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Zeitschrift für Analysis und ihre Anwendungen


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Volume 33, Issue 4, 2014, pp. 447–462
DOI: 10.4171/ZAA/1522

Published online: 2014-10-15

On $p_s(x)$-Laplacian Parabolic Problems with Non-Globally Lipschitz Forcing Term

Jacson Simsen[1], Mariza S. Simsen[2] and Marcos R. Teixeira Primo[3]

(1) Universidade Federal de Itajubá, Itajubá, Minas Gerais, Brazil
(2) Universidade Federal de Itajubá, Itajubá, Minas Gerais, Brazil
(3) Universidade Estadual de Maringá, Paraná, Brazil

In this work we prove continuity of solutions with respect to initial conditions and exponent parameters and we prove upper semicontinuity of a family of global attractors for problems of the form $$\frac{\partial u_s}{\partial t}-\textrm{div}(|\nabla u_{s}|^{p_s(x)-2}\nabla u_{s})+f(x,u_s)=g,$$ where $f:\Omega\times \mathbb R\to \mathbb R$ is a non-globally Lispchitz Carathéodory mapping, $g\in L^2(\Omega),$ $\Omega$ is a bounded smooth domain in $\mathbb{R}^{n},\;n\geq 1$ and $p_s(\cdot)\to p\;\mbox{in}\;L^\infty(\Omega)\; (p > 2\;\mbox{constant})$ as $s$ goes to infinity.

Keywords: Variable exponents, electrorheological fluids, $p_s(x)$-Laplacian, parabolic problems, global attractors, upper semicontinuity

Simsen Jacson, Simsen Mariza, Teixeira Primo Marcos: On $p_s(x)$-Laplacian Parabolic Problems with Non-Globally Lipschitz Forcing Term. Z. Anal. Anwend. 33 (2014), 447-462. doi: 10.4171/ZAA/1522