Positive steady state of a food chain system with diffusion
Jing Liu Xiaodong Liu Sining Zheng Yanping Lin
Conference Publications 2007, 2007(Special): 667-676 doi: 10.3934/proc.2007.2007.667
This paper deals positive steady state for predator-prey microorganisms in a flow reactor with diffusion and flow. The coexistence conditions for the predator-prey populations are established. The main method used here is the fixed point index theory.
keywords: Food chain; Predator-prey; Coexistence; Diffusion; Fixed point index; Steady state; Micro-organism; Nutrient.
A parabolic integro-differential equation arising from thermoelastic contact
Walter Allegretto John R. Cannon Yanping Lin
Discrete & Continuous Dynamical Systems - A 1997, 3(2): 217-234 doi: 10.3934/dcds.1997.3.217
In this paper we consider a class of integro-differential equations of parabolic type arising in the study of a quasi-static thermoelastic contact problem involving a critical parameter $\alpha$. For $\alpha <1$, the problem is first transformed into an equivalent standard parabolic equation with non-local and non-linear boundary conditions. Then the existence, uniqueness and continuous dependence of the solution upon the data are demonstrated via solution representation techniques and the maximum principle. Finally the asymptotic behavior of the solution as $ t \rightarrow \infty$ is examined, and we show that the non-local term has no impact on the asymptotic behavior for $ \alpha <1$. The paper concludes with some remarks on the case $\alpha >1$.
keywords: thermoelastic contact. Parabolic integro-differential non-local
Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows
Kun Wang Yinnian He Yanping Lin
Discrete & Continuous Dynamical Systems - B 2012, 17(5): 1551-1573 doi: 10.3934/dcdsb.2012.17.1551
In this article, we study the long time numerical stability and asymptotic behavior for the viscoelastic Oldroyd fluid motion equations. Firstly, with the Euler semi-implicit scheme for the temporal discretization, we deduce the global $H^2-$stability result for the fully discrete finite element solution. Secondly, based on the uniform stability of the numerical solution, we investigate the discrete asymptotic behavior and claim that the viscoelastic Oldroyd problem converges to the stationary Navier-Stokes flows if the body force $f(x,t)$ approaches to a steady-state $f_\infty(x)$ as $t\rightarrow\infty$. Finally, some numerical experiments are given to verify the theoretical predictions.
keywords: Oldroyd model Viscoelastic flows asymptotic analysis. $H^2-$stability long time behavior
Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations
Walter Allegretto Yanping Lin Zhiyong Zhang
Conference Publications 2009, 2009(Special): 11-23 doi: 10.3934/proc.2009.2009.11
In this paper we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and damping $$ \left\{\begin{array}{l} \psi_t = -(1-\alpha) \psi - \theta_x + \alpha \psi_{x x} + \psi\psi_x,                                                                                                            (E)\\ \theta_t = -(1-\alpha)\theta + \nu \psi_x + 2\psi\theta_x + \alpha \theta_{x x}, \end{array} \right. $$ with initial data converging to different constant states at infinity $$(\psi,\theta)(x,0)=(\psi_0(x), \theta_0(x)) \rightarrow (\psi_{\pm}, \theta_{\pm}) \ \ {as} \ \ x \rightarrow \pm \infty,                                                             (I) $$ where $\alpha$ and $\nu$ are positive constants such that $\alpha <1$, $\nu <4\alpha(1-\alpha)$. Under the assumption that $|\psi_+ - \psi_- |+| \theta_+ - \theta_-|$ is sufficiently small, we show that if the initial data is a small perturbation of the convection-diffusion waves defined by (11) which are obtained by the parabolic system (9), solutions to Cauchy problem (E) and (I) tend asymptotically to those convection-diffusion waves with exponential rates. We mainly propose a better asymptotic profile than that in the previous work by [13,3], and derive its decay rates by weighted energy method instead of considering the linearized structure as in [3].

