@article{pao_2018, title={Global attractors of some predator-prey reaction-diffusion systems with density-dependent diffusion and time-delays}, volume={464}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2018.03.076}, abstractNote={This paper deals with a two-species and a three-species predator–prey reaction diffusion systems where the diffusion coefficients are density dependent and time-delays are involved in the reaction functions. The diffusion terms are of porous medium type which are degenerate and the boundary conditions are of Neumann type. The aim of the paper is to investigate the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions of the system. This includes the existence of a unique positive classical solution, global attraction of a steady-state solution, and stability or instability of various semitrivial solutions and the positive steady-state solution.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, C. V.}, year={2018}, month={Aug}, pages={164–187} }
 @article{pao_ruan_2017, title={Dynamics of degenerate quasilinear reaction diffusion systems with nonnegative initial functions}, volume={263}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2017.08.024}, abstractNote={This paper is concerned with a system of quasilinear reaction–diffusion equations with density dependent diffusion coefficients and mixed quasimonotone reaction functions. The equations are allowed to be degenerate and the boundary conditions are of the nonlinear type. The main goals are to prove the existence and uniqueness of the weak solution between a pair of coupled upper and lower solutions; show that the weak solution evolves into the classical solution, and analyze the asymptotic behavior of the solution using quasi-solutions of the steady-state system. The general results are applied to a degenerate Lotka–Volterra competition model. Conditions are given for the solution to exist globally, to evolve into the classical solution, and to be attracted into a sector formed by quasi-solutions of the elliptic system. Especially for the Neumann problem we give a simple condition for the solution to converge to a unique constant steady-state solution which is a global attractor.}, number={11}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Pao, C. V. and Ruan, W. H.}, year={2017}, month={Dec}, pages={7709–7752} }
 @article{pao_he_2016, title={Numerical methods for coupled systems of quasilinear elliptic equations with nonlinear boundary conditions}, volume={6}, number={2}, journal={Journal of Applied Analysis and Computation}, author={Pao, C. V. and He, T. P.}, year={2016}, pages={543–581} }
 @article{pao_2016, title={Dynamics of food-chain models with density-dependent diffusion and ratio-dependent reaction function}, volume={433}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2015.05.075}, abstractNote={This paper is concerned with a 3-species and a 2-species food-chain reaction diffusion systems in a bounded domain where the diffusion coefficients may be density dependent and the reaction functions are ratio-dependent. These equations are quasilinear where the diffusion coefficients may be degenerate on the boundary of the domain. Three basic types of Dirichlet, Neumann and Robin boundary conditions are considered, and in each case some very simple conditions are obtained to ensure the dynamical behavior of the time-dependent solution in relation to some positive solutions or quasi-solutions of the steady-state problem, including the existence of these solutions. This dynamical behavior leads to the coexistence and global attractor of the food-chain systems. In the case of Neumann boundary condition sufficient conditions are given to ensure that the steady-state problem has a unique positive constant solution which is a global attractor of the time-dependent system.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, C. V.}, year={2016}, month={Jan}, pages={355–374} }
 @article{pao_ruan_2015, title={Existence and dynamics of quasilinear parabolic systems with time delays}, volume={258}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2015.01.010}, abstractNote={This paper is concerned with a coupled system of quasilinear parabolic equations where the effect of time delays is taken into consideration in the reaction functions of the system. The partial differential operators in the system may be degenerate and the reaction functions possess some mixed quasimonotone property, including quasimonotone nondecreasing functions. The aim of the paper is to show the existence and uniqueness of a global solution to the parabolic system, the existence of positive quasisolutions or maximal–minimal solutions of the corresponding elliptic system, and the asymptotic behavior of the solution of the parabolic system in relation to the quasisolutions or maximal–minimal solutions of the elliptic system. Applications are given to three reaction–diffusion models arising from mathematical biology and ecology where the diffusion coefficients are density dependent and are degenerate. This degenerate density-dependent diffusion leads to some interesting distinct asymptotic behavior of the time-dependent solution when compared with density-independent diffusion.}, number={9}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Pao, C. V. and Ruan, W. H.}, year={2015}, month={Mar}, pages={3248–3285} }
 @article{pao_wang_2016, title={Numerical methods for fourth-order elliptic equations with nonlocal boundary conditions}, volume={292}, ISSN={["1879-1778"]}, DOI={10.1016/j.cam.2015.07.018}, abstractNote={This paper is concerned with some numerical methods for a fourth-order semilinear elliptic boundary value problem with nonlocal boundary condition. The fourth-order equation is formulated as a coupled system of two second-order equations which are discretized by the finite difference method. Three monotone iterative schemes are presented for the coupled finite difference system using either an upper solution or a lower solution as the initial iteration. These sequences of monotone iterations, called maximal sequence and minimal sequence respectively, yield not only useful computational algorithms but also the existence of a maximal solution and a minimal solution of the finite difference system. Also given is a sufficient condition for the uniqueness of the solution. This uniqueness property and the monotone convergence of the maximal and minimal sequences lead to a reliable and easy to use error estimate for the computed solution. Moreover, the monotone convergence property of the maximal and minimal sequences is used to show the convergence of the maximal and minimal finite difference solutions to the corresponding maximal and minimal solutions of the original continuous system as the mesh size tends to zero. Three numerical examples with different types of nonlinear reaction functions are given. In each example, the true continuous solution is constructed and is used to compare with the computed solution to demonstrate the accuracy and reliability of the monotone iterative schemes.}, journal={JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}, author={Pao, C. V. and Wang, Yuan-Ming}, year={2016}, month={Jan}, pages={447–468} }
 @article{pao_2015, title={Dynamics of Lotka-Volterra competition reaction-diffusion systems with degenerate diffusion}, volume={421}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2014.07.070}, abstractNote={This paper is dealt with a class of Lotka–Volterra competition reaction–diffusion system and a competitor–competitor–mutualist system with density-dependent diffusion in a bounded domain under the three basic types of Dirichlet, Neumann and Robin boundary conditions. The governing equations for the competition system consist of an arbitrary number of degenerate quasilinear parabolic equations while the competitor–competitor–mutualist system involves three degenerate equations. The goal of the paper is to show: the existence of positive steady-state solutions or quasi-solutions, the existence and uniqueness of a classical global time-dependent solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions or quasi-solutions. The above goals are achieved under a very simple condition on the reaction rates of the reaction functions and these results yield a global attractor and the coexistence of the competing species. In the case of Neumann boundary condition the system has a unique constant positive steady-state solution which is a global attractor of all the competing species. The above conclusions lead to some interesting distinct dynamic behavior between degenerate quasilinear reaction–diffusion systems and the corresponding semilinear reaction–diffusion systems in which some or all of the competing species may be in extinction.}, number={2}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, C. V.}, year={2015}, month={Jan}, pages={1721–1742} }
