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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 679465, 30 pages
http://dx.doi.org/10.1155/2012/679465
Research Article

A Weak Comparison Principle for Reaction-Diffusion Systems

Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avenida Universidad s/n, 03202 Elche, Spain

Received 4 May 2012; Accepted 17 July 2012

Academic Editor: S. Romaguera

Copyright © 2012 José Valero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove a weak comparison principle for a reaction-diffusion system without uniqueness of solutions. We apply the abstract results to the Lotka-Volterra system with diffusion, a generalized logistic equation, and to a model of fractional-order chemical autocatalysis with decay. Moreover, in the case of the Lotka-Volterra system a weak maximum principle is given, and a suitable estimate in the space of essentially bounded functions 𝐿 is proved for at least one solution of the problem.

1. Introduction

Comparison results for parabolic equations and ordinary differential equations are well known in the literature (see, e.g., [14] among many others). One of the important applications of such kind of results is the theory of monotone dynamical systems, which leads to a more precise characterization of 𝜔-limit sets and attractors. In the last years, several authors have been working in this direction (see, e.g., [48] for the deterministic case, and [912] for the stochastic case). In all these papers, it is considered the classical situation where the initial-value problem possesses a unique solution.

However, the situation is more complicated when we consider a differential equation for which uniqueness of the Cauchy problem fails (or just it is not known to hold). Let us consider an abstract parabolic problem: 𝑑𝑢𝑑𝑡=𝐴(𝑡,𝑢(𝑡)),𝜏𝑡𝑇,𝑢(𝜏)=𝑢𝜏,(1.1) for which we can prove that for every initial data in the phase space 𝑋 (with a partial order ) there exists at least one solution.

If we try to compare solutions of (1.1) for two ordered initial data 𝑢1𝜏𝑢2𝜏, then we can consider a strong comparison principle and a weak one.

The strong version would imply the existence of a solution 𝑢1 with 𝑢1(𝜏)=𝑢1𝜏 such that 𝑢1(𝑡)𝑢2[],(𝑡)for𝑡𝜏,𝑇(1.2) for any solution 𝑢2 with 𝑢2(𝜏)=𝑢2𝜏, and, viceversa, the existence of a solution 𝑢2 with 𝑢2(𝜏)=𝑢2𝜏 such that (1.2) is satisfied for any solution 𝑢1 with 𝑢1(𝜏)=𝑢1𝜏. This kind of result is established in [13] for a delayed ordinary differential equations, defining then a multivalued order-preserving dynamical system.

The weak version of the comparison principle says that if 𝑢1𝜏𝑢2𝜏, then there exist two solutions 𝑢1,𝑢2 of (1.1) such that 𝑢1(𝜏)=𝑢1𝜏, 𝑢2(𝜏)=𝑢2𝜏, and (1.2) hold.

There is in fact an intermediate version of the comparison principle, which says that if we fix a solution 𝑢1 of (1.1) with 𝑢1(𝜏)=𝑢1𝜏, then there exists a solution 𝑢2 with 𝑢2(𝜏)=𝑢2𝜏 such that (1.2) is satisfied (and vice versa). This is proved in [14] for a differential inclusion generated by a subdifferential map.

In this paper, we establish a weak comparison principle for a reaction-diffusion system in which the nonlinear term satisfies suitable dissipative and growth conditions, ensuring existence of solutions but not uniqueness. This principle is applied to several well-known models in physics and biology. Namely, a weak comparison of solutions is proved for the Lotka-Volterra system, the generalized logistic equation and for a model of fractional-order chemical autocatalysis with decay. Moreover, in the case of the Lotka-Volterra system a weak maximum principle is given, and a suitable estimate in the space of essentially bounded functions 𝐿 is proved for at least one solution of the problem.

We note that in the papers [15, 16] the existence of a global attractor is proved for such kind of reaction-diffusion systems. In the near future, we will apply these results to obtain theorems concerning the structure of the global attractor.

2. Comparison Results for Reaction-Diffusion Systems

We shall denote by || and (,) the norm and scalar product in the space 𝑚, 𝑚1. Let 𝑑>0 be an integer and Ω𝑁 be a bounded open subset with smooth boundary. Consider the problem: 𝜕𝑢𝜕𝑡𝑎Δ𝑢+𝑓(𝑡,𝑢)=(𝑡,𝑥),(𝑡,𝑥)(𝜏,𝑇)×Ω,𝑢|𝑥𝜕Ω𝑢|=0,𝑡=𝜏=𝑢𝜏(𝑥),(2.1) where 𝜏,𝑇, 𝑇>𝜏, 𝑥Ω, 𝑢=(𝑢1(𝑡,𝑥),,𝑢𝑑(𝑡,𝑥)),𝑓=(𝑓1,,𝑓𝑑), 𝑎 is a real 𝑑×𝑑 matrix with a positive symmetric part (𝑎+𝑎𝑡)/2𝛽𝐼,𝛽>0, 𝐿2(𝜏,𝑇;(𝐿2(Ω))𝑑). Moreover, 𝑓=(𝑓1(𝑡,𝑢),,𝑓𝑑(𝑡,𝑢)) is jointly continuous on [𝜏,𝑇]×𝑑 and satisfies the following conditions: 𝑑𝑖=1||𝑓𝑖||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)𝐶11+𝑑𝑖=1||𝑢𝑖||𝑝𝑖,(𝑓(𝑡,𝑢),𝑢)𝛼𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐶2,(2.2) where 𝑝𝑖2, 𝛼,𝐶1,𝐶2>0.

Let 𝐻=(𝐿2(Ω))𝑑, 𝑉=(𝐻10(Ω))𝑑, and let 𝑉 be the dual space of 𝑉. By , 𝑉 we denote the norm by 𝐻 and 𝑉, respectively. For 𝑝=(𝑝1,,𝑝𝑑), we define the spaces 𝐿𝑝(Ω)=𝐿𝑝1(Ω)××𝐿𝑝𝑑(𝐿Ω),𝑝(𝜏,𝑇;𝐿𝑝(Ω))=𝐿𝑝1(𝜏,𝑇;𝐿𝑝1(Ω))××𝐿𝑝𝑑(𝜏,𝑇;𝐿𝑝𝑑(Ω)).(2.3) We take 𝑞=(𝑞1,,𝑞𝑑), where (1/𝑝𝑖)+(1/𝑞𝑖)=1.

We say that the function 𝑢() is a weak solution of (2.1) if 𝑢𝐿𝑝(𝜏,𝑇;𝐿𝑝(Ω))𝐿2(𝜏,𝑇;𝑉)𝐶([𝜏,𝑇];𝐻), 𝑑𝑢/𝑑𝑡𝐿2(𝜏,𝑇;𝑉)+𝐿𝑞(𝜏,𝑇;𝐿𝑞(Ω)), 𝑢(𝜏)=𝑢𝜏, and 𝑇𝜏𝑑𝑢𝑑𝑡,𝜉𝑑𝑡+𝑇𝜏Ω((𝑎𝑢),𝜉)𝑑𝑥𝑑𝑡+𝑇𝜏Ω(𝑓(𝑡,𝑢),𝜉)𝑑𝑥𝑑𝑡=𝑇𝜏Ω(,𝜉)𝑑𝑥𝑑𝑡,(2.4) for all 𝜉𝐿𝑝(𝜏,𝑇;𝐿𝑝(Ω))𝐿2(𝜏,𝑇;𝑉), where , denotes pairing in the space 𝑉+𝐿𝑞(Ω), and (𝑢,𝑣)=𝑑𝑖=1(𝑢𝑖,𝑣𝑖).

Under conditions (2.2), it is known [17, page 284] that for any 𝑢𝜏𝐻 there exists at least one weak solution 𝑢=𝑢(𝑡,𝑥) of (2.1), and also that the function 𝑡𝑢(𝑡)2 is absolutely continuous on [𝜏,𝑇] and (𝑑/𝑑𝑡)𝑢(𝑡)2=2𝑑𝑢/𝑑𝑡,𝑢 for a.a. 𝑡(𝜏,𝑇).

