Asymptotic Behavior and Lower Bounds for Semilinear Wave Equations in Hilbert Space with Applications
Howard Allen Levine, Amy C. Murray · SIAM Journal on Mathematical Analysis · 1975
In this paper the asymptotic behavior of abstract wave equations of the form $(1)\qquad \frac{{d^2 u}}{{dt^2 }} + A(t)u = \mathcal{F}(t,u,u_t )$ is discussed. The $A(t)$ are nonnegative symmetric operators defined on a dense subdomain D of a Hilbert space H and $u:(0,\infty ) \to D$ is a twice strongly continuously differentiable solution to (1). Under certain broad conditions on A and $\mathcal{F}$, it is shown that the energy $\mathcal{E}(t,u) \equiv \| {u(t)} \|^2 + \left\| {u_t (t)} \right\|^2 + \langle {u(t),A(t)u(t)} \rangle $ of the solution satisfies $\mathcal{E}(t,u) \geqq KE(\tau ,u)c^{ - \gamma f(t)} $ for some positive constants K, $\gamma $ unless $u \equiv 0$. Here $f(t) = t^c $ for some $c \geqq 0$ or $f(t) = \ln (t)$. Using these abstract results, lower bounds are obtained for solutions to the classical equations of linear elasticity with time dependent elasticities, for solutions to the Euler–Poisson–Darboux equation and for a nonlinear equation of motion for a transversely vibrating plate undergoing longitudinal stress.