Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2017: 1.89

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

Well-posedness and maximum principles for lattice reaction-diffusion equations

Antonín Slavík
  • Corresponding author
  • Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Petr Stehlík
  • Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 306 14 Plzeň, Czech Republic
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jonáš Volek
  • Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 306 14 Plzeň, Czech Republic
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-03-17 | DOI: https://doi.org/10.1515/anona-2016-0116

Abstract

Existence, uniqueness and continuous dependence results together with maximum principles represent key tools in the analysis of lattice reaction-diffusion equations. In this paper, we study these questions in full generality by considering nonautonomous reaction functions, possibly nonsymmetric diffusion and continuous, discrete or mixed time. First, we prove the local existence and global uniqueness of bounded solutions, as well as the continuous dependence of solutions on the underlying time structure and on initial conditions. Next, we obtain the weak maximum principle which enables us to get the global existence of solutions. Finally, we provide the strong maximum principle which exhibits an interesting dependence on the time structure. Our results are illustrated by the autonomous Fisher and Nagumo lattice equations and a nonautonomous logistic population model with a variable carrying capacity.

Keywords: Reaction-diffusion equation; lattice equation; existence and uniqueness; continuous dependence; maximum principle; time scale

MSC 2010: 34A33; 34A34; 34N05; 35A01; 35B50; 35F25; 39A14; 65M12

1 Introduction

The classical reaction-diffusion equation tu=kxxu+f(u) is a nonlinear partial differential equation frequently used to describe the evolution of numerous natural quantities (chemical concentrations, temperatures, populations, etc.). These phenomena combine a local dynamics (via the reaction function f) and a spatial dynamics (via the diffusion). It is well known that solutions to reaction-diffusion systems can exhibit rich behavior such as the existence of traveling waves or formation of spatial patterns [32].

Motivated by applications in biology, chemistry and kinematics [2, 10, 12, 19], various authors have considered the lattice reaction-diffusion equation (see [7, 8, 36, 37])

tu(x,t)=k(u(x+1,t)-2u(x,t)+u(x-1,t))+f(u(x,t)),x,t[0,),(1.1)

as well as the discrete reaction-diffusion equation (see [9, 8, 18])

u(x,t+1)-u(x,t)=k(u(x+1,t)-2u(x,t)+u(x-1,t))+f(u(x,t)),x,t0.(1.2)

Naturally, equations (1.1) and (1.2) are also interesting from the standpoint of numerical mathematics since they correspond to semi- or full discretization of the original reaction-diffusion equation [18].

The literature dealing with equations (1.1) and (1.2) studies mainly the dynamical properties such as the asymptotic behavior [5, 33, 34], existence of traveling wave solutions [9, 8, 10, 21, 35, 36, 37] and pattern formation [6, 7, 8], in particular for specific nonlinearities (e.g., the Fisher or Nagumo equation). A growing number of studies have dealt with those questions in nonautonomous cases [17, 24]. In this paper, we study (1.1)–(1.2) with a general time- and space-dependent nonlinearity f. Our focus lies on the existence, uniqueness, continuous dependence (both on the initial condition as well as on the underlying time structure/numerical discretization), and a priori bounds in the form of weak and strong maximum principles. Note that both continuous dependence and maximum principles are key assumptions in the proofs of the existence of traveling waves [21, 35]. Our goal is to explore and describe them in full generality.

In order to consider both (1.1) and (1.2) at once and motivated by convergence issues and continuous dependence of solutions on the time discretization, we use the language of the time scale calculus [4, 16]. We do not restrict ourselves to symmetric diffusion (see the following paragraph) and consider the nonautonomous reaction-diffusion processes

uΔ(x,t)=au(x+1,t)+bu(x,t)+cu(x-1,t)+f(u(x,t),x,t),x,t𝕋,(1.3)

where a,b,c, 𝕋 is a time scale, and the symbol uΔ denotes the delta derivative with respect to time. Our results are new even in the special cases 𝕋= (when uΔ becomes the partial derivative tu) and 𝕋= (when uΔ is the partial difference u(x,t+1)-u(x,t)).

If a=c and b=-2a, then (1.3) becomes the symmetric lattice reaction-diffusion equation. The asymmetric case ac, b=-(a+c) corresponds to the lattice reaction-advection-diffusion equation. Next, if c=0 and b=-a, or if a=0 and b=-c, then (1.3) reduces to the lattice reaction-transport equation. For more details and other special cases see [29, Section 1].

In Section 2, we formulate (1.3) as an abstract nonautonomous dynamic equation and prove the local existence of solutions. In comparison with the existing literature [5, 33, 34], we do not work in the Hilbert space 2() or in the weighted spaces δ2() but in the Banach space (); as explained in [12], this is a much more natural choice. We also prove the uniqueness of bounded solutions. In Section 3, we use techniques from the Kurzweil–Stieltjes integration theory to show the continuous dependence of solutions on the time scale (time discretization). In the special case, this implies the convergence of solutions of (1.2) to the solution of (1.1) as the time discretization step tends to zero. Following the ideas from [31] (which deals with initial-boundary-value problems on finite subsets of ), we provide weak maximum and minimum principles in Section 4. These a priori bounds, as usual, depend strongly on the time structure. Combined with the local existence results they enable us to prove the global existence of bounded solutions to (1.3). We illustrate our findings on the autonomous logistic and bistable nonlinearities (Fisher and Nagumo equations) and a nonautonomous logistic population model with a variable carrying capacity. Finally, in Section 5, we conclude with the strong maximum principle. In the linear case f0, the weak maximum principle was already proved in [29, Theorem 4.7], but the strong maximum principle is new even for linear equations.

2 Local existence and uniqueness of solutions

In this section, we study the local existence and global uniqueness of solutions to the initial-value problem

uΔ(x,t)=au(x+1,t)+bu(x,t)+cu(x-1,t)+f(u(x,t),x,t),x,t[t0,T]𝕋κ,u(x,t0)=ux0,x,

where {ux0}x is a bounded real sequence, a,b,c, 𝕋 is a time scale and t0,T𝕋. We use the notation [α,β]𝕋=[α,β]𝕋, α,β, and

[t0,T]𝕋κ={[t0,T]𝕋 if T is left-dense,[t0,T)𝕋 if T is left-scattered.

We impose the following conditions on the function f:××[t0,T]𝕋:

  • (H1)

    f is bounded on each set B××[t0,T]𝕋, where B is bounded.

  • (H2)

    f is Lipschitz-continuous in the first variable on each set B××[t0,T]𝕋, where B is bounded.

  • (H3)

    For each bounded set B and each choice of ε>0 and t[t0,T]𝕋 there exists a δ>0 such that if s(t-δ,t+δ)[t0,T]𝕋, then |f(u,x,t)-f(u,x,s)|<ε for all uB, x.

We begin with a local existence result. Given a function U:𝕋(), the symbol U(t)x denotes the x-th component of the sequence U(t), and should not be confused with the derivative of U with respect to x (which never appears in this paper).

Theorem 2.1 (Local existence).

Assume that the function f:R×Z×[t0,T]TR satisfies (H1)(H3). Then for each u0(Z) the initial-value problem (2.1) has a bounded local solution defined on Z×[t0,t0+δ]T, where δ>0 and δμ(t0). The solution is obtained by letting u(x,t)=U(t)x, where U:[t0,t0+δ]T(Z) is a solution of the abstract dynamic equation

UΔ(t)=Φ(U(t),t),U(t0)=u0,

with Φ:(Z)×[t0,T]T(Z) being given by

Φ({ux}x,t)={aux+1+bux+cux-1+f(ux,x,t)}x.

Proof.

Condition (H1) guarantees that Φ indeed takes values in (). Choose an arbitrary ρ>0 and denote

={u():u-u0ρ}andB=[infxux0-ρ,supxux0+ρ].

Note that if u,v, then ux,vxB for all x. If L is the Lipschitz constant for the function f on B××[t0,T]𝕋, we get

Φ(u,t)-Φ(v,t)a{ux+1-vx+1}x+b{ux-vx}x+c{ux-1-vx-1}x+{f(ux,x,t)-f(vx,x,t)}x(|a|+|b|+|c|)u-v+Lu-v.

This means that Φ is Lipschitz-continuous in the first variable on ×[t0,T]𝕋.

Next, we observe that Φ is bounded on ×[t0,T]𝕋. Indeed, let M be the boundedness constant for the function |f| on B××[t0,T]𝕋. For each u we have uxB for each x, and consequently

Φ(u,t)a{ux+1}x+b{ux}x+c{ux-1}x+{f(ux,x,t)}x(|a|+|b|+|c|)u+M(|a|+|b|+|c|)(u0+ρ)+M.