Denote 𝑟=(𝑟1,,𝑟𝑑),𝑟𝑖=max{1;𝑁(1/𝑞𝑖1/2)}. Any weak solution satisfies (𝑑𝑢/𝑑𝑡)𝐿𝑞(𝜏,𝑇;𝐻𝑟(Ω)) and 𝐿𝑞(0,𝑇;𝐻𝑟(Ω))=𝐿𝑞1(0,𝑇;𝐻𝑟1(Ω))××𝐿𝑞𝑑(0,𝑇;𝐻𝑟𝑑(Ω)).(2.5)

If, additionally, we assume that that 𝑓(𝑡,𝑢) is continuously differentiable with respect to 𝑢 for any 𝑡[𝜏,𝑇],𝑢𝑑, and 𝑓𝑢(𝑡,𝑢)𝑤,𝑤𝐶3(𝑡)|𝑤|2,𝑤,𝑢𝑑,(2.6) where 𝐶3()𝐿1(𝜏,𝑇), 𝐶3(𝑡)0, the weak solution of (2.1) is unique. Here, 𝑓𝑢 denotes the Jacobian matrix of 𝑓.

We consider also the following assumption: there exists 𝑅0>0 such that 𝑓𝑖(𝑡,𝑢)𝑓𝑖(𝑡,𝑣),(2.7) for any 𝑡[𝜏,𝑇] and any 𝑢,𝑣𝑑 such that 𝑢𝑖=𝑣𝑖 and 𝑢𝑗𝑣𝑗 if 𝑗𝑖, and |𝑢|,|𝑣|𝑅0, which means that the systems is cooperative in the ball with radius 𝑅0 centered at 0.

Consider the two problems: 𝜕𝑢𝜕𝑡𝑎Δ𝑢+𝑓1(𝑡,𝑢)=1(𝑡,𝑥),(𝑡,𝑥)(𝜏,𝑇)×Ω,𝑢|𝑥𝜕Ω𝑢|=0,𝑡=𝜏=𝑢𝜏(𝑥),(2.8)𝜕𝑢𝜕𝑡𝑎Δ𝑢+𝑓2(𝑡,𝑢)=2𝑢|(𝑡,𝑥),(𝑡,𝑥)(𝜏,𝑇)×Ω,𝑥𝜕Ω=0,𝑢|𝑡=𝜏=𝑢𝜏(𝑥),(2.9) where 𝑓𝑗 are jointly continuous on [𝜏,𝑇]×𝑑. Among conditions (2.2) and (2.6)-(2.7), we shall consider the following: 1(𝑡,𝑥)2(𝑡,𝑥),fora.a.(𝑡,𝑥),𝑓𝑖1(𝑡,𝑢)𝑓𝑖2(𝑡,𝑢),𝑡,𝑢.(2.10)

Lemma 2.1. If 𝑓𝑗 satisfy (2.2) and (2.10), then the constants 𝑝𝑖 have to be the same for 𝑓1 and 𝑓2.

Proof. Denote by 𝑝𝑗𝑖, 𝛼𝑗,𝐶𝑗1, and 𝐶𝑗2 the constants corresponding to 𝑓𝑗 in (2.2). By contradiction let, for example, 𝑝21>𝑝11. Take the sequence 𝑢𝑛=(𝑢1𝑛,0,,0), where 𝑢1𝑛+ as 𝑛. Then by (2.2), (2.10), and Young’s inequality, we have 𝛼2||𝑢1𝑛||𝑝21𝐶22𝑓2𝑡,𝑢𝑛,𝑢𝑛𝑓1𝑡,𝑢𝑛,𝑢𝑛=𝑓11𝑡,𝑢𝑛𝑢1𝑛𝐶11||𝑢1+1𝑛||𝑝11(𝑝111)/𝑝11𝑢𝑛1||𝑢𝐶1+1𝑛||𝑝11.(2.11) But 𝑝21>𝑝11 implies the existence of 𝑛 such that 𝛼2|𝑢1𝑛|𝑝21𝐶22>𝐶(1+|𝑢1𝑛|𝑝11), which is a contradiction. Hence, 𝑝21𝑝11.
Conversely, let 𝑝21<𝑝11. Then we take 𝑢𝑛=(𝑢1𝑛,0,,0) with 𝑢1𝑛 as 𝑛 so that 𝛼1||𝑢1𝑛||𝑝11𝐶12𝑓1𝑡,𝑢𝑛,𝑢𝑛𝑓2𝑡,𝑢𝑛,𝑢𝑛=𝑓12𝑡,𝑢𝑛𝑢1𝑛𝐶21||𝑢1+1𝑛||𝑝21(𝑝211)/𝑝21𝑢𝑛1||𝑢𝐶1+1𝑛||𝑝21.(2.12) As before, we obtain a contradiction, so 𝑝21=𝑝11.
Repeating similar arguments for the other 𝑝𝑗𝑖, we obtain that 𝑝1𝑖=𝑝2𝑖 for 𝑖=1,,𝑑.

We recall [15] that under conditions (2.2) any solution 𝑢() of (2.8) satisfies the inequality: 𝑢(𝑡)2+2𝛽𝑡𝑠𝑢(𝜏)2𝑑𝜏+𝛼𝑑𝑖=1𝑡𝑠𝑢𝑖(𝑟)𝑝𝑖𝐿𝑝𝑖(Ω)𝑑𝑟𝑢(𝑠)2+𝐶𝑡𝑠1(𝑟)2+1𝑑𝑟,(2.13) for some constant 𝐶>0. Of course, the same is valid for any solution of (2.9). From (2.13), for any 𝑇>𝜏 we obtain 𝑢(𝑡)2𝑢𝜏2+𝐶𝑇𝜏1(𝑟)2+1𝑑𝑟=𝐾2𝑢𝜏,𝜏,𝑇𝜏𝑡𝑇.(2.14)

We shall denote by 𝑢1(𝑡) the solution of (2.8) corresponding to the initial data 𝑢1𝜏 and by 𝑢2(𝑡) the solution of (2.9) corresponding to the initial data 𝑢2𝜏. Also, we take 𝑢+=max{𝑢,0} for 𝑢.

We obtain the following comparison result.

Theorem 2.2. Assume that 𝑓𝑗,𝑗 satisfy (2.2), (2.6), and (2.10). If 𝑢1𝜏𝑢2𝜏 and we suppose that 𝑓2 satisfies (2.7) with 𝑅202max{𝐾2(𝑢1𝜏,𝜏,𝑇),𝐾2(𝑢2𝜏,𝜏,𝑇)}, where 𝐾(𝑢𝑗𝜏,𝜏,𝑇) is taken from (2.14), we have 𝑢1(𝑡)𝑢2(𝑡), for all 𝑡[𝜏,𝑇].

Remark 2.3. The results remain valid if, instead, 𝑓1 satisfies (2.7) with 𝑅20𝐾2(𝑢1𝜏,𝑇).

Remark 2.4. If 𝑓2 satisfies (2.7) for an arbitrary 𝑅0>0 (i.e., in the whole space 𝑑), then the result is true for any initial data 𝑢1𝜏𝑢2𝜏.

