Local existence and uniqueness of weak solutions to fractional pseudo-parabolic equation with singular potential

Abstract

We establish the local existence and uniqueness of weak solutions for fractional pseudo-parabolic equation with singular potential by means of Galërkin method and contraction mapping theorem.

Introduction

In this work, we are concerned with the following initial boundary value problem of fractional pseudo-parabolic equation with singular potential

$$\left\{\begin{array}{cc}\hfill \frac{u_t}{{\left|x\right|}^{2s}}+{(-\Delta )}^s{u}_t+{(-\Delta )}^{s}u& ={\left|u\right|}^{p-2}u,(x,t)\in \Omega \times (0,T],\hfill \\ \hfill u(x,0)& =u_0\left(x\right),x\in \Omega ,\hfill \\ \hfill u(x,t)& =0,(x,t)\in ({\mathbb{R}}^N\setminus \Omega )\times (0,T],\hfill \end{array}\right.$$

where $\Omega$ is a bounded domain in ${\mathbb{R}}^N(N>2s)$ with smooth boundary $\partial \Omega$, and $2<p\le 2_s^{\ast }$, $2_s^{\ast }=\frac{2N}{N-2s}$ is critical exponent of fractional Sobolev embedding inequality, ${(-\Delta )}^s(0<s<1)$ is the spectral fractional Laplacian operator.

Recently, it is founded that the fractional Laplacian operator can describe practical problems better compared with the classical Laplacian operator, which consequently attracts considerable attention of mathematics and physicists. The fractional Laplacian operator was first proposed in observation of L$\acute{\mathrm{e}}$vy stationary diffusion process in physics, later also used to describe the phenomena such as plasma anomalous diffusion, stochastic analysis and fluid dynamics, etc.

For the nonlinear pseudo-parabolic equation:

$$\theta \left(x\right)u_t+k{(-\Delta )}^{\alpha }{u}_t+{(-\Delta )}^{s}u=f\left(u\right),$$

where the force term $f\left(u\right)=u^{p-1}$ or $f\left(u\right)={\left|u\right|}^{p-2}u$. The comprehensive physical backgrounds of pseudo-parabolic equation can refer to [1], [2], [3], [4], [5] and the references therein. Next we mention some related work for (1.2) as follows.

• When $\theta \left(x\right)=1$ with $k=0,s=1$, it is classical heat equation. The global existence, decay estimate, asymptotic behavior and blow-up behavior of solutions to Eq. (1.2) posed on bounded domain $\Omega$ has been widely discussed (see, e.g., [6], [7], [8], [9], [10]).

• When $\theta \left(x\right)=1$ with $k=1,\alpha =s=1$, (1.2) is called semilinear pseudo-parabolic equation. Such case was studied in [11], the authors reveal the relations between the initial data and the global well-posedness of the solution to the initial boundary value problem at three different initial energy levels under bounded domain. For such equation on $\Omega$ with general nonlinearity, Han [12] established finite time blowup properties. Other properties of solutions to the such integer order pseudo-parabolic equation, such as maximum principle, the influence of the inhomogeneous term on the asymptotic behavior was considered in [13], [14], [15].

• When $\theta \left(x\right)={\left|x\right|}^{-2}$ with $k=0,s=1$, Tan [16] studied the global existence and blow up of the solutions by using the Hardy inequality and potential well.

• When $\theta \left(x\right)={\left|x\right|}^{-\delta }(\delta >0)$ with $k=0,s=1$, Eq. (1.2) coupled with homogeneous Dirichlet boundary condition has been studied in [17], [18], [19], and the global existence and blow-up conditions were gotten by means of potential well method.

• When $\theta \left(x\right)={\left|x\right|}^{-\delta }(0\le \delta \le 1)$ with $k=1,\alpha =s=1$, Lian et al. [20] considered

  $$\frac{u_t}{{\left|x\right|}^{\delta }}-\Delta u_t-\Delta u={\left|u\right|}^{p-2}u,2<p<\frac{2N}{N-2},(x,t)\in \Omega \times (0,T),$$

  they obtained global existence, asymptotic behavior and blowup of solutions with three energy levels.

The above research are focusing on integer order equations, as for the related work concerning fractional order, we present as below.

