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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
##### Keywords:
small Cauchy data; counterexample
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