Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms

Pages: 2431 - 2453, Volume 37, Issue 5, May 2017      doi:10.3934/dcds.2017105

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Jorge Ferreira - EEIMVR - VCE - Universidade Federal Fluminense, Volta Redonda, RJ, Brazil (email)
Hermenegildo Borges de Oliveira - FCT - Universidade do Algarve, Faro, Portugal (email)

Abstract: In this work we study a system of parabolic reaction-diffusion equations which are coupled not only through the reaction terms but also by way of nonlocal diffusivity functions. For the associated initial problem, endowed with homogeneous Dirichlet or Neumann boundary conditions, we prove the existence of global solutions. We also prove the existence of local solutions but with less assumptions on the boundedness of the nonlocal terms. The uniqueness result is established next and then we find the conditions under which the existence of strong solutions is assured. We establish several bow-up results for the strong solutions to our problem and we give a criterium for the convergence of these solutions towards a homogeneous state.

Keywords:  Reaction-diffusion systems, nonlocal coupled diffusivity terms, local and global weak solutions, uniqueness, strong solutions, blow-up, asymptotic stability.
Mathematics Subject Classification:  Primary: 35K57, 35D30, 35D35; Secondary: 35B44, 35B35.

Received: June 2016;      Revised: December 2016;      Available Online: February 2017.