Finally, we claim that Φ is continuous on ×[t0,T]𝕋. To see this, consider an arbitrary ε>0 and a fixed pair (u,t)×[t0,T]𝕋. Let δ>0 be the corresponding number from (H3). Then for all (v,s)×[t0,T]𝕋 with u-v<ε and s(t-δ,t+δ)[t0,T]𝕋 we have

Φ(u,t)-Φ(v,s)Φ(u,t)-Φ(v,t)+Φ(v,t)-Φ(v,s)(|a|+|b|+|c|+L)u-v+{f(vx,x,t)-f(vx,x,s)}x(|a|+|b|+|c|+L+1)ε,

which proves that Φ is continuous at the point (u,t).

By [4, Theorem 8.16], the initial-value problem

UΔ(t)=Φ(U(t),t),U(t0)=u0,

has a local solution defined on [t0,t0+δ]𝕋, where δ>0 and δμ(t0). Letting u(x,t)=U(t)x, x, we see that u is a solution of the initial-value problem (2.1). ∎

Note that even in the linear case f0 the solutions of (2.1) are not unique in general (see, e.g., [29, Section 3]) and the uniqueness can be expected only in the class of bounded solutions. In the next theorem, we tackle this issue for an initial-value problem which generalizes (2.1).

Theorem 2.2.

Assume that φ:(Z)×Z×[t0,T]TR satisfies the following conditions:

  • (i)

    φ is bounded on each set ××[t0,T]𝕋 , where () is bounded.

  • (ii)

    φ is Lipschitz-continuous in the first variable on each set ××[t0,T]𝕋 , where () is bounded.

Then for each u0(Z) the initial-value problem

uΔ(x,t)=φ({u(x,t)}x,x,t),u(x,t0)=ux0,x,t[t0,T]𝕋κ,(2.2)

has at most one bounded solution u:Z×[t0,T]TR.

Proof.

Assume that u1, u2 are two bounded solutions that do not coincide on ×(t0,T]𝕋; let

t=inf{τ(t0,T]𝕋:u1(x,τ)u2(x,τ) for some x}.

We claim that u1(x,t)=u2(x,t) for every x. If t=t0, the statement is true. If t>t0 and t is left-dense, then the statement follows from the continuity of solutions with respect to the time variable. Finally, if t>t0 and t is left-scattered, then u1(x,ρ(t))=u2(x,ρ(t)), and the statement follows from the fact that u1Δ(x,ρ(t))=u2Δ(x,ρ(t)).

If t is right-scattered, then u1(x,t)=u2(x,t) and u1Δ(x,t)=u2Δ(x,t) imply u1(x,σ(t))=u2(x,σ(t)), a contradiction to the definition of t. Hence, t is right-dense. Since the functions Ui(τ)={ui(x,τ)}x, i{1,2}, τ[t0,T]𝕋, are bounded, their values are contained in a bounded set (). By the first assumption, there is a constant M0 such that |φ|M on ××[t0,T]𝕋. We have

ui(x,t2)-ui(x,t1)=t1t2uiΔ(x,τ)Δτ=t1t2φ(Ui(τ),x,τ)Δτ,i{1,2},t1,t2t0,x

(the last integral exists at least in the Henstock–Kurzweil sense; see [23, Theorem 2.3]). It follows that

|ui(x,t2)-ui(x,t1)||t2-t1|M,i{1,2},t1,t2t0,x,

and therefore

Ui(t2)-Ui(t1)|t2-t1|M,i{1,2},t1,t2t0,

i.e., the functions U1, U2 are continuous on [t0,T]𝕋.

By the second assumption, the mapping φ is Lipschitz-continuous in the first variable on ××[t0,T]𝕋; let L be the corresponding Lipschitz constant. Then

u1(x,r)-u2(x,r)=trφ(U1(τ),x,τ)-φ(U2(τ),x,τ)Δτ,rt,U1(r)-U2(r)trLU1(τ)-U2(τ)Δτ,rt

(the last integral exists since U1-U2 is continuous). Consequently, for each s[t,T]𝕋 we have

supτ[t,s]U1(τ)-U2(τ)(s-t)Lsupτ[t,s]U1(τ)-U2(τ).

Since t is right-dense, there is a point s[t,T]𝕋 with s>t and (s-t)L<1. Substituting this inequality into the previous estimate, we arrive at a contradiction. ∎

The uniqueness of bounded solutions to the initial-value problem (2.1) is now a simple consequence of the previous theorem.

Theorem 2.3 (Global uniqueness).

Assume that f:R×Z×[t0,T]TR satisfies (H1) and (H2). Then for each u0(Z) the initial-value problem (2.1) has at most one bounded solution u:Z×[t0,T]TR.

Proof.

Note that (2.1) is a special case of (2.2) with the function φ:()××[t0,T]𝕋 being given by

φ({ux}x,x,t)=aux+1+bux+cux-1+f(ux,x,t).

Hence, it is enough to verify that the two conditions in Theorem 2.2 are satisfied.

Given an arbitrary bounded set (), there exists a bounded set B such that u implies uxB, x. Hence, the first condition in Theorem 2.2 is an immediate consequence of (H1). To verify the second condition let L be the Lipschitz constant for the function f on B××[t0,T]𝕋. Then, for each pair of sequences u, v(), we have

|φ(u,x,t)-φ(v,x,t)|(|a|+|b|+|c|)u-v+|f(ux,x,t)-f(vx,x,t)|(|a|+|b|+|c|+L)u-v,

which means that φ is Lipschitz-continuous in the first variable on ××[t0,T]𝕋. ∎

3 Continuous dependence results

This section is devoted to the study of continuous dependence of solutions to abstract dynamic equations with respect to the choice of the time scale. The results are also applicable to (2.1), whose solutions (as we know from Theorem 2.1) are obtained from solutions to a certain abstract dynamic equation.

We begin by proving a continuous dependence theorem for the so-called measure differential equations, i.e., integral equations with the Kurzweil–Stieltjes integral (also known as the Perron–Stieltjes integral) on the right-hand side. For readers who are not familiar with this concept it is sufficient to know that the integral has the usual properties of linearity and additivity with respect to adjacent subintervals. The main advantage with respect to the Riemann–Stieltjes integral is that the class of Kurzweil–Stieltjes integrable functions is much larger. For example, if g:[a,b] has bounded variation, then the integral abf(t)dg(t) exists for each regulated function f:[a,b]X, where X is a Banach space (see [26, Proposition 15]).

The statement as well as the proof of the next theorem are closely related to [3, Theorem 5.1]; for more details, see Remark 3.3.

Theorem 3.1.

Let X be a Banach space and BX. Consider a sequence of nondecreasing left-continuous functions gn:[t0,T]R, nN0, such that gng0 on [t0,T]. Assume that Φ:B×[t0,T]X is Lipschitz-continuous in the first variable. Let xn:[t0,T]B, nN0, be a sequence of functions satisfying

xn(t)=xn(t0)+t0tΦ(xn(s),s)dgn(s),t[t0,T],n0,

and xn(t0)x0(t0). Suppose finally that the function sΦ(x0(s),s), s[t0,T], is regulated. Then xnx0 on [t0,T].

Proof.

Since gn(t0)g0(t0) and gn(T)g0(T), the sequences {gn(t0)}n=1 and {gn(T)}n=1 are necessarily bounded. Hence, there exists a constant M0 such that

vart[t0,T]gn(t)=gn(T)-gn(t0)M,n.

The Kurzweil–Stieltjes integral t0TΦ(x0(s),s)d(gn-g0)(s) exists because sΦ(x0(s),s) is regulated and gn-g0 has bounded variation. Since gn-g00, it follows from [22, Theorem 2.2] that

limnt0tΦ(x0(s),s)d(gn-g0)(s)=0

uniformly with respect to t[t0,T]. Thus, for an arbitrary ε>0 there exists an n0 such that

t0tΦ(x0(s),s)d(gn-g0)(s)ε,nn0,t[t0,T].

Moreover, the index n0 can be chosen in such a way that xn(t0)-x0(t0)ε for each nn0.

Consequently, the following inequalities hold for each nn0 and t[t0,T]:

xn(t)-x0(t)xn(t0)-x0(t0)+t0tΦ(xn(s),s)dgn(s)-t0tΦ(x0(s),s)dg0(s)ε+t0t(Φ(xn(s),s)-Φ(x0(s),s))dgn(s)+t0tΦ(x0(s),s)d(gn-g0)(s)2ε+t0tΦ(xn(s),s)-Φ(x0(s),s)dgn(s)2ε+Lt0txn(s)-x0(s)dgn(s),

where L is the Lipschitz constant for the function Φ. Using Grönwall’s inequality for the Kurzweil–Stieltjes integral (see, e.g., [25, Corollary 1.43]), we get

xn(t)-x0(t)2εeL(gn(t)-gn(t0))2εeLM,nn0,t[t0,T],

which completes the proof. ∎

We now use the relation between measure differential equations and dynamic equations to obtain a continuous dependence theorem for the latter type of equations. Since we need to compare solutions defined on different time scales (whose intersection might be empty), we introduce the following definitions.

Consider an interval [t0,T] and a time scale 𝕋 with t0𝕋, sup𝕋T. Let g𝕋:[t0,T] be given by

g𝕋(t)=inf{s[t0,T]𝕋:st},t[t0,T].

