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

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.