Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations

1998 ◽  
Vol 228 (1) ◽  
pp. 83-120 ◽  
Author(s):  
Thierry Cazenave ◽  
Fred B. Weissler
Keyword(s):  
Global Solutions ◽  
Heat Equations ◽  
Nonlinear Schrödinger ◽  
Self Similar

Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data

2019 ◽  
Vol 150 (2) ◽  
pp. 789-811
Author(s):  
Yūki Naito
Keyword(s):  
Initial Data ◽  
Existence Of Solutions ◽  
Global Solutions ◽  
Heat Equations ◽  
Complicated Manner ◽  
The Cauchy Problem ◽  
Self Similar ◽  
The One ◽  
Image Position ◽  
Similar Solutions

AbstractWe consider the Cauchy problem $$\left\{ {\matrix{ {u_t = \Delta u + u^p,\quad } \hfill & {x\in {\bf R}^N,\;t \leq 0,} \hfill \cr {u(x,0) = u_0(x),\quad } \hfill & {x\in {\bf R}^N,} \hfill \cr } } \right.$$where N > 2, p > 1, and u0 is a bounded continuous non-negative function in RN. We study the case where u0(x) decays at the rate |x|−2/(p−1) as |x| → ∞, and investigate the convergence property of the global solutions to the forward self-similar solutions. We first give the precise description of the relationship between the spatial decay of initial data and the large time behaviour of solutions, and then we show the existence of solutions with a time decay rate slower than the one of self-similar solutions. We also show the existence of solutions that behave in a complicated manner.

scholarly journals The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation

1991 ◽  
Vol 117 (3-4) ◽  
pp. 251-273 ◽  
Author(s):  
Thierry Cazenave ◽  
Fred B. Weissler
Keyword(s):  
Schrödinger Equation ◽  
Global Solution ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Global Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Structure Of Solutions ◽  
Nonlinear Schrödinger ◽  
Conformally Invariant

SynopsisWe study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

scholarly journals Full self-similar solutions of the subsonic radiative heat equations

2015 ◽  
Vol 22 (8) ◽  
pp. 082109 ◽  
Author(s):  
Tomer Shussman ◽  
Shay I. Heizler
Keyword(s):  
Radiative Heat ◽  
Heat Equations ◽  
Self Similar ◽  
Similar Solutions
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