+\frac{1}{p^{\prime}}=1.

We also introduce the following functional spaces

\mathcal{X}^{p}:=L^{p}(0,T;W^{2,p}(\Omega))\cap W^{1,p}(0,T;L^{p}(\Omega)).

(2.1)

We have the following classical embedding results (see, for instance, [LSU68, Lemma 3.3, p.80]): for $\displaystyle p,q\geqslant 1$ ,

\mathcal{X}^{p}\hookrightarrow L^{q}(0,T;L^{q}(\Omega))\quad\text{if}\quad\frac{1}{q}\geqslant\frac{1}{p}-\frac{2}{d+2},\quad\mathcal{X}^{p}\hookrightarrow L^{\infty}(0,T;L^{\infty}(\Omega))\quad\text{if}\quad p>\frac{d+2}{2},

(2.2)

\mathcal{X}^{p}\hookrightarrow L^{q}(0,T;W^{1,q}(\Omega))\quad\text{if}\quad\frac{1}{q}\geqslant\frac{1}{p}-\frac{1}{d+2},\quad\mathcal{X}^{p}\hookrightarrow L^{\infty}(0,T;W^{1,\infty}(\Omega))\quad\text{if}\quad p>d+2.

(2.3)

We also have, see for instance [LSU68, Lemma 3.4, p.82],

\mathcal{X}^{p}\hookrightarrow C^{0}([0,T];W^{2/p^{\prime},p}(\Omega)),

(2.4)

where $\displaystyle W^{\alpha,p}(\Omega)$ denotes the fractional Sobolev spaces (see, for instance, [LSU68, p.70]). We recall that functions in $\displaystyle W^{\alpha,p}(\Omega)$ admit a trace on $\displaystyle\partial\Omega$ if $\displaystyle\alpha>1/p$ . If $\displaystyle\alpha>1/p$ , we denote by $\displaystyle W_{0}^{\alpha,p}(\Omega)$ the subspace of functions $\displaystyle f\in W^{\alpha,p}(\Omega)$ such that $\displaystyle f=0$ on $\displaystyle\partial\Omega$ . We also write $\displaystyle W_{0}^{\alpha,p}(\Omega):=W^{\alpha,p}(\Omega)$ if $\displaystyle\alpha\leqslant 1/p$ . From [DD12, Corollary 4.53, p.216], we have

\displaystyle W^{2/p^{\prime},p}(\Omega)\hookrightarrow L^{\infty}(\Omega)\quad\text{if}\quad p>\frac{d+2}{2},

and thus

\mathcal{X}^{p}\hookrightarrow C^{0}([0,T];L^{\infty}(\Omega))\quad\text{if}\quad p>\frac{d+2}{2}.

(2.5)

We finish with some other classical results on the spaces $\displaystyle\mathcal{X}^{p}$ , for which we give a short proof for completeness.

Lemma 2.1.

The following statements hold.

1.

If $\displaystyle p>\frac{d+2}{2}$ , then $\displaystyle\mathcal{X}^{p}$ is an algebra.
2.

For any $\displaystyle N>1$ , $\displaystyle q>1$ , if

$\frac{1}{q}\left(1-\frac{1}{N}\right)<\frac{2}{2+d}$ (2.6)

then the embedding

$\mathcal{X}^{q}{\hookrightarrow}L^{Nq}((0,T)\times\Omega)\quad\text{is compact}.$ (2.7)

Proof.

For the first point, we consider $\displaystyle f,g\in\mathcal{X}^{p}$ . Then

\displaystyle\partial_{t}f,\ \partial_{t}g,\ \nabla^{2}f,\ \nabla^{2}g\in L^{p}(0,T;L^{p}(\Omega)),

and from (2.2) and (2.3)

\displaystyle f,\ g\in L^{\infty}(0,T;L^{\infty}(\Omega)),\quad\nabla f,\ \nabla g\in L^{2p}(0,T;L^{2p}(\Omega)).

We thus deduce that

\displaystyle\partial_{t}(fg),\ \nabla^{2}(fg)\in L^{p}(0,T;L^{p}(\Omega)).

For the second point, we can use (2.6) to consider $\displaystyle p>1$ such that

\frac{1}{qN}>\frac{1}{p}>\frac{1}{q}-\frac{2}{2+d}.

(2.8)

We thus deduce from (2.2) that

\displaystyle\mathcal{X}^{q}\hookrightarrow L^{p}((0,T)\times\Omega)\hookrightarrow L^{Nq}((0,T)\times\Omega)

and from the Hölder inequality, there exists $\displaystyle\theta\in(0,1)$ such that

\left\|f\right\|_{L^{Nq}((0,T)\times\Omega)}\leqslant\left\|f\right\|_{L^{q}((0,T)\times\Omega)}^{\theta}\left\|f\right\|_{L^{p}((0,T)\times\Omega)}^{1-\theta}\quad(f\in L^{p}((0,T)\times\Omega)).

(2.9)

From the Aubin-Lions lemma (see, for instance, [Sim87, Section 8, Corollary 4]), the embedding

\displaystyle\mathcal{X}^{q}\hookrightarrow L^{q}((0,T)\times\Omega)\quad\text{is compact}.

Consequently, if $\displaystyle\left(f_{n}\right)$ is a bounded sequence of $\displaystyle\mathcal{X}^{q}$ , it has a subsequence converging in $\displaystyle L^{q}((0,T)\times\Omega)$ and bounded in $\displaystyle L^{p}((0,T)\times\Omega)$ . From (2.9), this subsequence is converging in $\displaystyle L^{qN}((0,T)\times\Omega)$ . ∎

2.2 Linear heat equation

Let us first consider the linear nonhomogenenous heat equation

\left\{\begin{array}[]{ll}\partial_{t}y-\Delta y=g&\mathrm{in}\ (0,T)\times\Omega,\\ y=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ y(0,\cdot)=y_{0}&\mathrm{in}\ \Omega.\end{array}\right.

(2.10)

In this article, we need several definitions of solutions for (2.10):

Definition 2.2.

We introduce three concepts of solutions for (2.10).

1.

If $\displaystyle y_{0}\in W_{0}^{2/p^{\prime},p}(\Omega)$ and $\displaystyle g\in L^{p}((0,T)\times\Omega)$ , we say that $\displaystyle y\in\mathcal{X}^{p}$ is a strong solution of (2.10) if it satisfies (2.10) a.e. and in the trace sense.

If $\displaystyle y_{0}\in L^{2}(\Omega)$ and $\displaystyle g\in L^{2}(0,T;H^{-1}(\Omega))$ , we say that $\displaystyle y\in L^{2}(0,T;H_{0}^{1}(\Omega))\cap H^{1}(0,T;H^{-1}(\Omega))$ is a weak solution if

\int_{0}^{T}\langle\partial_{t}y(t,\cdot),\zeta(t,\cdot)\rangle)_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}dt+\int_{0}^{T}\int_{\Omega}\nabla y(t,x)\cdot\nabla\zeta(t,x)dtdx\\ =\int_{0}^{T}\langle g(t,\cdot),\zeta(t,\cdot)\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}dt\qquad\forall\zeta\in L^{2}(0,T;H_{0}^{1}(\Omega)),

(2.11)

and

y(0,\cdot)=y_{0}\ \text{in}\ L^{2}(\Omega).

(2.12)

If $\displaystyle y_{0}\in L^{1}(\Omega)$ and $\displaystyle g\in L^{1}((0,T)\times\Omega)$ , we say that $\displaystyle y\in L^{p}((0,T)\times\Omega)$ is a very weak solution of (2.10) if

\displaystyle\iint_{(0,T)\times\Omega}y(-\partial_{t}\zeta-\Delta\zeta)dtdx=\iint_{(0,T)\times\Omega}g\zeta dtdx+\int_{\Omega}y_{0}(x)\zeta(0,x)dx\qquad\forall\zeta\in C^{\infty}_{c}([0,T)\times\Omega).

We recall the following implications

\displaystyle\text{strong solution}\ \implies\text{weak solution}\ \implies\ \text{very weak solution},

and the reverse implications are also true assuming that $\displaystyle y$ is regular enough. We also note that the definition of weak solution is meaningful due to the continuous embedding (see, for instance, [Eva10, Theorem 3, p.303])

L^{2}(0,T;H_{0}^{1}(\Omega))\cap H^{1}(0,T;H^{-1}(\Omega))\hookrightarrow C^{0}([0,T];L^{2}(\Omega)).

(2.13)

We also state standard results for the well-posedness of (2.10) (see, for instance [Eva10, Theorems 3 and 4, pp.378-379], [LSU68, Theorem 7.1, p.181] and [LSU68, Theorem 9.1, p.341]):

Theorem 2.3.

The following well-posedness results hold.

For any $\displaystyle y_{0}\in L^{2}(\Omega)$ and $\displaystyle g\in L^{2}(0,T;H^{-1}(\Omega))$ , the equation (2.10) admits a unique weak solution $\displaystyle y$ and we have the estimate

\left\|y\right\|_{L^{2}(0,T;H_{0}^{1}(\Omega))\cap H^{1}(0,T;H^{-1}(\Omega))}\lesssim\left\|y_{0}\right\|_{L^{2}(\Omega)}+\left\|g\right\|_{L^{2}(0,T;H^{-1}(\Omega))}.

(2.14)

For $\displaystyle y_{0}\in L^{\infty}(\Omega)$ and $\displaystyle g\in L^{\infty}((0,T)\times\Omega)$ , the unique weak solution $\displaystyle y$ of (2.10) satisfies

\left\|y\right\|_{L^{\infty}((0,T)\times\Omega)}\lesssim\left\|y_{0}\right\|_{L^{\infty}(\Omega)}+\left\|g\right\|_{L^{\infty}((0,T)\times\Omega)}.

(2.15)

Assume $\displaystyle p\in(1,\infty)$ . For any $\displaystyle y_{0}\in W_{0}^{2/p^{\prime},p}(\Omega)$ and $\displaystyle g\in L^{p}((0,T)\times\Omega)$ , there exists a unique strong solution $\displaystyle y\in\mathcal{X}^{p}$ of (2.10) and we have the estimate

\left\|y\right\|_{\mathcal{X}^{p}}\lesssim\left\|y_{0}\right\|_{W^{2/p^{\prime},p}(\Omega)}+\left\|g\right\|_{L^{p}((0,T)\times\Omega)}.

(2.16)

2.3 Semi-linear heat equation

For $\displaystyle N\in\mathbb{N}$ , $\displaystyle N\geqslant 2$ , let us then consider the semi-linear heat equation

\left\{\begin{array}[]{ll}\partial_{t}y-\Delta y=y^{N}+g&\mathrm{in}\ (0,T)\times\Omega,\\ y=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ y(0,\cdot)=y_{0}&\mathrm{in}\ \Omega.\end{array}\right.

(2.17)

The space $\displaystyle\mathcal{W}$ is defined in (1.4). First we recall the definition of a weak solution for the system (2.17):

Definition 2.4.

