Global rough solution for $L^2$-critical semilinear heat equation in the negative Sobolev space

TL;DR

This paper establishes well-posedness for the $L^2$-critical semilinear heat equation with initial data in negative Sobolev spaces, extending previous results to broader initial conditions including some $L^p$ spaces with $p<2$.

Contribution

It proves local and global well-posedness for initial data in negative Sobolev spaces, including certain $L^p$ spaces with $p<2$, which was not previously known.

Findings

Existence of a positive constant $oldsymbol{ extit{ extepsilon}_0}$ for well-posedness.

Well-posedness holds for radial, support-away-from-origin initial data in $oldsymbol{ ext{dot}H^{- extit{ extepsilon}_0}}$.

Unconditional uniqueness and $L^2$-estimates as $t o 0$ and $t o o ext{infinity}$.

Abstract

In this paper, we consider the Cauchy global problem for the $L^2$-critical semilinear heat equations $\partial_t h=\Delta h\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p(\R^d)$ for some $p\ge 2$ or to subcritical Sobolev space $H^{s}(\R^d)$ with $s>0$. We here prove that there exists some positive constant $\varepsilon_0$ depending on $d$, such that the Cauchy problem is locally and globally well-posed for any initial data $h_0$ which is radial, supported away from origin and in the negative Sobolev space $\dot H^{-\varepsilon_0}(\R^d)$ including $L^p(\R^d)$ with certain $p<2$ as subspace. Furthermore, unconditional uniqueness, and $L^2$-estimate both as time $t\to0$ and $t\to +\infty$ were considered.

Equations

$$\{\begin{array}{rl}& {\partial }_{t}h=\Delta h\pm |h{|}^{\gamma -1}h,\\ & h(0,x)=h_0(x),x\in {\mathbb{R}}^d,\end{array}$$

$$h_{\lambda }(t,x):={\lambda }^{2/(\gamma -1)}h({\lambda }^{2}t,\lambda x),h_{\lambda }(0,x):={\lambda }^{2/(\gamma -1)}{h}_0(\lambda x),\lambda >0,$$

$$h_0(x)\mapsto {\lambda }^{2/(\gamma -1)}{h}_0(\lambda x).$$

$$p_c=\frac{d(\gamma -1)}{2}$$

$$\{\begin{array}{rl}& {\partial }_{t}h=\Delta h+\mu |h{|}^{\frac{4}{d}}h,\\ & h(0,x)=h_0(x),x\in {\mathbb{R}}^d,\end{array}$$

$$\varepsilon_{0}\in\begin{aligned} &\Big{[}0,\frac{d-1}{d+2}\Big{)},\ \ d\geq 2.\\ \end{aligned}$$

$$h\in C([0,\delta );L^2({\mathbb{R}}^d)+{\dot{H}}^{-{\epsilon }_0}({\mathbb{R}}^d))\cap L_{tx}^{\frac{2(2+d)}{d}}([0,\delta ]\times {\mathbb{R}}^d)$$

$$h=e^{t\Delta }{h}_0+w.$$

$$\Vert h(t){\Vert }_{L^2}\lesssim t^{-\frac{{\epsilon }_0}{2}}\Vert h_0{\Vert }_{{\dot{H}}^{-{\epsilon }_0}},t>0.$$

$$\Vert h_0{\Vert }_{{\dot{H}}^{-{\epsilon }_0}({\mathbb{R}}^d)}\lesssim \Vert h_0{\Vert }_{L^p({\mathbb{R}}^d)}$$

$$\widehat{P_{\le N}f}(\xi ):=\phi (\xi /N)\hat{f}(\xi ),\widehat{P_{N}f}(\xi ):=\phi (\xi /N)-\phi (2\xi /N)\hat{f}(\xi ),$$

$$P_{\le N}=\sum _{M\le N}{P}_{M}f;P_{>N}=\sum _{M>N}{P}_{M}f$$

$$\Vert P_{\ge N}f{\Vert }_{L_x^p({\mathbb{R}}^d)}\lesssim N^{-s}\Vert |\nabla {|}^s{P}_{\ge N}f{\Vert }_{L_x^p({\mathbb{R}}^d)},\Vert |\nabla {|}^s{P}_{\le N}f{\Vert }_{L_x^p({\mathbb{R}}^d)}\lesssim N^s\Vert |P_{\le N}f{\Vert }_{L_x^p({\mathbb{R}}^d)};$$

