FREE BOUNDARY PROBLEMS IN PDE

July 6-31, 2015

Workshop Schedule:

“偏微分方程的自由边界问题”

201576日至731

 第一周 （7月6日—7月12日） 7月6日 星期一 7月7日 星期二 7月8日 星期三 7月9日 星期四 7月10日 星期五 7月11日 星期六 7月12日 星期日 上午 9:30-11:30 胡钡 胡钡 胡钡 胡钡 下午 14:30-16:30 王存新 陶有山 陶有山 陈新富

 第二周（7月13日—7月19日） 7月13日 星期一 7月14日 星期二 7月15日 星期三 7月16日 星期四 7月17日 星期五 7月18日 星期六 7月19日 星期日 上午 9:30-11:30 胡钡 胡钡 Steve Cantrell （9-12pm） 胡钡 胡钡 下午 14:30-16:30 陈新富 娄本东 娄本东 娄本东 娄本东

 第三周（7月20日—7月26日） 7月20日 星期一 7月21日 星期二 7月22日 星期三 7月23日 星期四 7月24日 星期五 7月25日 星期六 7月26日 星期日 上午 9:30-11:30 胡钡 胡钡 Steve Cantrell (9-12pm) 胡钡 胡钡 下午 14:30-16:30 崔尚斌 崔尚斌 林建洋 林支桂 林支桂

 第四周（7月27日—7月31日） 7月27日 星期一 7月28日 星期二 7月29日 星期三 7月30日 星期四 7月31日 星期五 上午 9:30-11:30 胡钡 胡钡 全天会议 胡钡 胡钡 下午 14:30-16:30 林建洋

Nonlinear diffusion, resource matching and evolutionary implications

Steve Cantrell

University of Miami

Abstract: In these talks we will explore the evolutionary advantage that a nonlinear diffusive dispersal strategy obtains relative to a purely random dispersal strategy in bounded focal habitat as the nonlinear diffusion more and more closely matches the heterogeneous background resource pattern. In ecological terms, this means that the nonlinear diffusive strategy approximates one that leads to a so-called ideal free distribution. The first talk will focus on the larger mathematical and ecological-evolutionary context and the analysis of the single equation model with nonlinear diffusion. The second lecture will pick up where the first leaves off and will examine the competitive interaction of two ecologically identical competitors employing the two different dispersal strategies.

These lectures are based upon the following papers.

1. R.S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion equations, Journal of Differential Equations 245 (2008), 3687-3703

2. R.S. Cantrell, C. Cosner, Y. Lou and C. Xie,  Random dispersal versus fitness dependent dispersal, Journal of Differential Equations 254 (2013), 2905-2941

3. Y. Lou, Y. Tao and M. Winkler, Approximating the ideal free distribution in two species competition models with fitness-dependent dispersal, SIAM Journal on Mathematical Analysis 46 (2014), 1228-1262

Analysis of free boundary problems modeling tumor growth

（肿瘤生长自由边界问题的数学分析）

Shangbin Cui（崔尚斌）

Zhongshan University

Abstract: 描述肿瘤生长的偏微分方程自由边界问题，是1970年代最初出现、1990年之后蓬勃兴起的一个饶有兴味的数学课题，对于这类问题的严谨数学分析，需要应用反应扩散方程、非线性泛函分析、Fourier分析、微分几何等多个数学分支的理论和成果。在本讲座中我们将首先对肿瘤生长自由边界问题的生物学背景做简单介绍，然后以两类肿瘤生长模型为例，说明如何应用上述各领域的理论解决肿瘤生长自由边界问题，获得其解的长时间性态的有关信息的。

King-Yeung Lam (林建洋)

Ohio State University

Lecture 1  (722, 14:30-16:30)

Title: A mutation-selection model for evolution of random dispersal.