Proof. Let 𝑔2(𝑡,𝑢)=𝑓2(𝑡,𝑢)+𝐶3(𝑡)𝑢. The function 𝑔2(𝑡,) satisfies (2.6) with 𝐶30. For any 𝑢1,𝑢2𝑑 define 𝑣2(𝑢1,𝑢2) by 𝑣𝑖2=𝑢𝑖2,if𝑢𝑖1𝑢𝑖2,𝑢𝑖1,if𝑢𝑖1<𝑢𝑖2.(2.15) Note that 𝑢1𝑣2=(𝑢1𝑢2)+ and 𝑣2𝑢2=(𝑢2𝑢1)+, so 𝑣𝑖2𝑢𝑖2 for all 𝑖. For the function (𝑢1𝑢2)+, we can obtain by (2.10) and the mean value theorem that 12𝑑𝑢𝑑𝑡1𝑢2+2𝑢+𝑎1𝑢2+2𝑉Ω𝑓1𝑡,𝑢1𝑓2𝑡,𝑢2,𝑢1𝑢2+𝑑𝑥Ω𝑓2𝑡,𝑢1𝑓2𝑡,𝑣2,𝑢1𝑢2+𝑑𝑥Ω𝑓2𝑡,𝑣2𝑓2𝑡,𝑢2,𝑢1𝑢2+𝑑𝑥=Ω𝑔2𝑢𝑡,𝑣𝑡,𝑥,𝑢1,𝑢2𝑢1𝑣2,𝑢1𝑢2+𝑑𝑥+𝐶3(𝑢𝑡)1𝑢2+2Ω𝑓2𝑡,𝑣2𝑓2𝑡,𝑢2,𝑢1𝑢2+𝑑𝑥=Ω𝑔2𝑢𝑡,𝑣𝑡,𝑥,𝑢1,𝑢2𝑢1𝑢2+,𝑢1𝑢2+𝑑𝑥+𝐶3𝑢(𝑡)1𝑢2+2Ω𝑓2𝑡,𝑣2𝑓2𝑡,𝑢2,𝑢1𝑢2+𝑑𝑥𝐶3𝑢(𝑡)1𝑢2+2Ω𝑓2𝑡,𝑣2𝑓2𝑡,𝑢2,𝑢1𝑢2+𝑑𝑥.(2.16)
For all 𝑡, we have 𝑓2𝑡,𝑣2𝑓2𝑡,𝑢2,𝑢1𝑢2+=𝑖𝐽𝑓𝑖2𝑡,𝑣2𝑓𝑖2𝑡,𝑢2𝑢𝑖1𝑢𝑖2+,(2.17) where 𝑢𝑖1𝑢𝑖2>0, for 𝑖𝐽, and 𝑢𝑖1𝑢𝑖20 if 𝑖𝐽. For any 𝑖𝐽, we have that 𝑣𝑖2=𝑢𝑖2, and then by 𝑣2𝑢2, |𝑣2|2|𝑢1|2+|𝑢2|2, (2.14), and (2.7), we get 𝑓𝑖2𝑡,𝑣2𝑓𝑖2𝑡,𝑢20.(2.18)
By Gronwall’s lemma, we get 𝑢1(𝑡)𝑢2(𝑡)+2𝑢1𝜏𝑢2𝜏+2𝑒𝑡𝜏2𝐶3(𝑠)𝑑𝑠=0.(2.19) Thus (𝑢1𝑢2)+=0, which means that 𝑢𝑖1(𝑥,𝑡)𝑢𝑖2(𝑥,𝑡)0, for a.a. 𝑥Ω, and all 𝑖{1,,𝑑}, 𝑡[𝜏,𝑇].

Remark 2.5. In the scalar case, that is, 𝑑=1, condition (2.7) is trivially satisfied.

When condition (2.6) fails to be true, we will obtain a weak comparison principle.

Define a sequence of smooth functions 𝜓𝑘+[0,1] satisfying 𝜓𝑘(𝑠)=1,if0𝑠𝑘,0𝜓𝑘(𝑠)1,if𝑘𝑠𝑘+1,0,if𝑠𝑘+1.(2.20)

For every 𝑘1 we put 𝑓𝑖𝑘(𝑡,𝑢)=𝜓𝑘(|𝑢|)𝑓𝑖(𝑡,𝑢)+(1𝜓𝑘(|𝑢|))𝑔𝑖(𝑢), where 𝑔𝑖(𝑢)=|𝑢𝑖|𝑝𝑖2𝑢𝑖. Then 𝑓𝑘([𝜏,𝑇]×𝑑;𝑑) and for any 𝐴>0, sup𝑡[𝜏,𝑇]sup|𝑢|𝐴||𝑓𝑘||(𝑡,𝑢)𝑓(𝑡,𝑢)0,as𝑘.(2.21)

Let 𝜌𝜀𝑑+ be a mollifier, that is, 𝜌𝜖0(𝑑;), supp𝜌𝜖𝐵𝜖={𝑥𝑑|𝑥|<𝜖}, 𝑑𝜌𝜖(𝑠)𝑑𝑠=1 and 𝜌𝜖(𝑠)0 for all 𝑠𝑑. We define the functions 𝑓𝜖𝑘(𝑡,𝑢)=𝑑𝜌𝜖(𝑠)𝑓𝑘(𝑡,𝑢𝑠)𝑑𝑠.(2.22) Since for any 𝑘1𝑓𝑘 is uniformly continuous on [0,𝑇]×[𝑘1,𝑘+1], there exist 𝜖𝑘(0,1) such that for all 𝑢 satisfying |𝑢|𝑘, and for all 𝑠 for which |𝑢𝑠|<𝜖𝑘 we have sup𝑡[𝜏,𝑇]||𝑓𝑘(𝑡,𝑢)𝑓𝑘||1(𝑡,𝑠)𝑘.(2.23) We put 𝑓𝑘(𝑡,𝑢)=𝑓𝜖𝑘𝑘(𝑡,𝑢). Then 𝑓𝑘(𝑡,)(𝑑;𝑑), for all 𝑡[𝜏,𝑇],𝑘1.

For further arguments we need the following technical result [16, Lemma 2].

Lemma 2.6. Let 𝑓 satisfy (2.2). For all 𝑘1, the following statements hold: sup𝑡[𝜏,𝑇]sup|𝑢|𝐴||𝑓𝑘||(𝑡,𝑢)𝑓(𝑡,𝑢)0,as𝑘,𝐴>0,(2.24)𝑑𝑖=1||𝑓𝑘𝑖||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)𝐷11+𝑑𝑖=1||𝑢𝑖||𝑝𝑖,𝑓𝑘(𝑡,𝑢),𝑢𝛾𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐷2,𝑓(2.25)𝑘𝑢(𝑡,𝑢)𝑤,𝑤𝐷3(𝑘)|𝑤|2,𝑢,𝑤,(2.26) where 𝐷3(𝑘) is a nonnegative number, and the positive constants 𝐷1, 𝐷2𝐶2, 𝛾 do not depend on 𝑘.

Consider first the scalar case.

Theorem 2.7. Let 𝑑=1. Assume that 𝑓𝑗,𝑗 satisfy (2.2) and (2.10). If 𝑢1𝜏𝑢2𝜏, there exist two solutions 𝑢1,𝑢2 (of (2.8) and (2.9), resp.) such that 𝑢1(𝑡)𝑢2(𝑡) for all 𝑡[𝜏,𝑇].

