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
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., [1β4] 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 -limit sets and attractors. In the last years, several authors have been working in this direction (see, e.g., [4β8] for the deterministic case, and [9β12] 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:
for which we can prove that for every initial data in the phase space (with a partial order ) there exists at least one solution.
If we try to compare solutions of (1.1) for two ordered initial data , then we can consider a strong comparison principle and a weak one.
The strong version would imply the existence of a solution with such that
for any solution with , and, viceversa, the existence of a solution with such that (1.2) is satisfied for any solution with . 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 , then there exist two solutions of (1.1) such that , , and (1.2) hold.
There is in fact an intermediate version of the comparison principle, which says that if we fix a solution of (1.1) with , then there exists a solution with 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 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 , . Let be an integer and be a bounded open subset with smooth boundary. Consider the problem:
where , , , , is a real matrix with a positive symmetric part , . Moreover, is jointly continuous on and satisfies the following conditions:
where , .
Let , , and let be the dual space of . By , we denote the norm by and , respectively. For , we define the spaces
We take , where .
We say that the function is a weak solution of (2.1) if , , , and
for all , where denotes pairing in the space , and .
Under conditions (2.2), it is known [17, page 284] that for any there exists at least one weak solution of (2.1), and also that the function is absolutely continuous on and for a.a. .
Denote . Any weak solution satisfies and
If, additionally, we assume that that is continuously differentiable with respect to for any , and
where , , the weak solution of (2.1) is unique. Here, denotes the Jacobian matrix of .
We consider also the following assumption: there exists such that
for any and any such that and if , and , which means that the systems is cooperative in the ball with radius centered at .
Consider the two problems:
where are jointly continuous on . Among conditions (2.2) and (2.6)-(2.7), we shall consider the following:
Lemma 2.1. If satisfy (2.2) and (2.10), then the constants have to be the same for and .
Proof. Denote by , , and the constants corresponding to in (2.2). By contradiction let, for example, . Take the sequence , where as . Then by (2.2), (2.10), and Youngβs inequality, we have
But implies the existence of such that , which is a contradiction. Hence, . Conversely, let . Then we take with as so that
As before, we obtain a contradiction, so . Repeating similar arguments for the other , we obtain that for .
We recall [15] that under conditions (2.2) any solution of (2.8) satisfies the inequality:
for some constant . Of course, the same is valid for any solution of (2.9). From (2.13), for any we obtain
We shall denote by the solution of (2.8) corresponding to the initial data and by the solution of (2.9) corresponding to the initial data . Also, we take for .
We obtain the following comparison result.
Theorem 2.2. Assume that satisfy (2.2), (2.6), and (2.10). If and we suppose that satisfies (2.7) with , where is taken from (2.14), we have , for all .
Remark 2.3. The results remain valid if, instead, satisfies (2.7) with .
Remark 2.4. If satisfies (2.7) for an arbitrary (i.e., in the whole space ), then the result is true for any initial data .
Proof. Let . The function satisfies (2.6) with . For any define by
Note that and , so for all . For the function , we can obtain by (2.10) and the mean value theorem that
For all , we have
where , for , and if . For any , we have that , and then by , , (2.14), and (2.7), we get
By Gronwallβs lemma, we get
Thus , which means that , for a.a. , and all , .
Remark 2.5. In the scalar case, that is, , 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 satisfying
For every we put , where . Then and for any ,
Let be a mollifier, that is, , , and for all . We define the functions
Since for any is uniformly continuous on , there exist such that for all satisfying , and for all for which we have
We put . Then , for all .
For further arguments we need the following technical result [16, Lemma 2].
Lemma 2.6. Let satisfy (2.2). For all , the following statements hold:
where is a nonnegative number, and the positive constants , , do not depend on .
Consider first the scalar case.
Theorem 2.7. Let . Assume that satisfy (2.2) and (2.10). If , there exist two solutions (of (2.8) and (2.9), resp.) such that for all .
