Decay of solutions of the wave equation with localized nonlinear damping and trapped rays
The author proves some decay estimates of the energy of the wave equation governed by localized nonlinear dissipations in a bounded domain \(\Omega\) in which trapped rays may occur. The equation is of the form \(\partial_t^2 u-\Delta u+a g(\partial_t u)=0\) with \(a\geq 0\) and homogeneous Dirichlet boundary condition \(u=0\). The result applies to many choices for the function \(g\), e.g. \(g(s)=|s|^{p-1}s\) or \(g(s)=s/\sqrt{1+s^2}\). The approach is based on a comparison with the linear damped wave equation and an interpolation argument. The result extends to the nonlinear damped wave equation the well-known optimal logarithmic decay rate for the linear damped wave equation with regular initial data.
homogeneous Dirichlet boundary condition
scientific article; zbMATH DE number 5998568