Abstract: We consider a mutation-selection model of a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We focus on the asymptotic profile of positive steady state solutions. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. [Hastings, Theor. Pop. Biol. 24, 244-251, 1983] and [Dockery et al., J. Math. Biol. 37, 61-83, 1998] showed that for two competing species in spatially heterogeneous but temporally constant environment, the slower diffuser always prevails, if all other things are held equal. Our result suggests that their findings may hold for arbitrarily many traits. This is joint work with Yuan Lou (Renmin Univ. and Ohio State).

Lecture 2  (727, 14:30-16:30)

Title: On Global Dynamics of Competitive Systems in Ordered Banach Spaces

Abstract: A well-known result in [Hsu-Smith-Waltman, Trans. AMS (1996)] states that in a competitive semiflow defined on the product of two cones in respective Banach spaces, one of the following outcomes is possible for the two competitors: either there is at least one stable coexistence steady state, or else one of the exclusion states attracts all trajectories initiating in the order interval bounded by the two exclusion states. However, none of the exclusion states can be globally asymptotically stable if we broaden our scope to the entire positive cone. In this talk, we discuss two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of the exclusions states. Our results complement the counter example mentioned in the above paper and are frequently applicable in practice. This is joint work with Dan Munther (Cleveland State).

Spreading fronts of invasive species and diseases

（种群扩张和传染病蔓延的边沿）

Zhigui Lin （林支桂）

Yangzhou University

Abstract: This talk deals with a diffusive logistic model with a free boundary. We aim to use the dynamics of such a problem to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. We prove a spreading-vanishing dichotomy for this model, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or it fails to establish and dies out in the long run. Moreover, we show that when spreading occurs, for large time, the expanding front moves at a constant speed. This spreading speed is uniquely determined by an elliptic problem induced from the original model and less than the minimal wave speed of travelling wave front We also consider an SIS epidemic model with free boundary, this model describes the transmission of diseases. The behavior of positive solutions to a reaction-diffusion system in a radially symmetric domain are investigated. Sufficient conditions for the disease vanishing are given. Our result shows that the disease will spread to the whole area if $R_{0} \geq 1$, while if $R_{0}<1$, whether the disease is vanishing or spreading depends on the initial value of the infective.

Bendong Lou （娄本东）

Tongji University

Abstract:  Reaction-diffusion-advection equations with free boundaries can be used to describe the spreading of a new or invasive biological or chemical species in an environment with advection, with the free boundaries representing the expanding fronts. In this talk, I will introduce our recent studies on such a problem. First, I will present a derivation of the model by taking segregation limit in a competition system. Then I will introduce some useful comparison principles and zero number results. Finally, I will study the dynamics of the solutions, in particular, the influence of the advection parameter on the long time behaviour of the solutions. It turns out that, compared with the Cauchy problem and the problems in bounded domains, the free boundary problems admit some new and interesting results, which are expected to be used to explain the phenomena in biology.

Lecture 1. 带自由边界条件的反应扩散方程的推导

Lecture 2. 零点性质及其应用

Lecture 3. 带自由边界条件的反应扩散方程解的定性性质

Lecture 4. 对流对反应扩散方程自由边界问题的影响

Youshan Tao (陶有山)

Department of Applied Mathematics, Donghua University, Shanghai 200051

Lecture 1  (78, 14:30-16:30)

Title: On a fully parabolic chemotaxis-growth system

Abstract: This lecture will give an essential but simple proof of the boundedness of solutions to a parabolic-parabolic chemotaxis system with logistic source in multi-dimensional settings.

Lecture 2 (710, 14:30-15:30)

Title: Review on a chemotaxis-haptotaxis system

Abstract: This talk addresses a chemotaxis-haptotaxis model for cancer invasion, which describes the mutual interactions between cancer cells, enzymes and extracellular matrix. The system consists of two parabolic PDEs, one of which possesses two cross-diffusion terms reflecting the biased movements of cells due to chemotaxis and haptotaxis, coupled with an ODE. Inspired by some new observations or approaches toward this system, we could discuss the boundedness and asymptotic behavior of the solutions. This is a joint work with Michael Winkler (Paderborn).

