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Sharp existence results for self-similar solutions of semilinear wave equations. (English) Zbl 0967.35099
Summary: The existence of self-similar and asymptotically self-similar solutions of the nonlinear wave equation \(u_{tt}-\Delta u= f(u)\) with \(f(u)= \gamma|u|^{\alpha+1}\) or \(f(u)= \gamma|u|^\alpha u\) in \(\mathbb{R}^3\times \mathbb{R}^+\) for small Cauchy data is proven if \(\sqrt 2<\alpha< 2\). A counterexample is given which shows that the lower bound on \(\alpha\) is sharp.

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