We say that $\displaystyle y\in\mathcal{W}$ is a weak solution of (2.17) if

\int_{0}^{T}\langle\partial_{t}y(t,\cdot),\zeta(t,\cdot)\rangle)_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}dt+\int_{0}^{T}\int_{\Omega}\nabla y(t,x)\cdot\nabla\zeta(t,x)dtdx\\ =\int_{0}^{T}\int_{\Omega}y^{N}(t,x)\zeta(t,x)dtdx+\int_{0}^{T}\langle g(t,\cdot),\zeta(t,\cdot)\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}dt\qquad\forall\zeta\in L^{2}(0,T;H_{0}^{1}(\Omega)),

(2.18)

and

y(0,\cdot)=y_{0}\ \text{in}\ L^{2}(\Omega).

(2.19)

Let us state the following well-posedness result for (2.17) for small data. This result is standard, but we recall the proof for completeness.

Theorem 2.5.

Assume $\displaystyle p$ satisfies (1.5). There exists $\displaystyle\delta>0$ small enough such that for any $\displaystyle y_{0}\in L^{\infty}(\Omega)$ and $\displaystyle g\in L^{p}((0,T)\times\Omega)$ , satisfying

\left\|y_{0}\right\|_{L^{\infty}(\Omega)}+\left\|g\right\|_{L^{p}((0,T)\times\Omega)}\leqslant\delta,

(2.20)

the system (2.17) admits a unique weak solution. Moreover, we have

\left\|y\right\|_{\mathcal{W}}+\left\|y^{N}\right\|_{L^{\infty}((0,T)\times\Omega)}\lesssim\left\|y_{0}\right\|_{L^{\infty}(\Omega)}+\left\|g\right\|_{L^{p}((0,T)\times\Omega)}.

(2.21)

Proof.

First, we show that for any $\displaystyle F\in L^{\infty}((0,T)\times\Omega)$ , there exists a unique weak solution to the heat equation

\left\{\begin{array}[]{ll}\partial_{t}y-\Delta y=F+g&\mathrm{in}\ (0,T)\times\Omega,\\ y=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ y(0,\cdot)=y_{0}&\mathrm{in}\ \Omega.\end{array}\right.

(2.22)

In order to do this, we can write $\displaystyle y=y_{1}+y_{2}$ with

\left\{\begin{array}[]{ll}\partial_{t}y_{1}-\Delta y_{1}=F&\mathrm{in}\ (0,T)\times\Omega,\\ y_{1}=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ y_{1}(0,\cdot)=y_{0}&\mathrm{in}\ \Omega,\end{array}\right.\quad\left\{\begin{array}[]{ll}\partial_{t}y_{2}-\Delta y_{2}=g&\mathrm{in}\ (0,T)\times\Omega,\\ y_{2}=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ y_{2}(0,\cdot)=0&\mathrm{in}\ \Omega,\end{array}\right.

(2.23)

Applying 2.3, the above systems admit respectively a unique solution $\displaystyle y_{1}\in\mathcal{W}$ and $\displaystyle y_{2}\in\mathcal{X}^{p}$ and with the hypotheses on $\displaystyle p$ , we deduce from (2.2) that $\displaystyle\mathcal{X}^{p}\hookrightarrow\mathcal{W}$ . We conclude the existence and the uniqueness of a weak solution $\displaystyle y\in\mathcal{W}$ of (2.22) and we have the estimate

\left\|y\right\|_{L^{\infty}((0,T)\times\Omega)}\lesssim\left\|y_{0}\right\|_{L^{\infty}(\Omega)}+\left\|g\right\|_{L^{p}((0,T)\times\Omega)}+\left\|F\right\|_{L^{\infty}((0,T)\times\Omega)}.

(2.24)

We can thus define the following mapping

\mathcal{N}:L^{\infty}((0,T)\times\Omega)\to L^{\infty}((0,T)\times\Omega),\quad F\mapsto y^{N},

(2.25)

where $\displaystyle y$ is the unique weak solution to (2.22) and if $\displaystyle y_{0}$ and $\displaystyle g$ satisfy (2.20) and if we consider

B_{\delta}:=\{F\in L^{\infty}((0,T)\times\Omega)\ ;\ \left\|F\right\|_{L^{\infty}((0,T)\times\Omega)}\leqslant\delta\},

(2.26)

then we deduce from (2.24) that for $\displaystyle\delta>0$ small enough, $\displaystyle\mathcal{N}(B_{\delta})\subset B_{\delta}$ . We can also show a similar way that the restriction of $\displaystyle\mathcal{N}$ on $\displaystyle B_{\delta}$ is a strict contraction. The Banach fixed point yields the existence of a unique fixed point $\displaystyle F$ and the corresponding solution $\displaystyle y$ of (2.22) is a weak solution of (2.17).

For the uniqueness, we consider $\displaystyle y_{1},y_{2}\in\mathcal{W}$ two solutions of (2.17). Then, $\displaystyle y:=y_{1}-y_{1}$ satisfies (in a weak sense)

\left\{\begin{array}[]{ll}\partial_{t}{y}-\Delta{y}=y_{1}^{N}-{y}_{2}^{N}&\mathrm{in}\ (0,T)\times\Omega,\\ {y}=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ {y}(0,\cdot)=0&\mathrm{in}\ \Omega.\end{array}\right.

(2.27)

In particular, using that $\displaystyle y_{1},y_{2}\in L^{\infty}((0,T)\times\Omega)$ , we can write the standard energy estimate: for any $\displaystyle t\in[0,T]$ ,

\displaystyle\left\|y(t,\cdot)\right\|_{L^{2}(\Omega)}^{2}\leqslant\int_{0}^{t}\int_{\Omega}y\left(y_{1}^{N}-{y}_{2}^{N}\right)dsdx\lesssim\int_{0}^{t}\left\|y(s,\cdot)\right\|_{L^{2}(\Omega)}^{2}ds,

and we conclude with the Grönwall lemma. ∎

We now state some regularizing effects of (2.17).

Lemma 2.6.

Assume the same hypotheses of 2.5 and let us consider $\displaystyle y$ the corresponding weak solution of (2.17).

1.

If $\displaystyle g=0$ then for any $\displaystyle t\in(0,T]$ and for any $\displaystyle q>1$ , $\displaystyle y(t,\cdot)\in W^{2/q^{\prime},q}_{0}(\Omega)$ . Moreover, we have the estimate

$\left\|y(t,\cdot)\right\|_{W^{2/q^{\prime},q}(\Omega)}\lesssim_{t,q}\left\|y_{0}\right\|_{L^{\infty}(\Omega)}.$ (2.28)

In the general case, for any $\displaystyle t\in(0,T]$ , $\displaystyle y(t,\cdot)\in L^{\infty}(\Omega)$ and we have the estimate

\left\|y(t,\cdot)\right\|_{L^{\infty}(\Omega)}\lesssim_{t,q}\left\|y_{0}\right\|_{L^{\infty}(\Omega)}+\left\|g\right\|_{L^{p}((0,T)\times\Omega)}.

(2.29)

Proof.

Let us denote by $\displaystyle\theta$ the function $\displaystyle\theta(t)=t$ for $\displaystyle t\in\mathbb{R}$ . Then we deduce from (2.17) that

\displaystyle\left\{\begin{array}[]{ll}\partial_{t}(\theta y)-\Delta(\theta y)=y+\theta y^{N}&\mathrm{in}\ (0,T)\times\Omega,\\ \theta y=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ (\theta y)(0,\cdot)=0&\mathrm{in}\ \Omega\end{array}\right.

and from 2.5,

\displaystyle\left\|y+\theta y^{N}\right\|_{L^{\infty}((0,T)\times\Omega)}\lesssim\left\|y_{0}\right\|_{L^{\infty}(\Omega)}.

Applying 2.3, we deduce that $\displaystyle\theta y\in\mathcal{X}^{q}$ for any $\displaystyle q>1$ , and we conclude with (2.4).

The second point can be done similarly by using (2.5) and that $\displaystyle p$ satisfies (1.5). ∎

The above definition and properties can be extended to the parabolic system

\left\{\begin{array}[]{ll}\partial_{t}y_{1}-\Delta y_{1}=y_{1}^{N_{1}}+g&\mathrm{in}\ (0,T)\times\Omega,\\ \partial_{t}y_{2}-\Delta y_{2}=y_{1}^{N_{2}}+y_{2}^{N_{3}}&\mathrm{in}\ (0,T)\times\Omega,\\ y_{1}=y_{2}=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ y_{1}(0,\cdot)=y_{1,0},\quad y_{2}(0,\cdot)=y_{2,0}&\mathrm{in}\ \Omega.\end{array}\right.

(2.30)

More precisely, we have the following definition and well-posedness results:

Definition 2.7.

We say that $\displaystyle(y_{1},y_{2})\in\mathcal{W}\times\mathcal{W}$ is a weak solution of (2.30) if

\int_{0}^{T}\langle\partial_{t}y_{1}(t,\cdot),\zeta(t,\cdot)\rangle)_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}dt+\int_{0}^{T}\int_{\Omega}\nabla y_{1}(t,x)\cdot\nabla\zeta(t,x)dtdx\\ =\int_{0}^{T}\int_{\Omega}y_{1}^{N_{1}}(t,x)\zeta(t,x)dtdx+\int_{0}^{T}\langle g(t,\cdot),\zeta(t,\cdot)\rangle_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}dt\qquad\forall\zeta\in L^{2}(0,T;H_{0}^{1}(\Omega)),

\int_{0}^{T}\langle\partial_{t}y_{2}(t,\cdot),\zeta(t,\cdot)\rangle)_{H^{-1}(\Omega),H_{0}^{1}(\Omega)}dt+\int_{0}^{T}\int_{\Omega}\nabla y_{2}(t,x)\cdot\nabla\zeta(t,x)dtdx\\ =\int_{0}^{T}\int_{\Omega}y_{1}^{N_{2}}(t,x)\zeta(t,x)dtdx+\int_{0}^{T}\int_{\Omega}y_{2}^{N_{3}}(t,x)\zeta(t,x)dtdx\qquad\forall\zeta\in L^{2}(0,T;H_{0}^{1}(\Omega)),

and

(y_{1},y_{2})(0,\cdot)=(y_{1,0},y_{2,0})\ \text{in}\ L^{2}(\Omega)^{2}.

(2.31)

Theorem 2.8.

Assume $\displaystyle p$ satisfies (1.5). There exists $\displaystyle\delta>0$ small enough such that for any $\displaystyle(y_{1,0},y_{2,0})\in L^{\infty}(\Omega)^{2}$ and $\displaystyle g\in L^{p}((0,T)\times\Omega)$ , satisfying

\left\|(y_{1,0},y_{2,0})\right\|_{L^{\infty}(\Omega)^{2}}+\left\|g\right\|_{L^{p}((0,T)\times\Omega)}\leqslant\delta,

(2.32)

the system (2.30) admits a unique weak solution. Moreover, we have

\left\|y_{1}\right\|_{\mathcal{W}}+\left\|y_{2}\right\|_{\mathcal{W}}+\left\|y_{1}^{N_{1}}\right\|_{L^{\infty}((0,T)\times\Omega)}+\left\|y_{1}^{N_{2}}\right\|_{L^{\infty}((0,T)\times\Omega)}+\left\|y_{2}^{N_{3}}\right\|_{L^{\infty}((0,T)\times\Omega)}\\ \lesssim\left\|y_{0}\right\|_{L^{\infty}(\Omega)}+\left\|g\right\|_{L^{p}((0,T)\times\Omega)}.

(2.33)

3 Proof of 1.2

The goal of this part is to prove 1.2.

