An abstract nonlinear evolution problem: Energy decay

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.