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

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  • In the paper under study, we consider the following coupled non-degenerate Kirchhoff system

    $\begin{equation} \left \{ \begin{aligned} & y_{tt}-\mathtt{φ}\Big(\int_\Omega | \nabla y |^2\,dx\Big)\Delta y +\mathtt{α} \Delta \mathtt{θ} = 0, &{\rm{ in }}&\; \Omega \times (0, +\infty)\\ & \mathtt{θ}_t-\Delta \mathtt{θ}-\mathtt{β} \Delta y_t = 0, &{\rm{ in }}&\; \Omega \times (0, +\infty)\\ & y = \mathtt{θ} = 0,\; &{\rm{ on }}&\;\partial\Omega\times(0, +\infty)\\ & y(\cdot, 0) = y_0, \; y_t(\cdot, 0) = y_1,\;\mathtt{θ}(\cdot, 0) = \mathtt{θ}_0, \; \; &{\rm{ in }}&\; \Omega\\ \end{aligned} \right. \end{equation} \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

    where $ \Omega $ is a bounded open subset of $ \mathbb{R}^n $, $ \mathtt{α} $ and $ \mathtt{β} $ be two nonzero real numbers with the same sign and $ \mathtt{φ} $ is given by $ \mathtt{φ}(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[5,8].

    Mathematics Subject Classification: Primary: 35B40, 35B45, 35L70.


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