Each function x:[t0,T]𝕋X can be extended to a function x*:[t0,T]X by letting

x*(t)=x(g𝕋(t)),t[t0,T].(3.1)

Note that x* coincides with x on [t0,T]𝕋, and is constant on each interval (u,v], where (u,v)𝕋=. We will refer to x* as the piecewise constant extension of x, see Figure 1.

We are now ready to prove a theorem dealing with continuous dependence of solutions to abstract dynamic equations with respect to the choice of the time scale and initial condition.

Theorem 3.2 (Continuous dependence).

Let X be a Banach space and BX. Consider an interval [t0,T]R and a sequence of time scales {Tn}n=0 such that t0Tn and TTn for each nN0 and gTngT0 on [t0,T]. Denote

𝕋=n=0𝕋n¯.

Suppose that Φ:B×[t0,T]TX is continuous on its domain and Lipschitz-continuous with respect to the first variable. Let xn:[t0,T]TnB, nN0, be a sequence of functions satisfying

xnΔ(t)=Φ(xn(t),t),t[t0,T]𝕋nκ,n0,

and xn(t0)x0(t0). Then the sequence of piecewise constant extensions {xn*}n=1 is uniformly convergent to the piecewise constant extension x0* on [t0,T]. In particular, for every ε>0 there exists an n0N such that xn(t)-x0(t)<ε for all nn0, t[t0,T]Tn[t0,T]T0.

Proof.

According to the assumptions, we have

xn(t)=xn(t0)+t0tΦ(xn(s),s)Δs,t[t0,T]𝕋n,n0.

For each n0 let xn*:[t0,T]X be the piecewise constant extension of xn. Using the relation between Δ-integrals and Kurzweil–Stieltjes integrals (see [27, Theorem 5] or [11, Theorem 4.5]), we conclude that xn* satisfy

xn*(t)=xn*(t0)+t0tΦ(xn*(s),g𝕋n(s))dg𝕋n(s),t[t0,T],n0.(3.2)

Let Φ*:×[t0,T]X be given by

Φ*(x,t)=Φ(x,g𝕋(t)),x,t[t0,T].

Note that for each s[t0,T]𝕋n we have

Φ(xn*(s),g𝕋n(s))=Φ(xn*(s),s)=Φ(xn*(s),g𝕋(s))=Φ*(xn*(s),s).

Thus, by [11, Theorem 5.1], the integral equation (3.2) is equivalent to

xn*(t)=xn*(t0)+t0tΦ*(xn*(s),s)dg𝕋n(s),t[t0,T],n0.

Because x0 is continuous on [t0,T]𝕋0, its piecewise constant extension x0* is regulated on [t0,T] (see [27, Lemma 4]). Moreover, its one-sided limits at each point of [t0,T] are elements of (note that x0*([t0,T])=x0([t0,T]𝕋0) is compact because x0 is continuous and [t0,T]𝕋0 is compact). The function g𝕋 is the piecewise constant extension of the identity function from [t0,T]𝕋 to [t0,T]; therefore (again by [27, Lemma 4]), g𝕋 is regulated on [t0,T]. Consequently, the function s(x0*(s),g𝕋(s)) is also regulated on [t0,T], and its one-sided limits have values in ×[t0,T]𝕋. The continuity of Φ on ×[t0,T]𝕋 implies that sΦ(x0*(s),g𝕋(s))=Φ*(x0*(s),s) is regulated on [t0,T]. According to Theorem 3.1, we have xn*x0* on [t0,T]. ∎

The piecewise constant extension x*{x^{*}} (gray) of a function x (black); see (3.1).
Figure 1

The piecewise constant extension x* (gray) of a function x (black); see (3.1).

Remark 3.3.

The problem of continuous dependence of solutions to dynamic equations with respect to the choice of time scale has been studied by several authors; see, e.g., [1, 3, 13, 14, 15, 20]. Our approach is close to the one taken in [3] or [13]; it relies on the continuous dependence result for measure differential equations from Theorem 3.1, which is similar in spirit to [3, Theorem 5.1]. In this context, it seems appropriate to include a few remarks:

  • Although the statement of [3, Theorem 5.1] is essentially correct, the proof provided there is based on an erroneous estimate of the form t0tfndgn-t0tfndg0t0TMd(gn-g0), where fn, f0 are certain functions whose norm is bounded by M, and gn, g0 are nondecreasing.

  • The assumption that the Hausdorff distance between 𝕋n and 𝕋0 tends to zero is never used in the proof of [3, Theorem 5.1], and can be omitted. On the other hand, the assumption that the above-mentioned integral t0Tfndg0 exists is missing.

  • The result [3, Theorem 5.1] deals with measure functional differential equations; our Theorem 3.1 and its proof can be easily adapted to this type of equations.

The next result shows that each time scale can be approximated by a sequence of discrete time scales in such a way that the assumptions of Theorem 3.2 are satisfied. We introduce the following notation:

μ¯𝕋=maxt[t0,T)𝕋μ(t).

Theorem 3.4.

If T0R is a time scale with t0,TT0, there exists a sequence of discrete time scales {Tn}n=1 with TnT0, minTn=t0, maxTn=T, and such that gTngT0 on [t0,T].

Moreover, if μ¯T0=0, then limnμ¯Tn=0; otherwise, if μ¯T0>0, then the sequence {Tn}n=1 can be chosen so that μ¯Tn=μ¯T0 for all nN.

Proof.

We start by proving that for each ε>0 there exists a left-continuous nondecreasing step function gε:[t0,T] such that gε(t0)=t0, gε(T)=T, and gε-g𝕋0ε.

Given an ε>0, let t0=x0<x1<<xm=T be a partition of [t0,T] such that xi-xi-1ε, i{1,,m}. We begin the construction of the step function gε:[t0,T] by letting gε(T)=T. Then we proceed by induction in the backward direction and define gε on [xm-1,xm),,[x0,x1). At the same time, we are going to check that g𝕋0-gεε on these subintervals, and also ensure that gε(xi)=xi whenever xi𝕋0; this will guarantee that gε(t0)=t0.

Assume that gε is already defined at xi and we want to extend it to [xi-1,xi). We distinguish between two possibilities:

  • If 𝕋0[xi-1,xi)=, then, by the definition of g𝕋0, we have g𝕋0(t)=g𝕋0(xi) for each t[xi-1,xi). Let gε(t)=gε(xi), t[xi-1,xi). Then |gε(t)-g𝕋0(t)|=|gε(xi)-g𝕋0(xi)|ε, where the last inequality follows from the induction hypothesis.

  • If 𝕋0[xi-1,xi) is nonempty, let ti be its supremum. Define

    gε(xi-1)={xi-1 if xi-1𝕋0,ti if xi-1𝕋0,gε(t)={ti if t(xi-1,ti],gε(xi) if t(ti,xi).

    Note that ti might coincide with xi. In this case, we necessarily have xi𝕋0, and therefore, by the induction hypothesis, gε(xi)=xi; this guarantees that gε is left-continuous at xi.

    For each t[xi-1,ti] we have xi-1tg𝕋0(t)ti. Hence, there holds 0ti-g𝕋0(t)ti-xi-1ε, which in turn means that |gε(t)-g𝕋0(t)|ε. For each t(ti,xi) it follows from the definition of g𝕋0 that g𝕋0(t)=g𝕋0(xi), and therefore |gε(t)-g𝕋0(t)|=|gε(xi)-g𝕋0(xi)|ε.

Observe that the function gε constructed in this way has the property that gε(t)t, and observe that gε(t)=t implies t𝕋0.

Choosing ε=1/n, n, we get a sequence of left-continuous nondecreasing step functions {g1/n}n=1 such that g1/ng𝕋0 on [t0,T]. For each n consider the set

𝕋n={t[t0,T]:g1/n(t)=t}.

Clearly, t0 and T are elements of 𝕋n, and 𝕋n𝕋0. Moreover, 𝕋n is finite since g1/n is a step function and therefore its graph has only finitely many intersections with the graph of the identity function. Thus, 𝕋n is a discrete time scale. It follows from the definition of 𝕋n that g𝕋n=g1/n, and therefore g𝕋ng𝕋0 on [t0,T].

To prove the final part of the theorem, we distinguish between two cases:

  • Assume that μ¯𝕋0>0. Let y0=t0, and construct a sequence of points y1<<yk=T using the recursive formula

    yi=sup(yi-1,yi-1+μ¯𝕋0][t0,T]𝕋0.

    Since the graininess of 𝕋0 never exceeds μ¯𝕋0, the set whose supremum is being considered is never empty. Also, note that yi+1-yi-1μ¯𝕋0 (otherwise, the point yi+1 would have been chosen directly after yi-1). Thus, the recursive procedure always terminates by reaching the point yk=T for some k.

