Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity

By Youshan Tao. 2013

61 citations recorded by the source corpus.

This paper deals with the repulsion chemotaxis system $$ \left\{ \begin{array}{ll} u_t=\Delta u +\nabla \cdot (f(u)\nabla v), & x\in\Omega, \ t>0, \\ v_t=\Delta v +u-v, & x\in\Omega, \ t>0, \end{array} \right. $$ under homogeneous Neumann boundary conditions in a smooth bounded convex domain $\Omega\subset\mathbb{R}^n$ with $n\ge 3$, where $f(u)$ is the density-dependent chemotactic sensitivity function satisfying $$ f \in C^2([0, \infty)), f(0)=0, 0 0 $$ with some $K>0$ and $\alpha>0$. It is proved that under the assumptions that $0\not\equiv u_0\in C^0(\bar{\Omega})$ and $v_0\in C^1(\bar{\Omega})$ are nonnegative and that $\alpha<\frac{4}{n+2}$, the classical solutions to the above system are uniformly-in-time bounded. Moreover, it is shown that for any given $(u_0, v_0)$, the corresponding solution $(u, v)$ converges to $(\bar{u}_0, \bar{u}_0)$ as time goes to infinity, where $\bar{u}_0 :=\frac{1}{\Omega} f_{\Omega} u_0$.

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/a6ec10464f165222b2294bd39827f93f64b79aaa

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Citations in the corpus

Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals (2015, background): “also [33] for boundedness results in a repulsion system with general nonlinear cross-diffusion).”
A note on boundedness of solutions to a higher‐dimensional quasi–linear chemotaxis system with logistic source (2017, background): “[7] Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin.”
Global boundedness in a fully parabolic attraction–repulsion chemotaxis model (2015, background): “Finally, we mention that if the space dimension is three or four, then at least certain global weak solutions can be constructed for the full parabolic repulsion chemotaxis model [13] and that the boundedness of solutions to a repulsion chemotaxis system with general nonlinear cross-diffusion in hig”
Boundedness, blowup and critical mass phenomenon in competing chemotaxis (2016, unknown): “The results on the repulsive Keller–Segel model are very limited and a further study on such model was recently given in [36].”