Sharp well-posedness and blowup results for parabolic systems of the keller-segel type

We study two toy models obtained after a slight modification of the nonlinearity of the usual doubly parabolic Keller-Segel system. For these toy models, both consisting of a system of two parabolic equations, we establish that for data which are, in a suitable sense, smaller than tau/(ln tau)(3), where t is the diffusion parameter in the equation for the chemoattractant, we obtain global solutions. Moreover, for a class of data larger than t, we obtain the finite time blowup, in the whole space as well as in a bounded domain, with two different techniques. Thus, our analysis implies that our size condition on the initial data for the global existence of solutions is sharp, for large t, up to a logarithmic factor. These results show that global-in-time solutions can be obtained more easily with bigger diffusion coefficient t, similarly as is known for weaker nonlinear cross-diffusion terms compared to the strength of diffusion in the first equation.