期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：1980
卷号：77
期号：4
页码：1759-1760
DOI：10.1073/pnas.77.4.1759
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：The symbol {square} denotes the operator {partial}2/{partial}2 - {Delta} 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) {equiv} 0, if {square}u = F(utt) for x[isin] R3, t [≥] 0, provided u,ut,utt for t = 0 have compact support. Similarly v(x,t) {equiv} 0 if {square}v = F'(vt)vtt for x [isin] R3, t [≥]0, provided v,vt for t = 0 have compact support and satisfy {int}[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.
关键词：asymptotic behavior of solutions ; nonlinear equations and systems ; wave equation