Proof. For the functions 𝑓𝑗 we take the approximations 𝑓𝑘𝑗 (defined in Lemma 2.6), which satisfy (2.24)–(2.26), and consider the problems 𝜕𝑢𝜕𝑡𝑎Δ𝑢+𝑓𝑘𝑗(𝑡,𝑢)=𝑗(𝑡,𝑥),(𝑡,𝑥)(𝜏,𝑇)×Ω,𝑢|𝑥𝜕Ω𝑢|=0,𝑡=𝜏=𝑢𝜏,(2.27) for 𝑗=1,2. Problem (2.27) has a unique solution for any initial data 𝑢𝜏𝐻. In view of Lemma 2.1, the constant 𝑝 is the same for 𝑓1 and 𝑓2. We note that 𝑓1𝑘(𝑡,𝑢)=𝜓𝑘(|𝑢|)𝑓1(𝑡,𝑢)+1𝜓𝑘(|𝑢|)|𝑢|𝑝2𝑢𝜓𝑘(|𝑢|)𝑓2(𝑡,𝑢)+1𝜓𝑘(|𝑢|)|𝑢|𝑝2𝑢=𝑓2𝑘(𝑡,𝑢).(2.28) Then, it is clear that 𝑓𝑘1(𝑡,𝑢)𝑓𝑘2(𝑡,𝑢) for every (𝑡,𝑢).
By Theorem 2.2 we know that as 𝑢1𝜏𝑢2𝜏, we have 𝑢𝑘1(𝑡)𝑢𝑘2(𝑡), for all 𝑡[𝜏,𝑇], for the corresponding solutions of (2.27).
In view of Lemma 2.6, one can obtain in a standard way that (2.13) is satisfied for the solutions of (2.27) with a constant 𝐶 not depending on 𝑘 and replacing 𝛼 by 𝛾. Hence, the sequences 𝑢𝑘𝑗() are bounded in 𝐿(𝜏,𝑇;𝐻)𝐿2(𝜏,𝑇;𝑉)𝐿𝑝(𝜏,𝑇;𝐿𝑝(Ω)). It follows from (2.25) that 𝑓𝑗𝑘(,𝑢𝑘𝑗()) are bounded in 𝐿𝑞(𝜏,𝑇;𝐿𝑞(Ω)) and also that {(𝑑𝑢𝑘/𝑑𝑡)()} is bounded in 𝐿𝑞(𝜏,𝑇;𝐻𝑟(Ω)), where 𝑟𝑖=max{1;(1/21/𝑝𝑖)𝑁}. By the Compactness Lemma [18], we have that for some functions 𝑢𝑗=𝑢𝑗(𝑡,𝑥), 𝑗=1,2: 𝑢𝑘𝑗𝑢𝑗weaklystarin𝐿𝑢(𝜏,𝑇;𝐻),(2.29)𝑘𝑗𝑢𝑗in𝐿2(𝜏,𝑇;𝐻),𝑢𝑘𝑗(𝑡)𝑢𝑗𝑢(𝑡)in𝐻fora.a.𝑡(𝜏,𝑇),(2.30)𝑘𝑗(𝑡,𝑥)𝑢𝑗𝑢(𝑡,𝑥)fora.a.(𝑡,𝑥)(𝜏,𝑇)×Ω,(2.31)𝑘𝑗𝑢𝑗weaklyin𝐿2(𝜏,𝑇;𝑉),(2.32)𝑑𝑢𝑘𝑗𝑑𝑡𝑑𝑢𝑗𝑑𝑡weaklyin𝐿𝑞(𝜏,𝑇;𝐻𝑟𝑢(Ω)),(2.33)𝑘𝑗𝑢𝑗weaklyin𝐿𝑝(𝜏,𝑇;𝐿𝑝(Ω)).(2.34) Also, arguing as in [19, page 3037] we obtain 𝑢𝑘𝑗(𝑡)𝑢𝑗[].(𝑡)weaklyin𝐻𝑡𝜏,𝑇(2.35) Moreover, by (2.24) and (2.31) we have 𝑓𝑗𝑘(𝑡,𝑢𝑘(𝑡,𝑥))𝑓𝑗(𝑡,𝑢(𝑡,𝑥)) for a.a. (𝑡,𝑥)(𝜏,𝑇)×Ω and then the boundedness of 𝑓𝑗𝑘(,𝑢𝑘𝑗()) in 𝐿𝑞(𝜏,𝑇;𝐿𝑞(Ω)) implies that 𝑓𝑗𝑘(,𝑢𝑘𝑗()) converges to 𝑓(,𝑢()) weakly in 𝐿𝑞(𝜏,𝑇;𝐿𝑞(Ω)) [18]. It follows that 𝑢1(),𝑢2() are weak solutions of (2.8) and (2.9), respectively, with 𝑢1(𝜏)=𝑢1𝜏, 𝑢2(𝜏)=𝑢2𝜏.
Moreover, one can prove that 𝑢𝑘𝑗(𝑡)𝑢𝑗[].(𝑡)stronglyin𝐻𝑡𝜏,𝑇(2.36) Indeed, we define the functions 𝐽𝑗𝑘(𝑡)=𝑢𝑘𝑗(𝑡)2𝐶𝑡𝜏(𝑗(𝑠)2+1)𝑑𝑠, 𝐽𝑗(𝑡)=𝑢𝑗(𝑡)2𝐶𝑡𝜏(𝑗(𝑠)2+1)𝑑𝑠, which are nonincreasing in view of (2.13). Also, from (2.30) we have 𝐽𝑗𝑘(𝑡)𝐽𝑗(𝑡) for a.a. 𝑡(𝜏,𝑇). Then one can prove that limsup𝑘𝐽𝑗𝑘(𝑡)𝐽𝑗(𝑡) for all 𝑡[𝜏,𝑇] (see [15, page 623] for the details). Hence, limsup𝑘𝑢𝑘𝑗(𝑡)𝑢𝑗(𝑡). Together with (2.35) this implies (2.36) (see again [15, page 623] for more details).
Hence, passing to the limit we obtain 𝑢1(𝑡)𝑢2[].(𝑡),𝑡𝜏,𝑇(2.37)

Further, let us prove the general case for an arbitrary 𝑑.

Theorem 2.8. Assume that 𝑓𝑗,𝑗 satisfy (2.2) and (2.10). Also, suppose that either 𝑓1 or 𝑓2 satisfies (2.7) for an arbitrary 𝑅0>0. If 𝑢1𝜏𝑢2𝜏, there exist two solutions 𝑢1,𝑢2 (of (2.8) and (2.9), resp.) such that 𝑢1(𝑡)𝑢2(𝑡), for all 𝑡[𝜏,𝑇].

Proof. Let 𝑓1 be the function which satisfies (2.7). We take the approximations 𝑓𝑘1,𝑓𝑘2 (defined in Lemma 2.6), which satisfy (2.24)–(2.26). Then, we consider problems (2.27).
In view of Lemma 2.1, the constants 𝑝𝑖 are the same for 𝑓1 and 𝑓2. We note that 𝑓𝑖1𝑘(𝑡,𝑢)=𝜓𝑘(|𝑢|)𝑓𝑖1(𝑡,𝑢)+1𝜓𝑘||𝑢(|𝑢|)𝑖||𝑝𝑖2𝑢𝑖𝜓𝑘(|𝑢|)𝑓𝑖2(𝑡,𝑢)+1𝜓𝑘||𝑢(|𝑢|)𝑖||𝑝𝑖2𝑢𝑖=𝑓𝑖2𝑘(𝑡,𝑢).(2.38) Then, it is clear that 𝑓𝑘1(𝑡,𝑢)𝑓𝑘2(𝑡,𝑢) for every (𝑡,𝑢).
Using Lemma 2.6 it is standard to obtain estimate (2.14) with a constant 𝐶 not depending on 𝑘. Hence, the solutions 𝑢𝑘𝑗() of (2.27) satisfy 𝑢𝑘𝑗(𝑡)2𝑢𝑗𝜏2+𝐶𝑇𝜏𝑗(𝑟)2+1𝑑𝑟=𝐾2𝑢𝑗𝜏.,𝜏,𝑇(2.39) We note that 𝑓1𝑘(𝑡,𝑢)=𝑓1(𝑡,𝑢),(2.40) if |𝑢|𝑘, since in such a case 𝜓𝑘(|𝑢|)=1. Hence, if 𝑘22max{𝐾2(𝑢1𝜏,𝜏,𝑇),𝐾2(𝑢2𝜏,𝜏,𝑇)}, the functions 𝑓1𝑘 satisfy condition (2.7) with 𝑅0=𝑘. Therefore, for any 𝑡[𝜏,𝑇] and any 𝑢,𝑣𝑑 such that 𝑢𝑖=𝑣𝑖 and 𝑢𝑗𝑣𝑗 if 𝑗𝑖, and |𝑢|,|𝑣|𝑘1, we have 𝑓𝑘1(𝑡,𝑢)=𝑑𝜌𝜖𝑘(𝑠)𝑓1𝑘(𝑡,𝑢𝑠)𝑑𝑠𝑑𝜌𝜖𝑘(𝑠)𝑓1𝑘(𝑡,𝑣𝑠)𝑑𝑠=𝑓𝑘2(𝑡,𝑢).(2.41) Thus, if (𝑘1)22max{𝐾2(𝑢1𝜏,𝑇),𝐾2(𝑢2𝜏,𝑇)}, the functions 𝑓𝑘1 satisfy condition (2.7) with 𝑅0=𝑘1.
By Theorem 2.2 we know that as 𝑢1𝜏𝑢2𝜏, we have 𝑢𝑘1(𝑡)𝑢𝑘2(𝑡), for all 𝑡[𝜏,𝑇], 𝑘1+(2max{𝐾2(𝑢1𝜏,𝑇),𝐾2(𝑢2𝜏,𝑇)})1/2, for the corresponding solutions of (2.27).
Repeating the same proof of Theorem 2.7, we obtain that the sequences 𝑢𝑘1,𝑢𝑘2 converge (up to a subsequence) in the sense of (2.29)–(2.36) to the solutions 𝑢1,𝑢2 of problems (2.8) and (2.9), respectively. Also, it holds 𝑢1(𝑡)𝑢2[].(𝑡),𝑡𝜏,𝑇(2.42)

