Journal of Applied Mathematics

Volume 2014 (2014), Article ID 464205, 12 pages

http://dx.doi.org/10.1155/2014/464205

Research Article

On an Initial Boundary Value Problem for a Class of Odd Higher Order Pseudohyperbolic Integrodifferential Equations

Said Mesloub

Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 2 March 2014; Accepted 30 May 2014; Published 15 June 2014

Academic Editor: Yansheng Liu

Copyright © 2014 Said Mesloub. 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.

This paper is devoted to the study of the well-posedness of an initial boundary value problem for an odd higher order nonlinear pseudohyperbolic integrodifferential partial differential equation. We associate to the equation n nonlocal conditions and classical conditions. Upon some a priori estimates and density arguments, we first establish the existence and uniqueness of the strongly generalized solution in a class of a certain type of Sobolev spaces for the associated linear mixed problem. On the basis of the obtained results for the linear problem, we apply an iterative process in order to establish the well-posedness of the nonlinear problem.

1. Introduction

Classical and nonclassical and local and nonlocal initial boundary value problems for partial differential equations are widely studied and athre being studied nowadays. One of the most important and crucial tools to be applied to partial differential equations is functional analysis. It is the universal language of mathematics. No serious study in partial differential equations, mathematical physics, numerical analysis, mathematical economics, or control theory is conceivable without a broad solicitation to methods and results of the functional analysis and its applications.

The main objective of this research work is to develop one of the powerful methods of functional analysis, namely, the energy inequality method for a certain classes of partial differential equations with nonlocal constraints of convolution type in some functional spaces of Sobolev type. This method, based on the ideas of Petrovski [1], Leray [2], Garding [3], and presented on a method form by Dezin [4], was used to investigate and study different categories of mixed problems related to elliptic, parabolic, and hyperbolic equations [5–12], mixed equations [13–15], nonclassical equations [16, 17], and operational equations [18, 19], with classical conditions of types: Cauchy, Dirichlet, Neumann, and Robinson.

Mixed nonlocal problems are especially inspired from modern physics and technological sciences and they describe many physical and biological phenomena. That is in terms of applications, nonlocal mixed problems are widely applied in medical science, biological processes, chemical reaction diffusion, heat conduction processes, population dynamics, thermoelasticity, control theory, and in so many other domains of research. It is worth to mention that for these types of problems, we cannot measure the data directly on the boundary, but we only know the average value of the solution on the domain.

For second order parabolic equations with nonlocal conditions, the reader should refer to [20–23]. For hyperbolic equations and pseudoparabolic equations with purely or one integral conditions, the reader should refer to [24–31]. The reader could also refer to a recent paper dealing with a higher dimension Boussinesq equation with a purely nonlocal condition [32]. This paper is organized as follows. In Section 2, we pose and set the problem to be solved. In Section 3, we give some notations, introduce the functional frame, and state some important inequalities that will be used in the sequel. Section 4 is devoted to the proof of the uniqueness of the solution of the associated linear problem. In Section 5, we establish and prove the existence of solution of the posed associated linear problem. In the last Section, Section 6, we solve the nonlinear problem. On the basis of the results obtained in Sections 4 and 5, and by using an iterative process, we prove the existence and uniqueness of the solution of problem (1)–(6). Some proofs of Sections 3, 4, and 5 are given in Appendices A and B at the end of Section 6. At the end of the paper, we give a set of references.

2. Problem Setting

In the rectangle , where and , we consider the nonlinear higher order pseudohyperbolic differential equation of odd order

where

In (1), is a given function which will be specified later on and is a function satisfying the conditions

(H1) for all ,(H2),,, for all and all constants are strictly positive.

To (1), we associate the initial conditions

the Dirichlet boundary condition

the Neumann boundary conditions

and the nonlocal conditions

where the data functions and satisfy the compatibility conditions

In this paper, we are concerned with the proof of well-posedness of the nonlinear nonlocal initial boundary value problem (1)–(6) in some weighted Sobolev spaces.

The main tools used in our proofs are mainly based on some iterative processes, some priori bounds, and some density arguments.

3. Functional Framework, Notations, and Some Inequalities

For the investigation of problem (1)–(6), we need the following function spaces.

Let be the usual Hilbert space of square integrable functions and let [24] be the Hilbert space of Sobolev type constituted of functions if and of functions such that , if , with inner product

and with associated norm

Corollary 1. For all , one has the inequality

Proof. See Appendix A.

Corollary 2. For all , one has the inequalities

One denotes by the set of all abstract strongly measurable functions on into such that

The space is a Hilbert space having the inner product

One can write problem (1)–(6) in an operator form , where is an unbounded operator with domain , acting from a Banach space into a Hilbert space constructed as below. One defines the domain of the operator as the set

The space is the Banach space of functions verifying conditions (4)–(6) and having the norm

The space is the Hilbert of multivalued functions with finite norm

4. Uniqueness of Solution of the Associated Linear Problem

We first treat the following associated linear problem:

where is replaced by .

We establish a priori bound from which we deduce the uniqueness of solution of problem (17).

Theorem 3. If the coefficients satisfy condition (H1), then there exists a positive constant independent of such that

for all .

Proof. See Appendix B.

Proposition 4. The operator admits a closure.

Proof. See [26].

We denote by the closure of the operator and by the domain of definition of and define the strong solution of problem (17) as the solution of the operator equation .

Inequality (18) can be extended to

We can deduce from (19) that the strong solution of problem (17) is unique if it exists and depends continuously on and that the image of the operator coincides with the set .

