In this paper, we study the relationship between the long time behavior of a solution of the nonlinear heat equation on (where ) and the asymptotic behavior as of its initial value . In particular, we show that if the sequence of dilations converges weakly to as , then the rescaled solution converges uniformly on to along the subsequence , where is an appropriate flow. Moreover, we show there exists an initial value such that the set of all possible attainable in this fashion is a closed ball of a weighted space. The resulting “universal” solution is therefore asymptotically close along appropriate subsequences to all solutions with initial values in . These results are restricted to positive solutions in the case .
@article{ASNSP_2003_5_2_1_77_0, author = {Cazenave, Thierry and Dickstein, Fl\'avio and Weissler, Fred B.}, title = {Universal solutions of a nonlinear heat equation on $\mathbb {R}^N$}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {77--117}, publisher = {Scuola normale superiore}, volume = {Ser. 5, 2}, number = {1}, year = {2003}, mrnumber = {1990975}, zbl = {1170.35448}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_77_0/} }
TY - JOUR AU - Cazenave, Thierry AU - Dickstein, Flávio AU - Weissler, Fred B. TI - Universal solutions of a nonlinear heat equation on $\mathbb {R}^N$ JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2003 SP - 77 EP - 117 VL - 2 IS - 1 PB - Scuola normale superiore UR - http://www.numdam.org/item/ASNSP_2003_5_2_1_77_0/ LA - en ID - ASNSP_2003_5_2_1_77_0 ER -
%0 Journal Article %A Cazenave, Thierry %A Dickstein, Flávio %A Weissler, Fred B. %T Universal solutions of a nonlinear heat equation on $\mathbb {R}^N$ %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2003 %P 77-117 %V 2 %N 1 %I Scuola normale superiore %U http://www.numdam.org/item/ASNSP_2003_5_2_1_77_0/ %G en %F ASNSP_2003_5_2_1_77_0
Cazenave, Thierry; Dickstein, Flávio; Weissler, Fred B. Universal solutions of a nonlinear heat equation on $\mathbb {R}^N$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 77-117. http://www.numdam.org/item/ASNSP_2003_5_2_1_77_0/
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