$$\Vert |\nabla {|}^{\pm s}{P}_{\le N}f{\Vert }_{L_x^p({\mathbb{R}}^d)}\sim N^{\pm s}\Vert |P_{\le N}f{\Vert }_{L_x^p({\mathbb{R}}^d)};$$

$$\Vert P_{N}f{\Vert }_{L_x^q({\mathbb{R}}^d)}\lesssim N^{(\frac{d}{p}-\frac{d}{q})}\Vert f{\Vert }_{L_x^p({\mathbb{R}}^d)},\Vert P_{\le N}f{\Vert }_{L_x^q({\mathbb{R}}^d)}\lesssim N^{(\frac{d}{p}-\frac{d}{q})}\Vert f{\Vert }_{L_x^p({\mathbb{R}}^d)};$$

$$|{\varphi }_j|\le 1\text{mbox}and\text{mbox}dist(\text{supp}{\varphi }_1,\text{supp}{\varphi }_2)\ge A,$$

$$\displaystyle\bigl{\|}\phi_{1}P_{\leq N}(\phi_{2}f)\bigr{\|}_{L^{q}_{x}(\mathbb{R}^{d})}=\bigl{\|}\phi_{1}P_{\geq N}(\phi_{2}f)\bigr{\|}_{L^{q}_{x}(\mathbb{R}^{d})}\lesssim_{m}A^{-m+\frac{d}{q}-\frac{d}{p}}N^{-m}\|\phi_{2}f\|_{L^{p}_{x}(\mathbb{R}^{d})}.$$

$${\partial }_{t}h=\Delta h,h(0,x)=f(x),t>0,x\in {\mathbb{R}}^d,$$

$$\displaystyle\big{\|}e^{t\Delta}f\big{\|}_{L^{\infty}_{t}L^{p}_{x}(\mathbb{R}^{+}\times\mathbb{R}^{d})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{d})}.$$

$$\displaystyle\big{\|}\nabla e^{t\Delta}f\big{\|}_{L^{2}_{tx}(\mathbb{R}^{+}\times\mathbb{R}^{d})}$$

$$\displaystyle\big{\|}e^{t\Delta}f\big{\|}_{L^{\frac{2(2+d)}{d}}_{tx}(\mathbb{R}^{+}\times\mathbb{R}^{d})}$$

$$\displaystyle\Big{\|}\int_{0}^{t}e^{(t-s)\Delta}F(s)ds\Big{\|}_{L^{\infty}_{t}L^{2}_{x}\ \cap\ L^{\frac{2(2+d)}{d}}_{tx}\cap\ L^{2}_{t}\dot{H}^{1}_{x}(I\times\mathbb{R}^{d})}\lesssim\|F\|_{L^{\frac{2(2+d)}{d+4}}_{tx}(\mathbb{R}^{+}\times\mathbb{R}^{d})}.$$

$$(e^{t\Delta }f)(x)=(4\pi t{)}^{-d/2}{\int }_{{\mathbb{R}}^d}{e}^{-|x-y{|}^2/4t}f(y)dy,t>0.$$

$$\Vert e^{t\Delta }f{\Vert }_{L^q({\mathbb{R}}^d)}\lesssim t^{\frac{d}{2}(\frac{1}{q}-\frac{1}{p})}\Vert f{\Vert }_{L^p({\mathbb{R}}^d)},t>0.$$

$$\Big{\|}\Big{(}\int_{0}^{\infty}|\sqrt{t}\nabla e^{t\Delta}f|^{2}\frac{dt}{t}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}(\mathbb{R}^{d})}\lesssim\|f\|_{L^{2}(\mathbb{R}^{d})},$$

