#### 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.

*Remark 5. *Let us notice that for the slope goes to ; that is, the forbidden area from Remark 4 is smaller in the sense of inclusion and it is easier to satisfy assumption () if we decrease the time discretization step . We illustrate this fact in Figure 1.

Proposition 6. *Assume that . If then does not hold for any function .*

Note that the inequality is the necessary condition for the validity of maximum principles even in the linear case; see, for example, [13, Theorem 2.4].

* Proof. *If (i.e., ) then from (12) there should be a contradiction.

*Remark 7. *Notice that if then () implies that for all and . This situation corresponds to the case of the constant initial-boundary conditions and . From and from (9) there is

Now we state an auxiliary lemma which is crucial in the proof of the maximum principle.

Lemma 8. *Let , let function satisfy (), and let be the unique solution of (8). Then for all and for all *

*Proof. *For the sake of brevity, we only show that . The inequality can be proved in the same way.

Let , , be arbitrary. Then and trivially from the definition of (10) (recall that , i.e., ). If then we can estimate Thanks to the assumptions (12) and we get Therefore,

The weak maximum principle follows immediately.

Theorem 9. *Let be arbitrary, let function satisfy (), and let be the unique solution of (8). Then**holds for all and .*

* Proof. *From (10) to (11) we get for all . Immediately, Lemma 8 yields that (19) holds for all and .

*Remark 10. *If the reaction function does not satisfy the inequalities (12) we can find a counterexample that the maximum principle does not hold in general. For example, let us consider (8) with , , , and , . Let us assume that, for example, the latter inequality in (12) does not hold; that is, for some . Assuming without loss of generality that , then the maximum principle is straightforwardly violated since

In certain cases, the function could fail to satisfy () but could still provide a priori bounds for solutions of (8) if the following inequalities hold.()Let and let there exist and such that for all and such that and . In that case, we obtain a general version of the weak maximum principle (for the illustration of () see Figure 2).

Theorem 11. *Let be arbitrary, let function satisfy (), and let be the unique solution of (8). Then **holds for all and such that .*

* Proof. *For we have Now we can proceed analogously as in the proofs of Lemma 8 and Theorem 9 where we use () instead of (). We omit the details.

*Example 12. *The set of nonlinear reaction functions that could be considered in Theorem 9 or 11 includes, for example, (for the detailed analysis with see Section 8) (i) with ,(ii)the logistic function ,(iii)the bistable nonlinearity , ,(iv) where ,(v).

We state the following two claims that are direct corollaries of Theorem 9.

Corollary 13. *Assume that , are bounded. Let satisfy () for all . Then the unique solution of (8) is bounded.*

Corollary 14. *Assume that , , are nonnegative. Let satisfy () for all . Then the unique solution of (8) is nonnegative.*

#### 4. Discrete Reaction-Diffusion Equation: Strong Maximum Principle

As in the case of classical reaction-diffusion equation (3) (Theorem 3) we naturally turn our attention to strong maximum principles. Straightforwardly, the strong maximum principle does not hold in the discrete case in the sense of Theorem 3.

*Example 15. *Let us consider problem (8) with , , , and (note that ) and letThen from (9) we get Analogously, we can deduce that Consequently, the strong maximum principle does not hold.

Nonetheless, given the fact that the values of are given by (9), we can easily construct the domain of dependence of :and the domain of influence of :

Considering the following:()Let and let satisfy for all , :(a) when ,(b) when ,(c) and ,the weaker version of the strong maximum principle follows immediately.

Theorem 16. *Assume that the function satisfies () for all . Let be the unique solution of (8) and . *(1)*If (or ), then (or ) on .*(2)*If (or ), then (or ) on .*

* Proof. *Let us only focus on the former statement of the theorem; the latter could be proved in very similar way. We show that if the function satisfies () and for some , , , then . The rest follows by induction.

Assume by contradiction first that (the case follows easily). Using this assumption, (9), and Theorem 9 we can estimateThus, () yields Consequently, there has to be a contradiction.

If , then by the similar procedure as above we obtain Since in this case, () implies that Hence, a contradiction.

