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


Author: Howard A. Levine
Journal: Trans. Amer. Math. Soc. 192 (1974), 1-21
MSC: Primary 35L60; Secondary 47H15
DOI: https://doi.org/10.1090/S0002-9947-1974-0344697-2
MathSciNet review: 0344697
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0344697-2
Article copyright: © Copyright 1974 American Mathematical Society

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