A quasi-linear parabolic system of chemotaxis

By T. Senba, Takasi Suzuki. 2006

65 citations recorded by the source corpus.

We consider a quasi-linear parabolic system with respect to unknown functions u and v on a bounded domain of n-dimensional Euclidean space. We assume that the diffusion coefficient of u is a positive smooth function A(u), and that the diffusion coefficient of v is a positive constant. If A(u) is a positive constant, the system is referred to as so-called Keller-Segel system. In the case where the domain is a bounded domain of two-dimensional Euclidean space, it is shown that some solutions to Keller-Segel system blow up in finite time. In three and more dimensional cases, it is shown that solutions to so-called Nagai system blow up in finite time. Nagai system is introduced by Nagai. The diffusion coefficients of Nagai system are positive constants. In this paper, we describe that solutions to the quasi-linear parabolic system exist globally in time, if the positive function A(u) rapidly increases with respect to u.

Subfields: Modeling and Simulation, Molecular Biology, Oncology

View Paper: https://www.semanticscholar.org/paper/3a637261b5fe2753472e87166c3c58a00e0c1fd0

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Citations in the corpus

Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions (2011, background): “This result is optimal in view of the result in [16] guaranteeing global existence of bounded solutions to (1.”
Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system (2017, background): “For instance, quite a comprehensive understanding could be achieved in the case when D = D(u) > 0 and χ = χ(u) ≥ 0 are suitably smooth, independent of the signal concentration v = v(x, t) and exclusively depending on the population density u = u(x, t): Then, namely, known results indicate that essen”
Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux (2017, background): “3) admits the blow-up solutions under some technical assumptions (see [7, 18, 20,22,27,28] and references therein).”
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system (2011, unknown): “[32], [22], [23], [24], [9], [4], [19], [33] and also the survey [16]).”