We consider initial boundary value problem for a reaction-diffusion system with nonlinear and nonlocal boundary conditions and nonnegative initial data. We prove local existence, uniqueness, and nonuniqueness of solutions.

1. Introduction

In this paper we consider the following semilinear reaction-diffusion system with nonlinear nonlocal boundary conditions: where , is a bounded domain in   for   with smooth boundary . Here, are nonnegative Hölder continuous functions defined for , and and are nonnegative continuous functions defined for , , and . The initial data are nonnegative continuous functions satisfying the boundary conditions at .

In the past several decades, many physical phenomena have been formulated into nonlocal mathematical models. Initial boundary value problem for semilinear reaction-diffusion equations and systems with nonlocal boundary conditions has been analyzed by many authors (see, e.g., [112] and the references therein). Local and global existence, comparison principle, and various qualitative properties have been discussed.

We note that for the nonlinearities in (1) are non-Lipschitzian. The problem of uniqueness and nonuniqueness for different nonlinear parabolic equations and systems with non-Lipschitzian data has been addressed by several authors. See, for example, [1316] for equations and [1721] for systems. In particular, the authors of [16] have investigated the uniqueness of solution for a problem (1) with a single equation.

In [7] the authors have considered a problem (1) with and . They have proved a comparison principal and investigated the blowup properties of the positive solutions. The aim of this paper is to study the uniqueness of the nonnegative solution of the problem (1) for any .

The plan of this paper is as follows. In the next section we prove a comparison principle; an existence theorem of a local solution is given in Section 3; uniqueness of solutions with nontrivial initial data, uniqueness of solution with trivial initial datum for ; nonuniqueness of solution with trivial initial datum for are proved in Section 4.

2. Comparison Principle

Let us introduce the definitions of a subsolution and a supersolution. For the remainder of this paper we denote , .

Definition 1. A pair of nonnegative functions is called a subsolution of problem (1) in if and a pair of nonnegative functions is called a supersolution of problem (1) in if the reversed inequalities hold in (2).

Definition 2. A pair of functions is called a solution of (1) in if it is both a subsolution and a supersolution of problem (1) in .

Definition 3. We say that solution of (1) is positive in if and in .

To establish the uniqueness results we need a comparison principle. We prove it in a different way, not as in the work of [7].

Theorem 4. Let and be a nonnegative supersolution and a nonnegative subsolution of problem (1) in , respectively. Suppose that and or and in if . If and for , then and in .

Proof. Let be a nonnegative function such that . Then , where is the unit outward normal to the lateral boundary of . By the definition of a subsolution we have If we multiply (3) by and then integrate over for , we get
On the other hand, the supersolution satisfies (4) with reversed inequality. Set . Then we have where are nonnegative continuous functions if and positive continuous functions if in which satisfy the following equalities:
Obviously there exists a positive constant such that in . Since , are nonnegative and continuous functions, then there exists a constant such that , in and , respectively.
Consider the following backward problem in where . By the maximum principle for the heat equation , it is easy to show that on for some .
Let . Then from (5), we get where , and is the Lebesgue measure of . Consider the sequence of functions , , converging in to , defined as follows: Replacing by in (8) and passing to the limit as , we have Using a similar argument for the inequality ,  ,  , we get where , are positive constants. Adding (10) and (11), we have where .
Applying now Gronwall's inequality, we conclude that Since , then , .

3. Local Existence

Let be decreasing to 0 sequence such that . For let be the functions with the following properties: ; for ; as ; for .

Due to the nonlinearities in (1), the Lipschitz condition is not satisfied if , and thus we need to consider the following auxiliary problem:

Theorem 5. For small values of , (14) has a unique solution in .

Proof. We start the proof with the construction of a supersolution of (14). Let ,  . Denote . Introduce an auxiliary function    with the following properties: where  . Let be positive constants such that and Note that for and for . Obviously the pair of functions is a subsolution of problem (14). We show that is a supersolution of (14) in for if and if . We have for . On the other hand, we get that for . Similarly, we can show that To prove the existence of a solution of the problem (14) we introduce the set Clearly is a nonempty convex subset of .
Consider the following problem: where . Problem (22) has a nontrivial positive solution. Let us call . In order to show that has a fixed point in we verify that is a continuous mapping from into itself such that is relatively compact. Thanks to the comparison principle for (22) we have that maps into itself.
Let denote the Green's function for the heat equation given by with zero boundary condition. Then is a solution of (22) if and only if
We claim that is continuous. In fact, let be a sequence in converging to in . Denote . Then we see that where and Choosing so small that , we conclude that in as . The equicontinuity of follows from (24), (25) and the properties of the Green's function (see, e.g. [22]). The Ascoli-Arzela theorem guarantees the relative compactness of . Thus, we are able to apply the Schauder-Tychonoff fixed point theorem and conclude that has a fixed point in if is small. Since is a fixed point of , it is a solution of (14). Uniqueness of solution follows from a comparison principle for (14) which can be proved in a similar way as in the previous section.

Using Theorem 5, we can prove the following local existence theorem of a solution of problem (1).

Theorem 6. For small values of (1) has a maximal solution in .

Proof. Let . It is easy to show that is a supersolution of the problem (14) with . Then ,  . Using these inequalities and the continuation principle of solutions we deduce that the existence time of does not decrease as . Let , then and exist in for some .
Moreover, by dominated convergence theorem, satisfies the following equations: The interior regularity of follows from the continuity of in and the properties of the Green's function. Obviously satisfies (1). Let be any other solution of (1). Then by the comparison principle . Taking , we conclude that .

To prove the positiveness of nontrivial solutions we need the following definition.