Lecture 3 (710, 15:30-16:30)

Title: Boundedness and decay of classical solutions in a multi-dimensional chemotaxis-fluid system

Abstract: This talk addresses a newly proposed Keller-Segel-Navier-Stokes system modeling the phenomenon of broadcast spawning, such as the coral spawning, in which eggs release a chemical that attracts sperm. We study basic mathematical features of such a model for chemotaxis-fluid interaction. More precisely, under some explicit parameter conditions, the boundedness and decay of a classical solution to the corresponding initial-boundary problem is explored in two- and three-dimensional settings. This is a joint work with Michael Winkler (Paderborn).

Cunxin Wang (王存新)

Workshop on “PDEs with Applications”

July 29, 2015, 8:00am—5:20pm

Room 1405, Renmin University of China

 Morning Afternoon Chair:  Bei Hu Chair:  Zejia Wang 8:00-8:30,  Steve Cantrell 8:30-9:00,  Chris Cosner 9:00-9:30,  Renhao Cui 9:30-10:00,  Xinqi Gong 1:30-2:00,  Yingdong Liu 2:00-2:30,  Tianjia Ni 2:30-3:00,  Yaobin Ou 3:00-3:20,  Ningkui Sun Break Chair:  Steve Cantrell Chair:  Yuan Lou 10:10-10:40,  Litao Han 10:40-11:00,  Yaodang Huang 11:00-11:30,  Xiulan Lai 11:30-12:00,  King-Yeung Lam 3:30-4:00,  Yifu Wang 4:00-4:30,  Zejia Wang 4:30-4:50,  Yuan Wu 4:50-5:20,  Tian Xiang Lunch Break

Organizers:  Steve Cantrell (University of Miami & RUC), Bei Hu (University of Notre Dame & RUC), Yuan Lou (RUC & Ohio State), Zejia Wang (Jianxi Normal University)

Host:  Institute for Mathematical Sciences,  RUC

Title and abstract

Resident-Invader Dynamics in Infinite Dimensional Dynamical Systems

Steve Cantrell

University of Miami

Abstract: We discuss an extension of the resident-invader dynamics for similar strategies from adaptive dynamics to infinite dimensional contexts, including that of reaction-diffusion equations. This is joint work with Chris Cosner and King-Leung Lam.

The reduction principle, the ideal free distribution,

and the evolution of dispersal strategies
Chris Cosner

University of Miami

Abstract: The problem of understanding the evolution of dispersal has attracted much attention from mathematicians and biologists in recent years.  For reaction-diffusion models and their nonlocal and discrete analogues, in environments that vary in space but not in time, the strategy of not dispersing at all is often convergence stable within in many classes of strategies.  This is related to a “reduction principle” which states that that in general dispersal reduces population growth rates. However, when the class of feasible strategies includes strategies that generate an ideal free population distribution at equilibrium (all individuals have equal fitness, with no net movement), such strategies are known to be evolutionarily stable in various cases.  Much of the work in this area involves using ideas from dynamical systems theory and partial differential equations to analyze pairwise invasibility problems, which are motivated by ideas from adaptive dynamics and ultimately game theory. The talk will describe some past results and current work on these topics

Strong Allee Effect in a Diusive Predator-prey System with a Protection Zone

Renhao Cui

Renmin University of China

Abstract: In this talk we consider that a reaction diusion predator prey system with strong

Allee effect and a protection zone for the prey. Dynamics and steady state solutions of the

system are analyzed. In particular it is shown that the overexploitation phenomenon can be avoided if the Allee effect threshold is low and the protection zone is large.

Computational design of protein structure, function and interaction

Xinqi Gong （龚新奇）

Institute for Mathematical SciencesRenmin University of China

Abstract: Proteins perform specific biological functions with specific 3D structures, and usually interact with other partners or change their conformations to show different functions. But in many circumstances, it’s difficult and expensive for experiments to determine a protein’s basic information of structure, function or interaction. Here I will show how our computational algorithms help to design protein structure, function and interaction. Further importantly, biological experiments validate the power of our algorithms in three examples.

