Nonlinear scattering theory for a class of wave equations in \(H^{s}\)
This paper is devoted to the study of the existence of scattering operators for the equations of the form of (i) \(u_{tt}+m^2 u+(-\Delta)^k u=F(u)\) for \(k=1,2\); (ii) \(u_{tt}+(-\Delta)^k u=F(u)\), where \(k\in\mathbb N\); and (iii) \(iu_t+(-\Delta)^k u=F(u)\), where \(k\in\mathbb N\). In (i)-(iii) \(u(t,x)\) is a complex valued function of \((t,x)\) which belongs to \(\mathbb R\times\mathbb R^n\), \(m^2>0\), \(\Delta\) denotes the Laplace operator on \(\mathbb R^n\), \(F(u)=\sum D^\alpha f_\alpha(u)\), where \(f_\alpha(u)\) is a scalar nonlinear function. For \(k=1\), (i) represents the well-known nonlinear Klein-Gordon (NLKG) equation and for \(k=2\) it may be considered as the nonlinear beam (NLB) equation. When \(k=1\), (ii) and (iii) stands for the semilinear wave (NLW) and the nonlinear Schrödinger (NLS) equation, respectively. On the other hand, when \(k\geq 2\), (ii)-(iii) represent the higher order versions of the NLS and NLW equations, respectively. In this investigation, the author presents a unified way to study the scattering theory in \(H^s\) for the equations mentioned above, obtains new nonlinear mapping estimates in Besov spaces which can be applied to various nonlinear terms involved in (i)-(iii), discusses the existence of scattering operators with data in a band of \(H^s\) for Eqs. (i)-(iii), considers the regularity of scattering solutions for the NLS equation. In addition, on the basis of the regularity results, a spatial decaying estimate of solutions of the NLS equation is derived.
scientific article; zbMATH DE number 2093818