We first set

\rho_{0}(t):=\exp\left(-\frac{1}{t(T-t)}\right)\quad(t\in(0,T)),\quad\rho(0)=\rho(T)=0,

(3.1)

and

\rho(t):=\left\{\begin{array}[]{l}\exp\left(-\frac{1}{(T/2)^{2}}\right)\quad(t\in[0,T/2)),\\ \exp\left(-\frac{1}{t(T-t)}\right)\quad(t\in[T/2,T)),\end{array}\right.\quad\rho(T)=0.

(3.2)

Using 2.6, that is taking the control $\displaystyle h\equiv 0$ in $\displaystyle[0,T/2]\times\omega$ in order to benefit from the regularizing effect of the semi-linear heat equation (1.13), we see that it is sufficient to show the following result.

Theorem 3.1.

Assume $\displaystyle N,n\in\mathbb{N}$ , $\displaystyle N\geqslant 2$ and $\displaystyle T>0$ . Let us consider $\displaystyle p$ satisfying

p=(2n+1)(2k+1)+1,

(3.3)

with $\displaystyle k\in\mathbb{N}$ large enough so that

p>\frac{d+2}{2},

(3.4)

and $\displaystyle q\geqslant p^{\prime}$ satisfying (2.6). There exist $\displaystyle\delta>0$ and $\displaystyle m>0$ such that for any initial data $\displaystyle y_{0}\in W^{\frac{2}{q^{\prime}},q}_{0}(\Omega)$ with

\displaystyle\left\|y_{0}\right\|_{W^{\frac{2}{q^{\prime}},q}(\Omega)}\leqslant\delta,

there exists a control $\displaystyle h$ and a strong solution $\displaystyle y$ of (1.13) such that

\frac{y}{\rho^{m}}\in\mathcal{X}^{q},\quad y^{N}\in L^{q}(0,T;L^{q}(\Omega)),\quad h\in L^{\infty}(0,T;L^{\infty}(\Omega)),\quad\left(\frac{h}{\rho_{0}^{m}}\right)^{1/(2n+1)}\in\mathcal{X}^{p},\quad

(3.5)

together with the estimate

\left\|\frac{y}{\rho^{m}}\right\|_{\mathcal{X}^{q}}+\left\|\frac{y^{N}}{\rho^{m_{1}}}\right\|_{L^{q}(0,T;L^{q}(\Omega))}^{1/N}+\left\|\frac{h}{\rho_{0}^{m}}\right\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}+\left\|\left(\frac{h}{\rho_{0}^{m}}\right)^{1/(2n+1)}\right\|_{\mathcal{X}^{p}}^{2n+1}\lesssim\left\|y_{0}\right\|_{W^{2/q^{\prime},q}(\Omega)}.

(3.6)

In particular, $\displaystyle h$ satisfies (1.15) and (1.16) and $\displaystyle y$ satisfies (1.18).

3.1 Carleman estimate and $\displaystyle L^{2}$ observability inequality for the heat equation

The goal of this part is to deduce a weighted $\displaystyle L^{2}$ observability inequality for the heat equation from a Carleman estimate. We first recall a standard Carleman estimate for the heat equation that is due to Fursikov and Imanuvilov [FI96]. We start by introducing a nonempty domain $\displaystyle\omega_{0}$ such that $\displaystyle\chi_{\omega}>0$ on $\displaystyle\overline{\omega_{0}}\subset\omega$ . By using [FI96], see also [TW09, Theorem 9.4.3], there exists $\displaystyle\eta^{0}\in C^{2}(\overline{\Omega})$ satisfying

\eta^{0}>0\ \text{in}\ \Omega,\quad\eta^{0}=0\ \text{on}\ \partial\Omega,\quad\max_{\Omega}\eta^{0}=1,\quad\nabla\eta^{0}\neq 0\ \text{in}\ \overline{\Omega\setminus\omega_{0}}.

(3.7)

We then define the following functions:

\alpha(t,x)=\frac{\exp\left(4\lambda\right)-\exp\{\lambda(2+\eta^{0}(x))\}}{t(T-t)},\quad\xi(t,x)=\frac{\exp\{\lambda(2+\eta^{0}(x))\}}{t(T-t)}.

(3.8)

We can now state the Carleman estimate for the heat equation, see [FCG06, Lemma 1.3].

Theorem 3.2.

There exist $\displaystyle\lambda_{0},s_{0},C_{0}\in\mathbb{R}_{+}^{*}$ such that for any $\displaystyle\lambda\geqslant\lambda_{0}$ , $\displaystyle s\geqslant s_{0}(T+T^{2})$ , $\displaystyle\zeta\in\mathcal{X}^{2}$ with $\displaystyle\zeta=0$ on $\displaystyle(0,T)\times\partial\Omega$ ,

\iint_{(0,T)\times\Omega}s^{3}\lambda^{4}\xi^{3}e^{-2s\alpha}\left|\zeta\right|^{2}\ dxdt\\ \leqslant C_{0}\left(\iint_{(0,T)\times\Omega}e^{-2s\alpha}\left|\partial_{t}\zeta+\Delta\zeta\right|^{2}\ dxdt+\iint_{(0,T)\times\Omega}s^{3}\lambda^{4}\xi^{3}e^{-2s\alpha}\left|\chi_{\omega}\zeta\right|^{2}\ dxdt\right).

(3.9)

From the above result, one can obtain a similar estimate with weights depending only on time. We recall that $\displaystyle\rho_{0}$ and $\displaystyle\rho$ are defined in (3.1) and (3.2). We have that $\displaystyle\rho_{0},\rho\in W^{1,\infty}(0,T)\cap C^{0}([0,T])$ and

\rho_{0}\leqslant\rho\leqslant 1,\qquad\left|\left(\frac{\rho_{0}}{\rho}\right)^{\prime}\right|\lesssim 1.

(3.10)

Moreover, we have the following instrumental estimates

m_{1}<m_{2}\Rightarrow\left(\rho^{m_{2}}\leqslant\rho^{m_{1}},\ |(\rho^{m_{2}})^{\prime}|\lesssim\rho^{m_{1}}\right).

(3.11)

With the above notation, we can state the following corollary of 3.2.

Corollary 3.3.

Assume $\displaystyle r>1$ . Then, there exist $\displaystyle m_{0},M_{0}\in\mathbb{R}_{+}^{*}$ with

m_{0}<M_{0}<rm_{0},

(3.12)

such that for any $\displaystyle\zeta\in\mathcal{X}^{2}$ with $\displaystyle\zeta=0$ on $\displaystyle(0,T)\times\partial\Omega$ the following relation holds

\left\|\zeta(0,\cdot)\right\|_{L^{2}(\Omega)}+\left\|\rho^{M_{0}}\zeta\right\|_{L^{2}(0,T;L^{2}(\Omega))}\lesssim\left\|\rho^{m_{0}}\left(\partial_{t}\zeta+\Delta\zeta\right)\right\|_{L^{2}(0,T;L^{2}(\Omega))}+\left\|\rho_{0}^{m_{0}}\zeta\chi_{\omega}\right\|_{L^{2}(0,T;L^{2}(\Omega))}.

(3.13)

We want to highlight the fact that the dependence in space of the Carleman weights appearing in (3.9) has been removed in (3.13). Moreover, it is worth mentioning that the vanishing property at $\displaystyle t=T$ of the Carleman weights for the left-hand-side of (3.9) has been dropped. This is why one can make appeared the first left-hand-side of (3.13), that is the classical left-hand-side term for proving a $\displaystyle L^{2}$ observability inequality. The same remark applies for the first right-hand-side term of (3.9) to get the first right-hand-side term of (3.13). Finally, the fact that $\displaystyle m_{0}$ and $\displaystyle M_{0}$ are comparable is quantified in (3.12).

Proof of 3.3.

We consider $\displaystyle s_{0}$ and $\displaystyle\lambda_{0}$ from 3.2. Then, we deduce from (3.7) and (3.8) that for any $\displaystyle\lambda\geqslant\lambda_{0}$ , $\displaystyle s\geqslant s_{0}(T+T^{2})$ ,

\displaystyle 1\lesssim s^{3}\lambda^{4}\xi^{3}\lesssim(s\lambda^{2}\xi)^{3}\lesssim e^{s\lambda^{2}\frac{e^{3\lambda}}{t(T-t)}}.

Therefore combining these estimates with (3.7) and (3.8) and taking $\displaystyle s=s_{0}(T+T^{2})$ , we deduce that

\displaystyle\rho_{0}^{M_{0}}\lesssim s^{3}\lambda^{4}\xi^{3}e^{-s\alpha},\quad e^{-s\alpha}\lesssim s^{3}\lambda^{4}\xi^{3}e^{-s\alpha}\lesssim\rho_{0}^{m_{0}}

with

\displaystyle M_{0}:=s_{0}(T+T^{2})\left(e^{4\lambda}-e^{2\lambda}\right),\quad m_{0}:=s_{0}(T+T^{2})\left(e^{4\lambda}-e^{3\lambda}(1+\lambda^{2})\right).

We now fix $\displaystyle\lambda=\lambda_{0}$ large enough, so that (3.12) holds. Applying (3.9), we obtain

\left\|\rho_{0}^{M_{0}}\zeta\right\|_{L^{2}(0,T;L^{2}(\Omega))}\lesssim\left\|\rho_{0}^{m_{0}}\left(\partial_{t}\zeta+\Delta\zeta\right)\right\|_{L^{2}(0,T;L^{2}(\Omega))}+\left\|\rho_{0}^{m_{0}}\zeta\chi_{\omega}\right\|_{L^{2}(0,T;L^{2}(\Omega))}.

(3.14)

Using (3.1) and (3.2), the above relation yields

\left\|\rho^{M_{0}}\zeta\right\|_{L^{2}(T/2,T;L^{2}(\Omega))}\lesssim\left\|\rho_{0}^{m_{0}}\left(\partial_{t}\zeta+\Delta\zeta\right)\right\|_{L^{2}(0,T;L^{2}(\Omega))}+\left\|\rho_{0}^{m_{0}}\zeta\chi_{\omega}\right\|_{L^{2}(0,T;L^{2}(\Omega))}.

(3.15)

Let us consider $\displaystyle\chi_{T}\in C^{\infty}([0,T])$ , $\displaystyle\chi_{T}\equiv 1$ in $\displaystyle[0,T/2]$ , $\displaystyle\chi_{T}\equiv 0$ in $\displaystyle[3T/4,T]$ and $\displaystyle\left|\chi_{T}^{\prime}\right|\lesssim 1/T$ . Then

\left\{\begin{array}[]{rl}-\partial_{t}\left(\chi_{T}\zeta\right)-\Delta\left(\chi_{T}\zeta\right)=-\left(\chi_{T}\right)^{\prime}\zeta-\chi_{T}\left(\partial_{t}\zeta+\Delta\zeta\right)&\mathrm{in}\ (0,T)\times\Omega,\\ \left(\chi_{T}\zeta\right)=0&\text{on}\ (0,T)\times\partial\Omega,\\ \left(\chi_{T}\zeta\right)(T,\cdot)=0&\text{in}\ \Omega.\end{array}\right.