 @article{pao_2015, title={Dynamics of Lotka-Volterra cooperation systems governed by degenerate quasilinear reaction-diffusion equations}, volume={23}, ISSN={["1468-1218"]}, DOI={10.1016/j.nonrwa.2014.11.002}, abstractNote={This paper deals with a class of Lotka–Volterra cooperation system where the densities of the cooperating species are governed by a finite number of degenerate reaction–diffusion equations. Three basic types of Dirichlet, Neumann, and Robin boundary conditions and two types of reaction functions, with and without saturation, are considered. The aim of the paper is to show the existence of positive minimal and maximal steady-state solutions, including the uniqueness of the positive solution, the existence and uniqueness of a global time-dependent solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Some very simple conditions on the physical parameters for the above objectives are obtained. Also discussed is the finite-time blow up property of the time-dependent solution and the non-existence of positive steady-state solution for the system with Neumann boundary condition.}, journal={NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS}, author={Pao, C. V.}, year={2015}, month={Jun}, pages={47–60} }
 @article{pao_2014, title={A Lotka-Volterra cooperating reaction-diffusion system with degenerate density-dependent diffusion}, volume={95}, ISSN={["1873-5215"]}, DOI={10.1016/j.na.2013.09.015}, abstractNote={In the Lotka–Volterra cooperating reaction–diffusion system if the diffusion coefficients are constants then for a certain set of reaction rates in the reaction function the solution of the system blows up in finite time, and for another set of reaction rates, a unique global solution exists and converges to the trivial solution. However, if the diffusion coefficients are density-dependent then the dynamic behavior of the solution can be quite different. The aim of this paper is to investigate the global existence and the asymptotic behavior of the solution for a class of density-dependent cooperating reaction–diffusion systems where the diffusion coefficients are degenerate. It is shown that the time-dependent problem has a unique bounded global solution, and in addition to the trivial and semi-trivial solutions the corresponding steady-state problem has a positive maximal solution and a positive minimal solution. Moreover, the time-dependent solution converges to the maximal solution for one class of initial functions, and to the minimal solution for another class of initial functions. The above convergence property holds true for any reaction rates in the reaction function. Applications of the above results are given to a porous medium type of reaction–diffusion problem as well as other types of diffusion coefficients, including the finite sum and products of these functions.}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Pao, C. V.}, year={2014}, month={Jan}, pages={460–467} }
 @article{pao_chang_jau_2013, title={On a coupled system of reaction-diffusion-transport equations arising from catalytic converter}, volume={14}, ISSN={["1468-1218"]}, DOI={10.1016/j.nonrwa.2013.04.004}, abstractNote={This paper is concerned with two mathematical models which describe the transient behavior of a catalytic converter in automobile engineering. The first model consists of a coupled system of a heat-conduction equation and two integral equations while the second model involves only one integral equation. It is shown that for any nonnegative initial and boundary functions the three-equation model has a unique bounded global solution while the solution of the two-equation model blows up in finite time. The proof for the global existence and finite-time blow-up property of the solution is by the method of upper and lower solutions and its associated monotone iteration. This method can be used to develop computational algorithms for numerical solutions of the coupled systems.}, number={6}, journal={NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS}, author={Pao, C. V. and Chang, Yu-Hsien and Jau, Guo-Chin}, year={2013}, month={Dec}, pages={2152–2165} }
 @article{pao_ruan_2013, title={Quasilinear parabolic and elliptic systems with mixed quasimonotone functions}, volume={255}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2013.05.015}, abstractNote={This paper deals with a class of quasilinear parabolic and elliptic systems with mixed quasimonotone reaction functions. The boundary condition in the system may be Dirichlet, nonlinear, or a combination of these two types. The elliptic operators in the system are allowed to be degenerate. The aim is to show the existence and uniqueness of a classical solution to the parabolic system, the existence of maximal and minimal solutions or quasisolutions of the elliptic system, and the asymptotic behavior of the solution of the parabolic system. This consideration leads to a global attractor of the parabolic system as well as an one-sided stability of the maximal and minimal solutions. Applications of these results are given to three models arising from biology and ecology where diffusion coefficients are density-dependent and are degenerate. These applications exhibit quite distinct dynamical behavior of the population species between degenerate density-dependent diffusion and constant diffusion.}, number={7}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Pao, C. V. and Ruan, W. H.}, year={2013}, month={Oct}, pages={1515–1553} }
 @article{pao_chang_jau_2013, title={Numerical methods for a coupled system of differential equations arising from a thermal ignition problem}, volume={29}, ISSN={["0749-159X"]}, DOI={10.1002/num.21708}, abstractNote={AbstractThis article is concerned with monotone iterative methods for numerical solutions of a coupled system of a first‐order partial differential equation and an ordinary differential equation which arises from fast‐igniting catalytic converters in automobile engineering. The monotone iterative scheme yields a straightforward marching process for the corresponding discrete system by the finite‐difference method, and it gives not only a computational algorithm for numerical solutions of the problem but also the existence and uniqueness of a finite‐difference solution. Particular attention is given to the “finite‐time” blow‐up property of the solution. In terms of minimal sequence of the monotone iterations, some necessary and sufficient conditions for the blow‐up solution are obtained. Also given is the convergence of the finite‐difference solution to the continuous solution as the mesh size tends to zero. Numerical results of the problem, including a case where the continuous solution is explicitly known, are presented and are compared with the known solution. Special attention is devoted to the computation of the blow‐up time and the critical value of a physical parameter which determines the global existence and the blow‐up property of the solution. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013}, number={1}, journal={NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS}, author={Pao, C. V. and Chang, Yu-Hsien and Jau, Guo-Chin}, year={2013}, month={Jan}, pages={251–279} }
 @article{feng_pao_lu_2011, title={GLOBAL ATTRACTORS OF REACTION-DIFFUSION SYSTEMS MODELING FOOD CHAIN POPULATIONS WITH DELAYS}, volume={10}, ISSN={["1553-5258"]}, DOI={10.3934/cpaa.2011.10.1463}, abstractNote={In this paper, we study a reaction-diffusion system modeling the 
population dynamics of a four-species food chain with time delays. 