In the applications we need to generalize this theorem to the case where the constant 𝛼 can be negative. We shall do this when 𝑓1,𝑓2 have sublinear growth (i.e., 𝑝𝑖=2 for all 𝑖). Consider for (2.1) the following conditions: ||||𝑓(𝑡,𝑢)𝐶1(1+|𝑢|),(𝑓(𝑡,𝑢),𝑢)𝛼|𝑢|2𝐶2,(2.43) where 𝛼, and 𝐶1,𝐶2>0.

Let 𝑓1,𝑓2 satisfy (2.43) with constants 𝛼𝑗,𝐶𝑗1,𝐶𝑗2,𝑗=1,2. Then if min{𝛼1,𝛼2}0, we make in (2.1) the change of variable 𝑣=𝑒𝛽𝑡𝑢, where 𝛽>min{𝛼1,𝛼2}. Hence, multiplying (2.8) and (2.9) by 𝑒𝛽𝑡 we have 𝜕𝑣𝜕𝑡𝑎Δ𝑣+𝑒𝛽𝑡𝑓1𝑡,𝑒𝛽𝑡𝑣+𝛽𝑣=𝑒𝛽𝑡1(𝑡,𝑥),(𝑡,𝑥)(𝜏,𝑇)×Ω,𝑣|𝑥𝜕Ω=0,𝑣|𝑡=𝜏=𝑒𝛽𝜏𝑢𝜏(𝑥),(2.44)𝜕𝑣𝜕𝑡𝑎Δ𝑣+𝑒𝛽𝑡𝑓2𝑡,𝑒𝛽𝑡𝑣+𝛽𝑣=𝑒𝛽𝑡2(𝑡,𝑥),(𝑡,𝑥)(𝜏,𝑇)×Ω,𝑣|𝑥𝜕Ω=0,𝑣|𝑡=𝜏=𝑒𝛽𝜏𝑢𝜏(𝑥).(2.45)

It is easy to check that if 𝑣(𝑡) is a weak solution of (2.44), then 𝑢(𝑡)=𝑒𝛽𝑡𝑣(𝑡) is a weak solution of (2.8) (and the same is true, of course, for (2.45) and (2.9)). Conversely, if 𝑢(𝑡) is a weak solution of (2.8), then 𝑣(𝑡)=𝑒𝛽𝑡𝑢(𝑡) is a weak solution of (2.44).

The functions 𝑓𝑗(𝑡,𝑣)=𝑒𝛽𝑡𝑓𝑗(𝑡,𝑒𝛽𝑡𝑣)+𝛽𝑣 satisfy (2.2) with 𝑝𝑖=2 for all 𝑖. Indeed, ||𝑓𝑗||(𝑡,𝑣)𝑒𝛽𝑡𝐶𝑗11+𝑒𝛽𝑡𝐶|𝑣|+𝛽|𝑣|𝑗1𝑓(1+|𝑣|),𝑗(𝑡,𝑣),𝑣𝑒2𝛽𝑡𝑓𝑗𝑡,𝑒𝛽𝑡𝑣,𝑒𝛽𝑡𝑣+𝛽|𝑣|2𝛼𝑗+𝛽|𝑣|2𝐶𝑗2,(2.46) where 𝛼𝑗+𝛽>0.

Then, we obtain the following.

Theorem 2.9. Assume that 𝑓𝑗,𝑗 satisfy (2.43) and (2.10). Also, suppose that either 𝑓1 or 𝑓2 satisfies (2.7) for an arbitrary 𝑅0>0. If 𝑢1𝜏𝑢2𝜏, there exist two solutions 𝑢1,𝑢2 (of (2.8) and (2.9), resp.) such that 𝑢1(𝑡)𝑢2(𝑡), for all 𝑡[𝜏,𝑇].

Proof. We consider problems (2.44) and (2.45). In view of (2.46) 𝑓𝑗(𝑡,𝑣)=𝑒𝛽𝑡𝑓𝑗(𝑡,𝑒𝛽𝑡𝑣)+𝛽𝑣 satisfy (2.2). Also, defining 𝑗(𝑡,𝑥)=𝑒𝛽𝑡𝑗(𝑡,𝑥) it is clear that (2.10) holds. Finally, if, for example, 𝑓1 satisfies (2.7) for any 𝑅0>0, then it is obvious that for 𝑓1 is true as well.
Hence, by Theorem 2.8 there exist two solutions 𝑣1,𝑣2 (of (2.44) and (2.45), resp.), with 𝑣𝑗(𝜏)=𝑒𝛽𝜏𝑢𝑗𝜏 such that 𝑣1(𝑡)𝑣2(𝑡) for all 𝑡[𝜏,𝑇]. Thus 𝑢1(𝑡)=𝑒𝛽𝑡𝑣1(𝑡)𝑒𝛽𝑡𝑣2(𝑡)=𝑢2[],(𝑡),for𝑡𝜏,𝑇(2.47) and 𝑢1,𝑢2 are solutions of (2.8) and (2.9), respectively such that 𝑢𝑗(𝜏)=𝑢𝑗𝜏.

Remark 2.10. If 𝑓𝑗 satisfy (2.6), then the solutions 𝑢1,𝑢2 given in Theorem 2.9 are unique for the corresponding initial data.

3. Comparison for Positive Solutions

Denote 𝑑+={𝑢𝑑𝑢𝑖0}. Let us consider the previous results in the case where the solutions have to be positive. Consider now the following conditions. Thematrix𝑎isdiagonal,(3.1)𝑖(𝑡,𝑥)𝑓𝑖𝑡,𝑢1,,𝑢𝑖1,0,𝑢𝑖+1,,𝑢𝑑0,(3.2) for all 𝑖, a.e. (𝑡,𝑥)(𝜏,𝑇)×Ω and 𝑢𝑗0 if 𝑗𝑖. Obviously, in the scalar case these conditions just mean that (𝑡,𝑥)𝑓(𝑡,0)0,(3.3) for a.e. (𝑡,𝑥)(𝜏,𝑇)×Ω.

It is well known (see [16, Lemma 5] for a detailed proof) that if we assume conditions (2.2) only for 𝑢𝑑+, and also (2.6) and (3.1)-(3.2), then for any 𝑢𝜏0 there exists a unique weak solution 𝑢() of (2.1). Moreover, 𝑢() is such that 𝑢(𝑡)0 for all 𝑡[𝜏,𝑇].

