On the Cauchy problem for evolution $p(x)$-Laplace equation

Abstract

We consider the Cauchy problem for the equation

with measurable but possibly discontinuous variable exponent $p(x):{\mathbb{R}}^n\mapsto [p^{-},p^{+}]\subset (1,\infty )$. It is shown that for every $u(x,0)\in L^2({\mathbb{R}}^n)$ and $f\in L^2(S_T)$ the problem has at least one weak solution $u\in C^0([0,T];L_{loc}^2({\mathbb{R}}^n))\cap L^2(S_T)$, $|\nabla u{|}^{p(x)}\in L^1(S_T)$. We derive sufficient conditions for global boundedness of weak solutions and show that the bounded weak solution is unique.