(3.16)

By using the maximal regularity of the heat equation in $\displaystyle L^{2}$ i.e. 2.3 with $\displaystyle p=2$ to (3.16) and the Sobolev embedding (2.4) we deduce

\displaystyle\left\|\zeta(0,\cdot)\right\|_{L^{2}(\Omega)}+\left\|\zeta\right\|_{L^{2}(0,T/2;L^{2}(\Omega))}\lesssim\left\|{\partial_{t}\zeta+\Delta\zeta}\right\|_{L^{2}(0,3T/4;L^{2}(\Omega))}+\left\|\zeta\right\|_{L^{2}(T/2,3T/4;L^{2}(\Omega))},

and thus by using that $\displaystyle\rho(t)=\rho(T/2)$ in $\displaystyle(0,T/2)$ and $\displaystyle\rho(t)\geqslant\rho(3T/4)$ in $\displaystyle(0,3T/4)$ , we obtain

\displaystyle\left\|\zeta(0,\cdot)\right\|_{L^{2}(\Omega)}+\left\|\rho^{M_{0}}\zeta\right\|_{L^{2}(0,T/2;L^{2}(\Omega))}\lesssim\left\|\rho^{m_{0}}({\partial_{t}\zeta+\Delta\zeta})\right\|_{L^{2}(0,T;L^{2}(\Omega))}+\left\|\rho^{M_{0}}\zeta\right\|_{L^{2}(T/2,T;L^{2}(\Omega))}.

Combining this last estimate with (3.15) and (3.10), we deduce the expected observability inequality (3.13). ∎

3.2 A weighted $\displaystyle L^{p}$ observability inequality

The goal of this part is to deduce from the weighted $\displaystyle L^{2}$ observability inequality in 3.3 a weighted $\displaystyle L^{p}$ observability inequality for $\displaystyle p\geqslant 2$ , by applying maximal regularity results for the heat equation. More precisely, we show the following result:

Proposition 3.4.

Assume $\displaystyle p\geqslant 2$ and $\displaystyle r\in(1,p^{\prime})$ . Then, there exist $\displaystyle m_{0},m_{1}\in\mathbb{R}_{+}^{*}$ with

m_{0}<m_{1}<rm_{0},

(3.17)

such that for any $\displaystyle\zeta\in\mathcal{X}^{p}$ with $\displaystyle\zeta=0$ on $\displaystyle(0,T)\times\partial\Omega$ , the following relation holds

\left\|\zeta(0,\cdot)\right\|_{L^{p}(\Omega)}+\left\|\rho^{m_{1}}\zeta\right\|_{L^{p}(0,T;L^{p}(\Omega))}\lesssim\left\|\rho^{m_{0}}\left(\partial_{t}\zeta+\Delta\zeta\right)\right\|_{L^{p}(0,T;L^{p}(\Omega))}+\left\|\rho_{0}^{m_{0}}\zeta\chi_{\omega}\right\|_{L^{p}(0,T;L^{p}(\Omega))}.

(3.18)

The main difference between (3.18) and (3.13) is the $\displaystyle L^{p}$ framework. We want to highlight that $\displaystyle M_{0}$ of (3.12) has been transformed into $\displaystyle m_{1}$ of (3.17). Basically, the proof is as follows. By a bootstrap argument, we apply recursively maximal regularity results in $\displaystyle L^{r}$ , starting from $\displaystyle r=2$ together with Sobolev embeddings to obtain (3.18). During the induction process, $\displaystyle M_{0}$ becomes $\displaystyle M_{1}\in(M_{0},rm_{0})$ then $\displaystyle M_{2}\in(M_{1},rm_{0})$ , etc. to finally take the value $\displaystyle m_{1}\in(m_{0},rm_{0})$ .

Proof.

First, we apply 3.3 to obtain $\displaystyle m_{0},M_{0}\in\mathbb{R}_{+}^{*}$ satisfying (3.12) and such that (3.13) holds for any $\displaystyle\zeta\in\mathcal{X}^{2}$ with $\displaystyle\zeta=0$ on $\displaystyle(0,T)\times\partial\Omega$ . We then set $\displaystyle g:=-\partial_{t}\zeta-\Delta\zeta$ so that for any $\displaystyle M_{1}>0$ ,

\left\{\begin{array}[]{rl}-\partial_{t}\left(\rho^{M_{1}}\zeta\right)-\Delta\left(\rho^{M_{1}}\zeta\right)=-\left(\rho^{M_{1}}\right)^{\prime}\zeta+\rho^{M_{1}}g&\mathrm{in}\ (0,T)\times\Omega,\\ \left(\rho^{M_{1}}\zeta\right)=0&\text{on}\ (0,T)\times\partial\Omega,\\ \left(\rho^{M_{1}}\zeta\right)(T,\cdot)=0&\text{in}\ \Omega.\end{array}\right.

(3.19)

In particular, if we consider $\displaystyle M_{1}\in(M_{0},rm_{0})$ then by (3.12) and (3.11)

\displaystyle\left\lvert\left(\rho^{M_{1}}\right)^{\prime}\right\rvert\lesssim\rho^{M_{0}},\quad\rho^{M_{1}}\leqslant\rho^{m_{0}},

so that

\left\|\left(\rho^{M_{1}}\right)^{\prime}\zeta\right\|_{L^{2}(0,T;L^{2}(\Omega))}+\left\|\rho^{M_{1}}g\right\|_{L^{2}(0,T;L^{2}(\Omega))}\lesssim\left\|\rho^{M_{0}}\zeta\right\|_{L^{2}(0,T;L^{2}(\Omega))}+\left\|\rho^{m_{0}}g\right\|_{L^{2}(0,T;L^{2}(\Omega))}.

(3.20)

We can apply the maximal regularity result in $\displaystyle L^{2}$ , i.e. 2.3 with $\displaystyle p=2$ to (3.19), and use (3.20) and the $\displaystyle L^{2}$ observability inequality (3.13) to deduce

\left\|\rho^{M_{1}}\zeta\right\|_{\mathcal{X}^{2}}\lesssim\left\|\rho^{m_{0}}g\right\|_{L^{2}(0,T;L^{2}(\Omega))}+\left\|\rho_{0}^{m_{0}}\zeta\chi_{\omega}\right\|_{L^{2}(0,T;L^{2}(\Omega))}.

(3.21)

We then use the Sobolev embedding (2.2) to deduce

\mathcal{X}^{2}\hookrightarrow L^{q_{1}}(0,T;L^{q_{1}}(\Omega))

(3.22)

with $\displaystyle q_{1}\geqslant 2$ defined by

\displaystyle\text{if}\quad\frac{1}{2}-\frac{2}{2+d}\leqslant\frac{1}{p}\quad\text{then}\ q_{1}=p,\quad\text{else}\quad\frac{1}{q_{1}}=\frac{1}{2}-\frac{2}{2+d}.

Then, we consider $\displaystyle M_{2}\in(M_{1},rm_{0})$ so that from (3.11),

\displaystyle\left\lvert\left(\rho^{M_{2}}\right)^{\prime}\right\rvert\lesssim\rho^{M_{1}},\quad\rho^{M_{2}}\leqslant\rho^{m_{0}},

and with (3.22) and (3.21), we deduce

	$\displaystyle\displaystyle\left\\|\left(\rho^{M_{2}}\right)^{\prime}\zeta\right\\|_{L^{q_{1}}(0,T;L^{q_{1}}(\Omega))}+\left\\|\rho^{M_{2}}g\right\\|_{L^{q_{1}}(0,T;L^{q_{1}}(\Omega))}$	$\displaystyle\displaystyle\lesssim\left\\|\rho^{M_{1}}\zeta\right\\|_{\mathcal{X}^{2}}+\left\\|\rho^{m_{0}}g\right\\|_{L^{q_{1}}(0,T;L^{q_{1}}(\Omega))}$
		$\displaystyle\displaystyle\lesssim\left\\|\rho^{m_{0}}g\right\\|_{L^{q_{1}}(0,T;L^{q_{1}}(\Omega))}+\left\\|\rho_{0}^{m_{0}}\zeta\chi_{\omega}\right\\|_{L^{2}(0,T;L^{2}(\Omega))}.$		(3.23)

Now we apply 2.3 to

\left\{\begin{array}[]{rl}-\partial_{t}\left(\rho^{M_{2}}\zeta\right)-\Delta\left(\rho^{M_{2}}\zeta\right)=-\left(\rho^{M_{2}}\right)^{\prime}\zeta+\rho^{M_{2}}g&\mathrm{in}\ (0,T)\times\Omega,\\ \left(\rho^{M_{2}}\zeta\right)=0&\text{on}\ (0,T)\times\partial\Omega,\\ \left(\rho^{M_{2}}\zeta\right)(T,\cdot)=0&\text{in}\ \Omega,\end{array}\right.

(3.24)

with $\displaystyle p=q_{1}$ , and using (3.23), we obtain

\left\|\rho^{M_{2}}\zeta\right\|_{\mathcal{X}^{q_{1}}}\lesssim\left\|\rho^{m_{0}}g\right\|_{L^{q_{1}}(0,T;L^{q_{1}}(\Omega))}+\left\|\rho_{0}^{m_{0}}\zeta\chi_{\omega}\right\|_{L^{2}(0,T;L^{2}(\Omega))}.

(3.25)

If $\displaystyle q_{1}=p$ , then using $\displaystyle H^{1}(0,T)\hookrightarrow C^{0}([0,T])$ and $\displaystyle L^{p}(0,T;L^{p}(\Omega))\hookrightarrow L^{2}(0,T;L^{2}(\Omega))$ , we deduce from the above relation the desired observability inequality (3.18) with $\displaystyle m_{1}=M_{2}$ . Else, we have $\displaystyle q_{1}<p$ and we can repeat the argument, that is we use the Sobolev embedding (2.2) to deduce

\mathcal{X}^{q_{1}}\hookrightarrow L^{q_{2}}(0,T;L^{q_{2}}(\Omega))

(3.26)

with $\displaystyle q_{2}\geqslant q_{1}$ defined by

\displaystyle\text{if}\quad\frac{1}{q_{1}}-\frac{2}{2+d}\leqslant\frac{1}{p}\quad\text{then}\ q_{2}=p,\quad\text{else}\quad\frac{1}{q_{2}}=\frac{1}{q_{1}}-\frac{2}{2+d}=\frac{1}{2}-2\cdot\frac{2}{2+d}.

Taking $\displaystyle M_{3}\in(M_{2},rm_{0})$ , and proceeding as above, applying 2.3 with $\displaystyle p=q_{2}$ and using (3.26) and (3.25), we find

\displaystyle\left\|\rho^{M_{3}}\zeta\right\|_{\mathcal{X}^{q_{2}}}\lesssim\left\|\rho^{m_{0}}g\right\|_{L^{q_{2}}(0,T;L^{q_{2}}(\Omega))}+\left\|\rho_{0}^{m_{0}}\zeta\chi_{\omega}\right\|_{L^{2}(0,T;L^{2}(\Omega))}.

We can proceed by induction and since $\displaystyle 1/q_{n}$ decrease by $\displaystyle 2/(2+d)$ at each step, after a finite number of steps, we obtain $\displaystyle q_{n}=p$ and we deduce the desired observability inequality (3.18). ∎

3.3 Controllability of the heat equation with a source term in $\displaystyle L^{p^{\prime}}$

We use the above observability results to show, by a duality argument, the controllability of a linear system associated with (1.13):

\left\{\begin{array}[]{rl}\partial_{t}y-\Delta y=h\chi_{\omega}+F&\mathrm{in}\ (0,T)\times\Omega,\\ y=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\ y(0,\cdot)=y_{0}&\mathrm{in}\ \Omega.\end{array}\right.

