On the ultimate energy bound of solutions to some forced second order evolution equations with a general nonlinear damping operator

TL;DR

This paper establishes optimal bounds on the energy of solutions to certain forced second-order evolution equations with nonlinear damping, showing how the bounds depend on the forcing term and the damping's growth behavior.

Contribution

It provides the first comprehensive analysis of the ultimate energy bounds for solutions with nonlinear damping under general conditions, including optimal bounds for specific damping behaviors.

Findings

Energy bound is of order 1 + ||h||^4 in general

Quadratic growth bound is optimal when damping behaves like a power law

Lower bounds are obtained for anti-periodic forcing, showing optimality

Abstract

Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which ensure non-resonance, we estimate the ultimate bound of the energy of the general solution to the equation $\ddot{u}(t) + Au(t) + g(\dot{u}(t))=h(t),\quad t\in\mathbb{R}^+ ,$ where $A$ is a positive selfadjoint operator on a Hilbert space $H$ and $h$ is a bounded forcing term with values in $H$. In general the bound is of the form $ C(1+ ||h||^4)$ where $||h||$ stands for the $L^\infty$ norm of $h$ with values in $H$ and the growth of $g$ does not seem to play any role. If $g$ behaves lie a power for large values of the velocity, the ultimate bound has a quadratic growth with respect to $||h||$ and this result is optimal. If $h$ is anti periodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.