On blow up of solutions of nonlinear evolution equations

Korman, Philip

doi:10.1090/S0002-9939-1988-0938667-6

On blow up of solutions of nonlinear evolution equations
HTML articles powered by AMS MathViewer

by Philip Korman PDF

Proc. Amer. Math. Soc. 103 (1988), 189-197 Request permission

Abstract:

We give a complete description of domains of blow up for general second order inequalities, which allows us to obtain some new results on nonexistence of global solutions for nonlinear hyperbolic equations, both in ${R^n}$ and bounded domains.

References

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 112, 473–486. MR 473484, DOI 10.1093/qmath/28.4.473
Richard Bellman, Stability theory of differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0061235
Robert T. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z. 132 (1973), 183–203. MR 340799, DOI 10.1007/BF01213863
Robert T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z. 177 (1981), no. 3, 323–340. MR 618199, DOI 10.1007/BF01162066
Jack K. Hale, Ordinary differential equations, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. MR 0419901
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, DOI 10.1007/BF01647974
Stanley Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963), 305–330. MR 160044, DOI 10.1002/cpa.3160160307
Tosio Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math. 33 (1980), no. 4, 501–505. MR 575735, DOI 10.1002/cpa.3160330403
S. Klainerman and Gustavo Ponce, Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), no. 1, 133–141. MR 680085, DOI 10.1002/cpa.3160360106
Howard A. Levine, Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: the method of unbounded Fourier coefficients, Math. Ann. 214 (1975), 205–220. MR 385336, DOI 10.1007/BF01352106
Sigeru Mizohata, The theory of partial differential equations, Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. MR 0599580