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March  2004, 3(1): 25-40. doi: 10.3934/cpaa.2004.3.25

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

1. 

Department of Mathematics and Statistics, University of Surrey, GU2 7XH, United Kingdom

2. 

Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH, United Kingdom, United Kingdom

Received  December 2002 Revised  July 2003 Published  January 2004

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.

Citation: Michele V. Bartuccelli, K. B. Blyuss, Y. N. Kyrychko. Length scales and positivity of solutions of a class of reaction-diffusion equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 25-40. doi: 10.3934/cpaa.2004.3.25
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