Journal of Function Spaces

Journal of Function Spaces / 2021 / Article
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Inhomogeneous Nonlinear Partial Differential Problems: Existence and Non-Existence of Solutions

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Research Article | Open Access

Volume 2021 |Article ID 6674060 | https://doi.org/10.1155/2021/6674060

Said Mesloub, Faten Aldosari, "On a Nonhomogeneous Timoshenko System with Nonlocal Constraints", Journal of Function Spaces, vol. 2021, Article ID 6674060, 12 pages, 2021. https://doi.org/10.1155/2021/6674060

On a Nonhomogeneous Timoshenko System with Nonlocal Constraints

Academic Editor: Stanislav Hencl
Received16 Nov 2020
Accepted17 Feb 2021
Published28 Feb 2021

Abstract

Our main concern in this paper is to prove the well posedness of a nonhomogeneous Timoshenko system with two damping terms. The system is supplemented by some initial and nonlocal boundary conditions of integral type. The uniqueness and continuous dependence of the solution on the given data follow from some established a priori bounds, and the proof of the existence of the solution is based on some density arguments.

1. Introduction

Timoshenko [1] was the first who introduced a model describing the transverse vibration of a beam. More precisely, his research work concerns with the correction for shear of a differential equation for transverse vibrations of prismatic bars. This model was given by a system of two coupled hyperbolic partial differential equations complemented with some boundary conditions. where is the length of the beam in its equilibrium configuration. The function models the transverse displacement of the beam, and models the rotation angle of its filament. The coefficients , and are, respectively, the density, the polar moment of inertia of a cross section, the shear modulus, and the Young’s modulus of elasticity. In [2], the authors considered and proved some exponential decay results for a linear homogeneous Timoshenko system with a memory term of the form where . The same problem (2) was considered in [3] where the authors discussed the decay properties of the semigroup generated by a linear Timoshenko system with fading memory. In paper [4], the authors studied the exponential stability for the following Timoshenko system with two weak dampings.

In [5], the authors investigated the effect of both frictional and viscoelastic dampings. They considered in the domain the following system and proved some exponential and polynomial decay results. For more results concerning Timoshenko systems, we refer the reader to [615].

Motivated by the above systems, we consider a nonlocal initial boundary value problem for a nonhomogeneous Timoshenko system with memory term of type (2), complemented with boundary integral boundary conditions. The study of mixed problems with nonlocal conditions such as integral conditions goes back to the year 1963, when Cannon [16] used the potential method to investigate the existence and uniqueness of the solution of the heat equation subject to the specification of energy (integral constraint). This type of conditions arises mainly when the data cannot be measured directly on the boundary, but only their averages (weighted averages) are known. Due to their importance, physical significance (mean, total flux, total energy, etc.), and numerous applications in different fields of science and engineering, several authors extensively studied this type of problems, and we can cite, for example, [1724]. Some recent new results on this direction were obtained (see [25, 26]). In this work, a functional analysis method based on some a priori bounds and on the density of the range of the unbounded operator corresponding to the abstract formulation of the given problem is used to prove the well posedness of the posed problem. This work can be considered as a contribution to the development of the energy inequality method used to prove the well posedness of mixed problems with nonlocal conditions such as integral boundary conditions (see, for example, [17, 18, 2731]).

2. Formulation of the Problem and Function Spaces

In the bounded domain , we consider the initial boundary value problem for a nonhomogeneous Timoshenko system with a viscoelastic term of the form where , and are positive constants, , and are given functions, and is a twice differentiable function such that

The convolution term represents the memory effect with a real valued function of class

System (5) is supplemented with the initial conditions and the boundary integral conditions

This system of coupled hyperbolic equations represents a Timoshenko model for a thick beam of length where is the transverse displacement of the beam and is the rotation angle of the filament of the beam. The coefficients , and are, respectively, the density, the polar moment of inertia of a cross section, the shear modulus, and the Young modulus of elasticity. The integral conditions represent the averages (weighted averages) of the total transverse displacement of the beam and the rotation angle of the filament of the beam.

