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

AUTOR(ES)

John, Fritz

RESUMO

The symbol □ denotes the operator ∂2/∂2 - 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(utt) for x∈ R3, t ≥ 0, provided u,ut,utt for t = 0 have compact support. Similarly v(x,t) ≡ 0 if □v = F′(vt)vtt for x ∈ R3, 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.

Documentos Relacionados

• Blow-up of solutions of nonlinear wave equations in three space dimensions
• Solutions to the recurrence relation u n+1 = v n+1 + u n ⊗ v n in terms of Bell polynomials
• Solutions to the recurrence relation u n+1 = v n+1 + u n ⊗ v n in terms of Bell polynomials
• Optimal space maneuvers in three dimensions
• Lower bounds for the life-span of solutions of nonlinear wave equations in three dimensions