Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation

We study reaction-diffusion equations with a general reaction function f on one-dimensional lattices with continuous or discrete time u󸀠 x (or Δ t u x ) = k(u x−1 − 2u x + u x+1 ) + f(u x ), x ∈ Z. 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.


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   () = (, )):    () =  ( −1 () − 2  () +  +1 ()) +  (  ()) , or, for example, if nonoverlapping populations are considered, with both time and space variables being discrete (a discrete problem,  , := (, )): ,+ℎ −  , ℎ =  ( −1, − 2 , +  +1, ) +  ( , ) ,  ∈ Z,  ∈ {0, ℎ, 2ℎ, . ..} . ( 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 () = ( − )(1 − ), with  > 0 and  ∈ (0, 1) (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 reactiondiffusion 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 () = (1 −  2 ).
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 reactiondiffusion 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 () = (1 −  2 ), in Section 8.

Reaction-Diffusion Partial Differential Equation
In order to motivate and compare our results for the reactiondiffusion equations on discrete-space domains with the classical reaction-diffusion equation we briefly summarize few basic results for the following initial-boundary problem: where  : (, ) × R + × R → R is a reaction function and  : [, ] → R,   ,   : R + 0 → R are initial-boundary conditions satisfying () =   (0) and () =   (0).

Discrete Reaction-Diffusion Equation:
Weak Maximum Principles For  ∈ ℎN 0 , we define the following two numbers: For brevity of the following assertions we formulate the assumption in the reaction function : for all  ∈ (, ) Z ,  ∈ [0, ] ℎN 0 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 ℎ → 0+ the slope (2ℎ − 1)/ℎ 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 ℎ.
We illustrate this fact in Figure 1.Note that the inequality ℎ ≤ 1/2 is the necessary condition for the validity of maximum principles even in the linear case; see, for example, [13,Theorem 2.4].
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  = −1,  = 1,  ∈ N 0 , and () ≡ 0,   () ≡   () ≡ 0. Let us assume that, for example, the latter inequality in (12) does not hold; that is, for some  ∈ N 0 .Assuming without loss of generality that  = 0, 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  ∈ ℎN 0 and let there exist  ≥   and  ≤   such that The example of the function  that does not satisfy () but satisfies (  ) for some constants , .Such a function consequently provides a priori bounds for solutions of ( 8) in the sense of Theorem 11.
In that case, we obtain a general version of the weak maximum principle (for the illustration of (  ) see Figure 2).
Proof.For  = 0 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 (, , ) = (1 −  2 ) see Section 8) We state the following two claims that are direct corollaries of Theorem 9.
Corollary 13.Assume that   ,   are bounded.Let  satisfy () for all  > 0. Then the unique solution  of ( 8) is bounded.Corollary 14. Assume that ,   ,   are nonnegative.Let  satisfy () for all  > 0. Then the unique solution  of ( 8) is nonnegative.

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.
Nonetheless, given the fact that the values of (, ) are given by ( 9), we can easily construct the domain of dependence of ( 0 ,  0 ): and the domain of influence of ( 0 ,  0 ): Considering the following: Theorem 16.Assume that the function  satisfies (  ) for all  ∈ ℎN 0 .Let  be the unique solution of ( 8) and ( 0 ,  0 ) ∈ [, ] Z × ℎN 0 .
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 ( 0 ,  0 ) =   for some The rest follows by induction.

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,  : (, ) are initial-boundary conditions, and R + 0 := [0, +∞).Given the fact that (36) can be interpreted as a vector ODE, we can rewrite it as where u : 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 g is continuous on the rectangle
and satisfies the Lipschitz condition on ; that is, there exists  > 0 such that, for all holds.Then there exists  > 0 such that (37) has a unique solution u defined on [0, ].
We apply Theorem 17 to get the local existence for the semidiscrete reaction-diffusion equation (36).We use the following two assumptions: ( lip ) Let (, , ) be locally Lipschitz with respect to  on (, ) Z × R + 0 × R; that is, for all  0 ∈ (, ) Z ,  0 ∈ R + 0 , and  0 ∈ R there exist ,  > 0 and and  > 0 such that for all ( 0 ,  ) ) ) ) ) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (43) Thus, problem (36) can be rewritten in the vector form as follows: Assumptions ( cont ) and ( lip ) yield that the nonlinear function f is continuous and satisfies Lipschitz condition with respect to u on some rectangle .Since the term Au is linear and therefore Lipschitz with respect to u and  ∈  1 (R + 0 ) the assumptions of Theorem 17 are satisfied.Consequently, there exists  > 0 such that (44) has a unique solution on [0, ].
Remark 19.If we assume only ( cont ) 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.

