Coupled System Of Reaction-diffusion Equations Research Articles

Blow-up Properties of a Coupled System of Reaction-Diffusion Equations

This paper is concerned with a Coupled Reaction-diffusion system defined in a ball with homogeneous Dirichlet boundary conditions. Firstly, we studied the blow-up set showing that, under some conditions, the blow-up in this problem occurs only at a single point. Secondly, under some restricted assumptions on the reaction terms, we established the upper (lower) blow-up rate estimates. Finally, we considered the Ignition system in general dimensional space as an application to our results.

Cardiac memory phenomenon, time-fractional order nonlinear system and bidomain-torso type model in electrocardiology

Cardiac memory, also known as the Chatterjee phenomenon, refers to the persistent but reversible T-wave changes on ECG caused by an abnormal electrical activation pattern. After a period of abnormal ventricular activation in which the myocardial repolarization is altered and delayed (such as with artificial pacemakers, tachyarrhythmias with wide QRS complexes or ventricular pre-excitation), the heart remembers and mirrors its repolarization in the direction of the vector of "abnormally" activated QRS complexes. This phenomenon alters patterns of gap junction distribution and generates changes in repolarization seen at the level of ionic-channel, ionic concentrations, ionic-current gating and action potential. In this work, we propose a mathematical model of cardiac electrophysiology which takes into account cardiac memory phenomena. The electrical activity in heart through torso, which is dependent on the prior history of accrued heartbeats, is mathematically modeled by a modified bidomain system with time fractional-order dynamics (which are used to describe processes that exhibit memory). This new bidomain system, that I name "<i>it memory bidomain system</i>", is a degenerate nonlinear coupled system of reaction-diffusion equations in shape of a fractional-order differential equation coupled with a set of time fractional-order partial differential equations. Cardiac memory is represented via fractional-order capacitor (associate to capacitive current) and fractional-order cellular membrane dynamics. First, mathematical model is introduced. Afterward, results on generalized Gronwall inequality within the framework of coupled fractional differential equations are developed. Next, the existence and uniqueness of solution of state system are proved as well as stability result. Further, some preliminary numerical applications are performed to show that memory reproduced by fractional-order derivatives can play a significant role on key dependent electrical properties including APD, action potential morphology and spontaneous activity.

Blow-Up Rate Estimates for a System of Reaction-Diffusion Equations with Gradient Terms

This paper is concerned with the blow-up properties of Cauchy and Dirichlet problems of a coupled system of Reaction-Diffusion equations with gradient terms. The main goal is to study the influence of the gradient terms on the blow-up profile. Namely, under some conditions on this system, we consider the upper blow-up rate estimates for its blow-up solutions and for the gradients.

Time-varying delays in electrophysiological wave propagation along cardiac tissue and minimax control problems associated with uncertain bidomain type models

Motivated by topics and issues critical to human health, the problem studied in this work derives from the modeling and stabilizing control of electrical cardiac activity in order to maximize the efficiency and safety of treatment for cardiac disease. In this paper we consider nonlinear minimax control problems constrained by an uncertain modified bidomain model of cardiac tissue electrophysiology system, in order to take into account the influence of noises in data and time-delays in signal transmission. The state system is a degenerate nonlinear coupled system of reaction-diffusion equations in the shape of a set of delay differential equations coupled with a set of delay partial differential equations with multiple time-varying delays. The concept of our minimax control approach consists in setting the problem in the worst-case disturbances which leads to the game theory in which the controls and disturbances play antagonistic roles. The proposed strategy consists in controlling these instabilities by acting on certain data to maintain the system in a desired state. First, the mathematical model is introduced and its well-posedness is studied. Second, the minimax control problem is formulated. Afterwards the Frechet differentiability of nonlinear solution map from the couple control-disturbance input to the solution of state system is assessed as well as stability of the derived sensitive system. The existence of an optimal solution is proved and first-order necessary optimality conditions are established by using sensitivity and adjoint calculus.

Structure preserving reduced order modeling for gradient systems

Minimization of energy in gradient systems leads to formation of oscillatory and Turing patterns in reaction-diffusion systems. These patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state. In this paper, a reduced order model (ROM) is developed which preserves the gradient dissipative structure. The coupled system of reaction-diffusion equations are discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of ordinary differential equations (ODEs) are integrated in time by the average vector field (AVF) method, which preserves the energy dissipation of the gradient systems. The ROMs are constructed by the proper orthogonal decomposition (POD) with Galerkin projection. The nonlinear reaction terms are computed efficiently by discrete empirical interpolation method (DEIM). Preservation of the discrete energy of the FOMs and ROMs with POD-DEIM ensures the long term stability of the steady state solutions. Numerical simulations are performed for the gradient dissipative systems with two specific equations; real Ginzburg–Landau equation and Swift–Hohenberg equation. Numerical results demonstrate that the POD-DEIM reduced order solutions preserve well the energy dissipation over time and at the steady state.

Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model

A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other is not. A complete Lie symmetry classification, including a number of the cases characterised as being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications (tumour growth). Graphical representations of the solutions are provided and a biological interpretation is briefly addressed. The results are generalised on multi-dimensional case under the assumption of the radially symmetrical shape of the tumour.

