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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The blow-up surface for nonlinear wave equations with small spatial velocity


Authors: Avner Friedman and Luc Oswald
Journal: Trans. Amer. Math. Soc. 308 (1988), 349-367
MSC: Primary 35L70; Secondary 35B40
MathSciNet review: 946448
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Abstract: Consider the Cauchy problem for $ {u_{tt}} - {\varepsilon ^2}\Delta u = f(u)$ in space dimension $ \leqslant 3$ where $ f(u)$ is superlinear and nonnegative. The solution blows up on a surface $ t = {\phi _\varepsilon }(x)$. Denote by $ t = \phi (x)$ the blow-up surface corresponding to $ v'' = f(v)$. It is proved that $ \vert{\phi _\varepsilon }(x) - \phi (x)\vert \leqslant C{\varepsilon ^2}$, $ \vert\nabla ({\phi _\varepsilon }(x) - \phi (x))\vert \leqslant C{\varepsilon ^2}$ in a neighborhood of any point $ {x_0}$ where $ \phi ({x_0}) < \infty $.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0946448-7
PII: S 0002-9947(1988)0946448-7
Article copyright: © Copyright 1988 American Mathematical Society