Annali di Matematica Pura ed Applicata

, Volume 155, Issue 1, pp 243–260 | Cite as

Quenching, nonquenching, and beyond quenching for solution of some parabolic equations

  • Howard A. Levine
Article
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Summary

In this paper we examine the first initial boundary value problem for ut=uxx + ε(1 − u)−β,ɛ > 0,β > 0,on (0, 1) × (0, ∞) from the point of view of dynamical systems. We construct the set of stationary solutions, determine those which are stable, those which are not and show that there are solutions with initial data arbitrarily close to unstable stationary solutions which quench (reach one in finite time). We also examine the related problem ut=uxx, 0 <x < 1,t > 0;u(0,t)=0, ε(1 − u(1, t))−β. The set of stationary solutions for this problem, and the dynamical behavior of solutions of the time dependent problem are somewhat different.

Keywords

Dynamical System Initial Data Dynamical Behavior Stationary Solution Parabolic Equation 
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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1989

Authors and Affiliations

  • Howard A. Levine
    • 1
  1. 1.Department of MathematicsIowa State UniversityAmes

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