Superdiffusive blow-up with advection

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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
Issue number1-2
StatePublished - Mar 2012


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

ASJC Scopus subject areas

  • General Engineering
  • Discrete Mathematics and Combinatorics
  • Control and Optimization


Dive into the research topics of 'Superdiffusive blow-up with advection'. Together they form a unique fingerprint.

Cite this