Universal solutions of the heat equation on $\mathbb R^N$

Thierry Cazenave; Flávio Dickstein; Fred B. Weissler

doi:10.3934/dcds.2003.9.1105

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2003, Volume 9, Issue 5: 1105-1132. Doi: 10.3934/dcds.2003.9.1105

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Universal solutions of the heat equation on $\mathbb R^N$

1.
Laboratoire Jacques-Louis Lions, UMR CNRS 7598, Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris Cedex 05
2.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21944–970 Rio de Janeiro, R.J.
3.
LAGA, UMR CNRS 7539, Institut Galilée–Université Paris XIII, 99, Avenue J.-B. Clément, 93430 Villetaneuse

Received: October 2001

Revised: January 2003

Published: September 2003

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Abstract

In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the heat equation on $\R^N $ and the asymptotic behavior as $|x|\to \infty $ of its initial value $u_0$. In particular, we show that, for a fixed $0$<$\sigma$<$N$, if the sequence of dilations $\lambda _n^\sigma u_0(\lambda _n\cdot)$ converges weakly to $z(\cdot)$ as $\lambda _n\to \infty $, then the rescaled solution $t^{\frac{\sigma}{2}}$ $u(t, \cdot\sqrt t)$ converges uniformly on $\R^N $ to $e^\Delta z$ along the subsequence $t_n=\lambda _n^2$. Moreover, we show there exists an initial value $U_0$ such that the set of all possible $z$ attainable in this fashion is a closed ball $B$ of a weighted $L^\infty $ space. The resulting "universal" solution is therefore asymptotically close along appropriate subsequences to all solutions with initial values in $B$.

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Mathematics Subject Classification: 35K05, 35B40, 35B30; Secondary: 35B15, 47A20, 47D06.

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