Instability and nonexistence of global solutions to nonlinear wave equations of the form 𝑃𝑢💀💀=-𝐴𝑢+ℱ (𝓊)

For the equation in the title, let P and A be positive semidefinite operators (with P strictly positive) defined on a dense subdomain D ⊆ H D H, a Hilbert space. Let D be equipped with a Hilbert space norm and let the imbedding be continuous. Let F: D → H F: D H be a continuously differentiable gradient operator with associated potential function G G. Assume that (x, F (x) ) ≥ 2 (2 α + 1) G (x) (x, F (x) ) 2 (2 + 1) G (x) for all x ∈ D x D and some α > 0 > 0. Let E (0) = 1 2 (u 0, A u 0) + (v 0, P v 0) E (0) = 12 (u₀, Au₀) + (v₀, Pv₀) </inline-for