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Title:Blow-up in semilinear parabolic equations with weak diffusion
Author(s):Short, Mark; Kapila, Ashwani K.
Subject(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.
Issue Date:1997-12
Publisher:Department of Theoretical and Applied Mechanics. College of Engineering. University of Illinois at Urbana-Champaign
Series/Report:TAM R 874
Genre:Technical Report
Sponsor:Air Force Office of Scientific Research; National Science Foundation F 49620 96 1 0260 97/12
Rights Information:Copyright 1997 Board of Trustees of the University of Illinois
Date Available in IDEALS:2021-11-04

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  • Technical Reports - Theoretical and Applied Mechanics (TAM)
    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.

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