    In the construction of the function gε described at the beginning of this proof, we can always assume that the points y0,,yk are among x0,,xm. The construction then guarantees that gε(yi)=yi for each i{0,,k}. Consequently, the points y0,,yk are contained in all of the time scales 𝕋n, n, and

    μ¯𝕋nmax1ik(yi-yi-1)μ¯𝕋0.

    On the other hand, since 𝕋n𝕋0, we have μ¯𝕋0μ¯𝕋n, which in turn means that μ¯𝕋n=μ¯𝕋0.

  • Assume that μ¯𝕋0=0. If μ is the graininess function of an arbitrary time scale 𝕋 with min𝕋=t0 and sup𝕋T, observe that g𝕋(t+)-g𝕋(t)=μ(t) if t[t0,T)𝕋, and g𝕋(t+)-g𝕋(t)=0 if t[t0,T)𝕋. Hence, we have

    μ¯𝕋=supt[t0,T)𝕋μ(t)=supt[t0,T)(g𝕋(t+)-g𝕋(t)).

    Since g𝕋ng𝕋0 on [t0,T], the Moore–Osgood theorem implies that g𝕋n(t+)-g𝕋n(t)g𝕋0(t+)-g𝕋0(t) on [t0,T), and therefore

    limnμ¯𝕋n=limn(supt[t0,T)(g𝕋n(t+)-g𝕋n(t)))=supt[t0,T)(g𝕋0(t+)-g𝕋0(t))=μ¯𝕋0=0.

4 Weak maximum principle and global existence

A natural task in the analysis of diffusion-type equations is to establish the maximum principles. Given an initial condition u0(), let

m=infxux0,M=supxux0.

We introduce the following conditions, which will be useful for our purposes:

  • (H4)

    a,b,c are such that a,c0, b<0 and a+b+c=0.

  • (H5)

    b<0 and μ¯𝕋-1/b.

  • (H6)

    There exist r,R such that rmMR, and one of the following statements holds:

    • μ¯𝕋=0 and f(R,x,t)0f(r,x,t) for all x, t[t0,T]𝕋.

    • μ¯𝕋>0 and

      1+μ¯𝕋bμ¯𝕋(r-u)f(u,x,t)1+μ¯𝕋bμ¯𝕋(R-u)

      for all u[r,R], x, t[t0,T]𝕋.

Remark 4.1.

Let us notice the following:

  • If (H4)–(H5) are not satisfied, then the maximum principle does not hold even in the linear case with f0; see [29, Section 4].

  • (H6) defines forbidden areas that the function f(,x,t) cannot intersect for any x, t[t0,T]𝕋, similarly to [31] (see Figure 2).

  • If (H5) holds, there exists a function f satisfying (H6); indeed, the linear functions

    ψ1(u)=1+μ¯𝕋bμ¯𝕋(r-u)andψ2(u)=1+μ¯𝕋bμ¯𝕋(R-u)

    have identical nonpositive slopes, and the constant term of ψ1 is less than or equal to the constant term of ψ2. If μ¯𝕋=-1/b or r=R, then (H6) is equivalent to f(u,x,t)=0 for all u[r,R], x and t[t0,T]𝕋. Finally, if μ¯𝕋>-1/b and r<R, there does not exist any function satisfying (H6).

Illustration of (H6). The values r, R are chosen so that the function f⁢(⋅,x,t){f(\,\cdot\,,x,t)} does not intersect the gray forbidden areas. The slope of the boundary dashed lines is determined by the values of μ¯𝕋{\overline{\mu}_{\mathbb{T}}}.
Figure 2

Illustration of (H6). The values r, R are chosen so that the function f(,x,t) does not intersect the gray forbidden areas. The slope of the boundary dashed lines is determined by the values of μ¯𝕋.

If (H6) holds in the continuous case μ¯𝕋=0, the following lemma shows that (H6) is also satisfied for all sufficiently fine time scales (specifically, for almost all of the discrete approximating time scales 𝕋n from Theorem 3.4).

Lemma 4.2.

Assume that μ¯T=0 and (H2) and (H6) hold. Then there exists ε0>0 such that for all ε(0,ε0] the following inequalities hold:

1+εbε(r-u)f(u,x,t)1+εbε(R-u)for all u[r,R],x,t[t0,T].(4.1)

Proof.

Let L0 be the Lipschitz constant for the function f on the set [r,R]××[t0,T]. Then for all u[r,R], x and t[t0,T] we obtain

f(u,x,t)f(u,x,t)-f(R,x,t)|f(u,x,t)-f(R,x,t)|L|u-R|=L(R-u),f(u,x,t)f(u,x,t)-f(r,x,t)-|f(u,x,t)-f(r,x,t)|-L|u-r|=L(r-u).

Since L(r-u)f(u,x,t)L(R-u), the two inequalities in (4.1) will be satisfied if 1/ε+bL, i.e., for all ε(0,1/(L-b)]. ∎

The following lemma represents a weak maximum principle for time scales containing no right-dense points; it will be a key tool in the proof of the general weak maximum principle.

Lemma 4.3.

Assume that [t0,T)T does not contain any right-dense points, conditions (H4)(H6) hold and u:Z×[t0,T]TR is a solution of (2.1) with u0(Z). Then

ru(x,t)Rfor all x,t[t0,T]𝕋.(4.2)

Proof.

We show the statement via the induction principle [4, Theorem 1.7] in the variable t. For a fixed t[t0,T]𝕋 we have to distinguish among three cases:

  • For t=t0 we obtain from the definitions of m and M and from (H6) that

    rmu(x,t0)MRfor all x.

  • Let t(t0,T]𝕋 be left-dense and assume that ru(x,s)R for all s[t0,t)𝕋 and x. Then the continuity of the function u(x,) on [t0,T]𝕋 implies

    ru(x,t)=limst-u(x,s)Rfor all x.

  • Let t[t0,T)𝕋 be right-scattered, i.e., necessarily μ¯𝕋>0, and

    ru(x,t)Rfor all x.(4.3)

    We have to show that

    ru(x,t+μ𝕋(t))Rfor all x.(4.4)

    Notice that from (H5) and from the fact that μ¯𝕋μ𝕋(t)>0 we get

    01+μ¯𝕋bμ¯𝕋=1μ¯𝕋+b1μ𝕋(t)+b=1+μ𝕋(t)bμ𝕋(t).

    Consequently, (H6) yields

    1+μ𝕋(t)bμ𝕋(t)(r-u)f(u,x,t)1+μ𝕋(t)bμ𝕋(t)(R-u)for all u[r,R],x,t[t0,T]𝕋.(4.5)

    Let us prove the latter inequality in (4.4). Using the equation in (2.1), we obtain the estimate

    u(x,t+μ𝕋(t))=μ𝕋(t)au(x+1,t)+(1+μ𝕋(t)b)u(x,t)+μ𝕋(t)cu(x-1,t)+μ𝕋(t)f(u(x,t),x,t)μ𝕋(t)(a+c)R+(1+μ𝕋(t)b)u(x,t)+μ𝕋(t)f(u(x,t),x,t)(by (H4) and (4.3))=-μ𝕋(t)bR+(1+μ𝕋(t)b)u(x,t)+μ𝕋(t)f(u(x,t),x,t)(by (H4))-μ𝕋(t)bR+(1+μ𝕋(t)b)u(x,t)+(1+μ𝕋(t)b)(R-u(x,t))(by (4.3) and (4.5))=R

    for each x. The former inequality in (4.4) can be shown in a similar way.

We do not have to consider the case when t is right-dense since 𝕋 does not contain any such point. Therefore, the induction principle yields that (4.2) holds for all x, t[t0,T]𝕋. ∎

We now proceed to the general weak maximum principle for (2.1), where 𝕋 is an arbitrary time scale (i.e., allowing right-dense points). The basic idea of the proof is to use the continuous dependence results from Theorems 3.2 and 3.4 to approximate the solution of (2.1) on any time scale by solutions of (2.1) defined on discrete time scales, for which we can apply Lemma 4.3.

Theorem 4.4 (Weak maximum principle).

Assume that (H1)(H6) hold. If u:Z×[t0,T]TR is a bounded solution of (2.1), then

ru(x,t)Rfor all x,t[t0,T]𝕋.(4.6)

Proof.

From Theorems 2.1 and 2.3 we obtain that u has to be unique and U(t)={u(x,t)}x is the unique solution of the abstract initial-value problem

UΔ(t)=Φ(U(t),t),U(t0)=u0,

where Φ:()×[t0,T]𝕋() is given by

Φ({ux}x,t)={aux+1+bux+cux-1+f(ux,x,t)}x.

According to Theorem 3.4, there exists a sequence {𝕋n}n=1 of discrete time scales such that 𝕋n𝕋, min𝕋n=t0, max𝕋n=T, and g𝕋ng𝕋. Moreover, we have either μ¯𝕋=0 and μ¯𝕋n0, or μ¯𝕋n=μ¯𝕋 for all n. In any case, using (H5), we get the existence of an n0 such that

μ¯𝕋n-1bfor all n>n0.

