A Note on the Maximum Principle-based Approach for ISS Analysis of Higher Dimensional Parabolic PDEs with Variable Coefficients

Since the last decade, the ISS theory for infinite dimensional systems governed by partial differential equations (PDEs) has drawn much attention in the literature of PDE control. A comprehensive survey on this topic is presented in [18]. It is worth noting that the extension of the notion of ISS for finite dimensional systems originally introduced by Sontag in the late 1980’s to infinite dimensional systems with distributed in-domain disturbances is somehow straightforward, while the investigation on the ISS properties with respect to (w.r.t.) boundary disturbances is much more challenging. In recent years, different methods have been developed for ISS analysis of PDE systems with boundary disturbances, including, e.g.:

1. (i)

   the semigroup and admissibility methods for ISS of certain linear or nonlinear parabolic PDEs [3, 4, 5, 6, 7, 20];

2. (ii)

   the approach of spectral decomposition and finite-difference scheme for ISS of PDEs governed by Sturm-Liouville operators [8, 9, 10, 11, 12];

3. (iii)

   the Riesz-spectral approach for ISS of Riesz-spectral systems [15, 16];

4. (iv)

   the monotonicity-based method for ISS of certain nonlinear PDEs with Dirichlet boundary disturbances [17];

5. (v)

   the method of De Giorgi iteration for ISS of certain nonlinear PDEs with Dirichlet boundary disturbances [23, 25];

6. (vi)

   the application of variations of Sobolev embedding inequalities for ISS of certain nonlinear PDEs with Robin or Neumann boundary disturbances [24, 25, 26];

7. (vii)

   the maximum principle-based approach for ISS of certain nonlinear PDEs with different types of boundary conditions [27, 28].

Although a rapid progress on ISS theory has been obtained, it is still a challenging issue for ISS analysis of nonlinear PDE systems defined over higher dimensional domains with variable coefficients and different types of nonlinear boundary conditions. For example, the methods in (i) can be applied to certain linear or nonlienar PDEs, while it may be difficult to apply them to non-diagonal systems as the one given by, e.g., (2) and (3) of this paper. The methods in (ii) and (iii) are effective for ISS analysis of linear PDE systems over one dimensional domains. Whereas, these approaches may involve heavy computations for nonlinear PDEs or PDEs on multidimensional spatial domains. The methods in (iv)-(vi) are suitable for ISS analysis of parabolic PDEs with Dirichlet or Robin or Neumann boundary disturbances, while they cannot be used for PDEs with mixed boundary conditions.

It has been demonstrated in [27] and [28] that the method in (vii) is applicable for ISS analysis of certain nonlinear parabolic PDEs, with different types of boundary disturbances, or over higher dimensional domains. Therefore, the aim of this paper is put on the application of this approach to ISS analysis for a class of nonlinear parabolic PDEs defined over higher dimensional domains with variable coefficients under different types of nonlinear boundary conditions simultaneously.

The proposed method for achieving the ISS estimates of the solutions will be based on the the Lyapunov method and the maximum estimates for nonlinear parabolic PDEs with nonlinear boundary conditions. Specifically, we set up in the first step several maximum estimates of the solutions to the considered nonlinear parabolic PDEs with a nonlinear Robin or Dirichlet boundary condition by means of the weak maximum principle. In the second step, applying the technique of splitting as in [2, 23, 27, 28], we consider a nonlinear equation with the initial data free and establish the maximum estimate of the solution (denoted by $v$) according to the result obtained in the first step. By denoting the solution of the target system by $u$, then in the third step, we establish the $L^{2}$-estimate of $u-v$ by the Lyapunov method. Finally, the ISS estimate of the target system in $L^{2}$-norm, i.e., the estimate of $u$ in $L^{2}$-norm, is guaranteed by the maximum estimate of $v$ and the $L^{2}$-estimate of $u-v$. It’s worthy noting that combining with other approaches or techniques, the Lyapunov method was also applied for the ISS analysis of PDE systems in [4] by constructing non-coercive Lyapunov functions based on ISS characterizations devised in [19], and in [20, 21] and the literature mentioned in (iv)-(vii) by constructing coercive Lyapunov functions.

In the rest of the paper, Section 2 presents the problem statement, the basic assumptions, and the main result. By the weak maximum principle, some maximum estimates for nonlinear parabolic PDEs with nonlinear Robin and Dirichlet boundary conditions are proved respectively in Section 3. ISS analysis of nonlinear parabolic PDEs with different boundary conditions are detailed in Section 4. In order to illustrate the application of the approach presented in this paper, an example of ISS estimates for a parabolic equation with respectively a nonlinear Robin boundary condition and a nonlinear Dirichlet boundary condition is provided in Section 5, followed by some concluding remarks given in Section 6.

Notations: In this paper, $\mathbb{R}_{+}$ denotes the set of positive real numbers and $\mathbb{R}_{\geq 0}:={}{\{0\}}\cup\mathbb{R}_{+}$.

Let $B_{R}$ be a ball in $\mathbb{R}^{n}(n\geq 1)$ with the centre at $0$ and a radius $R>0$, i.e., $B_{R}=\{x\in\mathbb{R}^{n}||x|<R\}$. We denote by $\partial B_{R}$ and $\overline{B}_{R}$ the boundary and the closure of $B_{R}$, respectively. Denote by $|B_{R}|$ the $n$-dimensional Lebesgue measure of $B_{R}$, i.e., $|B_{R}|=\frac{\pi^{\frac{n}{2}}R^{n}}{\frac{n}{2}\Gamma(\frac{n}{2})}$ with the Gamma function $\Gamma(\cdot)$ defined on $\mathbb{R}$. Denote by $|\partial B_{R}|$ the $(n-1)$-dimensional Lebesgue measure of $\partial B_{R}$, i.e., $|\partial B_{R}|=\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}R^{n-1}$.

