## Existence and Uniqueness of Global Solutions for a Class of Semilinear Time Fractional Diffusion-Wave Equations

He Xinhai,, Liu Mei,, Yang Han,

School of Mathematics, Southwest Jiaotong University, Chengdu 611756

 基金资助: 国家自然科学基金.  11701477国家自然科学基金.  11971394

 Fund supported: the NSFC.  11701477the NSFC.  11971394

Abstract

The purpose of this paper is to study the Cauchy problem of a class of semilinear time fractional diffusion-wave equations. Based on the Lr-Lq estimates obtained from the corresponding linear problem, and combined with the global iteration method, the influence of the exponential of the nonlinear term on the global existence of the solutions is studied with small data, the existence and uniqueness of global solutions are proved under certain conditions of exponential.

Keywords： Time fractional diffusion-wave equation ; Cauchy problem ; Small data ; Global solution

He Xinhai, Liu Mei, Yang Han. Existence and Uniqueness of Global Solutions for a Class of Semilinear Time Fractional Diffusion-Wave Equations. Acta Mathematica Scientia[J], 2022, 42(6): 1705-1718 doi:

## 1 引言

$$$\left\{ \begin{array}{ll} \partial^{1+\alpha}_{t}u-\Delta u=|\nabla^\gamma u|^p, \quad &t\ge0, \quad x\in{{\Bbb R}} ^n, \\ u(0, x)=u_{0}, \quad u_{t}(0, x)=0, \quad& x\in{{\Bbb R}} ^n \end{array} \right.$$$

$$$\left\{ \begin{array}{ll} \partial^{1+\alpha}_{t}u-\Delta u=|\nabla^\gamma u|^p, \quad& t\ge0, \quad x\in{{\Bbb R}} ^n, \\ u(0, x)=u_{0}, \quad u_{t}(0, x)=u_{1}, \quad& x\in{{\Bbb R}} ^n, \\ \end{array} \right.$$$

$\alpha=0, \gamma=0$时, Fujita[1]研究了以下半线性热传导方程的柯西问题

$$$\left\{ \begin{array}{ll} \partial_{t}u-\Delta u=|u|^p, \quad &t\ge0, \quad x\in{{\Bbb R}} ^n, \\ u(0, x)=u_{0}, \quad & x\in{{\Bbb R}} ^n, \end{array} \right.$$$

$\alpha=0, \gamma=1$时, 对于粘性Hamilton-Jacobi方程

$$$\left\{ \begin{array}{ll} \partial_{t}u-\Delta u=|\nabla u|^p, \quad& t>0, \quad x\in{{\Bbb R}} ^n, \\ u(0, x)=u_{0}, \quad& x\in{{\Bbb R}} ^n, \end{array} \right.$$$

Ben-Artzi等[4]研究了解的整体存在性, 文献[56]先后在$1\le p\le\frac{n+2}{n+1}, \frac{n+2}{n+1}<p<2, p\ge2$等情况下讨论方程的解在$L^{1}$范数和$L^{q}(1\le q\le\infty)$范数下的长时间行为, 研究非线性项对于方程的影响程度.

$\alpha=1, \gamma=0$时, Kato[7]研究了以下半线性波动方程的双初值问题

$$$\left\{ \begin{array}{ll} \partial^{2}_{t}u-\Delta u=|u|^p, \quad &t\ge0, \quad x\in{{\Bbb R}} ^n, \\ u(0, x)=u_{0}, \quad u_{t}(0, x)=u_{1}, \quad &x\in{{\Bbb R}} ^n, \end{array} \right.$$$

$$$\left\{ \begin{array}{ll} \partial^{\alpha}_{t}u=\lambda^{2}\Delta u, \quad &t>0, \quad x\in{{\Bbb R}} ^n, \\ u(0, x)=u_{0}, \quad u_{t}(0, x)=u_{1}, \quad &x\in{{\Bbb R}} ^n, \end{array} \right.$$$

Zhang等[19]研究了时间分数阶扩散方程的柯西问题

$$$\left\{ \begin{array}{ll} \partial^{\alpha}_{t}u-\Delta u=|u|^{p-1}u, \quad &t>0, \quad x\in{{\Bbb R}} ^n, \\ u(0, x)=u_{0}, \quad &x\in{{\Bbb R}} ^n, \end{array} \right.$$$

$$$\tilde{p}=1+\frac{2}{n-2+2(1+\alpha)^{-1}}\quad\mbox{及}\quad \bar{p}=1+\frac{2}{n-2(1+\alpha)^{-1}}.$$$

