Superdiffusive blow-up with advection

C. M. Kirk*, W. E. Olmstead

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We examine the problem of a high-energy source localised within a superdiffusive medium with advection. This problem is modelled by a fractional diffusion equation with a nonlinear source term. Advection is introduced through a linear transport term that is proportional to the advection speed. In this paper we allow the medium to exhibit superdiffusive behaviour ranging from the classical (Gaussian) limit to the ballistic limit. We analyse the model to determine whether or not a thermal blow-up occurs. Specifically, it is shown that there exists a critical advection speed above which blow-up is avoided and below which blow-up is guaranteed. We also provide the asymptotic growth of the temperature near the time of blow-up.

Original languageEnglish (US)
Pages (from-to)93-102
Number of pages10
JournalInternational Journal of Dynamical Systems and Differential Equations
Volume4
Issue number1-2
DOIs
StatePublished - Mar 2012

Keywords

  • Advection
  • Nonlinear Volterra integral equations
  • Superdiffusion
  • Thermal blow-up

ASJC Scopus subject areas

  • Engineering(all)
  • Discrete Mathematics and Combinatorics
  • Control and Optimization

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