Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping

We study the rate of decay of solutions of the wave equation with localized nonlinear damping without any growth restriction and without any assumption on the dynamics. Providing regular initial data, the asymptotic decay rates of the energy functional are obtained by solving nonlinear ODE. Moreover, we give explicit uniform decay rates of the energy. More precisely, we find that the energy decays uniformly at last, as fast as 1/(ln(t+2))2−δ,∀δ>01/(ln(t+2))2−δ,∀δ>0, when the damping has a polynomial growth or sublinear, and for an exponential damping at the origin the energy decays at last, as fast as 1/(ln(ln(t+e2)))2−δ,∀δ>01/(ln(ln(t+e2)))2−δ,∀δ>0.