$$\big{\|}e^{t\Delta}f\big{\|}_{L^{\frac{2(2+d)}{d}}_{tx}(\mathbb{R}^{+}\times\mathbb{R}^{d})}\lesssim\big{\|}e^{t\Delta}f\big{\|}_{L^{\infty}_{t}L^{2}_{x}(\mathbb{R}^{+}\times\mathbb{R}^{d})}^{\frac{2}{d+2}}\big{\|}\nabla e^{t\Delta}f\big{\|}_{L^{2}_{tx}(\mathbb{R}^{+}\times\mathbb{R}^{d})}^{\frac{d}{d+2}}.$$

$$\frac{2}{p_1}+\frac{d}{r_1}=\frac{d}{2}+2+\frac{2}{p},1\le p_1\le 2,1<r_1\le 2,$$

$$\Big{\|}\int_{0}^{t}e^{(t-s)\Delta}F(s)ds\Big{\|}_{L^{p}_{t}L^{2}_{x}(\mathbb{R}^{+}\times\mathbb{R}^{d})}\lesssim\|F\|_{L_{t}^{p_{1}}L_{x}^{r_{1}}(\mathbb{R}^{+}\times\mathbb{R}^{d})}.$$

$$\displaystyle\Big{\|}\int_{0}^{t}e^{-(t-s)|\xi|^{2}}\widehat{F}(\xi,s)ds\Big{\|}_{L^{p}_{t}L^{2}_{\xi}(\mathbb{R}^{+}\times\mathbb{R}^{d})}\lesssim\|F\|_{L_{t}^{p_{1}}L_{x}^{r_{1}}(\mathbb{R}^{+}\times\mathbb{R}^{d})}.$$

$$\Big{\|}\int_{0}^{t}e^{-(t-s)|\xi|^{2}}\widehat{F}(\xi,s)ds\Big{\|}_{L^{p}(\mathbb{R}^{+})}\lesssim\Big{\|}|\xi|^{-(\frac{2}{p}+\frac{2}{p_{1}^{\prime}})}\widehat{F}(\xi,\cdot)\Big{\|}_{L_{t}^{p_{1}}(\mathbb{R}^{+})}.$$

Full text

1. Introduction

Consider the initial value problem for a semilinear heat equation:

[TABLE]

where $h(t,x)$ is an unknown real function defined on $\mathbb{R}^{+}\times\mathbb{R}^{d}$, $d\geq 2$, $\gamma>1$. The positive sign “+” in nonlinear term of (1.1) denotes focusing source, and the negative sign “-” denotes the defocusing one. The Cauchy problem (1.1) has been extensively studied in Lebesgue space $L^{p}(\mathbb{R}^{d})$ by many peoples, see e.g. [2, 3, 4, 6, 7, 10, 12, 13, 14, 15, 16, 18, 19, 21, 25, 26] and so on. The equation enjoys an interesting property of scaling invariance

[TABLE]

that is, if $h(t,x)$ is the solution of heat equation (1.1), then $h_{\lambda}(t,x)$ also does with the scaling data $\lambda^{2/\gamma}h_{0}(\lambda x)$. An important fact is that Lebsgue space $L^{p_{c}}(\mathbb{R}^{d})$ with $p_{c}=\frac{d(\gamma-1)}{2}$ is the only one invariant under the same scaling transform:

[TABLE]

If we consider the initial data $h_{0}\in L^{p}(\mathbb{R}^{d})$, then the scaling index

[TABLE]

plays a critical role on the local/global well-posedness of (1.1). Roughly speaking, one can divide the dynamics of (1.1) into the following three different regimes: (A) the subcritial case $p>p_{c}$, (B) the critical case $p=p_{c}$, (C) the supercritical case $p<p_{c}$. Specifically, In cases (A) and (B), i.e. $p\geq p_{c}$, when $p>\gamma$, Weissler in [25] proved the local existence and uniqueness of solution $h\in C([0,T);L^{q}(\mathbb{R}^{d}))\cap L^{\infty}_{loc}((0,T];L^{\infty}(\mathbb{R}^{d}))$. Later, Brezis and Cazenave [2] proved the unconditional uniquessness of Weissler’s solution. In double critical case $p=p_{c}=\gamma$ ( i.e. $p=\gamma=\frac{d}{d-2}$ ), the local conditional wellposedness of the problem (1.1) was due to Weissler in [26], but the unconditional uniqueness fails, see Ni-Sacks [16], Terraneo [22]. In the supercritical case $(C)$, i.e. $p<p_{c}$, it seems that there exists no local solution in any reasonable sense for some initial data $h_{0}\in L^{p}(\mathbb{R}^{d})$. In particular, in focusing case, there exists a nonnegative function $h_{0}\in L^{p}(\mathbb{R}^{d})$ such that the (1.1) does not admit any nonnegative classical $L^{p}$-solution in $[0,T)$ for any $T>0$, see e.g. Brezis and Cabré [1], Brezis and Cazenave [2], Haraux-Weissler[9] and Weissler [25, 26]. Also, one see book Quitnner-Souplet[17] for many related topics and references.

