$L^p$--$L^q$ decay estimates for wave equations with monotone time-dependent dissipation

Abstract

This expository article is intended to give an overview about recently achieved results on asymptotic properties of solutions to the Cauchy problem $u_{tt}-\Delta u+b(t)u_t =0,\qquad u(0,\cdot)=u_1,\quad \mathrm{D}_tu(0,\cdot)=u_2$ for a wave equation with time-dependent dissipation term. The results are based on structural properties of the Fourier multipliers representing its solution. The article explains the general philosophy behind the approach.