Length scales and positivity of solutions of a class of reaction-diffusion equations

Michele V. Bartuccelli; K. B. Blyuss; Y. N. Kyrychko

doi:10.3934/cpaa.2004.3.25

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2004, Volume 3, Issue 1: 25-40. Doi: 10.3934/cpaa.2004.3.25

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Length scales and positivity of solutions of a class of reaction-diffusion equations

1.
Department of Mathematics and Statistics, University of Surrey, GU2 7XH
2.
Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH

Received: December 2002

Revised: July 2003

Published: March 2004

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Abstract

In this paper, the sharpest interpolation inequalities are used to find a set of length scales for the solutions of the following class of dissipative partial differential equations

$u_{t}= -\alpha_{k}(-1)^{k} \nabla^{2k}u+\sum_{j=1}^{k-1}\alpha_{j} (-1)^{j}\nabla^{2j}u+\nabla^{2}(u^{m})+u(1-u^{2p}), $

with periodic boundary conditions on a one-dimensional spatial domain. The equation generalises nonlinear diffusion model for the case when higher-order differential operators are present. Furthermore, we establish the asymptotic positivity as well as the positivity of solutions for all times under certain restrictions on the initial data. The above class of equations reduces for some particular values of the parameters to classical models such as the KPP-Fisher equation which appear in various contexts in population dynamics.

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Mathematics Subject Classification: 35K57, 35B35.

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