1. Universal bounds for a class of second order evolution equations and applications
- Author
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Marina Ghisi, Massimo Gobbino, Alain Haraux, Dipartimento di Matematica [Pisa], University of Pisa - Università di Pisa, Dipartimento di Ingegneria Civile e Industriale (DICI), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Second order evolution equation ,Dissipation ,Universal bound ,01 natural sciences ,Kirchhoff equations ,Domain (mathematical analysis) ,010101 applied mathematics ,Nonlinear system ,35B40, 35L70, 35L90 ,Mathematics - Analysis of PDEs ,Decay estimates ,Bounded function ,FOS: Mathematics ,Initial value problem ,Restoring force ,Boundary value problem ,0101 mathematics ,[MATH]Mathematics [math] ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We consider a class of abstract second order evolution equations with a restoring force that is strictly superlinear at infinity with respect to the position, and a dissipation mechanism that is strictly superlinear at infinity with respect to the velocity. Under the assumption that the growth of the restoring force dominates the growth of the dissipation, we prove a universal bound property, namely that the energy of solutions is bounded for positive times, independently of the initial condition. Under a slightly stronger assumption, we show also a universal decay property, namely that the energy decays (as time goes to infinity) at least as a multiple of a negative power of $t$, again independent of the boundary conditions. We apply the abstract results to solutions of some nonlinear wave, plate and Kirchhoff equations in a bounded domain., 22 pages. Major rewriting of v1 with two new abstract results concerning asymptotic behavior. Applications to PDEs have been updated accordingly
- Published
- 2018
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