On the hardy-Hénon heat equation with an inverse square potential

Divyang G. Bhimani Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India divyang.bhimani@iiserpune.ac.in , Saikatul Haque Department of Mathematics, University of California
Los Angeles
CA 90095
USA
&
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211019, India saikatul@math.ucla.edu, saikatulhaque@hri.res.in and Masahiro Ikeda Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan & Center for Advanced Intelligence Project RIKEN, Japan. masahiro.ikeda@keio.jp, masahiro.ikeda@riken.jp

Abstract.

We study Cauchy problem for the Hardy-Hénon parabolic equation with an inverse square potential, namely,

\partial_{t}u-\Delta u+a|x|^{-2}u=|x|^{\gamma}F_{\alpha}(u),

where $a\geq-(\frac{d-2}{2})^{2},$ $\gamma\in{\mathbb{R}}$ , $\alpha>1$ and $F_{\alpha}(u)=\mu|u|^{\alpha-1}u,\mu|u|^{\alpha}$ or $\mu u^{\alpha}$ , $\mu\in\{-1,0,1\}$ . We establish sharp fixed time-time decay estimates for heat semigroups $e^{-t(-\Delta+a|x|^{-2})}$ in weighted Lebesgue spaces, which is of independent interest. As an application, we establish:

•

Local well-posedness (LWP) in scale subcritical and critical weighted Lebesgue spaces.
•

Small data global existence in critical weighted Lebesgue spaces.
•

Under certain conditions on $\gamma$ and $\alpha,$ we show that local solution cannot be extended to global one for certain initial data in the subcritical regime. Thus, finite time blow-up in the subcritical Lebesgue space norm is exhibited.
•

We also demonstrate nonexistence of local positive weak solution (and hence failure of LWP) in supercritical case for $\alpha>1+\frac{2+\gamma}{d}$ the Fujita exponent. This indicates that subcriticality or criticality are necessary in the first point above.

In summary, we establish a sharp dissipative estimate and addresses short and long time behaviors of solutions. In particular, we complement several classical results and shed new light on the dynamics of the considered equation.

Key words and phrases:

Hardy-Hénon equation, Inverse square potential, Dissipative estimate, well-posesness, finite time blow up

2000 Mathematics Subject Classification:

35K05, 35K67, 35B30, 35K57

1. Introduction

1.1. Fixed-time estimates for heat semigroup $e^{-t(-\Delta+a|x|^{-2})}$

Consider the linear heat equation associated with the inverse square potential, namely

\begin{cases}\partial_{t}u(t,x)+\mathcal{L}_{a}u(t,x)=0\\ u(0,x)=u_{0}(x)\end{cases}(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{d},

(1.1)

where $u(t,x)\in\mathbb{C}.$ In this paper, we assume that $a\geq a_{*}:=-(\frac{d-2}{2})^{2},d\geq 2,$ unless it is explicitly specified. The Schrödinger operator with inverse square potentials

{\mathcal{L}}_{a}=-\Delta+a|x|^{-2}

is initially defined with domain $C^{\infty}({\mathbb{R}}^{d}\setminus\{0\})$ . Then it is extended as an unbounded operator in weighted Lebesgue space $L_{s}^{q}({\mathbb{R}}^{d})$ that generates a positive semigroup $\{e^{-t{\mathcal{L}}_{a}}\}_{t\geq 0}$ provided $1<q<\infty$ and $\sigma_{-}<\frac{d}{q}+s<\sigma_{+}+2,$ where $\sigma_{-}$ , $\sigma_{+}$ defined by

\sigma_{\mp}=\sigma_{\mp}(d,a):=\frac{d-2}{2}\mp\frac{1}{2}\sqrt{(d-2)^{2}+4a}

(1.2)

are the roots of $s^{2}-(d-2)s-a=0$ , see [24, Theorems 3.2, 3.3]. Here, the weighted Lebesgue space $L_{s}^{q}({\mathbb{R}}^{d})$ is defined by the norm $\|f\|_{L^{q}_{s}}:=\||\cdot|^{s}f\|_{L^{q}}\ (s\in\mathbb{R}).$

The study of $\mathcal{L}_{a}$ is motivated from physics and mathematics spanning areas such as combustion theory, the Dirac equation with Coulomb potential, quantum mechanics and the study of perturbations of classic space-time metrics. See e.g. [30, 21, 8] and the references therein.

The aim of this article is to understand the dynamics of solutions of Hardy-Hénon heat equations (1.1) and (1.6) when a singular potential is present, in light of the research programme initiated by Zhang [32], Pinsky [27, 28], Ioku et al. in [17, 18], Ishige [19] and Ishige-Kawakami in [20], and Bhimani-Haque [3] (cf. [5, 6, 4]). We also note that there is a extensive literature on Hardy-Hénon heat equation without potential, i.e. (1.6) with $a=0,$ we refer to recent work of Chikami et al. in [10, 9] and the references therein, see also Remark 1.1.

We begin by stating our dissipative estimates in weighted Lebesgue spaces in the following theorem.

Theorem 1.1.

Let $\sigma_{-},\sigma_{+}$ be as defined in (1.2). Let $s_{1},s_{2}\in{\mathbb{R}}$ and $q_{1},q_{2}\in(1,\infty)$ . Then

\|e^{-t\mathcal{L}_{a}}f\|_{L^{q_{2}}_{s_{2}}}\leq Ct^{-\frac{d}{2}\left(\frac{1}{q_{1}}-\frac{1}{q_{2}}\right)-\frac{s_{1}-s_{2}}{2}}\|f\|_{L^{q_{1}}_{s_{1}}}\qquad\forall\ t>0,\ \forall\ f\in L^{q_{1}}_{s_{1}}({\mathbb{R}}^{d})

(1.3)

if and only if

\sigma_{-}<\frac{d}{q_{2}}+s_{2}\leq\frac{d}{q_{1}}+s_{1}<\sigma_{+}+2,

(1.4)

and

s_{2}\leq s_{1}.

(1.5)

Remark 1.1.

Theorem 1.1 deserve several comments.

(1)
The case $a=0$ : In this case $e^{-t\mathcal{L}_{0}}f=e^{t\Delta}f=k_{t}\ast f$ (where $k_{t}:=t^{-d/2}\exp(-\frac{|\cdot|^{2}}{4t})$ ) and $\sigma_{-}=0,\sigma_{+}+2=d$ .
- •
  
  subcase $s_{1},s_{2}=0$ : The sufficiency part (1.3) is a consequence of Young’s convolution inequality. See [25, Lemma 3.1]. This argument holds even if we replace strict inequities in (1.4) by equalities and thus $q_{1},q_{2}$ can take the extreme values $1,\infty$ .
- •
  
  subcase $s_{1}$ or $s_{2}\in{\mathbb{R}}\setminus\{0\}$ : For $q_{1}\leq q_{2}$ , this is due to Chikami-Ikeda-Taniguchi [9, Lemma 2.1]. Theorem 1.1 removes the assumption $q_{1}\leq q_{2}$ in [9, Lemma 2.1].
(2)
The case $a\in[a_{*},\infty)$ :
- •
  
  subcase $s_{1},s_{2}=0$ : In this subcase, the sufficiency part (1.3) is due to Ioku-Metafune-Sobajima-Spina [17, Theorem 5.1]. However, their method of proof is different than ours, which rely on embedding theorems and interpolation techniques. The
- •
  
  subcase $s_{1}$ or $s_{2}\in{\mathbb{R}}\setminus\{0\}$ : In this case, both necessity and sufficiency part of Theorem 1.1 is new. This is the main contribution of this article.
(3)

The power of $t$ in (1.3) is optimal which follow by a standard scaling argument, see Lemma 3.1.
(4)

Using Symmetry (in $x,y$ variable) of heat kernel $g_{a}(t,x,y)$ (see Subsection 2.2) associated with the operator $e^{-t{\mathcal{L}}_{a}}$ , it follows by duality and the relation $\sigma_{+}+2=d-\sigma_{-}$ that (1.3) holds for $(q_{1},s_{1},q_{2},s_{2})$ if and only if (1.3) holds for $(q_{2}^{\prime},-s_{2},q_{1}^{\prime},-s_{1})$ (here $q_{j}^{\prime}$ is the Hölder conjugate of $q_{j}$ ).
(5)

For $s_{1}=-\sigma_{-}$ , Theorem 1.1 holds even for end point cases $q_{1}\in\{1,\infty\}$ (hence allowing equality in the last strict inequality in (1.4)). For $s_{2}=\sigma_{-}$ , Theorem 1.1 holds even for end point cases $q_{2}\in\{1,\infty\}$ (hence allowing equality in the first strict inequality in (1.4)).
(6)

It is indispensable to consider weighted Lebesuge spaces in Theorem 1.1 in order to treat Hénon potential $|x|^{\gamma}\ (\gamma>0)$ while establishing well-posedness for (1.6).

1.2. Hardy-Hénon equations (HHE) with inverse-square potential

We consider (1.1) with an inhomogeneous power type nonlinearity:

\begin{cases}\partial_{t}u(t,x)+\mathcal{L}_{a}u(t,x)=|x|^{\gamma}F_{\alpha}(u(t,x))\\ u(x,0)=u_{0}(x)\end{cases}(t,x)\in[0,T)\times{\mathbb{R}}^{d},

(1.6)

where $\gamma\in{\mathbb{R}}$ , $T\in(0,\infty],$ and $\alpha>1$ and $u(x,t)\in\mathbb{R}$ or $u(x,t)\in\mathbb{C}.$ We assume that the non-linearity function $F_{\alpha}:{\mathbb{C}}\rightarrow{\mathbb{C}}$ satisfies the following conditions:

\displaystyle\begin{cases}|F_{\alpha}(z)-F_{\alpha}(w)|\leq C_{0}(|z|^{\alpha-1}+|w|^{\alpha-1})|z-w|&for\ z,w\in\mathbb{C}\\ F_{\alpha}(0)=0.\end{cases}

(1.7)

The typical examples of $F_{\alpha}$ would be

F_{\alpha}(z)=\mu|z|^{\alpha-1}z,\ \mu|z|^{\alpha}\text{ or }\ \mu z^{\alpha}\quad(\mu\in\mathbb{R}).

The potential $|x|^{\gamma}$ is called Hénon type if $\gamma>0$ and is called Sobolev type if $\gamma<0$ . The equation (1.6) with $\gamma<0$ is known as a Hardy parabolic equation, while that with $\gamma>0$ is known as a Hénon parabolic equation. Equation (1.6) is called Hardy-Hénon parabolic equation with an inverse square potential. The elliptic part of (1.6) when $a=0$ , i.e.

-\Delta u=|x|^{\gamma}|u|^{\alpha-1}u

was proposed by Hénon [14] as a model to study the rotating stellar systems and has been extensively studied in scientific community, see e.g. [12].

The equation (1.6) is invariant under the following scale transformation:

u_{\lambda}(t,x):=\lambda^{\frac{2+\gamma}{\alpha-1}}u(\lambda^{2}t,\lambda x),\quad\lambda>0.

More precisely, if $u$ is a solution to (1.6), then so is $u_{\lambda}$ with the rescaled initial data $\lambda^{\frac{2+\gamma}{\alpha-1}}u_{0}(\lambda x)$ . Then the following identity holds

\|u_{\lambda}(0)\|_{L^{q}_{s}}=\lambda^{-s+\frac{2+\gamma}{\alpha-1}-\frac{d}{q}}\|u_{0}\|_{L^{q}_{s}},\quad\lambda>0.

Hence, if $q$ and $s$ satisfy

s+\frac{d}{q}=\frac{2+\gamma}{\alpha-1},

then the identity $\|u_{\lambda}(0)\|_{L_{s}^{q}}=\|u_{0}\|_{L_{s}^{q}}$ holds for any $\lambda>0$ , i.e., the norm $\|u_{\lambda}(0)\|_{L^{q}_{s}}$ is invariant with respect to $\lambda$ . Denote

\tau=\tau(q,s,d):=s+\frac{d}{q}\quad\text{and}\quad\tau_{c}=\tau_{c}(\gamma,\alpha):=\frac{2+\gamma}{\alpha-1}.

(1.8)

We say Cauchy problem (1.6) scale

\displaystyle L_{s}^{q}-\begin{cases}subcritical&\text{if}\ \ \tau<\tau_{c}\\ critical&\text{if}\ \ \tau=\tau_{c}\\ supercritical&\text{if}\ \ \tau>\tau_{c}\end{cases}.

(1.9)

Remark 1.2.

For $\tau=\tau_{c},$ we get $s=\frac{2+\gamma}{\alpha-1}-\frac{d}{q}=:s_{c}(q,\gamma,\alpha,d)$ (often denoted by $s_{c}$ for shorthand). In particular, when $s=s_{c}=0,$ $\gamma\geq-2$ , we have $q=q_{c}:=\frac{d(\alpha-1)}{2+\gamma}=\frac{d}{\tau_{c}}.$ So $L^{\frac{d(\alpha-1)}{2+\gamma}}({\mathbb{R}}^{d})$ is the critical Lebesgue space without weight.

We recall the notion of well-posedness in the sense of Hadamard.

Definition 1.1 (well-posedness).

Let $T\in(0,\infty],s\in\mathbb{R}$ and $1\leq q\leq\infty.$

–

We say that $u$ is an $L_{s}^{q}$ -integral solution on $[0,T)$ to (1.6) if $u\in C([0,T);L^{q}_{s}({\mathbb{R}}^{d}))$ and satisfies

$u(t)=e^{-t\mathcal{L}_{a}}u_{0}+\int_{0}^{t}e^{-(t-\tau)\mathcal{L}_{a}}[|\cdot|^{\gamma}F_{\alpha}(u(\tau))]d\tau$ (1.10)

for any $t\in[0,T)$ . Maximum of such $T$ is denoted by $T_{m}$ .
–
Let $X,Y\subset\mathcal{S}^{\prime}({\mathbb{R}}^{d})$ be Banach spaces. Then (1.6) is called locally well-posed (in short LWP) from $X$ to $Y$ if, for each bounded $B\subset X$ , there exist $T>0$ and a Banach space $X_{T}\hookrightarrow C([0,T],Y)$ so that
1. (a)
  
  for all $u_{0}\in B$ , (1.6) has a unique integral solution $u\in X_{T}$
2. (b)
  
  $u_{0}\mapsto u$ is continuous from $(B,\|\cdot\|_{X})$ to $C([0,T],Y).$
If $X=Y$ we say (1.6) is locally well-posed in $X$ . If $T=\infty$ , then we say (1.6) is globally well-posed in $X$ .

Remark 1.3.

We briefly mention some history on several facets of (1.6). We define Fujita exponent by

\alpha_{F}=\alpha_{F}(d,\gamma,a)=1+\frac{(2+\gamma)^{+}}{\sigma_{+}+2}

which is often known to divide the existence and nonexistence of positive global solutions.

