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

Abstract:

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