If μ¯𝕋=0, it follows from Lemma 4.2 that n0 can be chosen in such a way that the inequalities

1+μ𝕋n(t)bμ𝕋n(t)(r-u)f(u,x,t)1+μ𝕋n(t)bμ𝕋n(t)(R-u)for all u[r,R],x,t[t0,T]𝕋n,

hold for each n>n0. If μ¯𝕋>0, the same inequalities hold for each n because of (H6) and the fact that μ¯𝕋n=μ¯𝕋.

Therefore, because 𝕋n are discrete time scales, Lemma 4.3 yields that the corresponding solutions un:×[t0,T]𝕋n of (2.1) satisfy

run(x,t)Rfor all x,t[t0,T]𝕋n,n>n0,

i.e., for Un(t)={un(x,t)}x we have

rinfxUn(t)xsupxUn(t)xRfor all t[t0,T]𝕋n,n>n0.(4.7)

Since the solution U is bounded, there is an S>0 such that U(t)S for each t[t0,T]𝕋. Let

={V():Vmax(|r|,|R|,S)}.

As in the proof of Theorem 2.1, one can show that the restriction of the mapping Φ to ×[t0,T]𝕋 is continuous on its domain and Lipschitz-continuous in the first variable. Therefore, if we let 𝕋0=𝕋, the assumptions of Theorem 3.2 are satisfied (recall that Un(t) for all t𝕋n and n>n0 from (4.7), and U(t) for all t𝕋 immediately from the definition of ), and hence Un*U* on [t0,T].

From the definition of the piecewise constant extension Un* and from (4.7) it is obvious that

rinfxUn*(t)xsupxUn*(t)xRfor all t[t0,T],n>n0.(4.8)

Since Un*U* on [t0,T], inequalities (4.8) imply

rinfxU*(t)xsupxU*(t)xRfor all t[t0,T].

Particularly, there has to be

rinfxU(t)xsupxU(t)xRfor all t[t0,T]𝕋,

which proves that (4.6) holds. ∎

Remark 4.5.

In connection with the previous theorem, we point out the following facts:

  • The classical maximum principle guarantees that mu(x,t)M, i.e., it corresponds to the case when r=m and R=M. However, for this choice of r and R, condition (H6) need not be satisfied. Choosing r<m and R>M, we can soften (H6), and obtain the weaker estimate ru(x,t)R.

  • An examination of the proofs of Lemma 4.3 and Theorem 4.4 reveals that if we are interested only in the upper bound u(x,t)R, it is sufficient to assume that a+b+c0. Symmetrically, to get the lower bound u(x,t)r, it is enough to suppose that a+b+c0.

As an application of the weak maximum principle, we obtain the following global existence theorem. Since we consider a general class of nonlinearities f, the result is new even in the special case 𝕋=.

Theorem 4.6 (Global existence).

If u0(Z) and (H1)(H6) hold, then (2.1) has a unique bounded solution u:Z×[t0,T]TR.

Moreover, the solution depends continuously on u0 in the following sense: For every ε>0 there exists a δ>0 such that if v0(Z), rvx0R for all xZ, and u0-v0<δ, then the unique bounded solution v:Z×[t0,T]TR of (2.1) corresponding to the initial condition v0 satisfies |u(x,t)-v(x,t)|<ε for all xZ, t[t0,T]T.

Proof.

We know from Theorems 2.1 and 2.3 that bounded solutions to (2.1) are unique, and that they correspond to solutions of the initial-value problem

UΔ(t)=Φ(U(t),t),t[t0,T]𝕋κ,U(t0)=u0,(4.9)

with Φ:()×[t0,T]𝕋() being given by

Φ({ux}x,t)={aux+1+bux+cux-1+f(ux,x,t)}x.

Thus, it is enough to prove that (4.9) has a solution on the whole interval [t0,T]𝕋.

Let 𝒮 be the set of all s[t0,T]𝕋 such that (4.9) has a solution on [t0,s]𝕋, and denote t1=sup𝒮. By Theorem 2.1, we have t1>t0. Let us prove that t1𝒮. The statement is obvious if t1 is a left-scattered maximum of 𝒮; therefore, we can assume that t1 is left-dense. It follows from the definition of t1 that (4.9) has a solution U defined on [t0,t1)𝕋. According to the weak maximum principle, the solution U takes values in the bounded set ={u():ruxR for each x}. As in the proof of Theorem 2.1, one can show that Φ is continuous on its domain and Lipschitz-continuous in the first variable and bounded on ×[t0,T]𝕋; let C be the boundedness constant for Φ. Since U is a solution of (4.9), we have

U(t)=U(t0)+t0tΦ(U(s),s)Δs(4.10)

for each t[t0,t1)𝕋. Note also that U(s1)-U(s2)C|s1-s2| for all s1,s2[t0,t1)𝕋. Thus, the Cauchy condition for the existence of the limit U(t1-)=limst1-U(s) is satisfied. If we extend U to [t0,t1]𝕋 by letting U(t1)=U(t1-), we see that (4.10) holds also for t=t1. Since the mapping sΦ(U(s),s) is continuous on [t0,t1]𝕋, it follows that U is a solution of (4.9) on [t0,t1]𝕋, i.e., t1𝒮.

If t1<T, we can use Theorem 2.1 to extend the solution U from [t0,t1]𝕋 to a larger interval. However, this contradicts the fact that t1=sup𝒮. Hence, the only possibility is t1=T, and the proof of the existence is complete.

To obtain continuous dependence of the solution on the initial condition, it is enough to show the following statement: If un for n, unu0 in () and Un:[t0,T]𝕋() is the unique solution of the initial-value problem

UnΔ(t)=Φ(Un(t),t),t[t0,T]𝕋κ,Un(t0)=un,

then UnU on [t0,T]𝕋. Since we know that the solutions Un in fact take values in , the statement is an immediate consequence of Theorem 3.2 where we take 𝕋n=𝕋 for each n0. ∎

Let us illustrate the application of the weak maximum principle and the global existence theorem on the following special cases of (2.1).

Example 4.7.

Consider the logistic nonlinearity f(u,x,t)=λu(1-u), u, x, t[t0,T]𝕋, where λ>0 is a parameter. In this case, problem (2.1) becomes a Fisher-type reaction-diffusion equation:

uΔ(x,t)=au(x+1,t)+bu(x,t)+cu(x-1,t)+λu(x,t)(1-u(x,t)),x,t[t0,T]𝕋κ,u(x,t0)=ux0,x.

Obviously, the function f satisfies (H1)–(H3). Suppose that a,c0, b<0, a+b+c=0, and μ¯𝕋-1/b, i.e., (H4) and (H5) hold. Consider an arbitrary nonnegative initial condition u0(), i.e., m0. We now distinguish between the cases μ¯𝕋=0 and μ¯𝕋>0:

  • If μ¯𝕋=0, let r=min(m,1) and R=max(M,1). Then f(R,x,t)0 and f(r,x,t)0, i.e., (H6) holds and there exists a unique global solution u of (4.11). Moreover, the solution u satisfies ru(x,t)R for all x and t[t0,T]𝕋. In particular, nonnegative initial conditions always lead to nonnegative solutions.

  • If μ¯𝕋>0, Lemma 4.2 together with the analysis of the previous case guarantee that (H6) holds with r=min(m,1) and R=max(M,1) whenever μ¯𝕋 is sufficiently small. For example, if M1, consider the linear functions

    ψ1(u)=1+μ¯𝕋bμ¯𝕋(r-u)andψ2(u)=1+μ¯𝕋bμ¯𝕋(R-u)

    from (H6). We have ψ1(u)0f(u,x,t) for u[r,R], i.e., the first inequality in (H6) is satisfied. The graphs of ψ2 and f(,x,t) meet at the point (1,0). Therefore, the second inequality f(u,x,t)ψ2(u) in (H6) will be satisfied for u[r,R] if and only if fu(1,x,t)ψ2(1), i.e., if and only if -λ-(1/μ¯𝕋+b). The last condition is equivalent to λ-b1/μ¯𝕋, which holds if μ¯𝕋1/(λ-b) (note that b<0<λ). Under these assumptions, condition (H6) holds and there exists a unique bounded global solution u of (4.11). Moreover, the solution u satisfies m=ru(x,t)R=1 for all x and t[t0,T]𝕋.

Example 4.8.

Consider the so-called bistable nonlinearity f(u,x,t)=λu(1-u2), u, x, t[t0,T]𝕋, where λ>0. In this case, problem (2.1) becomes a Nagumo-type reaction-diffusion equation:

uΔ(x,t)=au(x+1,t)+bu(x,t)+cu(x-1,t)+λu(x,t)(1-u(x,t)2),x,t[t0,T]𝕋κ,u(x,t0)=ux0,x.