5. Solvability of the Associated Linear Problem

Theorem 5. Assume that conditions H1 and H2 are hold. Then problem (17) admits a unique strong solution satisfying , and , depend continuously on the given data and verify

Proof. Since is closed and , then in order to prove the existence of the strong solution, we have to show that . We first prove it in the following special case:

Theorem 6. If conditions of Theorem 3 are satisfied and for , we have

for all , then vanishes almost everywhere in .

Proof. We first define the function by the relation

We now consider the equation

and define by

Relations (23) and (24) imply that is in , where .

We now have

The following lemma shows that given by (25) is in , where .

Lemma 7. If conditions of Theorem 6 are satisfied, then the function defined by the relations (23) and (24) has -derivatives up to third order which included are in .

Proof. See Appendix B.

We now continue to prove Theorem 6. We replace given by (25) in (21) to get

Straight forward successive integration by parts of the two terms in (26) gives

Substitution of (27) into (26) yields

By dropping the second term on the left-hand side (28) and by using conditions H1 and H2, we obtain

We now consider the two elementary inequalities

Combination of inequalities (29)-(30) leads to

where

We now introduce a new function defined by , then , and , and we have

If we choose such that , then for all , inequality (33) implies that

Inequality (34) can be written in the form of

where

It follows from (35) that from which it follows that almost everywhere in . By reiterating the same procedure, we deduce that a.e., in . We now continue the proof of Theorem 5.

We consider a function in . The function satisfies

If we pick an element in , equality (37) becomes

By virtue of Theorem 6, we deduce that , and (37) then takes the form

It follows from (39) that , . This results from the fact that the quantities and vanish independently and that the set of values of the trace operators and is dense in .

6. The Nonlinear Problem

On the basis of the results obtained for the linear case, we are now able to establish the existence and uniqueness results for the nonlinear problem (1)–(6).

Observe that the function solves the problem

where

whenever and are, respectively, solutions of the problems

The function satisfies the condition

for all .

According to Theorem 5, problem (43) has a unique solution depending continuously on , . It remains to prove that problem (40) has a unique weak solution.

Consider the inner product

with , such that , , , , , , , , , and .

By using the above conditions on and , we can write (45) in the form of

On the other hand, we have

It follows from (46) and (47) that

where

Definition 8. One calls a function a weak solution of problem (40) if (48) and conditions ,, are satisfied.

One now considers the following iterated problems:

Theorem 5 asserts that each problem (50) admits a unique solution . By setting , one gets the following mixed iterated problem:

where

Theorem 9. Assume that condition (44) holds then there exists a positive constant such that the solution of problem (51) satisfies the inequality

where .

Proof. By considering the scalar product in , of the partial differential equation in (51) and the intgrodifferential operator and by using initial and boundary conditions in (51), we obtain

It is easy to show the elementary inequality

Combination of (54) and (55) after discarding the second term on the left-hand side of (54) and using Cauchy inequality lead to

On the other hand, we have

Combining inequalities (56) and (57) and using (11), we obtain

By applying Gronwall’s lemma (see [22]) to inequality (58), we have

Integration of both sides of (59) with respect to over , yields

Inequality (60) implies that the series converges if . It is obvious that the sequence defined by

converges to a limit function which must satisfy (48) and conditions , .

It is obvious that from the partial differential equation in (50) we have

and we also have

Equality (63) gives

By using conditions on , evaluation of the right-hand side of (62) gives

Combination of (62) and (65) leads to

Application of Cauchy Shwartz to the two terms of the right-hand side of (66) gives

It follows from (66)-(67) that

On the other hand we have

Now taking into account inequalities (68) and (69) and passing to limit inequality (64) as , we obtain

which is exactly inequality (48). Now since , then , and we conclude that , , almost everywhere.

We now prove the uniqueness of solution of problem (40).

Theorem 10. Assume that condition (44) is fulfilled, then the initial boundary value problem (40) admits a unique solution.

Proof. Suppose that are two solutions of problem (40), then and satisfies

where

As we have proceeded in the proof of Theorem 9, we consider the scalar product in of the differential equation in (71) and the operator , we obtain

where .

Since it is assumed that , then it follows that . Therefore . Hence the uniqueness of solution of problem (40) is in.

Appendices

A.

Proof of Corollary 1. We have

Consequently,

Proof of Theorem 3. We consider the scalar product in of the differential equation in problem (17) and the integrodifferential operator

where and , we obtain

We separately consider the integrals in the right-hand side of (A.4) and we integrate by parts and taking into account boundary and initial conditions in (17), we obtain

Substitution of (A.5) into (A.4) yields

By Corollary 1, we have

If we discard the first term in (A.6), and by using (A.7), we obtain

By virtue of the elementary inequality

and (A.8), we have

where

Let

then it follows from Gronwall’s lemma (see [3]) that

The right-hand side in (A.13) is independent of , hence replacing the left-hand side by its upper bound with respect to over , we obtain (18), with , and Theorem 3 follows.

B.

Proof of Lemma 7. We have

It is obvious that the first and second terms on the right-side of (B.1) are in . To prove that the term , we use the -averaging operators of the form

where , , .

If we apply the operators and to (23), we obtain

It follows from (B.3) that

Now by using the properties of the -averaging operators , conditions (H1) and (H2), and Corollary 2, we see from (B.4) that

where

Since in as and the norm of in is bounded, we conclude that .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no. RGP-VPP-117.

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