(3.27)

In order to control the above system, we fix $\displaystyle p\in 2\mathbb{N}^{*}$ and we consider $\displaystyle m_{0}$ and $\displaystyle m_{1}$ as in 3.4. Then, we introduce

\mathcal{Y}_{0}:=\{\zeta\in C^{\infty}([0,T]\times\overline{\Omega})\ ;\ \zeta=0\ \text{on}\ (0,T)\times\partial\Omega\},

(3.28)

and we define the following norm for $\displaystyle\zeta\in\mathcal{Y}_{0}$ ,

\left\|\zeta\right\|_{\mathcal{Y}}:=\left\|\rho^{m_{0}}(\partial_{t}\zeta+\Delta\zeta)\right\|_{L^{p}(0,T;L^{p}(\Omega))}+\left\|\rho_{0}^{m_{0}}\zeta\chi_{\omega}\right\|_{L^{p}(0,T;L^{p}(\Omega))}.

(3.29)

The fact that it is a norm is a consequence of the weighted $\displaystyle L^{p}$ observability inequality (3.18). We denote by $\displaystyle\mathcal{Y}$ the completion of $\displaystyle\mathcal{Y}_{0}$ with respect to the norm $\displaystyle\|\cdot\|_{\mathcal{Y}}$ .

First, we have the following result that roughly states that a function $\displaystyle\zeta\in\mathcal{Y}$ belongs to some suitable weighted $\displaystyle\mathcal{X}^{p}$ spaces.

Lemma 3.5.

Assume $\displaystyle m>m_{1}$ . Then, for any $\displaystyle\zeta\in\mathcal{Y}$ ,

\left\|\rho_{0}^{m}\zeta\right\|_{\mathcal{X}^{p}}\lesssim\left\|\rho^{m}\zeta\right\|_{\mathcal{X}^{p}}\lesssim\left\|\zeta\right\|_{\mathcal{Y}}.

(3.30)

Proof.

Using $\displaystyle m>m_{1}$ , (3.17) and (3.11), we have

\left|\left(\rho^{m}\right)^{\prime}\right|\lesssim\rho^{m_{1}},\quad\rho^{m}\lesssim\rho^{m_{0}}.

(3.31)

Now, if $\displaystyle\zeta\in\mathcal{Y}$ , then

\left\{\begin{array}[]{rl}-\partial_{t}\left(\rho^{m}\zeta\right)-\Delta\left(\rho^{m}\zeta\right)=-\left(\rho^{m}\right)^{\prime}\zeta+\rho^{m}\left(-\partial_{t}\zeta-\Delta\zeta\right)&\mathrm{in}\ (0,T)\times\Omega,\\ \left(\rho^{m}\zeta\right)=0&\text{on}\ (0,T)\times\partial\Omega,\\ \left(\rho^{m}\zeta\right)(T,\cdot)=0&\text{in}\ \Omega.\end{array}\right.

(3.32)

Combining the observability inequality (3.18) and (3.31), we deduce

\displaystyle\left\|-\left(\rho^{m}\right)^{\prime}\zeta+\rho^{m}\left(-\partial_{t}\zeta-\Delta\zeta\right)\right\|_{L^{p}(0,T;L^{p}(\Omega))}\lesssim\left\|\zeta\right\|_{\mathcal{Y}}.

Applying the maximal regularity result 2.3 on (3.32) and using the above relation, we deduce the second estimate in (3.30). For the first estimate, we use (3.10) to obtain that $\displaystyle\left\|\rho_{0}/\rho\right\|_{W^{1,\infty}(0,T)}\lesssim 1$ and this allows us to conclude the proof. ∎

We now introduce some functional spaces: for $\displaystyle m>0$ and $\displaystyle p\in[1,\infty]$ , we set

L^{p}_{m}(0,T;L^{p}(\Omega)):=\left\{f\in L^{p}(0,T;L^{p}(\Omega))\ ;\ \frac{f}{\rho^{m}}\in L^{p}(0,T;L^{p}(\Omega))\right\},

(3.33)

L^{p}_{m,0}(0,T;L^{p}(\Omega)):=\left\{f\in L^{p}(0,T;L^{p}(\Omega))\ ;\ \frac{f}{\rho_{0}^{m}}\in L^{p}(0,T;L^{p}(\Omega))\right\},

(3.34)

endowed with the following norm

\left\|f\right\|_{L^{p}_{m}(0,T;L^{p}(\Omega))}:=\left\|\frac{f}{\rho^{m}}\right\|_{L^{p}(0,T;L^{p}(\Omega))},\quad\left\|f\right\|_{L^{p}_{m,0}(0,T;L^{p}(\Omega))}:=\left\|\frac{f}{\rho_{0}^{m}}\right\|_{L^{p}(0,T;L^{p}(\Omega))}.

(3.35)

Let us consider, for any $\displaystyle y_{0}\in L^{p^{\prime}}(\Omega)$ and $\displaystyle F\in L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))$ , the functional $\displaystyle J=J_{y_{0},F}$ defined as follows:

J(\zeta):=\frac{1}{p}\iint_{(0,T)\times\Omega}\rho^{m_{0}p}(-\partial_{t}\zeta-\Delta\zeta)^{p}dtdx+\frac{1}{p}\iint_{(0,T)\times\Omega}\rho_{0}^{m_{0}p}\zeta^{p}\chi_{\omega}^{p}dtdx\\ -\iint_{(0,T)\times\Omega}F\zeta dtdx-\int_{\Omega}y_{0}(x)\zeta(0,x)dx.

(3.36)

Using the $\displaystyle L^{p}$ observability inequality (3.18), we can check that $\displaystyle J\in C^{1}(\mathcal{Y};\mathbb{R})$ is a strictly convex and coercive functional on $\displaystyle\mathcal{Y}$ . In particular, $\displaystyle J$ admits a unique minimum $\displaystyle\overline{\zeta}$ . We can thus define, for $\displaystyle y_{0}\in L^{p^{\prime}}(\Omega)$ and $\displaystyle F\in L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))$ , the following maps

\mathcal{M}_{1}(y_{0},F):=\overline{\zeta},\quad\mathcal{M}_{2}(y_{0},F):=\rho^{m_{0}p}(-\partial_{t}\overline{\zeta}-\Delta\overline{\zeta})^{p-1},\quad\mathcal{M}_{3}(y_{0},F):=-\rho_{0}^{m_{0}p}\chi_{\omega}^{p-1}\overline{\zeta}^{p-1}.

(3.37)

Proposition 3.6.

Assume $\displaystyle p\in 2\mathbb{N}^{*}$ and $\displaystyle r\in(1,p^{\prime})$ and let us consider $\displaystyle m_{0}$ and $\displaystyle m_{1}$ given by 3.4. For any $\displaystyle y_{0}\in L^{p^{\prime}}(\Omega)$ and $\displaystyle F\in L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))$ , let us set

\overline{\zeta}=\mathcal{M}_{1}(y_{0},F),\quad y=\mathcal{M}_{2}(y_{0},F),\quad h=\mathcal{M}_{3}(y_{0},F).

(3.38)

Existence of a solution. We have $\displaystyle y\in L^{p^{\prime}}_{m_{0}}(0,T;L^{p^{\prime}}(\Omega))$ and $\displaystyle h\in L^{p^{\prime}}_{m_{0},0}(0,T;L^{p^{\prime}}(\Omega))$ , together with the estimates

\left\|\overline{\zeta}\right\|_{\mathcal{Y}}^{p}\lesssim\left\|F\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}^{p^{\prime}}+\left\|y_{0}\right\|_{L^{p^{\prime}}(\Omega)}^{p^{\prime}},

(3.39)

\left\|y\right\|_{L^{p^{\prime}}_{m_{0}}(0,T;L^{p^{\prime}}(\Omega))}+\left\|h\right\|_{L^{p^{\prime}}_{m_{0},0}(0,T;L^{p^{\prime}}(\Omega))}\lesssim\left\|F\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}+\left\|y_{0}\right\|_{L^{p^{\prime}}(\Omega)}.

(3.40)

Moreover, $\displaystyle y$ is the very weak solution of (3.27) associated with $\displaystyle F$ , $\displaystyle h$ and $\displaystyle y_{0}$ in the sense of 2.2.

Odd behavior of the control. The control $\displaystyle h$ satisfies $\displaystyle h^{1/(p-1)}\in\mathcal{X}^{p}$ and

h^{1/(p-1)}(0,\cdot)=h^{1/(p-1)}(T,\cdot)=0\quad\text{in}\ \Omega,

(3.41)

together with the estimate

\left\|h^{1/(p-1)}\right\|_{\mathcal{X}^{p}}\lesssim\left\|F\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}^{1/(p-1)}+\left\|y_{0}\right\|_{L^{p^{\prime}}(\Omega)}^{1/(p-1)}.

(3.42)

Regularity of the solution. Assume that $\displaystyle y_{0}\in W^{2/p,p^{\prime}}(\Omega)$ and that $\displaystyle y_{0}=0$ on $\displaystyle\partial\Omega$ if $\displaystyle p=2$ . Then for any $\displaystyle m<m_{0}$ , $\displaystyle y/\rho^{m}\in\mathcal{X}^{p^{\prime}}$ together with the estimate

\left\|\frac{y}{\rho^{m}}\right\|_{\mathcal{X}^{p^{\prime}}}\lesssim\left\|F\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}+\left\|y_{0}\right\|_{W^{2/p,p^{\prime}}(\Omega)}.

(3.43)

In particular, $\displaystyle y(T,\cdot)=0$ .

The first point will be obtained from Euler-Lagrange equation. The odd behavior of the control, i.e. (3.42), remarking that $\displaystyle p-1$ is odd, comes from the identification of $\displaystyle h$ in (3.38), (3.37) and from a weighted $\displaystyle\mathcal{X}^{p}$ estimate of $\displaystyle\overline{\zeta}$ . Finally, the regularity result on the solution comes from a maximal parabolic regularity result. Note that if $\displaystyle p\neq 2$ , then $\displaystyle p\geqslant 4$ and $\displaystyle p^{\prime}<3/2$ so that we do not need to impose the compatibility condition $\displaystyle y_{0}=0$ on $\displaystyle\partial\Omega$ .

Proof of 3.6.

We start by writing the Euler-Lagrange equation for $\displaystyle J$ at $\displaystyle\overline{\zeta}$ to obtain

\iint_{(0,T)\times\Omega}\rho^{m_{0}p}(-\partial_{t}\overline{\zeta}-\Delta\overline{\zeta})^{p-1}(-\partial_{t}\zeta-\Delta\zeta)dtdx+\iint_{(0,T)\times\Omega}\rho_{0}^{m_{0}p}\chi_{\omega}^{p}\overline{\zeta}^{p-1}\zeta dtdx\\ =\iint_{(0,T)\times\Omega}F\zeta dtdx+\int_{\Omega}y_{0}(x)\zeta(0,x)dx\quad(\zeta\in\mathcal{Y}).

