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European Mathematical Society Publishing House
2016-09-19 17:05:24
Portugaliae Mathematica
Port. Math.
PM
0032-5155
1662-2758
General
10.4171/PM
http://www.ems-ph.org/doi/10.4171/PM
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2008)
72
2015
2
On the Cauchy problem for evolution $p(x)$-Laplace equation
Stanislav
Antontsev
Universidade de Lisboa, LISBOA, PORTUGAL
Sergey
Shmarev
Universidad de Oviedo, OVIEDO, SPAIN
Nonlinear parabolic equation, variable nonlinearity, $p(x)$-Laplace, Cauchy problem
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.
Partial differential equations
125
144
10.4171/PM/1961
http://www.ems-ph.org/doi/10.4171/PM/1961