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 $\Box u+\lambda|u|^{p-1}u=0$ in two or three space dimensions under the assumption $p_0(n) <p <1+4/(n-1)$, where $p_0(n)$ is the larger root of the equation $(n-1)p^2-(n+1)p-2=0$. 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 $(A(t))_{t \ge 0}$ and $(B(t))_{t \ge 0}$ be two families of closed operators satisfying the Acquistapace--Terreni conditions or the Kato--Tanabe conditions, or assumptions of maximal regularity, and let $(U(t,s))_{t > s \ge 0}$ and $(V(t,s))_{t > s\ge0}$ be the associated evolution families. We obtain some estimates for $ \| {U(t,s) - V(t,s)} \| $ in terms of $ \| {A(\tau)^{-1} - B(\tau)^{-1}} \| $ for $s \le \tau \le t$. We deduce some results showing that if $ \| {A(\tau)^{-1} - B(\tau)^{-1}} \| \to 0$ sufficiently quickly as $\tau \to \infty$, then $U$ and $V$ have similar asymptotic behaviour.