(1)
By taking $a=\gamma=0$ and $F_{\alpha}(z)=z^{\alpha}$ in (1.6), we get classical heat equation

$\displaystyle\partial_{t}u-\Delta u=u^{\alpha},\quad u(0)=u_{0}.$ (1.11)

We recall following known results for (1.11):
1. (a)
  
  Let $q_{c}$ be as in Remark 1.2. If $q\geq q_{c}$ and $q>1$ or $q>q_{c}$ and $q\geq 1,$ Weissler [1] proved the existence of a unique local solution $u\in C([0,T),L^{q}({\mathbb{R}}^{d}))\cap L^{\infty}_{loc}(0,T],L^{\infty}({\mathbb{R}}^{d})).$ Later on, Brezis-Cazenave [7] proved the unconditional uniqueness of Weissler’s solutions.
2. (b)
  
  If $q<q_{c}$ , there are indications that there exists no (local) solution in any reasonable weak sense, see [1, 7, 31]. Moreover, it is known that uniqueness is lost for the initial data $u_{0}=0$ and for $1+\frac{1}{d}<q<\frac{d+2}{d-2}$ , see [13].
3. (c)
  
  Fujita [11] proved, for $1<\alpha<\alpha_{F}(d,0,0),$ (1.11) has no global solution (i.e. every solution blows up in finite time in $L^{\infty}-$ norm), whereas for $\alpha>\alpha_{F}(d,0,0),$ classical solution is global for small data.
(2)

Taking $a=0,F_{\alpha}(z)=z|z|^{\alpha-1}$ in (1.6), we get classical Hardy-Hénon heat equation

$\partial_{t}u-\Delta u=|x|^{\gamma}u|u|^{\alpha-1},\quad u(0)=u_{0}.$ (1.12)

In this case, Chikami et al. in [9] introduced weighted Lebesuge space $L^{q}_{s}({\mathbb{R}}^{d})$ to treat potential $|x|^{\gamma},$ and establish well-posedness results. Later, Chikami et al. in [10] generalize these results in weighted Lorentz spaces. In this paper, we could establish analogue of these results in the presence of potential, i.e. for (1.6) with $a\neq 0$ and relaxed conditions on other parameters $\gamma,\alpha,q,s$ . See Remarks 1.4 and 1.7 below.
(3)
Several authors considered (1.6) with some mild restriction on external potential:

$\displaystyle\partial_{t}u-\Delta u-V(x)u=b(x)u^{\alpha},\quad u(0)=u_{0},$ (1.13)

and showed sharp contrast between existence of classical global solution and finite time blow-up in $L^{\infty}-$ norm by finding appropriate Fujita exponent. We recall some of them here:
1. (a)
  
  Let $V(x)=\frac{a}{|x|^{2}}$ and $b\in C^{\beta}({\mathbb{R}})$ ( $\beta\in(0,1]$ ) with $b(x)\sim|x|^{\gamma}$ for large $|x|$ . In this case, for $1<\alpha\leq\alpha_{F}(\gamma,d,a),$ Pinsky [28, p.153] proved (1.6) does not posses global solution for any $u_{0}>0,$ and establish classical global solutions for $\alpha>\alpha_{F}(\gamma,d).$ See [28, p.153], [27, Theorem 1].
2. (b)
  
  Let $d\geq 3,$ $\alpha=\alpha_{F}(d,0)$ or $1$ , and $V(x)=\frac{a}{1+|x|^{b}}\ (b>0)$ in (1.13). In this case, Zhang [32] found Fujita exponents under certain conditions on $a,b$ . Later, Ishige [19, Theorems 1.1, 1.2] considered $d\geq 2,$ and potential $V(x)=\frac{a}{|x|^{2}}$ with $a>0,$ and $b=1$ and determined the Fujita exponent $\alpha_{F}(d,0,a)$ . See also recent work of Ishige and Kawakami in [20].

1.3. Dynamics of HHE with inverse square potential

We are now ready to state our well-posedness result in the following theorem.

Theorem 1.2 (Well-posedness: subcritical and critical case).

Let $q\in(1,\infty)$ and $\sigma_{-},\sigma_{+}$ be as defined in (1.2). Let

\gamma\in\begin{cases}(-2,\infty)&\text{if}\ a\leq 0\\ {\mathbb{R}}&\text{if}\ a>0\end{cases}

(1.14)

and $\alpha$ satisfies

\displaystyle\alpha\in\begin{cases}\ \quad\left(1,1+\frac{\gamma+2}{\sigma_{-}}\right)&\text{if}\ a\leq 0\\ \left(1+\max\left(\frac{\gamma+2}{\sigma_{-}},0\right),\infty\right)&\text{if}\ a>0\end{cases}.

(1.15)

Let $s\geq\frac{\gamma}{\alpha-1}$ , $s>\sigma_{-}-\frac{d}{\alpha}$ and $\tau,\tau_{c}$ be as in (1.8) and satisfy

\sigma_{-}<\tau<\sigma_{+}+2\qquad\text{and}\qquad\tau\leq\tau_{c}.

(1.16)

Then Cauchy problem (1.6) is locally well-posed in $L_{s}^{q}({\mathbb{R}}^{d}),$ and for the critical case we also have small data global existence. In the subcritical case, if we impose further restriction

q>\alpha\qquad\text{and}\qquad\tau<\frac{\sigma_{+}+2+\gamma}{\alpha},

(1.17)

then one has uniqueness in $C([0,T_{m}),L_{s}^{q}({\mathbb{R}}^{d}))$ .

(a) The case

d=3

a=-\frac{15}{64}

\gamma=\frac{1}{10}

and

s\geq\frac{1}{\alpha-1},s>\frac{3}{8}-\frac{3}{\alpha}

(b) The case

d=3

a=\frac{3}{4}

\gamma=1

and

s\geq\frac{1}{\alpha-1},s>-\frac{1}{2}-\frac{3}{\alpha}

Figure 1. Local well-posedness in

L_{s}^{q}({\mathbb{R}}^{d})

occurs in the deep & medium dark region by Theorem 1.2 (only the boundary

\tau=\tau_{c}

is included). Uniqueness in mere

L_{s}^{q}({\mathbb{R}}^{d})

is guaranteed by Theorem 1.2 (furthermore part) in the open deep dark region. No LWP in the unbounded lightest regionby Theorem 1.4.

Theorem 1.2 is new for $a\neq 0$ and $\gamma>0.$ Up to now, we could not know the well-posedness of (1.6) with $\gamma>0$ in the mere $L^{q}-$ spaces but in weighted $L^{q}_{s}-$ spaces. See Remark 1.1(6). We prove Theorem 1.2 via fixed point argument. To this end, the main new ingredient required is our fixed-time estimate established in Theorem 1.1.

Remark 1.4.

We have several comments on Theorem 1.2.

–

Theorem 1.2 recover results mentioned in Remark 1.3(1a) and is the main part of a detailed well-posedness Theorem 4.1.
–

For $a=0$ and $\tau=\tau_{c}$ , we have from (1.16) that $\tau_{c}<d\Longleftrightarrow\alpha>\alpha_{F}$ . In this case, Theorem 1.2 along with below Theorem 4.1, recover [9, Theorem 1.4] and remove the assumption $q\geq\alpha$ and allows $s=\frac{\gamma}{\alpha-1}$ . See Remark 1.3(1c).
–

For $a=0,$ Theorem 1.2 eliminate technical hypothesis (1.13) and $\alpha>\alpha_{F}$ from [9, Theorem 1.13] in the subcritical case.
–

Assume $s=0,\gamma<0$ . Then for $a=0$ Theorem 1.2 recovers [2, Theorem 1.1] and for $a\neq 0$ Theorem 1.2 recovers [3, Theorem 1.1].
–

For $V(x)=\frac{a_{*}}{|x|^{2}}$ and $d\geq 3$ in (1.13), Ioku and Ogawa [18, Theorem 1.4] proved small data global existence for $1+\frac{4}{d+2}<\alpha<1+\frac{4}{d-2}$ . Theorem 1.2 relaxes this assumption and prove the result for any $\alpha>\alpha_{F}$ (note that $\alpha_{F}<1+\frac{4}{d+2}$ for $d\geq 2$ ). See Remark 1.5.
–

In the subcritical case with assumption (1.17), Theorem 1.2 shows uniqueness of solution in $C([0,T_{m}),L_{s}^{q}({\mathbb{R}}^{d})).$ While [9, Theorem 1.13] established uniqueness for (1.12) in a proper subset of $C([0,T_{m}),L_{s}^{q}({\mathbb{R}}^{d}))$ . See Remark 4.7.
–

For detail comments on hypotheses of Theorem 1.2, see Remarks 4.3, 4.5, 4.6.

We now strengthen and complement Theorem 1.2 by establishing following result.

Theorem 1.3 (Finite time blow-up for large data in the subcritical case).

Assume that $\tau\leq\tau_{c}.$ Let $d,\gamma,\alpha,q,s$ be as in Theorem 1.2 (so local wellposedness for (1.6) holds). Let $F_{\alpha}$ satisfies $F_{\alpha}(z)=z^{\alpha}$ for $z\geq 0$ ¹¹1for example $F_{\alpha}(z)=\mu|z|^{\alpha-1}z,\mu|z|^{\alpha}$ or $\mu z^{\alpha}$ . Further assume

d+\gamma<\begin{cases}\quad\alpha d&\text{ if }a=0\\ \alpha(d-2)&\text{ if }a\neq 0\end{cases}.

(1.18)

Then there exists initial data $u_{0}\in L_{s}^{q}({\mathbb{R}}^{d})$ such that $T_{m}(u_{0})<\infty$ . Moreover if $\tau<\tau_{c}$ , one has a unique blow-up solution to (1.6) with initial data $u_{0}$ in the following sense: there exist a unique solution $u$ of (1.6) defined on $[0,T_{m})$ such that

T_{m}<\infty\quad\text{and}\quad\lim_{t\uparrow T_{m}}\|u(t)\|_{L_{s}^{q}}=\infty.

Remark 1.5.

We have several comments for Theorem 1.3.

–

For the critical case $\tau=\tau_{c}$ , similar blowup happens in a Kato norm: If $T_{m}<\infty$ , one would have $\|u\|_{\mathcal{K}_{k,s}^{p,q}(T_{m})}=\infty$ for certain choice of $(k,p)$ . See Section 4 for definition of Kato norm.
–

Take $\gamma=s=0$ in (1.8), and so $\tau<\tau_{c}\Leftrightarrow q>\frac{d(\alpha-1)}{2}$ . Weissler [31] established blow-up solution for (1.11) in $L^{q}(\mathbb{R}^{d}).$ Theorem 1.3 is compatible with this classical result.
–

For $V(x)=\frac{a_{*}}{|x|^{2}},$ $u_{0}\in L^{\frac{d(\alpha-1)}{2}}(\mathbb{R}^{d})$ with $\alpha\leq 1+\frac{4}{d+2}$ , Ioku and Ogawa [18] pointed out that (1.13) have blow-up solution in finite time in $L^{\infty}-$ norm. However, we are not aware of any previous results on finite time blow-up solution in $L^{q}_{s}-$ norm for $a,s,\gamma\neq 0$ and $q\neq\infty.$ Thus Theorem 1.3 is new.

–

Assume $d\geq 3$ ,

\begin{cases}1+\frac{\gamma+2}{d-2}<\alpha<1+\frac{\gamma+2}{\sigma_{-}}\ \quad&\text{for}\ a\leq 0\\ 1+\frac{\gamma}{d}<\alpha<\infty\ \quad&\text{for}\ a=0\\ 1+\max(\frac{\gamma+2}{\sigma_{-}},\frac{\gamma+2}{d-2})<\alpha<\infty\quad&\text{for}\ a>0\end{cases}

and the hypothesis on $\gamma,q,s$ from Theorem 1.2. Let $F(z)=|z|^{\alpha}$ or $|z|^{\alpha-1}z$ or $z^{\alpha}$ . Then Theorem 1.3 reveals that, there exists data in $L_{s}^{q}({\mathbb{R}}^{d})$ such that the local solution established in Theorem 1.2 cannot be extend to global in time. In the critical case, it also says that small data assumption in Theorem 1.2 is essentially optimal to establish global existence.

Definition 1.2 (weak solution).

Let $u_{0}\in L^{1}_{loc}({\mathbb{R}}^{d})$ , then we say a function $u$ is a weak solution to (1.6) if $u\in L^{\alpha}((0,T),(L^{\alpha}_{\frac{\gamma}{\alpha}})_{loc}({\mathbb{R}}^{d}))$ and satisfies the equation (1.6) in the distributional sense, i.e.

	$\displaystyle\int_{{\mathbb{R}}^{d}}$	$\displaystyle u(T^{\prime},x)\eta(T^{\prime},x)\,dx-\int_{{\mathbb{R}}^{d}}u_{0}(x)\eta(0,x)\,dx$
		$\displaystyle=\int_{[0,T^{\prime}]\times{\mathbb{R}}^{d}}u(t,x)(\partial_{t}\eta+\Delta\eta-a\|x\|^{-2}\eta)(t,x)+\|x\|^{\gamma}F_{\alpha}(u(t,x))\,\eta(t,x)\,dx\,dt$		(1.19)

for all $T^{\prime}\in[0,T]$ and for all $\eta\in C^{1,2}([0,T]\times{\mathbb{R}}^{d})$ such that $\operatorname{supp}\eta(t,\cdot)$ is compact. The time $T$ is said to be the maximal existence time, which is denoted by $T_{m}^{w}$ , if the weak solution cannot be extended beyond $[0,T).$

Remark 1.6.

Proceeding as [16, Proposition 3.1] it follows that $L_{s}^{q}$ -integral solutions are weak solution. In that case $T_{m}\leq T_{m}^{w}$ .

We shall now turn our attention to supercritical case. In this case, we show that there exists positive initial data in $L^{q}_{s}({\mathbb{R}}^{d})$ that do not generate a (weak) local solution to (1.6). Specifically, we have the following theorem.

Theorem 1.4 (Nonexistence of local positive weak solution in supercritical case).

Let $d\in\mathbb{N}$ , $a,\gamma\in\mathbb{R}$ , $\alpha$ satisfy (1.18) and

\alpha>\alpha_{F}(d,\gamma,0)=1+\frac{(2+\gamma)^{+}}{d}.

(1.20)

Assume that $F_{\alpha}$ satisfies $F_{\alpha}(z)=z^{\alpha}$ for $z\geq 0$ , $q\in[1,\infty]$ , $s\in{\mathbb{R}}.$ Let $\tau,\tau_{c}$ be as in (1.8) and satisfy $\tau<\tau_{c}.$ Then there exists an initial data $u_{0}\in L^{q}_{s}({\mathbb{R}}^{d})$ such that (1.6) with $u(0)=u_{0}$ has no positive local weak solution.

Remark 1.7.

–

For $a=0=\gamma,$ Theorem 1.4 recovers results mentioned in Remark 1.3(1b).
–

For $a=0,\gamma>-2$ , condition $\alpha>\alpha_{F}(d,\gamma,0)$ (1.20) implies $d+\gamma<\alpha d$ in (1.18). Thus, in this case, Theorem 1.4 recovers [9, Theorem 1.16].
–

Theorem 1.4 implies failure of LWP in super-critical case. Theorem 1.4 tells if $\alpha$ satisfies (1.20) then the sub criticality or criticality condition is necessary in Theorem 1.2.