Obviously, the function f satisfies (H1)–(H3). Suppose that a,c0, b<0, a+b+c=0, and μ¯𝕋-1/b, i.e., (H4) and (H5) hold. Consider an arbitrary initial condition u0(). Again, we distinguish between the cases μ¯𝕋=0 and μ¯𝕋>0:

  • If μ¯𝕋=0, let

    r={min(m,-1) if m<0,min(m,1) if m0,R={max(M,-1) if M0,max(M,1) if M>0.

    Then f(R,x,t)0 and f(r,x,t)0, i.e., (H6) holds and there exists a unique bounded global solution u of (4.12). Moreover, the solution u satisfies ru(x,t)R for all x and t[t0,T]𝕋. In particular, nonnegative/nonpositive initial conditions always lead to nonnegative/nonpositive solutions.

  • If μ¯𝕋>0, Lemma 4.2 together with the analysis of the previous case guarantee that (H6) holds whenever μ¯𝕋 is sufficiently small. For example, if u01, one can follow the computations from [31, Section 8] to conclude that there exists a unique global solution u of (4.12) satisfying

    u(x,t){[-1,1]if μ¯𝕋1/(2λ-b),if 1/(2λ-b)<μ¯𝕋2/(λ-2b),

    where

    R~=2λμ¯𝕋(1/3+(1+2bμ¯𝕋)/3λμ¯𝕋)3/21+bμ¯𝕋.

    We have no a priori bounds for μ¯𝕋>2/(λ-2b).

Example 4.9.

Consider the nonautonomous nonlinearity f(u,x,t)=λu(d(x,t)-u), u, x, t[t0,T]𝕋, where λ>0 and d:×[t0,T]𝕋. In this case, problem (2.1) has the form

uΔ(x,t)=au(x+1,t)+bu(x,t)+cu(x-1,t)+λu(x,t)(d(x,t)-u(x,t)),x,t[t0,T]𝕋κ,u(x,t0)=ux0,x.

This equation can be interpreted as the logistic population model where the carrying capacity d depends on position and time. Assume that d has the following properties:

  • d is bounded.

  • For each choice of ε>0 and t[t0,T]𝕋 there exists a δ>0 such that if s(t-δ,t+δ)[t0,T]𝕋, then |d(x,t)-d(x,s)|<ε for all x.

Then the function f satisfies (H1)–(H3). Indeed, let D be the boundedness constant for |d|. If B is bounded, it is contained in a ball of radius ρ centered at the origin. Consequently, for all u,vB, x, t,s[t0,T]𝕋, we get the estimates

|f(u,x,t)|λ|u|(|d(x,t)|+|u|)λρ(D+ρ),|f(u,x,t)-f(v,x,t)|=λ|(u-v)(d(x,t)-u-v)|λ|u-v|(D+2ρ),|f(u,x,t)-f(u,x,s)|=λ|u(d(x,t)-d(x,s))|λρ|d(x,t)-d(x,s)|,

which imply that (H1)–(H3) hold.

As an example, let us mention the model of population dynamics with a shifting habitat, which was described by Hu and Li in [17]. There, the authors considered problem (4.13) with 𝕋=, a=c, b=-2a (i.e., symmetric diffusion), and d(x,t)=e(x-γt), where γ>0 and e: is continuous, nondecreasing, and bounded. It follows that e is uniformly continuous on : Given an ε>0, there exists a δ>0 such that |t1-t2|<δ implies |e(t1)-e(t2)|<ε. Thus, we get

|d(x,t)-d(x,s)|=|e(x-γt)-e(x-γs)|<ε

whenever |t-s|<δ/γ and x; this shows that d satisfies our assumptions. (We remark that some of the results presented in [17] can be found in our earlier paper [28]. In particular, the fundamental solution of the linear lattice diffusion equation was derived in [28, Example 3.1], and [17, Corollary 2.1] is a consequence of our superposition principle from [28, Theorem 2.2].)

Another simple example is obtained by letting d(x,t)=e(t), where e: is a continuous periodic function; this choice corresponds to a population model with a periodically changing habitat. Since e is necessarily bounded and uniformly continuous on , it is obvious that d satisfies our assumptions.

Suppose now that a,c0, b<0, a+b+c=0, and μ¯𝕋-1/b, i.e., (H4) and (H5) hold. For simplicity, let us restrict ourselves to the case when d is a positive function, and let

dmin=inf(x,t)×[t0,T]𝕋d(x,t),dmax=sup(x,t)×[t0,T]𝕋d(x,t).

Consider an arbitrary nonnegative initial condition u0(), i.e., m0. Take r=min(m,dmin) and R=max(M,dmax). Then f(r,x,t)0 and f(R,x,t)0 for all x and t[t0,T]𝕋. This means that (H6) holds if μ¯𝕋=0, or (by Lemma 4.2) if μ¯𝕋 is positive and sufficiently small. In these cases, problem (4.13) possesses a unique global solution u, and ru(x,t)R for all x and t[t0,T]𝕋.

5 Strong maximum principle

In the rest of the paper, we focus on the strong maximum principle for (2.1). We need the following stronger versions of (H4)–(H6):

  • (H4)

    a, b, c are such that a,c>0, b<0 and a+b+c=0.

  • (H5)

    b<0 and μ¯𝕋<-1/b.

  • (H6)

    There exist r,R such that rmMR, and the following statements hold for all x and t[t0,T]𝕋:

    • f(R,x,t)0f(r,x,t).

    • If μ¯𝕋>0, then

      f(u,x,t)>1+μ¯𝕋bμ¯𝕋(r-u)for all u(r,R].

    • If μ¯𝕋>0, then

      f(u,x,t)<1+μ¯𝕋bμ¯𝕋(R-u)for all u[r,R).

The next lemma analyzes the situation when a solution of (2.1) attains its maximum at a left-scattered point.

Lemma 5.1.

Assume that (H1), (H2), (H3), (H4¯), (H5¯), and (H6¯) hold, and u:Z×[t0,T]TR is a bounded solution of (2.1). If u(x¯,t¯){r,R} for some x¯Z and a left-scattered point t¯(t0,T]T, then u(x,ρT(t¯))=u(x¯,t¯) for each x{x¯-1,x¯,x¯+1}.

Proof.

We consider the case when u(x¯,t¯)=R; the case u(x¯,t¯)=r can be treated in a similar way. Denote s¯=ρ𝕋(t¯). We have

u(x¯,t¯)=μ𝕋(s¯)au(x¯+1,s¯)+(1+μ𝕋(s¯)b)u(x¯,s¯)+μ𝕋(s¯)cu(x¯-1,s¯)+μ𝕋(s¯)f(u(x¯,s¯),x¯,s¯).

By the weak maximum principle (which holds because (H4¯)–(H6¯) imply (H4)–(H6)), the values of u cannot exceed R. If at least one of the values u(x¯+1,s¯), u(x¯-1,s¯) is smaller than R and u(x¯,s¯)=R, then

u(x¯,t¯)<(H4¯)μ𝕋(s¯)(a+c)R+(1+μ𝕋(s¯)b)R+μ𝕋(s¯)f(R,x¯,s¯)=(H4¯)R+μ𝕋(s¯)f(R,x¯,s¯)(H6¯)R,

which contradicts the fact that u(x¯,t¯)=R. If u(x¯,s¯)<R, then

u(x¯,t¯)μ𝕋(s¯)(a+c)R+(1+μ𝕋(s¯)b)u(x¯,s¯)+μ𝕋(s¯)f(u(x¯,s¯),x¯,s¯)<μ𝕋(s¯)(a+c)R+(1+μ𝕋(s¯)b)u(x¯,s¯)+μ𝕋(s¯)1+μ¯𝕋bμ¯𝕋(R-u(x¯,s¯))(by (H4¯) and (H6¯))μ𝕋(s¯)(a+c)R+(1+μ𝕋(s¯)b)u(x¯,s¯)+(1+μ𝕋(s¯)b)(R-u(x¯,s¯))=R(by (H4¯)),

which is a contradiction again. Thus, the only possibility is that

u(x¯+1,s¯)=u(x¯,s¯)=u(x¯-1,s¯)=R,

as desired. ∎

We now turn our attention to the case when the maximum is attained at a left-dense point.

Lemma 5.2.

Assume that (H1), (H2), (H3), (H4¯), (H5¯), and (H6¯) hold, and u:Z×[t0,T]TR is a bounded solution of (2.1). If u(x¯,t¯){r,R} for some x¯Z and a left-dense point t¯(t0,T]T, then u(x,t)=u(x¯,t¯) for all xZ and t[t0,t¯]T.

Proof.

We consider the case when u(x¯,t¯)=R; the case u(x¯,t¯)=r can be treated in a similar way. We begin by proving that

u(x¯,t)=Rfor all t[t0,t¯]𝕋.(5.1)

Assume that there exists a s¯[t0,t¯)𝕋 such that u(x¯,s¯)<R. Let L0 be the Lipschitz constant for f on the set [r,R]××[t0,T]𝕋. Choose a partition s¯=s0<s1<<sk=t¯ such that s0,,sk𝕋 and for each i{1,,k} we have either si-si-1<1/(L-b) or si=σ𝕋(si-1). We will use induction with respect to i to show that u(x¯,si)<R for each i{0,,k}; this will be a contradiction to the fact that u(x¯,sk)=u(x¯,t¯)=R.