(3.44)

Taking $\displaystyle\zeta=\overline{\zeta}$ in the above relation and using Young’s inequality and the $\displaystyle L^{p}$ observability inequality (3.18), we deduce (3.39).

Then, (3.38) and (3.37) yield

\displaystyle\left|\frac{y}{\rho^{m_{0}}}\right|^{p^{\prime}}=\rho^{m_{0}p}\left|\partial_{t}\overline{\zeta}+\Delta\overline{\zeta}\right|^{p},\quad\left|\frac{h}{\rho_{0}^{m_{0}}}\right|^{p^{\prime}}=\left|\rho_{0}^{m_{0}}\chi_{\omega}\overline{\zeta}\right|^{p},

and we deduce (3.40) from (3.39).

Moreover, (3.44) and (3.38) imply

\iint_{(0,T)\times\Omega}y(-\partial_{t}\zeta-\Delta\zeta)dtdx=\iint_{(0,T)\times\Omega}\chi_{\omega}h\zeta dtdx+\iint_{(0,T)\times\Omega}F\zeta dtdx\\ +\int_{\Omega}y_{0}(x)\zeta(0,x)dx\quad(\zeta\in C^{\infty}_{c}([0,T)\times\Omega)),

(3.45)

that is $\displaystyle y$ is the very weak solution to (3.27) associated with the control $\displaystyle h$ , $\displaystyle F$ and $\displaystyle y_{0}$ in the sense of 2.2.

For the second point, from (3.38) and (3.37), we have

h^{1/(p-1)}=-\rho_{0}^{p^{\prime}m_{0}}\overline{\zeta}\chi_{\omega},

(3.46)

and since $\displaystyle p^{\prime}m_{0}>m_{1}$ , we can apply 3.5 with $\displaystyle m=p^{\prime}m_{0}$ and we deduce (3.42) from (3.39) and (3.30). Since $\displaystyle p^{\prime}m_{0}>m_{1}$ , there exists $\displaystyle r>0$ such that $\displaystyle p^{\prime}m_{0}-r>m_{1}$ , we then obtain (3.41) because

\displaystyle\left\|\rho_{0}^{-r}h^{1/(p-1)}\right\|_{C([0,T];W^{2/p^{\prime},p}(\Omega))}\lesssim\left\|\rho_{0}^{-r}h^{1/(p-1)}\right\|_{\mathcal{X}^{p}}\lesssim\left\|\rho_{0}^{p^{\prime}m_{0}-r}\overline{\zeta}\right\|_{\mathcal{X}^{p}}\lesssim\left\|\zeta\right\|_{\mathcal{Y}}.

Finally, for the last point, we write the system satisfied by $\displaystyle y/\rho^{m}$ :

\left\{\begin{array}[]{rl}\partial_{t}\left(\frac{y}{\rho^{m}}\right)-\Delta\left(\frac{y}{\rho^{m}}\right)=\frac{h}{\rho^{m}}\chi_{\omega}+\frac{F}{\rho^{m}}-m\frac{\rho^{\prime}}{\rho^{m+1}}y&\mathrm{in}\ (0,T)\times\Omega,\\[11.38109pt] \frac{y}{\rho^{m}}=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\[8.53581pt] \frac{y}{\rho^{m}}(0,\cdot)=\frac{y_{0}}{\rho^{m}(0)}&\mathrm{in}\ \Omega.\end{array}\right.

(3.47)

By using $\displaystyle m<m_{0}<m_{1}$ , (3.10), (3.11) and (3.40), we have

\displaystyle\left\|\frac{h}{\rho^{m}}\chi_{\omega}+\frac{F}{\rho^{m}}-m\frac{\rho^{\prime}}{\rho^{m+1}}y\right\|_{L^{p^{\prime}}(0,T;L^{p^{\prime}}(\Omega))}\lesssim\left\|F\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}+\left\|y_{0}\right\|_{L^{p^{\prime}}(\Omega)}

Applying 2.3 to (3.47) with the above estimate, we deduce the regularity estimate on $\displaystyle y$ , i.e. (3.43). ∎

3.4 $\displaystyle L^{\infty}$ bound on the control and $\displaystyle L^{q}$ estimate of the nonlinearity

From now on, we assume $\displaystyle r\in(1,p^{\prime})$ and we assume that $\displaystyle m_{0}$ and $\displaystyle m_{1}$ are given by 3.4 with this $\displaystyle r$ . In particular they satisfy (3.17) which yields

\displaystyle 0<m_{0}p-m_{1}(p-1)<m_{0}.

First we have the following result on the control $\displaystyle h$ .

Lemma 3.7.

Assume $\displaystyle p$ satisfies (3.3) and (3.4). Then for any

0\leqslant m<m_{0}p-m_{1}(p-1),

(3.48)

the control $\displaystyle h$ given by (3.38) satisfies $\displaystyle h^{1/(2n+1)}\in\mathcal{X}^{p}$ and $\displaystyle h\in L_{m,0}^{\infty}(0,T;L^{\infty}(\Omega))$ with the estimate

\left\|h\right\|_{L_{m,0}^{\infty}(0,T;L^{\infty}(\Omega))}+\left\|\left(\frac{h}{\rho_{0}^{m}}\right)^{1/(2n+1)}\right\|_{\mathcal{X}^{p}}^{2n+1}\lesssim\left\|F\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}+\left\|y_{0}\right\|_{L^{p^{\prime}}(\Omega)}.

(3.49)

In the above result, $\displaystyle p$ has to be sufficiently large to get that $\displaystyle\mathcal{X}^{p}$ is an algebra, and this enables us to get that $\displaystyle h^{1/(2n+1)}$ is sufficiently smooth, because $\displaystyle p-1=(2n+1)(2k+1)$ , as expected in (1.16).

Proof of 3.7.

Since $\displaystyle p$ satisfies (3.4), we can apply 2.1 and deduce that $\displaystyle\mathcal{X}^{p}$ is an algebra. On the other hand, from 3.6, $\displaystyle h^{1/(2n+1)}\in\mathcal{X}^{p}$ , we can thus conclude by using that

\displaystyle h^{1/(2n+1)}=\left(h^{1/(p-1)}\right)^{2k+1}.

Now, from (3.38) and (3.37), we can write

\frac{h}{\rho_{0}^{m}}=-\left(\rho_{0}^{(m_{0}p-m)/(p-1)}\chi_{\omega}\overline{\zeta}\right)^{p-1}.

(3.50)

If $\displaystyle m$ satisfies (3.48), then $\displaystyle(m_{0}p-m)/(p-1)>m_{1}$ , we can apply 3.5 and use (3.50), (3.10), (3.39) to obtain

\left\|\frac{h}{\rho_{0}^{m}}\right\|_{\mathcal{X}^{p}}\lesssim\left\|\left(\rho_{0}^{(m_{0}p-m)/(p-1)}\chi_{\omega}\overline{\zeta}\right)^{p-1}\right\|_{\mathcal{X}^{p}}\lesssim\left\|\rho_{0}^{(m_{0}p-m)/(p-1)}\chi_{\omega}\overline{\zeta}\right\|_{\mathcal{X}^{p}}^{p-1}\\ \lesssim\left\|\overline{\zeta}\right\|_{\mathcal{Y}}^{p-1}\lesssim\left\|F\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}+\left\|y_{0}\right\|_{L^{p^{\prime}}(\Omega)}.

We obtain (3.49) by using that $\displaystyle\mathcal{X}^{p}$ is an algebra and (2.2). ∎

Proposition 3.8.

Let $\displaystyle N\in\mathbb{N}^{*}$ , $\displaystyle N\geqslant 2$ and assume $\displaystyle p,q$ satisfying $\displaystyle q\geqslant p^{\prime}$ , (2.6), (3.3) and (3.4). Let us consider $\displaystyle m_{0}$ and $\displaystyle m_{1}$ given by 3.4 with

r:=\frac{p}{p-1+\frac{1}{N}}\in(1,p^{\prime}).

(3.51)

For any $\displaystyle y_{0}\in W^{\frac{2}{q^{\prime}},q}_{0}(\Omega)$ and $\displaystyle F\in L^{q}_{m_{1}}(0,T;L^{q}(\Omega))$ , and for any $\displaystyle m$ satisfying (3.48), $\displaystyle y$ defined by (3.38) satisfies $\displaystyle y/\rho^{m}\in\mathcal{X}^{q}$ and $\displaystyle y^{N}\in L^{q}_{m_{1}}(0,T;L^{q}(\Omega))$ with the estimates

\left\|\frac{y}{\rho^{m}}\right\|_{\mathcal{X}^{q}}\lesssim\left\|F\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}+\left\|y_{0}\right\|_{W^{2/q^{\prime},q}(\Omega)},

(3.52)

\left\|y^{N}\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}\lesssim\left(\left\|F\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}+\left\|y_{0}\right\|_{W^{2/q^{\prime},q}(\Omega)}\right)^{N}.

(3.53)

The goal of the above result is to get an appropriate $\displaystyle L^{q}$ bound on the nonlinearity, this would be a first step in order to prove the local null-controllability of the semi-linear heat equation.

Proof.

We define $\displaystyle q_{1}$ as follows

\text{if}\quad\frac{1}{q}\leqslant\frac{1}{p^{\prime}}-\frac{2}{d+2},\ \text{then}\ q_{1}=q,\ \text{else}\ \frac{1}{q_{1}}=\frac{1}{p^{\prime}}-\frac{2}{d+2}.

(3.54)

In both cases, we have $\displaystyle q\geqslant q_{1}$ and $\displaystyle 1/q^{\prime}\geqslant 1/q_{1}^{\prime}$ .

We deduce from (3.43) and the Sobolev embedding (2.2) that for any $\displaystyle\widetilde{m}<m_{0}$ ,

\left\|y\right\|_{L^{q_{1}}_{\widetilde{m}}(0,T;L^{q_{1}}(\Omega))}\lesssim\left\|F\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}+\left\|y_{0}\right\|_{W^{2/p,p^{\prime}}(\Omega)}.

(3.55)

We then consider $\displaystyle m$ satisfying (3.48). We have in particular $\displaystyle m<m_{0}<m_{1}$ and we can write

\left\{\begin{array}[]{rl}\partial_{t}\left(\frac{y}{\rho^{m}}\right)-\Delta\left(\frac{y}{\rho^{m}}\right)=\frac{h}{\rho^{m}}\chi_{\omega}+\frac{F}{\rho^{m}}-m\frac{\rho^{\prime}}{\rho^{m+1}}y&\mathrm{in}\ (0,T)\times\Omega,\\[11.38109pt] \frac{y}{\rho^{m}}=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\[8.53581pt] \frac{y}{\rho^{m}}(0,\cdot)=\frac{y_{0}}{\rho^{m}(0)}&\mathrm{in}\ \Omega.\end{array}\right.