The paper is organized as follows. In Section 2, we gather some general tools which will be used later. In Section 3, we prove Theorem 1.1. In Section 4 we establish wellposedness results. In Section 5, we prove Theorems 1.3 and 1.4.

2. Preliminaries

Notations: The symbol $\alpha\wedge\beta$ means $\min(\alpha,\beta)$ whereas $\alpha\vee\beta$ mean $\max(\alpha,\beta)$ . By $a^{+}$ we denote $a\vee 0$ . The notation $A\lesssim B$ means $A\leq cB$ for some universal constant $c>0.$ By $A\gtrsim B$ we mean $B\lesssim A$ . By $A\sim B$ we mean $A\lesssim B$ and $A\gtrsim B$ .

We shortly denote unweighed Lebesgue space norm by $\|f\|_{L^{p}}=\|f\|_{p}.$ The Schwartz space is denoted by $\mathcal{S}(\mathbb{R}^{d})$ , and the space of tempered distributions is denoted by $\mathcal{S^{\prime}}(\mathbb{R}^{d}).$ For $s\in{\mathbb{R}}$ and $q\in[1,\infty]$ , we introduce the weighted local Lebesgue space $L^{q}_{s,loc}({\mathbb{R}}^{d})$ given by

L^{q}_{s,loc}({\mathbb{R}}^{d}):=\left\{f\in L^{0}({\mathbb{R}}^{d})\,;\,f|_{K}\in L^{q}_{s}({\mathbb{R}}^{d}),\ \forall K\subset{\mathbb{R}}^{d},\ K\ \text{compact}\right\}

where $L^{0}({\mathbb{R}}^{d})$ is the set of measurable functions on ${\mathbb{R}}^{d}$ .

2.1. Lorentz space

The Lorentz space is the space of all complex-valued measurable functions $f$ such that $\|f\|_{L^{p,q}({\mathbb{R}}^{d})}<\infty$ where $\|f\|_{L^{p,q}({\mathbb{R}}^{d})}$ is defined by

\|f\|_{L^{p,q}({\mathbb{R}}^{d})}:=p^{\frac{1}{q}}\left\|t\mu\{|f|>t\}^{\frac{1}{p}}\right\|_{L^{q}\left((0,\infty),\frac{dt}{t}\right)}

with $0<p<\infty$ , $0<q\leq\infty$ and $\mu$ denotes the Lebesgue measure on ${\mathbb{R}}^{d}$ . Therefore

\|f\|_{p,q}:=\|f\|_{L^{p,q}({\mathbb{R}}^{d})}=\begin{cases}p^{1/q}\left(\int_{0}^{\infty}t^{q-1}\mu\{|f|>t\}^{\frac{q}{p}}dt\right)^{1/q}\quad\text{for }q<\infty\\ \sup_{t>0}t\mu\{|f|>t\}^{\frac{1}{p}}\quad\quad\quad\text{for }q=\infty.\end{cases}

Let us gather some useful results on Lorentz spaces relevant to subsequent our proofs.

Lemma 2.1 (Lemmata 2.2, 2.5 in [26]).

Let $1\leq p\leq\infty$ , $1\leq q_{1},q_{2}\leq\infty$ . Then

(1)

$\|f\|_{p,p}\sim\|f\|_{p}$ , the usual Lebesgue $p$ -norm.
(2)

$\|f\|_{p,q_{2}}\lesssim\|f\|_{p,q_{1}}$ if $q_{1}\geq q_{2}$ .
(3)

$|\cdot|^{-b}\in L^{\frac{d}{b},\infty}({\mathbb{R}}^{d})$ for $b>0$ .

Lemma 2.2 (Theorems 2.6, 3.4 in [26]).

We have the following inequalities in Lorentz spaces:

(1)

(Hölder’s inequality) Let $\frac{1}{r}=\frac{1}{r_{0}}+\frac{1}{r_{1}}\in[0,1)$ and $s\geq 1$ is such that $\frac{1}{s}\leq\frac{1}{s_{0}}+\frac{1}{s_{1}}$ . Then $\|fg\|_{L^{r,s}}\leq r^{\prime}\|f\|_{L^{r_{0},s_{0}}}\|g\|_{L^{r_{1},s_{1}}}$ .
(2)

(Young’s inequality) Let $\frac{1}{r}=\frac{1}{r_{0}}+\frac{1}{r_{1}}-1\in(0,1]$ and $s\geq 1$ is such that $\frac{1}{s}\leq\frac{1}{s_{0}}+\frac{1}{s_{1}}$ . Then $\|f\ast g\|_{L^{r,s}}\leq 3r\|f\|_{L^{r_{0},s_{0}}}\|g\|_{L^{r_{1},s_{1}}}$ .

2.2. Heat kernel estimate

Let $g_{a}$ be the symmetric (in $x,y$ variable) heat kernel associated with the operator ${\mathcal{L}}_{a}$ , i.e.

e^{-t\mathcal{L}_{a}}f(x)=\int_{\mathbb{R}^{d}}g_{a}(t,x,y)f(y)dy\quad(t>0)

see [24, Proposition 3.6.]. Then we have the following bounds for $g_{a}$ :

Theorem A (see Theorem 6.2 in [23]).

Let $\sigma_{-},\sigma_{+}$ be as defined in (1.2). Let $d\geq 2$ , $a\geq a_{*}$ . Then there exist $c_{1},c_{2}>0$ such that for any $t>0$ and $x,y\in{\mathbb{R}}^{d}\backslash\{0\}$ , the following estimate holds:

\big{(}1\vee\frac{\sqrt{t}}{|x|}\big{)}^{\sigma_{-}}\big{(}1\vee\frac{\sqrt{t}}{|y|}\big{)}^{\sigma_{-}}t^{-\frac{d}{2}}e^{-\frac{|x-y|^{2}}{c_{1}t}}\lesssim\ g_{a}(t,x,y)\ \lesssim\big{(}1\vee\frac{\sqrt{t}}{|x|}\big{)}^{\sigma_{-}}\big{(}1\vee\frac{\sqrt{t}}{|y|}\big{)}^{\sigma_{-}}t^{-\frac{d}{2}}e^{-\frac{|x-y|^{2}}{c_{2}t}}.

3. Dissipative estimates in weighted Lebesgue spaces

In order to prove Theorem 1.1, we first show it is enough to prove for $t=1$ (Lemma 3.1), then using a duality (Lemma 3.2) we show it is enough to prove for $s_{1}\geq 0$ . Then we crucially use a known heat kernel estimate (Theorem A) to achieve the desired result.

Lemma 3.1.

Let $1\leq q_{1},q_{2}\leq\infty$ , and $s_{1},s_{2}\in\mathbb{R}.$ Then $e^{-\mathcal{L}_{a}}$ is bounded from $L^{q_{1}}_{s_{1}}(\mathbb{R}^{d})$ into $L^{q_{2}}_{s_{2}}(\mathbb{R}^{d})$ if and only if $e^{-t\mathcal{L}_{a}}$ is bounded from $L^{q_{1}}_{s_{1}}(\mathbb{R}^{d})$ into $L^{q_{2}}_{s_{2}}(\mathbb{R}^{d})$ with

\|e^{-t\mathcal{L}_{a}}\|_{L^{q_{1}}_{s_{1}}\to L^{q_{2}}_{s_{2}}}=t^{-\frac{d}{2}(\frac{1}{q_{1}}-\frac{1}{q_{2}})-\frac{s_{1}-s_{2}}{2}}\|e^{-\mathcal{L}_{a}}\|_{L^{q_{1}}_{s_{1}}\to L^{q_{2}}_{s_{2}}}

(3.1)

for any $t>0$ .

Proof.

It is enough to show (3.1) if $e^{-\mathcal{L}_{a}}$ is bounded from $L^{q_{1}}_{s_{1}}(\mathbb{R}^{d})$ into $L^{q_{2}}_{s_{2}}(\mathbb{R}^{d})$ , since the converse is trivial. The proof is based on the scaling argument. Let $f\in L^{q_{1}}_{s_{1}}(\mathbb{R}^{d})$ . Since

(e^{-t\mathcal{L}_{a}}f)(x)=\left(e^{-\mathcal{L}_{a}}(f(t^{\frac{1}{2}}\cdot))\right)(t^{-\frac{1}{2}}x),

(e^{-\mathcal{L}_{a}}f)(x)=\left(e^{-t\mathcal{L}_{a}}(f(t^{-\frac{1}{2}}\cdot))\right)(t^{\frac{1}{2}}x),

for $t>0$ and $x\in\mathbb{R}^{d}$ , we have

\|e^{-t\mathcal{L}_{a}}f\|_{L^{q_{2}}_{s_{2}}}\leq t^{-\frac{d}{2}(\frac{1}{q_{1}}-\frac{1}{q_{2}})-\frac{s_{1}-s_{2}}{2}}\|e^{-\mathcal{L}_{a}}\|_{L^{q_{1}}_{s_{1}}\to L^{q_{2}}_{s_{2}}}\|f\|_{L^{q_{1}}_{s_{1}}},

\|e^{-\mathcal{L}_{a}}f\|_{L^{q_{2}}_{s_{2}}}\leq t^{\frac{d}{2}(\frac{1}{q_{1}}-\frac{1}{q_{2}})+\frac{s_{1}-s_{2}}{2}}\|e^{-t\mathcal{L}_{a}}\|_{L^{q_{1}}_{s_{1}}\to L^{q_{2}}_{s_{2}}}\|f\|_{L^{q_{1}}_{s_{1}}}.

Hence, (3.1) is proved. ∎

Lemma 3.2.

Let $q_{1},q_{2}\in(1,\infty)$ and $s_{1},s_{2}\in{\mathbb{R}}$ and $A=\{x\in{\mathbb{R}}^{d}:|x|\geq 1\}$ . Let $k(x,y)=k(y,x)$ for $x,y\in A$ and for $x\in A$ set $Tf(x)=\int_{A}k(x,y)f(y)dy$ . Then

\|Tf\|_{L^{q_{2}}_{s_{2}}(A)}\leq C\|f\|_{L^{q_{1}}_{s_{1}}(A)}\text{ for all }f

if and only if

\|Tf\|_{L^{q_{1}^{\prime}}_{-s_{1}}(A)}\leq C\|f\|_{L^{q_{2}^{\prime}}_{-s_{2}}(A)}\text{ for all }f.

Proof.

Note that

$\displaystyle\\|Tf\\|_{L^{q_{1}^{\prime}}_{-s_{1}}(A)}$	$\displaystyle=$	$\displaystyle\sup_{\\|g\\|_{L_{s_{1}}^{q_{1}}}\leq 1}\|\int_{A}\int_{A}k(x,y)f(y)dyg(x)dx\|$
	$\displaystyle=$	$\displaystyle\sup_{\\|g\\|_{L_{s_{1}}^{q_{1}}}\leq 1}\|\int_{A}\int_{A}k(x,y)g(x)dxf(y)dy\|$
	$\displaystyle=$	$\displaystyle\sup_{\\|g\\|_{L_{s_{1}}^{q_{1}}}\leq 1}\|\int_{A}(Tg)(y)f(y)dy\|\leq\sup_{\\|g\\|_{L_{s_{1}}^{q_{1}}}\leq 1}\\|Tg\\|_{L_{s_{2}}^{q_{2}}}\\|f\\|_{L_{-s_{2}}^{q_{2}^{\prime}}}$
	$\displaystyle\leq$	$\displaystyle c\sup_{\\|g\\|_{L_{s_{1}}^{q_{1}}}\leq 1}\\|g\\|_{L_{s_{1}}^{q_{1}}}\\|f\\|_{L_{-s_{2}}^{q_{2}^{\prime}}}=c\\|f\\|_{L_{-s_{2}}^{q_{2}^{\prime}}(A)}$

This completes the proof. ∎

Proof of Theorem 1.1 (Sufficiency part).

Assume that (1.4) and (1.5) hold. In view of Lemma 3.1 it is enough to prove the case $t=1$ i.e.

\|e^{-\mathcal{L}_{a}}f\|_{L^{q_{2}}_{s_{2}}}\lesssim\|f\|_{L^{q_{1}}_{s_{1}}}.

(3.2)

For $x\in{\mathbb{R}}^{d}$ and $f\in L_{s_{1}}^{q_{1}}({\mathbb{R}}^{d})$ applying Theorem A we achieve

|e^{-\mathcal{L}_{a}}f(x)|\lesssim\ \left(1\vee\frac{1}{|x|}\right)^{\sigma_{-}}\int_{{\mathbb{R}}^{d}}\left(1\vee\frac{1}{|y|}\right)^{\sigma_{-}}G(x-y)|f(y)|dy.

(3.3)

where $G(x):=e^{-\frac{|x|^{2}}{c_{2}}}$ with $c_{2}$ as in Theorem A. Set

1_{\geq 1}(x):=\begin{cases}0\text{ for }|x|<1\\ 1\text{ for }|x|\geq 1\end{cases}\qquad\text{and}\qquad 1_{<1}:=1-1_{\geq 1}.

Then using $e^{-\mathcal{L}_{a}}f=1_{\geq 1}e^{-\mathcal{L}_{a}}f+1_{<1}e^{-\mathcal{L}_{a}}f$ and (3.3) we have

$\displaystyle\\|e^{-\mathcal{L}_{a}}f\\|_{L^{q_{2}}_{s_{2}}}$	$\displaystyle\leq$	$\displaystyle\\|e^{-\mathcal{L}_{a}}f(x)\\|_{L^{q_{2}}_{s_{2}}{(\|x\|\geq 1)}}+\\|e^{-\mathcal{L}_{a}}f(x)\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}$
	$\displaystyle\lesssim$	$\displaystyle\\|\int_{{\mathbb{R}}^{d}}\big{(}1\vee\frac{1}{\|y\|}\big{)}^{\sigma_{-}}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|\geq 1)}}$
		$\displaystyle+\ \\|\|x\|^{-\sigma_{-}}\int_{{\mathbb{R}}^{d}}\big{(}1\vee\frac{1}{\|y\|}\big{)}^{\sigma_{-}}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}.$

Splitting the integrations in $y$ variable we obtain

$\displaystyle\\|e^{-\mathcal{L}_{a}}f\\|_{L^{q_{2}}_{s_{2}}}$	$\displaystyle\lesssim$	$\displaystyle\\|\int_{\|y\|\geq 1}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|\geq 1)}}$
		$\displaystyle+\ \\|\int_{\|y\|<1}\|y\|^{-\sigma_{-}}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|\geq 1)}}$
		$\displaystyle+\ \\|\|x\|^{-\sigma_{-}}\int_{\|y\|\geq 2}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}$
		$\displaystyle+\ \\|\|x\|^{-\sigma_{-}}\int_{\|y\|<2}\|y\|^{-\sigma_{-}}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}=:I+\textit{II}+\textit{III}+\textit{IV}.$

Now we show that each of these terms is dominated by $\|f\|_{L^{q_{1}}_{s_{1}}}$ which would prove (3.2) to conclude the proof.