Evaluating the Impact of Test-and-Treat on the HIV Epidemic among MSM in China Using a Mathematical Model

Litao Han

Renmin University of China

Abstract: Background: Various studies have modeled the impact of test-and-treat policies on the HIV epidemics worldwide. However, few modeling studies have taken into account China’s context. To understand the potential effect of test-and-treat on the HIV epidemic among MSM in China, we developed a mathematical model to evaluate the impact of the strategy. Method: Based on the natural history of the CD4 count of people living with HIV (PLHIV), we constructed a dynamic compartmental model of HIV transmission among Chinese MSM to project the number of new HIV infection and prevalence over 10 years. We predicted the annual number of new HIV infections and the total number of MSM living with HIV and AIDS (based on Beijing data) between 2010 and 2022 under the following conditions: (1) current practice (testing rate of 50% and ART coverage of 39%); (2) both testing rate and ART coverage increasing to 70% in 2013; (3) both testing rate and ART coverage increasing to 90% in 2013; and (4) testing rate and ART coverage increasing by 5% each year until 90% since 2013. Results: Based on our model, if HIV test-and-treat policy was implemented among Chinese MSM, the total number of new HIV infections over 10 years (2013-2022) would be reduced by 50.6-70.9% compared with the current practice. When ART coverage for PLHIV increases to 57.5%, the turning point would occur on the curve of HIV new infections by 2015. A 25% reduction in annual number of new HIV infections by 2015 may be achieved if the testing rate increases from 50% to 70% and treatment coverage for PLHIV increases to 55%. Conclusion: Implementation of the test-and-treat strategy may significantly reduce new HIV infections among MSM in China. Great efforts need to be made to scale up HIV testing and ART coverage among Chinese MSM.

Upper bound of blowup time for a semilinear parabolic system

with potentials and large initial data

Yaodan Huang

Xian Jiaotong University

Modeling HIV-1 virus dynamics with both cell free virus infection

and cell-to-cell transmission

Xiulan Lai

Renmin University of China

AbstractIn this talk, we propose a mathematical model to consider these two modes of viral infection and spread, direct cell-to-cell transfer of HIV-1 and virus-to-cell infection, in which infection age is also incorporated. By a rigorous analysis of the model, we show that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number, which is identified explicitly. The formula for the basic reproduction number of our model reveals that the basic reproduction number of a model that neglects either the cell-to-cell spread or

virus-to-cell infection might be under-evaluated.

On Global Dynamics of Competitive Systems in Ordered Banach Spaces

King-Yueng Lam

Ohio State University

Abstract: A well-known result in [Hsu-Smith-Waltman, Trans. AMS (1996)] states that in a competitive semiflow defined on the product of two cones in respective Banach spaces, one of the following outcomes is possible for the two competitors: either there is at least one stable coexistence steady state, or else one of the exclusion states attracts all trajectories initiating in the order interval bounded by the two exclusion states. However, none of the exclusion states can be globally asymptotically stable if we broaden our scope to the entire positive cone. In this talk, we discuss two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of the exclusions states. Our results complement the counter example mentioned in the above paper and are frequently applicable in

practice. This is joint work with Dan Munther (Cleveland State).

Yingdong Liu

Beijing Jiaotong University

Abstract: 利用分支理论证明了一类SEIS传染病模型的非常数平衡解的存在性，并计算了分支方向，由此刻画了此类传染病模型的门槛现象。

Laplacian on fractals

Tianjia Ni

Renmin University of China

Abstract: On Euclidean space, diffusion of heat is described by the heat equation. If the heat transport is undertaken in some disordered media such as porous rocks, the heat equation on Euclidean space is not suitable any more. Fractal is considered as an appropriate model for geometrical structure of most disordered media. However, the differential operator Laplacian, which is used in heat equation, on fractal can not be easily defined as on Euclidean space, since fractals may not have smooth structures. In this talk, we will define the Laplacian on a particular class of fractals. The main tool is the generalized Kakutani fixed point theorem.

