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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  &gt; 0 \alpha &gt; 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 ) &gt; E ( 0 ) \mathcal {G}({u_0}) &gt; E(0)      , then      lim t →  T −  ( u , P u ) = + ∞  {\lim _{t \to {T^ - }}}(u,Pu) = + \infty       for some      T &gt; ∞  T &gt; \infty      . If      ( u 0 , P v 0 ) &gt; 0 , 0 &gt; E ( 0 ) −  G ( u 0 ) &gt; α  ( u 0 , P v 0 ) 2 / 4 ( 2 α  + 1 ) ( u 0 , P u 0 ) ({u_0},P{v_0}) &gt; 0,0 &gt; E(0) - \mathcal {G}({u_0}) &gt; \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 ) &gt; 0 ({u_0},P{v_0}) &gt; 0       and      α  ( u 0 , P v 0 ) 2 / 4 ( 2 α  + 1 ) ( u 0 , P u 0 ) ≤  E ( 0 ) −  G ( u 0 ) &gt; 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}) &gt; \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.

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