Nonlinear wave equation with potential

Abstract

We study the Cauchy problem for $u_{tt}-\Delta u+V(x)|u|^{p-1}u=0$ with $x\in R^{n}$. The function $V(x)$ is positive and regular. The exponent $p$ is subcritical or critical. By the aid of Shatah-Struwe technique (cf.[7]), we prove the existence of the global classical solution with suitable hypotheses on $V(x):V(x)>0,3\leq n\leq 7$ or $V(x)=|x|^{2}$, $n=3$. To approach this second case we cannot follow directly the argument used in [7]: we need and prove weighted nonlinear estimates in Besov spaces.