scholarly journals Global Structure of Self-Similar Solutions in a Semilinear Parabolic Equation

2000 ◽  
Vol 244 (2) ◽  
pp. 348-368 ◽  
Author(s):  
Munemitsu Hirose ◽  
Eiji Yanagida
Keyword(s):  
Parabolic Equation ◽  
Global Structure ◽  
Semilinear Parabolic Equation ◽  
Self Similar ◽  
Similar Solutions ◽  
Semilinear Parabolic

scholarly journals Convergence to self-similar solutions for a semilinear parabolic equation

2008 ◽  
Vol 21 (3) ◽  
pp. 703-716 ◽  
Author(s):  
Marek Fila ◽  
◽  
Michael Winkler ◽  
Eiji Yanagida ◽  
◽  
...  
Keyword(s):  
Parabolic Equation ◽  
Semilinear Parabolic Equation ◽  
Self Similar ◽  
Similar Solutions ◽  
Semilinear Parabolic

Regional blow-up for a higher-order semilinear parabolic equation

2001 ◽  
Vol 12 (5) ◽  
pp. 601-623 ◽  
Author(s):  
MANUELA CHAVES ◽  
VICTOR A. GALAKTIONOV
Keyword(s):  
Parabolic Equation ◽  
Evolution Equations ◽  
Blow Up ◽  
Semilinear Parabolic Equation ◽  
General Integral ◽  
Superlinear Growth ◽  
Self Similar Solution ◽  
Self Similar ◽  
Semilinear Parabolic ◽  
Hamilton Jacobi Equation

We study the blow-up behaviour of solutions of a 2mth order semilinear parabolic equation[formula here]with a superlinear function q(u) for |u| Gt; 1. We prove some estimates on the asymptotic blow-up behaviour. Such estimates apply to general integral evolution equations. We answer the following question: find a continuous function q(u) with a superlinear growth as u → ∞ such that the parabolic equation exhibits regional blow-up in a domain of finite non-zero measure. We show that such a regional blow-up can occur for q(u) = u|ln|u‖2m. We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t → T− is described by the self-similar solution[formula here]of the complex Hamilton–Jacobi equation[formula here].

Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient term

1999 ◽  
Vol 129 (6) ◽  
pp. 1291-1307 ◽  
Author(s):  
S. Snoussi ◽  
S. Tayachi ◽  
F. B. Weissler
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Asymptotic Behaviour ◽  
Global Solutions ◽  
Gradient Term ◽  
Semilinear Parabolic Equation ◽  
Small Initial Data ◽  
Self Similar ◽  
Semilinear Parabolic

We study the existence and the asymptotic behaviour of global solutions of the semilinear parabolic equation u(0) = ϧwhere a, b ∈ℝ, q > 1, p > 1. Forq=2p/(p+1) and ½ 1(p-1)>1 (equivalently, q > (n + 2)/(n + 1)), we prove the existence of mild global solutions for small initial data with respect to some norm. Some of those solutions are asymptotically self-similar.

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