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Nonexistence of global solutions of □u=F(utt) and of □v=F'(vt)vtt in three space dimensions

John, Fritz (1980) Nonexistence of global solutions of □u=F(utt) and of □v=F'(vt)vtt in three space dimensions. Proceedings of the National Academy of Sciences of the United States of America, 77 (4). pp. 1759-1760. ISSN 0027-8424. http://resolver.caltech.edu/CaltechAUTHORS:JOHpnas80b

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Abstract

The symbol □ denotes the operator δ2/δ2- Δ in three space dimensions, and F denotes a function with F(0) = F'(0) = 0, inf F" > 0. It is shown that u(x,t) ≡ 0, if □u = F(u tt) for xεR3, t ≥ 0, provided u, ut, utt for t = 0 have compact support. Similarly v(x,t) ≡ 0 if □v = F'(vt)vtt for xεR3, t ≥ 0, provided v,vt for t = 0 have compact support and satisfy ∫[vt-F(vt)]dx ≥ 0. This shows that the global existence theorem proved by S. L. Klainerman [(1980) Commun. Pure Appl. Math. 33, in press] in more than five space dimensions is not valid for three dimensions. The theorems also imply instability at rest of certain hyperelastic materials.


Item Type:Article
Additional Information:© 1980 by the National Academy of Sciences. Contributed by Fritz John, January 10, 1980. This article represents work performed at the Courant Institute of Mathematical Sciences, supported by the National Science Foundation under Grant NSF-MCS-76-05721, and at the California Institute of Technology as a Sherman Fairchild Distinguished Scholar. The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Subject Keywords:asymptotic behavior of solutions; nonlinear equations and systems; wave equation
Record Number:CaltechAUTHORS:JOHpnas80b
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:JOHpnas80b
Alternative URL:http://www.pnas.org/cgi/content/abstract/77/4/1759
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8826
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:20 Sep 2007
Last Modified:14 Nov 2014 19:20

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