Global existence to a reaction–diffusion equation through a self‐similar‐like upper solution

A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation ?u_t+Δu+u^p=0 in ?ⁿ×[0, ∞), for exponents p?3 and space dimensions n?3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright © 2009 John Wiley & Sons, Ltd.     

Solving the reaction–diffusion equations with nonlocal boundary conditions based on reproducing kernel space

The reaction–diffusion equations with initial condition and nonlocal boundary conditions are discussed in this article. A reproducing kernel space is constructed, in which an arbitrary function satisfies the initial condition and nonlocal boundary conditions of the reaction‐diffusion equations. Based on the reproducing kernel space, a new algorithm for solving the reaction–diffusion equations with initial condition and nonlocal boundary conditions is presented. Some examples are displayed to demonstrate the validity and applicability of the proposed method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Reaction–diffusion systems for the microscopic cellular model of the cardiac electric field

The paper deals with a mathematical model for the electric activity of the heart at microscopic level. The membrane model used to describe the ionic currents is a generalization of the phase‐I Luo–Rudy, a model widely used in 2‐D and 3‐D simulations of the action potential propagation. From the mathematical viewpoint the model is made up of a parabolic reaction diffusion system coupled with an ODE system. We derive existence and some regularity results. Copyright © 2006 John Wiley & Sons, Ltd.     

Spike patterns in a reaction–diffusion ODE model with Turing instability

We explore a mechanism of pattern formation arising in processes described by a system of a single reaction–diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions between cellular processes and diffusing growth factors. We focus on the model of early carcinogenesis proposed by Marciniak‐Czochra and Kimmel, which is an example of a wider class of pattern formation models with an autocatalytic non‐diffusing component. We present a numerical study showing emergence of periodic and irregular spike patterns because of diffusion‐driven instability. To control the accuracy of simulations, we develop a numerical code on the basis of the finite‐element method and adaptive mesh grid. Simulations, supplemented by numerical analysis, indicate a novel pattern formation phenomenon on the basis of the emergence of nonstationary structures tending asymptotically to a sum of Dirac deltas. Copyright © 2013 John Wiley & Sons, Ltd.     

On the well‐posedness for Keller–Segel system with fractional diffusion

In this paper, we study the Cauchy problem for the Keller–Segel system with fractional diffusion generalizing the Keller–Segel model of chemotaxis for the initial data (u₀,v₀) in critical Fourier‐Herz spaces with q ∈ [2, ∞], where 1 < α ≤ 2. Making use of some estimates of the linear dissipative equation in the frame of mixed time‐space spaces, the Chemin ‘mono‐norm method’, the Fourier localization technique and the Littlewood–Paley theory, we get a local well‐posedness result and a global well‐posedness result with a small initial data. In addition, ill‐posedness for ‘doubly parabolic’ models is also studied. Copyright © 2011 John Wiley & Sons, Ltd.     

Lyapunov functionals for reaction–diffusion equations with memory

We consider a reaction‐diffusion equation in which the usual diffusion term also depends on the past history of the diffusion itself. This equation has been analysed by several authors, with an emphasis on the longtime behaviour of the solutions. In this respect, the first results have been obtained by using the past history approach. They show that the equation, subject to a suitable boundary condition, defines a dissipative dynamical system which possesses a global attractor. A similar theorem has been recently proved by Chepyzhov and Miranville, using a different method based on the notion of trajectory attractors. In addition, those authors provide sufficient conditions that ensure the existence of a Lyapunov functional. Here we show that a similar result can be demonstrated within the past history approach, with less restrictive conditions. Copyright © 2005 John Wiley & Sons, Ltd.     

Iterative operator‐splitting methods for nonlinear differential equations and applications

In this article, we consider iterative operator‐splitting methods for nonlinear differential equations with bounded and unbounded operators. The main feature of the proposed idea is the embedding of Newton's method for solving the split parts of the nonlinear equation at each step. The convergence properties of such a mixed method are studied and demonstrated. We confirm with numerical applications the effectiveness of the proposed scheme in comparison with the standard operator‐splitting methods by providing improved results and convergence rates. We apply our results to deposition processes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1026–1054, 2011     

An upwind finite volume scheme and its maximum‐principle‐preserving ADI splitting for unsteady‐state advection‐diffusion equations