### 3 定理的证明

$$$\int_{0}^{t}(t-\tau)^{-a}(1+\tau)^{-b}{\rm d}\tau\lesssim\left\{ \begin{array}{ll} (1+t)^{-a}, \qquad\quad\qquad& a<1<b, \\ (1+t)^{-a}\log(1+t), \quad &a<1=b, \\ (1+t)^{-a+1-b}, \qquad\quad & a, b<1. \end{array} \right.$$$

$$$G_{1+\alpha, \beta}(t, x):={\cal F}^{-1}\left(E_{1+\alpha, \beta}(-t^{1+\alpha}|\xi|^{2})\right),$$$

$$$E_{1+\alpha, \beta}(z)=\sum\limits_{k=0}^{\infty}\frac{z^{k}}{\Gamma(k+\alpha k+\beta)}$$$

$$$\|\nabla^{\gamma}G_{1+\alpha, \beta}\|_{L^{q}}\lesssim t^{-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})-\frac{(1+\alpha)\gamma}{2}},$$$

$$$\frac{n}{2}(1-\frac{1}{q})+\frac{\gamma}{2}<\left\{ \begin{array}{ll} 1, \quad\beta=1, 2, \\ 2, \quad\beta=1+\alpha. \end{array} \right.$$$

$$$\|\nabla^{a}u\|_{L^{q}}\lesssim\|\nabla^{\sigma}u\|_{L^{m}}^{\theta_{a, \sigma}(q, m)}\|u\|_{L^{m}}^{1-\theta_{a, \sigma}(q, m)},$$$

### 3.1 定理2.1的证明

$$$\left\{ \begin{array}{ll} \partial^{1+\alpha}_{t}u-\Delta u=0, \quad &t>0, \quad x\in{{\Bbb R}} ^n, \\ u(0, x)=u_{0}, \quad u_{t}(0, x)=u_{1}, \quad &x\in{{\Bbb R}} ^n. \end{array} \right.$$$

$$$\|u\|_{L^{q}}\lesssim t^{-\frac{n(1+\alpha)}{2}(\frac{1}{r}-\frac{1}{q})}\|u_{0}\|_{L^{r}}+t^{1-\frac{n(1+\alpha)}{2}(\frac{1}{r}-\frac{1}{q})}\|u_{1}\|_{L^{r}}, \quad t>0,$$$

$$$\|\nabla u\|_{L^{q}}\lesssim (1+t)^{-\frac{1+\alpha}{2}}t^{-\frac{n(1+\alpha)}{2}(\frac{1}{r}-\frac{1}{q})}\|u_{0}\|_{H^{1, r}}+t^{-\frac{n(1+\alpha)}{2}(\frac{1}{r}-\frac{1}{q})+\frac{1-\alpha}{2}}\|u_{1}\|_{L^{r}}, \quad t>0,$$$

$$$\left\{ \begin{array}{ll} \|u\|_{L^{q}}\lesssim\|u_{0}\|_{L^{q}}, \qquad\qquad\quad &0\le t\le 1, \\ \|u\|_{L^{q}}\lesssim t^{-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})}\|u_{0}\|_{L^{1}}, \quad &t>1. \end{array} \right.$$$

$$$\|u\|_{L^{q}}\lesssim (1+t)^{-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})}\|u_{0}\|_{L^{1}\cap L^{q}}, \quad t\ge0.$$$

$$$\left\{ \begin{array}{ll} \|\nabla u\|_{L^{q}}\lesssim(1+t)^{-\frac{1+\alpha}{2}}\|u_{0}\|_{H^{1, q}}, \qquad\quad\qquad & 0\le t\le 1, \\ \|\nabla u\|_{L^{q}}\lesssim(1+t)^{-\frac{1+\alpha}{2}}t^{-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})}\|u_{0}\|_{H^{1, 1}}, \quad & t>1. \end{array} \right.$$$

$$$\|\nabla u\|_{L^{q}}\lesssim(1+t)^{-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})-\frac{1+\alpha}{2}} \|u_{0}\|_{H^{1, 1}\cap H^{1, q}}, \quad t\ge0.$$$

