Uniqueness for Keller-Segel-type chemotaxis models

By José Antonio Carrillo, S. Lisini, Edoardo Mainini. 2012

50 citations recorded by the source corpus.

We prove uniqueness in the class of integrable and bounded nonnegative solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our proof works for the fully parabolic KS model, it includes the classical parabolic-elliptic KS equation as a particular case, and it can be generalized to nonlinear diffusions in the particle density equation as long as the diffusion satisfies the classical McCann displacement convexity condition. The strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and the above-the-tangent characterizations of displacement convexity. As a consequence, the displacement convexity of the free energy functional associated to the KS system is obtained from its evolution for bounded integrable initial data.

Subfields: Computational Theory and Mathematics, Modeling and Simulation, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/49283590c46e28a42bc07054d32beb960399b58a

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Citations in the corpus

Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case (2013, background): “3 improves the uniqueness result proved in [20] in the class of solutions f ∈ C([0, T ]; L(1)2(R (2))) ∩ L((0, T ) × R(2)) which can be built under the additional assumption f0 ∈ L(R(2)) (see also [24] where a uniqueness result is established for a related model).”
Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane (2014, background): “). To the best of our knowledge, the contraction property for the distance between two solutions of (1.1)-(1.2) was previously proved in [7] for the parabolic-elliptic radially symmetric case, and in [15], in the context of the gradient ow formulation of (1.1)-(1.2), for initial data u 0 2(L1 \L1)(R”