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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 $| x | \to \infty$ of its initial value $u_{0}$ . In particular, we show that if the sequence of dilations $λ_{n}^{2 / α} u_{0} (λ_{n} \cdot)$ converges weakly to $z (\cdot)$ as $λ_{n} \to \infty$ , then the rescaled solution $t^{1 / α} u (t, \cdot \sqrt{t})$ 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...

Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields

Thierry Cazenave; Jalal Shatah; A. Shadi Tahvildar-Zadeh — 1998

Annales de l'I.H.P. Physique théorique

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A remark on Schrödinger's equation with a short range potential

Asymptotically periodic solutions for a class of nonlinear coupled oscillators.

Solutions self-similaires de l'équation de Schrödinger non-linéaire

Some stability properties for minimal solutions of $- Δ u = λ g (u)$ .

Équations d'évolution avec non linéarité logarithmique

The Cauchy problem for the nonlinear Schrödinger Equation in H1.

Universal solutions of a nonlinear heat equation on $ℝ^{N}$

Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields