Existence and blowing up for a system of the drift-diffusion equation in $R^2$

Abstract

We discuss the existence of the blow-up solution for multi-component parabolic-elliptic drift-diffusion model in two space dimensions. We show that the local existence, uniqueness and well posedness of a solution in the weighted $L^2$ spaces. Moreover, we prove that if the initial data satisfies a threshold condition, the corresponding solution blows up in a finite time. This is a system case for the blow up result of the chemotactic and drift-diffusion equation proved by Nagai [22] and Nagai-Senba-Suzuki [24] and gravitational interaction of particles by Biler-Nadzieja [4], [5]. We generalize the result in Kurokiba-Ogawa [17] for multi-component problem and give a sufficient condition for the finite time blow up of the solution.