For i=0, we know that u(x¯,s0)=u(x¯,s¯)<R. By the weak maximum principle (which holds because (H4¯)–(H6¯) imply (H4)–(H6)), the values of u cannot exceed R. If i{0,,k-1} is such that si+1=σ𝕋(si), then the induction hypothesis u(x¯,si)<R and Lemma 5.1 imply that u(x¯,si+1)<R. Otherwise, we have si+1-si<1/(L-b). For each t[si,si+1)𝕋 we get

(u(x¯,t)-R)Δ=au(x¯+1,t)+bu(x¯,t)+cu(x¯-1,t)+f(u(x¯,t),x¯,t)(a+c)R+bu(x¯,t)+f(u(x¯,t),x¯,t)-f(R,x¯,t)+f(R,x¯,t)(by (H4¯) and Theorem 4.4)-b(R-u(x¯,t))+f(u(x¯,t),x¯,t)-f(R,x¯,t)(by (H4¯) and (H6¯))-b(R-u(x¯,t))+|f(u(x¯,t),x¯,t)-f(R,x¯,t)|-b(R-u(x¯,t))+L|u(x¯,t)-R|=(b-L)(u(x¯,t)-R)(by Theorem 4.4).

Notice that 1+μ𝕋(t)(b-L)>0 for all t[si,si+1)𝕋. Therefore, Grönwall’s inequality [4, Theorem 6.1] yields

u(x¯,si+1)-R(u(x¯,si)-R)<0eb-L(si+1,si)>0<0,

which completes the proof by induction and confirms that (5.1) holds.

Let us prove that u(x¯±1,t)=R for all t[t0,t¯]𝕋. Assume that there exists a t[t0,t¯]𝕋 such that at least one of the values u(x¯±1,t) is smaller than R. The fact that u(x¯,) is a constant function on [t0,t¯]𝕋 implies that uΔ(x¯,t)=0 (note that if t=t¯, then t is necessarily left-dense). On the other hand,

uΔ(x¯,t)=au(x¯+1,t)+bu(x¯,t)+cu(x¯-1,t)+f(u(x¯,t),x¯,t)<(a+b+c)R+f(R,x¯,t)0,

i.e., uΔ(x¯,t)<0, a contradiction.

Once we know that u(x¯±1,t)=R for all t[t0,t¯]𝕋, it follows by induction with respect to x that u(x,t)=R for all x and t[t0,t¯]𝕋. ∎

With the help of the previous two lemmas, we derive the strong maximum principle.

Theorem 5.3 (Strong maximum principle).

Assume that (H1), (H2), (H3), (H4¯), (H5¯), and (H6¯) hold with r=mM=R and u:Z×[t0,T]TR is a bounded solution of (2.1). If u(x¯,t¯){r,R} for some x¯Z and t¯(t0,T]T, then the following statements hold:

  • (a)

    If [t0,t¯]𝕋 contains only isolated points, i.e., t0=ρ𝕋k(t¯) for some k , and

    𝒟(x¯,t¯)={(x,t)×[t0,t¯]𝕋:t=ρ𝕋j(t¯),j=0,,k, and x=x¯±i,i=0,,j},

    then u(x,t)=u(x¯,t¯) for all (x,t)𝒟(x¯,t¯).

  • (b)

    Otherwise, if [t0,t¯]𝕋 contains a point which is not isolated, then u is constant on ×[t0,T]𝕋.

Remark 5.4.

In order to prevent any confusion, we emphasize that the fact whether a point is isolated or not is considered with respect to the time scale interval [t0,t¯]𝕋, not the entire time scale 𝕋. In other words, the statement distinguishes between the cases in which the interval [t0,t¯]𝕋 is a finite set (part (a)) or at least countable (part (b)).

Proof of Theorem 5.3.

We consider the case when u(x¯,t¯)=R; the case u(x¯,t¯)=r can be treated in a similar way. We prove the statement by analyzing two different cases: Case (1): Let there be a left-dense point in [t0,t¯]𝕋. Denote

𝒫ld={t[t0,t¯]𝕋:t is left-dense}

and tld=sup𝒫ld. Given the definition of the supremum and the fact that 𝕋 is a closed set, we obtain tld𝕋. To show that tld is left-dense let us assume by contradiction that tld is left-scattered. Thus, tld𝒫ld and immediately from the definition of the supremum we get a contradiction. From the proofs of Lemmas 5.1 and 5.2 we obtain that u(x¯,t)=R for all t[t0,t¯]𝕋, and particularly u(x¯,tld)=R. Furthermore, since tld is left-dense, Lemma 5.2 yields that

u(x,t)=Rfor all x,t[t0,tld]𝕋.(5.2)

There remains to prove the statement for t[tld,T]𝕋. From (5.2) we get that u(x,t0)=ux0=R for all x, and thus r=m=M=R. Consequently, since (H6) holds with r=m=M=R, Theorem 4.4 (weak maximum principle) yields that

Ru(x,t)R,i.e.,u(x,t)=Rfor all x,t[t0,T]𝕋.

Case (2): Let us assume that [t0,t¯]𝕋 does not contain any left-dense point. Subcase (i): If [t0,t¯)𝕋 does not contain any right-dense point, i.e., [t0,T]𝕋 contains only isolated points, then part (a) of the theorem follows immediately from Lemma 5.1. Subcase (ii): Let there exist a right-dense point in [t0,t¯)𝕋. Denote

𝒫rd={t[t0,t¯)𝕋:t is right-dense}

and trd=sup𝒫rd. From the fact that t¯ is left-scattered and from the definition of the supremum we obtain trd<t¯. Moreover, since 𝕋 is closed, there is trd𝕋. Further, we show that trd is right-dense as well. Indeed, let us assume that trd is right-scattered, i.e., trd𝒫rd. Then trd is an unattained supremum of 𝒫rd and there exists a sequence {tn}n=1𝒫rd such that tntrd. This would imply that trd is left-dense, a contradiction. Thus, trd is right-dense.

From the definition of trd, the sequence of predecessors of t¯, namely

{ρ𝕋j(t¯)}j=1(trd,t¯]𝕋,

is well-defined and satisfies ρ𝕋j(t¯)trd. Let us assume that x is arbitrary but fixed, i.e., x=x¯+i0 or x=x¯-i0 for some i00. We consider the case x=x¯+i0; the other case is similar. Lemma 5.1 implies that for all ji0 there is

u(x,ρ𝕋j(t¯))=u(x¯+i0,ρ𝕋j(t¯))=R.

Then the continuity of the function u(x,) yields that

R=limju(x,ρ𝕋j(t¯))=u(x,trd),

and since x is arbitrary, there is u(x,trd)=R for all x.

Now we prove that u(x,t)=R for x and t[t0,trd]𝕋. We use the backward induction principle in the variable t (see [4, Theorem 1.7 and Remark 1.8]):

  • Above we have shown that for t=trd there is u(x,trd)=R for all x.

  • Let t(t0,trd]𝕋 be left-scattered and u(x,t)=R for all x. Then Lemma 5.1 immediately implies that u(x,ρ𝕋(t))=R for all x.

  • Let t[t0,trd)𝕋 be right-dense and u(x,s)=R for all x and s(t,trd]𝕋. Then again from the continuity of the functions u(x,) we obtain

    R=limst+u(x,s)=u(x,t)for all x.

  • We do not have to consider the case when t(t0,trd]𝕋 is left-dense, since we assume that [t0,trd]𝕋 does not contain any such point.

The backward induction principle implies that u(x,t)=R for all x and t[t0,trd]𝕋.

Finally, it remains to prove that u(x,t)=R for x and t[trd,T]𝕋. Since u(x,t0)=ux0=R for all x, there is r=m=M=R and, analogously to above, we can use Theorem 4.4 (weak maximum principle) to show that

Ru(x,t)R,i.e.,u(x,t)=Rfor all x,t[t0,T]𝕋.

Corollary 5.5.

Assume that (H1), (H2), (H3), (H4¯), (H5¯), and (H6¯) hold with r=mM=R. Suppose that u:Z×[t0,T]TR is a bounded solution of (2.1). If there is a point td[t0,T)T that is not isolated and if the initial condition u0 is not constant, then

r<u(x,t)<Rfor all x,t(td,T]𝕋.

Proof.

Assume by contradiction that there exist x¯, t¯(td,T]𝕋 such that u(x¯,t¯){r,R}. Since td[t0,t¯)𝕋 and td is not isolated, part (b) of Theorem 5.3 yields that u is constant on ×[t0,T]𝕋, a contradiction to the assumption that u0 is not constant. ∎

The following remarks explain why the original conditions (H4)–(H6) are not sufficient to establish the strong maximum principle, and had to be replaced by their stronger counterparts (H4¯)–(H6¯).

