Open AccessNonlinear nonlocal reaction-diffusion problem with local reaction
Author(s) -
Anı́bal Rodrı́guez-Bernal,
Silvia Sastre-Gómez
Publication year - 2022
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2021170
Subject(s) - mathematics , reaction–diffusion system , smoothing , type (biology) , nonlinear system , diffusion , pure mathematics , laplace operator , focus (optics) , term (time) , operator (biology) , sign (mathematics) , metric space , mathematical analysis , physics , ecology , chemistry , repressor , quantum mechanics , gene , transcription factor , optics , biology , thermodynamics , biochemistry , statistics
In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.