On uniqueness for ODEs Arising in Blow-up Asymptotics for Nonlinear Heat Equations

2002 ◽  
Vol 2 (3) ◽  
Author(s):  
Manuela Chaves ◽  
Victor A. Galaktionov
Keyword(s):  
Blow Up ◽  
Reaction Diffusion ◽  
Similarity Solutions ◽  
Heat Equations ◽  
Nonlinear Ordinary Differential Equations ◽  
Singular Behaviour ◽  
Nonlinear Heat Equations ◽  
Self Similar ◽  
Similar Solutions ◽  
Diffusion Absorption

AbstractWe study uniqueness for nonlinear ordinary differential equations arising in constructing blow-up and extinction self-similar solutions of various reaction-diffusion- absorption equations. Such particular similarity solutions describe the asymptotic singular behaviour of wide classes of general solutions of nonlinear heat equations. We prove that under some monotonicity assumptions, such similarity profiles are unique.

On Uniqueness and Stability of Self-Similar Blow-Up Solutions of Nonlinear Heat Equations: Evolution Approach

2003 ◽  
Vol 05 (03) ◽  
pp. 329-348 ◽  
Author(s):  
Manuela Chaves ◽  
Victor A. Galaktionov
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Domain Of Attraction ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Nonlinear Parabolic ◽  
Linear Parabolic Equations ◽  
Self Similar ◽  
Similar Solutions ◽  
Evolution Approach

We present evolution arguments of studying uniqueness and asymptotic stability of blow-up self-similar solutions of second-order nonlinear parabolic equations from combustion and filtration theory. The analysis uses intersection comparison techniques based on the Sturm Theorem on zero set for linear parabolic equations. We show that both uniqueness and stability of similarity ODE profiles are directly related to the asymptotic structure of their domain of attraction relative to the corresponding parabolic evolution.

Self-similar boundary blow-up for higher-order quasilinear parabolic equations

2005 ◽  
Vol 135 (6) ◽  
pp. 1195-1227 ◽  
Author(s):  
V. A. Galaktionov ◽  
A. E. Shishkov
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Similarity Solutions ◽  
Energy Estimates ◽  
Quasilinear Parabolic Equations ◽  
Asymptotic Results ◽  
Propagation Of Singularities ◽  
Self Similar ◽  
Similar Solutions ◽  
Quasilinear Parabolic

We study evolution properties of boundary blow-up for 2mth-order quasilinear parabolic equations in the case where, for homogeneous power nonlinearities, the typical asymptotic behaviour is described by exact or approximate self-similar solutions. Existence and asymptotic stability of such similarity solutions are established by energy estimates and contractivity properties of the rescaled flows.Further asymptotic results are proved for more general equations by using energy estimates related to Saint-Venant's principle. The established estimates of propagation of singularities generated by boundary blow-up regimes are shown to be sharp by comparing with various self-similar patterns.

Convergence of blow-up solutions of nonlinear heat equations in the supercritical case

1999 ◽  
Vol 129 (6) ◽  
pp. 1197-1227 ◽  
Author(s):  
J. Matos
Keyword(s):  
Parabolic Equation ◽  
Heat Equation ◽  
Convergence Theorem ◽  
Nonlinear Parabolic Equation ◽  
Blow Up ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
Nonlinear Parabolic ◽  
Nonlinear Heat Equations ◽  
Self Similar

In this paper, we study the blow-up behaviour of the radially symmetric non-negative solutions u of the semilinear heat equation with supercritical power nonlinearity up (that is, (N – 2)p> N + 2). We prove the existence of non-trivial self-similar blow-up patterns of u around the blow-up point x = 0. This result follows from a convergence theorem for a nonlinear parabolic equation associated to the initial one after rescaling by similarity variables.

Blow-up similarity solutions of the fourth-order unstable thin film equation

2007 ◽  
Vol 18 (2) ◽  
pp. 195-231 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV ◽  
J. R. KING
Keyword(s):  
Thin Film ◽  
Fourth Order ◽  
Blow Up ◽  
Similarity Solutions ◽  
Diffusion Term ◽  
Thin Film Equation ◽  
Self Similar ◽  
Image Position ◽  
General Aspects ◽  
Similar Solutions

We study blow-up behaviour of solutions of the fourth-order thin film equationwhich contains a backward (unstable) diffusion term. Our main goal is a detailed study of the case of the first critical exponentwhereN≥ 1 is the space dimension. We show that the free-boundary problem with zero contact angle and zero-flux conditions admits continuous sets (branches) of blow-up self-similar solutions. For the Cauchy problem inRN×R+, we detect compactly supported blow-up patterns, which have infinitely many oscillations near interfaces and exhibit a “maximal” regularity there. As a key principle, we use the fact that, for small positiven, such solutions are close to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equationwhich are better understood and have been studied earlier [19]. We also discuss some general aspects of formation of self-similar blow-up singularities for other values ofp.

Sign in / Sign up

Export Citation Format

Share Document