Under Dirichlet and Neumann boundary conditions, we discuss the 
existence of a positive global attractor which demonstrates the 
presence of a positive steady state and the permanence effect in the 
ecological system. Sufficient conditions on the interaction rates 
are given to ensure the persistence of all species in the food 
chain. For the case of Neumann boundary condition, we further obtain 
the uniqueness of a positive steady state, and in such case the 
density functions converge uniformly to a constant solution. 
Numerical simulations of the food-chain models are also given to 
demonstrate and compare the asymptotic behavior of the 
time-dependent density functions.}, number={5}, journal={COMMUNICATIONS ON PURE AND APPLIED ANALYSIS}, author={Feng, Wei and Pao, C. V. and Lu, Xin}, year={2011}, month={Sep}, pages={1463–1478} }
 @article{pao_lu_2010, title={BLOCK MONOTONE ITERATIVE METHOD FOR SEMILINEAR PARABOLIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS}, volume={47}, ISSN={["0036-1429"]}, DOI={10.1137/090748706}, abstractNote={Two block monotone iterative schemes, called Jacobi and Gauss-Seidel monotone iterations, are presented for numerical solutions of a class of semilinear parabolic equations under nonlinear boundary conditions by the finite difference method. These iteration schemes extend the method for semilinear elliptic boundary value problems to parabolic equations, including a comparison result between them. It is shown that by using an upper solution and a lower solution as initial iterations each of the iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite difference system. Some error estimates and a convergence theorem are given, and various sufficient conditions for the construction of upper and lower solutions are obtained. Numerical results are presented for some physical model problems, including some problems with known continuous solutions and two problems with L-shaped and trapezoidal domains.}, number={6}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Pao, C. V. and Lu, Xin}, year={2010}, pages={4581–4606} }
 @article{pao_2010, title={EIGENVALUE PROBLEMS OF A DEGENERATE QUASILINEAR ELLIPTIC EQUATION}, volume={40}, ISSN={["0035-7596"]}, DOI={10.1216/rmj-2010-40-1-305}, number={1}, journal={ROCKY MOUNTAIN JOURNAL OF MATHEMATICS}, author={Pao, C. V.}, year={2010}, pages={305–311} }
 @article{pao_wang_2010, title={Nonlinear fourth-order elliptic equations with nonlocal boundary conditions}, volume={372}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2010.07.027}, abstractNote={This paper is concerned with a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of Rn. Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method.}, number={2}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, C. V. and Wang, Yuan-Ming}, year={2010}, month={Dec}, pages={351–365} }
 @article{pao_ruan_2010, title={Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition}, volume={248}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2009.12.011}, abstractNote={Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i=1,…,N, and the boundary condition is ui=0. Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology.}, number={5}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Pao, C. V. and Ruan, W. H.}, year={2010}, month={Mar}, pages={1175–1211} }
 @article{pao_2009, title={Singular reaction diffusion equations of porous medium type}, volume={71}, ISSN={["1873-5215"]}, DOI={10.1016/j.na.2009.01.122}, abstractNote={Abstract A class of reaction-diffusions equations with nonlinear boundary conditions and porous medium type of diffusion is investigated by the method of upper and lower solutions. The diffusion term in the equation includes the so-called slow diffusion ( m > 1 ) , fast diffusion ( 0 m 1 ) and super diffusion ( m ≤ 0 ) , and the reaction term may involve a singular function. The aim of this paper is to show the existence of a global time-dependent solution, the existence of positive minimal and maximal steady-state solutions, including the uniqueness of the positive solution, and the asymptotic behavior of the time-dependent solution in relation to the positive steady-state solutions. Of particular concern is the convergence of the time-dependent solution to a unique positive steady-state solution and the global attraction property of the reaction-diffusion problem. Applications are given to several specific problems which are of considerable interest in the current literature.}, number={5-6}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Pao, C. V.}, year={2009}, month={Sep}, pages={2033–2052} }
 @article{pao_wang_2008, title={Numerical solutions of a three-competition Lotka-Volterra system}, volume={204}, ISSN={["1873-5649"]}, DOI={10.1016/j.amc.2008.06.057}, abstractNote={This paper is concerned with finite difference solutions of a Lotka–Volterra reaction–diffusion system with three-competing species. The reaction–diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference system for the time-dependent solution and its asymptotic behavior in relation to the corresponding steady-state problem. Three monotone iterative schemes for the computation of the time-dependent solution are presented, and the sequences of iterations are shown to converge monotonically to a unique positive solution. Also discussed is the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions. A simple condition on the competing rate constants is obtained, which ensures that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges either to a unique positive steady-state solution or to one of the semitrivial steady-state solutions. The above results lead to the coexistence and permanence of the competing system as well as computational algorithms for numerical solutions. Some numerical results from these computational algorithms are given. All the conclusions for the reaction–diffusion equations are directly applicable to the finite difference solution of the corresponding ordinary differential system.}, number={1}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Pao, C. V. and Wang, Yuan-Ming}, year={2008}, month={Oct}, pages={423–440} }
 @article{chang_jau_pao_2008, title={Blowup and global existence of solutions for a catalytic converter in interphase heat-transfer}, volume={9}, DOI={10.1016/j.nonrwa.2007.01.002}, abstractNote={Abstract In this paper we investigate the blowup property and global existence of a solution for a coupled system of first-order partial differential equation and ordinary differential equation which arises from a catalytic converter in automobile engineering. It is shown, in terms of a single physical parameter σ , that a unique bounded global solution exists if σ σ and the solution blows up in finite time if σ > σ ¯ , where σ σ ¯ . Various estimates for σ ¯ and its associate blow-up time T * are explicitly given. The value of T * can be used to estimate the ignition time and ignition length of the ignition system which is an important concern in automobile engineering.}, number={3}, journal={Nonlinear Analysis. Real World Applications}, author={Chang, Y. H. and Jau, G. C. and Pao, C. V.}, year={2008}, pages={822–829} }
