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

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_Ω| \nabla y |^2\,dx\Big)Δy +\upalpha Δ\uptheta=0, &\mbox{ in }&\; Ω\times (0, +\infty)\\ &\displaystyle \uptheta_t-Δ\uptheta-\upbeta Δy_t =0, &\mbox{ in }&\; Ω\times (0, +\infty)\\ &\displaystyle y=\uptheta=0,\; &\mbox{ on }&\;\partialΩ\times(0, +\infty)\\ %&\displaystyle y=0,\; &\mbox{ on }&\;\partialΩ\times(0, +\infty)\\ %&\displaystyle \partial_νy=0, &\mbox{ on }&\;Γ_1\times(0, +\infty)\\ &\displaystyle y(\cdot, 0)=y_0, \; y_t(\cdot, 0)=y_1,\;\uptheta(\cdot, 0)=\uptheta_0, \; \; &\mbox{ in }&\; Ω\\ \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}.