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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'' &gt; 0. It is shown that u(x,t) identical with 0, if squareu = F(u(tt)) for xin R(3), t &gt;/= 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 &gt;/=0, provided v,v(t) for t = 0 have compact support and satisfy integral[v(t) - F(v(t))]dx &gt;/= 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.

proceedings of the national academy of sciences of the united states of america

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