System Of Semilinear Parabolic Equations Research Articles

DECOMPOSITION OF TRAVELING WAVES PROBLEMS

In the article, the traveling waves problem for singularly perturbed systems of semilinear parabolic equations is considered. An effective method for the order reduction of singularly perturbed systems is proposed. The obtained mathematical results are used to study traveling waves both for abstract partial differential equations and for a specific model that can arise in physics problems, chemistry, and biology.

Generalization of the Trotter–Daletsky formula for systems of the "reaction–diffusion" type

An iterative method for constructing a solution to the Cauchy problem for a system of parabolic equations with a nonlinear potential has been proposed and substantiated. The method is based on the Trotter–Daletsky formula, generalized for a nonlinear perturbation of an elliptic operator. The idea of generalization is the construction of a composition of the semigroup generated by the Laplacian and the phase flow corresponding to a system of ordinary differential equations. A computational experiment performed for a two-dimensional system of semilinear parabolic equations of the “reaction–diffusion” type confirms estimates for the convergence of iterations established in the proof of this formula. Obtained results suggest the feasibility of an unconventional approach to modeling dynamic systems with distributed parameters.

Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations

A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. The MBP plays a crucial role in understanding the physical meaning and the wellposedness of the mathematical model. Investigation on numerical algorithms with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations, since the violation of MBP may lead to nonphysical solutions or even blow-ups of the algorithms. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge–Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen–Cahn type of equations. To our best knowledge, this is the first time to present a fourth-order linear numerical method preserving the MBP. In addition, convergence of these numerical schemes is proved theoretically and verified numerically, as well as their efficiency by simulations of 2D and 3D long-time evolutional behaviors. Numerical experiments are also carried out for a model which is not a typical gradient flow as the Allen–Cahn type of equations.

On Global Existence of Solutions of Initial Boundary Value Problem for a System of Semilinear Parabolic Equations with Nonlinear Nonlocal Neumann Boundary Conditions

We establish conditions for the existence and nonexistence of global solutions of initial boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal Neumann boundary conditions. We show that these conditions are determined by the behavior of the problem coefficients as t→∞.

Feedback stabilization with one simultaneous control for systems of parabolic equations

In this work controlled systems of semilinear parabolic equations are considered. Only one control is acting in both equations and it is distributed in a subdomain. Local feedback stabilization is studied. The approach is based on approximate controllability for the linearized system and the use of an appropriate norm obtained from a Lyapunov equation. Applications to reaction-diffusion systems are discussed.

Preserving invariance properties of reaction–diffusion systems on stationary surfaces

We propose and analyse a novel surface finite element method that preserves the invariant regions of systems of semilinear parabolic equations on closed compact surfaces in $\mathbb{R}^3$ under discretisation. We also provide a fully-discrete scheme by applying the implicit-explicit (IMEX) Euler method in time. We prove the preservation of the invariant rectangles of the continuous problem under spatial and full discretizations. For scalar equations, these results reduce to the well-known discrete maximum principle. Furthermore, we prove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are provided to support the theoretical findings. In particular we provide examples in which, in the absence of lumping, the numerical solution violates the invariant region leading to blow-up due to the nature of the kinetics.

The spreading fronts in a mutualistic model with delay

This paper is concerned with a system of semilinear parabolic equations with two free boundaries describing the spreading fronts of the invasive species in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that two free boundaries tend monotonically to finite or infinite limits at the same time, and the free boundary problem admits a global slow solution with unbounded free boundaries if the intra-specific competitions are strong, while if the intra-specific competitions are weak, there exist the blowup solution and global fast solution.

On the existence of global solutions of a system of semilinear parabolic equations with nonlinear nonlocal boundary conditions

We establish conditions for the existence and nonexistence of global solutions of an initial–boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal boundary conditions. The results depend on the behavior of variable coefficients as t→∞.

Attractivity in Time-Periodic Parabolic Systems and Application to Reaction–Diffusion Neural Networks

This paper is concerned with the global attractivity for a system of semilinear parabolic equations with homogeneous Dirichlet conditions, in a bounded Lipschitz domain. The coefficients are time-periodic and a delay appears in the nonlinear reaction terms. Our approach uses the Lyapunov method and new average estimates for the Halanay differential inequality. The efficiency and novelty of the introduced criteria are illustrated by applications to reaction–diffusion neural networks.

Blow‐up phenomena for a system of semilinear parabolic equations with nonlinear boundary conditions

AbstractThis paper deals with the blow‐up phenomena for a system of parabolic equations with nonlinear boundary conditions. We show that under some conditions on the nonlinearities, blow‐up occurs at some finite time. We also obtain upper and lower bounds for the blow‐up time when blow‐up occurs. Copyright © 2014 John Wiley & Sons, Ltd.

