Global existence for the dissipative wave equations with space–time dependent potential

Abstract

This research paper aims at studying the long time behavior of solutions for wave equations with focusing nonlinearity.We consider pcr(n,α,β)<p<(n+2)/(n−2), and separable potential a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)−α and η(t)∼(1+t)−β satisfying conditions (3) and (4) which are given below. The main result is that if the exponent p lies between pcr:=1+4(β+1)2(n−α)(β+1)−β(2−α)<p<n−2n+2, the solutions are global for sufficiently small data. Also, the precise decay estimates for the energy, L2 and Lp+1 norms of solutions in the region of global solutions have been found.