Advances in Differential Equations

Convergence to self-similar solutions for a parabolic-elliptic system of drift-diffusion type in $\mathbb R^2$

Toshitaka Nagai

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Abstract

We consider the Cauchy problem for a parabolic-elliptic system of drift-diffusion type in $\mathbb R^2$, modeling chemotaxis and self-attracting particles, with $L^1$-initial data. Under the assumption that the total mass of nonnegative initial data is less than $8\pi$, by using similarity arguments, it is shown that the nonnegative solution converges to a radially symmetric self-similar solution at rate $o(t^{-1+1/p})$ in the $L^p$-norm $(1\le p\le\infty)$ as time goes to infinity.

Article information

Source
Adv. Differential Equations Volume 16, Number 9/10 (2011), 839-866.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355703178

Mathematical Reviews number (MathSciNet)
MR2850755

Zentralblatt MATH identifier
1227.35118

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35K15: Initial value problems for second-order parabolic equations 35K55: Nonlinear parabolic equations

Citation

Nagai, Toshitaka. Convergence to self-similar solutions for a parabolic-elliptic system of drift-diffusion type in $\mathbb R^2$. Adv. Differential Equations 16 (2011), no. 9/10, 839--866. https://projecteuclid.org/euclid.ade/1355703178.


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