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IMA Journal of Applied Mathematics Advance Access originally published online on October 27, 2007
IMA Journal of Applied Mathematics 2007 72(6):854-864; doi:10.1093/imamat/hxm055
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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.

Spatial decay in a cross-diffusion problem

L. E. Payne{dagger}

Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA

J. C. Song{ddagger}

Department of Applied Mathematics, Hanyang University, Ansan, Gyeonggi-do 426-791, Korea

{dagger} Email: lep8{at}cornell.edu

{ddagger} Corresponding author. Email: jcsong{at}hanyang.ac.kr

Received on May 30, 2007; Accepted on September 17, 2007

In this paper, the authors investigate the decay of end effects for a cross-diffusion problem defined on a semi-infinite cylindrical region. With homogeneous Dirichlet or Neumann conditions prescribed on the lateral surface of the cylinder, it is shown that for fixed finite time and under certain restrictions on the coefficients, solutions decay point-wise as the distance d from the finite end of the cylinder tends to infinity at least of order e–kd2. Under less restrictive conditions, it is shown that solutions decay in L2 at least as fast as e–kd. In both cases, k is a computable function of time.

Keywords: cross-diffusive problem; spatial decay; energy bounds.


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