On uniform decay of coupled wave equation of Kirchhoff type subject to memory condition on the boundary
The author considers in \(Q=(0,\infty)\times \Omega, \)where \(\Omega\) can be non-simple-connected, a mixed problem for a system of two coupled Kirchhoff type equations of second order. The boundary of \(\Omega\) is divided in two parts \(\Gamma_0\) and \(\Gamma_1\). On \(\Gamma_0\) the condition \[ u=v=\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0 \tag{1} \] is posed and on \(\Gamma_1\) a condition with a memory effect. The author claims the existence of global solutions and uniform decay of the energy. Condition (1) looks like exessive because if \(\Gamma_1\) is empty, then one has Cauchy data for the Laplace equation on \(\Gamma_0\). In this case, the problem is overdetermined.
scientific article; zbMATH DE number 2167864