Remark 5.6.

(H4) is too weak for the strong maximum principle; we need the constants a,c to be strictly positive. Indeed, let us consider the linear transport equation

ut(x,t)=-u(x,t)+u(x-1,t),x,t[0,T],u(x,0)={1,x0,0,x<0,

i.e., the initial-value problem (2.1) with a=0, b=-1, c=1, and f0. Then the unique bounded solution is given by (see [30, Corollary 4.3])

u(x,t)={j=0xtjj!e-t,x0,t[0,T],0,x<0,t[0,T].

Thus, the strong maximum principle does not hold.

Remark 5.7.

To see that (H5) does not suffice, consider the time scale 𝕋=0 and the following linear equation (f0):

uΔ(x,t)=12u(x+1,t)-u(x,t)+12u(x-1,t),x,t0,

which corresponds to (2.1) with a=c=12, b=-1 and f0. This equation holds if and only if

u(x,t+1)=12u(x+1,t)+12u(x-1,t),x,t0.

For the initial condition

u(x,0)={1,x is even,0,x is odd,

we obtain

u(x,1)={0,x is even,1,x is odd,

which violates the strong maximum principle.

Remark 5.8.

Finally, let a,b,c be an arbitrary triple satisfying (H4¯), and 𝕋=μ0={0,μ,2μ,}, where μ>0 satisfies (H5¯). Consider problem (2.1) with

ux0={1,x0,0,x=0,andf(u,x,t)=(b+1μ)(1-u).

We have m=0 and M=1. For r=0 and R=1 the function f satisfies (H6), but not (H6¯). Using (2.1), we calculate

u(0,μ)=μau(1,0)+(1+μb)u(0,0)+μcu(-1,0)+μf(u(0,0),0)=μ(a+c)+(1+μb)=(H4¯)1.

Therefore, u(0,μ)=1=R, but u(0,0)=0, which contradicts the strong maximum principle.

References

  • [1]

    L. Adamec, A note on continuous dependence of solutions of dynamic equations on time scales, J. Difference Equ. Appl. 17 (2011), 647–656.  CrossrefWeb of ScienceGoogle Scholar

  • [2]

    J. Bell, Some threshold results for models of myelinated nerves, Math. Biosci. 54 (1981), 181–190.  CrossrefGoogle Scholar

  • [3]

    M. Bohner, M. Federson and J. G. Mesquita, Continuous dependence for impulsive functional dynamic equations involving variable time scales, Appl. Math. Comput. 221 (2013), 383–393.  Web of ScienceGoogle Scholar

  • [4]

    M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.  Google Scholar

  • [5]

    T. Caraballo, F. Morillas and J. Valero, Asymptotic behaviour of a logistic lattice system, Discrete Contin. Dyn. Syst. 34 (2014), no. 10, 4019–4037.  CrossrefGoogle Scholar

  • [6]

    S.-N. Chow, Lattice dynamical systems, Dynamical Systems, Lecture Notes in Math. 1822, Springer, Berlin (2003), 1–102.  Google Scholar

  • [7]

    S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst. 42 (1995), 746–751.  CrossrefGoogle Scholar

  • [8]

    S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations 149 (1998), 248–291.  CrossrefGoogle Scholar

  • [9]

    S.-N. Chow and W. Shen, Dynamics in a discrete Nagumo equation: Spatial topological chaos, SIAM J. Appl. Math. 55 (1995), 1764–1781.  CrossrefGoogle Scholar

  • [10]

    T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Phys. D 67 (1993), 237–244.  CrossrefGoogle Scholar

  • [11]

    M. Federson, J. G. Mesquita and A. Slavík, Basic results for functional differential and dynamic equations involving impulses, Math. Nachr. 286 (2013), no. 2–3, 181–204.  CrossrefWeb of ScienceGoogle Scholar

  • [12]

    A. Feintuch and B. Francis, Infinite chains of kinematic points, Automatica J. IFAC 48 (2012), no. 5, 901–908.  CrossrefGoogle Scholar

  • [13]

    M. Friesl, A. Slavík and P. Stehlík, Discrete-space partial dynamic equations on time scales and applications to stochastic processes, Appl. Math. Lett. 37 (2014), 86–90.  Web of ScienceCrossrefGoogle Scholar

  • [14]

    B. M. Garay, S. Hilger and P. E. Kloeden, Continuous dependence in time scale dynamics, Proceedings of the Sixth International Conference on Difference Equations (Augsburg 2001), CRC Press, Boca Raton (2004), 279–287.  Google Scholar

  • [15]

    N. T. Ha, N. H. Du, L. C. Loi and D. D. Thuan, On the convergence of solutions to dynamic equations on time scales, Qual. Theory Dyn. Syst. (2015), 10.1007/s12346-015-0166-8.  Web of ScienceGoogle Scholar

  • [16]

    S. Hilger, Analysis on measure chains – A unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18–56.  CrossrefGoogle Scholar

  • [17]

    C. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations 259 (2015), 1967–1989.  Web of ScienceCrossrefGoogle Scholar

  • [18]

    H. Hupkes and E. Van Vleck, Travelling waves for complete discretizations of reaction diffusion systems, J. Dynam. Differential Equations 28 (2016), 955–1006.  CrossrefGoogle Scholar

  • [19]

    J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math. 47 (1987), 556–572.  CrossrefGoogle Scholar

  • [20]

    P. E. Kloeden, A Gronwall-like inequality and continuous dependence on time scales, Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday, Kluwer Academic, Dordrecht (2003), 645–659.  Google Scholar

  • [21]

    J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, Dynamical Systems, Lecture Notes in Math. 1822, Springer, Berlin (2003), 231–298.  Google Scholar

  • [22]

    G. A. Monteiro and M. Tvrdý, Generalized linear differential equations in a Banach space: Continuous dependence on a parameter, Discrete Contin. Dyn. Syst. 33 (2013), no. 1, 283–303.  Web of ScienceGoogle Scholar

  • [23]

    A. Peterson and B. Thompson, Henstock–Kurzweil delta and nabla integrals, J. Math. Anal. Appl. 323 (2006), 162–178.  CrossrefGoogle Scholar

  • [24]

    C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Springer, Berlin, 2010.  Web of ScienceGoogle Scholar

  • [25]

    Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, River Edge, 1992.  Google Scholar

  • [26]

    Š. Schwabik, Abstract Perron–Stieltjes integral, Math. Bohem. 121 (1996), 425–447.  Google Scholar

  • [27]

    A. Slavík, Dynamic equations on time scales and generalized ordinary differential equations, J. Math. Anal. Appl. 385 (2012), 534–550.  Web of ScienceCrossrefGoogle Scholar

  • [28]

    A. Slavík and P. Stehlík, Explicit solutions to dynamic diffusion-type equations and their time integrals, Appl. Math. Comput. 234 (2014), 486–505.  Web of ScienceGoogle Scholar

  • [29]

    A. Slavík and P. Stehlík, Dynamic diffusion-type equations on discrete-space domains, J. Math. Anal. Appl. 427 (2015), no. 1, 525–545.  CrossrefWeb of ScienceGoogle Scholar

  • [30]

    P. Stehlík and J. Volek, Transport equation on semidiscrete domains and Poisson–Bernoulli processes, J. Difference Equ. Appl. 19 (2013), no. 3, 439–456.  Web of ScienceCrossrefGoogle Scholar

  • [31]

    P. Stehlík and J. Volek, Maximum principles for discrete and semidiscrete reaction-diffusion equation, Discrete Dyn. Nat. Soc. 2015 (2015), Article ID 791304.  Web of ScienceGoogle Scholar

  • [32]

    V. Volpert, Elliptic Partial Differential Equations: Volume 2 Reaction-Diffusion Equations, Springer Monogr. Math. 104, Springer, Basel, 2014.  Google Scholar

  • [33]

    B. Wang, Dynamics of systems of infinite lattices, J. Differential Equations 221 (2006), 224–245.  CrossrefGoogle Scholar

  • [34]

    B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl. 331 (2007), 121–136.  Web of ScienceCrossrefGoogle Scholar

  • [35]

    H. F. Weinberger, Long time behavior of a class of biological models, SIAM J. Math. Anal. 13 (1982), 353–396.  CrossrefGoogle Scholar

  • [36]

    B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations 96 (1992), 1–27.  CrossrefGoogle Scholar

  • [37]

    B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher’s equation, J. Differential Equations 105 (1993), 46–62.  CrossrefGoogle Scholar

About the article

Received: 2016-05-23

Revised: 2016-10-03

Accepted: 2017-02-16

Published Online: 2017-03-17


Funding Source: Grantová Agentura České Republiky

Award identifier / Grant number: GA15-07690S

All three authors acknowledge the support by the Czech Science Foundation, grant number GA15-07690S.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 303–322, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0116.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[2]
Petr Stehlík and Petr Vaněk
Linear Algebra and its Applications, 2017, Volume 531, Page 64

Comments (0)

Please log in or register to comment.
Log in