Exponential decay for solutions of  coupled nonlinear viscoelastic  and heat system

Karima Allab; Akram Ben Aissa; Mounir Bahlil

Karima Allab University of Sciences and Technology Houari Boumediene, Algiers 16111, Algeria
Akram Ben Aissa Higher Institute of transport and Logistics of Sousse, University of Sousse, Tunisia
Mounir Bahlil Laboratory of Analysis and Control of Partial Differential Equations,\\ Djillali Liabes University, P. O. Box 89, Sidi Bel Abbes 22000, Algeria

Abstract

Following the paper of Ben Aissa\cite{akram} dealing with internal stability of coupled non-degenrate Kirchhoff system via heat conduction, this manuscript concerns with the problem of the following nonlinear viscoelastic equation
$$|y_t|^ly_{tt}-\gamma \Delta y_{tt}-\Delta y +\alpha \Delta \theta=0$$ coupled to linear heat equation in a bounded domain. So under a clever adaptation of the well-known Fadeo-Galerkin approximation method, we establish existence and uniqueness of solution to this coupled system. Meanwhile, using the piecewise multiplied method, we show the exponenetial decay of the solution