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
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.
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
A posteriori error analysis for FEM of American options
Walter Allegretto Yanping Lin Ningning Yan
Discrete & Continuous Dynamical Systems - B 2006, 6(5): 957-978 doi: 10.3934/dcdsb.2006.6.957
In this paper, we present the a posteriori error analysis for the finite element approximation of American option valuation problems. We introduce an efficient and reliable error estimator both for the semi discrete and fully discrete backward Euler scheme.
keywords: American option a posteriori error analysis. adaptive finite element methods

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