Concentration Phenomenon in Some Non-Local Equation

Abstract

We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the following integro-differential equation $$\partial\_t u(t, x) = \left(a(x) -- \int\_{\Omega} k(x, y)u(t, y) dy\right ) u(t, x) + \int\_{\Omega} m(x, y)[u(t, y) -- u(t, x)] dy\quad \text{ for}\quad (t, x) $\in$ \mathbb{R}\_{+} \times \Omega,$$ together with the initial condition $u(0, \cdot) = u0 \quad \text{ in }\quad \Omega$. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function u represents the density of individuals characterized by the trait, the domain of trait values $\Omega$ is a bounded subset of $\mathbb{R}^N$ , the kernels $k$ and $m$ respectively account for the competition between individuals and the mutations occurring in every generation, and the function a represents a growth rate. When the competition is independent of the trait, we construct a positive stationary solution which belongs to the space of Radon measures on $\Omega$. Moreover, when this '' stationary '' measure is regular and bounded, we prove its uniqueness and show that, for any non negative initial datum in $L^{\infty} (\Omega) \cap L^1 (\Omega)$, the solution of the Cauchy problem converges to this limit measure in $L^2 (\Omega)$. We also construct an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. These numerical simulations seem to reveal some dependence of the limit measure with respect to the initial datum.