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.