Self-similar and asymptotically self-similar solutions of nonlinear wave equations
The paper deals with the global small data Cauchy problem for the semilinear wave equation \[ u_{tt}-\Delta u= \gamma|u|^\alpha u,\quad x\in\mathbb{R}^3,\quad t\geq 0. \] The main interest is in self-similar solutions, that is in solutions \(u(t,x)\) such that \(u(t,x)= \lambda^{2/\alpha} u(\lambda t,\lambda x)\) for any \(\lambda> 0\). Global existence, uniqueness and asymptotic behavior of solutions are studied. Similar results and techniques for nonlinear Schrödinger and heat equations are due to \textit{Th. Cazenave} and \textit{F. B. Weissler} [Math. Z. 228, No. 1, 83-120 (1998; Zbl 0916.35109); Nonlinear Differ. Equ. Appl. 5, No. 3, 355-365 (1998)].
global existence, uniqueness and asymptotic behavior
scientific article; zbMATH DE number 1423132