Reaction-diffusion-advecton systems arising from mathematical biology modeling chemotaxis - Part I of III
For Part II, see MS-Tu-E-17
For Part III, see MS-We-D-17

Date: August 11
Time: 13:30--15:30
Room: 205B

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Xiang, Tian (Renmin Univ. of China)

Abstract: As with all living organisms, single cells and bacteria sense and respond to the environment where they live. The primary way these organisms achieve this is through the phenomenon of chemotaxis. Chemotaxis is the oriented movement of cells and organisms along chemical gradients, as a response to gradients of the concentration of chemical substances. It plays a significant role in many biological fields, and chemotaxis models have been successfully applied to the aggregation patterns in bacteria, slime molds, skin pigmentation patterns, angiogenesis in tumor progression and wound healing and many other examples. Therefore, a huge number of works, both theoretical and experimental, have been devoted to exploring and hence understanding the mechanistic basis of chemotaxis.

In 1953, Patlak contributed the first mathematical idea to model chemotaxis. In 1970s, Keller and Segel introduced a classical and important chemotaxis model ( a advection-diffusion type parabolic-parabolic quasi-linear PDE systems) to describe the aggregation process of cellular slime mold by chemical attractions. These pioneering works have initiated an intensive mathematical investigation of the (Patlak-)Keller-Segel model and chemotaxis models have become one of the best study models in mathematical biology over the last 40 years.

Despite its simple looking, the Keller-Segel model exhibits the phenomenon of cell aggregation, which is usually modeled by time-dependent solutions blowing up in finite or infinite time. Thus, the issue whether or not the solutions of the proposed chemotaxis models are globally bounded or blow-up becomes the main concern in studying K-S type models. It is a very active research subject; up to now, there are at least 5 beautiful survey papers, Horstsmann [1,2], Hillen and Painter [3], Wang [4] and Blanchet [5], where one is provided with a broad survey on the progress of various chemotaxis models as well as with a rich selection of references. The key phenomena are: no blow-up in 1-D, except in some extreme nonlinear diffusion models, critical mass blow-up in 2-D, and generic blow-up in $\geq 3$-D, a breakthrough made in Winkler [6].

Chemotaxis phenomenon has been also successfully applied to other equations, for instance, Navier-Stokes equations, see [7] for a glimpse.

Thus, in our mini-symposium, our group topics center mainly on reaction-diffusion-advecton systems modeling chemotaxis arising from mathematical biology. We bring together active researchers to share and discuss their very recent results on boundedness versus blow-up, critical mass blow-up, global existences, stability and large time behavior so as to understand more insights on the mechanism of chemotaxis. This mini-symposium will definitely stimulate more inspirations.
[1] D. Horstman, From 1970 until now: the Keller-Segal model in chaemotaxis and its consequence I, Jahresber DMV, 105 (2003), 103--165.
[2] D. Horstman, From 1970 until now: the Keller-Segal model in chaemotaxis and its consequence II, Jahresber DMV, 106 (2003), 51--69.
[3]T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183--217.
[4] Z. A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), 601--641.
[5] A. Blanchet, On the Parabolic-Elliptic Patlak-Keller-Segel System in Dimension 2 and Higher, preprint, arXiv:1109.1543
[6] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl. 100 (2013), 748--767.
[7] R. J. Duan and Z.Y. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion. Int. Math. Res. Not. IMRN 2014, no. 7, 1833-1852.

Boundedness and global existence for the Keller-Segel chemotaxis systems with/without growth
Xiang, Tian (Renmin Univ. of China)

The Existence and Stability of Nontrivial Steady States for S-K-T Competition Model with Cross-Diffusion
Qian, Xu (Beijing union Univ.)

Spiky steady states of a chemotaxis model with singular sensitivity
Li, Huicong (East China Normal Univ.)

Xiang, Zhaoyin (Univ. of Electronic Sci. & Tech. of China)


Code: Type-Date-Time-Room No.
Type : IL=Invited Lecture, SL=Special Lectures, MS=Minisymposia, IM=Industrial Minisymposia, CP=Contributed Papers, PP=Posters
Date: Mo=Monday, Tu=Tuesday, We=Wednesday, Th=Thursday, Fr=Friday
Time : A=8:30-9:30, B=10:00-11:00, C=11:10-12:10, BC=10:00-12:10, D=13:30-15:30, E=16:00-18:00, F=19:00-20:00, G=12:10-13:30, H=15:30-16:00
Room No.: TBA