Relatively Compact Orbits and Compact Attractors for a Class of Nonlinear Evolution Equations

Abstract

In the present paper we consider the nonlinear evolution equation u′+Au∋G(u), where A:D(A)⫅X→X is m-accretive with (I+λA)−1 compact for some λ>0, and $$G:\overline {D(A)} \to X$$ is continuous, and we prove that the orbit $$\{ u(t);t \in \mathbb{R}_ + \}$$ is relatively compact if and only if u is uniformly continuous, and both u and G^u are bounded on $$\mathbb{R}_ +$$ . In the same spirit, we derive conditions for orbits of bounded sets to have compact attractors. Some consequences and an example from age-structured population dynamics illustrate the effectiveness of the abstract result.