Front-like entire solutions for equations with convection

E. C.M. Crooks*, Je Chiang Tsai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)


We construct families of front-like entire solutions for problems with convection, both for bistable and monostable reaction-diffusion-convection equations, and, via vanishing-viscosity arguments, for bistable and monostable balance laws. The unified approach employed is inspired by ideas of Chen and Guo and based on a robust method using front-dependent sub and supersolutions. Unlike convectionless problems, the equations studied here lack symmetry between increasing and decreasing travelling waves, which affects the choice of sub and supersolutions used. Our entire solutions include both those that behave like two fronts coming together and annihilating as time increases, and, for bistable equations, those that behave like two fronts merging to propagate like a single front. We also characterise the long-time behaviour of each family of entire solutions, which in the case of solutions constructed from a monostable front merging with a bistable front answers a question that was open even for reaction-diffusion equations without convection.

Original languageEnglish
Pages (from-to)1206-1249
Number of pages44
JournalJournal of Differential Equations
Issue number4
Publication statusPublished - 2012 Aug 15
Externally publishedYes


  • Annihilating fronts
  • Bistable reaction
  • Entire solution
  • Merging fronts
  • Monostable reaction
  • Reaction-diffusion-convection equation
  • Travelling front
  • Vanishing-viscosity

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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