On the other hand, if we assume these conditions except (2.6), then there exists at least one weak solution 𝑢() of (2.1) such that 𝑢(𝑡)0 for all 𝑡[𝜏,𝑇] [16, Theorem 4]. Moreover, we can prove the following.

Lemma 3.1. Assume conditions (2.2), (2.6) only for 𝑢𝑑+, and also (3.1)-(3.2). Then there exists a weak solution 𝑢() of (2.1), which is unique in the class of solutions satisfying 𝑢(𝑡)0 for all 𝑡[𝜏,𝑇].

Proof. Let 𝑢1,𝑢2 be two solutions with 𝑢𝑖(𝜏)=𝑢𝜏, 𝑖=1,2 such that 𝑢𝑖(𝑡)0 for all 𝑡. Denote 𝑤(𝑡)=𝑢1(𝑡)𝑢2(𝑡). Then in a standard way by the mean value theorem, we obtain 12𝑑(𝑑𝑡𝑤𝑡)2Ω𝑓𝑡,𝑢1(𝑡,𝑥)𝑓𝑡,𝑢2(𝑡,𝑥),𝑤(𝑡,𝑥)𝑑𝑥=Ω𝑓𝑢𝑡,𝑣𝑡,𝑥,𝑢1,𝑢2𝑤(𝑡,𝑥),𝑤(𝑡,𝑥)𝑑𝑥𝐶3(𝑡)𝑤(𝑡)2,(3.4) where 𝑣(𝑡,𝑥,𝑢1,𝑢2)𝐿(𝑢1(𝑡,𝑥),𝑢2(𝑡,𝑥))={𝛼𝑢1(𝑡,𝑥)+(1𝛼)𝑢2(𝑡,𝑥)𝛼0,1]} so that 𝑣(𝑡,𝑥,𝑢1,𝑢2)0. The uniqueness follows from Gronwall’s lemma

We prove also a result, which is similar to Lemma 2.1. Denote by 𝑝𝑗𝑖, 𝛼𝑗,𝐶𝑗1, and 𝐶𝑗2 the constants corresponding to 𝑓𝑗 in (2.2) for problems (2.8) and (2.9), respectively. Arguing as in the proof of Lemma 2.1 we obtain the following lemma.

Lemma 3.2. If 𝑓𝑗 satisfy (2.2) and (2.10) for 𝑢+𝑑, then 𝑝1𝑖𝑝2𝑖 for all 𝑖.

Theorem 3.3. Let 𝑓𝑗,𝑗 satisfy (2.6) and (3.1)-(3.2). Assume that 𝑓𝑗,𝑗 satisfy (2.2) and (2.10) for 𝑢+𝑑. If 0𝑢1𝜏𝑢2𝜏 and one supposes that 𝑓2 satisfies (2.7) for 𝑢𝑑+ with 𝑅202max{𝐾2(𝑢1𝜏,𝜏,𝑇),𝐾2(𝑢2𝜏,𝜏,𝑇)}, where 𝐾(𝑢𝑗𝜏,𝜏,𝑇) is taken from (2.14), one has 0𝑢1(𝑡)𝑢2(𝑡) for all 𝑡[𝜏,𝑇], where 𝑢1(),𝑢2() are the solutions corresponding to 𝑢1𝜏 and 𝑢2𝜏, respectively.

Proof. As the solutions 𝑢1(), 𝑢2() corresponding to 𝑢1𝜏 and 𝑢2𝜏 are nonnegative, repeating exactly the same steps of the proof of Theorem 2.2 we obtain the desired result.

Remark 3.4. The results remain valid if, instead, 𝑓1 satisfies (2.7) with the same 𝑅0.

Remark 3.5. If 𝑓2 satisfies (2.7) for an arbitrary 𝑅0>0 (i.e., in the whole space 𝑑), then the result is true for any initial data 0𝑢1𝜏𝑢2𝜏.

We shall need also the following modification of Theorem 3.3.

Theorem 3.6. Let 𝑓𝑗,𝑗 satisfy (2.6) and (3.1)-(3.2). Assume that 𝑓𝑗,𝑗 satisfy (2.2) and (2.10) for 𝑢𝑑+. Let 0𝑢1𝜏𝑢2𝜏. One supposes that 𝑓2 satisfies 𝑓𝑖2(𝑡,𝑢)𝑓𝑖2(𝑡,𝑣)𝜀,(3.5) for any 𝑡[𝜏,𝑇] and any 𝑢,𝑣𝑑+ such that 𝑢𝑖=𝑣𝑖 and 𝑢𝑗𝑣𝑗 if 𝑗𝑖, and |𝑢|,|𝑣|𝑅0 with 𝑅202max{𝐾2(𝑢1𝜏,𝜏,𝑇),𝐾2(𝑢2𝜏,𝜏,𝑇)}, where 𝐾(𝑢𝑗𝜏,𝜏,𝑇) is taken from (2.14).
Then there exists a constant 𝐶(𝜏,𝑇) such that 𝑢1(𝑡)𝑢2(𝑡)+[],𝐶(𝜏,𝑇)𝜀,𝑡𝜏,𝑇(3.6) where 𝑢1(),𝑢2() are the solutions corresponding to 𝑢1𝜏 and 𝑢2𝜏, respectively.

Proof. Arguing as in the proof of Theorem 2.2 we obtain the inequality 12𝑑𝑢𝑑𝑡1𝑢2+2𝑢+𝑎1𝑢2+2𝑉𝐶3𝑢(𝑡)1𝑢2+2Ω𝑓2𝑡,𝑣2𝑓2𝑡,𝑢2,𝑢1𝑢2+𝑑𝑥,(3.7) where 𝑣2 is defined in (2.15).
Using (2.17), 𝑣2𝑢2,|𝑣2|2|𝑢1|2+|𝑢2|2, (2.14), and (3.5), we get 𝑓𝑖2𝑡,𝑣2𝑓𝑖2𝑡,𝑢2𝜀.(3.8) Thus 𝑑𝑢𝑑𝑡1𝑢2+22𝐶3(𝑢𝑡)1𝑢2+2+2𝜀Ω𝑖𝐽𝑢𝑖1𝑢𝑖2+𝑑𝑥2𝐶3𝑢(𝑡)+11𝑢2+2+𝐾𝜀2,(3.9) for some constant 𝐾>0. By Gronwall’s lemma, we get 𝑢1(𝑡)𝑢2(𝑡)+2𝑢1𝜏𝑢2𝜏+2𝑒𝑡𝜏(2𝐶3(𝑠)+1)𝑑𝑠+𝐾𝜀2𝑡𝜏𝑒𝑡𝑟(2𝐶3(𝑠)+1)𝑑𝑠𝑑𝑟𝐶2(𝜏,𝑇)𝜀2.(3.10)

Let us consider now the multivalued case. We will obtain first some auxiliary statements.

We shall define suitable approximations. For any 𝑛1 we put 𝑓𝑖𝑛(𝑡,𝑢)=𝜓𝑛(|𝑢|)𝑓𝑖(𝑡,𝑢)+(1𝜓𝑛(|𝑢|))𝑔𝑖(𝑡,𝑢), where 𝑔𝑖(𝑡,𝑢)=|𝑢𝑖|𝑝𝑖2𝑢𝑖+𝑓𝑖(𝑡,0,,0), and 𝜓𝑛 was defined in (2.20). Then 𝑓𝑛([𝜏,𝑇]×𝑑;𝑑) and for any 𝐴>0, sup𝑡[𝜏,𝑇]sup|𝑢|𝐴||𝑓𝑛||(𝑡,𝑢)𝑓(𝑡,𝑢)0,as𝑛.(3.11) We will check first that 𝑓𝑛 satisfy conditions (2.2) for 𝑢𝑑+, where the constants do not depend on 𝑛.

