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# Portugaliae Mathematica

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**Volume 72, Issue 2-3, 2015, pp. 125–144**

**DOI: 10.4171/PM/1961**

Published online: 2015-07-23

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

Stanislav Antontsev^{[1]}and Sergey Shmarev

^{[2]}(1) Universidade de Lisboa, Portugal

(2) Universidad de Oviedo, Spain

We consider the Cauchy problem for the equation \[ \text{$u_{t}-\operatorname{div} \left( a(x,t) |\nabla u|^{p(x)-2}\nabla u\right) =f(x,t)$ in $S_{T}=\mathbb{R}^{n}\times(0,T)$} \] 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.

*Keywords: *Nonlinear parabolic equation, variable nonlinearity, $p(x)$-Laplace, Cauchy problem

Antontsev Stanislav, Shmarev Sergey: On the Cauchy problem for evolution $p(x)$-Laplace equation. *Port. Math.* 72 (2015), 125-144. doi: 10.4171/PM/1961