The weak solutions of a doubly nonlinear parabolic equation related to the $p(x)$-Laplacian

Abstract

Abstract A nonlinear degenerate parabolic equation related to the $p(x)$p(x)-Laplacian $$ {u_{t}}= \operatorname{div} \bigl({b(x)} { \bigl\vert {\nabla a(u)} \bigr\vert ^{p(x) - 2}}\nabla a(u) \bigr)+\sum _{i=1}^{N}\frac{\partial b_{i}(u)}{ \partial x_{i}}+c(x,t) -b_{0}a(u) $$ut=div(b(x)|∇a(u)|p(x)−2∇a(u))+∑i=1N∂bi(u)∂xi+c(x,t)−b0a(u) is considered in this paper, where $b(x)|_{x\in \varOmega }>0$b(x)|x∈Ω>0, $b(x)|_{x \in \partial \varOmega }=0$b(x)|x∈∂Ω=0, $a(s)\geq 0$a(s)≥0 is a strictly increasing function with $a(0)=0$a(0)=0, $c(x,t)\geq 0$c(x,t)≥0 and $b_{0}>0$b0>0. If $\int _{\varOmega }b(x)^{-\frac{1}{p ^{-}-1}}\,dx\leq c$∫Ωb(x)−1p−−1dx≤c and $\vert \sum_{i=1}^{N}b_{i}'(s) \vert \leq c a'(s)$|∑i=1Nbi′(s)|≤ca′(s), then the solutions of the initial-boundary value problem is well-posedness. When $\int _{\varOmega }b(x)^{-(p(x)-1)}\,dx<\infty $∫Ωb(x)−(p(x)−1)dx<∞, without the boundary value condition, the stability of weak solutions can be proved.