Universal solutions of a nonlinear heat equation on N
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, p. 77-117

In this paper, we study the relationship between the long time behavior of a solution u(t,x) of the nonlinear heat equation u t -Δu+|u| α u=0 on N (where α>0) and the asymptotic behavior as |x| of its initial value u 0 . In particular, we show that if the sequence of dilations λ n 2/α u 0 (λ n ·) converges weakly to z(·) as λ n , then the rescaled solution t 1/α u(t,·t) converges uniformly on N to 𝒰(1)z along the subsequence t n =λ n 2 , where 𝒰(t) 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 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 α<2/N.

Classification:  35K55,  35B40
@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},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {1},
     year = {2003},
     pages = {77-117},
     zbl = {1170.35448},
     mrnumber = {1990975},
     language = {en},
     url = {http://www.numdam.org/item/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, Serie 5, Volume 2 (2003) no. 1, pp. 77-117. http://www.numdam.org/item/ASNSP_2003_5_2_1_77_0/

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