Estimate for IV: Using boundedness of $G$ and changing the order of integration and Hölder’s inequality we obtain

IV	$\displaystyle\lesssim$	$\displaystyle\\|\|x\|^{-\sigma_{-}}\int_{\|y\|<2}\|y\|^{-\sigma_{-}}\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}(\|x\|<1)}$
	$\displaystyle=$	$\displaystyle\\|\|x\|^{-\sigma_{-}}\\|_{L^{q_{2}}_{s_{2}}(\|x\|<1)}\int_{\|y\|<2}\|y\|^{-\sigma_{-}}\|f(y)\|dy$
	$\displaystyle=$	$\displaystyle\\|\|x\|^{-\sigma_{-}}\\|_{L^{q_{2}}_{s_{2}}(\|x\|<1)}\int_{\|y\|<2}\|y\|^{-\sigma_{-}-s_{1}}\|y\|^{s_{1}}\|f(y)\|dy$
	$\displaystyle\leq$	$\displaystyle\\|\|x\|^{-\sigma_{-}}\\|_{L^{q_{2}}_{s_{2}}(\|x\|<1)}\\|\|y\|^{-\sigma_{-}-s_{1}}\\|_{L^{q_{1}^{\prime}}(\|y\|<2)}\\|\|y\|^{s_{1}}f(y)\\|_{q_{1}}\ \lesssim\ \\|f\\|_{L^{q_{1}}_{s_{1}}}$

where in the last step we have used the hypothesis

	$\displaystyle(s_{2}-\sigma_{-})q_{2}+d>0$	$\displaystyle\Longleftrightarrow$	$\displaystyle\sigma_{-}<s_{2}+\frac{d}{q_{2}},$
	$\displaystyle(-\sigma_{-}-s_{1})q_{1}^{\prime}+d>0$	$\displaystyle\Longleftrightarrow$	$\displaystyle s_{1}+\frac{d}{q_{1}}<d-\sigma_{-}=\sigma_{+}+2.$

Estimate for III: Note that for $|x|<1$ , $|y|\geq 2$ we have $|x-y|\geq|y|-|x|\geq\frac{1}{2}|y|$ , and as $G$ is radially decreasing we have $G(x-y)\leq G(\frac{y}{2})$ therefore

III	$\displaystyle\leq$	$\displaystyle\\|\|x\|^{-\sigma_{-}}\int_{\|y\|\geq 2}G(\frac{y}{2})\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}$
	$\displaystyle=$	$\displaystyle\\|\|x\|^{-\sigma_{-}}\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}\int_{\|y\|\geq 2}G(\frac{y}{2})\|y\|^{-s_{1}}\|y\|^{s_{1}}\|f(y)\|dy$
	$\displaystyle\leq$	$\displaystyle\\|\|x\|^{-\sigma_{-}}\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}\\|G(\frac{y}{2})\|y\|^{-s_{1}}\\|_{L^{q_{1}^{\prime}}(\|y\|\geq 2)}\\|\|y\|^{s_{1}}f(y)\\|_{q_{1}}\ \lesssim\ \\|f\\|_{L^{q_{1}}_{s_{1}}}$

where in the last step we have used the hypothesis $\sigma_{-}<s_{2}+\frac{d}{q_{2}}$ as in the estimate for IV and the fact that $G$ is Schwartz class function.

Estimate for II: We claim that $\||x|^{s_{2}}G(x-y)\|_{L^{q_{2}}{(|x|\geq 1)}}\lesssim 1$ uniformly for all $|y|<1$ . In fact when $s_{2}\leq 0$ we have $\||x|^{s_{2}}G(x-y)\|_{L^{q_{2}}{(|x|\geq 1)}}\leq\|G(x-y)\|_{L^{q_{2}}{(|x|\geq 1)}}\leq\|G\|_{q_{2}}$ for all $y$ . On the other hand when $s_{2}>0$ , using $|x|^{s_{2}}\lesssim|x-y|^{s_{2}}+|y|^{s_{2}}$ , for $|y|<1$ we have

$\displaystyle\\|\|x\|^{s_{2}}G(x-y)\\|_{L^{q_{2}}{(\|x\|\geq 1)}}$	$\displaystyle\lesssim$	$\displaystyle\\|\|x-y\|^{s_{2}}G(x-y)\\|_{L^{q_{2}}{(\|x\|\geq 1)}}+\\|\|y\|^{s_{2}}G(x-y)\\|_{L^{q_{2}}{(\|x\|\geq 1)}}$
	$\displaystyle\leq$	$\displaystyle\\|\|\cdot\|^{s_{2}}G\\|_{q_{2}}+\\|G(x-y)\\|_{L^{q_{2}}{(\|x\|\geq 1)}}$
	$\displaystyle\leq$	$\displaystyle\\|\|\cdot\|^{s_{2}}G\\|_{q_{2}}+\\|G\\|_{q_{2}}.$

This proves the claim. Then

II	$\displaystyle=$	$\displaystyle\\|\|x\|^{s_{2}}\int_{\|y\|<1}\|y\|^{-\sigma_{-}}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}{(\|x\|\geq 1)}}$
	$\displaystyle\leq$	$\displaystyle\\|\int_{\|y\|<1}\\|\|x\|^{s_{2}}G(x-y)\\|_{L^{q_{2}}{(\|x\|\geq 1)}}\|y\|^{-\sigma_{-}}\|f(y)\|dy$
	$\displaystyle\lesssim$	$\displaystyle\\|\int_{\|y\|<1}\|y\|^{-\sigma_{-}-s_{1}}\|y\|^{s_{1}}\|f(y)\|dy$
	$\displaystyle\leq$	$\displaystyle\\|\|y\|^{-\sigma_{-}-s_{1}}\\|_{L^{q_{1}^{\prime}}(\|y\|<1)}\\|\|y\|^{s_{1}}f(y)\\|_{q_{1}}\ \lesssim\ \\|f\\|_{L^{q_{1}}_{s_{1}}}$

using the claim above and the hypothesis $s_{1}+\frac{d}{q_{1}}<\sigma_{+}+2$ as in the estimate for IV.

Estimate for $I$ : Let us treat $I$ case by case.

Case $s_{1}=s_{2}=0$ By hypothesis $\frac{1}{p}:=1+\frac{1}{q_{2}}-\frac{1}{q_{1}}\in[0,1]$ . Then using Young’s inequality

\displaystyle I=\|\int_{|y|\geq 1}G(x-y)|f(y)|dy\|_{L^{q_{2}}{(|x|\geq 1)}}\leq\|G\ast(1_{\geq 1}|f|)\|_{q_{2}}\leq\|G\|_{p}\|f\|_{q_{1}}\lesssim\ \|f\|_{L^{q_{1}}_{0}}.

Case $0=s_{2}<s_{1}$ If $\frac{1}{q_{2}}=\frac{1}{q_{1}}+\frac{s_{1}}{d}$ , then using Young’s and Holder’s inequalities in Lorentz spaces i.e. Lemma 2.2 we have

\displaystyle I\ \lesssim\ \|G\|_{1,q_{2}}\|1_{\geq 1}f\|_{q_{2},\infty}\ \lesssim\ \||\cdot|^{-s_{1}}\|_{\frac{d}{s_{1}},\infty}\||\cdot|^{s_{1}}f\|_{q_{1},\infty}\ \lesssim\ \|f\|_{L^{q_{1}}_{s_{1}}}.

If $\frac{1}{q_{2}}<\frac{1}{q_{1}}+\frac{s_{1}}{d}$ , then by using Lemma 3.3 (1) we choose $\frac{1}{p_{0}},\frac{1}{p_{1}},\frac{1}{p_{2}}\in[0,1]$ so that

1+\frac{1}{q_{2}}=\frac{1}{p_{0}}+\frac{1}{p_{1}}\qquad\frac{1}{p_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{1}},\qquad\frac{1}{p_{2}}<\frac{s_{1}}{d}.

(3.4)

and then using Young’s and Holder’s inequalities we achieve

	$\displaystyle I$	$\displaystyle\leq$	$\displaystyle\\|G\\|_{p_{0}}\\|1_{\geq 1}f\\|_{p_{1}}$
		$\displaystyle\lesssim$	$\displaystyle\\|1_{\geq 1}\|\cdot\|^{-s_{1}}\\|_{p_{3}}\\|\|\cdot\|^{s_{1}}f\\|_{q_{1}}\ \lesssim\ \\|f\\|_{L^{q_{1}}_{s_{1}}}.$

Case $0<s_{2}=s_{1}$ Using $|x|^{s_{2}}\lesssim|x-y|^{s_{2}}+|y|^{s_{2}}$ and Young’s inequality

$\displaystyle I$	$\displaystyle=$	$\displaystyle\\|\|x\|^{s_{2}}\int_{\|y\|\geq 1}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}{(\|x\|\geq 1)}}$
	$\displaystyle\lesssim$	$\displaystyle\\|\int_{\|y\|\geq 1}\|x-y\|^{s_{2}}G(x-y)\|f(y)\|dy\\|_{q_{2}}+\\|\int_{\|y\|\geq 1}G(x-y)\|y\|^{s_{2}}\|f(y)\|dy\\|_{q_{2}}$
	$\displaystyle=$	$\displaystyle\\|(\|\cdot\|^{s_{2}}G)\ast(1_{\geq 1}\|f\|)\\|_{q_{2}}+\\|G\ast(1_{\geq 1}\|\cdot\|^{s_{2}}\|f\|)\\|_{q_{2}}:=Ia+Ib$

Note that we have $\frac{d}{q_{2}}<\frac{d}{q_{1}}+{s_{1}}$ as $s_{2}>0$ and (1.4) is assumed. Then by choosing $\frac{1}{p_{0}},\frac{1}{p_{1}},\frac{1}{p_{2}}\in[0,1]$ satisfying (3.4) and using Young’s and Holder’s inequalities we achieve

	$\displaystyle Ia$	$\displaystyle\leq$	$\displaystyle\\|\|\cdot\|^{s_{2}}G\\|_{p_{0}}\\|1_{\geq 1}f\\|_{p_{1}}$
		$\displaystyle\lesssim$	$\displaystyle\\|1_{\geq 1}\|\cdot\|^{-s_{1}}\\|_{p_{2}}\\|\|\cdot\|^{s_{1}}f\\|_{q_{1}}\ \lesssim\ \\|f\\|_{L^{q_{1}}_{s_{1}}}.$

By hypothesis $\frac{1}{p}:=1+\frac{1}{q_{2}}-\frac{1}{q_{2}}\in[0,1]$ then

\displaystyle Ib

\displaystyle\leq

\displaystyle\|G\|_{p}\|1_{\geq 1}|\cdot|^{s_{2}}f\|_{q_{1}}\lesssim\ \|f\|_{L^{q_{1}}_{s_{1}}}.

Case $0<s_{2}<s_{1}$ Since $\frac{d}{q_{2}}<\frac{d}{q_{1}}+{s_{1}}$ we proceed as in above case and prove the estimate for $Ia$ . Now with the assumption $s_{2}<s_{1}$ using Lemma 3.3 (2) we choose $\frac{1}{p_{3}},\frac{1}{p_{4}},\frac{1}{p_{5}}\in[0,1]$ satisfying

1+\frac{1}{q_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}},\qquad\frac{1}{p_{4}}=\frac{1}{p_{5}}+\frac{1}{q_{2}},\qquad\frac{1}{p_{5}}<\frac{s_{1}-s_{2}}{d}

(3.5)

and obtain

	$\displaystyle Ib$	$\displaystyle\leq$	$\displaystyle\\|G\\|_{p_{3}}\\|1_{\geq 1}\|\cdot\|^{s_{2}}f\\|_{p_{4}}$
		$\displaystyle\lesssim$	$\displaystyle\\|1_{\geq 1}\|\cdot\|^{s_{2}-s_{1}}\\|_{p_{5}}\\|\|\cdot\|^{s_{1}}f\\|_{q_{1}}\ \lesssim\ \\|f\\|_{L^{q_{1}}_{s_{1}}}.$

Case $s_{2}<0<s_{1}$ If $\frac{d}{q_{2}}+s_{2}<\frac{d}{q_{1}}+{s_{1}}$ , by Lemma 3.3 (3) we choose $\frac{1}{p_{6}},\cdots,\frac{1}{p_{10}}\in[0,1]$ satisfying

\frac{1}{q_{2}}=\frac{1}{p_{6}}+\frac{1}{p_{7}},\qquad 1+\frac{1}{p_{7}}=\frac{1}{p_{8}}+\frac{1}{p_{9}},\qquad\frac{1}{p_{9}}=\frac{1}{p_{10}}+\frac{1}{q_{1}},\qquad\frac{1}{p_{6}}<-\frac{s_{2}}{d},\qquad\frac{1}{p_{10}}<\frac{s_{1}}{d}

(3.6)

so that

$\displaystyle I$	$\displaystyle=$	$\displaystyle\\|\|x\|^{s_{2}}\int_{\|y\|\geq 1}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}{(\|x\|\geq 1)}}$
	$\displaystyle\leq$	$\displaystyle\\|\|x\|^{s_{2}}\\|_{L^{p_{6}}(\|x\|\geq 1)}\\|G\ast(1_{\geq 1}\|f\|)\\|_{L^{p_{7}}{(\|x\|\geq 1)}}$
	$\displaystyle\leq$	$\displaystyle\\|\|x\|^{s_{2}}\\|_{L^{p_{6}}(\|x\|\geq 1)}\\|G\\|_{p_{8}}\\|1_{\geq 1}f\\|_{p_{9}}$
	$\displaystyle\leq$	$\displaystyle\\|\|x\|^{s_{2}}\\|_{L^{p_{6}}(\|x\|\geq 1)}\\|G\\|_{p_{8}}\\|\|y\|^{-s_{1}}\\|_{L^{p_{10}}(\|y\|\geq 1)}\\|\|\cdot\|^{s_{1}}f\\|_{q_{1}}\ \lesssim\\|f\\|_{L^{q_{1}}_{s_{1}}}.$

If $\frac{d}{q_{2}}+s_{2}=\frac{d}{q_{1}}+{s_{1}}$ , then we claim $0<\frac{-s_{2}}{d}<1$ . We need to show $-s_{2}<d$ i.e. $s_{2}>-d$ . Infact if $s_{2}\leq-d$ , then $\frac{d}{q_{1}}+{s_{1}}=\frac{d}{q_{2}}+s_{2}\leq\frac{d}{q_{2}}-d<0$ , a contradiction as $q_{1},s_{1}>0$ . Next we claim $0<\frac{1}{q_{1}}+\frac{s_{1}}{d}<1$ . This is because $0<\frac{1}{q_{1}}+\frac{s_{1}}{d}=\frac{1}{q_{2}}+\frac{s_{2}}{d}<\frac{1}{q_{2}}<1$ using $s_{2}<0$ .
Above claim shows $\frac{1}{p_{12}}:=\frac{s_{1}}{d}+\frac{1}{q_{1}}\in(0,1)$ , then we have $\frac{1}{q_{2}}=\frac{1}{-d/s_{2}}+\frac{1}{p_{12}}$ . Therefore

$\displaystyle I$	$\displaystyle=$	$\displaystyle\\|\|x\|^{s_{2}}\int_{\|y\|\geq 1}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}{(\|x\|\geq 1)}}$
	$\displaystyle\leq$	$\displaystyle\\|\|x\|^{s_{2}}\\|_{\frac{d}{-s_{2}},\infty}\\|G\ast(1_{\geq 1}\|f\|)\\|_{p_{12},q_{2}}$
	$\displaystyle\leq$	$\displaystyle\\|\|x\|^{s_{2}}\\|_{\frac{d}{-s_{2}},\infty}\\|G\\|_{1,q_{2}}\\|1_{\geq 1}f\\|_{p_{12},\infty}$
	$\displaystyle\leq$	$\displaystyle\\|\|x\|^{s_{2}}\\|_{\frac{d}{-s_{2}},\infty}\\|G\\|_{1,q_{2}}\\|\|y\|^{-s_{1}}\\|_{\frac{d}{s_{1}},\infty}\\|\|\cdot\|^{s_{1}}f\\|_{q_{1},\infty}\ \lesssim\\|f\\|_{L^{q_{1}}_{s_{1}}}.$

Case $s_{2}\leq s_{1}\leq 0$ Follows from duality Lemma 3.2 and the above cases. This completes the proof. ∎

Proof of Theorem 1.1 (Necessity part).

