Existence and nonexistence of positive radial solutions for quasilinear systems
Haiyan Wang
Conference Publications 2009, 2009(Special): 810-817 doi: 10.3934/proc.2009.2009.810
The paper deals with the existence and nonexistence of positive radial solutions for the weakly coupled quasilinear system div$( | \nabla u|^{p-2}\nabla u ) + \lambda f(v)=0$, div $( | \nabla v|^{p-2}\nabla v ) + \lambda g(u)=0$ in $\B$, and $\u =v=0$ on $\partial B,$ where $p>1$, $B$ is a finite ball, $f$ and $g$ are continuous and nonnegative functions. We prove that there is a positive radial solution for the problem for various intevals of $\lambda$ in sublinear cases. In addition, a nonexistence result is given. We shall use fixed point theorems in a cone.
keywords: p-Laplace operator positive radial solution cone
Spatial dynamics for a model of epidermal wound healing
Haiyan Wang Shiliang Wu
Mathematical Biosciences & Engineering 2014, 11(5): 1215-1227 doi: 10.3934/mbe.2014.11.1215
In this paper, we consider the spatial dynamics for a non-cooperative diffusion system arising from epidermal wound healing. We shall establish the spreading speed and existence of traveling waves and characterize the spreading speed as the slowest speed of a family of non-constant traveling wave solutions. We also construct some new types of entire solutions which are different from the traveling wave solutions and spatial variable independent solutions. The traveling wave solutions provide the healing speed and describe how wound healing process spreads from one side of the wound. The entire solution exhibits the interaction of several waves originated from different locations of the wound. To the best of knowledge of the authors, it is the first time that it is shown that there is an entire solution in the model for epidermal wound healing.
keywords: non-cooperative diffusion systems epidermal wound healing. entire solution Traveling waves spreading speed
Partial differential equations with Robin boundary condition in online social networks
Guowei Dai Ruyun Ma Haiyan Wang Feng Wang Kuai Xu
Discrete & Continuous Dynamical Systems - B 2015, 20(6): 1609-1624 doi: 10.3934/dcdsb.2015.20.1609
In recent years, online social networks such as Twitter, have become a major source of information exchange and research on information diffusion in social networks has been accelerated. Partial differential equations are proposed to characterize temporal and spatial patterns of information diffusion over online social networks. The new modeling approach presents a new analytic framework towards quantifying information diffusion through the interplay of structural and topical influences. In this paper we develop a non-autonomous diffusive logistic model with indefinite weight and the Robin boundary condition to describe information diffusion in online social networks. It is validated with a real dataset from an online social network, The simulation shows that the logistic model with the Robin boundary condition is able to more accurately predict the density of influenced users. We study the bifurcation, stability of the diffusive logistic model with heterogeneity in distance. The bifurcation and stability results of the model information describe either information spreading or vanishing in online social networks.
keywords: diffusive logistic model indefinite weight Robin boundary condition. Bifurcation online social networks stability
Positive radial solutions for quasilinear equations in the annulus
Haiyan Wang
Conference Publications 2005, 2005(Special): 878-885 doi: 10.3934/proc.2005.2005.878
The paper deals with the existence of positive radial solutions for the quasilinear system $\textrm{ div} \left ( | \nabla u_i|^{p-2}\nabla u_i \right ) + f^i(u_1,...,u_n)=0,\; p>1, R_1 <|x| < R_2,\;u_i(x)=0,$ on $|x|=R_1$ and $R_2$, $i=1,...,n$, $x \in \mathbb{R}^N.$ $f^i$, $i=1,...,n,$ are continuous and nonnegative functions. Let $\vect{u}=(u_1,...,u_n),$ $\varphi(t)=|t|^{p-2}t,$ $f_0^i =\lim_{\norm{\vect{u}} \to 0} \frac{f^i(\vect{u})}{\var(\norm{\vect{u}})},$ $f_{\infty}^i =\lim_{\norm{\vect{u}} \to \infty} \frac{f^i(\vect{u})}{\var(\norm{\vect{u}})}$, $i=1,...,n,$ $\vect{f}=(f^1,...,f^n),$ $\vect{f}_0=\sum_{i=1}^n f_0^i$ and $\vect{f}_{\infty}=\sum_{i=1}^n f_{\infty}^i$. We prove that $\vect{f}_0 =0$ and $\vect{f}_{\infty}=\infty$ (superlinear) guarantee the existence of positive radial solutions for the system. We shall use fixed point theorems in a cone.
keywords: p-Laplacian positive radial solution. fixed index theorem
Convex solutions of boundary value problem arising from Monge-Ampère equations
Shouchuan Hu Haiyan Wang
Discrete & Continuous Dynamical Systems - A 2006, 16(3): 705-720 doi: 10.3934/dcds.2006.16.705
In this paper we study an eigenvalue boundary value problem which arises when seeking radial convex solutions of the Monge-Ampère equations. We shall establish several criteria for the existence, multiplicity and nonexistence of strictly convex solutions for the boundary value problem with or without an eigenvalue parameter.