For any $T>0$, let $Q_{T}=B_{R}\times(0,T)$ and $\partial_{p}Q_{T}=(\partial B_{R}\times(0,T))\cup(\overline{B}_{R}\times\{{}{0}\})$.

We use $\|\cdot\|$ to denote the norm $\|\cdot\|_{L^{2}(B_{R})}$ in $L^{2}(B_{R})$.

Let $\mathcal{K}=\{\gamma:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}|\ \gamma(0)=0,\gamma$ is continuous, strictly increasing$\}$; $\mathcal{K}_{\infty}=\{\theta\in\mathcal{K}|\ \lim\limits_{s\rightarrow\infty}\theta(s)=\infty\}$; $\mathcal{L}=\{\gamma:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}|\ \gamma$ is continuous, strictly decreasing, $\lim\limits_{s\rightarrow\infty}\gamma(s)=0\}$; $\mathcal{K}\mathcal{L}=\{\mu:\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}|\ \mu\in\mathcal{K},\forall t\in\mathbb{R}_{\geq 0}$, and $\mu(s,\cdot)\in\mathcal{L},\forall s\in{\mathbb{R}_{+}}\}$.

Given the following functions:

we consider the following nonlinear parabolic equation with variable coefficients:

where $L_{t}[u]:=u_{t}-\text{div}\ (a\nabla u)+\textbf{b}\cdot\nabla u+cu$ with $\textbf{b}:=(b_{1},b_{2},\ldots,b_{n})$, $N[u]:=h(\cdot,\cdot,u)$ is the nonlinear term of the equation, $\mathscr{B}[u]$ is given by:

or

or

with $\bm{\nu}=\frac{1}{R}x=\frac{1}{R}(x_{1},x_{2},\ldots,x_{n})$, which is the unit outer normal vector at the point $x\in\partial B_{R}$.

In general, $f$ and $d$ represents the distributed in-domain disturbance and boundary disturbance, respectively. (3), (4) and (5) represent the nonlinear Robin, Neumann and Dirichlet boundary condition, respectively.

Throughout this paper, without special statements, we always denote by $x,t$ respectively the first and second variable (if any) of the functions $a,b_{i}\ (i=1,2,\ldots,n),c,h,f,d,\phi$. Moreover, we always assume that $a,b_{i}\ (i=1,2,\ldots,n),c,h,\psi,f,d,\phi$ are given by (1) and satisfy for some $\underline{a},\overline{a},\underline{b},\overline{b},\underline{c}\in\mathbb{R}_{+}$:

where $C_{\text{Trace}}$ is the best constant of the trace embedding inequality given by the Trace Theorem in the appendix, and

for all $(x,t)\in B_{R}\times\mathbb{R}_{+}$ and all $u,v,w\in\mathbb{R}$ with $u<v$.

Furthermore, we impose the following compatibility condition:

Let $\mathbb{Y}:=C^{2}(\overline{B}_{R}\times\mathbb{R}_{\geq 0};\mathbb{R})$, $\mathbb{D}_{0}:=\{d\in\mathbb{Y}|d(\cdot,0)=0\ \text{on}\ \partial B_{R}\}$ and $\mathbb{U}_{0}:=\{u\in\mathbb{Y}|\mathscr{B}[u](\cdot,0)=d(\cdot,0)\ \text{on}\ \partial B_{R}\ \text{for}\ d\in\mathbb{D}_{0}\}$.

The main result of this paper is stated in the following theorem.

Proof of Theorem 1 We proceed on the proof in the following 3 steps.

(i) We establish an EISS estimate of the solution to (2) with the nonlinear Robin boundary condition (3).

Let $v\in{}{C^{2,1}(\overline{Q}_{T})}$ be the unique solution of the following parabolic equation:

According to Proposition 3, we have

where $R_{0}=\frac{R}{2}+\frac{1}{\underline{c}R}(\overline{a}n+R\overline{a}+R\overline{b})$.

Let $w=u-v$. It is obvious that $w$ satisfies:

Multiplying (11) with $w$ and integrating by parts, we have

Applying the formula of integration by parts, the Trace Theorem (see the appendix) and by (6b), we have

By (7b) and (7a), we always have

Thus, we obtain by (6a), (6c) and (6d)

which gives $\|w(\cdot,T)\|^{2}\leq\|\phi\|^{2}\e^{-\big{(}2\underline{c}-\overline{b}\left(1+2C^{2}_{\text{Trace}}\right)\big{)}T}.$

Finally, we have

(ii) We establish an EISS estimate of the solution to (2) with the nonlinear Neumann boundary condition (4). Let $v\in{}{C^{2,1}(\overline{Q}_{T})}$ be the unique solution of the following parabolic equation:

According to Proposition 3, we have

where $R_{0}=\frac{R}{2}+\frac{1}{\underline{c}R}(\overline{a}n+R\overline{a}+R\overline{b})$.

Let $w=u-v$, which satisfies:

By Young’s inequality, the Trace Theorem and (6a), it follows that

Proceeding in the same way as in (i), we get

where we choose $\varepsilon>0$ small enough such that $\lambda=2\underline{c}-\overline{b}(1+2C^{2}_{\text{Trace}})-2\varepsilon C^{2}_{\text{Trace}}>0$ and $\overline{b}C^{2}_{\text{Trace}}-\underline{a}-\varepsilon C^{2}_{\text{Trace}}>0$ due to (6d).