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IMA Journal of Applied Mathematics Advance Access originally published online on September 25, 2007
IMA Journal of Applied Mathematics 2007 72(5):540-555; doi:10.1093/imamat/hxm026
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Cross-diffusion influence on the non-linear L2-stability analysis for a Lotka–Volterra reaction–diffusion system of PDEs

J. N. Flavin{dagger}

Department of Mathematical Physics, National University of Ireland, Galway, Ireland

S. Rionero

Department of Mathematics and Applications, University of Naples Federico II, Naples, Italy

{dagger} Email: james.flavin{at}nuigalway.ie

Received on September 1, 2006; Accepted on March 12, 2007

The paper considers non-linear stability and instability of the zero solutions of a pair of reaction–diffusion partial differential equations, which incorporate both self-diffusion and cross-diffusion, and non-linear source/forcing terms depending on the dependent variables. Both Dirichlet and Neumann boundary conditions are considered. Lyapunov functionals are used to obtain the stability and instability criteria.

Keywords: stability; cross-diffusion; reaction-diffusion systems.


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