Assume that (1.3) hold. Let $g=\exp({-\frac{|\cdot|^{2}}{c_{1}}})$ where $c_{1}$ as in Theorem A.

Necessity of $\sigma_{-}<s_{2}+\frac{d}{q_{2}}$ , $s_{1}+\frac{d}{q_{1}}<\sigma_{+}+2$ : Let $f$ be supported in $B(0,1)$ and equal to $|\cdot|^{\theta}$ in $B(0,\frac{1}{2})$ with $\theta>\max(-s_{1}-\frac{d}{q_{1}},\sigma_{-}-d,0)$ . Then $f\in L_{s_{1}}^{q_{1}}({\mathbb{R}}^{d})$ and hence by hypothesis (1.3), we have $e^{-{\mathcal{L}}_{a}}f\in L_{s_{2}}^{q_{2}}({\mathbb{R}}^{d})$ . On the other hand for $|x|\leq 1$

	$\displaystyle[e^{-{\mathcal{L}}_{a}}f](x)$	$\displaystyle\geq$	$\displaystyle\int_{\|y\|\leq 1/2}g_{a}(1,x,y)f(y)dy$
		$\displaystyle\gtrsim$	$\displaystyle\|x\|^{-\sigma_{-}}\int_{\|y\|\leq 1/2}\|y\|^{-\sigma_{-}+\theta}g(x-y)dy\sim\|x\|^{-\sigma_{-}}.$

where we have used Theorem A in the second step and $\theta>\sigma_{-}-d$ in the last step. Since $e^{-{\mathcal{L}}_{a}}f\in L_{s_{2}}^{q_{2}}({\mathbb{R}}^{d})$ , we must have $\sigma_{-}<s_{2}+\frac{d}{q_{2}}$ . Using symmetry of heat kernel see (4) in Remark 1.1. it follows that $s_{1}+\frac{d}{q_{1}}<\sigma_{+}+2$ . This proof is a major modification made to [22, Section 4] where $q_{1}=q_{2}$ , $s_{1}=s_{2}=0$ was treated.

Necessity of $s_{2}+\frac{d}{q_{2}}\leq s_{1}+\frac{d}{q_{1}}$ : Let $0\neq f\in L^{2}\cap L_{s_{1}}^{q_{1}}.$ If $s_{2}+\frac{d}{q_{2}}>s_{1}+\frac{d}{q_{1}}$ , then using (1.3), we have $e^{-t\mathcal{L}_{a}}f\to 0$ in $L^{q_{2}}_{s_{2}}$ (and hence pointwise a.e.) as $t\to 0$ . Since $f\in L^{2}$ , using semigroup property, we have $e^{-t\mathcal{L}_{a}}f\to f$ in $L^{2}$ as $t\to 0$ . Thus $f=0$ which is a contraction.

Necessity of $s_{2}\leq s_{1}$ : We prove this by modifying the proof in case $a=0$ in [29, Remark 10]. Let $\varphi\in L_{s_{1}}^{q_{1}}$ be a smooth non-negative function with support in $B(0,1)$ and take $f_{\tau}=\varphi(\cdot-\tau x_{0})$ with $|x_{0}|=1$ . Then for $\tau>2$ and $|x|\geq 1$

$\displaystyle[e^{-{\mathcal{L}}_{a}}f_{\tau}](x)$	$\displaystyle\geq$	$\displaystyle\int_{\|y\|\geq 1}g_{a}(1,x,y)f_{\tau}(y)dy$
	$\displaystyle\gtrsim$	$\displaystyle\int_{\|y\|\geq 1}g(x-y)f_{\tau}(y)dy$
	$\displaystyle=$	$\displaystyle\int g(x-y)f_{\tau}(y)dy=(g\ast f_{\tau})(x)=(g\ast\varphi)(\cdot-\tau x_{0})$

where we have used Theorem A in the second step, the fact $B(0,1)\cap\rm{supp}(f_{\tau})=\emptyset$ in the third step. Now $\||\cdot|^{s_{2}}(g\ast\varphi)(\cdot-\tau x_{0})\|_{q_{2}}=\||\cdot+\tau x_{0}|^{s_{2}}(g\ast\varphi)\|_{q_{2}}=\tau^{s_{2}}\||\frac{\cdot}{\tau}+x_{0}|^{s_{2}}(g\ast\varphi)\|_{q_{2}}$ and $\||\cdot|^{s_{1}}f\|_{q_{1}}=\tau^{s_{1}}\||\frac{\cdot}{\tau}+x_{0}|^{s_{1}}\varphi\|_{q_{1}}$ . Therefore for $\tau>2$ we have from (1.3) that

\tau^{s_{2}-s_{1}}\left\|\left|\frac{\cdot}{\tau}+x_{0}\right|^{s_{2}}(g\ast\varphi)\right\|_{q_{2}}\lesssim\left\|\left|\frac{\cdot}{\tau}+x_{0}\right|^{s_{1}`}\varphi\right\|_{q_{1}}

but $\||\frac{\cdot}{\tau}+x_{0}|^{s_{2}}(g\ast\varphi)\|_{q_{2}}\to\|g\ast\varphi\|_{q_{2}}$ and $\||\frac{\cdot}{\tau}+x_{0}|^{s_{1}`}\varphi\|_{q_{1}}\to\|\varphi\|_{q_{1}}$ as $\tau\to\infty$ . Therefore we must have $s_{2}\leq s_{1}$ . ∎

Lemma 3.3.

There exists $p_{0},\cdots,p_{10}\in[1,\infty]$ so that

(1)

if $0<s_{1}$ , $\frac{d}{q_{2}}<\frac{d}{q_{1}}+{s_{1}}$ hold, then (3.4) is satisfied,
(2)

if $s_{2}<s_{1}$ holds, then (3.5) is satisfied,
(3)

if $s_{2}<0<s_{1}$ , $\frac{d}{q_{2}}+s_{2}<\frac{d}{q_{1}}+{s_{1}}$ hold, then (3.6) is satisfied.

Proof.

(1) Choose

\frac{1}{p_{0}}\in\big{(}\max(\frac{1}{q_{2}},1+\frac{1}{q_{2}}-\frac{1}{q_{1}}-\frac{s_{1}}{d}),\min(1,1+\frac{1}{q_{2}}-\frac{1}{q_{1}})\big{)}.

The last interval in nonempty as $q_{1},q_{2}\in(1,\infty)$ , $s_{1}>0$ and $\frac{d}{q_{2}}<\frac{d}{q_{1}}+{s_{1}}$ . Now set $\frac{1}{p_{1}}=1+\frac{1}{q_{2}}-\frac{1}{p_{0}}$ , $\frac{1}{p_{2}}=1+\frac{1}{q_{2}}-\frac{1}{p_{0}}-\frac{1}{q_{1}}$ then (3.4) is satisfied.

(2) Proof is similar to (1), only $s_{1}$ is replaced by $s_{1}-s_{2}$ . Choose

\frac{1}{p_{3}}\in\big{(}\max(\frac{1}{q_{2}},1+\frac{1}{q_{2}}-\frac{1}{q_{1}}-\frac{s_{1}-s_{2}}{d}),\min(1,1+\frac{1}{q_{2}}-\frac{1}{q_{1}})\big{)}.

The last interval in nonempty as $q_{1},q_{2}\in[1,\infty]$ , $s_{1}-s_{2}>0$ and $\frac{1}{q_{2}}<\frac{1}{q_{1}}+\frac{s_{1}-s_{2}}{d}$ . Now set $\frac{1}{p_{4}}=1+\frac{1}{q_{2}}-\frac{1}{p_{3}}$ , $\frac{1}{p_{5}}=1+\frac{1}{q_{2}}-\frac{1}{p_{3}}-\frac{1}{q_{1}}$ then (3.5) is satisfied.

(3) Note that $1+\frac{1}{q_{2}}+\frac{s_{2}}{d}-\frac{1}{q_{1}}-\frac{s_{1}}{d}<1$ , then choose

\frac{1}{p_{8}}\in(\max(1+\frac{1}{q_{2}}+\frac{s_{2}}{d}-\frac{1}{q_{1}}-\frac{s_{1}}{d},1-\frac{1}{q_{1}}-\frac{s_{1}}{d},\frac{1}{q_{2}}+\frac{s_{2}}{d},0),\min(1+\frac{1}{q_{2}}-\frac{1}{q_{1}},1)).

Then choose

\frac{1}{p_{7}}\in(\max(\frac{1}{q_{2}}+\frac{s_{2}}{d},\frac{1}{p_{8}}+\frac{1}{q_{1}}-1,0),\min(\frac{1}{p_{8}}+\frac{1}{q_{1}}+\frac{s_{1}}{d}-1,\frac{1}{q_{2}},\frac{1}{p_{8}})).

Set

\frac{1}{p_{6}}=\frac{1}{q_{2}}-\frac{1}{p_{7}},\qquad\frac{1}{p_{9}}=1+\frac{1}{p_{7}}-\frac{1}{p_{8}},\qquad\frac{1}{p_{10}}=1+\frac{1}{p_{7}}-\frac{1}{p_{8}}-\frac{1}{q_{1}}

so that equalities in (3.6) are satisfied. ∎

4. Local and small data global well-posedness

In this section we prove the well-posedness in critical and subcritical case i.e. when $\tau\leq\tau_{c}$ (recall that $\tau=\frac{d}{q}+s$ and $\tau_{c}=\frac{2+\gamma}{\alpha-1}$ ). In order to prove Theorem 1.2, we introduce the Kato space depending on four parameters $(p,q,k,s)$ .

Definition 4.1 (Kato space).

Let $k,s\in{\mathbb{R}}$ and $p,q\in[1,\infty]$ , set $\beta=\beta(d,k,s,p,q):=\frac{1}{2}(s+\frac{d}{q}-k-\frac{d}{p})$ . Then the Kato space $\mathcal{K}^{p,q}_{k,s}(T)$ is defined by

\mathcal{K}^{p,q}_{k,s}(T):=\left\{u:[0,T)\to L_{k}^{p}({\mathbb{R}}^{d}):\|u\|_{\mathcal{K}^{p,q}_{k,s}(T^{\prime})}<\infty\ \text{for any }T^{\prime}\in(0,T)\right\}

endowed with the norm

\|u\|_{\mathcal{K}^{p,q}_{k,s}(T)}:=\sup_{0\leq t<T}t^{\beta}\|u(t)\|_{L^{p}_{k}}.

Remark 4.1.

In [9], Kato space with three parameter was used. This is basically $\mathcal{K}^{q,q}_{k,s}(T)$ when one puts $p=q$ in Definition 4.1. This restriction didn’t allow authors in [9] to consider the case $1\leq q<\alpha$ .

By Theorem 1.1, we immediately get the following result (in fact these results are equivalent):

Lemma 4.1.

Let $k,s\in{\mathbb{R}}$ and $p,q\in(1,\infty)$ . Then

\|e^{-t\mathcal{L}_{a}}f\|_{\mathcal{K}^{p,q}_{k,s}}\leq C\|f\|_{L^{q}_{s}},\qquad\forall f\in L^{q}_{s}({\mathbb{R}}^{d})

if and only if

k\leq s\ \ \ \text{and}\ \ \sigma_{-}<\frac{d}{p}+k\leq\frac{d}{q}+s<\sigma_{+}+2.

(4.1)

Recall that by solution we meant integral solution and therefore, we introduce a nonlinear mapping $\mathcal{J}$ given by

\mathcal{J}_{\varphi}[u](t):=e^{-t\mathcal{L}_{a}}\varphi+\int_{0}^{t}e^{-(t-\tau)\mathcal{L}_{a}}F_{\alpha}(u(\tau))d\tau.

A fixed point of this map would essentially be a solution to (1.6). Next using Lemma 4.2, we establish the nonlinear estimates in Kato spaces with appropriate conditions on the parameters.

Proposition 4.1 (Nonlinear estimate, sub-critical & critical case).

Let $\alpha>1$ , $\gamma\in{\mathbb{R}}$ satisfy (1.14) , (1.15). Let $s\in{\mathbb{R}}$ and $q\in(1,\infty)$ satisfy

\tau=s+\frac{d}{q}\leq\tau_{c}=\frac{2+\gamma}{\alpha-1}.

(4.2)

Let $k,p$ satisfy

\frac{\gamma}{\alpha-1}\leq k,\qquad\alpha<p<\infty

(4.3)

\frac{s+\gamma}{\alpha}\leq k

(4.4)

\sigma_{-}<k+\frac{d}{p}<\frac{\sigma_{+}+2+\gamma}{\alpha}

(4.5)

\frac{1}{\alpha}(\frac{d}{q}+s+\gamma)<\begin{cases}\frac{d}{p}+k\leq\tau\quad\text{ if }\ \tau<\tau_{c}\\ \frac{d}{p}+k<\tau\quad\text{ if }\ \tau=\tau_{c}\end{cases}.

(4.6)

Then for any $u,v\in\mathcal{K}^{p,q}_{k,s}(T)$ we have

\begin{rcases}\|\mathcal{J}_{\varphi}[u]-\mathcal{J}_{\varphi}[v]\|_{\mathcal{K}^{p,q}_{k,s}(T)}\\ \|\mathcal{J}_{\varphi}[u]-\mathcal{J}_{\varphi}[v]\|_{\mathcal{K}^{q,q}_{s,s}(T)}\end{rcases}\lesssim T^{\frac{\alpha-1}{2}(\tau_{c}-\tau)}(\|u\|_{\mathcal{K}^{p,q}_{k,s}(T)}^{\alpha-1}+\|v\|_{\mathcal{K}^{p,q}_{k,s}(T)}^{\alpha-1})\|u-v\|_{\mathcal{K}^{p,q}_{k,s}(T)}.

