Title: Superdiffusive blow-up with advection

Authors: C.M. Kirk; W.E. Olmstead

Addresses: Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407, USA. ' Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA

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.

Keywords: superdiffusion; advection; thermal blow-up; nonlinear Volterra integral equations; high energy sources; modelling.

DOI: 10.1504/IJDSDE.2012.045997

International Journal of Dynamical Systems and Differential Equations, 2012 Vol.4 No.1/2, pp.93 - 102

Available online: 19 Mar 2012 *

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