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

### Stanislav Antontsev

Universidade de Lisboa, Portugal### Sergey Shmarev

Universidad de Oviedo, Spain

## Abstract

We consider the Cauchy problem for the equation

$u_{t}−div(a(x,t)∣∇u∣_{p(x)−2}∇u)=f(x,t)inS_{T}=R_{n}×(0,T)$

with measurable but possibly discontinuous variable exponent $p(x):R_{n}↦[p_{−},p_{+}]⊂(1,∞)$. It is shown that for every $u(x,0)∈L_{2}(R_{n})$ and $f∈L_{2}(S_{T})$ the problem has at least one weak solution $u∈C_{0}([0,T];L_{loc}(R_{n}))∩L_{2}(S_{T})$, $∣∇u∣_{p(x)}∈L_{1}(S_{T})$. We derive sufficient conditions for global boundedness of weak solutions and show that the bounded weak solution is unique.

## Cite this article

Stanislav Antontsev, Sergey Shmarev, On the Cauchy problem for evolution $p(x)$-Laplace equation. Port. Math. 72 (2015), no. 2/3, pp. 125–144

DOI 10.4171/PM/1961