Instability and nonexistence of global solutions to nonlinear wave equations of the form 𝑃𝑢𝑡𝑡=-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(2α + 1)G(x) for all x ∈ D x ∈ D and some α > 0 α > 0 . Let E ( 0 ) = 1 2 [ ( u 0 , A u 0 ) + ( v 0 , P v 0 ) ] E(0) = 1/2[({u_0},A{u_0}) + ({v_0},P{v_0})] </inline-for