Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction

TL;DR

This paper proves the well-posedness and exponential decay of solutions for a coupled Kirchhoff heat conduction system, using multiplier techniques and integral inequalities, contributing to the understanding of stability in nonlinear PDE systems.

Contribution

It establishes existence and exponential decay of solutions for a coupled non-degenerate Kirchhoff system with heat conduction, a novel result in this context.

Findings

Existence of solutions is proven.

Solutions exhibit exponential decay over time.

Method employs multiplier technique and integral inequalities.

Abstract

In the paper under study, we consider the following coupled non-degenerate Kirchhoff system \begin{equation}\label{P} \left \{ \begin{aligned} &\displaystyle y_{tt}-\upvarphi\Big(\int_\Omega | \nabla y |^2\,dx\Big)\Delta y +\upalpha \Delta \uptheta=0, &\mbox{ in }&\; \Omega \times (0, +\infty)\\ &\displaystyle \uptheta_t-\Delta \uptheta-\upbeta \Delta y_t =0, &\mbox{ in }&\; \Omega \times (0, +\infty)\\ &\displaystyle y=\uptheta=0,\; &\mbox{ on }&\;\partial\Omega\times(0, +\infty)\\ %&\displaystyle y=0,\; &\mbox{ on }&\;\partial\Omega\times(0, +\infty)\\ %&\displaystyle \partial_\nu y=0, &\mbox{ on }&\;\Gamma_1\times(0, +\infty)\\ &\displaystyle y(\cdot, 0)=y_0, \; y_t(\cdot, 0)=y_1,\;\uptheta(\cdot, 0)=\uptheta_0, \; \; &\mbox{ in }&\; \Omega\\ \end{aligned} \right. \end{equation} where $\Omega$ is a bounded open subset of $\mathbb{R}^n$, $\upalpha$ and $\upbeta$ be two nonzero real numbers with the same sign and $\upvarphi$ is given by $\upvarphi(s)= \mathfrak{m}_0+\mathfrak{m}_1s$ with some positive constants $\mathfrak{m}_0$ and $\mathfrak{m}_1$. So we prove existence of solution and establish its exponential decay. The method used is based on multiplier technique and some integral inequalities due to Haraux and Komornik\cite{H1,KOM}.