Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source

By C. Yang, Xinru Cao, Zhaoxin Jiang, Sining Zheng. 2015

92 citations recorded by the source corpus.

This paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of logistic type $ u_t=\nabla\cdot(\phi(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+g(u)$, $\tau v_t=\Delta v-v+u$ in $\Omega\times (0,T)$, subject to nonnegative initial data and homogeneous Neumann boundary condition, where $\Omega$ is smooth and bounded domain in $\mathbb{R}^n$, $n\ge 2$, $\phi$ and $g$ are smooth and positive functions satisfying $ks^p\le\phi$ when $s\ge s_0>1$, $g(s) \le as - \mu s^2$ for $s>0$ with $g(0)\ge0$ and constants $a\ge 0$, $\tau,\chi,\mu>0$. It was known that the model without the logistic source admits both bounded and unbounded solutions, identified via the critical exponent $\frac{2}{n}$. On the other hand, the model is just a critical case with the balance of logistic damping and aggregation effects, for which the property of solutions should be determined by the coefficients involved. In the present paper it is proved that there is $\theta_0>0$ such that the problem admits global bounded classical solutions, regardless of the size of initial data and diffusion whenever $\frac{\chi}{\mu}<\theta_0$. This shows the substantial effect of the logistic source to the behavior of solutions.

Subfields: Modeling and Simulation, Molecular Biology, Oncology

View Paper: https://www.semanticscholar.org/paper/b34bfa22ca9a0b452d0e8b01744dc3ba197f045c

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Citations in the corpus

A note on boundedness of solutions to a higher‐dimensional quasi–linear chemotaxis system with logistic source (2017, background): “[15] showed that global bounded solutions exist whenever β = 1 and χ b is sufficiently small.”
How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system? (2016, methodology): “Motivated by the works [34, 44, 53, 63], we attempt to provide a quantitative description of the competition between logistic damping and other ingredients, especially, chemotactic aggregation, and, in particular, we aim to find a full picture on how the lower bound μ0 of the logistic damping rate μ”
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], ”
Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity (2017, methodology): “The main ingredient in this paper heavily relies on the maximal Sobolev regularity with time weighted function, as obtained in [4, 18].”