Bounded solutions to an energy subcritical non-linear wave equation on [formula omitted

Abstract

In this work we consider an energy subcritical semi-linear wave equation (3<p<5){∂t2u−Δu=ϕ(x)|u|p−1u,(x,t)∈R3×R;u|t=0=u0∈H˙sp(R3);∂tu|t=0=u1∈H˙sp−1(R3); where sp=3/2−2/(p−1) and the function ϕ:R3→[−1,1] is a radial continuous function with a limit at infinity. We prove that unless the elliptic equation −ΔW=ϕ(x)|W|p−1W has a nonzero radial solution W∈C2(R3)∩H˙sp(R3), any radial solution u with a finite uniform upper bound on the critical Sobolev norm ‖(u(⋅,t),∂tu(⋅,t))‖H˙sp×H˙sp(R3) for all t in the maximal lifespan must be a global solution in time and scatter.