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The degenerate drift-diffusion system with the Sobolev critical exponent

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  • 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.

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


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