keywords: strictly convex solution fixed index theorem. boundary value problem Monge-Ampère equation
Spreading speeds and traveling waves for non-cooperative integro-difference systems
Haiyan Wang Carlos Castillo-Chavez
Discrete & Continuous Dynamical Systems - B 2012, 17(6): 2243-2266 doi: 10.3934/dcdsb.2012.17.2243
The study of spatially explicit integro-difference systems when the local population dynamics are given in terms of discrete-time generations models has gained considerable attention over the past two decades. These nonlinear systems arise naturally in the study of the spatial dispersal of organisms. The brunt of the mathematical research on these systems, particularly, when dealing with cooperative systems, has focused on the study of the existence of traveling wave solutions and the characterization of their spreading speed. Here, we characterize the minimum propagation (spreading) speed, via the convergence of initial data to wave solutions, for a large class of non cooperative nonlinear systems of integro-difference equations. The spreading speed turns out to be the slowest speed from a family of non-constant traveling wave solutions. The applicability of these theoretical results is illustrated through the explicit study of an integro-difference system with local population dynamics governed by Hassell and Comins' non-cooperative competition model (1976). The corresponding integro-difference nonlinear systems that results from the redistribution of individuals via a dispersal kernel is shown to satisfy conditions that guarantee the existence of minimum speeds and traveling waves. This paper is dedicated to Avner Friedman as we celebrate his immense contributions to the fields of partial differential equations, integral equations, mathematical biology, industrial mathematics and applied mathematics in general. His leadership in the mathematical sciences and his mentorship of students and friends over several decades has made a huge difference in the personal and professional lives of many, including both of us.
keywords: biological invasion traveling wave Minimum speed dispersal integro-difference systems.
Response of yeast mutants to extracellular calcium variations
Pamela A. Marshall Eden E. Tanzosh Francisco J. Solis Haiyan Wang
Discrete & Continuous Dynamical Systems - B 2009, 12(2): 439-453 doi: 10.3934/dcdsb.2009.12.439
We study, both experimentally and through mathematical modeling, the response of wild type and mutant yeast strains to systematic variations of extracellular calcium abundance. We extend a previously developed mathematical model (Cui and Kaandorp, Cell Calcium, 39, 337 (2006))[3], that explicitly considers the population and activity of proteins with key roles in calcium homeostasis. Modifications of the model can directly address the responses of mutants lacking these proteins. We present experimental results for the response of yeast cells to sharp, step-like variations in external $Ca^{++}$ concentrations. We analyze the properties of the model and use it to simulate the experimental conditions investigated. The model and experiments diverge more markedly in the case of mutants laking the Pmc1 protein. We discuss possible extensions of the model to address these findings.
keywords: ion storage. calcium homeostasis extracellular signaling yeast cells
Traveling waves of diffusive predator-prey systems: Disease outbreak propagation
Xiang-Sheng Wang Haiyan Wang Jianhong Wu
Discrete & Continuous Dynamical Systems - A 2012, 32(9): 3303-3324 doi: 10.3934/dcds.2012.32.3303
We study the traveling waves of reaction-diffusion equations for a diffusive SIR model. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Our proof is based on Schauder fixed point theorem and Laplace transform.
keywords: Schauder fixed point theorem Laplace transform. Traveling waves SIR model
Some recent developments on linear determinacy
Carlos Castillo-Chavez Bingtuan Li Haiyan Wang
Mathematical Biosciences & Engineering 2013, 10(5&6): 1419-1436 doi: 10.3934/mbe.2013.10.1419
The process of invasion is fundamental to the study of the dynamics of ecological and epidemiological systems. Quantitatively, a crucial measure of species' invasiveness is given by the rate at which it spreads into new open environments. The so-called ``linear determinacy'' conjecture equates full nonlinear model spread rates with the spread rates computed from linearized systems with the linearization carried out around the leading edge of the invasion. A survey that accounts for recent developments in the identification of conditions under which linear determinacy gives the ``right" answer, particularly in the context of non-compact and non-cooperative systems, is the thrust of this contribution. Novel results that extend some of the research linked to some the contributions covered in this survey are also discussed.
keywords: ecology population biology Dispersal integer difference integral equations nonlinear reaction diffusion difference equations.
Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian
Guowei Dai Ruyun Ma Haiyan Wang
Communications on Pure & Applied Analysis 2013, 12(6): 2839-2872 doi: 10.3934/cpaa.2013.12.2839
In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight. By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$-Laplacian with the sign-changing weight. In particular, we show that there exist two simple, isolated, principal eigenvalues $\lambda_0^+$ and $\lambda_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(\lambda_0^\nu,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $C_\nu^{+}$ and $C_\nu^{-}$, consisting of the continuum $C_\nu$ emanating from $(\lambda_0^\nu, 0)$, where $\nu\in\{+,-\}$. As an application of the above result, we study the existence of one-sign solutions for a class of quasilinear periodic boundary problems with the sign-changing weight. Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $\lambda$ are also studied.
keywords: One-sign solutions. Eigenvalues Unilateral global bifurcation Periodic $p$-Laplacian

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