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

Jacson Simsen; Mariza S. Simsen; Marcos R. Teixeira Primo

doi:10.4171/zaa/1522

JournalszaaVol. 33, No. 4pp. 447–462

On $p_{s} (x)$ -Laplacian Parabolic Problems with Non-Globally Lipschitz Forcing Term

Jacson Simsen
Universidade Federal de Itajubá, Itajubá, Minas Gerais, Brazil
Mariza S. Simsen
Universidade Federal de Itajubá, Itajubá, Minas Gerais, Brazil
Marcos R. Teixeira Primo
Universidade Estadual de Maringá, Paraná, Brazil

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Abstract

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} - div (∣\nabla u_{s} ∣^{p_{s} (x) - 2} \nabla u_{s}) + f (x, u_{s}) = g,

where $f : Ω \times R \to R$ is a non-globally Lispchitz Carathéodory mapping, $g \in L^{2} (Ω),$ $Ω$ is a bounded smooth domain in $R^{n}, n \geq 1$ and $p_{s} (\cdot) \to p in L^{\infty} (Ω) (p > 2 constant)$ as $s$ goes to infinity.

Cite this article

Jacson Simsen, Mariza S. Simsen, Marcos R. Teixeira Primo, On $p_{s} (x)$ -Laplacian Parabolic Problems with Non-Globally Lipschitz Forcing Term. Z. Anal. Anwend. 33 (2014), no. 4, pp. 447–462

DOI 10.4171/ZAA/1522