keywords: decay rate energy method diffusion waves a priori estimates Evolution equations
Existence and long time behaviour of solutions to obstacle thermistor equations
Walter Allegretto Yanping Lin Shuqing Ma
Discrete & Continuous Dynamical Systems - A 2002, 8(3): 757-780 doi: 10.3934/dcds.2002.8.757
In this paper we introduce an obstacle thermistor system. The existence of weak solutions to the steady-state systems and capacity solutions to the time dependent systems are obtained by a penalized method under reasonable assumptions for the initial and boundary data. At the same time, we prove that there exists a uniform absorbing set for nonnegative initial data in $L_2(\Omega)$. Finally for smooth initial data a global attractor to the system is obtained by a series of Campanato space arguments.
keywords: capacity solutions Obstacle thermistor equations global attractors.
Stabilized finite element method for the non-stationary Navier-Stokes problem
Yinnian He Yanping Lin Weiwei Sun
Discrete & Continuous Dynamical Systems - B 2006, 6(1): 41-68 doi: 10.3934/dcdsb.2006.6.41
In this article, a locally stabilized finite element formulation of the two-dimensional Navier-Stokes problem is used. A macroelement condition which provides the stability of the $Q_1-P_0$ quadrilateral element and the $P_1-P_0$ triangular element is introduced. Moreover, the $H^1$ and $L^2$-error estimates of optimal order for finite element solution $(u_h,p_h)$ are analyzed. Finally, a uniform $H^1$ and $L^2$-error estimates of optimal order for finite element solution $(u_h,p_h)$ is obtained if the uniqueness condition is satisfied.
keywords: stabilized finite element uniform error estimate. Navier-Stokes problem
Hölder continuous solutions of an obstacle thermistor problem
Walter Allegretto Yanping Lin Shuqing Ma
Discrete & Continuous Dynamical Systems - B 2004, 4(4): 983-997 doi: 10.3934/dcdsb.2004.4.983
In this paper we consider a thermistor problem with a current source, i.e., a nonlocal boundary condition. The electric potential is unknown on part of the boundary, but the current through it is known. We apply a decomposition technique and transform the equation satisfied by the potential into two elliptic problems with usual boundary conditions. The unique solvability of the initial boundary value problem is achieved.
keywords: existence uniqueness. current driven Obstacle thermistor problem
On the box method for a non-local parabolic variational inequality
Walter Allegretto Yanping Lin Shuqing Ma
Discrete & Continuous Dynamical Systems - B 2001, 1(1): 71-88 doi: 10.3934/dcdsb.2001.1.71
In this paper we study a box scheme (or finite volume element method) for a non-local nonlinear parabolic variational inequality arising in the study of thermistor problems. Under some assumptions on the data and regularity of the solution, optimal error estimates in the $H^1$-norm are attained.
keywords: Box methods non-local thermistor and variational inequality.
Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients
Walter Allegretto Liqun Cao Yanping Lin
Discrete & Continuous Dynamical Systems - A 2008, 20(3): 543-576 doi: 10.3934/dcds.2008.20.543
In this paper we discuss initial-boundary problems for second order parabolic equations with rapidly oscillating coefficients in a bounded convex domain. The asymptotic expansions of the solutions for problems with multiple spatial and temporal scales are presented in four different cases. Higher order corrector methods are constructed and associated explicit convergence rates obtained.
keywords: finite element method Homogenization asymptotic expansion boundary layer parabolic equation with rapidly oscillating coefficients linear system of differential equation with real periodic coefficients.
Analysis of a biosensor model
Walter Allegretto Yanping Lin Zhiyong Zhang
Communications on Pure & Applied Analysis 2008, 7(3): 677-698 doi: 10.3934/cpaa.2008.7.677
In this paper we consider a biosensor model in $R^3$, consisting of a coupled parabolic differential equation with Robin boundary condition and an ordinary differential equation. Theoretical analysis is done to show the existence and uniqueness of a Holder continuous solution based on a maximum principle, weak solution arguments. The long-time convergence to a steady state is also discussed as well as the system situation. Next, a finite volume method is applied to the model to obtain an approximate solution. Drawing in part on the analytical results given earlier, we establish the existence, stability and error estimates for the approximate solution, and derive $L^2$ spatial norm convergence properties. Finally, some illustrative numerical simulation results are presented.
keywords: stability. finite volume method Robin boundary condition Biosensor model

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