We develop an upwind finite volume (UFV) scheme for unsteady‐state advection‐diffusion partial differential equations (PDEs) in multiple space dimensions. We apply an alternating direction implicit (ADI) splitting technique to accelerate the solution process of the numerical scheme. We investigate and analyze the reason why the conventional ADI splitting does not satisfy maximum principle in the context of advection‐diffusion PDEs. Based on the analysis, we propose a new ADI splitting of the upwind finite volume scheme, the alternating‐direction implicit, upwind finite volume (ADFV) scheme. We prove that both UFV and ADFV schemes satisfy maximum principle and are unconditionally stable. We also derive their error estimates. Numerical results are presented to observe the performance of these schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 211–226, 2003     

Existence of global solutions for a three‐species predator‐prey model with cross‐diffusion

In this short note, we study a strongly coupled system of partial differential equations which models the dynamics of a two‐predator‐one‐prey ecosystem in which the prey exercises defense switching and the predators collaboratively take advantage of the prey's strategy. We prove the existence of global strong solutions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A second‐order linearized difference scheme for a strongly coupled reaction‐diffusion system

This article deals with the numerical solution to some models described by the system of strongly coupled reaction–diffusion equations with the Neumann boundary value conditions. A linearized three‐level scheme is derived by the method of reduction of order. The uniquely solvability and second‐order convergence in L₂‐norm are proved by the energy method. A numerical example is presented to demonstrate the accuracy and efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Strong well‐posedness of a three phase problem with nonlinear transmission condition

We prove existence and uniqueness of strong solutions to a quasilinear parabolic‐elliptic system modelling an ionic exchanger. This chemical system consists of three phases connected with nonlinear boundary conditions. The most interesting difficulty of our problem manifests in the nonlinear transmission condition, as almost all quantities are non‐linearly involved in this boundary equation. Our approach is based on the contraction mapping principle, where maximal L_p‐regularity of the associated linear problem is used to obtain a fixed point equation of the starting problem. Copyright © 2012 John Wiley & Sons, Ltd.     

A weak solution approach to a reaction–diffusion system modeling pattern formation on seashells

We investigate a reaction–diffusion system proposed by H. Meinhardt as a model for pattern formation on seashells. We give a new proof for the existence of a local weak solution for general initial conditions and parameters upon using an iterative approach. Furthermore, the solution is shown to exist globally for suitable initial data. The behavior of the solution in time and space is illustrated through numerical simulations. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamical transition for S‐K‐T biological competing model with cross‐diffusion

The main objective of this paper is to study the dynamical transition for the S‐K‐T biological competition system with cross‐diffusion. Based on the spectral analysis, the principle of exchange of stabilities conditions for eigenvalues are obtained. By using the dynamical transition theory, 2 different types of dynamical transition for the S‐K‐T model are also derived. In addition, an example is given to illustrate our main results.     

An efficient high‐order algorithm for solving systems of reaction‐diffusion equations

An efficient higher‐order finite difference algorithm is presented in this article for solving systems of two‐dimensional reaction‐diffusion equations with nonlinear reaction terms. The method is fourth‐order accurate in both the temporal and spatial dimensions. It requires only a regular five‐point difference stencil similar to that used in the standard second‐order algorithm, such as the Crank‐Nicolson algorithm. The Padé approximation and Richardson extrapolation are used to achieve high‐order accuracy in the spatial and temporal dimensions, respectively. Numerical examples are presented to demonstrate the efficiency and accuracy of the new algorithm. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 340–354, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10012     

A Lyapunov functional for a triangular reaction–diffusion system with nonlinearities of exponential growth

The aim of this work is to study the global existence of solutions to a triangular system of reaction–diffusion equations, which describes epidemiological or chemical situations. On the basis of the construction of a suitable Lyapunov functional, we show that for any initial data, classical global solutions exist even when the nonlinearities are of exponential growth. Copyright © 2012 John Wiley & Sons, Ltd.     

The four‐dimensional Martínez Alonso–Shabat equation: nonlinear self‐adjointness and conservation laws

We show that the four‐dimensional Martínez Alonso–Shabat equation is nonlinearly self‐adjoint with differential substitution and the required differential substitution is just the admitted adjoint symmetry and vice versa. By means of computer algebra system, we obtain a number of local and nonlocal symmetries admitted by the equations under study. Then such symmetries are used to construct conservation laws of the equation under study and its reductions. Copyright © 2016 John Wiley & Sons, Ltd.     

Blow‐up solutions for localized reaction‐diffusion equations with variable exponents

This paper deals with radial solutions to localized reaction‐diffusion equations with variable exponents, subject to homogeneous Dirichlet boundary conditions. The global existence versus blow‐up criteria are studied in terms of the variable exponents. We proposed that the maximums of variable exponents are the key clue to determine blow‐up classifications and describe blow‐up rates for positive solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.