Instability and nonexistence of global solutions to nonlinear wave equations of the form 𝑃𝑢_{𝑡𝑡}=-𝐴𝑢+ℱ(𝓊)

Abstract

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 \subseteq H , a Hilbert space. Let D be equipped with a Hilbert space norm and let the imbedding be continuous. Let F : D → H \mathcal {F}:D \to H be a continuously differentiable gradient operator with associated potential function G \mathcal {G} . Assume that ( x , F ( x ) ) ≥ 2 ( 2 α + 1 ) G ( x ) (x,\mathcal {F}(x)) \geq 2(2\alpha + 1)\mathcal {G}(x) for all x ∈ D x \in D and some α > 0 \alpha > 0 . Let E ( 0 ) = 1 2 [ ( u 0 , A u 0 ) + ( v 0 , P v 0 ) ] E(0) = \tfrac {1}{2}[({u_0},A{u_0}) + ({v_0},P{v_0})] where u 0 = u ( 0 ) , v 0 = u t ( 0 ) {u_0} = u(0),{v_0} = {u_t}(0) and u : [ 0 , T ) → D u:[0,T) \to D be a solution to the equation in the title. The following statements hold: If G ( u 0 ) > E ( 0 ) \mathcal {G}({u_0}) > E(0) , then lim t → T − ( u , P u ) = + ∞ {\lim _{t \to {T^ - }}}(u,Pu) = + \infty for some T > ∞ T > \infty . If ( u 0 , P v 0 ) > 0 , 0 > E ( 0 ) − G ( u 0 ) > α ( u 0 , P v 0 ) 2 / 4 ( 2 α + 1 ) ( u 0 , P u 0 ) ({u_0},P{v_0}) > 0,0 > E(0) - \mathcal {G}({u_0}) > \alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0}) and if u exists on [ 0 , ∞ ) [0,\infty ) , then ( u,Pu ) grows at least exponentially. If ( u 0 , P v 0 ) > 0 ({u_0},P{v_0}) > 0 and α ( u 0 , P v 0 ) 2 / 4 ( 2 α + 1 ) ( u 0 , P u 0 ) ≤ E ( 0 ) − G ( u 0 ) > 1 2 ( u 0 , P v 0 ) 2 / ( u 0 , P u 0 ) \alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0}) \leq E(0) - \mathcal {G}({u_0}) > \tfrac {1}{2}{({u_0},P{v_0})^2}/({u_0},P{u_0}) and if the solution exists on [ 0 , ∞ ) [0,\infty ) , then ( u,Pu ) grows at least as fast as t 2 {t^2} . A number of examples are given.