Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source

Year 2023, Volume: 11 Issue: 3, 112 - 128, 02.09.2023

Carlos Raposo , Adriano Cattai , Octavio Vera , Ganesh Ch. Goraın , Ducival Pereira

https://doi.org/10.36753/mathenot.1084371

Cited By: 1

Abstract

This manuscript deals with global solution, polynomial stability and blow-up behavior at a finite time for the nonlinear system $$ \left\{ \begin{array}{rcl} & u'' - \Delta_{p} u + \theta + \alpha u' = \left\vert u\right\vert ^{p-2}u\ln \left\vert u\right\vert \\ &\theta' - \Delta \theta = u' \end{array} \right. $$ where $\Delta_{p}$ is the nonlinear $p$-Laplacian operator, $ 2 \leq p < \infty$. Taking into account that the initial data is in a suitable stability set created from the Nehari manifold, the global solution is constructed by means of the Faedo-Galerkin approximations. Polynomial decay is proven for a subcritical level of initial energy. The blow-up behavior is shown on an instability set with negative energy values.

Keywords

Global solution, blow-up, thermoelastic system of p-Laplacian type, logarithmic source

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