 @article{pao_2007, title={Numerical methods for quasi-linear elliptic equations with nonlinear boundary conditions}, volume={45}, ISSN={["1095-7170"]}, DOI={10.1137/060653640}, abstractNote={The purpose of this paper is to give a numerical treatment for a class of quasi‐linear elliptic equations under nonlinear boundary conditions, including the three basic types of linear boundary conditions. The quasi‐linear equation is discretized by the finite difference method, and the method of upper‐lower solutions and its associated monotone iteration are used to compute the solutions of the finite difference system. This method leads to monotone iterative schemes for the computation of numerical solutions as well as some comparison results among the monotone iterative schemes. It also leads to the existence of a maximal and a minimal finite difference solution, including the uniqueness of the solution, and the convergence of the finite difference solution to the corresponding continuous solution. Applications are given to two physical problems in heat conduction and combustion theory, and numerical results for the heat‐conduction problem are given, and are compared with the known true continuous solution.}, number={3}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Pao, C. V.}, year={2007}, pages={1081–1106} }
 @article{pao_2007, title={Monotone iterative methods for numerical solutions of nonlinear integro-elliptic boundary problems}, volume={186}, ISSN={["1873-5649"]}, DOI={10.1016/j.amc.2006.08.074}, abstractNote={Abstract The aim of this paper is to obtain various monotone iterative schemes for numerical solutions of a class of nonlinear nonlocal reaction–diffusion–convection equations under linear boundary conditions. The boundary-value problem under consideration is discretized into a system of nonlinear algebraic equations by the finite difference method, and the iterative schemes are given for the finite difference system using upper and lower solutions as the initial iterations. The construction of the monotone sequences and the definition of upper and lower solutions depend on the quasimonotone property of the reaction function, and the iterative schemes are presented for each of the three types of quasimonotone functions. An application of the monotone iterations, including some numerical results, is given to a modified logistic diffusive equation where the kernel in the integral term may be positive, negative, or changing sign in its domain.}, number={2}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Pao, C. V.}, year={2007}, month={Mar}, pages={1624–1642} }
 @article{pao_ruan_2007, title={Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions}, volume={333}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2006.10.005}, abstractNote={The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a two-component competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, C. V. and Ruan, W. H.}, year={2007}, month={Sep}, pages={472–499} }
 @article{pao_2007, title={Quasilinear parabolic and elliptic equations with nonlinear boundary conditions}, volume={66}, ISSN={["1873-5215"]}, DOI={10.1016/j.na.2005.12.007}, abstractNote={This paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both Dirichlet and nonlinear Neumann boundary conditions. The equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient. The consideration of the class of equations is motivated by some heat-transfer problems where the heat capacity and thermal conductivity are both temperature dependent. The aim of the paper is to show the existence and uniqueness of a global time-dependent solution of the parabolic problem, existence of maximal and minimal steady-state solutions of the elliptic problem, including conditions for the uniqueness of a solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Applications are given to some heat-transfer problems and an extended logistic reaction–diffusion equation.}, number={3}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Pao, C. V.}, year={2007}, month={Feb}, pages={639–662} }
 @article{pao_2007, title={The global attractor of a competitor-competitor-mutualist-reaction-diffusion system with time delays}, volume={67}, ISSN={["1873-5215"]}, DOI={10.1016/j.na.2006.09.027}, abstractNote={The aim of this paper is to investigate the asymptotic behavior of time-dependent solutions of a three-species reaction–diffusion system in a bounded domain under a Neumann boundary condition. The system governs the population densities of a competitor, a competitor–mutualist and a mutualist, and time delays may appear in the reaction mechanism. It is shown, under a very simple condition on the reaction rates, that the reaction–diffusion system has a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the positive steady-state solution. An immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable. Moreover, the state–state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion. All the conclusions for the time-delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays.}, number={9}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Pao, C. V.}, year={2007}, month={Nov}, pages={2623–2631} }
 @article{wang_pao_2006, title={Time-delayed finite difference reaction-diffusion systems with nonquasimonotone functions}, volume={103}, ISSN={["0945-3245"]}, DOI={10.1007/s00211-006-0685-y}, number={3}, journal={NUMERISCHE MATHEMATIK}, author={Wang, YM and Pao, CV}, year={2006}, month={May}, pages={485–513} }
 @article{li_pao_qiao_2007, title={A Finite Difference Method and Analysis for 2D Nonlinear Poisson–Boltzmann Equations}, volume={30}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/s10915-005-9019-y}, DOI={10.1007/s10915-005-9019-y}, number={1}, journal={Journal of Scientific Computing}, publisher={Springer Science and Business Media LLC}, author={Li, Zhilin and Pao, C. V. and Qiao, Zhonghua}, year={2007}, month={Jan}, pages={61–81} }
 @article{pao_2005, title={Global attractor of coupled difference equations and applications to Lotka-Volterra systems}, ISSN={["1687-1847"]}, DOI={10.1155/ade.2005.57}, abstractNote={This paper is concerned with a coupled system of nonlinear difference equations which is a discrete approximation of a class of nonlinear differential systems with time delays. The aim of the paper is to show the existence and uniqueness of a positive solution and to investigate the asymptotic behavior of the positive solution. Sufficient conditions are given to ensure that a unique positive equilibrium solution exists and is a global attractor of the difference system. Applications are given to three basic types of Lotka-Volterra systems with time delays where some easily verifiable conditions on the reaction rate constants are obtained for ensuring the global attraction of a positive equilibrium solution.}, number={1}, journal={ADVANCES IN DIFFERENCE EQUATIONS}, author={Pao, C. V.}, year={2005}, pages={57–79} }
 @article{pao_2005, title={Stability and attractivity of periodic solutions of parabolic systems with time delays}, volume={304}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2004.09.014}, abstractNote={This paper is concerned with the existence, stability, and global attractivity of time-periodic solutions for a class of coupled parabolic equations in a bounded domain. The problem under consideration includes coupled system of parabolic and ordinary differential equations, and time delays may appear in the nonlinear reaction functions. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement using Schauder fixed point theorem, while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme. The results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system. Applications are also given to three Lotka–Volterra reaction diffusion model problems, and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions.}, number={2}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, CV}, year={2005}, month={Apr}, pages={423–450} }
