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

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

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