Solutions of evolutionary ${\varvec{p(x)}}$-Laplacian equation based on the weighted variable exponent space

Abstract

The paper studies the equation

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

with the boundary degeneracy due to $a(x)\mid _{x\in \partial \Omega }=0$, and $\Omega \subset \mathbb {R}^{N}$, where N is a positive integer. By the theory of the weighted variable exponent Sobolev spaces, the well posedness of weak solutions of this equation is discussed. The novelty of our results lies in the fact that under certain conditions, if a(x) satisfies $\int \nolimits _{\Omega }a^{-\frac{1}{p(x)-1}}\mathrm{d}x<\infty$, the global stability of weak solutions can be established without any boundary value condition. While $\int \nolimits _{\Omega }a^{-\frac{1}{p(x)-1}}\mathrm{d}x=\infty$, the local stability of weak solutions can be obtained without any boundary value condition.