Abstract

In this paper, we study the relationship between the long time behavior of a solution $u$ of the nonlinear heat equation $u_t-\Delta u+|u|^\alpha u=0$ on $\mathbb{R}^N$ and the asymptotic behavior as $|x|\rightarrow \infty $ of its initial value $u_0$. In particular, we show that if the sequence of dilations $\lambda _n^{2/\alpha } u_0$ converges weakly to $z$ as $\lambda _n\rightarrow \infty $, then the rescaled solution $t^{1/\alpha } u$ converges uniformly on $\mathbb{R}^N $ to ${\mathcal{U}}z$ along the subsequence $t_n=\lambda _n^2$, where ${\mathcal{U}}$ is an appropriate flow. 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 $B$. These results are restricted to positive solutions in the case $\alpha <2/N$