Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source

By Xie Li, Zhaoyin Xiang. 2015

64 citations recorded by the source corpus.

In this paper, we investigate the quasilinear Keller-Segel equations (q-K-S): \[ \left\{ \begin{split} &n_t=\nabla\cdot\big(D(n)\nabla n\big)-\nabla\cdot\big(\chi(n)\nabla c\big)+\mathcal{R}(n), \qquad x\in\Omega,\,t>0,\\ &\varrho c_t=\Delta c-c+n, \qquad x\in\Omega,\,t>0, \end{split} \right. \] under homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$. For both $\varrho=0$ (parabolic-elliptic case) and $\varrho>0$ (parabolic-parabolic case), we will show the global-in-time existence and uniform-in-time boundedness of solutions to equations (q-K-S) with both non-degenerate and degenerate diffusions on the non-convex domain $\Omega$, which provide a supplement to the dichotomy boundedness vs. blow-up in parabolic-elliptic/parabolic-parabolic chemotaxis equations with degenerate diffusion, nonlinear sensitivity and logistic source. In particular, we improve the recent results obtained by Wang-Li-Mu (2014, Disc. Cont. Dyn. Syst.) and Wang-Mu-Zheng (2014, J. Differential Equations).

Subfields: Cancer Research, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/6e133356756a8482754504521ea4faae11582d7e

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Citations in the corpus

Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation (2018, background): “1), inter alia accounting for nonlinear cell diffusion ([51], [6], [41], [55], [58]), for variants in the cross-diffusive interaction ([53], [20], [4], [39], [57]) or also in the signal evolution, or even containing couplings to further mechanisms and components such as haptotactic cues ([5], [40], ”
How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system? (2016, background): “So far, there are a couple of existing ways to handle these boundary integrals, see [22, 29, 44, 64], for instance.”
Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux (2017, background): “On the other hand, for general m ∈ R, the existence of global bounded solutions can be ensured by m + α > 2 − 2 d (see [5,6,16,19,22,27] and references therein).”