General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type. (Q976613)

The decay rate of the solution to the hyperbolic problem given by \(u_{tt}-\Delta u+F(x,t,u,\nabla u)=0\) in \(\Omega \times \mathbb R_+\), \(u=0\) on \(\Gamma _0\), \(u+\int _0^t g(t-s)\frac {\partial u}{\partial \nu }(s)\,\text ds=0\) on \(\Gamma _1\times \mathbb R_+\), \(u(x,0)=u^0(x)\), \(u'(x,0)=u^1(x)\) in \(\Omega \) is investigated. Here \(\Omega \) is a bounded region whose boundary is partitioned into disjoint sets \(\Gamma _0\), \(\Gamma _1\). The decay of the solution is due to dissipativity of the boundary condition which includes memory effect.