University of Illinois Urbana-Champaign

Blow-up in semilinear parabolic equations with weak diffusion

Short, Mark; Kapila, Ashwani K.

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https://hdl.handle.net/2142/112579

Description

Title
Blow-up in semilinear parabolic equations with weak diffusion
Author(s)
  • Short, Mark
  • Kapila, Ashwani K.
Issue Date
1997-12
Keyword(s)
  • Finite Time Blow Up
  • Semilinear Parabolic Equations
  • Weak Diffusion
Abstract
Finite time blow-up in the semilinear reactive-diffusive parabolic equation ϕt = μϕzz + eϕ is examined in the limit of weak diffusion μ « 1. for a Cauchy initial value problem with ϕ(x,t = 0) = ϕi(x) in which ϕi(x) possesses a smooth global maximum. An asymptotic descrip tion of the evolution of ϕ is obtained from the initial time through blow-up using singular perturbation techniques. Near blow-up, an exact self-similar focusing structure for ϕ, identical to that previously associated with non-diffusive thermal runaway, is shown to be appropriate. However, in an exponentially small layer close to the blow-up time, the focusing structure must be modified to ensure a uniformly valid solution. This modification uncovers the asymptotically self-similar focusing structure previously recognized for blow-up in equations of the form ϕt = μϕzz + eϕ In contrast to previous studies, however, the structure arises here as a natural consequence of removing the non-uniformity in the expansions which occurs exponentially close to blow-up when the effects of diffusion have t o be reinstated . Identical weakdiffusion limit asymptotics can be applied to a variety of semilinear or quasilinear parabolic equations that exhibit finite time blow-up in order to reveal the associated focusing structure.
Publisher
Department of Theoretical and Applied Mechanics. College of Engineering. University of Illinois at Urbana-Champaign
Series/Report Name or Number
  • TAM R 874
  • 1997-6035
ISSN
0073-5264
Type of Resource
text
Language
eng
Permalink
http://hdl.handle.net/2142/112579
Sponsor(s)/Grant Number(s)
Air Force Office of Scientific Research; National Science Foundation F 49620 96 1 0260 97/12
Copyright and License Information
Copyright 1997 Board of Trustees of the University of Illinois

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Technical Reports - Theoretical and Applied Mechanics (TAM) PRIMARY
TAM technical reports include manuscripts intended for publication, theses judged to have general interest, notes prepared for short courses, symposia compiled from outstanding undergraduate projects, and reports prepared for research-sponsoring agencies.