Differential and Integral Equations

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

Masaki Kurokiba

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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.

Article information

Source
Differential Integral Equations, Volume 27, Number 5/6 (2014), 425-446.

Dates
First available in Project Euclid: 3 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.die/1396558090

Mathematical Reviews number (MathSciNet)
MR3189526

Zentralblatt MATH identifier
1340.35121

Subjects
Primary: 35K15: Initial value problems for second-order parabolic equations 35K55: Nonlinear parabolic equations 35Q60: PDEs in connection with optics and electromagnetic theory 78A35: Motion of charged particles

Citation

Kurokiba, Masaki. Existence and blowing up for a system of the drift-diffusion equation in $R^2$. Differential Integral Equations 27 (2014), no. 5/6, 425--446. https://projecteuclid.org/euclid.die/1396558090


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