Partial differential equations
A note on the variational analysis of the parabolic–parabolic Keller–Segel system in one spatial dimension
Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 849-854.

We prove the existence of global-in-time weak solutions to a version of the parabolic–parabolic Keller–Segel system in one spatial dimension. If the coupling of the system is suitably weak, we prove the convergence of those solutions to the unique equilibrium with an exponential rate. Our proofs are based on an underlying gradient flow structure with respect to a mixed Wasserstein-L2 distance.

Nous prouvons l'existence de solutions faibles globales en temps d'une variante du système de Keller–Segel parabolique–parabolique à une dimension spatiale. Si le couplage du système est assez faible, nous prouvons la convergence de ces solutions vers l'équilibre univoque à une vitesse exponentielle. Nos preuves reposent sur une structure de flux de gradient dans l'espace produit des espaces Wasserstein et L2.

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DOI: 10.1016/j.crma.2015.06.014
Zinsl, Jonathan 1

1 Zentrum für Mathematik, Technische Universität München, 85747 Garching, Germany
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Zinsl, Jonathan. A note on the variational analysis of the parabolic–parabolic Keller–Segel system in one spatial dimension. Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 849-854. doi : 10.1016/j.crma.2015.06.014. http://www.numdam.org/articles/10.1016/j.crma.2015.06.014/

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