林可,西南财经大学

Large time behavior in a higher-dimensional attraction-repulsion chemotaxis system 

李青,首都师范大学

Stability of steady states for SKT competition system with cross-diffusion

Monotonicity of principal eigenvalue for elliptic operators with incompressible flow

Title and Abstract

    Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flow

边东芬

北京理工大学

E-mail address: biandongfen@bit.edu.cn

Abstract: This paper is devoted to studying the temperature-dependent incompressible nematic liquid crystal flows in a bounded domain . The global well-posedness for small perturbation around the trivial equilibrium state is established. This is a joint work with Dr. Yao Xiao.

    Global Solvability and Boundedness to a Coupled Chemotaxis-Fluid Model with Arbitrary Porous Medium Diffusion

金春花

华南师范大学

E-mail address: jinchhua@126.com

Abstract: We consider a coupled chemotaxis-fluid model with zero-flux boundary and no-slip boundary. It is shown that for any large initial datum, for any $m>0$, $\alpha>0$, the problem admits a global weak solution, which is uniformly bounded. On the basis of this, the stability of the steady states also be discussed.

Global stabilization of the full attraction-repulsion chemotaxis model

金海洋 

华南理工大学

  E-mail address: mahyjin@scut.edu.cn

Abstract: In this talk, we are concerned with the full attraction-repulsion chemotaxis system with different diffusion coefficients of the two chemical signal.  By constructing some appropriate Lyapunov functions, we establish the boundedness and asymptotical behaviors of solutions to the system with large initial data in a two-dimensional bounded domain under the homogeneous Neumann boundary conditions.  This is a joint work with Prof. Zhian Wang.

    Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source

Abstract: The purpose of this talk is to mathematically understand the effect of the attractive signal, the repulsive one and the logistic source on the global boundedness and asymptotic behavior of solutions to a fully parabolic attraction--repulsion chemotaxis system with logistic source in the higher-dimensional setting.

Stability of steady states for SKT competition system with cross-diffusion

首都师范大学

E-mail address: qingli_324@163.com

Abstract: In this talk, we shall concerned with a quasilinear reaction diffusion system with cross diffusion, which was first proposed by Shigeseda, Kawasaki and Teramoto in 1979 for investigating the spatial segregation of two competing species under inter- and intra-species population pressures. I will talk about some resent research progress on the existence and stability of some types of nontrivial steady states for the SKT competition model as one of the cross diffusion rate is large enough, which may correspond to some new pattern formation induced by cross diffusion. This is a joint work with Professor Yaping Wu.

Large time behavior in a higher-dimensional attraction-repulsion chemotaxis system

 E-mail address: linke@swufe.edu.cn

Abstract: In this talk we consider the long time behavior of solutions to an attraction-repulsion chemotaxis system in a bounded domain under zero-flux boundary conditions if repulsion dominates over attraction. It is known that under the assumption that repulsion balances/cancels attraction, for any suitably regular initial data all solutions of this problem will be global and bounded. We further show that if the degradation rate of a repulsive signal is smaller than the attractive one or the repulsion is suitable strong, these global solutions converge to the steady state for arbitrarily large initial data.

Monotonicity of principal eigenvalue for elliptic operators with incompressible flow

E-mail address: liushuangnqkg@ruc.edu.cn

Abstract: In this talk, we establish the monotonicity of the principal eigenvalue ,  as a function of the advection amplitude , for the elliptic operator  with incompressible flow , subject to Dirichlet, Robin and Neumann boundary conditions. As a consequence, the limit of  as always exists and is finite for  Robin boundary conditions.  These results  answer some open questions raised by Berestycki, Hamel and Nadirashvili (CMP 2005). This is a joint work with Prof. Yuan Lou.