Remark 4.2.

Note that $\|v\|_{\mathcal{K}^{q,q}_{s,s}(T)}=\sup_{0\leq t<T}\|v(t)\|_{L_{s}^{q}}$ .

Remark 4.3.

First inequality in (4.5), last inequality in (4.6) and (4.2) imposes the condition $\sigma_{-}<\tau_{c}$ . This is equivalent with

\sigma_{-}<\frac{2+\gamma}{\alpha-1}\Longleftrightarrow\begin{cases}\alpha<1+\frac{2+\gamma}{\sigma_{-}}\quad\text{if }\sigma_{-}>0\\ 0<2+\gamma\quad\text{if }\sigma_{-}=0\\ \alpha>1+\frac{2+\gamma}{\sigma_{-}}\quad\text{if }\sigma_{-}<0,\end{cases}

(4.7)

which is confirmed by (1.14), (1.15) (using the fact $\sigma_{-}>0\Leftrightarrow a<0$ and $\sigma_{-}<0\Leftrightarrow a>0$ and $\sigma_{-}=0$ if $a=0$ ).

Remark 4.4.

Note that (4.5) imposes the condition

\sigma_{-}<\frac{\sigma_{+}+2+\gamma}{\alpha}\Longleftrightarrow\begin{cases}\alpha<\frac{\sigma_{+}}{\sigma_{-}}+\frac{2+\gamma}{\sigma_{-}}\ \text{if }\sigma_{-}>0\\ 0<\sigma_{+}+2+\gamma\ \text{if }\sigma_{-}=0\\ \alpha>\frac{\sigma_{+}}{\sigma_{-}}+\frac{2+\gamma}{\sigma_{-}}\ \text{if }\sigma_{-}<0\end{cases}

and this is implied by (4.7) as $\frac{\sigma_{+}}{\sigma_{-}}>1$ for $\sigma_{-}>0$ and $\frac{\sigma_{+}}{\sigma_{-}}<0$ for $\sigma_{-}<0$ .

Before proving Proposition 4.1, we prove a technical lemma as an application of Theorem 1.1.

Lemma 4.2.

Assume $\alpha\geq 1$ and let $p\in(\alpha,\infty)$ , $r\in(1,\infty)$ , $l,k\in{\mathbb{R}}$ and

\sigma_{-}<\frac{d}{r}+l,\quad\frac{d}{p}+k<\frac{\sigma_{+}+2+\gamma}{\alpha},\qquad\gamma\leq\alpha k-l+\min(\frac{\alpha d}{p}-\frac{d}{r},0).

(4.8)

then for $t>0$ and $\varphi,\psi\in L_{k}^{p}({\mathbb{R}}^{d})$ we have

\|e^{-t{\mathcal{L}}_{a}}[|\cdot|^{\gamma}|\{\varphi|^{\alpha-1}\varphi)-|\psi|^{\alpha-1}\psi\}]\|_{L_{l}^{r}}\lesssim t^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{r})-\frac{\alpha k-l-\gamma}{2}}(\|\varphi\|_{L^{p}_{k}}^{\alpha-1}+\|\psi\|_{L^{p}_{k}}^{\alpha-1})\|\varphi-\psi\|_{L^{p}_{k}}.

Proof.

Note that (4.8) is equivalent with

\sigma_{-}<\frac{d}{r}+l\leq\frac{d}{p/\alpha}+\alpha k-\gamma<\sigma_{+}+2,\qquad l\leq\alpha k-\gamma.

By Theorem 1.1, with $s_{2}=l,s_{1}=\alpha k-\gamma,q_{2}=r,q_{1}=\frac{p}{\alpha}$ we obtain

			$\displaystyle\\|e^{-t{\mathcal{L}}_{a}}[\|\cdot\|^{\gamma}\|\{\varphi\|^{\alpha-1}\varphi)-\|\psi\|^{\alpha-1}\psi\}]\\|_{L_{l}^{r}}$
		$\displaystyle\lesssim$	$\displaystyle t^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{r})-\frac{\alpha k-\gamma-l}{2}}\\|\|\cdot\|^{\gamma}(\|\varphi\|^{\alpha-1}\varphi-\|\psi\|^{\alpha-1}\psi)\\|_{L_{\alpha k-\gamma}^{\frac{p}{\alpha}}}$
		$\displaystyle=$	$\displaystyle t^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{r})-\frac{\alpha k-l-\gamma}{2}}\\|\|\cdot\|^{\alpha k}(\|\varphi\|^{\alpha-1}+\|\psi\|^{\alpha-1})\|\varphi-\psi\|\\|_{\frac{p}{\alpha}}$
		$\displaystyle=$	$\displaystyle t^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{r})-\frac{\alpha k-l-\gamma}{2}}\\|[(\|\cdot\|^{k}\|\varphi\|)^{\alpha-1}+(\|\cdot\|^{k}\|\psi\|)^{\alpha-1}][\|\cdot\|^{k}(\varphi-\psi)]\\|_{\frac{p}{\alpha}}.$

By using $\frac{\alpha}{p}=\frac{\alpha-1}{p}+\frac{1}{p}$ and Holder’ inequality, the above quantity is dominated by

			$\displaystyle t^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{r})-\frac{\alpha k-l-\gamma}{2}}\\|(\|\cdot\|^{k}\|\varphi\|)^{\alpha-1}+(\|\cdot\|^{k}\|\psi\|)^{\alpha-1}\\|_{L^{\frac{p}{\alpha-1}}}\\|\|\cdot\|^{k}(\varphi-\psi)\\|_{L^{p}}$
		$\displaystyle\lesssim$	$\displaystyle t^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{r})-\frac{\alpha k-l-\gamma}{2}}(\\|\varphi\\|_{L_{k}^{p}}^{\alpha-1}+\\|\psi\\|_{L_{k}^{p}}^{\alpha-1})\\|\varphi-\psi\\|_{L_{k}^{p}}$

which completes the proof. ∎

Proof of Proposition 4.1.

Let us first establish two claims:
Claim I: Let $\beta=\beta(d,k,s,p,q)$ be as in Definition 4.1. Then

\beta\alpha<1

(4.9)

Proof of Claim I: Note that $s+\frac{d}{q}=\tau\leq\tau_{c}=\frac{2+\gamma}{\alpha-1}$ implies $(s+\frac{d}{q})\alpha-2\leq s+\frac{d}{q}+\gamma$ . First inequality in (4.6) says $s+\frac{d}{q}+\gamma<(\frac{d}{p}+k)\alpha$ . Thus

(s+\frac{d}{q})\alpha-2<(\frac{d}{p}+k)\alpha\Longleftrightarrow\left[\frac{d}{q}+s-\frac{d}{p}-k\right]\alpha<2\Longleftrightarrow\eqref{AC1}.

Claim II:

\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{q})+\frac{\alpha k-\gamma-s}{2}<1

(4.10)

Proof of Claim II: For the subcritical case $\tau<\tau_{c}$ we have Proof of claim:

$\displaystyle\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{q})+\frac{\alpha k-\gamma-s}{2}$	$\displaystyle\leq$	$\displaystyle\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{p})+\frac{\alpha k-\gamma-k}{2}$
	$\displaystyle=$	$\displaystyle\frac{1}{2}(\frac{d}{p}+k)(\alpha-1)-\frac{\gamma}{2}$
	$\displaystyle\leq$	$\displaystyle\frac{1}{2}(\frac{d}{q}+s)(\alpha-1)-\frac{\gamma}{2}<1;$

where in the first and third inequalities we used $\frac{d}{p}+k\leq\tau=\frac{d}{q}+s$ and in the last step we used $\tau<\tau_{c}$ . Proof for the ease $\tau=\tau_{c}$ , we only need to make the first, third nonstrict inequalities by strict inequalities (using $\frac{d}{p}+k<\tau=\frac{d}{q}+s$ ) and last strict inequality by equality (using $\tau=\tau_{c}$ ). This proves Claim II.

Now note that (4.3), (4.5) implies (4.8) for $(p,r,l,s)=(p,p,k,k)$ . By Lemma 4.2 with $(p,r,l,s)=(p,p,k,k)$ and (1.7) we have

	$\displaystyle\\|\mathcal{J}_{\varphi}[u]-\mathcal{J}_{\varphi}[v]\\|_{L^{p}_{k}}$	(4.11)
$\displaystyle\lesssim$	$\displaystyle\int_{0}^{t}\\|e^{-(t-\tau)\mathcal{L}_{a}}[\|x\|^{\gamma}(\|u\|^{\alpha-1}u-\|v\|^{\alpha-1}v)(\tau)]\\|_{L^{p}_{k}}d\tau$
$\displaystyle\lesssim$	$\displaystyle\int_{0}^{t}(t-\tau)^{-\frac{d(\alpha-1)}{2p}-\frac{1}{2}\{(\alpha-1)k-\gamma\}}(\\|u(\tau)\\|_{L^{p}_{k}}^{\alpha-1}+\\|v(\tau)\\|_{L^{p}_{k}}^{\alpha-1})\\|u(\tau)-v(\tau)\\|_{L^{p}_{k}}d\tau$
$\displaystyle\lesssim$	$\displaystyle(\\|u\\|_{\mathcal{K}^{p,q}_{k,s}(T)}^{\alpha-1}+\\|v\\|_{\mathcal{K}^{p,q}_{k,s}(T)}^{\alpha-1})\\|u-v\\|_{\mathcal{K}^{p,q}_{k,s}(T)}\int_{0}^{t}(t-\tau)^{-\frac{d(\alpha-1)}{2p}-\frac{1}{2}\{(\alpha-1)k-\gamma\}}\tau^{-\beta\alpha}d\tau,$

where the last inequality is due to the fact $u,u\in\mathcal{K}^{p,q}_{k,s}(T)$ . Recall $\tau_{c}=\frac{2+\gamma}{\alpha-1}$ and $B(x,y):=\int_{0}^{1}\tau^{x-1}(1-\tau)^{y-1}d\tau$ is convergent if $x,y>0$ . Taking (4.2), (4.6), (4.9) into account, note that the last time-integral in (4.11) is bounded by

			$\displaystyle t^{1-\frac{d(\alpha-1)}{2p}-\frac{1}{2}\{(\alpha-1)k-\gamma\}-\alpha\beta}\int_{0}^{1}(1-\tau)^{-\frac{d(\alpha-1)}{2p}-\frac{1}{2}\{(\alpha-1)k-\gamma\}}\tau^{-\alpha\beta}d\tau$
		$\displaystyle=$	$\displaystyle t^{\frac{(\alpha-1)}{2}\left(\tau_{c}-\tau\right)}t^{-\beta}B\left(\frac{(\alpha-1)}{2}\big{(}\tau_{c}-\frac{d}{p}-k\big{)},1-\alpha\beta\right)<\infty.$

This together with (4.11) implies the first part of the result.

Note that (4.5), (4.6) implies (4.8) for $(p,r,l,s)=(p,q,k,s).$ So by Lemma 4.2 with $(p,r,l,s)=(p,q,k,s),$ we have

	$\displaystyle\\|\mathcal{J}_{\varphi}[u](t)-\mathcal{J}_{\varphi}[v](t)\\|_{L^{q}_{s}}$	(4.12)
$\displaystyle\lesssim$	$\displaystyle\int_{0}^{t}\\|e^{-(t-\tau)\mathcal{L}_{a}}[\|\cdot\|^{-\gamma}(\|u\|^{\alpha-1}u-\|v\|^{\alpha-1}v)(\tau)]\\|_{L^{q}_{s}}d\tau$
$\displaystyle\lesssim$	$\displaystyle\int_{0}^{t}(t-\tau)^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{q})-\frac{\alpha k-\gamma-s}{2}}(\\|u(\tau)\\|_{L^{p}_{k}}^{\alpha-1}+\\|v(\tau)\\|_{L^{p}_{k}}^{\alpha-1})\\|u(\tau)-v(\tau)\\|_{L^{p}_{k}}d\tau$
$\displaystyle\lesssim$	$\displaystyle(\\|u\\|_{\mathcal{K}^{p,q}_{k,s}(T)}^{\alpha-1}+\\|v\\|_{\mathcal{K}^{p,q}_{k,s}(T)}^{\alpha-1})\\|u-v\\|_{\mathcal{K}^{p,q}_{k,s}(T)}\int_{0}^{t}(t-\tau)^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{q})-\frac{\alpha k-\gamma-s}{2}}\tau^{-\alpha\beta}d\tau.$

The last integral is bounded by

			$\displaystyle t^{1-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{q})-\frac{\alpha k-\gamma-s}{2}-\alpha\beta}\int_{0}^{1}(1-\tau)^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{q})-\frac{\alpha k-\gamma-s}{2}}\tau^{-\alpha\beta}d\tau$		(4.13)
		$\displaystyle=$	$\displaystyle t^{\frac{(\alpha-1)}{2}(\tau_{c}-\tau)}\int_{0}^{1}(1-\tau)^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{q})-\frac{\alpha k-\gamma-s}{2}}\tau^{-\alpha\beta}d\tau,$		(4.13)

which is finite in view of (4.9) and (4.10). Now (4.12) and (4.13) implies the second part of the result. ∎

Remark 4.5 (Hypotheses of Proposition 4.1).

•

Condition (4.2) and last inequality in (4.6) are used to make sure the beta functions $B(x,y)$ is finite for various choices of $x,y$ .
•

Conditions in (4.3), (4.5) (4.6) are used to invoke Lemma 4.2 with $(p,r,l,s)=(p,p,k,k)$ and with $(p,r,l,s)=(p,q,k,s)$ .

In the next result, we prove that there exists parameter $p,k$ such that (4.1) in Lemma 4.1 and (4.3), (4.5), (4.6) in Proposition 4.1 are satisfied.

Lemma 4.3.

Assume (1.14), (1.15). Let $\frac{\gamma}{\alpha-1}\leq s$ , $\sigma_{-}-\frac{d}{\alpha}<s$ and $q\in(1,\infty)$ satisfy $\sigma_{-}<\frac{d}{q}+s<\sigma_{+}+2$ . Then there exist $k\in{\mathbb{R}}$ and $p\in(\alpha,\infty)$ satisfying hypothesis (4.1) of Lemma 4.1, and hypotheses (4.3), (4.5), (4.6) of Proposition 4.1. If we further assume $\tau<\tau_{c}$ , (1.17), we can choose $p=q$ and $k=s$ .

Proof.

We need

\sigma_{-}<\frac{d}{p}+k<\frac{d}{q}+s<-\gamma+(\frac{d}{p}+k)\alpha<\sigma_{+}+2,

(4.14)

and

\frac{s+\gamma}{\alpha}\leq k\leq s.

