An abstract nonlinear evolution problem: Energy decay

TL;DR

This paper investigates energy decay in a nonlinear evolution equation with possibly nonlinear damping, extending previous work by employing a modified perturbed energy method under certain assumptions.

Contribution

It extends existing results on energy decay for abstract evolution equations to include nonlinear damping, using a novel modified perturbed energy technique.

Findings

Established energy decay estimates for the nonlinear evolution problem.

Extended previous linear damping results to nonlinear damping cases.

Introduced a new modified perturbed energy method for analysis.

Abstract

In this paper, we consider energy decay estimates for the following nonlinear evolution problem $$\begin{split} [P(u_t(t))]_t + A u(t) + B(t , x , u_t(t)) =0,\quad t\in J=(0,\infty), \end{split}$$ under suitable assumptions on the self-adjoint operators $P$ and $B$ and the divergence operator $A$. The work extends the earlier work of Marcati, Nakao, Levin and Pucci on abstract evolution equations to the case with possibly nonlinear damping. The technique also differs from earlier techniques. Here, we make use of a modified perturbed energy technique.