Decay on expanding spheres as \(t\to +\infty\) of the solutions of semilinear wave equations (Q1826024)

The paper is devoted to asymptotic properties of finite energy solutions of semilinear wave equations \(u_{tt}-\Delta u+q(x,t)| u|^{p- 1}u=0,\) when \(x\in R^ n\), \(n\geq 3\), \(1\leq p<1+4/(n-2)\), and \(q\geq 0\). The solutions are studied in a modified forward lightcone defined by \(| x| \leq (1-\epsilon)(t-M),\) \(t\geq M\) for \(0<\epsilon <1\) and \(M>0.\) Extending results and methods by C. Morawetz and W. A. Strauss it is shown that, with suitable assumptions upon q, the total energy in the region \(| x| \leq (1-\epsilon)(t-M)\) is \(O(t^{-2})\) as \(t\to \infty\). Conditions on q are given which ensure the boundedness of the total energy. The results are combined to prove exponential decay in the modified lightcone under suitable assumptions upon q and for \(n\geq 3\) and odd.