Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1988-0946448-7
MathSciNet review: 946448
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $.


References [Enhancements On Off] (What's this?)

  • [1] L. A. Caffarelli and A. Friedman, Differentiality of the blow-up curve for one dimensional nonlinear wave equations, Arch. Rational Mech. Anal. 91 (1985), 83-98. MR 802832 (87d:35089)
  • [2] -, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc. 297 (1986), 223-241. MR 849476 (87h:35215)
  • [3] R. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z. 132 (1973), 183-203. MR 0340799 (49:5549)
  • [4] -, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z. 177 (1981), 323-340. MR 618199 (82i:35120)
  • [5] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), 235-268. MR 535704 (80i:35114)
  • [6] H. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $ P{u_{tt}} = - Au + F(u)$, Trans. Amer. Math. Soc. 192 (1974), 1-21. MR 0344697 (49:9436)
  • [7] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math. 32 (1980), 501-505. MR 575735 (82f:35128)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35L70, 35B40

Retrieve articles in all journals with MSC: 35L70, 35B40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0946448-7
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society