Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 791304, 13 pages

http://dx.doi.org/10.1155/2015/791304

## Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation

Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitni 8, 30614 Pilsen, Czech Republic

Received 24 March 2015; Accepted 29 July 2015

Academic Editor: Cengiz Çinar

Copyright © 2015 Petr Stehlík and Jonáš Volek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study reaction-diffusion equations with a general reaction function on one-dimensional lattices with continuous or discrete time , . We prove weak and strong maximum and minimum principles for corresponding initial-boundary value problems. Whereas the maximum principles in the semidiscrete case (continuous time) exhibit similar features to those of fully continuous reaction-diffusion model, in the discrete case the weak maximum principle holds for a smaller class of functions and the strong maximum principle is valid in a weaker sense. We describe in detail how the validity of maximum principles depends on the nonlinearity and the time step. We illustrate our results on the Nagumo equation with the bistable nonlinearity.

#### 1. Introduction

Reaction-diffusion equation (sometimes called FKPP equation, which abbreviates Fisher, Kolmogorov, Petrovsky, Piskounov) serves as a nonlinear model to describe a class of (biological, chemical, economic, and so forth) phenomena in which two factors are combined. Firstly, the diffusion process causes the concentration of a substance (animals, wealth, and so forth) to spread in space. Secondly, a local reaction leads to dynamics based on the concentration values.

For the sake of applications and correctness of numerical procedures it makes sense to consider partially or fully discretized reaction-diffusion equation. In certain situations (e.g., spatially structured environment) it is natural to study reaction-diffusion equations with discretized space variable and continuous time (we refer to it as a semidiscrete problem and use ):or, for example, if nonoverlapping populations are considered, with both time and space variables being discrete (a discrete problem, ):Examples of such phenomena are chemical reactions related to crystal formation, see Cahn [1], or myelinated nerve axons, see Bell and Cosner [2] and Keener [3]. Existence and nonexistence of travelling waves in those models have been recently studied in Chow [4], Chow et al. [5], and Zinner [6] mostly with the cubic (or bistable, double-well) nonlinearities of the form , with and (this special case of FKPP equation is being referred to as Nagumo equation). In contrast, various reaction functions have been proposed in models without spatial interaction, for example, Xu et al. [7].

Motivated by these facts, we allow for a general form of the reaction function in this paper (i.e., we do not restrict ourselves to cubic nonlinearities). We prove a priori estimates for discrete reaction-diffusion equation (2) and then use Euler method to show their validity for semidiscrete reaction-diffusion equation (1). Whereas the maximum principles in the semidiscrete case exhibit similar features to those of continuous reaction-diffusion model (i.e., they hold under similar assumptions), in the discrete case the weak maximum principle holds for a smaller class of functions and the strong maximum principle is only valid in a weaker sense involving the domain of dependence. Finally, we use the maximum principles to get the global existence of solutions of the initial-boundary problem for the semidiscrete case (1). All our results are illustrated in detail in Nagumo equations with a symmetric bistable nonlinearity; that is, we consider problems (1)-(2) with .

Our motivation is twofold. First, maximum principles could be used to obtain comparison principles (Protter and Weinberger [8]), which in turn could serve as a valuable tool in the study of traveling waves, for example, Bell and Cosner [2]. Moreover, similarly as in the case of (non)existence of traveling wave solutions for Nagumo equations, it has been shown that discrete and semidiscrete structures influence the validity of maximum principles in a significant way. Even the simplest one-dimensional linear problems require additional assumptions on the step size; see Mawhin et al. [9] and Stehlík and Thompson [10]. In the case of partial difference and semidiscrete equations, the strong influence of the underlying structure on maximum principles has been described in the linear case for transport equation in Stehlík and Volek [11] and for diffusion-type equations in Slavík and Stehlík [12] and Friesl et al. [13] (interestingly, the proofs of maximum principles in this case are based on product integration; see Slavík [14]). Finally, simple maximum principles for nonlinear transport equations on semidiscrete domains have been presented in Volek [15].

In the classical case, maximum principles for diffusion (and parabolic) equations go back to Picone [16] and Levi [17]. Strong maximum principles were later established by Nirenberg [18] and a survey of various versions and applications could be found in a classical monograph Protter and Weinberger [8].

This paper is segmented in the following way. In Section 2, we briefly summarize results for the classical reaction-diffusion equation. Next, we prove weak and strong maximum principles for the discrete case (2) (Sections 3 and 4). In the case of the initial-boundary value problem for the semidiscrete equation (1) we provide local existence results (Section 5) and maximum principles (Section 6) which we consequently apply to get global existence of solutions in Section 7. Our results are then applied to the Nagumo equation with a symmetric bistable nonlinearity, that is, problems (1)-(2) with , in Section 8.

#### 2. Reaction-Diffusion Partial Differential Equation

In order to motivate and compare our results for the reaction-diffusion equations on discrete-space domains with the classical reaction-diffusion equation we briefly summarize few basic results for the following initial-boundary problem:where is a reaction function and , are initial-boundary conditions satisfying and .

The following existence and uniqueness result for (3) can be found, for example, in [19, page 298].

Theorem 1. *Let be arbitrary and let be uniformly Hölder continuous in and and Lipschitz in for . Then for all Hölder continuous initial-boundary conditions , , problem (3) has a unique bounded solution which is defined on .*

We define the following two numbers for the brevity:

For the linear diffusion equation (i.e., (3) with ) the maximum principle is proved, for example, in [8, Chapter 3.1]. For the nonlinear problem (3) (i.e., ) the following weak maximum principle holds (see [19, Theorem 1]).

Theorem 2. *Let be arbitrary and let be uniformly Hölder continuous in and and Lipschitz in for and assume that**Let be a continuous solution of (3) with Hölder continuous initial-boundary conditions , , . Then **holds for all .*

Moreover, the strong maximum principle also holds (see [19, Theorem 2]).

Theorem 3. *Let the assumptions of Theorem 2 be satisfied and let be a solution of (3) on . If (or ) for some then *

*3. Discrete Reaction-Diffusion Equation: Weak Maximum Principles*

*Let us consider the initial-boundary value problem for the discrete reaction-diffusion equation (which could be obtained, e.g., by Euler discretization of (3)):where is a reaction function, are initial-boundary conditions, , , and (for brevity, we assume the space discretization step , but all our results are easily extendable to an arbitrary step if we use the diffusion constant instead of ; we discuss this in detail in a specific example at the end of Section 8).*

*Straightforwardly, problem (8) has a unique solution which is defined in , since is uniquely given byFor , we define the following two numbers:For brevity of the following assertions we formulate the assumption in the reaction function : ( D)Let and let satisfy for all , and .*

*Remark 4. *The inequalities (12) imply that for all fixed and the graph of function does not intersect the forbidden area depicted in Figure 1.