Global existence versus blow up for some models of interacting particles

We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst-Planck and the Debye-Hukel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and oscillating complex-valued initial data, using a method by S. Montgomery-Smith.Comment: Colloq. Math. (to appear

Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms

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.CAPES [BEX 2478-12-8]; MEC/MCTI/CAPES/CNPq/FAPs, Brazil [71/2013, 88881.030388/2013-01]; Fundacao para a Ciencia e a Tecnologia, Portugal [UID/MAT/04561/2013-2015]info:eu-repo/semantics/publishedVersio