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]).
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 ().
Proof.We prove the latter inequality in (49) by contradiction.
Now we have to distinguish between two cases.
(i) If there does not exist any subsequence {   } ∞ =1 ⊂ {  } ∞ =1 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 ( cont ), which implies boundedness of the function  on (, ) We show that we get a contradiction with ( lip ) in this case.Since the interval (, ) Z is bounded there exists a convergent subsequence Let ,  > 0, and  > 0 be arbitrary.Then we can find l ∈ N sufficiently large such that If we put x :=   l , t :=   l , û :=   l , and ũ :=   and ( x, t, ũ) is the rectangle from assumption ( lip ) with given  > 0 and  > 0 then ( x, t, û), ( x, t, ũ) ∈ ( x, t, ũ).Now from (51), (52), and ( sign ) we can estimate a contradiction with ( lip ).
Remark 23.The assumption ( sign ) 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 ℎ → 0+ (see Remark 5 and Figure 1).Moreover, it is equivalent to the assumption for classical PDEs; see (5).
Theorem 24.Let  > 0 be arbitrary, let  satisfy ( cont ), ( lip ), and (  ), and let  be a solution of ( 36 Proof.We prove that for all  ∈ (, ) Z ,  ∈ [0, ] there is (, ) ≤   .The first inequality in (54) can be proved similarly.Let us assume by contradiction that there exist   ∈ (, ) Z and   ∈ (0, ] such that From the continuity of the solution  there exist  0 ∈ (, ) Z and  0 ∈ [0,   ) such that (56) Let us analyze the new initial-boundary value problem (36) with the initial condition (,  0 ) at time  0 .Let us understand this problem as the initial value problem for the vector ODE (44) with the initial condition at time  0 .
From ( cont ), ( lip ) we know that f(, u) is continuous and Lipschitz on some rectangle .From ( cont ) we also get that f is bounded by some constant  > 0 on .Therefore, Theorem 21 implies that for sufficiently small discretization steps ℎ > 0 and for sufficiently small interval [ 0 ,  0 + ] the Euler polygons y (ℎ) () converge uniformly to the unique solution u() on [ 0 ,  0 + ].
If the assumption ( sign ) is not satisfied but the nonlinear function  satisfies the following: (  sign ) Let  > 0 be arbitrary and let there exist  ≥   and  ≤   such that for all  ∈ (, ) Z ,  ∈ [0, ], then we can state the following generalized weak maximum principle.
Theorem 25.Let  > 0 be arbitrary, let  satisfy ( cont ), ( lip ), and (   ), and let  be a solution of ( 36 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 ,  : [, ] → R be continuous functions and let  be differentiable on (, ).If then Further, we need the following auxiliary lemma.
The strong maximum principle for (36) follows immediately.

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 g be continuous and let u be a solution of (37) defined on [0, ).Then one says [0, ) is a maximal interval of existence for u if there does not exist an  1 >  and a solution w defined on [0,  1 ) such that u() = w() for  ∈ [0, ).
In the following we apply the extendability theorem (see [20,Theorem 8.33]).
Theorem 30.Let g be continuous and let u be a solution of (37) defined on [0, ).Then  can be extended to a maximal interval of existence [0, ), 0 <  ≤ ∞.Furthermore, there is either a contradiction with the maximum principle in Theorem 24 (which holds thanks to ( cont ), ( lip ), and ( sign )).
Therefore, there has to be  = ∞; that is, the solution u() is defined on the interval [0, ∞) and from ( lip ) it has to be unique.
Remark 32.All nonlinear functions  listed in Example 12 can be considered in Theorems 24 and 31.However, it is worth noting that we have additional assumptions on the nonlinearity in the semidiscrete case (conditions ( cont ) and ( lip )).Thus, for non-Lipschitz functions (e.g., with  ∈ [0, 1)), we only get the maximum principles in discrete case.On the other hand, in discrete case, the validity of maximum principle depends strongly on the interaction between the discretization step ℎ and the nonlinearity  (see (12); we illustrate this dependence in detail in Section 8).
Let us finish with the two corollaries that are immediate consequences of Theorems 24 and 31.

Application: Discrete and Semidiscrete Nagumo Equation
In this section we apply the results of this paper to the most common nonlinearity occurring in the connection with the reaction-diffusion equation, the bistable/double-well nonlinearity.For simplicity, we consider only the symmetric case and use interval [−1, 1] so that our arguments for positive values can be directly reproduced for the negative ones; that is, we study Throughout this section we assume that the initial-boundary conditions ,   ,   are such that   = −1 and   = 1 (or possibly   ≥ −1 and   ≤ 1) for all  > 0.
Consequently, we can apply Theorem 11 whenever  ≤ √ 2, which is equivalent to If we choose  >  * , then we can easily observe that (−) lies above the tangent line and therefore Theorem 11 cannot be applied (see Figure 3(d)).
Since we intentionally chose a symmetric , we can repeat the same argument on the lower bound of solutions of (78).
To sum up, depending on values of  and ℎ we obtain the following bounds for the solution of (78) In the light gray area, the bounds follow from Theorem 9.In the dark gray area, the bounds are implied by Theorem 11 and  is given by (81).In the white area we have no bounds on solutions.

𝑢 (𝑥, 𝑡)
one could get the same bounds as in (83) by replacing  with /ℎ 2  .The dependence of regions of maximum principles' validity on time and space discretization steps ℎ  and ℎ  for  > 0 is depicted in Figure 4(b) (notice that very small values of ℎ  are necessary for small ℎ  ).

Final Remarks
In this paper, we studied a priori bounds for solutions of initial-boundary value problems related to discrete and semidiscrete diffusion.Our main motivation for the initialboundary problems was the direct comparison with the classical results (Theorems 2 and 3).However, note that, in the discrete case, the results would be identical if we dealt with an initial problem on Z.On the other hand, in the semidiscrete or classical case, even the solutions of linear diffusion equations are not necessarily bounded (see, e.g., [12]).
Similarly, the ideas of this paper could be easily extended to a general reaction-diffusion-type equation ( (86)

Figure 4 :
Figure 4: Bounds of solutions of the discrete Nagumo equation (78) with   = −1 and   = 1 and their dependence on the values of  and ℎ (a) and on the values of space and time discretization steps ℎ  and ℎ = ℎ  for a fixed  > 0 (b).In the light gray area, the bounds follow from Theorem 9.In the dark gray area, the bounds are implied by Theorem 11 and  is given by (81).In the white area we have no bounds on solutions.