Universal solutions of the heat equation on $\mathbb R^N$
Thierry Cazenave, Flávio Dickstein, Fred B. Weissler · Discrete and Continuous Dynamical Systems · 2003
In this paper, we study the relationship between the long timebehavior of a solution $u(t,x)$ of the heat equation on $\R^N $ andthe 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 therescaled solution $t^{\frac{\sigma}{2}}$ $u(t, \cdot\sqrt t)$convergesuniformly on $\R^N $ to $e^\Delta z$ along the subsequence $t_n=\lambda_n^2$. Moreover, we show there exists an initial value $U_0$ suchthat the set of all possible $z$ attainable in this fashion is aclosed ball $B$ of a weighted $L^\infty $ space. The resulting'universal' solution is therefore asymptotically close alongappropriate subsequences to all solutions with initial values in$B$.