Lemma 3.7. Let 𝑓 satisfy (2.2) for 𝑢𝑑+. For all 𝑛1 one has 𝑑𝑖=1||𝑓𝑖𝑛||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)𝐷11+𝑑𝑖=1||𝑢𝑖||𝑝𝑖,𝑓𝑛(𝑡,𝑢),𝑢𝛾𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐷2,(3.12) for 𝑢+𝑑, where the positive constants 𝐷1, 𝐷2, and 𝛾 do not depend on 𝑛.
If |𝑢|>𝑛+1, then for any 𝑤𝑑 one has 𝑓𝑛𝑢(𝑡,𝑢)𝑤,𝑤0.(3.13) Moreover, if 𝑓, satisfy (3.2), then 𝑓𝑛, also satisfies this condition.

Proof. In view of (2.2) we get 𝑓𝑛(𝑡,𝑢),𝑢=𝜓𝑛(|𝑢|)(𝑓(𝑡,𝑢),𝑢)+1𝜓𝑛(|𝑢|)(𝑔(𝑢),𝑢)𝜓𝑛𝛼(|𝑢|)𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐶2+1𝜓𝑛(|𝑢|)𝑑𝑖=1||𝑢𝑖||𝑝𝑖+1𝜓𝑛(|𝑢|)𝑑𝑖=1𝑓𝑖(𝑡,0,,0)𝑢𝑖𝜓𝑛(|𝑢|)𝛼𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐶2+1𝜓𝑛1(|𝑢|)2𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐾11𝜓𝑛(|𝑢|)𝑑𝑖=1||𝑓𝑖||(𝑡,0,,0)𝑝𝑖/(𝑝𝑖1)𝛼𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐶2𝐾1𝐶1,(3.14) where 𝛼=min{1/2,𝛼}, for some constant 𝐾1>0. Also, 𝑑𝑖=1||𝑓𝑖𝑛||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)𝐾2𝑑𝑖=1||𝑓𝑖||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)+𝑑𝑖=1||𝑔𝑖||(𝑢)𝑝𝑖/(𝑝𝑖1)𝐾3𝐶11+𝑑𝑖=1||𝑢𝑖||𝑝𝑖+𝑑𝑖=1||𝑢𝑖||𝑝𝑖+𝑑𝑖=1||𝑓𝑖||(𝑡,0,,0)𝑝𝑖/(𝑝𝑖1)𝐾4𝑑𝑖=1||𝑢𝑖||𝑝𝑖,+1(3.15) for some constant 𝐾4>0. Thus, for 𝐷1=𝐾4, 𝐷2=𝐶2+𝐾1𝐶1, 𝛼=min{1/2,𝛼} we have (3.12) for the functions 𝑓𝑛.
Moreover, if |𝑢|>𝑛+1, then for any 𝑤𝑑, 𝑓𝑛𝑢=𝑔(𝑡,𝑢)𝑤,𝑤𝑢=(𝑡,𝑢)𝑤,𝑤𝑑𝑖=1𝑝𝑖||𝑢1𝑖||𝑝𝑖2𝑤2𝑖0.(3.16)
Finally, if (3.2) is satisfied, then 𝑖(𝑡,𝑥)𝑓𝑖𝑛(𝑡,𝑢)=𝜓𝑛(|𝑢|)𝑖(𝑡,𝑥)𝑓𝑖+(𝑡,𝑢)1𝜓𝑛(|𝑢|)𝑖(𝑡,𝑥)𝑓𝑖(𝑡,0,,0)0,(3.17) for all 𝑖, a.e. (𝑡,𝑥)(𝜏,𝑇)×Ω and 𝑢 such that 𝑢𝑖=0 and 𝑢𝑗0 if 𝑗𝑖.

Let 2𝑞𝑖𝑝𝑖, 𝑖=1,,𝑑. We define also the following approximations 𝑓𝑖𝑛(𝑡,𝑢)=𝜓𝑛(|𝑢|)𝑓𝑖(𝑡,𝑢)+(1𝜓𝑛(|𝑢|))̃𝑔𝑖(𝑡,𝑢), where ̃𝑔𝑖(𝑡,𝑢)=|𝑢𝑖|𝑝𝑖2𝑢𝑖+|𝑢𝑖|𝑞𝑖2𝑢𝑖+𝑓𝑖(𝑡,0,,0). Then (3.11) holds. We check that 𝑓𝑛 satisfy conditions (2.2) for 𝑢𝑑+, where the constants do not depend on 𝑛.

Lemma 3.8. Let 𝑓 satisfy (2.2) for 𝑢𝑑+. For all 𝑛1 one has 𝑑𝑖=1|||𝑓𝑖𝑛|||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)𝐷11+𝑑𝑖=1||𝑢𝑖||𝑝𝑖,𝑓𝑛(𝑡,𝑢),𝑢𝛾𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐷2,(3.18) for 𝑢+𝑑, where the positive constants 𝐷1, 𝐷2, and 𝛾 do not depend on 𝑛.
If |𝑢|>𝑛+1, then for any 𝑤𝑑 one has 𝑓𝑛𝑢(𝑡,𝑢)𝑤,𝑤0.(3.19)
Moreover, if 𝑓, satisfy (3.2), then 𝑓𝑛, also satisfy this condition.

Proof . In view of (3.12), we have 𝑓𝑛=𝑓(𝑡,𝑢),𝑢𝑛+(𝑡,𝑢),𝑢1𝜓𝑛(|𝑢|)𝑑𝑖=1||𝑢𝑖||𝑞𝑖𝛾𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐷2,𝑑𝑖=1|||𝑓𝑖𝑛|||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)𝐾1𝑑𝑖=1||𝑓𝑖𝑛||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)+𝑑𝑖=1||𝑢𝑖||𝑝𝑖(𝑞𝑖1)/(𝑝𝑖1)𝐾1𝐷11+𝑑𝑖=1||𝑢𝑖||𝑝𝑖+𝑑𝑖=1||𝑢𝑖||𝑞𝑖𝐾21+𝑑𝑖=1||𝑢𝑖||𝑝𝑖,(3.20) where we have used that 𝑝𝑖𝑞𝑖 implies 𝑝𝑖/(𝑝𝑖1)𝑞𝑖/(𝑞𝑖1). Finally, (3.19) and condition (3.2) are proved in the same way as in Lemma 3.7.

For every 𝑛1 consider the sequence 𝑓𝜀𝑛(𝑡,𝑢) defined by 𝑓𝜖𝑛(𝑡,𝑢)=𝑑𝜌𝜖(𝑠)𝑏𝑛(𝑡,𝑢𝑠)𝑑𝑠, where either 𝑏𝑛=𝑓𝑛 or 𝑏𝑛=𝑓𝑛, defined before. Since any 𝑏𝑛 are uniformly continuous on [𝜏,𝑇]×[𝑘1,𝑘+1], for any 𝑘1, there exist 𝜖𝑘,𝑛(0,1) such that for all 𝑢 satisfying |𝑢|𝑘, and for all 𝑠 for which |𝑢𝑠|<𝜖𝑘,𝑛 we have sup𝑡[𝜏,𝑇]||𝑏𝑛(𝑡,𝑢)𝑏𝑛||1(𝑡,𝑠)𝑘.(3.21) We put 𝑓𝑘𝑛(𝑡,𝑢)=𝑓𝜖𝑘,𝑛𝑛(𝑡,𝑢). Then, 𝑓𝑘𝑛(𝑡,)(𝑑;𝑑) for all 𝑡[𝜏,𝑇],𝑘,𝑛1. Since for any compact subset 𝐴𝑑 and any 𝑛 we have 𝑓𝑘𝑛𝑏𝑛 uniformly on [𝜏,𝑇]×𝐴, we obtain the existence of a sequence 𝛿𝑛𝑘(0,1) such that 𝛿𝑛𝑘0, as 𝑘, and |𝑓𝑛𝑘𝑖(𝑡,𝑢)𝑏𝑖𝑛(𝑡,𝑢)|𝛿𝑛𝑘, for any 𝑖,𝑛 and any 𝑢 satisfying |𝑢|𝑛+2. We define the function 𝐹𝑘𝑛=(𝐹𝑛𝑘1,,𝐹𝑛𝑘𝑑) given by 𝐹𝑛𝑘𝑖(𝑡,𝑢)=𝑓𝑛𝑘𝑖(𝑡,𝑢)𝑝𝛿𝑛𝑘,(3.22) where 𝑝.