• When $\theta \left(x\right)=1$ with $k=0,0<s<1$, Fu and Pucci [21] proved the global existence and exponential decay, as well as the finite time blowup.

• When $\theta \left(x\right)=1$ with $k>0,\alpha =1,s>0$, Jin et al. [22] investigated

  $$u_t-k\Delta u_t+{(-\Delta )}^{s}u={\left|u\right|}^{p+1},x\in {\mathbb{R}}^N,t>0,k>0,s>0,p>\frac{4s}{N},$$

  they established the global existence and time-decay rates for small-amplitude solutions to the Cauchy problem.

• When $\theta \left(x\right)=1$ with $k>0,0<\alpha =s<1$, Cheng and Fang [23] observed the equation

  $$u_t+k{(-\Delta )}^s{u}_t+{(-\Delta )}^{s}u=u^p,p>0,x\in {\mathbb{R}}^N,t>0$$

  supplemented with the sufficiently smooth and nonnegative initial data. They use the Green’s function to express the solution, and get the pointwise estimate and exponential decay.

Motivated by the previous work, we focus on fractional order with singular potential posed in the bounded domain, specifically, $\theta \left(x\right)={\left|x\right|}^{-2s}(0<s<1)$ with $k=1,0<\alpha =s<1$. We aim to obtain the Local existence and uniqueness of Eqs. (1.1).

Denote $\mathcal{H}=C([0,T];H_0^s\left(\Omega \right))$ endowed with the norm ${\Vert u\Vert }_{\mathcal{H}}^2={max}_{t\in [0,T]}{\Vert {(-\Delta )}^{s∕2}u\Vert }_2^2$. And due to the singularity existing in the Eqs. (1.1), we introduce ${\rho }_n=min\{{\left|x\right|}^{-2s},n\}$.

Definition 1.1 Weak Solution

A function $u=u(x,t)$ is referred to as a weak solution of Eqs. (1.1) on ${\Omega }_T=\Omega \times [0,T]$, if $u\in L^{\infty }([0,T];H_0^s\left(\Omega \right))$ with $u_t\in L^2([0,T];H_0^s\left(\Omega \right))$ satisfying

(i) for any $\eta \in H_0^s\left(\Omega \right),t\in [0,T]$,

$$(\frac{u_t}{{\left|x\right|}^{2s}},\eta )+({(-\Delta )}^{s∕2}u,{(-\Delta )}^{s∕2}\eta )+({(-\Delta )}^{s∕2}{u}_t,{(-\Delta )}^{s∕2}\eta )=({\left|u\right|}^{p-2}u,\eta );$$

(ii)${\int }_0^t{\int }_{\Omega }\frac{u_{\tau }}{{\left|x\right|}^{2s}}dxd\tau <\infty$ and $u_0\left(x\right)\in H_0^s\left(\Omega \right)$.

Firstly, we establish a key lemma as follows.

Lemma 1.1

For every $T>0$, for every known $u_n\in \mathcal{H}$ and every initial data $u_{n0}\in C_0^{\infty }\left(\Omega \right)$, $u_0\in H_0^s\left(\Omega \right)$, there exists a unique solution

$$v_n\in \mathcal{H}suchthat{\dot{v}}_n\in L^2([0,T];H_0^s\left(\Omega \right)),$$

which solves

$$\left\{\begin{array}{cc}\hfill {\rho }_n\left(x\right){\dot{v}}_n+{(-\Delta )}^s{\dot{v}}_n+{(-\Delta )}^s{v}_n& ={\left|u_n\right|}^{p-2}{u}_n,(x,t)\in \Omega \times (0,T],\hfill \\ \hfill v_n(x,t)& =0,(x,t)\in ({\mathbb{R}}^N\setminus \Omega )\times [0,T],\hfill \\ \hfill v_n(x,0)& =u_{n0},x\in \Omega ,\hfill \end{array}\right.$$

where $\dot{v_n}≔\frac{\partial v_n}{\partial t}$, and $2<p\le 2_s^{\ast }=\frac{2N}{N-2s}(N>2s)$.

Based upon above lemma, by applying the Banach fixed point theorem, we arrive at the following result.

Theorem 1.1

For $2<p\le 2_s^{\ast }$ and $T>0$, there exists a unique solution $u$ of problem (1.1) over $[0,T]$.