American Institute of Mathematical Sciences

July  2011, 16(1): 57-71. doi: 10.3934/dcdsb.2011.16.57

Finite to infinite steady state solutions, bifurcations of an integro-differential equation

 1 Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh 2 Department of Mathematics and Maxwell Institute, Heriot-Watt University, Edinburgh, United Kingdom, United Kingdom 3 Department of Mathematics, University of Strathclyde, Glasgow, United Kingdom

Received  April 2010 Revised  November 2010 Published  April 2011

We consider a bistable integral equation which governs the stationary solutions of a convolution model of solid--solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is increased to examine the transition from an uncountably infinite number of steady states to three for the continuum limit of the semi--discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.
Citation: Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57
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