Lemma 3.9. Let 𝑓 satisfy (2.2) for 𝑢𝑑+. For all 𝑛,𝑘1 we have 𝑑𝑖=1||𝐹𝑛𝑘𝑖||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)𝐷31+𝑑𝑖=1||𝑢𝑖||𝑝𝑖,𝐹𝑘𝑛(𝑡,𝑢),𝑢𝜈𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐷4,(3.23) for 𝑢𝑑+, where the positive constants 𝐷3,𝐷4, and   𝜈 do not depend neither on 𝑛 nor 𝑘.
Moreover, if 𝑓, satisfy (3.2), then 𝐹𝑘𝑛, also satisfy this condition if |𝑢|𝑛+2.

Proof. Since 𝑓𝑛 satisfy (3.12) and 𝑓𝑛 satisfies (3.18), we have 𝑑𝑖=1||𝐹𝑛𝑘𝑖||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)𝑅1𝑑𝑖=1||𝑓𝑛𝑘𝑖||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)+1𝑅1𝑑𝑖=1𝑑𝜌𝜖𝑘(𝑠)𝑑𝑠1/(𝑝𝑖1)𝑑𝜌𝜖𝑘||𝑏(𝑠)𝑖𝑛||(𝑡,𝑢𝑠)𝑝𝑖/(𝑝𝑖1)𝑑𝑠+1𝑅2𝑑𝑖=1𝑑𝜌𝜖𝑘||𝑢(𝑠)1+𝑖𝑠𝑖||𝑝𝑖𝑑𝑠+1𝑅3𝑑𝑖=1𝑑𝜌𝜖𝑘(||𝑢𝑠)𝑖||𝑝𝑖+𝜖𝑝𝑖𝑘𝑑𝑠+1𝑅41+𝑑𝑖=1||𝑢𝑖||𝑝𝑖,(3.24) for some constant 𝑅4>0.
On the other hand, 𝐹𝑘𝑛=(𝑡,𝑢),𝑢𝑑𝜌𝜖𝑘𝑏(𝑠)𝑛(𝑡,𝑢𝑠),𝑢𝑠𝑑𝑠+𝑑𝜌𝜖𝑘𝑏(𝑠)𝑛(𝑡,𝑢𝑠),𝑠𝑑𝑠𝑝𝛿𝑛𝑛𝑘𝑖=1𝑢𝑖𝑑𝜌𝜖𝑘𝛾(𝑠)𝑑𝑖=1||𝑢𝑖𝑠𝑖||𝑝𝑖𝐷2𝑑𝑠𝑑𝜌𝜖𝑘(𝑠)𝑑𝑖=1𝛾2𝐷1||𝑏𝑖𝑛||(𝑡,𝑢𝑠)𝑝𝑖/(𝑝𝑖1)+𝑅5||𝑠𝑖||𝑝𝑖𝑑𝑠𝑝𝛿𝑛𝑛𝑘𝑖=1𝑢𝑖𝑑𝜌𝜖𝑘𝛾(𝑠)𝑑𝑖=1||𝑢𝑖𝑠𝑖||𝑝𝑖𝐷2𝑑𝑠𝑑𝜌𝜖𝑘𝛾(𝑠)2𝑑𝑖=1||𝑢𝑖𝑠𝑖||𝑝𝑖+𝑅6𝑑𝑠𝑝𝛿𝑛𝑛𝑘𝑖=1𝑢𝑖𝛾2𝑑𝜌𝜖𝑘(𝑠)𝑑𝑖=1||𝑢𝑖𝑠𝑖||𝑝𝑖𝑑𝑠𝑅7𝑝𝛿𝑛𝑛𝑘𝑖=1𝑢𝑖𝜈𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝑅8,(3.25) for some constants 𝜈,𝑅8>0, where in the last inequality we have used that for some 𝐷>0, ||𝑢𝑖||𝑝𝑖=||𝑢𝑖𝑠𝑖+𝑠𝑖||𝑝𝑖||𝑢𝐷𝑖𝑠𝑖||𝑝𝑖+||𝑠𝑖||𝑝𝑖||𝑢𝐷𝑖𝑠𝑖||𝑝𝑖+𝜖𝑝𝑖𝑘.(3.26) Hence, (3.23) holds.
In view of Lemmas 3.7 and 3.8 the functions 𝑏𝑛, satisfy (3.2). Hence, |𝑓𝑛𝑘𝑖(𝑡,𝑢)𝑏𝑖𝑛(𝑡,𝑢)|𝛿𝑛𝑘, for any 𝑖,𝑛 and any 𝑢 satisfying |𝑢|𝑛+2, implies that 𝑖(𝑡,𝑥)𝐹𝑛𝑘𝑖(𝑡,𝑢)=𝑖(𝑡,𝑥)𝑓𝑛𝑘𝑖(𝑡,𝑢)+𝑝𝛿𝑛𝑘𝑖(𝑡,𝑥)𝑏𝑖𝑛(𝑡,𝑢)0,(3.27) for 𝑢 such that 𝑢𝑖=0, 𝑢𝑗0, 𝑗𝑖, and |𝑢|𝑛+2.

Define a smooth function 𝜙𝑛+[0,1] satisfying 𝜙𝑛(𝑠)=1,if0𝑠𝑛+1+𝛾,0𝜙𝑛(𝑠)1,if𝑛+1+𝛾𝑠𝑛+2,0,if𝑠𝑛+2,(3.28) where 0<𝛾<1 is fixed. Let 𝑙𝑘𝑛(𝑡,𝑢) be given by 𝑙𝑘𝑛(𝑡,𝑢)=𝜙𝑛(|𝑢|)𝐹𝑘𝑛(𝑡,𝑢)+1𝜙𝑛𝑏(|𝑢|)𝑛(𝑡,𝑢).(3.29) Since for any compact subset 𝐴𝑑 and any 𝑛 we have 𝑓𝑘𝑛𝑏𝑛 uniformly on [𝜏,𝑇]×𝐴 as 𝑘, it is clear that sup𝑡[𝜏,𝑇]sup|𝑢|𝐴||𝑙𝑘𝑛(𝑡,𝑢)𝑏𝑛||(𝑡,𝑢)0,as𝑘.(3.30)

Lemma 3.10. Let 𝑓 satisfy (2.2) for 𝑢𝑑+. For all 𝑛,𝑘1 we have 𝑑𝑖=1||𝑙𝑛𝑘𝑖||(𝑡,𝑢)𝑝𝑖/(𝑝𝑖1)𝐷51+𝑑𝑖=1||𝑢𝑖||𝑝𝑖,𝑙𝑘𝑛(𝑡,𝑢),𝑢𝜆𝑑𝑖=1||𝑢𝑖||𝑝𝑖𝐷6,(3.31) for 𝑢𝑑+, where the positive constants 𝐷5,𝐷6, and 𝜆 do not depend neither on 𝑛 nor 𝑘. Also, 𝑙𝑘𝑛𝑢(𝑡,𝑢)𝑤,𝑤𝐷7(𝑘,𝑛)|𝑤|2,𝑢,𝑤,(3.32) where 𝐷7(𝑘,𝑛) is a nonnegative number.
Moreover, if 𝑓, satisfy (3.2), then 𝑙𝑘𝑛, also satisfies this condition.

Proof. The inequalities given in (3.31) are an easy consequence of (3.12), (3.18), and (3.23).
On the other hand, if 𝑢 is such that 𝑢𝑖=0, 𝑢𝑗0,