Existence and blowing up for a system of the drift-diffusion equation in \(R^2\). (English) Zbl 1340.35121

Summary: We discuss the existence of the blow-up solution for multicomponent 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 T. Nagai [J. Inequal. Appl. 6, No. 1, 37–55 (2001; Zbl 0990.35024)] and T. Nagai et al. [Hiroshima Math. J. 30, No. 3, 463–497 (2000; Zbl 0984.35079)] and gravitational interaction of particles by P. Biler and T. Nadzieja [Colloq. Math. 66, No. 2, 319–334 (1994; Zbl 0817.35041); Adv. Differ. Equ. 3, No. 2, 177–197 (1998; Zbl 0952.35008)]. We generalize the result in [the author and T. Ogawa, Differ. Integral Equ. 16, No. 4, 427–452 (2003; Zbl 1161.35432)] for multi-component problem and give a sufficient condition for the finite time blow up of the solution.


35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35Q60 PDEs in connection with optics and electromagnetic theory
78A35 Motion of charged particles
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B44 Blow-up in context of PDEs