Modified scattering for a wave equation with weak dissipation

Abstract

We consider the Cauchy problem for the weakly dissipative wave equation $$ \square u+\frac\mu{1+t} u_t=0 $$ with parameter $\mu\ge2$. Based on the explicit representations of solutions provided in [Math. Meth. Appl. Sci. 2004; {\bf 27}:101-124] sharp decay estimates for data from a dense subspace of the energy space are derived. Furthermore, sharpness is discussed in terms of a modified scattering theory.