Well-posedness and stability for Bresse-Timoshenko type systems with thermodi usion effects and nonlinear damping

Nonlinear Bresse-Timoshenko beam model with thermal, mass diffusion and theormoelastic effects is studied. We state and prove the well-posedness of problem. The global existence and uniqueness of solution is proved by using the classical Faedo-Galerkin approximations along with two a priori estimates. We prove an exponential stability estimate under assumption (2.3)1 and polynomial decay rate for solution under (2.3)2, by using a multiplier technique combined with an appropriate Lyapuniv functions.


Introduction and position of problem
In the present paper, a nonlinear Bresse-Timoshenko system with thermodiffusion effects is considered. The beam is modeled by the following system here L is the distance between the ends of the center line of the beam. The function C denotes the concentration of diffusive material in the elastic body. The constant h > 0 is the diffusion coefficient, is a measure of the thermo-diffusion effect. To simplify the system, we use the next relation between chemical potential P and the concentration of the diffusion material C Here is a measure of the diffusive effect, we put Substitute in (1.1), the problem becomes (1. 2) The aim of this paper is to study the system (1.2) with following initial data where ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , θ 0 , P 0 are given functions, and the Dirichlet boundary conditions ϕ (x, t) = ψ (x, t) = θ (x, t) = P(x, t) = 0, x = 0, L, t > 0. (1.4) In engineering practice, when solving problems of the dynamics of composite mechanical structures, which are various kinds of connections, questions arise on determining the characteristics of natural vibrations of such coupled systems. Note that problems related to the category of non-classical problems of mathematical physics, when we talk about the combination of elements, the behavior of which is described by equations of different type. This causes certain difficulties in solving them, therefore, in practice, models of real structures are used, simplified by introducing additional hypotheses and assumptions into consideration. We mention som references dealing with dynamics of engineering structures and non-classical problems of mathematical physics [7,8,16,17]. This new kind of problem is due to a mixture of Timoshenko system [20] and Bresse system or the curved beam [9]. The coupled system from where one gets the Bresse-Timoshenko comes from Elishakoff [11] by combining d'Alembert's principle for dynamic equilibrium from Timoshenko hypothesis, resulting the coupled system (1.5) One most famous thermoelasticity is the Cattaneo's law, which is unable to account for some physical properties and it cannot answer all questions, its uses are limited, this let us think to couple the fields of strain, temperature, and mass diffusion according to the Gurtin-Pinkin model. The stabilization of the Bresse-Timoshenko model is studied only by few authors. When G ≡ 0, the problem (1.2) has been studied in [5], where a new Timoshenko system with thermal and mass diffusion effects according to the Gurtin-Pinkin model is proposed. The authors proved global well-posedness of system by using the semigroup theory and also the quasistability. Despite the fact that a sufficient number of works have been devoted to the study of natural vibrations of a Breese-Timoshenko beam, the problem of determining qualitative properties with thermal, mass diffusion and theormoelastic effects remains unsolved. [2,3,10,13,19].
In [6], the authors studied stability of thermoviscoelastic Bresse beam system. The exponential decay of energy is proved and implicit Euler type scheme based on finite differences in time and finite elements in spaces is introduced to show that the discrete energy decreases in time and an error estimates are obtained.
Without thermodiffusion effects, in [13], Feng and al., considered a Bresse-Timoshenko type system with time-dependent delay terms and In both systems (1.6) and (1.7), the authors used an appropriate Lyapunov functional to prove an exponential decay results. (See [1][2][3]19]). The present article is a logical continuation of works [5,10,13] for nonlinear case with thermal, mass diffusion and thermoelastic effects. The rest of work is organized as follows: In section 2, we recall some preliminaries and assumptions. In section 3, we state and prove the well-posedness of solution. In section 4, we prove the main stability result in both cases where H is linear and nonlinear.

Preliminary
We assume that the symmetric matrix is positive definite, and thus for all θ, P rP 2 + cθ 2 + 2dPθ > 0. (2.2) In recent years, there has been an increase in interest in the use of nonlinear properties. The value of the nonlinearity is influenced by nonlinear damping. It is associated with the development of a wave process of diffusion of the fundamental wave by waves that are far from it in frequency. To date, such nonlinear processes have not been studied fairly well in thermodiffusion effects. The function G ∈ C 1 (R, R) is assumed to be a non-decreasing function (can be taken as G(y) = |y| m−2 y, m ≥ 2) such that there exist ε, c 1 , c 2 > 0 and a convex increasing function H ∈ C 2 (R + , R + ) satisfying 1)H(0) = 0 and H is linear 3) The energy of solution is defined as Proof. Multiplying the equations of (1.2) by ∂ t ϕ, ∂ t ψ, θ, P respectively, using integration by parts, and (1.4), we get Then, taking the derivative (1.2) 1 , we get Now substituting (2.7) in (1.2) 2 using integration by parts and summing, then by using (2.3), we obtain E is decreasing.
We introduce the following Hilbert spaces

Well-posedness of problem
In this section, we prove the existence and the uniqueness of global solution for system (1.2)-(1.4) by using the Faedo-Galerkin method. If the initial data In addition, the solution (ϕ, ∂ t ϕ, ∂ tt ϕ, ψ, θ, P) depends continuously on the initial data in H × L 2 (0, L) × L 2 (0, L). In particular, problem (1.2)-(1.4) has a unique weak solution.

The first a priori estimate
Multiplying equations of (3.2) by ∂ t g jn , ∂ t h jn , ∂ t f jn and ∂ t k jn respectively and using Now integrating (3.5) and by using (2.3) 1 , we have Thus, there exists a positive constant C independent on n such that By (2.1) and (3.9), we have Then t n = T , for all T > 0.
First, we need to introduce an auxiliary Lemmas. Let Proof. Direct computation using integration by parts, we get Owing to Young and Poincare's inequalities, we obtain (4.5).

Conclusions
Our research falls within the scope of the modern interests, it is considered among the issues that have wide applications in modern science and engineering related to the energy systems. The importance of this research, although it is theoretical, lies in the following: 1. There are several generalizations and contributions that are very important in terms of the system itself. We proposed a system related to a large number of Bresse-Timoshenko type with the presence of three different types of damping, each one has functionality and physical properties, and we look at the overlapping of these three terms. 2. The great importance lies in the presence of a non-linear sources, which makes the problem have a very wide applications and importan in terms of applications in modern science. 3. Qualitatively, we proposed a new tools to study the asymptotic behavior of solutions commensurate with the existence of nonlinear term after proving the existence of the solution using a usual method. We found a new decay rate of system's energ, although the system's energy decreased according to a very general rate that includes all previous results and more than that, so, to our knowledge, there is no generalization more than this.