 @article{pao_2005, title={Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion}, volume={60}, ISSN={["0362-546X"]}, DOI={10.1016/j.na.2004.10.008}, abstractNote={The aim of this paper is to investigate the existence and method of construction of solutions for a general class of strongly coupled elliptic systems by the method of upper and lower solutions and its associated monotone iterations. The existence problem is for nonquasimonotone functions arising in the system, while the monotone iterations require some mixed monotone property of these functions. Applications are given to three Lotka–Volterra model problems with cross-diffusion and self-diffusion which are some extensions of the classical competition, prey–predator, and cooperating ecological systems. The monotone iterative schemes lead to some true positive solutions of the competition system, and to quasisolutions of the prey–predator and cooperating systems. Also given are some sufficient conditions for the existence of a unique positive solution to each of the three model problems.}, number={7}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Pao, CV}, year={2005}, month={Mar}, pages={1197–1217} }
 @article{pao_2003, title={Accelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems}, volume={46}, DOI={10.1016/S0898-1221(03)00381-X}, number={37905}, journal={Computers & Mathematics With Applications}, author={Pao, C. V.}, year={2003}, pages={1535–1544} }
 @article{pao_lu_2003, title={Block monotone iterations for numerical solutions of fourth-order nonlinear elliptic boundary value problems}, volume={25}, ISSN={["1095-7197"]}, DOI={10.1137/S1064827502409912}, abstractNote={This paper is concerned with monotone iterative methods for numerical solutions of a class of nonlinear fourth-order elliptic boundary value problems in a two-dimensional domain. The boundary value problem is discretized by the finite difference method, and two iterative processes, called block Jacobi and block Gauss-Seidel monotone iterations, are presented for the computation of solutions of the finite difference system using either an upper solution or a lower solution as the initial iteration. It is shown that the sequence of iterations converges monotonically to a maximal solution or a minimal solution if the nonlinear function is quasi-monotone nondecreasing. A sufficient condition is given to ensure that the maximal and minimal solutions coincide and their common value is the unique solution of the finite difference system. Similar results are obtained for quasi-monotone nonincreasing functions. An analytical comparison relation between the block Jacobi and block Gauss--Seidel monotone iterations is obtained. It is also shown that the finite difference solution converges to the continuous solution as the mesh size tends to zero. Numerical results by the block monotone iterative schemes are given for two model problems and are compared with the known analytical solutions for accuracy. Also compared are the rates of convergence of the two types of block monotone iterations as well as similar types of pointwise monotone iterations.}, number={1}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Pao, CV and Lu, X}, year={2003}, pages={164–185} }
 @article{pao_2003, title={Finite difference reaction-diffusion systems with coupled boundary conditions and time delays (vol 272, pg 407, 2002)}, volume={288}, number={2}, journal={Journal of Mathematical Analysis and Applications}, author={Pao, C. V.}, year={2003}, pages={870} }
 @article{pao_2003, title={Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays}, volume={281}, ISSN={["0022-247X"]}, DOI={10.1016/S0022-247X(03)00033-7}, abstractNote={This paper is concerned with three 3-species time-delayed Lotka–Volterra reaction–diffusion systems and their corresponding ordinary differential systems without diffusion. The time delays may be discrete or continuous, and the boundary conditions for the reaction–diffusion systems are of Neumann type. The goal of the paper is to obtain some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady-state solution for each of the three model problems. These conditions involve only the reaction rate constants and are independent of the diffusion effect and time delays. The result of global asymptotic stability implies that each of the three model systems coexists, is permanent, and the trivial and all semitrivial solutions are unstable. Our approach to the problem is based on the method of upper and lower solutions for a more general reaction–diffusion system which gives a common framework for the 3-species model problems. Some global stability results for the 2-species competition and prey–predator reaction–diffusion systems are included in the discussion.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, CV}, year={2003}, month={May}, pages={186–204} }
 @article{pao_2004, title={Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays}, volume={5}, ISSN={["1468-1218"]}, DOI={10.1016/S1468-1218(03)00018-X}, abstractNote={In the Lotka–Volterra competition system with N-competing species if the effect of dispersion and time-delays are both taken into consideration, then the densities of the competing species are governed by a coupled system of reaction–diffusion equations with time-delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to the competing rate constants to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the competing system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays, and the conclusions for the reaction–diffusion system are directly applicable to the corresponding ordinary differential system.}, number={1}, journal={NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS}, author={Pao, CV}, year={2004}, month={Feb}, pages={91–104} }
 @article{pao_2003, title={Global attractor of a coupled finite difference reaction diffusion system with delays}, volume={288}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2003.08.011}, abstractNote={In the study of asymptotic behavior of solutions for reaction diffusion systems, an important concern is to determine whether and when the system has a global attractor which attracts all positive time-dependent solutions. The aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays. Sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady-state system. Applications are given to three types of Lotka–Volterra reaction diffusion models, where time-delays may appear in the opposing species.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, CV}, year={2003}, month={Dec}, pages={251–273} }
 @article{pao_2002, title={Convergence of solutions of reaction-diffusion systems with time delays}, volume={48}, ISSN={["0362-546X"]}, DOI={10.1016/S0362-546X(00)00189-9}, number={3}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Pao, CV}, year={2002}, month={Jan}, pages={349–362} }
