Nonexistence of global solutions of squareu = F(u) and of squarev = F'(v)v in three space dimensions.
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.
Courant Institute, New York University, New York, New York 10012.
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John