Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 1105 - 1132, Volume 9, Issue 5, September 2003      doi:10.3934/dcds.2003.9.1105

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Thierry Cazenave - Laboratoire Jacques-Louis Lions, UMR CNRS 7598, Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris Cedex 05, France (email)
Flávio Dickstein - Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21944–970 Rio de Janeiro, R.J., Brazil (email)
Fred B. Weissler - LAGA, UMR CNRS 7539, Institut Galilée–Université Paris XIII, 99, Avenue J.-B. Clément, 93430 Villetaneuse, France (email)

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 $B$.

Keywords:  Heat equation, asymptotic behavior, dilation properties, asymptotically self-similar solutions, chaos, almost periodic functions.
Mathematics Subject Classification:  35K05, 35B40, 35B30; Secondary: 35B15, 47A20, 47D06.

Received: October 2001;      Revised: January 2003;      Available Online: June 2003.