Proof. For the functions we take the approximations (defined in Lemma 2.6), which satisfy (2.24)β(2.26), and consider the problems
for . Problem (2.27) has a unique solution for any initial data . In view of Lemma 2.1, the constant is the same for and . We note that
Then, it is clear that for every . By Theorem 2.2 we know that as , we have , for all , 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 not depending on and replacing by . Hence, the sequences are bounded in . It follows from (2.25) that are bounded in and also that is bounded in , where . By the Compactness Lemma [18], we have that for some functions , :
Also, arguing as in [19, page 3037] we obtain
Moreover, by (2.24) and (2.31) we have for a.a. and then the boundedness of in implies that converges to weakly in [18]. It follows that are weak solutions of (2.8) and (2.9), respectively, with , . Moreover, one can prove that
Indeed, we define the functions , , which are nonincreasing in view of (2.13). Also, from (2.30) we have for a.a. . Then one can prove that for all (see [15, page 623] for the details). Hence, . Together with (2.35) this implies (2.36) (see again [15, page 623] for more details). Hence, passing to the limit we obtain
Further, let us prove the general case for an arbitrary .
Theorem 2.8. Assume that satisfy (2.2) and (2.10). Also, suppose that either or satisfies (2.7) for an arbitrary . If , there exist two solutions (of (2.8) and (2.9), resp.) such that , for all .
Proof. Let be the function which satisfies (2.7). We take the approximations (defined in Lemma 2.6), which satisfy (2.24)β(2.26). Then, we consider problems (2.27). In view of Lemma 2.1, the constants are the same for and . We note that
Then, it is clear that for every . Using Lemma 2.6 it is standard to obtain estimate (2.14) with a constant not depending on . Hence, the solutions of (2.27) satisfy
We note that
if , since in such a case . Hence, if , the functions satisfy condition (2.7) with . Therefore, for any and any such that and if , and , we have
Thus, if , the functions satisfy condition (2.7) with . By Theorem 2.2 we know that as , we have , for all , , for the corresponding solutions of (2.27). Repeating the same proof of Theorem 2.7, we obtain that the sequences converge (up to a subsequence) in the sense of (2.29)β(2.36) to the solutions of problems (2.8) and (2.9), respectively. Also, it holds
In the applications we need to generalize this theorem to the case where the constant can be negative. We shall do this when have sublinear growth (i.e., for all ). Consider for (2.1) the following conditions:
where , and .
Let satisfy (2.43) with constants . Then if , we make in (2.1) the change of variable , where . Hence, multiplying (2.8) and (2.9) by we have
It is easy to check that if is a weak solution of (2.44), then is a weak solution of (2.8) (and the same is true, of course, for (2.45) and (2.9)). Conversely, if is a weak solution of (2.8), then is a weak solution of (2.44).
The functions satisfy (2.2) with for all . Indeed,
where .
Then, we obtain the following.
Theorem 2.9. Assume that satisfy (2.43) and (2.10). Also, suppose that either or satisfies (2.7) for an arbitrary . If , there exist two solutions (of (2.8) and (2.9), resp.) such that , for all .
Proof. We consider problems (2.44) and (2.45). In view of (2.46) satisfy (2.2). Also, defining it is clear that (2.10) holds. Finally, if, for example, satisfies (2.7) for any , then it is obvious that for is true as well. Hence, by Theorem 2.8 there exist two solutions (of (2.44) and (2.45), resp.), with such that for all . Thus
and are solutions of (2.8) and (2.9), respectively such that .
Remark 2.10. If satisfy (2.6), then the solutions given in Theorem 2.9 are unique for the corresponding initial data.
3. Comparison for Positive Solutions
Denote . Let us consider the previous results in the case where the solutions have to be positive. Consider now the following conditions.
for all , a.e. and if . Obviously, in the scalar case these conditions just mean that
for a.e. .
It is well known (see [16, Lemma 5] for a detailed proof) that if we assume conditions (2.2) only for , and also (2.6) and (3.1)-(3.2), then for any there exists a unique weak solution of (2.1). Moreover, is such that for all .
On the other hand, if we assume these conditions except (2.6), then there exists at least one weak solution of (2.1) such that for all [16, Theorem 4]. Moreover, we can prove the following.
Lemma 3.1. Assume conditions (2.2), (2.6) only for , and also (3.1)-(3.2). Then there exists a weak solution of (2.1), which is unique in the class of solutions satisfying for all .
Proof. Let be two solutions with , such that for all . Denote . Then in a standard way by the mean value theorem, we obtain
where so that . The uniqueness follows from Gronwallβs lemma
We prove also a result, which is similar to Lemma 2.1. Denote by , , and the constants corresponding to 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 satisfy (2.2) and (2.10) for , then for all .
Theorem 3.3. Let satisfy (2.6) and (3.1)-(3.2). Assume that satisfy (2.2) and (2.10) for . If and one supposes that satisfies (2.7) for with , where is taken from (2.14), one has for all , where are the solutions corresponding to and , respectively.