Our aim is to study the well posedness of the solution of problems (5), (9), and (10). That is, on the basis of some a priori bounds and on the density of the range of the operator generated by the problem under consideration, we prove the existence, uniqueness, and continuous dependence of the solution on the given data of problems (5), (9), and (10). We now introduce some function spaces needed throughout the sequel. Let be the Hilbert space of square integrable functions on with scalar product and norm, respectively.

We also use the space on the interval , whose definition is analogous to the space on Let be the space obtained by completion of the space of real continuous functions with compact support in the interval with respect to the inner product where for every fixed The associated norm is We denote by with the set of all continuous functions with norm and the set of functions with norm

To obtain a priori estimates for the solution, we write down our problems (5), (9), and (10) in its operator form: with , , and , where

The operator is an unbounded operator of domain of definition consisting of elements such that belong to and verify initial and boundary conditions (9) and (10). The operator is acting from the Banach space into the Hilbert space , where is the Banach space obtained by completing with respect to the norm and is the Hilbert space consisting of vector-valued functions for which the norm is finite. The functions are continuous on the interval with values in and having continuous derivatives on with values in Hence, the mappings are defined and continuous on the Banach space

3. A Priori Estimate and Its Consequences

In this section, we establish an energy inequality from which we deduce the uniqueness and continuous dependence of solution of problems (5), (9), and (10) on the given data.

Theorem 1. For any function , the following a priori estimate holds where with is a positive constant independent of given by equation (41) below.

Proof. Define the integrodifferential operators and where and consider the identity where

The standard integration by parts of each term in (21) and conditions (9) and (10) give and in the same manner, we have

Substituting equalities (22)-(30) into (21), we obtain

By using the Cauchy -inequality, the last six terms in the right-hand side of (31) can be estimated as follows:

If we let we then infer from (31)-(37) that

By discarding the last three terms on the left-hand side of (39) and using the poincare type inequality [27], we have where

Application of Gronwall’s lemma (see [28]) to (40) gives

The right-hand side of inequality (42) does not depend on . By taking the supremum with respect to over , we get

Then, the a priori estimate (19) follows with .

At the moment, we do not have any information about the range of the operator except that , and we must extend so that inequality (43) holds for the extension and its range is the whole space . In this regard, we prove the following.

Proposition 2. The unbounded operator admits a closure with domain of definition

Proof. Let be a sequence such that where Then, we must show that and That is, and

Equality ((44)) implies that where is the space of distributions on By the continuity of derivation of then (47) implies in Then, from (45) it follows that in . Therefore, in . By virtue of the uniqueness of the limit in , the identities (48) and (50) lead to and

Similarly, we have from (45)

We observe from (44) and the obvious inequalities that

We conclude from (51) and (53) and the uniqueness of the limit in that and In the same manner, we show that and

Definition 3. The solution of the equation is called a strong solution of problems (5), (9), and (10).

The energy inequality (19) can be extended to

The previous a priori bound shows that the operator is injective and that is continuous from the range onto from which we assert that if a strong solution of problems (5), (9), and (10) exists, it is unique and depends continuously on the initial data and the free terms and

Corollary 4. The set is closed and

4. Solvability of the Posed Problem

Here is the main result of the paper.

Theorem 5. Problems (5), (9), and (10) admit a unique strong solution satisfying

Moreover, the solution and its time derivative depend continuously on the data , that is,

Proof. It follows from Corollary 4 that in order to prove the existence of the strongly generalized solution of problems (5), (9), and (10), it is sufficient to show that the range of the operator is everywhere dense in the space ; that is, the operator is injective. To this end, we first prove the density in the following special case.

Theorem 6. If for some function and for elements , we have then a.e in .

Proof. Since relation (57) holds for any element of , we take an element with special form given by