In this paper, we mainly concerned with the local and global existence of solution for some supercritical initial data $h_{0}\in L^{p}(\mathbb{R}^{d})$ by $p<p_{c}$ and more generally, initial data in $\dot{H}^{-\epsilon}$. For simplicity, we only consider the Cauchy problem for the $L^{2}$-critical semilinear heat equations,

[TABLE]

That is, $p_{c}=2$ ( i.e $\gamma=1+\frac{4}{d}$ ), we will prove that there exists some positive constant $\varepsilon_{0}$ depending on $d$, such that the Cauchy problem is locally and globally wellposed for any initial data $h_{0}$ is radial, supported away from origin and in the negative Sobolev space $\dot{H}^{-\varepsilon_{0}}(\mathbb{R}^{d})$, which includes certain $L^{p}$-space with $p<p_{c}=2$ as a subspace (see Remark 1.1 below). We remark that, at present the the range of $\epsilon_{0}$ in the following theorem may not be optimal to local and global existence of solution of the problem (1.2). On the other hand, we also mention that a result in Brezis and Freidman[3] implies that the problem (1.2) has no any solution ( even weak one) with a Dirac initial data $\delta$, which is in $H^{-\epsilon}(\mathbb{R}^{d})$ for any $s>d/2$.

Theorem 1.1**.**

Let $\mu=\pm 1$ and

[TABLE]

Suppose that $h_{0}\in\dot{H}^{-\varepsilon_{0}}(\mathbb{R}^{d})$ is a radial initial data satisfying $\mbox{supp }h_{0}\subset\{x:|x|\geq 1\}.$ Then there exists a time $\delta=\delta(h_{0})>0$ and a unique strong solution

[TABLE]

to the equation (1.2) with the initial data $h_{0}$. Moreover, the following two statements hold:

• (1).

If $d>4$, then the solution $h$ is unique in the following sense that there exists a unique function $w$ in $C([0,\delta],L^{2}(\mathbb{R}^{d}))$ such that

[TABLE]

• (2).

If $\|h_{0}\|_{\dot{H}^{-\varepsilon_{0}}(\mathbb{R}^{d})}$ is small enough, then the solution is global in time and satisfies the following decay estimate for $d\geq 4$,

[TABLE]

Remark 1.1**.**

If $h_{0}\in L^{p}$ for some $p<2$, then there exists some $\epsilon_{0}>0$ such that $h_{0}\in\dot{H}^{-\varepsilon_{0}}(\mathbb{R}^{d})$ and

[TABLE]

by the Sobolev embedding estimate ( see e.g. Lemma 3.1 below ). Thus, Theorem 1.1 shows that the solution $h$ of the equation (1.2) exists locally for any radial and supported away from zero initial datum $h_{0}$ in $L^{p}(\mathbb{R}^{d})$ as p\in\Big{(}\frac{d^{2}+4d-2}{2d^{2}+2d},2\Big{)} and $d\geq 2.$

Remark 1.2**.**

It seems that the restriction $d>4$ is necessary for unconditional uniqueness. In fact, when $d=4$, the uniqueness problem is related to the “double critical” case ( i.e. $p=p_{c}=\gamma=\frac{d}{d-2}=2$ ). It was well-known that the unconditional uniqueness failed by Ni-Sacks [16] and Brezis and Cazenave [2].