(3.11)及(3.13)式同时成立需满足

$$$\frac{n}{2}(1-\frac{1}{q})<\frac{1}{2}.$$$

$$$u=u^{lin}+Nu,$$$

$$$\|Nu\|_{L^{q}}\lesssim\int_{0}^{t}(t-\tau)^{\alpha}(t-\tau)^{-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})}\|f(\tau, x)\|_{L^{1}}{\rm d}\tau.$$$

$$$\|Nu\|_{L^{q}}\lesssim\left\{ \begin{array}{ll} K(1+t)^{\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})}, \qquad\qquad\quad&\tilde{\eta}>1, \\ K(1+t)^{\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})}\log(1+t), \quad &\tilde{\eta}=1, \\ K(1+t)^{\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})+1-\tilde{\eta}}, &\tilde{\eta}<1. \end{array} \right.$$$

$$$\|\nabla Nu\|_{L^{q}}\lesssim\left\{ \begin{array}{ll} K(1+t)^{\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})-\frac{1+\alpha}{2}}, &\tilde{\eta}>1, \\ K(1+t)^{\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})-\frac{1+\alpha}{2}}\log(1+t), \quad &\tilde{\eta}=1, \\ K(1+t)^{\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})-\frac{1+\alpha}{2}+1-\tilde{\eta}}, &\tilde{\eta}<1. \end{array} \right.$$$

(i) $n\ge2$时, 要满足(3.14)式, 则需使$q<1+\frac{1}{n-1}.$

$\begin{eqnarray} \|u\|_{X}&=&\sup\limits_{t\ge0}(1+t)^{-\alpha}\Big\{\|u\|_{L^{1}}+(1+t)^{\frac{n(1+\alpha)}{2} (1-\frac{1}{p})}\|u\|_{L^{p}}{}\\ &&+(1+t)^{\frac{1+\alpha}{2}}\|\nabla u\|_{L^{1}}+(1+t)^{\frac{n(1+\alpha)}{2}(1-\frac{1}{p})+\frac{1+\alpha}{2}}\|\nabla u\|_{L^{p}}\Big\}. \end{eqnarray}$

${\rm (a)}\; \forall u\in X$, 有$\|\Phi u\|_{X}\lesssim\|u_{0}\|_{H^{1, 1}\cap H^{1, p}}+\|u\|_{X}^{p}$,

${\rm (b)}\; \forall u, v\in X$, 有$\|Nu-Nv\|_{X}\lesssim\|u-v\|_{X}(\|u\|_{X}^{p-1}+\|v\|_{X}^{p-1}).$

$\begin{eqnarray} \||\nabla^{\gamma}u|^{p}\|_{L^{1}}&\le&\|\nabla^{\gamma}u\|_{L^{p}}^{p}\lesssim \|\nabla u\|_{L^{p}}^{p\theta_{\gamma, 1}(p, p)}\| u\|_{L^{p}}^{p[1-\theta_{\gamma, 1}(p, p)]}{}\\ &\lesssim&(1+t)^{[\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{p})-\frac{1+\alpha}{2}]\gamma p}\|u\|_{X}^{\gamma p}(1+t) ^{[\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{p})](1-\gamma)p}\|u\|_{X}^{(1-\gamma)p}{}\\ &=&(1+t)^{[\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{p})-\frac{(1+\alpha)\gamma}{2}]p}\|u\|_{X}^{p}. \end{eqnarray}$

$K=C\|u\|_{X}^{p}$ ($C$为一正常数), $\tilde{\eta}=\frac{n(1+\alpha)}{2}(p-1)+\frac{(1+\alpha)\gamma p}{2}-\alpha p$, 当且仅当$p>\tilde{p}$时, $\tilde{\eta}>1.$由式(3.18)及(3.19)式有