Proof. As the solutions , corresponding to and 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, satisfies (2.7) with the same .
Remark 3.5. If satisfies (2.7) for an arbitrary (i.e., in the whole space ), then the result is true for any initial data .
We shall need also the following modification of Theorem 3.3.
Theorem 3.6. Let satisfy (2.6) and (3.1)-(3.2). Assume that satisfy (2.2) and (2.10) for . Let . One supposes that satisfies
for any and any such that and if , and with , where is taken from (2.14). Then there exists a constant such that
where are the solutions corresponding to and , respectively.
Proof. Arguing as in the proof of Theorem 2.2 we obtain the inequality
where is defined in (2.15). Using (2.17), , (2.14), and (3.5), we get
Thus
for some constant . By Gronwallβs lemma, we get
Let us consider now the multivalued case. We will obtain first some auxiliary statements.
We shall define suitable approximations. For any we put , where , and was defined in (2.20). Then and for any ,
We will check first that satisfy conditions (2.2) for , where the constants do not depend on .
Lemma 3.7. Let satisfy (2.2) for . For all one has
for , where the positive constants , , and do not depend on . If , then for any one has
Moreover, if satisfy (3.2), then also satisfies this condition.
Proof. In view of (2.2) we get
where , for some constant . Also,
for some constant . Thus, for , , we have (3.12) for the functions . Moreover, if , then for any ,
Finally, if (3.2) is satisfied, then
for all , a.e. and such that and if .
Let , . We define also the following approximations , where . Then (3.11) holds. We check that satisfy conditions (2.2) for , where the constants do not depend on .
Lemma 3.8. Let satisfy (2.2) for . For all one has
for , where the positive constants , , and do not depend on . If , then for any one has
Moreover, if satisfy (3.2), then also satisfy this condition.
Proof . In view of (3.12), we have
where we have used that implies . Finally, (3.19) and condition (3.2) are proved in the same way as in Lemma 3.7.
For every consider the sequence defined by , where either or , defined before. Since any are uniformly continuous on , for any , there exist such that for all satisfying , and for all for which we have
We put . Then, for all . Since for any compact subset and any we have uniformly on , we obtain the existence of a sequence such that , as , and , for any and any satisfying . We define the function given by
where .
Lemma 3.9. Let satisfy (2.2) for . For all we have
for , where the positive constants , and ββ do not depend neither on nor . Moreover, if satisfy (3.2), then also satisfy this condition if .
Proof. Since satisfy (3.12) and satisfies (3.18), we have
for some constant . On the other hand,
for some constants , where in the last inequality we have used that for some ,
Hence, (3.23) holds. In view of Lemmas 3.7 and 3.8 the functions satisfy (3.2). Hence, , for any and any satisfying , implies that
for such that , , , and .
Define a smooth function satisfying
where is fixed. Let be given by
Since for any compact subset and any we have uniformly on as , it is clear that
Lemma 3.10. Let satisfy (2.2) for . For all we have
for , where the positive constants , and do not depend neither on nor . Also,
where is a nonnegative number. Moreover, if satisfy (3.2), then 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 is such that , , , then in view of Lemmas 3.7, 3.8, and 3.9 we have
as , for . Hence, so that condition (3.2) holds. It is also clear that is continuously differentiable with respect to for any and . We obtain the existence of such that (3.32) holds. Indeed, if , then so that
If , then so that by (3.13), (3.19) we have . Finally, if , we have
where we have used similar arguments as in (3.34), (3.13), and (3.19) and also that
for any satisfying and any .
Now we are ready to obtain the weak comparison principle for positive solutions.
Theorem 3.11. Let satisfy (3.1)-(3.2). Assume that satisfy (2.2) and (2.10) for . One supposes that either or satisfies (2.7) for and for an arbitrary . If , there exist two solutions of (2.8) and (2.9), respectively, with , such that for all .