Asymptotics of eigenvalues of the Jacobi matrix of a system of semilinear parabolic equations

We consider the stationary spatially homogeneous solutions of a system of semilinear parabolic equations in a bounded domain with Neumann boundary conditions. It is well known that the stability of such solutions is related to the signs of the real parts of the eigenvalues of the linearized operator composed of the Jacobi matrix of the dynamical system and the differential operator generated by a diffusion process. We obtain the asymptotics of these eigenvalues. We also study the special case in which the diffusion operator is described by matrices containing Jordan blocks, which corresponds to the case of cross diffusion.

A note on blow-up analysis for a system of semilinear parabolic equations

We consider the behavior of the positive solutions to a semilinear parabolic system u t = Δ u + e p v u α and v t = Δ v + u q e β v with null Dirichlet boundary conditions. We prove that the solutions will blow up in finite time and establish the exact blow-up rates under some conditions.

Global existence and blowup of solutions to a free boundary problem for mutualistic model

This article is concerned with a system of semilinear parabolic equations with a free boundary, which arises in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained. The asymptotic behavior of the free boundary problem is studied. Our results show that the free problem admits a global slow solution if the inter-specific competitions are strong, while if the inter-specific competitions are weak there exist the blowup solution and global fast solution.

Blowup rate estimate for a system of semilinear parabolic equations

In this paper, we study the blowup rate estimate for a system of semilinear parabolic equations. The blowup rate depends on whether the two components of the solution of this system blow up simultaneously or not.

A moving-boundary problem for concrete carbonation: Global existence and uniqueness of weak solutions

This paper deals with a one-dimensional coupled system of semi-linear parabolic equations with a kinetic condition on the moving boundary. The latter furnishes the driving force for the moving boundary. The main result is a global existence and uniqueness theorem of positive weak solutions. The system under consideration is modelled on the so-called carbonation of concrete – a prototypical chemical-corrosion process in a porous solid – concrete – which incorporates slow diffusive transport, interfacial exchange between wet and dry parts of the pores and, in particular, a fast reaction in thin layers, here idealized as a moving-boundary surface in the solid. We include simulation results showing that the model captures the qualitative behaviour of the carbonation process.

A free boundary problem for a predator–prey model

This article is concerned with a system of semilinear parabolic equations with a free boundary, which arises in a predator–prey ecological model. The conditions for the existence and uniqueness of a classical solution are obtained. The evolution of the free boundary problem is studied. It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time.

Time delayed parabolic system in a two-species competitive model with stage structure

A two-species competitive model with stage structure is discussed. The dynamics of coupled system of semilinear parabolic equations with time delays are investigated. Results on the local and global stabilities of the axial equilibria and positive equilibrium are given. Our results show that the introduction of diffusion does not affect the permanence and extinction of the species though the introduction of stage structure brings negative effect on it.

Existence and Stability of Periodic Quasisolutions in Nonlinear Parabolic Systems with Discrete Delays

The existence of periodic quasisolutions and the dynamics for a couple system of semilinear parabolic equations with discrete time delays and periodic coefficients are investigated using the method of upper and lower solutions. It is shown that if the reaction function in the system possesses a mixed quasimonotone property and the periodic boundary value system has a pair of coupled T-upper and lower solutions then the periodic boundary value problem has a pair of periodic quasisolutions and the sector between the quasisolutions is an attractor of the delayed periodic parabolic system. Under some additional conditions the periodic quasisolutions are exactly true periodic solutions of the periodic boundary value system. Finally, three models arising from ecology are given to illustrate the obtained results.

Non–existence of global solutions to anisotropic Fujita–type systems of semilinear parabolic equations

Two different techniques with which to construct subsolutions to anisotropic Fujita–type systems of semilinear parabolic equations on bounded or unbounded domains are developed. The first makes use of the principal eigenfunction of an associated boundary–value problem (bounded domains) or the fundamental solution of the ‘backward’ heat equation (unbounded domains) to cast the problem into a set of integral inequalities. The second is a generalization of a technique developed by Weissler. A series of theorems concerning global non–existence of solutions is established by applying these techniques.

On the cauchy problem for certain system of semilinear parabolic equations

In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following system of parabolic equations: $$\left\{ \begin{gathered} \psi _t = - (\sigma - \alpha )\psi - \sigma \theta _x + \alpha \psi _{xx} , \hfill \\ \theta _t = - (1 - \beta )\theta + v\psi _x + (\psi \theta )_x + \beta \theta _{xx} . \hfill \\ \end{gathered} \right.$$