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Universal solutions of a nonlinear heat equation on ℝ N

Thierry Cazenave;  Flávio Dickstein;  Fred B. Weissler — 2003

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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 - Δ u + | u | α u = 0 on ℝ 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 ℝ 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...

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