Finally, it is worth mentioning that in the defocusing case, the smallness restriction on the initial datum in the statement (2) is not necessary for global existence. Indeed, we have $h(\delta)\in L^{2}(\mathbb{R}^{d})$, then it follows by considering the solution from $t=\delta$. Moreover, it is easy to find a large class of $h_{0}$ satisfying the conditions of theorem above. As described in Remark 1.1, our result shows that the solution $h$ of the equation (1.2) exists globally on $\mathbb{R}+$, for any the initial datum $h_{0}$ in $L^{p}(\mathbb{R}^{d})$ with some $p<2$, which is radial and supported away from zero.

The paper is organized as follows: In Section 2, we will list several useful lemmas about Littlewood-Paley theory, and space-time estimates for the solution of linear heat equation. Then in Section 3, we will give the proof of the main results, respectively.

2. Preliminary

2.1. Littlewood-Paley multipliers and related inequalities

Throughout this paper, we write $A\lesssim B$ to signify that there exists a constant $c$ such that $A\leq cB$, while we denote $A\sim B$ when $A\lesssim B\lesssim A$. We first define the Littlewood-Paley projection multiplier. Let $\varphi(\xi)$ be a fixed real-valued radially symmetric bump function adapted to the ball $\{\xi\in\mathbb{R}^{d}:\ |\xi|\leq 2\}$ which equals 1 on the ball $\{\xi\in\mathbb{R}^{d}:\ |\xi|\leq 1\}$. Define a dyadic number to any number $N\in 2^{\mathbf{Z}}$ of the form $N=2^{j}$ where $j\in\mathbf{Z}$ ( the integer set). For each dyadic number $N$, we define the the Fourier multipliers

[TABLE]

where $\hat{f}$ denotes the Fourier transform of $f$. Moreover, define $P_{>N}=I-P_{\leq N}$ and $P_{<N}=P_{\leq N}-P_{N}$, etc. In particular, we have the telescoping expansion:

[TABLE]

where $M$ ranges over dyadic numbers. It was well-known that the Littlewood-Paley operators satisfy the following useful Bernstein inequalities with $s>0$ and $1\leq p\leq q\leq\infty$ ( see e.g. Tao [23] ):

[TABLE]

[TABLE]

[TABLE]

Moreover, we also have the following mismatch estimate, see e.g. [11].

Lemma 2.1** (Mismatch estimates).**

Let $\phi_{1}$ and $\phi_{2}$ be smooth functions obeying

[TABLE]

for some large constant $A$. Then for $m>0$, $N\geq 1$ and $1\leq p\leq q\leq\infty$,

[TABLE]

2.2. Space-time estimates of linear heat equation

Let $e^{t\Delta}$ denote the heat semigroup on $\mathbb{R}^{d}$. Then for suitable function $f$, $e^{t\Delta}f$ solves the linear heat equation

[TABLE]

and the solution satisfies the following fundamental space-time estimates:

Lemma 2.2**.**

Let $f\in L^{p}(\mathbb{R}^{d})$ for $1\leq p\leq\infty$, then

[TABLE]

Moreover, let $I\subset\mathbb{R}^{+}$, then for $f\in L^{2}(\mathbb{R}^{d})$ and $F\in L^{\frac{2(2+d)}{d+4}}_{tx}(\mathbb{R}^{+}\times\mathbb{R}^{d})$,

[TABLE]

[TABLE]

[TABLE]

We can give some remarks on the inequalities $\eqref{lem:Lp-bound}-\eqref{Lpq}$ above as follows:

(i). The estimate (2.1) is classical and immediately follows from the Younger inequality by the following heat kernel integral:

[TABLE]

More generally, for all $1\leq p\leq q\leq\infty$, the following (decay) estimates hold:

[TABLE]

(ii). The estimate (2.2) is equivalent to a kind of square-function inequality on $L^{2}(\mathbb{R}^{d})$, which can be reformulated as

[TABLE]

which follows directly by the Plancherel’s theorem, and also holds in the $L^{p}(\mathbb{R}^{d})$ for $1<p<\infty$ ( see e.g. Stein[20, p. 27-46] ).

