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Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms

The first author was partially supported by the Research Project CAPES -Grant BEX 2478-12-8 and MEC/MCTI/CAPES/CNPq/FAPs no. 71/2013, Grant 88881.030388/2013-01, Brazil. The second author was partially supported by Fundação para a Ciência e a Tecnologia, UID/MAT/04561/2013-2015, Portugal.
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  • 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 blow-up results for the strong solutions to our problem and we give a criterium for the convergence of these solutions towards a homogeneous state.

    Mathematics Subject Classification: Primary: 35K57, 35D30, 35D35; Secondary: 35B44, 35B35.


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