Global solutions and exponential decay for a nonlinear coupled system of beam equations of Kirchhoff type with memory in a domain with moving boundary

M. L. Santos, UFPA, Para, Brazil
U. R. Soares, UFPA, Para, Brazil

E. J. Qualitative Theory of Diff. Equ., No. 9. (2007), pp. 1-24.

Abstract: In this paper we prove the exponential decay in the case $n>2$, as time goes to infinity, of regular solutions for a nonlinear coupled system of beam equations of Kirchhoff type with memory and weak damping
\begin{eqnarray*}
&&u_{tt}+\Delta^2 u-M(||\nabla u||^2_{L^2(\Omega_t)}+||\nabla
v||^2_{L^2(\Omega_t)})\Delta u\\
&&+\int^{t}_{0}g_1(t-s)\Delta u(s)ds
+\alpha u_{t}+h(u-v)=0 \quad \mbox{in} \quad \hat{Q},\\
&&v_{tt}+\Delta^2 v-M(||\nabla u||^2_{L^2(\Omega_t)}+||\nabla
v||^2_{L^2(\Omega_t)})\Delta v \\
&&+\int^{t}_{0}g_2(t-s)\Delta v(s)ds + \alpha v_{t}-h(u-v)=0 \quad
\mbox{in} \quad \hat{Q}
\end{eqnarray*}
in a non cylindrical domain of $\R^{n+1}$ $(n\ge1)$ under suitable hypothesis on the scalar functions $M$, $h$, $g_1$ and $g_2$, and where $\alpha$ is a positive constant. We show that such dissipation is strong enough to produce uniform rate of decay. Besides, the coupling is nonlinear which brings up some additional difficulties, which plays the problem interesting. We establish existence and uniqueness of regular solutions for any $n\ge 1$.

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