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doi: 10.3934/dcdss.2021106
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## Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction

 UR Analysis and Control of PDE's, UR 13ES64, Higher Institute of transport and Logistics of Sousse, University of Sousse, Tunisia

Received  April 2021 Revised  July 2021 Early access September 2021

In the paper under study, we consider the following coupled non-degenerate Kirchhoff system
 \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. \ \ \ \ \ \ \ \ \ \ \ \ \ (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].
Citation: Akram Ben Aissa. Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021106
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