In the case of nonconstant time discretization we can follow similar techniques and consider () (eventually, () or ()) with (or for ); see Remark 5.

#### 5. Semidiscrete Reaction-Diffusion Equation: Local Existence

In this section we study the local existence of the following initial-boundary value problem on semidiscrete domains:where denotes the time derivative, is a reaction function, , are initial-boundary conditions, and .

Given the fact that (36) can be interpreted as a vector ODE, we can rewrite it aswhere , is continuous and .

Naturally, we use the well-known result of Picard and Lindelöf to get the local existence for the initial value problem (37) (see [20, Theorem 8.13]).

Theorem 17. *Assume that is continuous on the rectangle **and satisfies the Lipschitz condition on ; that is, there exists such that, for all **holds. Then there exists such that (37) has a unique solution defined on .*

We apply Theorem 17 to get the local existence for the semidiscrete reaction-diffusion equation (36). We use the following two assumptions:()Let be continuous in for all .()Let be locally Lipschitz with respect to on ; that is, for all , , and there exist and * *and such that for all there is

Theorem 18. *Let satisfy () and (). Then there exists such that (36) has a unique solution defined on .*

* Proof. *Since the space variable is from a finite set problem (36) corresponds to the following vector ODE:coupled with the initial condition Thus, problem (36) can be rewritten in the vector form as follows:Assumptions () and () yield that the nonlinear function is continuous and satisfies Lipschitz condition with respect to on some rectangle . Since the term is linear and therefore Lipschitz with respect to and the assumptions of Theorem 17 are satisfied. Consequently, there exists such that (44) has a unique solution on .

*Remark 19. *If we assume only () then we can apply the Peano theorem [20, Theorem 8.27] instead of Theorem 17 to get the local existence of solutions of (36) which need not be unique.

#### 6. Semidiscrete Reaction-Diffusion Equation: Maximum Principles

Having the local existence and uniqueness we focus on the maximum principles for (36). In the following analysis we approximate the solution of (36) by the solutions of the discrete problem (8) which arises from (36) by the explicit (*Euler*) discretization of the time variable.

First, we define the Euler polygon (see [21, I.7]).

*Definition 20. *Let be a discretization step. Consider the initial value problem (37) on the interval where , . Define the subdivision of interval as the set of points , , and for define Then the continuous function defined byis called* Euler polygon*.

The following statement sums up the convergence of Euler method (see [21, I.7, Theorem 7.3 and I.9, page 54]).

Theorem 21. *Let and let be continuous, satisfying Lipschitz condition on **and let be bounded by a constant on . If then the following hold: *(a)*for the Euler polygons converge uniformly to a continuous function on ,*(b)* and it is the unique solution of (37) on .*

We define the bounds of initial-boundary conditions similarly as in the discrete problem:

Before we state the weak maximum principle we describe the connection between the discretization of (36) and the assumption ().

Lemma 22. *Let and let satisfy (), (), and *()* for all **, *.* Then there exists such that for all **holds for all , , and .*

* Proof. *We prove the latter inequality in (49) by contradiction. The former inequality can be proved in the same way. Let us assume that for all there exist , , , and such that Therefore, there exist sequences and (we denote , , ) such thatFirst, we observe that if for some we get a contradiction with (). Thus, we can assume that for all .

Now we have to distinguish between two cases. (i)If there does not exist any subsequence such that then the right-hand side of inequality in (51) goes to infinity. Hence, from (51) also goes to infinity. This yields a contradiction with (), which implies boundedness of the function on .(ii)Let there exist a subsequence such that . We show that we get a contradiction with () in this case. Since the interval is bounded there exists a convergent subsequence such that . Analogically, since is bounded there also exists a convergent subsequence such that . Let , and be arbitrary. Then we can find sufficiently large such that If we put , , , and and is the rectangle from assumption () with given and then . Now from (51), (52), and () we can estimate a contradiction with ().

*Remark 23. *The assumption () defines the forbidden area for a reaction function in the same way as (). However, this area is reduced to a pair of half-lines. Let us notice that it is the limit case of forbidden areas for the discrete case if (see Remark 5 and Figure 1). Moreover, it is equivalent to the assumption for classical PDEs; see (5).

