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

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 $\mathrm{\Omega }$ is a bounded open subset of ${\mathbb{R}}^n$, $\text{upalpha}$ and $\text{upbeta}$ be two nonzero real numbers with the same sign and $\text{upvarphi}$ is given by $\upvarphi(s)= \mathfrak{m}_0+\mathfrak{m}_1s$withsomepositiveconstants$\mathfrak{m}_0$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}.