 @article{pao_2002, title={Finite difference reaction-diffusion systems with coupled boundary conditions and time delays}, volume={272}, ISSN={["1096-0813"]}, DOI={10.1016/S0022-247X(02)00145-2}, abstractNote={This paper is concerned with finite difference solutions of a coupled system of reaction–diffusion equations with nonlinear boundary conditions and time delays. The system is coupled through the reaction functions as well as the boundary conditions, and the time delays may appear in both the reaction functions and the boundary functions. The reaction–diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference equations for both the time-dependent problem and its corresponding steady-state problem. This investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions, monotone convergence of the time-dependent solution to a maximal or a minimal steady-state solution for quasimonotone functions, and local and global attractors of the time-dependent system, including the convergence of the time-dependent solution to a unique steady-state solution. Also discussed are some computational algorithms for numerical solutions of the steady-state problem when the reaction function and the boundary function are quasimonotone. All the results for the coupled reaction–diffusion equations are directly applicable to systems of parabolic–ordinary equations and to reaction–diffusion systems without time delays.}, number={2}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, CV}, year={2002}, month={Aug}, pages={407–434} }
 @article{pao_2001, title={Finite difference solutions of reaction diffusion equations with continuous time delays}, volume={42}, ISSN={["1873-7668"]}, DOI={10.1016/S0898-1221(01)00165-1}, abstractNote={This paper is an extension of the monotone iterative methods for finite difference equations with discrete time delays to a class of nonlinear finite difference system with continuous time delays. The system under consideration is a finite difference approximation of a class of reaction diffusion equations with continuous time delays in the nonlinear reaction under either Dirichlet or Neumann-Robin boundary conditions. Various monotone iterative schemes, which depend on the property of the nonlinear reaction mechanism, are developed for the computation of numerical solutions. It is shown by the method of upper and lower solutions that the two sequences obtained from each iterative scheme converge monotonically from above and below, respectively, to a unique solution of the finite difference system. Applications are given to two model problems known as the diffusive logistic equation and the Fisher's diffusion equation in population genetics.}, number={3-5}, journal={COMPUTERS & MATHEMATICS WITH APPLICATIONS}, author={Pao, CV}, year={2001}, pages={399–412} }
 @article{pao_2001, title={Numerical methods for fourth-order nonlinear elliptic boundary value problems}, volume={17}, ISSN={["1098-2426"]}, DOI={10.1002/num.1016}, abstractNote={AbstractThe aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth‐order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence‐comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two‐point boundary‐value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347–368, 2001}, number={4}, journal={NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS}, author={Pao, CV}, year={2001}, month={Jul}, pages={347–368} }
 @article{pao_2001, title={Numerical methods for nonlinear integro-parabolic equations of Fredholm type}, volume={41}, ISSN={["0898-1221"]}, DOI={10.1016/S0898-1221(00)00325-4}, abstractNote={This paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction-diffusion-convection equations under either linear or nonlinear boundary conditions. The discrete approximation of the problem is based on the finite-difference method, and the computation of the finite-difference solution is by the method of upper and lower solutions. Three types of quasi-monotone reaction functions are considered and for each type, a monotone iterative scheme is obtained. Each of these iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite-difference system. This monotone convergence leads to an existence-uniqueness theorem as well as a computational algorithm for the computation of the solution. An error estimate between the computed approximations and the true finite-difference solution is obtained for each iterative scheme. These error estimates are given in terms of the strength of the reaction function and the effect of diffusion-convection, and are independent of the true solution. Applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm.}, number={7-8}, journal={COMPUTERS & MATHEMATICS WITH APPLICATIONS}, author={Pao, CV}, year={2001}, month={Apr}, pages={857–877} }
 @article{pao_2001, title={Numerical methods for time-periodic solutions of nonlinear parabolic boundary value problems}, volume={39}, ISSN={["0036-1429"]}, DOI={10.1137/S0036142999361396}, abstractNote={This paper is devoted to a numerical analysis of periodic solutions of a finite difference system which is a discrete version of a class of nonlinear reaction-diffusion-convection equations under nonlinear boundary conditions. Three monotone iterative schemes for the finite difference system are presented, and it is shown by the method of upper and lower solutions that the sequence of iterations from each of these iterative schemes converges monotonically to either a maximal periodic solution or a minimal periodic solution depending on whether the initial iteration is an upper solution or a lower solution. A comparison theorem for the various monotone sequences is given. It is also shown that the maximal and minimal periodic solutions of the finite difference system converge to the corresponding maximal and minimal periodic solutions of the reaction-diffusion-convection equation as the mesh size decreases to zero. Some error estimates between the theoretical and the computed iterations for each of the three iterative schemes are obtained, and a discussion on the numerical stability of these schemes is given. Also given are some numerical results of a logistic reaction diffusion problem.}, number={2}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Pao, CV}, year={2001}, month={Jul}, pages={647–667} }
 @article{pao_2001, title={Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions}, volume={136}, ISSN={["0377-0427"]}, DOI={10.1016/S0377-0427(00)00614-2}, abstractNote={The purpose of this paper is to present some iterative methods for numerical solutions of a class of nonlinear reaction–diffusion equations with nonlocal boundary conditions. Using the finite-difference method and the method of upper and lower solutions we present some monotone iterative schemes for both the time-dependent and the steady-state finite-difference systems. Each monotone iterative scheme gives a computational algorithm for numerical solutions and an existence-comparison theorem for the corresponding finite-difference system. The existence-comparison theorems are used to investigate the asymptotic behavior of the discrete time-dependent solution in relation to the discrete maximal and minimal solutions of the steady-state problem. Numerical results are given to a model problem where the solution of the continuous problem is explicitly known and its values at the mesh points are used to compare with the numerical solutions obtained by the monotone iterative schemes.}, number={1-2}, journal={JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}, author={Pao, CV}, year={2001}, month={Nov}, pages={227–243} }
 @article{pao_2001, title={Quenching problem of a functional parabolic equation}, volume={8}, number={1}, journal={Journal of High Energy Physics}, author={Pao, C. V.}, year={2001}, pages={89–98} }
 @article{pao_2001, title={Reaction diffusion systems with time delays in a half-space domain}, volume={47}, ISSN={["0362-546X"]}, DOI={10.1016/S0362-546X(01)00551-X}, number={7}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Pao, CV}, year={2001}, month={Aug}, pages={4365–4375} }
