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

## TL;DR

This paper studies the long-term behavior of solutions to a damped second order evolution equation, revealing a dichotomy based on the number of Fourier components, with implications for energy distribution and decay.

## Contribution

It introduces a novel analysis of energy quantization and weak turbulence phenomena in solutions, highlighting the impact of Fourier component count on asymptotic behavior.

## Key findings

- Solutions with finitely many Fourier components resemble linear solutions without damping.
- Solutions with infinitely many Fourier components tend to zero weakly but not strongly.
- The energy limit depends solely on the number of Fourier components.

## Abstract

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 Fouriercomponents are asymptotic to solutions of the linearized equationwithout damping, and exhibit some sort of equipartition of theenergy among the components. Solutions with infinitely manyFourier components tend to zero weakly but not strongly. We showalso that the limit of the energy of solutions depends only on thenumber of their Fourier components.The proof of our results is inspired by the analysis of asimplified model which we devise through an averaging procedure,and whose solutions exhibit the same asymptotic properties as thesolutions to the original equation.