Finding the exact decay rate of all solutions to some second order evolution equations with dissipation

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• 作者：Marina Ghisi^a ; ^{marina.ghisi@unipi.it" class="auth_mail" title="E-mail the corresponding author} ; Massimo Gobbino^a ; ^{massimo.gobbino@unipi.it" class="auth_mail" title="E-mail the corresponding author} ; Alain Haraux^b ; ^{haraux@ann.jussieu.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35L71 ; 35L90 ; 35B40
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 November 2016
• 年：2016
• 卷：271
• 期：9
• 页码：2359-2395
• 全文大小：591 K

文摘

We consider an abstract second order evolution equation with damping. The “elastic” term is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We investigate the asymptotic behavior of solutions.

We prove the coexistence of slow solutions and fast solutions. Slow solutions live close to the kernel of A, and decay as negative powers of t as solutions of the first order equation obtained by neglecting the operator A and the second order time-derivatives in the original equation. Fast solutions live close to the range of A and decay exponentially as solutions of the linear homogeneous equation obtained by neglecting the nonlinear terms in the original equation.

The abstract results apply to semilinear dissipative hyperbolic equations.