Abstract

We develop partial differential equation (PDE) methods to study the dynamics of pattern formation in partial integro-differential equations (PIDEs) defined on a spatially extended domain. Our primary focus is on scalar equations in two spatial dimensions. These models arise in a variety of neuronal modeling problems and also occur in material science. We first derive a PDE which is equivalent to the PIDE. We then find circularly symmetric solutions of the resultant PDE; the linearization of the PDE around these solutions provides a criterion for their stability. When a solution is unstable, our analysis predicts the exact number of peaks that form to comprise a multipeak solution of the full PDE. We illustrate our results with specific numerical examples and discuss other systems for which this technique can be used.

MSC codes

  1. 34B15
  2. 34C23
  3. 93C15
  4. 34C11

Keywords

  1. pattern formation
  2. integro-differential equation
  3. PDE
  4. nonlocal

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Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 487 - 516
ISSN (online): 1536-0040

History

Published online: 7 August 2006

MSC codes

  1. 34B15
  2. 34C23
  3. 93C15
  4. 34C11

Keywords

  1. pattern formation
  2. integro-differential equation
  3. PDE
  4. nonlocal

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