1. Introduction

Over the previous years nonclassical problems for partial differential equations have been widely used for a description a number of phenomena in modern physics and technology. Nonclassical problems with nonlocal conditions include relations between boundary values of an unknown solution and its derivatives and their values at internal points of a domain. Nonlocal problems with integral conditions which are naturally generalization of discrete nonlocal conditions can be considered as mathematical models the processes with inaccessible boundary.

One of the initial works devoted to problems with integral condition for second-order partial differential equations is the paper of Cannon ^[2] where the nonlocal problem was used for modeling of heat conduction process. Later, nonlocal problems with integral conditions were investigated in ^{[1, 3, 4, 6]}. Note here some resent results ^{[7, 8, 9, 10]}.

It was found that the choice of the method of research depends considerably on the kind of nonlocal conditions.

Usually, we understand nonlocal condition of the II kind as correlations connecting values of the solution, and possibly, its derivatives on some inner manifold and at boundary points. If these correlations do not contain values of the solution on the boundary, then they are said to be nonlocal conditions of the I kind ^[11]. In the case of one spatial variable difficulties generated by the I kind integral condition can be avoided through reducing the problem to the problem with the II kind integral condition, and hence, in such case nonlocal problems with the II kind conditions are of considerable interest.

Motivated by this, in the present article we study a nonlocal problem with the II kind integral condition for parabolic equation. The main aim of the paper is to demonstrate a nonlinear functional analysis technique which is applicable for investigation of such nonlocal problems.

2. Preliminaries

Let . Consider the equation

(1)

and set a problem: determine a solution of the equation (1) in satisfying the initial condition

(2)

the boundary condition

(3)

and the nonlocal condition

(4)

Definition. Following ^[5], we define as a Hilbert space which consists of all elements of such that with the norm:

Definition. A function is said to be a solution of the problem (1)-(4) if , , and for all the function satisfies the integral identity

(5)

3. Results

Theorem. Let , , , and , . Then there exists a unique solution of the problem (1)-(4).

The proof of the theorem is organized as follows. First we obtain energy estimates which imply uniqueness of the solution. To prove the existence we use a parameter continuation method.

We take an arbitrary number and consider the problem (1)-(3) with the nonlocal condition

(6)

Assume that a solution of the problem (1)-(3), (6) exists. First we multiply (1) by and integrate over , . As a result after integration by parts we obtain the equality

(7)

Our next aim is to derive estimates of the right-hand side part of (7). Applying the -inequality to the first, second and third terms of (7) we have

(8)

Observe further that for a solution of (1)-(3), (6) the following estimates are valid.

(9)

(10)

(11)

[p. 77],

where the constants are such that

is arbitrary and defines by .

From the inequalities (8)-(11) we obtain

(12)

Second, we multiply (1) we multiply (1) by and integrate over , . After integration by parts we have

Applying the estimates (9)-(11) we obtain

Choose . Then

(13)

It follows from (12), (13) that

Let . Then we obtain

(14)

where , ,

It follows from (14) that

Now by Gronwall’s lemma we conclude that

and hence,

(15)

From (14) it also follows that

Therefore, the inequality (15) implies that

where defines by and does not depend on the chosen , when , that is

(16)

Assume that the problem (1)-(3), (6) has two different solution and . Then the function is a solution of the problem (1)-(3), (6) with . From the estimate (16) it immediately follows that , and hence, a.e. in .

To prove the existence part we shall apply the parameter continuation method.

Let be an arbitrary number. Denote by the set of such for which the problem (1)-(3) with nonlocal condition (6)

has a solution in .

If such a set is non-empty, open and closed then , and therefore, there exists a solution of the problem (1)-(3), (6) for any .

First, we note that is non-empty. Indeed, for the case the problem (1)-(3), (6) becomes

It is possible to see that there exists a solution of this problem [^[5], pp. 273-276].

Our next aim is to prove that the set is closed. To this end we consider a sequence which converges to some as . We shall show that .

Indeed, to each there corresponds a function that satisfies the integral identity (5) and conditions (2), (3), (6). Thus, the estimate (16) is also valid for , that is , where the constant does not depend on and .

Therefore, as is a Hilbert space, so there exists a subsequence and a function such that

weakly in and

a.e. on .

Thereafter, letting in the equalities

it is easy to see that is a solution of the problem (1)-(3), (6) and hence, .

This implies that is closed.

The next step is to prove that the set is open. To this end we take and shall prove that for small enough .

We introduce an operator in the following way. Let and be a solution of the integral equation

(17)

For every fixed the equation (17) is Volterra equation with respect to the function . We note that as the function belongs to and the kernel is bounded, so there exists a unique solution of (17) such that for every fixed .

Observe that

Therefore the solution of (17) is represented as a sum of two functions

where the first function is an element of and second one is bounded. Hence, .

Also it is easy to prove that . Indeed, we formally differentiate (17) with respect to and consider as a solution of the integral equation

with a bounded kernel and right-hand side part from . Therefore, repeating the above arguments we see that .

Next, we differentiate (17) with respect to and obtain that can be defined as

(18)

By the properties of the functions , , it means that . Now we differentiate (18) with respect to and get

(19)

Hence, using the properties of , , we conclude that .

Therefore, the solution of the Volterra equation .

Next, we define a function as a solution of the problem

(20)

(21)

(22)

(23)

where ,

By the assumption, , and hence, there exists the solution of the problem (21)-(23).

Now consider the function . It is easy to see that satisfies (1)-(3) and

(24)

Therefore, is a solution of the problem (1), (3), (24). We shall consider this function a mapping : .

Now our aim is to prove that the introduced operator is a contraction mapping for small enough . To this end, we consider functions , and let , be the corresponding solutions of the integral equation (18) and , be the solutions of the problem (21)-(24). We define , .

Let us denote , , , . Then is a solution of the problem (20)-(23) with and .

Following the proof of the estimate (16) we obtain

(25)

where the constant depends on , and depends on , , .

The next step is to obtain an estimate of each norm on the right-hand side of (25). Observe that (17) implies

Applying the Cauchy inequality we have

(26)

Now by Gronwall’s lemma we conclude that

(27)

and hence,

Therefore,

(28)

Observe further that

Applying the Cauchy inequality and (27) we obtain

(29)

Using the Gronwall’s lemma for (29) we obtain

This inequality implies that

(30)

where .

Now it remains to obtain an estimate for . To this end first we find an estimate for . From (19) it follows that , which implies

Thus, , where is such that .

Now we consider (20) and conclude that

Therefore,

and hence,

(31)

where , .

It immediately follows from (28), (30), (31) that

Finally, we conclude that

where depends on , , , .

Now if is small enough, then and hence, the operator is a contraction mapping. Taking into account that the space is complete we conclude that there exists a unique fixed point which satisfies (1)-(3) and the condition

(32)

Therefore, for small enough the problem (1)-(3), (32) has a solution .

This implies that , and hence the set is open.

Finally, we conclude that , and the problem (1)-(3), (6) has a solution for any , that is, there exists a solution of the problem (1)-(3), (6) for , and hence, the existence part for the problem (1)-(4) has been proved.

4. Conclusion

In the present research the investigation of the nonclassical problem with the second type integral condition has been demonstrated. While dealing with such problems the main question is about a choice of the most powerful and suitable research method. This choice essentially depends on the kind of a nonlocal condition. In this article we have shown that a nonlinear functional analysis method can be applied to study nonlocal problems and existence of the introduced solution has been proved by a parameter continuation method. To prove the uniqueness part we used an a priori estimate method.

Acknowledgement

The author would like to thank Ludmila S. Pulkina for her insistent support during this research.

References

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