 @article{pao_2002, title={Time delayed parabolic systems with coupled nonlinear boundary conditions}, volume={130}, ISSN={["1088-6826"]}, DOI={10.1090/S0002-9939-01-06319-5}, abstractNote={The aim of this paper is to show the existence and uniqueness of a solution for a system of time-delayed parabolic equations with coupled nonlinear boundary conditions. The time delays are of discrete type which may appear in the reaction function as well as in the boundary function. The approach to the problem is by the method of upper and lower solutions for nonquasimonotone functions.}, number={4}, journal={PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Pao, CV}, year={2002}, pages={1079–1086} }
 @inbook{pao_2000, title={Dynamics of a Volterra-Lotka competition model with diffusion and time delays, integral and integro differential equations}, volume={2}, booktitle={Integral and integrodifferential equations: Theory, method and applications (Series in mathematical analysis and applications ; v.2 )}, publisher={Amsterdam: Gordon and Breach Science Publishers}, author={Pao, C. V.}, editor={Agarwal, R. P. and O'Regan, D.Editors}, year={2000}, pages={269–277} }
 @article{wang_pao_2000, title={Finite difference reaction-diffusion equations with nonlinear diffusion coefficients}, volume={85}, ISSN={["0029-599X"]}, DOI={10.1007/s002110000140}, number={3}, journal={NUMERISCHE MATHEMATIK}, author={Wang, JH and Pao, CV}, year={2000}, month={May}, pages={485–502} }
 @article{pao_2000, title={On fourth-order elliptic boundary value problems}, volume={128}, number={4}, journal={Proceedings of the American Mathematical Society}, author={Pao, C. V.}, year={2000}, pages={1023–1030} }
 @article{pao_2000, title={Periodic solutions of parabolic systems with time delays}, volume={251}, ISSN={["1096-0813"]}, DOI={10.1006/jmaa.2000.7045}, abstractNote={Existence of maximal and minimal periodic solutions of a coupled system of parabolic equations with time delays and with nonlinear boundary conditions is discussed. The proof of the existence theorem is based on the method of upper and lower solutions and its associated monotone iterations. This method is constructive and can be used to develop a computational algorithm for numerical solutions of the periodic-parabolic system. An application is given to a competitor-competitor-mutualist model which consists of a coupled system of three reaction-diffusion equations with time delays.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, CV}, year={2000}, month={Nov}, pages={251–263} }
 @article{pao_2000, title={Periodic solutions of systems of parabolic equations in unbounded domains}, volume={40}, ISSN={["1873-5215"]}, DOI={10.1016/s0362-546x(00)85031-2}, number={1-8}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Pao, CV}, year={2000}, pages={523–535} }
 @article{pao_2000, title={Quenching problem of a reaction-diffusion equation with time delay}, volume={41}, ISSN={["0362-546X"]}, DOI={10.1016/S0362-546X(98)00269-7}, number={1-2}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Pao, CV}, year={2000}, month={Jul}, pages={133–142} }
 @article{pao_1999, title={Numerical analysis of coupled systems of nonlinear parabolic equations}, volume={36}, ISSN={["0036-1429"]}, DOI={10.1137/S0036142996313166}, abstractNote={This paper is concerned with numerical solutions of a general class of coupled nonlinear parabolic equations by the finite difference method. Three monotone iteration processes for the finite difference system are presented, and the sequences of iterations are shown to converge monotonically to a unique solution of the system, including an existence-uniqueness-comparison theorem. A theoretical comparison result for the various monotone sequences and an error analysis of the three monotone iterative schemes are given. Also given is the convergence of the finite difference solution to the continuous solution of the parabolic boundary-value problem. An application to a reaction-diffusion model in chemical engineering and combustion theory is given.}, number={2}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Pao, CV}, year={1999}, month={Mar}, pages={393–416} }
 @article{pao_1999, title={Numerical methods for systems of nonlinear parabolic equations with time delays}, volume={240}, ISSN={["1096-0813"]}, DOI={10.1006/jmaa.1999.6619}, abstractNote={Abstract The purpose of this paper is to investigate some numerical aspects of a class of coupled nonlinear parabolic systems with time delays. The system of parabolic equations is discretized by the finite difference method which yields a coupled system of nonlinear algebraic equations. The mathematical analysis of the nonlinear system is by the method of upper and lower solutions and its associated monotone iterations. Three monotone iterative schemes are presented and it is shown that the sequence of iterations from each one of these iterative schemes converges monotonically to a unique solution of the finite difference system. A theoretical comparison result for the various monotone sequences and error estimates for the three monotone iterative schemes are obtained. It is also shown that the finite difference solution converges to the classical solution of the parabolic system as the mesh size decreases to zero.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, CV}, year={1999}, month={Dec}, pages={249–279} }
 @article{pao_1999, title={Periodic solutions of parabolic systems with nonlinear boundary conditions}, volume={234}, ISSN={["1096-0813"]}, DOI={10.1006/jmaa.1999.6412}, abstractNote={This paper is concerned with the existence and stability of periodic solutions for a coupled system of nonlinear parabolic equations under nonlinear boundary conditions. The approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for parabolic initial-boundary value problems. A sufficient condition for the stability of a periodic solution is also given. These results are applied to three model problems arising from chemical kinetics, ecology, and population biology.}, number={2}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, CV}, year={1999}, month={Jun}, pages={695–716} }
 @article{yang_pao_1999, title={Positive solutions and dynamics of some reaction diffusion models in HIV transmission}, volume={35}, ISSN={["0362-546X"]}, DOI={10.1016/S0362-546X(97)00672-X}, number={3}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Yang, ZP and Pao, CV}, year={1999}, month={Feb}, pages={323–341} }
 @article{pao_1998, title={Accelerated monotone iterative methods for finite difference equations of reaction-diffusion}, volume={79}, ISSN={["0029-599X"]}, DOI={10.1007/s002110050340}, number={2}, journal={NUMERISCHE MATHEMATIK}, author={Pao, CV}, year={1998}, month={Apr}, pages={261–281} }
 @article{pao_1998, title={Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions}, volume={88}, DOI={10.1016/S0377-0427(97)00215-X}, abstractNote={This paper is concerned with some dynamical property of a reaction-diffusion equation with nonlocal boundary condition. Under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function, the asymptotic behavior of the time-dependent solution is characterized in relation to a finite or an infinite set of constant steady-state solutions. This characterization is determined solely by the initial function and it leads to the stability and instability of the various steady-state solutions. In the case of finite constant steady-state solutions, the time-dependent solution blows up in finite time when the initial function in greater than the largest constant solution. Also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain.}, number={1998}, journal={Journal of Computational and Applied Mathematics}, author={Pao, C. V.}, year={1998}, pages={225–238} }