(iii). The estimate (2.3) can be obtained by interpolation between the (2.1) and (2.2):

[TABLE]

(iv). The estimate (2.4) consists of the three same type inequalities with the different norms $L^{\infty}_{t}L^{2}_{x}$, $L^{\frac{2(2+d)}{d}}_{tx}$ and $L^{2}_{t}\dot{H}^{1}_{x}$ on the left side. As shown in (iii) above, the second norm $L^{\frac{2(2+d)}{d}}_{tx}$ can be controlled by interpolation between $L^{\infty}_{t}L^{2}_{x}$ and $L^{2}_{t}\dot{H}^{1}_{x}$. Because of similarity of their proofs, we can give a proof to the first one, which is the special case of the following lemma. It is worth to noting that when $p<\infty$, the estimate is $L^{2}$-subcritical.

Lemma 2.3**.**

Let $2\leq p\leq\infty$, and the pair $(p_{1},r_{1})$ satisfy

[TABLE]

then

[TABLE]

Proof.

By Plancherel’s theorem, it is equivalent that

[TABLE]

Since by the Young inequality of the convolution on $\mathbb{R}^{+}$, for any $1\leq p_{1}\leq p\leq\infty$,

[TABLE]

Note that $p_{1}\leq 2\leq p$, thus by Minkowski’s inequality, Plancherel’s theorem, Sobolev’s embedding we obtain

[TABLE]

which gives the desired estimate (2.6). ∎

Finally, we also need the following maximal $L^{p}$-regularity result for the heat flow. See Lemarie-Rieusset’s book [5, P.64] for example.

Lemma 2.4**.**

Let $p\in(1,\infty),q\in(1,\infty)$, and let $T\in(0,\infty]$, then the operator $A$ defined by

[TABLE]

is bounded from $L^{p}((0,T),L^{q}(\mathbb{R}^{d}))$ to $L^{p}((0,T),L^{q}(\mathbb{R}^{d}))$.

3. Proof of Theorem 1.1

In this section, we will divide several subsection to finish the proof of Theorem 1.1. For the end, we first establish a supercritical estimate on the linear heat flow in the following subsection.

3.1. A supercritical estimate on the linear heat flow

Let us recall the following radial Sobolev embedding, see [24] for example.

Lemma 3.1**.**

Let $\alpha,q,p,s$ be the parameters which satisfy

[TABLE]

with

[TABLE]

Moreover, let at most one of the following equalities hold:

[TABLE]

Then the radial Sobolev embedding inequality holds:

[TABLE]

Lemma 3.2**.**

For any $q>2$ and any \gamma\in\big{(}\frac{1}{2}-\frac{3}{q},1-\frac{4}{q}\big{)}, suppose that the radial function $f\in H^{\gamma}(\mathbb{R}^{d})$ satisfying

[TABLE]

then

[TABLE]

Proof.

By Lemma 2.1, we have

[TABLE]

Let $\alpha=\frac{d}{2}-s>0$ and $s\in(\frac{1}{2},1)$, then by Lemma 3.1 we have

[TABLE]

where the first inequality above has used the condition $\mbox{supp }f\subset\{x:|x|\geq 1\}$. Thus we get that

[TABLE]

Interpolation between this last estimate and (2.2), gives our desired estimates. ∎

3.2. Local theory and global criterion

We use $\chi_{\leq a}$ for $a\in\mathbb{R}^{+}$ to denote the smooth function

[TABLE]

and set $\chi_{\geq a}=1-\chi_{\leq a}$.

Now write

[TABLE]

where

[TABLE]

Then we will first claim that $w_{0}\in L^{2}(\mathbb{R}^{d})$, and

[TABLE]

Note that w_{0}=\chi_{\leq\frac{1}{2}}\big{(}P_{\geq N}h_{0}\big{)}+P_{<N}h_{0}. Firstly, we give the following estimate on the first part, which is a consequence of Lemma 2.1.

Lemma 3.3**.**

Let $h_{0}$ be the function satisfying the hypothesis in Theorem 1.1, then

[TABLE]

Proof.

