The blow-up boundary for nonlinear wave equations
Authors:
Luis A. Caffarelli and Avner Friedman
Journal:
Trans. Amer. Math. Soc. 297 (1986), 223-241
MSC:
Primary 35L70
DOI:
https://doi.org/10.1090/S0002-9947-1986-0849476-3
MathSciNet review:
849476
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Consider the Cauchy problem for a nonlinear wave equation in
space dimensions,
, with
superlinear and nonnegative. It is well known that, in general, the solution blows up in finite time. In this paper it is shown, under some assumptions on the Cauchy data, that the blow-up set is a space-like surface
with
continuously differentiable.
- [1] Robert T. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z. 132 (1973), 183–203. MR 0340799, https://doi.org/10.1007/BF01213863
- [2] Robert T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z. 177 (1981), no. 3, 323–340. MR 618199, https://doi.org/10.1007/BF01162066
- [3] Fritz John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), no. 1-3, 235–268. MR 535704, https://doi.org/10.1007/BF01647974
- [4] Howard A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form 𝑃𝑢_{𝑡𝑡}=-𝐴𝑢+\cal𝐹(𝑢), Trans. Amer. Math. Soc. 192 (1974), 1–21. MR 0344697, https://doi.org/10.1090/S0002-9947-1974-0344697-2
- [5] Tosio Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math. 33 (1980), no. 4, 501–505. MR 575735, https://doi.org/10.1002/cpa.3160330403
Retrieve articles in Transactions of the American Mathematical Society with MSC: 35L70
Retrieve articles in all journals with MSC: 35L70
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1986-0849476-3
Article copyright:
© Copyright 1986
American Mathematical Society