Scattering theory for semilinear wave equations with small data in two space dimensions
We consider a scattering problem for the semilinear wave equation \[ u_{tt}-\Delta u=F(u),\qquad (x,t)\in\mathbb{R}^ n\times\mathbb{R},\tag{1} \] where \(F(u)=\lambda| u|^ p\) or \(\lambda| u|^{p-1} u\), \(\lambda\in\mathbb{R}\), \(p>1\) and \(n=2\). We compare the asymptotic behavior as \(t\to\pm\infty\) of the solution of (1) with those of suitable solutions of the Cauchy problem for the free wave equation \(u_{tt}-\Delta u=0\), \((x,t)\in\mathbb{R}^ n\times\mathbb{R}\), in the sense of energy norm. The aim of this paper is to prove that if \(p>(3+\sqrt{17})/2\), the scattering operator for (1) exists for smooth and small data in two space dimensions.
scientific article; zbMATH DE number 221221
scientific article