Decay estimates for dissipative wave equations in exterior domains

Abstract

Consider the initial boundary value problem for the linear dissipative wave equation (□+ ∂ t ) u=0 in an exterior domain Ω⊂ R N . Using the so-called cut-off method together with local energy decay and L 2 decays in the whole space, we study decay estimates of the solutions. In particular, when N⩽3, we derive L p decays with p⩾1 of the solutions. Next, as an application of the decay estimates for the linear equation, we consider the global solvability problem for the semilinear dissipative wave equations (□+ ∂ t ) u= f( u) with f( u)=| u| α+1 ,| u| α u in an exterior domain.