The degenerate drift-diffusion system with the Sobolev critical exponent

Pages: 875 - 886, Volume 4, Issue 4, August 2011

doi:10.3934/dcdss.2011.4.875       Abstract        References        Full Text (407.6K)       Related Articles

T. Ogawa - Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan (email)

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:  Degenerate drift-diffusion, Sobolev critical, blow-up, global weak solution.
Mathematics Subject Classification:  Primary: 35K15, 35K55, 35Q60; Secondary: 78A35.

Received: September 2009;      Revised: December 2009;      Published: November 2010.

 References