Nonexistence of global solutions of □ u = F ( u tt ) and of □ v = F ′( v t ) v tt in three space dimensions

Abstract

The symbol square denotes the operator partial differential(2)/ partial differential(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) identical with 0, if squareu = F(u(tt)) for xin R(3), t >/= 0, provided u,u(t),u(tt) for t = 0 have compact support. Similarly v(x,t) identical with 0 if squarev = F'(v(t))v(tt) for x in R(3), t >/=0, provided v,v(t) for t = 0 have compact support and satisfy integral[v(t) - F(v(t))]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.