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Publication 08-CNA-15

Contraction in $L^1$ and large time behavior for a system arising in chemical reactions and molecular motors

Michel Chipot
Universitaet Zurich
Angewandte Mathematik
Zurich CH-8057
Switzerland
m.m.chipot@math.unizh.ch

Danielle Hilhorst
CNRS and Université de Paris-Sud
Laboratoire de Mathématiques
91405 Orsay, Cédex, France
Danielle.Hilhorst@math.u-psud.fr

David Kinderlehrer
Carnegie Mellon University
Department of Mathematical Sciences
Pittsburgh, PA 15213
davidk@andrew.cmu.edu

Michał Ołech
Uniwersytetu Wrocławskiego
Instytut Matematyczny
pl. Grunwaldzki 2/4
50-384 Wroclaw, Polska
olech@math.uni.wroc.pl
and
CNRS


Abstract: We prove a contraction in $L^1$ property for the solutions of a nonlinear reaction-diffusion system whose special cases include intercellular transport as well as reversible chemical reactions. Assuming the existence of stationary solutions we show that the solutions stabilize as $t$ tends to infinity. Moreover, in the special case of linear reaction terms, we prove the existence and the uniqueness (up to a multiplicative constant) of the stationary solution.

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