American Institute of Mathematical Sciences

September  2003, 9(5): 1105-1132. doi: 10.3934/dcds.2003.9.1105

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, France 2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21944–970 Rio de Janeiro, R.J., Brazil 3 LAGA, UMR CNRS 7539, Institut Galilée–Université Paris XIII, 99, Avenue J.-B. Clément, 93430 Villetaneuse, France

Received  October 2001 Revised  January 2003 Published  June 2003

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 $B$.
Citation: Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Universal solutions of the heat equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1105-1132. doi: 10.3934/dcds.2003.9.1105
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