Proof. For let us consider the approximations , defined before with , where the last inequality follows from Lemma 3.2. Then by Lemmas 3.7 and 3.8 we have that satisfy (2.2) for with constants not depending on . Also, (3.13), (3.19) hold and (3.2) is satisfied in both cases. Moreover, by (2.10) for , we have
if . As explained before, we can choose a sequence such that , as , and , for any and any satisfying . Further, we consider the functions
By Lemma 3.9 we know that satisfy (2.2) for with constants not depending neither on nor , and condition (3.2) for , as well. Moreover, by (3.37) we have
if and . Suppose, for example, that satisfies condition (2.7) for . For any and any such that and if , , we have
Hence, (3.5) is satisfied with and . Now, we will define the following functions:
By Lemma 3.10, these functions satisfy (2.2) for with constants not depending neither on nor , inequality (3.32), and condition (3.2). In view of (3.37), (3.39), and , if , we obtain
and then if ,
On the other hand, since if , (3.40) implies
for any and any such that and if . Thus, (3.5) is satisfied with and . We consider now the follwoing problems:
where and . In view of Lemma 3.10 and [16, Lemma 5], problem (3.45) has a unique weak solution such that for all . Let be the solutions of (3.45) corresponding to the initial data , where . Using Lemma 3.10 it is standard to obtain estimate (2.14) with a constant not depending neither on nor . Then, the solutions of (3.45) satisfy
Thus, since satisfy condition (3.5) with , by Theorem 3.6 we know that as , we have
for all , all and . Arguing as in the proof of Theorem 2.7 we obtain that for the sequence converges (up to a subsequence) in the sense of (2.29)β(2.36) to a solution of problem (3.46) with initial data . In particular, as we have
Fix and take any . Denote by the set
By (3.48) as we have
Hence, (3.49) implies
Define the sequence
Then, it is clear that as . We note also that weakly in , as for any , (3.51) gives
Therefore, strongly in so that for a.a. . Since , for a.a. , we obtain
Arguing again as in the proof of Theorem 2.8 we obtain that the sequences converge (up to a subsequence) in the sense of (2.29)β(2.36) to the solutions of problems (2.8) and (2.9), respectively. Also, it holds
As in the previous section, we shall generalize this theorem to the case where the constant can be negative. We note that if satisfy (3.2), then , also satisfy (3.2). Arguing as in Theorem 2.9, we obtain the following.
Theorem 3.12. Let satisfy (3.1)-(3.2). Assume that satisfy (2.43) and (2.10) for . Also, suppose that either or satisfies (2.7) for for an arbitrary . If , there exist two solutions (of (2.8) and (2.9), resp.), with , such that , for all .
But in this case we can consider another interesting situation, as the values and are not necessarily equal.
Let satisfy (2.2) for , whereas satisfy (2.43) for , with constants . Then if , we make in (2.1) the change of variable , where . We obtain problems (2.44)-(2.45).
If is a weak solution of (2.44), then is a weak solution of (2.8) (and the same is true, of course, for (2.45) and (2.9)).
The function satisfies (2.2) for with the same as . Indeed, as , we get
Then, we obtain the following.
Theorem 3.13. Let satisfy (3.1)-(3.2). Let satisfy (2.2) for , whereas satisfy (2.43) for , with constants . Assume that satisfy (2.10). Also, suppose that either or satisfies (2.7) for for an arbitrary . If , there exist two solutions (of (2.8) and (2.9), resp.), with , such that for all .
Proof. We consider problems (2.44) and (2.45). In view of (2.46) and (3.57), satisfy (2.2) for with , and satisfies (2.2) for with the same as . Also, defining it is clear that (2.10) and (3.2) hold. Finally, if, for example, satisfies (2.7) for for any , then it is obvious that for is true as well. Hence, by Theorem 3.11 there exist two solutions (of (2.44) and (2.45), resp.), with such that for all . Thus,
and are solutions of (2.8) and (2.9), respectively such that .
Remark 3.14. If satisfy (2.6), then the solutions given in Theorems 3.12 and 3.13 are unique for the corresponding initial data.
Remark 3.15. All the result proved so far are true if instead of Dirichlet boundary conditions we consider Neumann boundary conditions
where is the unit outward normal. In such a case, the space will be . The proofs remain the same.
4. Applications
We shall apply now the previous results to some model of physical and biological interest.
4.1. The Lotka-Volterra System
We also study the Lotka-Volterra system with diffusion:
with either Dirichlet or Neumann boundary conditions, where and the functions are positive and continuous. Also, are positive constants and . The initial data belongs to .
In this case, the functions are given by
Uniqueness of the Cauchy problem for this system has been proved only if we consider solutions confined in an invariant region (e.g., in a parallelepiped when the parameters do not depend on ) (see [20, 21]). However, in the general case for initial data just in it is an open problem so far.
