A variational problem for the spatial segregation of reaction-diffusion systems

Abstract

In this paper we study a class of stationary states for reaction-diffusion systems of k ≥ 3 densities having disjoint supports. For a class of segregation states governed by a variational principle we prove existence and provide conditions for uniqueness. Some qualitative properties and the local regularity both of the densities and of their free boundaries are established in the more general context of a functional class characterized by differential inequalities.

Coexistence and segregation for strongly competing species in special domains

“…This is precisely the assumption used in [8]. Condition (ND) is stronger and essentially means that u 0 i is a local nondegenerate minimizer of the energy on B i (see also [3]). As a model for f i we can consider logistic type nonlinearities f i (x, s) = λ(s − |s| p−1 s), p > 1.…”

“…The link between the differential inequalities (3) and population dynamics is reinforced by considering another class of segregation states between species, governed by a minimization principle rather than strong competition-diffusion. In [3] (see also [2,4]), the following energy functional was considered:…”

“…As a matter of fact, since all the asymptotic states of the Lotka-Volterra system have to satisfy (2)-(3), the existence of such a solution is necessary to ensure that all the species survive under strong competition. It has to be stressed that in [3,5], the strict positivity is guaranteed by assuming positive boundary values for each component, in the form u i = φ i on ∂Ω with φ i > 0 on a set of positive (N − 1)-measure.…”

“…For instance, if Ω is convex, it is shown in [19] that two competing species cannot coexist under strong competition. On the other hand, the main variational procedure leading to solutions of (3), that is, the minimization of the internal energy J , may fail under Dirichlet homogeneous conditions, since it in general provides a k-tuple of the form (0, . .…”

Regularity and long-time behavior of nonlocal heat flows

“…Another hint about the proper way to formulate a gradient flow corresponding to problem (P*) is provided by the body of work relating problems such as (P) to limiting problems of singularly perturbed elliptic systems (see [7,9,10]). In particular, Conti-Terracini-Verzini [11,12,13,15] related singularly perturbed systems to optimal partition problems for nonlinear eigenvalues and Nehari's problem, and established Lipschitz continuity of the limiting solutions, as well as regularity of the free interfaces in two dimensions. CaffarelliLin [7] studied the minimization problem…”