Collapsing heat waves

William R. Derrick, Leonid V. Kalachev, Joseph A. Cima

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In certain combustion models, an initial temperature profile will develop into a combustion wave that will travel at a specific wave speed. Other initial profiles do not develop into such waves, but die out to the ambient temperature. There exists a clear demarcation between those initial conditions that evolve into combustion waves and those that do not. This is sometimes called a watershed initial condition. In this paper we will show that there may be numerous exact watershed conditions to the initial-Neumann-boundary value problem ut = D ux x + e- 1 / u - σ (u - α), with ux (0, t) = ux (1, t) = 0, on I = [0, 1]. They are composed from the positive non-constant solutions of D vx x + e- 1 / v - σ (v - α) = 0, with vx (0) = vx (1) = 0, for small values of D. We will give easily verifiable conditions for when combustion waves arise and when they do not.

Original languageEnglish
Pages (from-to)612-624
Number of pages13
JournalMathematical and Computer Modelling
Issue number5-6
StatePublished - Sep 2007


  • Bifurcation
  • Combustion
  • Domain of attraction
  • Parabolic equation


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