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.

scholarly journals Very singular solutions to a nonlinear parabolic equation with absorption II. Uniqueness

2004 ◽  
Vol 134 (1) ◽  
pp. 39-54 ◽  
Author(s):  
Saïd Benachour ◽  
Herbert Koch ◽  
Philippe Laurençot
Keyword(s):  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Singular Solution ◽  
Singular Solutions ◽  
Nonlinear Parabolic ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions ◽  
Very Singular Solution

We prove the uniqueness of the very singular solution to when 1 < p < (N + 2)/(N + 1), thus completing the previous result by Qi and Wang, restricted to self-similar solutions.

On backward self-similar blow-up solutions to a supercritical semilinear heat equation

2010 ◽  
Vol 140 (4) ◽  
pp. 821-831 ◽  
Author(s):  
Noriko Mizoguchi
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Blow Up ◽  
Radial Solution ◽  
Similar Solution ◽  
Semilinear Heat Equation ◽  
Blowing Up ◽  
Self Similar Solution ◽  
Self Similar ◽  
Image Position

We are concerned with a Cauchy problem for the semilinear heat equationthen u is called a backward self-similar solution blowing up at t = T. Let pS and pL be the Sobolev and the Lepin exponents, respectively. It was shown by Mizoguchi (J. Funct. Analysis257 (2009), 2911–2937) that k ≡ (p − 1)−1/(p−1) is a unique regular radial solution of (P) if p > pL. We prove that it remains valid for p = pL. We also show the uniqueness of singular radial solution of (P) for p > pS. These imply that the structure of radial backward self-similar blow-up solutions is quite simple for p ≥ pL.

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.

Minimal blow-up asymptotics of quasilinear heat equations

2001 ◽  
Vol 131 (6) ◽  
pp. 1297-1321
Author(s):  
M. Chaves ◽  
Victor A. Galaktionov
Keyword(s):  
Heat Equation ◽  
Blow Up ◽  
Asymptotic Properties ◽  
Basin Of Attraction ◽  
Boundary Function ◽  
Similar Solution ◽  
Heat Equations ◽  
Blowing Up ◽  
Self Similar Solution ◽  
Self Similar

We study the asymptotic properties of blow-up solutions u = u(x, t) ≥ 0 of the quasilinear heat equation , where k(u) is a smooth non-negative function, with a given blowing up regime on the boundary u(0, t) = ψ(t) > 0 for t ∈ (0, 1), where ψ(t) → ∞ as t → 1−, and bounded initial data u(x, 0) ≥ 0. We classify the asymptotic properties of the solutions near the blow-up time, t → 1−, in terms of the heat conductivity coefficient k(u) and of boundary data ψ(t); both are assumed to be monotone. We describe a domain, denoted by , of minimal asymptotics corresponding to the data ψ(t) with a slow growth as t → 1− and a class of nonlinear coefficients k(u).We prove that for any problem in S11−, such a blow-up singularity is asymptotically structurally equivalent to a singularity of the heat equation ut = uxx described by its self-similar solution of the form u*(x, t) = −ln(1 − t) + g(ξ), ξ = x/(1 − t)1/2, where g solves a linear ordinary differential equation. This particular self-similar solution is structurally stable upon perturbations of the boundary function and also upon nonlinear perturbations of the heat equation with the basin of attraction .

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