A one-dimensional logistic like equation with nonlinear and nonlocal diffusion: Strong convergence to equilibrium

Ducrot, Arnaud; Manceau, David

doi:10.1090/proc/14971

A one-dimensional logistic like equation with nonlinear and nonlocal diffusion: Strong convergence to equilibrium
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by Arnaud Ducrot and David Manceau PDF

Proc. Amer. Math. Soc. 148 (2020), 3381-3392 Request permission

Abstract:

We consider a one-dimensional logistic like equation with a nonlinear and nonlocal diffusion term with periodic boundary conditions. In this note we focus on smooth nonlocal kernels exhibiting a repulsive effect, that is expressed in terms of the positivity of their Fourier transforms. Here we describe the large time behaviour of the solutions for a large class of initial data. We roughly prove that the nontrivial solutions converge strongly to the unique spatially homogeneous equilibrium, by providing a refined study of the properties of the characteristic curves associated to the problem.

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