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) (x, x) for all x D and some > 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 (2 + 1) G (x) (x, F (x) ) for all x D and some > 0. If u: 0, T D is a solution to Puₓₓ = - Au + F (u) with u (0) = u₀, vₜ (0) = v₀, and if G (u₀) > 12 (u₀, Au₀) + (v₀, Pv₀) - (/ (2 + 1) ) (u₀, Pu₀), then, for some T <, ₓ ₓ^ - (u, Pu) = +. An analogous result holds for weak solutions to this equation and for the damped equation Puₓₓ + Auₜ + Au = F (u), where A is a positive semidefinite linear operator.