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Nonlinear nonlocal reaction-diffusion problem with local reaction

  • * Corresponding author: Aníbal Rodríguez-Bernal

    * Corresponding author: Aníbal Rodríguez-Bernal 

The first author is supported by the projects Projects MTM2016-75465 and PID2019-103860GB-I00, MINECO, Spain and Grupo CADEDIF GR58/08, Grupo 920894. Second author is partially supported by projects MTM2016-75465, MTM2017-83391 and Grupo CADEDIF GR58/08, Grupo 920894, Junta de Andalucía FQM-131. ICMAT is partially supported by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO)

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  • 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.

    Mathematics Subject Classification: Primary: 37L15, 45G, 45M05, 45M10, 45M20, 45P05.

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