Lack of symmetry in linear determinacy due to convective effects in reaction-diffusion-convection problems

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Published: Mar 30, 2016
DOI: https://doi.org/10.5556/j.tkjm.47.2016.1891
Keywords:
Travelling wave linear determinacy convection reaction-diffusion spreading speed

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Ameera Al-Kiffai
Elaine Crooks

Abstract

This paper is concerned with linear determinacy in monostable reaction- diffusion-convection equations and co-operative systems. We present sufficient conditions for minimal travelling-wave speeds (equivalent to spreading speeds) to equal values obtained from linearisations of the travelling-wave problem about the unstable equilibrium. These conditions involve both reaction and convection terms. We present separate conditions for non-increasing and non-decreasing travelling waves, called `right' and `left' conditions respectively, because of the asymmetry in propagation caused by the convection terms. We also give a necessary condition on the reaction term for the existence of convection terms such that both the right and left conditions can be satisfied simultaneously. Examples show that our sufficient conditions for linear determinacy are not necessary and compare these conditions in the scalar case with alternative conditions observed in Malaguti-Marcelli [15] and Benguria-Depassier-Mendez [3]. We also illustrate, for both an equation and a system, the existence of reaction and (non-trivial) convection terms for which the right and left linear determinacy conditions are simultaneously satisfied. An example is given of an equation which is right but not left linearly determinate.

Article Details

How to Cite
Al-Kiffai, A., & Crooks, E. (2016). Lack of symmetry in linear determinacy due to convective effects in reaction-diffusion-convection problems. Tamkang Journal of Mathematics, 47(1). https://doi.org/10.5556/j.tkjm.47.2016.1891
Issue
Vol. 47 No. 1 (2016)
Section
Special Issue
Author Biographies

Ameera Al-Kiffai

Department ofMathematics, College of Science, Swansea University, Swansea SA2 8PP, U.K. College of Education for Girls, Kufa University, Al-Najaf Al-Ashraf, Iraq.

Elaine Crooks

Department ofMathematics, College of Science, Swansea University, Swansea SA2 8PP, U.K.

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