 @article{pao_1998, title={Dynamics of a finite difference system of reaction diffusion equations with time delay}, volume={4}, ISSN={["1563-5120"]}, DOI={10.1080/10236199808808124}, abstractNote={The purpose of this paper is to investigate the dynamics of a nonlinear finite difference system which approximates a class of nonlinear reaction-diffusion equations with time delay. It is shown that for one class of initila vectors the solution n n of the finite difference system converges to the maximal solution of the corresponding “steady-state” problem, while for another class of initial vectors it converges to the minimal solution. When the maximal and minimal solutions coincide it yields a unique solution u ∗ of the steady-state problem, and the solution n n converges to u ∗ as n→∞ for every initial vector in a sector. A sufficient condition for the uniqueness of u ∗ is given. The above convergence results are applied to a reaction-diffusion problem with three different types of reaction functions.}, number={1}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Pao, CV}, year={1998}, pages={1–11} }
 @article{pao_1998, title={Monotone iterations for numerical solutions of reaction-diffusion-convection equations with time delay}, volume={14}, DOI={10.1002/(sici)1098-2426(199805)14:3<339::aid-num4>3.0.co;2-n}, abstractNote={In this article we use the monotone method for the computation of numerical solutions of a nonlinear reaction-diffusion-convection problem with time delay. Three monotone iteration processes for a suitably formulated finite-difference system of the problem are presented. It is shown that the sequence of iteration from each of these iterative schemes converges from either above or below to a unique solution of the finite-difference system without any monotone condition on the nonlinear reaction function. An analytical comparison result among the three processes of iterations is given. Also given is the application of the iterative schemes to some model problems in population dynamics, including numerical results of a model problem with known analytical solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 339–351, 1998}, number={3}, journal={Numerical Methods for Partial Differential Equations}, author={Pao, C. V.}, year={1998}, pages={339–351} }
 @article{pao_1998, title={Monotone methods for a finite difference system of reaction diffusion equation with time delay}, volume={36}, ISSN={["0898-1221"]}, DOI={10.1016/S0898-1221(98)80007-2}, abstractNote={The aim of this paper is to present some monotone iterative schemes for computing the solution of a system of nonlinear difference equations which arise from a class of nonlinear reaction-diffusion equations with time delays. The iterative schemes lead to computational algorithms as well as existence, uniqueness, and upper and lower bounds of the solution. An application to a diffusive logistic equation with time delay is given.}, number={10-12}, journal={COMPUTERS & MATHEMATICS WITH APPLICATIONS}, author={Pao, CV}, year={1998}, pages={37–47} }
 @article{pao_1998, title={Parabolic systems in unbounded domains I. Existence and dynamics}, volume={217}, ISSN={["0022-247X"]}, DOI={10.1006/jmaa.1997.5706}, abstractNote={Abstract In this paper we investigate the existence, uniqueness, and asymptotic behavior of a solution for a class of coupled nonlinear parabolic equations in a general unbounded domain that includes the whole space R n, the exterior of a bounded domain, and a half space in R n. The asymptotic behavior of the solution is with respect to a pair of quasi-solutions of the corresponding elliptic system, and when these two quasi-solutions coincide the solution of the parabolic system converges to a unique solution of the elliptic system.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, CV}, year={1998}, month={Jan}, pages={129–160} }
 @article{pao_1998, title={Parabolic systems in unbounded domains. II. Equations with time delays}, volume={225}, ISSN={["0022-247X"]}, DOI={10.1006/jmaa.1998.6051}, abstractNote={Abstract This paper extends the results of an earlier article concerning the existence, uniqueness, comparison, and dynamics problems for a coupled system of parabolic equations in unbounded domains, including the whole space R n. The present extension is for a system of functional parabolic-ordinary equations where the “reaction function” is allowed to depend on the unknown function with time delay, which may be discrete or continuous, finite or infinite. Applications are made to some model problems, with emphasis on the convergence fo the time-dependent solution to a steady-state solution and the decay property of the solution as |x| → ∞.}, number={2}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, CV}, year={1998}, month={Sep}, pages={557–586} }
 @misc{pao_1998, title={Theory and applications of partial functional differential equations. By Jianhong Wu.}, volume={40}, number={3}, journal={SIAM Review}, author={Pao, C. V.}, year={1998}, pages={746–747} }
 @article{pao_1997, title={Dynamics of reaction-diffusion systems and applications}, volume={30}, ISSN={["0362-546X"]}, DOI={10.1016/S0362-546X(97)00350-7}, abstractNote={The dynamics of a system of arbitrary number of nonlinear parabolic equations in terms of the quasisolutions of the corresponding system of elliptic equations is discussed. The conclusions for the general system are used to study the coexistence and permanence of a three-species predator-prey model and a food-chain model arising from ecology.}, number={6}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Pao, CV}, year={1997}, month={Dec}, pages={3371–3377} }
 @inbook{pao_1997, title={Dynamics of weakly coupled parabolic systems with nonlocal boundary conditions}, volume={5}, booktitle={Advances in nonlinear dynamics}, publisher={Australia: Gordon and Breach}, author={Pao, C. V.}, editor={S. Sivasundaram and Martynyuk, A. A.Editors}, year={1997}, pages={319–327} }
 @article{pao_1997, title={Systems of parabolic equations with continuous and discrete delays}, volume={205}, ISSN={["1096-0813"]}, DOI={10.1006/jmaa.1996.5177}, abstractNote={Abstract In this paper we investigate the global existence and the dynamics of a coupled system of nonlinear parabolic equations where the nonlinear “reaction function” may depend on both continuous (infinite or finite) and discrete delays. It is shown that if the reaction function is locally Lipschitz continuous and the system possesses a pair of coupled upper and lower solutions then there exists a unique global solution to the system without any quasimonotone condition on the reaction function. For systems with mixed quasimonotone reaction functions we use the monotone method to establish more dynamic property of the parabolic system in terms of the quasisolutions of the corresponding elliptic system. This approach yields a (global) attractor of the parabolic system, and under some additional conditions this attractor leads to the existence and asymptotic stability of a solution of the elliptic system. Application to three model problems in population dynamics and chemical reactions is given.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Pao, CV}, year={1997}, month={Jan}, pages={157–185} }