By the support property of $h_{0}$, we may write

[TABLE]

By Lemma 2.1 and Bernstein’s inequality, we have

[TABLE]

Moreover, since $P_{\geq N}=I-P_{<N}$ and $M>2N$, we obtain

[TABLE]

where $P_{\geq\frac{1}{8}M}(\chi_{\geq\frac{9}{10}})$ denotes the high frequency truncation of the bump function $\chi_{\geq\frac{9}{10}}$.

Note that

[TABLE]

Hence, we have

[TABLE]

Therefore, taking summation, we obtain

[TABLE]

Inserting (3.6) and (3.7) into (3.5), we prove the lemma. ∎

Moreover, by the Bernstein estimate,

[TABLE]

Then this last estimate combining with Lemma 3.3 gives (3.3).

Second, we claim that

[TABLE]

Indeed,

[TABLE]

Hence, we only consider the latter term. By Sobolev’s embedding and Hölder’s inequality, we have

[TABLE]

Hence (3.8) follows from Lemma 3.3.

We denote

[TABLE]

Then $v_{L}$ is globally existence, and by Plancherel’s theorem and (3.8)

[TABLE]

Moreover, let $\epsilon$ be a sufficiently small positive constant, then we claim that

[TABLE]

Indeed, let $\gamma=-\frac{d-1}{d+2}+\epsilon$, then by Lemma 3.2,

[TABLE]

Note that

[TABLE]

For the former term, since $\gamma<-\varepsilon_{0}$, by Bernstein’s inequality,

[TABLE]

So we only need to estimate the latter term. Let $q$ be the parameter satisfying

[TABLE]

then $q>1$. Since $\gamma<0$, by Sobolev’s and Hölder’s inequalities,

[TABLE]

Furthermore, by Lemma 3.3,

[TABLE]

Combining the last two estimates above, we obtain

[TABLE]

This gives (3.10).

Now we denote $w=h-v_{L}$, then $w$ is the solution of the following equation,

[TABLE]

The following lemma is the local well-posedness and global criterion of the Cauchy problem (3.11).

Lemma 3.4**.**

There exists $\delta>0$, such that for any $h_{0}$ satisfying the hypothesis in Theorem 1.1 and $w_{0}=h_{0}-v_{0}$, the Cauchy problem (3.11) is well-posed on the time interval $[0,\delta]$, and the solution

[TABLE]

Furthermore, let $T^{*}$ be the maximal lifespan, and suppose that

[TABLE]

then $T^{*}=+\infty$. In particular, if $\|h_{0}\|_{\dot{H}^{-\varepsilon_{0}}(\mathbb{R}^{d})}\ll 1$, then $T^{*}=+\infty$.

Proof.

For local well-posedness, we only show that the solution $w\in L^{\infty}_{t}L^{2}_{x}([0,\delta]\times\mathbb{R}^{d})\cap L^{\frac{2(2+d)}{d}}_{tx}([0,\delta]\times\mathbb{R}^{d})\cap L^{2}_{t}\dot{H}^{1}_{x}([0,\delta]\times\mathbb{R}^{d})$ for some $\delta>0$. Indeed, the local well-posedness with the lifespan $[0,\delta)$ is then followed by the standard fixed point argument. By Duhamel’s formula, we have

[TABLE]

Then by Lemma 2.2, for any $t_{*}\leq\delta$,

[TABLE]

Note that

[TABLE]

let \eta_{0}=(\frac{4}{d}+1)\big{(}\frac{d-1}{d+2}-\varepsilon_{0}-\epsilon\big{)}>0, then using (3.10), we obtain

[TABLE]

Noting that either $\|h_{0}\|_{\dot{H}^{-\varepsilon_{0}}(\mathbb{R}^{d})}\ll 1$, or choosing $\delta$ small enough and $N$ large enough, we have

[TABLE]

then by the continuity argument, we

[TABLE]

Further, by Lemma 2.2 again,

[TABLE]

Hence, using (3.10) and (3.12), we obtain

[TABLE]

for some C=C(N,\big{\|}h_{0}\big{\|}_{\dot{H}^{-\varepsilon_{0}}(\mathbb{R}^{d})})>0.