Application of He's Homotopy-perturbation Method to Nonlinear Coupled Systems of Reaction-diffusion Equations

Application of He's Homotopy-perturbation Method to Nonlinear Coupled Systems of Reaction-diffusion Equations

Large Time Behavior and Uniqueness of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations

where N ≥ 1, N∗ = {1, 2, . . . N}, d ≥ 1, pi > 0(i ∈ N∗) and ui,0(i ∈ N∗) are nonnegative bounded and continuous functions. Throughout this paper we mean uN+i = ui, uN+i,0 = ui,0, pN+i = pi for each i ∈ Z and u = (u1, u2, · · · , uN), u0 = (u1,0, u2,0, · · · , uN,0). Problem (1) has a nonnegative and bounded solution at least locally in time (see Theorem 2.1). For any given initial value u0, let T ∗ = T ∗(u0) be the maximal existence time of the solution. If T ∗ = ∞, it is called a global solution. On the other hand, if T ∗ < ∞, there exists i ∈ N∗ such that

Blow-up and Large Time Behavior of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations

Blow-up and Large Time Behavior of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations

The Life Span of Blow-up Solutions for a Weakly Coupled System of Reaction-Diffusion Equations

We consider the weakly coupled system of reaction-diffusion equations \[ u_t=\Delta u+|x|^{\sigma_1}v^p, \quad v_t=\Delta v+|x|^{\sigma_2}u^q , \] \[ u(x,0)=\lambda^\mu\varphi(x), \quad v(x,0)=\lambda^\nu\psi(x) \] where $x\in\mathbf{R}^N$, $t>0$, $p$, $q>1$ and $0\leq\sigma_1<N(p-1)$, $0\leq\sigma_2<N(q-1)$. The existence of solutions, blow-up conditions, and global solutions of the above equations are studied by Mochizuki-Huang. In this paper, we consider the estimate of maximal existence time of blow-up solutions in $I^{\delta_1}\times I^{\delta_2}$ as $\lambda$ goes to 0 or $\infty$.

An efficient computational method for simulations of reaction-diffusion processes

Coupled systems of reaction-diffusion equations are usually solved using a unified time step determined by the most restrictive time scale in the system. We demonstrate in this paper that this time step could be excessively small in many situations, and discuss a more efficient time integration method that does not require linearization procedure and uses different time steps for different equations in a coupled system, depending on their respective time scales. The basic idea is to decouple the system of equations and then apply different time steps to different equations with various time scales. This will also eliminate the need for linearization. Numerical results from a test problem will be discussed to demonstrate significant improvement in computational efficiency.

Tikhonov regularization for an inverse source problem for a coupled system of reaction-diffusion equations

Tikhonov regularization for an inverse source problem for a coupled system of reaction-diffusion equations

Competitive exclusion and persistence in models of resource and sexual competition

We study a combined mathematical model of resource and sexual competition. The population dynamics in this model is analyzed through a coupled system of reaction-diffusion equations. It is shown that strong sexual competition and low birth rate lead to competitive exclusion of the biological species. If sexual competition is weak, then the persistence of the species is possible, depending on the initial density functions and the growth rates of the species. When sexual competition affects both species, persistence and competitive exclusion results are also obtained in terms of the ecological data in the model.

On the dynamics of a reaction-diffusion system

This paper considers application of the decomposition method to the dynamics of a coupled system of reaction-diffusion equations arising in ecology.

A predator-prey reaction-diffusion system with nonlocal effects

We consider a predator-prey system in the form of a coupled system of reaction-diffusion equations with an integral term representing a weighted average of the values of the prey density function, both in past time and space. In a limiting case the system reduces to the Lotka Volterra diffusion system with logistic growth of the prey. We investigate the linear stability of the coexistence steady state and bifurcations occurring from it, and expressions for some of the bifurcating solutions are constructed. None of these bifurcations can occur in the degenerate case when the nonlocal term is in fact local.

Positive Steady-State Solutions of a Competing Reaction-Diffusion System with Large Cross-Diffusion Coefficients

This paper deals with a coupled system of reaction-diffusion equations modeling the competition of two species with cross diffusion. We discuss the existence of positive steady-state solutions in relation to the intrinsic birth ratesa1,a2of species and the cross diffusion coefficients β12,β21. Our results show that positive steady-state solutions exist ifa1anda2lie in certain range, or if β12and β21are sufficiently large.

Coupled system of reaction-diffusion equations and applications in carrier facilitated diffusion

IN RECENT years, the reaction-diffusion systems have been studied and applied extensively to various models arisen from biology and chemistry. In [3] Witta Ebel introduced the concept of a mobile carrier combining reversibly with a substrate as a possible mechanism for facilitated transport across biological membranes (cf. [7]). The equations for this model in a onedimensional domain are given by the system

Asymptotic behavior of a reaction-diffusion system in bacteriology

This paper is concerned with the asymptotic behavior of the solution for a coupled system of reaction-diffusion equations which describes the bacteria growth and the diffusion of histidine and buffer concentrations. Under the basic boundary condition of Neumann type or mixed type the coupled system can have infinitely many steady-state solutions. The present paper gives some explicit information on the asymptotic limit of the time-dependent solution in relation to these steady states. This information exhibits some rather distinct properties of the solutions between the Neumann boundary problem and the Dirichlet or mixed boundary problem.

Models of genetic control by repression with time delays and spatial effects

Two models for cellular control by repression are developed in this paper. The models use standard theory from compartmental analysis and biochemical kinetics. The models include time delays to account for the processes of transcription and translation and diffusion to account for spatial effects in the cell. This consideration leads to a coupled system of reaction-diffusion equations with time delays. An analysis of the steady-state problem is given. Some results on the existence and uniqueness of a global solution and stability of the steady-state problem are summarized, and numerical simulations showing stability and periodicity are presented. A Hopf bifurcation result and a theorem on asymptotic stability are given for the limiting case of the models without diffusion.