Applying 2.3 on the above equation and using (3.49) and (3.55) with $\displaystyle\tilde{m}\in(m,m_{0})$ together with (3.10), (3.11), we deduce

\left\|\frac{y}{\rho^{m}}\right\|_{\mathcal{X}^{q_{1}}}\lesssim\left\|h\right\|_{L_{m}^{\infty}(0,T;L^{\infty}(\Omega))}+\left\|F\right\|_{L^{q_{1}}_{m_{1}}(0,T;L^{q_{1}}(\Omega))}+\left\|\frac{\rho^{\prime}}{\rho^{m+1}}y\right\|_{L^{q_{1}}(0,T;L^{q_{1}}(\Omega))}\\ \lesssim\left\|F\right\|_{L^{q_{1}}_{m_{1}}(0,T;L^{q_{1}}(\Omega))}+\left\|y_{0}\right\|_{W^{2/q_{1}^{\prime},q_{1}}(\Omega)}+\left\|y\right\|_{L^{q_{1}}_{\widetilde{m}}(0,T;L^{q_{1}}(\Omega))}\\ \lesssim\left\|F\right\|_{L^{q_{1}}_{m_{1}}(0,T;L^{q_{1}}(\Omega))}+\left\|y_{0}\right\|_{W^{2/q^{\prime},q}(\Omega)}.

(3.56)

We can proceed by induction, using again (2.2), and since the corresponding sequence $\displaystyle 1/q_{n}$ decreases by $\displaystyle 2/(d+2)$ (see (3.54)) at each step, we obtain after a finite number of steps that for any $\displaystyle m$ satisfying (3.48), we have

\left\|\frac{y}{\rho^{m}}\right\|_{\mathcal{X}^{q}}\lesssim\left\|F\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}+\left\|y_{0}\right\|_{W^{2/q^{\prime},q}(\Omega)}.

(3.57)

Using that $\displaystyle q$ satisfies (2.6) so that the Sobolev embedding (2.7) holds, we deduce that

\left\|\frac{y^{N}}{\rho^{Nm}}\right\|_{L^{q}((0,T)\times\Omega)}\lesssim\left(\left\|F\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}+\left\|y_{0}\right\|_{W^{2/q^{\prime},q}(\Omega)}\right)^{N}.

(3.58)

From (3.17) and (3.51), we have

\frac{m_{1}}{N}<m_{0}p-m_{1}(p-1)

so that we can take $\displaystyle m=m_{1}/N$ in (3.57), (3.58) and we deduce (3.53). ∎

3.5 A Schauder fixed-point argument

Let us consider the hypotheses of 3.8 and assume $\displaystyle y_{0}\in W_{0}^{2/q^{\prime},q}(\Omega)$ . Then, using the conclusion of 3.8, we can define the mapping

\mathcal{N}:L^{q}_{m_{1}}(0,T;L^{q}(\Omega))\to L^{q}_{m_{1}}(0,T;L^{q}(\Omega)),\quad F\mapsto y^{N},

(3.59)

where $\displaystyle y=\mathcal{M}_{2}(y_{0},F)$ . Moreover, using (3.53), we deduce that if $\displaystyle R_{0}:=\left\|y_{0}\right\|_{W^{2/q^{\prime},q}(\Omega)}$ is small enough, then the closed set

B_{R_{0}}:=\left\{F\in L^{q}_{m_{1}}(0,T;L^{q}(\Omega))\ ;\ \left\|F\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}\leqslant R_{0}\right\}

(3.60)

is invariant by $\displaystyle\mathcal{N}$ .

Proposition 3.9.

The mapping $\displaystyle\mathcal{N}:B_{R_{0}}\to B_{R_{0}}$ defined above is continuous and $\displaystyle\mathcal{N}(B_{R_{0}})$ is relatively compact into $\displaystyle B_{R_{0}}$ .

Proof.

Let us consider a sequence $\displaystyle\left(F_{n}\right)_{n}$ of $\displaystyle B_{R_{0}}$ . We write $\displaystyle y_{n}=\mathcal{M}_{2}(y_{0},F_{n})$ . Then we can use (3.52) to obtain that $\displaystyle\left(y_{n}/\rho^{(m_{1}/N)}\right)_{n}$ is bounded in $\displaystyle\mathcal{X}^{q}$ . Applying 2.1, we deduce that, up to a subsequence,

\displaystyle\frac{y_{n}}{\rho^{(m_{1}/N)}}\to\frac{y}{\rho^{(m_{1}/N)}}\quad\text{in}\ L^{qN}((0,T)\times\Omega),

for some $\displaystyle y\in L^{qN}_{m_{1}/N}(0,T;L^{qN}(\Omega))$ . We deduce that $\displaystyle\mathcal{N}(B_{R_{0}})$ is relatively compact into $\displaystyle B_{R_{0}}$ .

To show the continuity of $\displaystyle\mathcal{N}$ , we consider $\displaystyle F_{1},F_{2}\in B_{R_{0}}$ and we write (see (3.37) and (3.38)) for $\displaystyle i=1,2$ ,

\displaystyle\overline{\zeta}_{i}:=\mathcal{M}_{1}(y_{0},F_{i}),\quad y_{i}:=\mathcal{M}_{2}(y_{0},F_{i}),\quad h_{i}:=\mathcal{M}_{3}(y_{0},F_{i}).

From the Euler-Lagrange equation (3.44) for $\displaystyle J_{y_{0},F_{1}}$ and $\displaystyle J_{y_{0},F_{2}}$ , we deduce

\iint_{(0,T)\times\Omega}\rho^{m_{0}p}\left[(-\partial_{t}\overline{\zeta}_{1}-\Delta\overline{\zeta}_{1})^{p-1}-(-\partial_{t}\overline{\zeta}_{2}-\Delta\overline{\zeta}_{2})^{p-1}\right](-\partial_{t}\zeta-\Delta\zeta)dtdx\\ +\iint_{(0,T)\times\Omega}\rho_{0}^{m_{0}p}\chi_{\omega}^{p}\left(\overline{\zeta}_{1}^{p-1}-\overline{\zeta}_{2}^{p-1}\right)\zeta dtdx=\iint_{(0,T)\times\Omega}\left(F_{1}-F_{2}\right)\zeta dtdx\quad(\zeta\in\mathcal{Y}).

(3.61)

In the above relation, we take $\displaystyle\zeta=\overline{\zeta}_{1}-\overline{\zeta}_{2}$ in the above relation and we combine it with the observability inequality (3.18) and with the relation

\displaystyle(x_{1}-x_{2})^{p}\lesssim(x_{1}^{p-1}-x_{2}^{p-1})(x_{1}-x_{2})\quad(x_{1},x_{2}\in\mathbb{R}),

to deduce

\left\|\overline{\zeta}_{1}-\overline{\zeta}_{2}\right\|_{\mathcal{Y}}^{p}\lesssim\left\|F_{1}-F_{2}\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}^{p^{\prime}}.

(3.62)

Moreover, using that

\displaystyle\left|x_{1}^{p-1}-x_{2}^{p-1}\right|\lesssim\left|x_{1}-x_{2}\right|(\left|x_{1}\right|^{p-2}+\left|x_{2}\right|^{p-2})\quad(x_{1},x_{2}\in\mathbb{R}),

we obtain from (3.10)

\left|\frac{h_{1}-h_{2}}{\rho^{m}}\right|=\left(\frac{\rho_{0}}{\rho}\right)^{m}\left|\left(\rho^{(m_{0}p-m)/(p-1)}\chi_{\omega}\overline{\zeta}_{1}\right)^{p-1}-\left(\rho^{(m_{0}p-m)/(p-1)}\chi_{\omega}\overline{\zeta}_{2}\right)^{p-1}\right|\\ \lesssim\left|\rho^{(m_{0}p-m)/(p-1)}\left(\overline{\zeta}_{1}-\overline{\zeta}_{2}\right)\right|\left(\left(\rho^{(m_{0}p-m)/(p-1)}\overline{\zeta}_{1}\right)^{p-2}+\left(\rho^{(m_{0}p-m)/(p-1)}\overline{\zeta}_{2}\right)^{p-2}\right).

Thus, if $\displaystyle m$ satisfies (3.48), the above relation combined with (3.10), (3.4) that guarantees that $\displaystyle\mathcal{X}^{p}$ is an algebra and 3.5 yields

\left\|h_{1}-h_{2}\right\|_{L^{\infty}_{m}(0,T;L^{\infty}(\Omega))}\lesssim\left\|\rho^{(m_{0}p-m)/(p-1)}\left(\overline{\zeta}_{1}-\overline{\zeta}_{2}\right)\right\|_{\mathcal{X}^{p}}\left(\left\|\rho^{(m_{0}p-m)/(p-1)}\overline{\zeta}_{1}\right\|_{\mathcal{X}^{p}}^{p-2}+\left\|\rho^{(m_{0}p-m)/(p-1)}\overline{\zeta}_{2}\right\|_{\mathcal{X}^{p}}^{p-2}\right)\\ \lesssim\left\|\overline{\zeta}_{1}-\overline{\zeta}_{2}\right\|_{\mathcal{Y}}\left(\left\|\overline{\zeta}_{1}\right\|_{\mathcal{Y}}^{p-2}+\left\|\overline{\zeta}_{2}\right\|_{\mathcal{Y}}^{p-2}\right).

Therefore, using (3.39) and (3.62), we find

\left\|h_{1}-h_{2}\right\|_{L^{\infty}_{m}(0,T;L^{\infty}(\Omega))}\\ \lesssim\left\|F_{1}-F_{2}\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}^{1/(p-1)}\left(\left\|F_{1}\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}+\left\|F_{2}\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}+\left\|y_{0}\right\|_{L^{p^{\prime}}(\Omega)}\right)^{(p-2)/(p-1)}.

(3.63)

Note that $\displaystyle y_{1}-y_{2}$ satisfies the following system

\left\{\begin{array}[]{rl}\partial_{t}\left(\frac{y_{1}-y_{2}}{\rho^{m}}\right)-\Delta\left(\frac{y_{1}-y_{2}}{\rho^{m}}\right)=\frac{h_{1}-h_{2}}{\rho^{m}}\chi_{\omega}+\frac{F_{1}-F_{2}}{\rho^{m}}-m\frac{\rho^{\prime}}{\rho^{m+1}}(y_{1}-y_{2})&\mathrm{in}\ (0,T)\times\Omega,\\[11.38109pt] \frac{y}{\rho^{m}}=0&\mathrm{on}\ (0,T)\times\partial\Omega,\\[8.53581pt] \frac{y}{\rho^{m}}(0,\cdot)=\frac{y_{0}}{\rho^{m}(0)}&\mathrm{in}\ \Omega.\end{array}\right.

Now, we follow the same proof as in 3.8 and we use that $\displaystyle m=m_{1}/N$ satisfies (3.48) to deduce from (3.63) that

\left\|\frac{y_{1}-y_{2}}{\rho^{m_{1}/N}}\right\|_{\mathcal{X}^{q}}\lesssim\left\|F_{1}-F_{2}\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}\\ +R_{0}^{(p-2)/(p-1)}\left\|F_{1}-F_{2}\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}^{1/(p-1)}.

(3.64)

We then write

\displaystyle\left|\frac{y_{1}^{N}-y_{2}^{N}}{\rho^{m_{1}}}\right|\lesssim\frac{\left|y_{1}-y_{2}\right|}{\rho^{m_{1}/N}}\frac{\left|y_{1}\right|^{N-1}+\left|y_{2}\right|^{N-1}}{\rho^{m_{1}(N-1)/N}}

so that from Hölder’s inequality, we have

\displaystyle\left\|y_{1}^{N}-y_{2}^{N}\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}\lesssim\left\|y_{1}-y_{2}\right\|_{L^{qN}_{m_{1}/N}(0,T;L^{qN}(\Omega))}\left(\left\|y_{1}\right\|_{L^{qN}_{m_{1}/N}(0,T;L^{qN}(\Omega))}^{N-1}+\left\|y_{1}\right\|_{L^{qN}_{m_{1}/N}(0,T;L^{qN}(\Omega))}^{N-1}\right).