(4.15)

Now (4.14) follows if we chosse $\frac{d}{p}+k$ so that

\max(\sigma_{-},\frac{\tau+\gamma}{\alpha})<\frac{d}{p}+k<\min(\frac{\sigma_{+}+2+\gamma}{\alpha},\tau).

Choose $k$ such that

\max(\sigma_{-}-\frac{d}{\alpha},\frac{s+\gamma}{\alpha})<k<\min(\frac{\sigma_{+}+2+\gamma}{\alpha},s)

so that (4.15) is satisfied. Then choose $p$ so that

\max(\sigma_{-}-k,\frac{\tau+\gamma}{\alpha}-k,0)<\frac{d}{p}<\min(\frac{\sigma_{+}+2+\gamma}{\alpha}-k,\frac{d}{q}+s-k,\frac{d}{\alpha})

which is possible as $\sigma_{-}<\frac{\sigma_{+}+2+\gamma}{\alpha}$ as a consequence of (1.15). This completes the proof.

The furthermore more part is clear. ∎

As we are done with linear estimate Lemma 4.1 and nonlinear estimate 4.1 and existence of parameter $p,k$ we are in a position to prove the following well-posedness result which implies Theorem 1.2.

Theorem 4.1 (Local well-posedness in the subcritical weighted Lebesgue space).

Let $\alpha>1$ , $\gamma\in{\mathbb{R}}$ satisfy (1.14) , (1.15). Let $s\in{\mathbb{R}}$ , $q\in(1,\infty)$ satisfy the subcriticality condition defined in (1.9) and

\frac{\gamma}{\alpha-1}\leq s,\qquad\sigma_{-}-\frac{d}{\alpha}<s.

(4.16)

Let $k\in{\mathbb{R}}$ and $p\in(\alpha,\infty)$ satisfy hypothesis (4.1) of Lemma 4.1, and hypotheses (4.3), (4.5), (4.6) of Proposition 4.1. Then the Cauchy problem (1.6) is locally well-posed in $L^{q}_{s}({\mathbb{R}}^{d})$ for arbitrary data $u_{0}\in L_{s}^{q}({\mathbb{R}}^{d})$ . More precisely, the following assertions hold.

(1)

(Existence) For any $u_{0}\in L^{q}_{s}({\mathbb{R}}^{d}),$ there exist a positive number $T$ and an $L^{q}_{s}({\mathbb{R}}^{d})$ -integral solution $u$ to (1.6) satisfying

$\|u\|_{\mathcal{K}^{p,q}_{k,s}(T)}\leq 2\|e^{-t\mathcal{L}_{a}}u_{0}\|_{\mathcal{K}^{p,q}_{k,s}(T)}.$

Moreover, the solution can be extended to the maximal interval $[0,T_{m})$ .
(2)

(Uniqueness in ${\mathcal{K}}^{p,q}_{k,s}(T)$ ) Let $T>0.$ If $u,v\in{\mathcal{K}}^{p,q}_{k,s}(T)$ satisfy (1.10) with $u(0)=v(0)=u_{0}$ , then $u=v$ on $[0,T].$

(3)

(Continuous dependence on initial data) For any initial data $\varphi$ and $\psi$ in $L^{q}_{s}({\mathbb{R}}^{d}),$ let $T(\varphi)$ and $T(\psi)$ be the corresponding existence time given by part (1). Then there exists a constant $C$ depending on $\varphi$ and $\psi$ such that the corresponding solutions $u$ and $v$ satisfy

\|u-v\|_{L^{\infty}(0,T;L^{q}_{s})\cap{\mathcal{K}}^{p,q}_{k,s}(T)}\leq CT^{\frac{\alpha-1}{2}(\tau_{c}-\tau)}\|u_{0}-v_{0}\|_{L^{q}_{s}}

for $T<\min\{T(u_{0}),T(v_{0})\}.$

(4)

(Blow-up criterion in subcritical case $\tau<\tau_{c}$ )) If $T_{m}<\infty,$ then $\displaystyle\lim_{t\uparrow T_{m}}\|u(t)\|_{L^{q}_{s}}=\infty.$ Moreover, the following lower bound of blow-up rate holds: there exists a positive constant $C$ independent of $t$ such that

$\|u(t)\|_{L^{q}_{s}}\gtrsim{(T_{m}-t)^{-\frac{\alpha-1}{2}(\tau_{c}-\tau)}}$ (4.17)

for $t\in(0,T_{m})$ .
(5)

(Blow-up criterion in critical case $\tau=\tau_{c}$ ) If $u$ is an $L^{q}_{s}({\mathbb{R}}^{d})$ -integral solution constructed in the assertion (1) and $T_{m}<\infty,$ then $\|u\|_{\mathcal{K}^{p,q}_{k,s}(T_{m})}=\infty.$
(6)

(Small data global existence in critical case $\tau=\tau_{c}$ ) There exists $\epsilon_{0}>0$ depending only on $d,\gamma,\alpha,q$ and $s$ such that if $u_{0}\in\mathcal{S}^{\prime}({\mathbb{R}}^{d})$ satisfies $\|e^{-t\mathcal{L}_{a}}u_{0}\|_{\mathcal{K}^{p,q}_{k,s}}<\epsilon_{0}$ (or $\|u_{0}\|_{L_{s}^{q}}<\epsilon_{0}$ in view of Lemma 4.1), then $T_{m}=\infty$ and $\|u\|_{\mathcal{K}^{p,q}_{k,s}}\leq 2\epsilon_{0}$ .

Proof of Theorem 4.1.

Existence in Kato space $\mathcal{K}_{k,s}^{p,q}(T)$ : Define

B_{M}^{T}:=\{u\in\mathcal{K}_{k,s}^{p,q}(T):\|u\|_{\mathcal{K}_{k,s}^{p,q}(T)}\leq M\}

with the metric

d(u,v)=:\|u-v\|_{\mathcal{K}_{k,s}^{p,q}(T)}.

Then by Lemma 4.1, Proposition 4.1, for $u,v\in B_{M}^{T}$ we have

\displaystyle\|\mathcal{J}_{u_{0}}[u]\|_{\mathcal{K}_{k,s}^{p,q}(T)}

\displaystyle\leq

\displaystyle\|e^{-t{\mathcal{L}}_{a}}u_{0}\|_{\mathcal{K}_{k,s}^{p,q}(T)}+cT^{\frac{\alpha-1}{2}(\tau_{c}-\tau)}M^{\alpha}

(4.18)

and

			$\displaystyle\\|\mathcal{J}_{u_{0}}[u]-\mathcal{J}_{v_{0}}[v]\\|_{\mathcal{K}_{k,s}^{p,q}(T)}$		(4.19)
		$\displaystyle\leq$	$\displaystyle\\|e^{-t{\mathcal{L}}_{a}}(u_{0}-v_{0})\\|_{\mathcal{K}_{k,s}^{p,q}(T)}+cT^{\frac{\alpha-1}{2}(\tau_{c}-\tau)}M^{\alpha-1}\\|u-v\\|_{\mathcal{K}^{p,q}_{k,s}(T)}$		(4.19)

Subcritical case $\tau<\tau_{c}$ : Using (4.18), (4.19) and choosing $M=2\|e^{-t{\mathcal{L}}_{a}}u_{0}\|_{\mathcal{K}_{k,s}^{p,q}(T)}$ and $T>0$ small enough so that $cT^{\frac{\alpha-1}{2}(\tau_{c}-\tau)}M^{\alpha-1}\leq\frac{1}{2}$ , we find $\mathcal{J}_{u_{0}}$ is a contraction in $B_{M}^{T}$ sub-critical case (and hence we have existence of unique solution $u\in B_{M}^{T}$ ). This proves (1), (2).

Critical case $\tau=\tau_{c}$ : Note that using a density argument we have

\lim_{T\to 0}\|e^{-t{\mathcal{L}}_{a}}u_{0}\|_{\mathcal{K}_{k,s}^{p,q}(T)}=0.

Thus we choose $T>0$ so that $M:=2\|e^{-t{\mathcal{L}}_{a}}u_{0}\|_{\mathcal{K}_{k,s}^{p,q}(T)}$ and $cM^{\alpha-1}<\frac{1}{2}$ where $c$ as in (4.18). Then by using (4.18), (4.19) for $u,v\in B_{M}^{T}$ we have

\displaystyle\|\mathcal{J}_{u_{0}}[u]\|_{\mathcal{K}_{k,s}^{p,q}(T)}

\displaystyle\leq

\displaystyle\|e^{-t{\mathcal{L}}_{a}}u_{0}\|_{\mathcal{K}_{k,s}^{p,q}(T)}+cM^{\alpha}\leq\frac{M}{2}+\frac{M}{2}=M

and

\displaystyle\|\mathcal{J}_{u_{0}}[u]-\mathcal{J}_{v_{0}}[v]\|_{\mathcal{K}_{k,s}^{p,q}(T)}

\displaystyle\leq

\displaystyle\|e^{-t{\mathcal{L}}_{a}}(u_{0}-v_{0})\|_{\mathcal{K}_{k,s}^{p,q}(T)}+\frac{1}{2}\|u-v\|_{\mathcal{K}^{p,q}_{k,s}(T)}

Thus $\mathcal{J}_{u_{0}}$ is a contraction in $B_{M}^{T}$ . This proves (1).

Solution is in $C([0,T),L_{s}^{q}({\mathbb{R}}^{d}))$ : Using Lemma 4.1, Proposition 4.1

\displaystyle\|\mathcal{J}_{u_{0}}[u](t)\|_{L^{q}_{s}}

\displaystyle\lesssim

\displaystyle\|u_{0}\|_{L^{q}_{s}}+M^{\alpha}T^{\frac{\alpha-1}{2}(\tau_{c}-\tau)}

and

\displaystyle\|\mathcal{J}_{u_{0}}[u](t)-\mathcal{J}_{v_{0}}[v](t)\|_{L^{q}_{s}}

\displaystyle\lesssim

\displaystyle\|u_{0}-v_{0}\|_{L^{q}_{s}}+M^{\alpha-1}\|u-v\|_{\mathcal{K}^{p,q}_{k,s}(T)}T^{\frac{\alpha-1}{2}(\tau_{c}-\tau)}.

Since $\mathcal{J}_{u_{0}}[u]=u$ , solution is indeed in $L^{\infty}([0,T);L_{s}^{q}({\mathbb{R}}^{d}))$ rest of the results follows as in classical case.

Uniqueness, continuous dependency, blow-up, small data global existence are usual as in classical case. ∎

Remark 4.6.

The hypothesis (1.16), (1.17) are to make a integral functional map contraction in Kato spaces. On the other hand conditions on $\alpha$ in (1.15) are to make sure there exists $\tau$ satisfying (1.16).

Proof of Theorem 1.2.

The result follows from Theorem 4.1 and Lemma 4.3. For the furthermore part, as in Lemma 4.3 we are choosing $p=q,k=s$ , the uniqueness part follows from the uniqueness of fixed point in $B_{T}^{M}\subset\mathcal{K}^{q,q}_{s,s}(T)$ and Remark 4.2. ∎

Remark 4.7.

In [9], Kato space $\mathcal{K}^{q,q}_{s,s}(T)$ was not used and hence the they did not achieve uniqueness in mere $C([0,T_{m}),L^{q}_{s}({\mathbb{R}}^{d}))$ .

5. Finite time blow-up and nonexistence results

In this section we establish that in the sub-critical and critical case there exists initial data for which solution established by Theorem 1.2 cannot be extended globally in time. Then blow-up alternative (see Theorem 4.1) implies solution must blow-up in finite time. On the other hand for super-critical case, we shall prove that there exists data such that no local weak (hence integral) solution exists.

Before proving the above two we establish the following important lemma which will be used in both he proofs.

Lemma 5.1.

Assume (1.18). Let $u$ be a non-negative weak solution on $[0,T)$ to (1.6) with initial data $u_{0}$ . Let $\phi\in C^{\infty}_{0}(\mathbb{R}^{d},[0,1])$ be such that $\phi=1$ on $B_{1/2}$ and supported in $B_{1}$ . Then for $l\geq\max(3,\frac{2\alpha}{\alpha-1})$ we have

\int_{|x|<\sqrt{T}}u_{0}(x)\phi^{l}\left(\frac{x}{\sqrt{T}}\right)\,dx\lesssim T^{-\frac{2+\gamma}{2(\alpha-1)}+\frac{d}{2}}.

Proof.

Let

\psi_{T}(t,x)=\eta(\frac{t}{T})\phi(\frac{x}{\sqrt{T}}).

where $\eta\in C^{\infty}_{0}({\mathbb{R}},[0,1])$ is such that $\eta=1$ on $B_{1/2}$ and supported in $B_{1}$ . We note that for $l\geq 3$ we have $\psi_{T}^{l}\in C^{1,2}([0,T)\times{\mathbb{R}}^{d})$ and the estimate

	$\displaystyle\partial_{t}\psi_{T}^{l}(t,x)\|+\|\Delta\psi_{T}^{l}(t,x)\|$	$\displaystyle\lesssim$	$\displaystyle T^{-1}\psi_{T}^{l-2}(t,x)$		(5.1)
		$\displaystyle\lesssim$	$\displaystyle T^{-1}\psi_{T}^{\frac{l}{\alpha}}(t,x)$		(5.1)

by choosing

l\geq\frac{2\alpha}{\alpha-1}\Longleftrightarrow\frac{l}{\alpha}\leq l-2.

We define a function $I:[0,T)\rightarrow{\mathbb{R}}_{\geq 0}$ given by

I(T):=\int_{[0,T)\times\{|x|<\sqrt{T}\}}|x|^{\gamma}u(t,x)^{\alpha}\,\psi_{T}^{l}(t,x)\,dtdx.

We note that $I(T)<\infty$ , since $u\in L_{t}^{\alpha}(0,T;L^{\alpha}_{\frac{\gamma}{\alpha},loc}({\mathbb{R}}^{d}))$ . By using the weak form (1.2), non-negativity of $u$ , the above estimate (5.1), Hölder’s inequality and Young’s inequality, the estimates hold:

$\displaystyle I(T)+\int_{\|x\|<\sqrt{T}}u_{0}(x)\phi^{l}\left(\frac{x}{\sqrt{T}}\right)\,dx$	$\displaystyle=$	$\displaystyle\left\|\int_{[0,T)\times\{\|x\|<\sqrt{T}\}}u(\partial_{t}\psi_{T}^{l}+\Delta\psi_{T}^{l}+a\|x\|^{-2}\psi_{T}^{l})\,dt\,dx\right\|$	(5.2)
	$\displaystyle\leq$	$\displaystyle\int_{[0,T)\times\{\|x\|<\sqrt{T}\}}(CT^{-1}+\|a\|\|x\|^{-2})\|u\|\psi_{T}^{\frac{l}{\alpha}}dtdx$
	$\displaystyle\leq$	$\displaystyle CI(T)^{\frac{1}{\alpha}}K(T)^{\frac{1}{\alpha^{\prime}}}$
	$\displaystyle\leq$	$\displaystyle\frac{1}{2}I(T)+CK(T),$

where $1=\frac{1}{\alpha}+\frac{1}{\alpha^{\prime}}$ , i.e., $\alpha^{\prime}=\frac{\alpha}{\alpha-1}$ and $K(T)$ is defined by

K(T):=\int_{[0,T)\times\{|x|<\sqrt{T}\}}\{T^{-\alpha^{\prime}}|x|^{-\frac{\gamma\alpha^{\prime}}{\alpha}}+|a|^{\alpha^{\prime}}|x|^{-\left(2+\frac{\gamma}{\alpha}\right)\alpha^{\prime}}\}dxdt\sim T^{{-\frac{2+\gamma}{2(\alpha-1)}+\frac{d}{2}}}.