Theorem 24. *Let be arbitrary, let satisfy (), (), and (), and let be a solution of (36) defined on . Then**holds for all , .*

* Proof. *We prove that for all , there is . The first inequality in (54) can be proved similarly. Let us assume by contradiction that there exist and such that From the continuity of the solution there exist and such that (a) for all and ,(b),(c)there exists such that

Let us analyze the new initial-boundary value problem (36) with the initial condition at time . Let us understand this problem as the initial value problem for the vector ODE (44) with the initial condition at time .

From (), () we know that is continuous and Lipschitz on some rectangle . From () we also get that is bounded by some constant on . Therefore, Theorem 21 implies that for sufficiently small discretization steps and for sufficiently small interval the Euler polygons converge uniformly to the unique solution on .

Notice that the node points of Euler polygons are the solutions of (8). From (), (), and () and from Lemma 22 the assumption (*D*) is satisfied (recall that is sufficiently small) and therefore, from Theorem 9But if converge uniformly to and on for all then there has to be a contradiction with (56).

If the assumption () is not satisfied but the nonlinear function satisfies the following:()Let be arbitrary and let there exist and such that for all , , then we can state the following generalized weak maximum principle.

Theorem 25. *Let be arbitrary, let satisfy (), (), and (), and let be a solution of (36) defined on . Then **holds for all , .*

* Proof. *The statement can be proved in the similar way as Lemma 22 and Theorem 24.

As in the previous sections we want to establish the strong maximum principle. First, we recall the well-known Grönwall’s inequality (see, e.g., [20, Corollary 8.62]).

Lemma 26. *Let be continuous functions and let be differentiable on . If **then *

Further, we need the following auxiliary lemma.

Lemma 27. *Let be arbitrary, let satisfy (), (), and (), and let be a solution of (36) defined on . If (or ) for some and then *

* Proof. *We prove the statement with . The case with is similar. First, the weak maximum principle (Theorem 24) holds; that is, for all and . Suppose by contradiction that there exists such thatWithout loss of generality let be sufficiently small so that the function is uniformly Lipschitz in on with Lipschitz constant (follows from ()). With the help of these facts and also from () we can estimatefor all . If we denote then the function satisfies on . If we substitute then the function satisfies the following differential inequality: Therefore, Grönwall’s inequality (Lemma 26) implies that and hence on , a contradiction.

The strong maximum principle for (36) follows immediately.

Theorem 28. *Let be arbitrary, let satisfy (), (), and (), and let be a solution of (36) defined on . If (or ) for some and then *

* Proof. *Lemma 27 yields that for all . If (or ) at some then applying () the following has to be satisfied:a contradiction with the fact that the function is constant and equal to on . Therefore, functions and are also constant and equal to on . Then we can continue inductively in to the boundary points or . The case with is similar.

#### 7. Semidiscrete Reaction-Diffusion Equation: Global Existence

In this section we combine the local existence and uniqueness and the maximum principle to obtain the global existence of solution of (36).

Once again, we use known results from the theory of ordinary differential equations. First, we define the maximal interval of existence (see [20, Definition 8.31]).

*Definition 29. *Let be continuous and let be a solution of (37) defined on . Then one says is a* maximal interval of existence* for if there does not exist an and a solution defined on such that for .

In the following we apply the extendability theorem (see [20, Theorem 8.33]).

Theorem 30. *Let be continuous and let be a solution of (37) defined on . Then can be extended to a maximal interval of existence , . Furthermore, there is either *

This theorem enables us to conclude with the global existence for (36).

Theorem 31. *Let satisfy (), (), and () for all . Then (36) has a unique solution defined on which satisfies *

*for all .*

* Proof. *Let us understand problem (36) as the initial value problem for the vector ODE (44). From Theorem 18 there exists a uniquely determined local solution . From Theorem 30 the solution can be extended to a maximal interval of existence which is open from right and either If for then for all there have to exist and such that If we put (, are functions on and therefore bounded on ) then a contradiction with the maximum principle in Theorem 24 (which holds thanks to (), (), and ()).

Therefore, there has to be ; that is, the solution is defined on the interval and from (