Combining this relation with the Sobolev embedding (2.2), (3.52), (3.64), we deduce that

\left\|y_{1}^{N}-y_{2}^{N}\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}\lesssim\left(\left\|F_{1}-F_{2}\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}+R_{0}^{(p-2)/(p-1)}\left\|F_{1}-F_{2}\right\|_{L^{p^{\prime}}_{m_{1}}(0,T;L^{p^{\prime}}(\Omega))}^{1/(p-1)}\right)\\ \times\left(\left\|F_{1}\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}+\left\|F_{2}\right\|_{L^{q}_{m_{1}}(0,T;L^{q}(\Omega))}+\left\|y_{0}\right\|_{W^{2/q^{\prime},q}(\Omega)}\right)^{N-1}

(3.65)

which implies the continuity of $\displaystyle\mathcal{N}$ . ∎

Remark 3.10.

In the above proof, let us remark that we show that the mapping $\displaystyle\mathcal{N}:B_{R_{0}}\to B_{R_{0}}$ is $\displaystyle\alpha$ -Hölder continuous with $\displaystyle\alpha=1/(p-1)$ (see (3.65)). It is not clear if this mapping is Lipschitz continuous or if we can show that for $\displaystyle R_{0}$ small enough it is contractive. As a consequence, in the proof of 3.1, we do not apply the Banach fixed-point theorem (as it can be done with the method proposed in [LTT13]) and we use instead the Schauder fixed-point theorem.

We are now in a position to prove 3.1.

Proof of 3.1.

From 3.9, if $\displaystyle R_{0}:=\left\|y_{0}\right\|_{W^{2/q^{\prime},q}(\Omega)}$ is small enough, then the mapping $\displaystyle\mathcal{N}:B_{R_{0}}\to B_{R_{0}}$ defined by (3.59) is continuous, where $\displaystyle B_{R_{0}}$ is the closed convex set defined by (3.60). Moreover, $\displaystyle\mathcal{N}(B_{R_{0}})$ is relatively compact in $\displaystyle B_{R_{0}}$ and we can thus apply the Schauder fixed point theorem to deduce the existence of a fixed point $\displaystyle F\in B_{R_{0}}$ . Setting $\displaystyle y=\mathcal{M}_{2}(y_{0},F)$ and $\displaystyle h=\mathcal{M}_{3}(y_{0},F)$ , we can apply 3.6, 3.7 and 3.8 and obtain that $\displaystyle h$ satisfies (1.16), that $\displaystyle y$ is the strong solution of (1.13) associating with $\displaystyle h$ and $\displaystyle y_{0}$ and that for any $\displaystyle m$ satisfying (3.48), $\displaystyle y/\rho^{m}\in\mathcal{X}^{q}$ , $\displaystyle y^{N}\in L^{q}_{m_{1}}(0,T;L^{q}(\Omega))$ , $\displaystyle h^{1/(2n+1)}\in\mathcal{X}^{p}$ , $\displaystyle h\in L_{m,0}^{\infty}(0,T;L^{\infty}(\Omega))$ together with the estimates (3.6). ∎

4 Proof of 1.1

The goal of this part is to prove the local null-controllability of (1.3).

Proof of 1.1.

As explained in the introduction, the proof is divided into two steps.

Step 1: control of the first equation in $\displaystyle(0,T/2)$ .

First we apply 2.5: there exists $\displaystyle\widetilde{\delta}>0$ small enough such that if

\left\|y_{2,0}\right\|_{L^{\infty}(\Omega)}\leqslant\widetilde{\delta},\quad\left\|g\right\|_{L^{\infty}((0,T/2)\times\Omega)}\leqslant\widetilde{\delta},

(4.1)

the system

\left\{\begin{array}[]{rl}\partial_{t}y_{2}-\Delta y_{2}=y_{2}^{N_{3}}+g&\mathrm{in}\ (0,T/2)\times\Omega,\\ y_{2}=0&\mathrm{on}\ (0,T/2)\times\partial\Omega,\\ y_{2}(0,\cdot)=y_{2,0}&\mathrm{in}\ \Omega,\end{array}\right.

(4.2)

admits a unique weak solution in the sense of 2.4. Now we apply 1.2 to

\left\{\begin{array}[]{rl}\partial_{t}y_{1}-\Delta y_{1}=y_{1}^{N_{1}}+h\chi_{\omega}&\mathrm{in}\ (0,T/2)\times\Omega,\\ y_{1}=0&\mathrm{on}\ (0,T/2)\times\partial\Omega,\\ y_{1}(0,\cdot)=y_{1,0}&\mathrm{in}\ \Omega.\end{array}\right.

(4.3)

There exists $\displaystyle\delta>0$ such that for any $\displaystyle y_{1,0}\in L^{\infty}(\Omega)$ with

\left\|y_{1,0}\right\|_{L^{\infty}(\Omega)}\leqslant\delta,

(4.4)

there exists a control $\displaystyle h\in L^{\infty}(0,T/2;L^{\infty}(\Omega))$ such that $\displaystyle y_{1}(T/2,\cdot)=0$ and

\displaystyle\left\|y_{1}\right\|_{L^{\infty}(0,T/2;L^{\infty}(\Omega))}\lesssim\left\|y_{1,0}\right\|_{L^{\infty}(\Omega)}.

Assuming (1.7) with $\displaystyle\delta>0$ possibly smaller, we have that

\displaystyle g:=y_{1}^{N_{2}}\in L^{\infty}((0,T)\times\Omega)

satisfies (4.1) so that we have obtained at this step a control $\displaystyle h\in L^{\infty}(0,T/2;L^{\infty}(\Omega))$ , such that (1.3) admits a weak solution $\displaystyle(y_{1},y_{2})$ in $\displaystyle(0,T/2)$ and $\displaystyle y_{1}(T/2,\cdot)=0$ . By using 2.6, $\displaystyle y_{2,T/2}:=y_{2}(T/2,\cdot)$ satisfies

\displaystyle\left\|y_{2,T/2}\right\|_{L^{\infty}(\Omega)}\lesssim\delta.

Step 2: control of the second equation in $\displaystyle(T/2,T)$ through a fictitious odd control.

By taking $\displaystyle\delta>0$ possibly smaller, we can apply 1.2 to

\left\{\begin{array}[]{rl}\partial_{t}y_{2}-\Delta y_{2}=H\chi_{\omega}+y_{2}^{N_{3}}&\mathrm{in}\ (T/2,T)\times\Omega,\\ y_{2}=0&\mathrm{on}\ (T/2,T)\times\partial\Omega,\\ y_{2}(T/2,\cdot)=y_{2,T/2}&\mathrm{in}\ \Omega.\end{array}\right.

(4.5)

We deduce the existence of a control $\displaystyle H$ such that $\displaystyle y_{2}(T,\cdot)=0$ and such that

\displaystyle H^{1/N_{2}}\in L^{p}(T/2,T;W^{2,p}(\Omega))\cap W^{1,p}(T/2,T;L^{p}(\Omega)),\quad H^{1/N_{2}}(T/2,\cdot)=H^{1/N_{2}}(T,\cdot)=0.

We then set, in $\displaystyle(T/2,T)$ ,

\displaystyle y_{1}:=\left(H\chi_{\omega}\right)^{1/N_{2}},\quad h:=\partial_{t}y_{1}-\Delta y_{1}-y_{1}^{N_{1}}\in L^{p}((T/2,T)\times\Omega).

Concatenating $\displaystyle y_{1}$ , $\displaystyle y_{2}$ and $\displaystyle h$ between the two steps, we can check that $\displaystyle h\in L^{p}((0,T)\times\Omega)$ , that $\displaystyle(y_{1},y_{2})$ is the weak solution of (1.3) and that (1.9) holds. This concludes the proof of 1.1. ∎

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	$\displaystyle\displaystyle\left\\|\left(\rho^{M_{2}}\right)^{\prime}\zeta\right\\|_{L^{q_{1}}(0,T;L^{q_{1}}(\Omega))}+\left\\|\rho^{M_{2}}g\right\\|_{L^{q_{1}}(0,T;L^{q_{1}}(\Omega))}$	$\displaystyle\displaystyle\lesssim\left\\|\rho^{M_{1}}\zeta\right\\|_{\mathcal{X}^{2}}+\left\\|\rho^{m_{0}}g\right\\|_{L^{q_{1}}(0,T;L^{q_{1}}(\Omega))}$
		$\displaystyle\displaystyle\lesssim\left\\|\rho^{m_{0}}g\right\\|_{L^{q_{1}}(0,T;L^{q_{1}}(\Omega))}+\left\\|\rho_{0}^{m_{0}}\zeta\chi_{\omega}\right\\|_{L^{2}(0,T;L^{2}(\Omega))}.$		(3.23)

Null-controllability of cascade reaction-diffusion systems with odd coupling terms

Abstract

1 Introduction

1.1 Main results

Theorem 1.1.

Theorem 1.2.

1.2 Outline of the paper

2 Well-posedness and regularity results for the heat equation

2.1 Functional spaces

Lemma 2.1.

Proof.

2.2 Linear heat equation

Definition 2.2.

Theorem 2.3.

2.3 Semi-linear heat equation

Definition 2.4.

Theorem 2.5.

Proof.

Lemma 2.6.

Proof.

Definition 2.7.

Theorem 2.8.

3 Proof of 1.2

Theorem 3.1.

3.1 Carleman estimate and L2\displaystyle L^{2} observability inequality for the heat equation

Theorem 3.2.

Corollary 3.3.

Proof of 3.3.

3.2 A weighted Lp\displaystyle L^{p} observability inequality

Proposition 3.4.

Proof.

3.3 Controllability of the heat equation with a source term in Lp′\displaystyle L^{p^{\prime}}

Lemma 3.5.

Proof.

Proposition 3.6.

Proof of 3.6.

3.4 L∞\displaystyle L^{\infty} bound on the control and Lq\displaystyle L^{q} estimate of the nonlinearity

Lemma 3.7.

Proof of 3.7.

Proposition 3.8.

Proof.

3.5 A Schauder fixed-point argument

Proposition 3.9.

Proof.

Remark 3.10.

Proof of 3.1.

4 Proof of 1.1

Proof of 1.1.

Step 1: control of the first equation in (0,T/2)\displaystyle(0,T/2).

Step 2: control of the second equation in (T/2,T)\displaystyle(T/2,T) through a fictitious odd control.

References

3.1 Carleman estimate and $\displaystyle L^{2}$ observability inequality for the heat equation

3.2 A weighted $\displaystyle L^{p}$ observability inequality

3.3 Controllability of the heat equation with a source term in $\displaystyle L^{p^{\prime}}$

3.4 $\displaystyle L^{\infty}$ bound on the control and $\displaystyle L^{q}$ estimate of the nonlinearity

Step 1: control of the first equation in $\displaystyle(0,T/2)$ .

Step 2: control of the second equation in $\displaystyle(T/2,T)$ through a fictitious odd control.