Asymptotic Estimates for the Parabolic-Elliptic Keller-Segel Model in the Plane

By J. Campos, Jean Dolbeault. 2012

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We investigate the large-time behavior of the solutions of the two-dimensional Keller-Segel system in self-similar variables, when the total mass is subcritical, that is less than 8 π after a proper adimensionalization. It was known from previous works that all solutions converge to stationary solutions, with exponential rate when the mass is small. Here we remove this restriction and show that the rate of convergence measured in relative entropy is exponential for any mass in the subcritical range, and independent of the mass. The proof relies on symmetrization techniques, which are adapted from a paper of Diaz et al. and allow us to establish uniform estimates for L p norms of the solution. Exponential convergence is obtained by the mean of a linearization in a space which is defined consistently with relative entropy estimates and in which the linearized evolution operator is self-adjoint. The core of proof relies on several new spectral gap estimates which are of independent interest.

Subfields: Finance, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/cb99874fb93d3f9755fe6de7f22de12d87e31b3c

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Citations in the corpus

Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, methodology): “We should point out that the result of (v) depends on the variant of the spectral study [17] in Appendix A.”
Uniqueness for Keller-Segel-type chemotaxis models (2012, background): “]. The same techniques could eventually be used to get local in time bounded solutions for all masses, although such a result is not present in the literature. Let us also mention the recent preprint [17] in which the authors actually show that the L∞-norm of the solution decays in time like for the”
Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case (2013, methodology): “In that last work [17], the uniform exponential stability (with optimal rate) of the linearized rescaled equation is established in L(2)(G) by the mean of the analysis of the associated linear operator in a well chosen (equivalent) Hilbert norm.”
Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime (2020, background): “(As to the corresponding asymptotic analysis we refer to [6, 10, 13].”
Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane (2014, methodology): “in the case of the non-linear heat equation [17] and of a convection-diusion equation [18]. This is also the case for the two dimensional parabolic-elliptic Keller-Segel system with = 0 and M8ˇ(see [7, 8, 9, 10, 13, 14]). The case of the doubly parabolic Keller-Segel system with "= 1, has been ”