Definition 7. We say that a function has the property () if there exist and , such that ,  , and as .

Remark 8. Note that if a nonnegative function has no the property () then in for some .

Theorem 9. Let either or be a nontrivial function in . Supposing that and are nontrivial functions for any and ,   has the property () if ,  and   has the property () if . Let be a supersolution of (1) in . Then is positive in for .

Proof. Suppose for definiteness that is a nontrivial function. We show at first that in for . We have then by strong maximum principle a minimum of should be attained in on a parabolic boundary. Thus, in , otherwise, there would be a contradiction with the initial datum. We show that on . Let there exist a point such that . But for . By boundary conditions (1) and assumption for we have for . This contradiction shows that on , and therefore in for .
Now we show the positiveness of . If is a nontrivial function, then in for by previous arguments. If we suppose that there exists a constant such that in since otherwise we can use the arguments from the beginning of the proof again. But this is a contradiction with the second equation in (1) since in and has the property (). Hence, we conclude that in for .

4. Uniqueness and Nonuniqueness

As a simple consequence of Theorem 4 and Theorem 9 we get the first uniqueness result for problem (1).

Theorem 10. Let problem (1) have a solution in with nonnegative initial datum for and with positive initial datum under conditions and are nontrivial functions for any and . Then solution of (1) is unique in .

Now we show nonuniqueness of solutions of problem (1) with trivial initial datum for .

Theorem 11. Let , . Suppose that the maximal solution of problem (1) exists in . Assume that at least one of the following conditions is fulfilled: Then the maximal solution of problem (1) is nontrivial in .

Proof. In the local existence theorem we constructed a maximal solution of (1) in the following way: , , where is some positive supersolution of (1). To prove the theorem we construct a nontrivial nonnegative subsolution of some problem with trivial initial datum. By the comparison principle we conclude that , and therefore maximal solution is a nontrivial solution.
Consider at first the case when and for some and . Then there exists a neighborhood of in and such that for .
Consider the following problem: where is a bounded nontrivial nonnegative continuous function which satisfies a boundary condition. By the strong maximum principle for , where .
Let . Denote that where . Note that if or . After simple calculations we obtain Similarly we can get that Then and are subsolution and supersolution, respectively, of the following problem: By comparison principle for (38) we conclude that , and hence ,  , for .
Now consider the case when and for any and some ,  . We will consider the following problem: where will be defined later. Construct a subsolution of (39) using the change of variables in a neighborhood of as in [23]. Let . We denote by the inner unit normal to at the point . Since is smooth it is well known that there exists such that the mapping given by defines new coordinates in a neighborhood of in . A straightforward computation shows that, in these coordinates, applied to a function which is independent of the variable , evaluated at a point is given by where denote the principal curvatures of at .
Under the made assumption there exists such that for , where is some neighborhood of in . Let and assume that and . For points in of coordinates define and extend as zero to the whole . Using (40), we get that for sufficiently small values of . It is clear that It remains to verify the validity of the inequality for and . Here is Jacobian of the change of variables. Estimating the integral I in the right-hand side of (44) we get where constant does not depend on ; we obtain that (44) is true if is small enough.
By comparison principle for (39) we conclude that , and, respectively, ,   for .
The proof in the case and for any and some is similar.

It is easy to get from Theorem 9 and the proof of Theorem 11 the following statement.

Corollary 12. Let the conditions of Theorem 11 hold with . Suppose also that and are nontrivial functions for any and ,    has the property () if (32) is realized, and has the property () if (33) is realized. Then maximal solution is positive in for .

Under the conditions of Corollary 12 for some class of the coefficients and ,   , we can prove the uniqueness of solution for (1) with trivial initial datum which is positive for all positive times as long as it exists.

Theorem 13. Let the conditions of Corollary 12 hold. Suppose also that there exists such that for the functions , ,   are nondecreasing with respect to .
Then there exists exactly one solution of problem (1) which is positive in for .

Proof. Suppose that there exists different from solution of (1) with trivial initial datum which is a positive in for . Denote . Due to the conditions of the theorem it is easy to see that is positive supersolution of (1) with trivial initial datum in for any . By Theorem 4 we have for every . Passing to the limit as we get for . Hence in .

Note that by Theorem 10, the solution of (1) is unique if . Now we specify our uniqueness result in the case .

Theorem 14. Let the conditions of Corollary 12 fulfill only or . Then the solution of (1) is unique.

Proof. To prove the uniqueness of the solution if , it suffices to show that if is any solution of (1), then where is a maximal solution of (1).
First, consider the case when . Let Then satisfies for some . By Theorem 13, there exists a unique solution of the following problem: such that for for some . Let . In a similar way as in Theorems 13 and 4 we can prove that . Put . Now, use an elementary inequality, which is recalled for instance in [20], where ,  , and . Then we obtain We show that in . In fact, otherwise, by Theorem 9 there exists such that in . Suppose that for some . Then we get for and . This is a contradiction since in for some , and .
If for any and some we can obtain a contradiction by another way. Indeed, for and and we get again a contradiction since in , and .
Since in by comparison principle with arguments of Theorems 13 and 4 we conclude that in . This implies (46) and completes the proof for the first case.
Now suppose that . Assume, for example, that . Then, as in the first case, we introduce the functions . We use the following relations: where are nonnegative continuous functions which are between and ,  , respectively, in . Then we have that a pair of functions satisfies relations
Further develop the arguments, as in the first case, only for ,  . Using the linearization of terms with powers greater than 1 in the equations and boundary conditions of (1) as above we can prove the theorem for the remaining cases in a similar way.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors would like to thank the referees for their valuable comments and suggestions regarding the original paper.