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# Mathematical Research Letters

## Volume 10 (2003)

### Number 4

### Asymptotic behavior of nonlinear diffusions

Pages: 551 – 557

DOI: http://dx.doi.org/10.4310/MRL.2003.v10.n4.a13

#### Authors

#### Abstract

We describe the asymptotic behavior as $t\to \infty$ of the solution of $u_t=\Delta_p u$ in $\R^N$, for $(2N+1)/(N+1)\le p <N$ and non-negative, integrable initial data. Optimal rates in $L^q$, $q=2-1/(p-1)$ for the convergence towards a self-similar profile corresponding to a solution with Dirac distribution initial data are found. They are connected with optimal constants for a Gagliardo-Nirenberg inequality.