Global classical solutions to vacuum free boundary problem of full Navier-Stokes equations with large initial data

Yaobin Ou

Renmin University of China

Abstract: In this talk, I`ll present a recent result on the the free boundary problem of  Navier-Stokes equations for viscous ideal polytropic and heat-conducting fluids, when the density connects to the vacuum continuously. The global existence of classical solution to the problem with large initial data was established in this work.

Blow-up and Asymptotic Behavior of the Reaction-diffusion Equations

with Free Boundaries

Ningkui Sun

Tongji University

Abstract: We study the reaction-diffusion equations

in a varying domain , where g(t) and h(t) are two free boundaries.

We obtain the blowup-transition-vanishing trichotomy result. When blow-up

happens, we deduce that the blow-up set is a compact subset of initial domain

and the two free boundaries keep bounded.

Boundedness in a chemotaxis--haptotaxis model

with remodeling of non-diffusible attractant

Yifu Wang

Beijing Institute of Science and Technology

Abstract: This paper is concerned with the cancer invasion model

in a bounded smooth domain  with zero-flux boundary conditions, where  and  are positive parameters. As compared to previous mathematical studies, the novelty here consists of allowing for positive values of , reflecting processes of self-remodeling of the extracellular matrix. It is shown that under appropriate regularity assumption on the initial data , the corresponding initial-boundary problem possesses a unique classical solution which is global in time and bounded. This paper develops some Lp-estimate techniques for the full model of

Chaplain and Lolas.

Travelling Wave for Credit Rating Migration Problem

Yuan Wu

Tongji University

Abstract: In this paper, a free boundary model for pricing a corporate bond with credit rating migration is proposed. The existence, uniqueness and regularity of the solution for the model are obtained. With reasonable assumptions, traveling wave solution is obtained, and the original problem is convergent to the traveling wave

solution.

Bifurcation for a free boundary problem modeling tumor growth with inhibitors

Zejia Wang

Jiangxi Normal University

AbstractIn this talk, we deal with a free boundary problem modeling tumor growth with inhibitors. This problem has a unique radially symmetric stationary solution with radius $r=R_s$. The tumor aggressiveness is modeled by a positive tumor aggressiveness parameter $\mu$. It is shown that there exist a positive integer $m^{**}\in\mathbb R$ and a sequence of $\mu_m$, such that for each $\mu_m(m>m^{**})$, symmetry-breaking solutions bifurcate from the radially symmetric stationary solutions.

On a class of Keller-Segel chemotaxis systems with cross-diffusion

Tian Xiang

Renmin University of China

AbstractWe study a class of Keller–Segel chemotaxis systems with cross-diffusion. By using the entropy dissipation method and assuming mainly the chemotactic sensitivity separates the cell density and the chemical signal, we first establish the existence of global weak solutions with the effects of cross diffusion included in ≤3-D. Then we show there is a critical cross diffusion rate δc such that no patterns may be expected for δ≥δc, while patterns are formed for δ<δc and their stability is also derived. In particular, in 1-D, patterns are always formed whenever δ<δc and the chemotactic coefficient is larger than an expressible bifurcation value, and there is another critical cross diffusion rate δccsuch that cells with cross-diffusion rate δ(δcc) are stable, while, for cells with δ<δc to be stable, their degradation rate must be less than a threshold value. Hence, in some sense, cross-diffusion is harmful to enable pattern formation, while it is helpful to stabilize the cells once patterns are formed. Finally, we show that the cross diffusion plays a role in regularizing the cell aggregation phenomenon for large chemotactic coefficient. Our results provide global dynamics and insights on how the biological parameters, especially, the cross diffusion, affect pattern formations