The degenerate drift-diffusion system with the Sobolev critical exponent

T. Ogawa

doi:10.3934/dcdss.2011.4.875

Article Contents

2011, Volume 4, Issue 4: 875-886. Doi: 10.3934/dcdss.2011.4.875

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The degenerate drift-diffusion system with the Sobolev critical exponent

T. Ogawa^1,

1.
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578

Received: September 2009

Revised: December 2009

Published: August 2011

Abstract / Introduction Related Papers Cited by

Abstract

We consider the drift-diffusion system of degenerated type. For $n\ge 3$,

$\partial_t \rho -\Delta \rho^\alpha + \kappa\nabla\cdot (\rho \nabla \psi ) =0, t>0, x \in R^n,$

$-\Delta \psi = \rho, t>0, x \in R^n,$

$\rho(0,x) = \rho_0(x)\ge 0, x \in R^n,$

where $\alpha>1$ and $\kappa=1$. There exists a critical exponent that classifies the global behavior of the weak solution. In particular, we consider the critical case $\alpha_*=\frac{2 n}{n+2}=(2^*)'$, where the Talenti function $U(x)$ solving $-2^*\Delta U^{\frac{n-2}{n+2}}=U$ in $R^n$ classifies the global existence of the weak solution and finite blow-up of the solution.

Keywords:

Mathematics Subject Classification: Primary: 35K15, 35K55, 35Q60; Secondary: 78A35.

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