scholarly journals Forward self-similar solution with a moving singularity for a semilinear parabolic equation

2010 ◽  
Vol 26 (1) ◽  
pp. 313-331 ◽  
Author(s):  
Shota Sato ◽  
◽  
Eiji Yanagida ◽  
Keyword(s):  
Parabolic Equation ◽  
Similar Solution ◽  
Semilinear Parabolic Equation ◽  
Self Similar Solution ◽  
Self Similar ◽  
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].

On blow-up and degeneracy for the semilinear heat equation with source

1990 ◽  
Vol 115 (1-2) ◽  
pp. 19-24 ◽  
Author(s):  
V. A. Galaktionov
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Semilinear Parabolic Equation ◽  
Quasilinear Parabolic Equations ◽  
Open Problems ◽  
Semilinear Heat Equation ◽  
Self Similar Solution ◽  
Self Similar ◽  
Semilinear Parabolic ◽  
Hamilton Jacobi Equation

SynopsisThe asymptotic behaviour of the solution of the semilinear parabolic equation ut = uxx + (1 + u)ln2(l + u) for t > 0, x ∊[−π, π ], ux(t, ± π) = 0 for t > 0 and u(0, x) = u0(x) ≧ 0 in [−π, π], which blows up at a finite time T0, is investigated. It is proved that for some two-parametric set of initial functions u0 the behaviour of u(t, x) near t = T0 is described by the approximate self-similar solution va(t, x) = exp {(T0 −t)−1 cos2 (x/2)} − 1, satisfying the first order nonlinear Hamilton–Jacobi equation vt, = (vx)2 /(1 + v) + (1 + v) ln2 (1 + v). Some open problems of degeneracy near a finite blow-up time for other semilinear or quasilinear parabolic equations with source ut, = Δu + (1 + u) lnβ (1 + u) (β >1), ut, = Δu + uβ(β > l), ut = Δu + eu; ut = ∇. (lnσ(1 + u)∇u)+ (1 + u)lnβ(1 + u) (σ > 0, β > 1) are discussed.

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

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.

Sign in / Sign up

Export Citation Format

Share Document