Suppose that

[TABLE]

then if $T^{*}<+\infty$, we have

[TABLE]

Hence, $w$ exists on $[0,T^{*}]$, and $w(T^{*})\in L^{2}(\mathbb{R}^{d})$. Hence, using the local theory obtained before from time $T^{*}$, the lifespan can be extended to $T^{*}+\delta$, this is contradicted with the definition of the maximal lifespan $T^{*}$. Hence, $T^{*}=+\infty$. ∎

3.3. Uniqueness

Here we adopt the argument in [15], where the main tool is the the maximal $L^{p}$-regularity of the heat flow. Let $h_{1},h_{2}$ be two distinct solutions of (1.2) with the same initial data $h_{0}$, and write

[TABLE]

By the Duhamel formula, we have

[TABLE]

Denote $w=w_{1}-w_{2}$, then $w$ obeys

[TABLE]

Note that there exists an absolute constant $C>0$ such that

[TABLE]

Then by the positivity of the heat kernel, we have

[TABLE]

Then we get that for $2\leq p<\infty$, $\tau\in(0,\delta]$,

[TABLE]

For the first term in the right-hand side above, using Lemma 2.3 and choosing $p$ large enough, we have

[TABLE]

where we have chose $(p_{1},r_{1})$ that

[TABLE]

(Note that $d>4$ and $p$ is large, we have that $p_{1}\in(1,2),r_{1}\in(1,2)$). Hence, by Hölder’s inequality, we obtain that

[TABLE]

For the second term in the right-hand side above, using Lemma 2.4,

[TABLE]

Since $d>4$, by Sobolev’s embedding, we further have

[TABLE]

Collection the estimates above, we obtain that

[TABLE]

where

[TABLE]

By (3.10) and Lemma 2.2, we have

[TABLE]

Further, since $w_{1},w_{2}\in C([0,\delta],L^{2}(\mathbb{R}^{d}))$, we get

[TABLE]

Hence, choosing $\tau$ small enough and from (3.13), we obtain that $w\equiv 0$ on $t\in[0,\tau)$. By iteration, we have $w_{1}\equiv w_{2}$ on $[0,\delta]$. This proves the first statement (1) in Theorem 1.1.

3.4. $L^{2}$-estimates

In this subsection, we prove the second statement (2) in Theorem 1.1.

Firstly, by Lemma 3.4, when $\|h_{0}\|_{\dot{H}^{-\varepsilon_{0}}(\mathbb{R}^{d})}\ll 1$, we immediately have the global existence of the solution for the both cases $\mu=\pm 1$. However, in the defocusing case ($\mu=1$). the smallness of $\|h_{0}\|_{\dot{H}^{-\varepsilon_{0}}(\mathbb{R}^{d})}\ll 1$ can be cancelled. In fact, note that $h=v_{L}+w$ and

[TABLE]

Hence, from Lemma 3.4, we have $h(\delta)\in L^{2}(\mathbb{R}^{d})$. Let $I=[0,T^{*})$ be the maximal lifespan of the solution $h$ of the Cauchy problem (1.2). Then from the $L^{2}$ estimate of the solution (by inner producing with $h$ in (1.2)), we have

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This gives the uniform boundedness of \big{\|}h\big{\|}_{L^{\frac{2(2+d)}{d}}_{tx}(I\times\mathbb{R}^{d})} and thus \big{\|}w\big{\|}_{L^{\frac{2(2+d)}{d}}_{tx}(I\times\mathbb{R}^{d})}. Then by the global criteria given in Lemma 3.4, we have $T^{*}=+\infty$.

Secondly, we consider the time estimate of the solution ($\mu=\pm 1$). When $t\leq 1$, it follows from (3.14) and Lemma 3.4, that

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So it remains to show the decay estimate when $t>1$. By Duhamel’s formula, we have

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Similar as (3.14), we have

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Then using the estimate above and Lemma 2.5, we further have

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In the last step we have used the fact $d\geq 4$ such that $\frac{2d}{d+4}\geq 1$.

Now we denote

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Fixing $T>1$, then for any $t\in(1,T]$,

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Thus we obtain that

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By the continuity argument, we get

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Since the estimate is independent on $T$, we give that

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Therefore, we obtain that

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This proves the second statement (2) in Theorem 1.1.