We are concerned with the mathematical analysis of a predator--prey system in heterogeneous domains. We first give a global existence result for the problem with spatially variable coefficients. For highly heterogeneous systems, using homogenization techniques we derive a simpler model with constant coefficients yielding the macroscopic dynamic of the predator--prey system. In this process, standard Lotka--Volterra functional responses to predation are preserved, while Holling type II responses are transformed into unusual nonlocal nonlinearities.

We study scattering theory and self-similar solutions for the nonlinear wave equation $$ in two or three space dimensions under the assumption $$, where $$ is the larger root of the equation $$. The relation between the theory of scattering and that of self-similar solutions is considered from the point of view of asymptotically free solutions and asymptotically self-similar solutions.

We study the stability of nonlinear dissipative dynamical systems. New results concerning the convergence of a solution to a critical point are given in various situations.

Let $$ and $$ be two families of closed operators satisfying the Acquistapace--Terreni conditions or the Kato--Tanabe conditions, or assumptions of maximal regularity, and let $$ and $$ be the associated evolution families. We obtain some estimates for $$ in terms of $$ for $$. We deduce some results showing that if $$ sufficiently quickly as $$, then $$ and $$ have similar asymptotic behaviour.