General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type. (English) Zbl 1212.35270
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.
Reviewer: Ivan Straškraba (Praha)
MSC:
35L05 | Wave equation |
35L70 | Second-order nonlinear hyperbolic equations |
74Dxx | Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) |
93D20 | Asymptotic stability in control theory |