$\Phi$是由$X$到自身的映射.对于$(b)$压缩性的证明运用以下不等式

$\begin{eqnarray} \||\nabla^{\gamma}u|^{p}-|\nabla^{\gamma}v|^{p}\|_{L^{1}}&\le& \||\nabla^{\gamma}u-\nabla^{\gamma}v|(|\nabla^{\gamma}u|^{p-1}+|\nabla^{\gamma}v|^{p-1})\|_{L^{1}}{}\\ &\le&\|\nabla^{\gamma}u-\nabla^{\gamma}v\|_{L^{p}}\||\nabla^{\gamma}u|^{p-1}+|\nabla^{\gamma}v|^{p-1}\|_{L^{\frac{p}{p-1}}}{}\\ &\le&\|\nabla^{\gamma}(u-v)\|_{L^{p}}(\|\nabla^{\gamma}u\|_{L^{p}}^{p-1}+\|\nabla^{\gamma}v\|_{L^{p}}^{p-1}){}\\ &\lesssim&(1+t)^{[\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{p})-\frac{(1+\alpha)\gamma}{2}]p}\|u-v\|_{X}(\|u\|_{X}^{p-1}+\|v\|_{X}^{p-1}).{\qquad} \end{eqnarray}$

$K=C\|u-v\|_{X}(\|u\|_{X}^{p-1}+\|v\|_{X}^{p-1})$, 又因为

$$$\|Nu-Nv\|_{X}\lesssim\|u-v\|_{X}(\|u\|_{X}^{p-1}+\|v\|_{X}^{p-1}).$$$

$1<p<\tilde{p}$, 考虑初值$u_{0}$变为$\varepsilon u_{0}$的情况, $\varepsilon$充分小, 重复以上$(a)$证明步骤, 可以得到

$$$\|u^{lin}+Nu\|_{X}\le C(\varepsilon+(1+t)^{1-\tilde{\eta}}\|u\|_{X}^{p}).$$$

$$$\|u^{lin}+Nu\|_{X}\le C(\varepsilon+(1+t)^{1-\tilde{\eta}}\|u\|_{X}^{p}) \le C(\varepsilon+(1+t)^{1-\tilde{\eta}}(C_{0}\varepsilon)^{p})\le2C\varepsilon.$$$

$p=\tilde{p}$, 同样由(3.18)及(3.19)式可得

$$$\|u^{lin}+Nu\|_{X}\le C(\varepsilon+\log(1+t)\|u\|_{X}^{p}) \le C(\varepsilon+\log(1+t)(C_{0}\varepsilon)^{p})\le2C\varepsilon.$$$

(ii) $n=1$时, 对$\forall q\ge1,$

$$$\frac{1}{2}(1-\frac{1}{q})<\frac{1}{2}$$$

$$$\|u\|_{L^{q}}\lesssim (1+t)^{-\frac{1+\alpha}{2}(1-\frac{1}{q})}\|u_{0}\|_{L^{1}\cap L^{q}}, \quad t\ge0,$$$

$$$\|\nabla u\|_{L^{q}}\lesssim(1+t)^{-\frac{1+\alpha}{2}(1-\frac{1}{q})-\frac{1+\alpha}{2}}\|u_{0}\|_{H^{1, 1} \cap H^{1, q}}, \quad t\ge0.$$$

$1\le q\le p<\infty$, 在$p>\tilde{p}$时整体解的存在性证明与$n\ge2$时类似, 在$1<p\le\tilde{p}$时局部解的存在区间估计也同理可得.

### 3.2 定理2.2的证明

$$$\frac{n}{2}(\frac{1}{r}-\frac{1}{q})<\frac{1}{2}$$$

$$$1-\frac{n(1+\alpha)}{2}(\frac{1}{r}-\frac{1}{q})>\frac{1-\alpha}{2}>0$$$

$$$\|\nabla^{\gamma}Nu\|_{L^{q}}\le\int_{0}^{t}(t-\tau)^{\alpha}\|\nabla^{\gamma}G_{1+\alpha, 1+\alpha}(t-\tau, x)\|_{L^{r_{0}}}\|f(\tau, x)\|_{L^{r_{1}}}{\rm d}\tau,$$$

$$$\|\nabla^{\gamma}Nu\|_{L^{q}}\lesssim\int_{0}^{t}(t-\tau)^{\alpha}(t-\tau)^{-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})-\frac{(1+\alpha)\gamma}{2}}\|f(\tau, x)\|_{L^{1}}{\rm d}\tau.$$$

$$$-\alpha+\frac{n(1+\alpha)}{2}(1-\frac{1}{q})+\frac{(1+\alpha)\gamma}{2}<1 ,$$$

$$$\|\nabla^{\gamma}Nu\|_{L^{q}}\lesssim\left\{ \begin{array}{ll} K(1+t)^{\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})-\frac{(1+\alpha)\gamma}{2}}, \qquad\qquad\quad&\bar{\eta}>1, \\ K(1+t)^{\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})-\frac{(1+\alpha)\gamma}{2}}\log(1+t), \quad&\bar{\eta}=1, \\ K(1+t)^{\alpha-\frac{n(1+\alpha)}{2}(1-\frac{1}{q})-\frac{(1+\alpha)\gamma}{2}+1-\bar{\eta}}, & \bar{\eta}<1. \end{array} \right.$$$

