Asymptotic estimates for the spatial segregation of competitive systems

Abstract

For a class of population models of competitive type, we study the asymptotic behavior of the positive solutions as the competition rate tends to infinity. We show that the limiting problem is a remarkable system of differential inequalities, which defines the functional class S in (2). By exploiting the regularity theory recently developed in Conti et al. (Indiana Univ. Math. J., to appear) for the elements of functional classes of the form S, we provide some qualitative and regularity property of the limiting configurations. Besides, for the case of two competing species, we obtain a full description of the limiting states and we prove some quantitative estimates for the rate of convergence. Finally, we prove some new Liouville-type results which allow to have uniform regularity estimates of the solutions.

Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition

“…Our result relies upon the blow-up technique (section 3) and suitable Liouville-type theorems (section 2). Such a strategy has been already adopted by some of the authors in [9] in proving uniform Hölder estimates for competitiondiffusion systems with Lotka-Volterra type of interactions. The arguments there, however, though helpful in the present situation, need to be complemented with some new ideas, including a proper use of the Almgren's frequency formula [1].…”

Resonance Phenomena in a Singular Perturbation Problem in the Case of Exchange of Stabilities

“…We have that V + is even, (V + ) x > 0, x > 0, and V + > x, x ≥ 0. We then seek other solutions of (1.11) in the form v = V + − W, and find that W has to satisfy: 12) with W beingˆ /ε 2 3 -periodic in z. We will show, in Proposition 3.1, that (1.12) has a one-dimensional, positive, even solution w such that V (x) := V + (x) − w(x) < |x|, x ∈ R, solves (1.11).…”

“…(Sketch) It is well known that problem (3.20) has an even solution V + such that V + (x) > |x|, x ∈ R, and (V + ) x (x) > 0, x > 0. This can be proven by the method of upper and lower solutions, see [12,24,26,29,38]. Actually, by a theorem of [7], this is the unique non-negative solution in R 2 of the elliptic equation of (1.11).…”

“…Convergence, in the singular limit, to the maximum or minimum of a family of functions, as in (1.5) or (1.6), typically occurs in systems of nonlinear elliptic equations with competition, and describes phase separation, as the repulsive interaction tends to infinity. These include systems of Lotka-Voltera type [12], and systems arising in the Hartree-Fock theory of a mixture of Bose-Einstein condensates [41]. Actually, the original problem that led the authors of [8] to study (1.1) was a coupled system of this type.…”

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…”