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Asymptotic behavior of solutions to a chemotaxis-logistic model with transitional end-states.

Asymptotic behavior of solutions to a chemotaxis-logistic model with transitional end-states.

Authors :
Zeng, Yanni
Zhao, Kun
Source :
Journal of Differential Equations. Nov2022, Vol. 336, p1-43. 43p.
Publication Year :
2022

Abstract

We study Cauchy problem of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. Our Cauchy data connect two different end-states for the chemical signal while the cell density takes its typical carrying capacity at the far fields. We are interested in the time-asymptotic behavior of the solution. We show that in the borderline, the component representing the chemical signal converges to a permanent, diffusive background wave, which connects the two end-states monotonically. On the other hand, the cell component converges to the spatial derivative of a heat kernel. The asymptotic solution has explicit formulation and is common to all solutions sharing the same end-states. Optimal L 2 and L ∞ convergence rates are obtained. We first convert the model into a 2 × 2 hyperbolic-parabolic system via inverse Hopf-Cole transformation. Then we apply Chapman-Enskog expansion to identify the asymptotic solution. After extracting the asymptotic solution, we use a variety of analytic tools to study the remainder and obtain optimal rates. These include time-weighted energy method, spectral analysis, Green's function estimate and iterations. Our results apply to a general class of Cauchy data for the model and for its transformed system. In particular, our results apply to large data solutions. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
00220396
Volume :
336
Database :
Academic Search Index
Journal :
Journal of Differential Equations
Publication Type :
Academic Journal
Accession number :
158727119
Full Text :
https://doi.org/10.1016/j.jde.2022.07.013