The last equality holds only when (1.18) holds. Now from (5.2), we have

\int_{|x|<\sqrt{T}}u_{0}(x)\phi^{l}\left(\frac{x}{\sqrt{T}}\right)\,dx\lesssim K(T)\sim T^{-\frac{2+\gamma}{2(\alpha-1)}+\frac{d}{2}}

which completes the proof. ∎

5.1. Finite time blow-up in critical and subcritical case

The proof of this theorem is based on the arguments of [15, Proposition 2.2, Theorem 2.3] where lifespan of solution for nonlinear Schrödinger equation is studied.

Proof of Theorem 1.3.

Let $\lambda>0$ be a parameter. We take an initial data $u_{0}$ as $\lambda f$ , where $f:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}_{\geq 0}$ is given by

f(x):=\begin{cases}|x|^{-\beta}\quad&|x|\leq 1,\\ 0&\text{otherwise}\end{cases}

(5.3)

with $\beta$ satisfying

\beta<\min\left\{s+\frac{d}{q},d\right\}.

(5.4)

Then we see $u_{0}\in L^{q}_{s}({\mathbb{R}}^{d})$ and hence by Theorem 1.2, we can define the maximal existence time $T_{m}=T_{m}(u_{0})=T_{m}(\lambda f)$ . Moreover the solution with initial data $\lambda f$ would be nonnegative as heat kernel is so. Since $T_{m}(\lambda f)\leq T_{m}^{w}(\lambda f)$ , it follows from a change of variable and then Lemma 5.1 that for any $0<T<T_{m}(\lambda f)$

$\displaystyle\lambda T^{\frac{d-\beta}{2}}\int_{\|y\|<1/\sqrt{T}}\|y\|^{-\beta}\phi^{l}(y)dy$	$\displaystyle=$	$\displaystyle\lambda\int_{\|x\|<1}\|x\|^{-\beta}\phi^{l}(\frac{x}{\sqrt{T}})dx$
	$\displaystyle\leq$	$\displaystyle\lambda\int_{{\mathbb{R}}^{d}}\|x\|^{-\beta}\phi^{l}(\frac{x}{\sqrt{T}})dx$
	$\displaystyle=$	$\displaystyle\lambda\int_{\|x\|<\sqrt{T}}\|x\|^{-\beta}\phi^{l}(\frac{x}{\sqrt{T}})dx\leq CT^{-\frac{2+\gamma}{2(\alpha-1)}+\frac{d}{2}}$

which implies

\lambda\leq CL_{T}^{-1}T^{\frac{\beta}{2}-\frac{2+\gamma}{2(\alpha-1)}}

(5.5)

where $L_{T}=\int_{|y|<1/\sqrt{T}}|y|^{-\beta}\phi^{l}(y)dy$ .

Claim: There exists $\lambda_{0}$ such that if $\lambda>\lambda_{0}$ , then $T_{m}(\lambda f)\leq 4$ .
Indeed, on the contrary we assume that $T_{m}(\lambda_{j}f)>4$ for a sequence $\lambda_{j}\to\infty$ . Since $\beta<d$ , we have $L_{T}<\infty$ . The following estimates hold:

\displaystyle\lambda_{j}\leq CL_{4}^{-1}4^{\frac{\beta}{2}-\frac{2+\gamma}{2(\alpha-1)}}<\infty

which a contradiction and hence the claim is established.

Let $\lambda>\lambda_{0}$ and $0<T<T_{m}(\lambda f)\leq 4$ then again using (5.5)

\lambda\leq CL_{T}^{-1}T^{\frac{\beta}{2}-\frac{2+\gamma}{2(\alpha-1)}}\leq CL_{4}^{-1}T^{\frac{\beta}{2}-\frac{2+\gamma}{2(\alpha-1)}}

as $L_{T}$ is decreasing in $T$ . By (5.4) and the fact $\tau\leq\tau_{c}$ we have $\kappa:=\frac{2+\gamma}{2(\alpha-1)}-\frac{\beta}{2}>0$ and so for all $T\in(0,T_{m}(\lambda f)$

T\leq c\lambda^{-\frac{1}{\kappa}}

which implies $T_{m}(\lambda f)\leq c\lambda^{-\frac{1}{\kappa}}$ .

Then the result follows from blowup criterion in Theorem 4.1(4). First point in Remark 1.5 follows from Theorem 4.1(5). ∎

5.2. Nonexistece of weak solution in the supercritical case

In this subsection we give a proof of Theorem 1.4. We only give a sketch of the proof. For the details, we refer to [15, Proposition 2.4, Theorem 2.5] where nonlinear Schrödinger equation is studied.

Proof of Theorem 1.4.

Let $T\in(0,1)$ . Suppose that the conclusion of Theorem 1.4 does not hold. Then there exists a positive weak solution $u$ on $[0,T)$ to (1.6) (See Definition 1.2) with any initial data $u_{0}$ in particular for $f$ given by (5.3) with $\beta$ satisfying

\frac{2+\gamma}{\alpha-1}<\beta<\min\left\{s+\frac{d}{q},d\right\}.

(5.6)

Note that such choice is possible as $\tau>\tau_{c}$ and (1.20) i.e. $\alpha>\alpha_{F}(d,\gamma)$ . Now (5.6) implies $u_{0}\in L^{q}_{s}({\mathbb{R}}^{d})\cap L^{1}_{loc}({\mathbb{R}}^{d})$ . For $T<1$ we have using Lemma 5.1

\displaystyle\int_{|x|<\sqrt{T}}u_{0}(x)\phi^{l}\left(\frac{x}{\sqrt{T}}\right)\,dx

\displaystyle=T^{-\frac{\beta-d}{2}}\int_{|y|<1}|y|^{-\beta}\phi^{l}(y)\,dx=CT^{-\frac{\beta-d}{2}}.

(5.7)

Combining Lemma 5.1 and (5.7), we obtain

0<C\leq T^{\frac{\beta}{2}-\frac{2+\gamma}{2(\alpha-1)}}\to 0\quad\text{as }T\to 0

which leads to a contradiction, as $\beta$ satisfies

\frac{\beta}{2}-\frac{2+\gamma}{2(\alpha-1)}>0\quad\text{i.e.}\quad\beta>\frac{2+\gamma}{\alpha-1}.

This completes the proof. ∎

Acknowledgement: S Haque is thankful to DST–INSPIRE (DST/INSPIRE/04/2022/001457) & USIEF–Fulbright-Nehru fellowship for financial support. S Haque is also thankful to Harish-Chandra Research Institute & University of California, Los Angeles for their excellent research facilities.

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$\displaystyle\\|Tf\\|_{L^{q_{1}^{\prime}}_{-s_{1}}(A)}$	$\displaystyle=$	$\displaystyle\sup_{\\|g\\|_{L_{s_{1}}^{q_{1}}}\leq 1}\|\int_{A}\int_{A}k(x,y)f(y)dyg(x)dx\|$
	$\displaystyle=$	$\displaystyle\sup_{\\|g\\|_{L_{s_{1}}^{q_{1}}}\leq 1}\|\int_{A}\int_{A}k(x,y)g(x)dxf(y)dy\|$
	$\displaystyle=$	$\displaystyle\sup_{\\|g\\|_{L_{s_{1}}^{q_{1}}}\leq 1}\|\int_{A}(Tg)(y)f(y)dy\|\leq\sup_{\\|g\\|_{L_{s_{1}}^{q_{1}}}\leq 1}\\|Tg\\|_{L_{s_{2}}^{q_{2}}}\\|f\\|_{L_{-s_{2}}^{q_{2}^{\prime}}}$
	$\displaystyle\leq$	$\displaystyle c\sup_{\\|g\\|_{L_{s_{1}}^{q_{1}}}\leq 1}\\|g\\|_{L_{s_{1}}^{q_{1}}}\\|f\\|_{L_{-s_{2}}^{q_{2}^{\prime}}}=c\\|f\\|_{L_{-s_{2}}^{q_{2}^{\prime}}(A)}$

$\displaystyle\\|e^{-\mathcal{L}_{a}}f\\|_{L^{q_{2}}_{s_{2}}}$	$\displaystyle\leq$	$\displaystyle\\|e^{-\mathcal{L}_{a}}f(x)\\|_{L^{q_{2}}_{s_{2}}{(\|x\|\geq 1)}}+\\|e^{-\mathcal{L}_{a}}f(x)\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}$
	$\displaystyle\lesssim$	$\displaystyle\\|\int_{{\mathbb{R}}^{d}}\big{(}1\vee\frac{1}{\|y\|}\big{)}^{\sigma_{-}}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|\geq 1)}}$
		$\displaystyle+\ \\|\|x\|^{-\sigma_{-}}\int_{{\mathbb{R}}^{d}}\big{(}1\vee\frac{1}{\|y\|}\big{)}^{\sigma_{-}}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}.$

$\displaystyle\\|e^{-\mathcal{L}_{a}}f\\|_{L^{q_{2}}_{s_{2}}}$	$\displaystyle\lesssim$	$\displaystyle\\|\int_{\|y\|\geq 1}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|\geq 1)}}$
		$\displaystyle+\ \\|\int_{\|y\|<1}\|y\|^{-\sigma_{-}}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|\geq 1)}}$
		$\displaystyle+\ \\|\|x\|^{-\sigma_{-}}\int_{\|y\|\geq 2}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}$
		$\displaystyle+\ \\|\|x\|^{-\sigma_{-}}\int_{\|y\|<2}\|y\|^{-\sigma_{-}}G(x-y)\|f(y)\|dy\\|_{L^{q_{2}}_{s_{2}}{(\|x\|<1)}}=:I+\textit{II}+\textit{III}+\textit{IV}.$

$\displaystyle\\|\|x\|^{s_{2}}G(x-y)\\|_{L^{q_{2}}{(\|x\|\geq 1)}}$	$\displaystyle\lesssim$	$\displaystyle\\|\|x-y\|^{s_{2}}G(x-y)\\|_{L^{q_{2}}{(\|x\|\geq 1)}}+\\|\|y\|^{s_{2}}G(x-y)\\|_{L^{q_{2}}{(\|x\|\geq 1)}}$
	$\displaystyle\leq$	$\displaystyle\\|\|\cdot\|^{s_{2}}G\\|_{q_{2}}+\\|G(x-y)\\|_{L^{q_{2}}{(\|x\|\geq 1)}}$
	$\displaystyle\leq$	$\displaystyle\\|\|\cdot\|^{s_{2}}G\\|_{q_{2}}+\\|G\\|_{q_{2}}.$

			$\displaystyle\\|e^{-t{\mathcal{L}}_{a}}[\|\cdot\|^{\gamma}\|\{\varphi\|^{\alpha-1}\varphi)-\|\psi\|^{\alpha-1}\psi\}]\\|_{L_{l}^{r}}$
		$\displaystyle\lesssim$	$\displaystyle t^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{r})-\frac{\alpha k-\gamma-l}{2}}\\|\|\cdot\|^{\gamma}(\|\varphi\|^{\alpha-1}\varphi-\|\psi\|^{\alpha-1}\psi)\\|_{L_{\alpha k-\gamma}^{\frac{p}{\alpha}}}$
		$\displaystyle=$	$\displaystyle t^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{r})-\frac{\alpha k-l-\gamma}{2}}\\|\|\cdot\|^{\alpha k}(\|\varphi\|^{\alpha-1}+\|\psi\|^{\alpha-1})\|\varphi-\psi\|\\|_{\frac{p}{\alpha}}$
		$\displaystyle=$	$\displaystyle t^{-\frac{d}{2}(\frac{\alpha}{p}-\frac{1}{r})-\frac{\alpha k-l-\gamma}{2}}\\|[(\|\cdot\|^{k}\|\varphi\|)^{\alpha-1}+(\|\cdot\|^{k}\|\psi\|)^{\alpha-1}][\|\cdot\|^{k}(\varphi-\psi)]\\|_{\frac{p}{\alpha}}.$

On the hardy-Hénon heat equation with an inverse square potential

Abstract.

Key words and phrases:

2000 Mathematics Subject Classification:

1. Introduction

1.1. Fixed-time estimates for heat semigroup e−t​(−Δ+a​|x|−2)e^{-t(-\Delta+a|x|^{-2})}

Theorem 1.1.

Remark 1.1.

1.2. Hardy-Hénon equations (HHE) with inverse-square potential

Remark 1.2.

Definition 1.1 (well-posedness).

Remark 1.3.

1.3. Dynamics of HHE with inverse square potential

Theorem 1.2 (Well-posedness: subcritical and critical case).

Remark 1.4.

Theorem 1.3 (Finite time blow-up for large data in the subcritical case).

Remark 1.5.

Definition 1.2 (weak solution).

Remark 1.6.

Theorem 1.4 (Nonexistence of local positive weak solution in supercritical case).

Remark 1.7.

2. Preliminaries

2.1. Lorentz space

Lemma 2.1 (Lemmata 2.2, 2.5 in [26]).

Lemma 2.2 (Theorems 2.6, 3.4 in [26]).

2.2. Heat kernel estimate

Theorem A (see Theorem 6.2 in [23]).

3. Dissipative estimates in weighted Lebesgue spaces

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Proof of Theorem 1.1 (Sufficiency part).

Proof of Theorem 1.1 (Necessity part).

Lemma 3.3.

Proof.

4. Local and small data global well-posedness

Definition 4.1 (Kato space).

Remark 4.1.

Lemma 4.1.

Proposition 4.1 (Nonlinear estimate, sub-critical & critical case).

Remark 4.2.

Remark 4.3.

Remark 4.4.

Lemma 4.2.

Proof.

Proof of Proposition 4.1.

Remark 4.5 (Hypotheses of Proposition 4.1).

Lemma 4.3.

Proof.

Theorem 4.1 (Local well-posedness in the subcritical weighted Lebesgue space).

Proof of Theorem 4.1.

Remark 4.6.

Proof of Theorem 1.2.

Remark 4.7.

5. Finite time blow-up and nonexistence results

Lemma 5.1.

Proof.

5.1. Finite time blow-up in critical and subcritical case

Proof of Theorem 1.3.

5.2. Nonexistece of weak solution in the supercritical case

Proof of Theorem 1.4.

References

1.1. Fixed-time estimates for heat semigroup $e^{-t(-\Delta+a|x|^{-2})}$