Quantization of energy and weakly turbulent profiles of solutions to some damped second-order evolution equations

Advances in Nonlinear Analysis

article

Marina Ghisi¹  Massimo Gobbino¹  Alain Haraux² 

[1] Dipartimento di Matematica, Università degli Studi di Pisa;Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie

关键词: Dissipative hyperbolic equations;    nonlinear damping;    decay rate;    weak turbulence;    equipartition of the energy;   

DOI  :  10.1515/anona-2017-0181

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We consider a second-order equation with a linear “elastic” part and a nonlinear damping term depending on a power of the norm of the velocity. We investigate the asymptotic behavior of solutions, after rescaling them suitably in order to take into account the decay rate and bound their energy away from zero. We find a rather unexpected dichotomy phenomenon. Solutions with finitely many Fourier components are asymptotic to solutions of the linearized equation without damping and exhibit some sort of equipartition of the total energy among the components. Solutions with infinitely many Fourier components tend to zero weakly but not strongly. We show also that the limit of the energy of the solutions depends only on the number of their Fourier components. The proof of our results is inspired by the analysis of a simplified model, which we devise through an averaging procedure, and whose solutions exhibit the same asymptotic properties as the solutions to the original equation.

【 授权许可】

CC BY