Some Additional Remarks on the Nonexistence of Global Solutions to Nonlinear Wave Equations

Let P and A be symmetric linear operators defined on a dense domain D ⊂ H, a (real) Hilbert space. Let (x,Av) λ (x,x) for all x ∈ D and some λ > 0 and $(x,Px) > 0$ for all x ∈ D, x ≠ 0. Let D have a Hilbert space structure and let the embedding be continuous. Let F:D → H be a nonlinear gradient operator and let G be a potential associated with F. Suppose that 2(2α + 1)G(x) (x,F(x)) for all x ∈ D and some α > 0. If u:[0,T] → D is a solution to Puₜₜ = - Au + F(u) with u(0) = u₀ ,vₜ (0) = v₀, and if G(u₀ ) > 1/2[(u₀ ,Au₀ ) + (v₀ ,Pv₀ )] - (λ α /(2α + 1))(u₀ ,Pu₀ ), then, for some T < ∞, lim t → T^ - (u,Pu) = + ∞. An analogous result holds for weak solutions to this equation and for the damped equation Puₜₜ + Auₜ + Au = F(u), where A is a positive semidefinite linear operator.