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 $u\in C^{0}([0,T];L^{2} _{loc}(\mathbb{R}^{n}))\cap L^{2}(S_{T})$, $|\nabla u|^{p(x)}\in L^{1}(S_{T} )$. We derive sufficient conditions for global boundedness of weak solutions and show that the bounded weak solution is unique.