System (4.1) satisfies conditions (2.2) with for [16, page 263]. Also, it is clear that (3.1)-(3.2) hold.
We shall compare with the following system:
which is a system of three uncoupled logistic equations. The functions are given by
It is easy to see that system (4.3) satisfies conditions (2.2) with for. Also, it is clear that (3.1)-(3.2) hold, and that condition (2.7) is trivially satisfied. Moreover,
for all , and so that (2.10) holds for .
Also, we have
for all , and , where .
Thus, by Theorem 3.11, Lemma 3.1, and Remark 3.15, we obtain the following theorem.
Theorem 4.1. If , there exists a solution for (4.1) with such that , for all , where is the unique solution with of (4.3) in the class of solutions satisfying for all .
This theorem says that there exists at least one solution for the Lotka-Volterra system which is dominated by the unique nonnegative solution of the uncoupled logistic system (4.3).
Further, we shall compare with the uncoupled linear system
Hence,
Obviously, system (4.7) satisfies conditions (2.43) for. Also, it is clear that (3.1)-(3.2) and condition (2.7) are trivially satisfied. Moreover,
for all , and so that (2.10) holds for . Also, we have
for all , and , where .
Thus, by Theorem 3.13 and Remark 3.15, we have the following.
Theorem 4.2. If , there exist a solution for (4.1) with such that for all , where is the unique solution of (4.7) with .
Let us consider now the autonomous case, that is, and , with Diriclet boundary conditions.
By the changes of variable , system (4.7) becomes
with initial data .
The operator in the space with domain is sectorial [22]. Moreover, since the eigenvalues of are , we have that the minimum eigenvalue is strictly greater than . Denote by the analytic semigroup generated by the operator . Then, is the unique solution of (4.11) with .
It is well known [22] that the operator generates a scale of interpolation spaces with the norm , , where with continuous embedding. Take . Then, by Theoremββ1.4.3 in [22] we obtain
Since this is true for every , we obtain that
We note that with continuous embedding if . Since the constants are bounded for in compact sets, we obtain the existence of such that
where . Then the unique solution of (4.7) with satisfies
where .
Joining (4.15) and Theorem 4.2, we obtain the following result.
Theorem 4.3. There exist at least one solution for the autonomous system (4.1) with and Dirichlet boundary conditions such that
where and .
We shall obtain also a weak maximum principle for the autonomous Lotka-Volterra system with Dirichlet boundary conditions.
By the maximum principle for the heat equation it is well known (see [23]) that for any the unique solution of (4.11) satisfies
Hence, the unique solution of (4.7) satisfies
By (4.18) and Theorem 4.2, we obtain the following weak maximum principle.
Theorem 4.4. There exist at least one solution for the autonomous system (4.1) with and Dirichlet boundary conditions such that for any ,
In particular, if , then
where .
4.2. A Model of Fractional-Order Chemical Autocatalysis with Decay
Consider the following scalar problem:
where , , , and . The initial data belongs to . This equation models an isothermal chemical autocatalysis (see [24]). In [24], the authors study the travelling waves of the equation in the case where with Neumann boundary conditions at . The variable is nonnegative, since it represents a chemical concentration.
The functions are given by , . Clearly, conditions (2.2) hold (with) for . In this case (3.1)-(3.2) and (2.7) are trivially satisfied.
We take and apply Theorem 3.11 and Remark 3.15 to obtain the following.
Theorem 4.5. If , there exist solutions for (4.21) with such that for all .
4.3. A Generalized Logistic Equation
Consider the following scalar problem:
where , , , , and . The initial data belongs to .
This kind of nonlinearities for the logistic equation (instead of the classical ) has been considered in [25, Chapter 11].
The functions are given by , . Clearly, conditions (2.2) hold (with ) for . In this case, (3.1)-(3.2) and (2.7) are trivially satisfied.
We take and apply Theorem 3.11 and Remark 3.15 to obtain the following.
Theorem 4.6. If , there exist two solutions for (4.22) with such that for all .
Acknowledgments
This work was been partially supported by the Ministerio de Ciencia e InnovaciΓ³n, Projects MTM2011-22411 and MTM2009-11820, by the ConsejerΓa de InnovaciΓ³n, Ciencia y Empresa (Junta de AndalucΓa), Grant P07-FQM-02468, and by the ConsejerΓa de Cultura y EducaciΓ³n (Comunidad AutΓ³noma de Murcia), Grant 08667/PI/08.
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