(i) $n\ge2$时, 要满足(3.37)式, 则需使$q<1+\frac{2-\gamma}{n-2+\gamma}$. 对任意$q\in[1, p], p<1+\frac{2-\gamma}{n-2+\gamma}$. 定义

$$$\begin{array}{ll} \|u\|_{X}=\sup\limits_{t\ge0}(1+t)^{-1}\{(1+t)^{\frac{(1+\alpha)\gamma}{2}}\|\nabla^{\gamma}u\|_{L^{1}}+(1+t)^{\frac{n(1+\alpha)}{2}(1-\frac{1}{p})+\frac{(1+\alpha)\gamma}{2}}\|\nabla^{\gamma}u\|_{L^{p}}\}. \end{array}$$$

(a) $\forall u\in X$, 有$\|\Psi u\|_{X}\lesssim\|u_{0}\|_{H^{\gamma, 1}\cap H^{\gamma, p}}+\|u_{1}\|_{L^{1}\cap L^{p}}$,

(b) $\forall u, v\in X$, 有$\|Nu-Nv\|_{X}\lesssim\|u-v\|_{X}(\|u\|_{X}^{p-1}+\|v\|_{X}^{p-1}).$任取$u\in X$, 由(3.36)式有

$$$\begin{array}{ll} \|u^{lin}\|_{X}\lesssim\|u_{0}\|_{H^{\gamma, 1}\cap H^{\gamma, p}}+\|u_{1}\|_{L^{1}\cap L^{p}}. \end{array}$$$

$$$\begin{array}{ll} \||\nabla^{\gamma}u|^{p}\|_{L^{1}}=\|\nabla^{\gamma}u\|_{L^{p}}^{p}\lesssim (1+t)^{-[-1+\frac{n(1+\alpha)}{2}(1-\frac{1}{p})+\frac{(1+\alpha)\gamma}{2}]p}\|u\|_{X}^{p}. \end{array}$$$

$K=C\|u\|_{X}^{p}, \bar{\eta}=\frac{n(1+\alpha)}{2}(p-1)+\frac{(1+\alpha)\gamma p}{2}-p$, 当$p>\hat{p}:=1+\frac{4(1+\alpha)^{-1}-\gamma}{n+\gamma-2(1+\alpha)^{-1}}$时, $\bar{\eta}>1$. 由(3.41)式有

$$$\|Nu\|_{X}\lesssim\|u\|_{X}^{p}.$$$

$$$\|Nu\|_{X}\lesssim\|u\|_{X}^{p}.$$$

$1<p<\bar{p}$, 考虑初值$(u_{0}, u_{1})$变为$(\varepsilon u_{0}, \varepsilon u_{1})$的情况, $\varepsilon$充分小, 重复以上证明步骤, 可以得到

$$$\|u^{lin}+Nu\|_{X}\le C^{'}(\varepsilon+(1+t)^{\alpha-\bar{\eta}}\|u\|_{X}^{p}).$$$

$$$\|u^{lin}+Nu\|_{X}\le C^{'}(\varepsilon+(1+t)^{\alpha-\bar{\eta}}\|u\|_{X}^{p})\le C^{'}(\varepsilon+(1+t)^{\alpha-\bar{\eta}}(C_{1}\varepsilon)^{p})\le2C^{'}\varepsilon.$$$

(ii) $n=1$时, 可以看到对$\forall q\ge1,$条件

$$$\frac{1}{2}(1-\frac{1}{q})+\frac{\gamma}{2}<1, \quad\gamma\in(0, 1)$$$

$$$\|\nabla^{\gamma}u^{lin}\|_{L^{q}}\lesssim (1+t)^{1-\frac{1+\alpha}{2}(1-\frac{1}{q})-\frac{(1+\alpha)\gamma}{2}}(\|u_{0}\|_{H^{\gamma, 1}\cap H^{\gamma, q}}+\|u_{1}\|_{L^{1}\cap L^{q}}), \quad t\ge0.$$$

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