Solutions of evolutionary equation based on the anisotropic variable exponent Sobolev space

Abstract

In this paper, we are concerned with the equation

$$\begin{aligned} u_t= \sum _{i=1}^N\frac{\partial }{\partial x_i} \left( a_i(x)\left| u_{x_i} \right| ^{p_i(x) - 2} u_{x_i} \right) +\sum _{i=1}^N\frac{\partial b_i(u)}{\partial x_i},\ \ (x,t) \in \Omega \times (0,T), \end{aligned}$$

where $a_i(x)|_{x\in \partial \Omega }=0$ and $a_i(x)|_{x\in \Omega }>0$. By the theory of anisotropic variable exponent Sobolev spaces, we study the well-posedness of weak solutions of this equation. Since $a_i(x)$ is degenerate on the boundary, the stability of weak solutions may be established without any boundary value condition. The main feature which distinguishes this paper from other related works lies in the fact that we propose a novel analytical method to deal with the stability of weak solutions.