Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework

By Chao Deng, Tong Li. 2012

53 citations recorded by the source corpus.

Abstract We investigate global strong solution to a 3-dimensional parabolic–hyperbolic system arising from the Keller–Segel model. We establish the global well-posedness and asymptotic behavior in the energy functional setting. Precisely speaking, if the initial difference between cell density and its mean is small in L 2 , and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in H 1 , then they remain to be small in L 2 × H 1 for all time. Moreover, if the mean value of the initial cell density is smaller than some constant, then the cell density approaches its initial mean and the chemical concentration decays exponentially to zero as t goes to infinity. The proof relies on an application of Fourier analysis to a linearized parabolic–hyperbolic system and the smoothing effect of the cell density and the damping effect of the chemical concentration.

Subfields: Computational Theory and Mathematics, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/9f9fa64dace9eea9f6925bdefbf0e16595c49ea7

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Citations in the corpus

Boundary spike‐layer solutions of the singular Keller–Segel system: existence and stability (2019, background): “[3, 4, 6, 13, 21, 23–26]), global well-posedness and/or asymptotic behavior of solutions (see [5, 8, 19, 20, 22, 28, 33, 37, 42, 43] in one dimensional bounded or unbounded space and [7, 9, 18, 22, 32, 35, 39, 41] in multidimensional spaces) and boundary layer solutions [10–12].”