Stability for abstract evolution equations

In a recent paper [Comm. Pure Appl. Math. 49 (1996), 177-216], we studied the question of asymptotic stability for non-autonomous dissipative wave systems. Earlier work in the same direction is due to P. Marcati [J. Differential Equations 55 (1984), 30-58; Nonlinear Anal. 8 (1984), 237-252] and Nakao, who treated the case of abstract evolution equations in particular. In this paper we give a new asymptotic stability theorem which extends the analysis in the papers by Marcati and Nakao cited above by taking into account the techniques introduced in our paper cited above. We focus on abstract equations of the form (1.1) [P(u′(t))]′+A(u(t))+Q(t,u′(t))+F(u(t))=0, where A, F, P and Q are nonlinear operators on appropriate Banach spaces. We understand P to be the evolution operator, A as a differential operator of divergence form, Q as a damping term, and F as a restoring force. Concrete examples of (1.1) include the principal case of wave systems, where P=I, A=−Δ, and also, more generally, the p-Laplacian, where A=−Δp, p>1, as well as the polyharmonic operator A=(−Δ)L, L≥1. As an important feature of the present work, we allow the damping Q=Q(t,v) to be strongly non-autonomous in t and nonlinear in v. Our main theorem is given in Section 3, while Section 2 is devoted to preliminary results. In particular, in Section 2 we formulate a careful definition of the solution of (1.1), which clarifies and generalizes the corresponding definitions in the above-cited papers by Marcati and Nakao, and moreover resolves the principal difficulty in treating the abstract case, namely that an appropriate energy balance for (1.1) cannot be derived directly, but must instead be inferred from analogy with concrete equations and systems.