Global Solutions to the Coupled Chemotaxis-Fluids System in a 3D Unbounded Domain with Finite Depth

 E-mail address: yingping_peng@163.com

Abstract: we will investigate the global existence of solutions to a coupled chemotaxis-fluids system in a three dimensional unbounded domain with finite depth. In the chemotaxis-Navier-Stokes case, we establish the global existence and uniqueness of strong solutions around a constant state, while in the chemotaxis-Stokes case, we show the global existence of weak solutions for large initial cell density and velocity. Our proof is based on some uniform a priori estimates obtained by using the anisotropic Lp technique and the elliptic estimates. To the best of our knowledge, this is the first analytical work for the well-posedness of chemotaxis-fluids system in an unbounded domain with boundary. Our results are consistent with the experiment observation and numerical simulation.

Fisher-KPP equation with free boundaries and time-periodic advections

E-mail address: sunnk1987@163.com

Abstract: We consider the Fisher-KPP equation with free boundaries and time-periodic advections. We are interested in the long time behavior of solutions. When the advection is small, we give a vanishing-spreading dichotomy results; when the advection is medium-sized, we show a vanishing-transition-virtual spreading trichotomy result; when the advection is large, we prove that vanishing happens for all solutions. Moreover, we give the estimate for the spreading speed of the spreading/virtual spreading solutions. There is a joint work with Bendong Lou and Maolin Zhou.

Global boundedness of solutions to a chemotaxis—haptotaxis model with tissue remodeling

Abstract: This talk is concerned with a cancer invasion model comprising a strongly coupled PDE-ODE system in two and three space dimensions. The system consists of a parabolic equation describing cancer cell migration arising from a combination of chemotaxis and haptotaxis, a parabolic/elliptic equation describing the dynamics of matrix degrading enzymes (MDE), and an ODE describing the evolution and re-modeling of the extracellular matrix (ECM). We point out that this strongly coupled PDE-ODE setup presents new mathematical difficulties, which are overcome by developing new integral estimate techniques.We prove that the system admits a unique global classical solution which is uniformly bounded in time in the two-dimensional spatial setting at all cancer cell proliferation rates. We also prove that, in the case of three-dimensional convex spatial domain, when cancer cell proliferation is suitably small, the system also possesses a unique classical solution for appropriately small initial data. These results improve previously known ones.

On a two-species chemotaxis system with two chemicals and competitive kinetics

Abstract: This work is devoted to the  two-species chemotaxis system with two chemicals and Lotka-Volterra competitive kinetics. There are few results on chemotaxis system with two different chemicals. Recently, in the absence of population competition,  the global existence of the above system were established in the 2-dimensional case for parabolic-elliptic case  (Tao & Winkler, 2015,DCDS-B) and parabolic-parabolic case (Li & Wang, 2016, DCDS-B) respectively. The purpose of this work is firstly to give global existence and boundedness of solutions to the fully parabolic system with Lotka-Volterra competitive kinetics, and then, under some conditions on the parameters, to establish the large time behavior of solutions.

Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

赵相东

大连理工大学

E-mail address: Zhaoxd_521@163.com

Abstract: We consider a chemotaxis consumption system with singular sensitivity $u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v^\alpha}\nabla v)$, $v_t=\epsilon\Deltav-uv$ in a bounded domain $\Omega\subset\mathbb{R}^n$ with $\chi,\alpha,\epsilon>0$. It is proved that there exists a global classical solution with $n=1$. In particular, for any global classical solution $(u,v)$ to the case of $\alpha\ge 1$, it is shown that $v$ converses to $0$ in the $L^\infty$-norm as $t\rightarrow\infty$ with the decay rate established whenever $\epsilon\in(0,1)$ appropriately large, rather than those in the $L^p$-sense as generally obtained.

Persistence and boundedness of solutions for two-speices chemotaxis model with two signals

 E-mail address: zhengpan@cqupt.edu.cn

Abstract: In this talk, we consider a two-competing-species chemotaxis system with two different chemicals under homogeneous Neumann boundary conditions in a smooth bounded domain. Firstly, when , based on some a priori estimates and Moser-Alikakos iteration, it is shown that regardless of the size of initial data, for any positive parameters, the system possesses a unique globally bounded classical solution if $n=2$. On the other hand, when , relying on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that $\mu_{1}$ and $\mu_{2}$ are suitably large. Finally, under the noncompetition case, we study the persistence property of mass for this model.