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Self-similar and asymptotically self-similar solutions of nonlinear wave equations. (English) Zbl 0960.35067
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 Th. Cazenave and F. B. Weissler [Math. Z. 228, No. 1, 83-120 (1998; Zbl 0916.35109); Nonlinear Differ. Equ. Appl. 5, No. 3, 355-365 (1998)].

MSC:
35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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