Global existence and stability results for a nonlinear Timoshenko system of thermoelasticity of type III with delay

In this paper, we consider a nonlinear Timoshenko-type system of thermoelasticity of type III with delay: in which \(\rho_{1}\), \(\rho_{2}\), \(\rho_{3}\), K, b, k, β, γ, δ, \(\mu_{1}\), \(\mu _{2}\), τ are positive constants. In this system, \(\mu_{1}\psi_{t}\) represents a frictional damping, \(\mu _{2}\psi_{t}(x,t-\tau)\) represents a delay term, and \(f(\psi)\) is a forcing term. We impose the following initial and boundary conditions:

We start our literature review with the pioneer work of Messaoudi and Said-Houari [1]. The authors considered the system as follows: They obtained an exponential decay result under equal wave speeds (\(\frac{K}{\rho_{1}}=\frac{b}{\rho_{2}}\)). Later Messaoudi and Fareh [2] studied the case of nonequal speeds (\(\frac{K}{\rho_{1}}\neq\frac {b}{\rho_{2}}\)) and established a polynomial decay result. In [3], the author consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound as follows: They established general decay results for the cases of \(\mu=0\) and \(\mu\neq0\) with the constant \(\mu=(\tau-\frac{ \rho_{1}}{K\rho _{3}})(\frac{\rho_{2} }{b}-\frac{\rho_{1} }{K})-\frac{\tau\delta^{2}\rho _{1}}{bK\rho_{3}}\).

On the other hand, Timoshenko systems with delay term have attracted extensive attention, and the increasing complexity of their types makes research more significant. An arbitrarily small delay may be the source of instability, see [4, 5]. Racke [6] considered the following coupled system of linear thermoelastic equations with constant delays \(\tau_{1}\) and \(\tau_{2}\): He obtained that the solution of problem (1.4) is instable with any delay \(\tau_{1}>0\) or \(\tau_{2}>0\). In recent years constant delay τ has been extended to the time-varying function \(\tau(t)\) in the thermoelastic equations, see [7]. Also, in [8] Nicaise and Pignotti studied the initial-boundary value problem of wave equation with boundary distributed delay as follows: They proved an exponential stability result for system (1.5) with the condition and when the boundary distributed delay term in the above system is replaced by the internal feedback \(\int_{\tau_{1}}^{\tau_{2}}a(x)\mu (s)u_{t}(t-s)\,ds\) with \(a(x)\) satisfying some suitable conditions. They also obtained that the energy of solution is exponentially decaying to zero under the condition We refer the reader to [9–14] for more analogous results. Kafini et al. [15], considered a one-dimensional Timoshenko-type system They proved the well-posedness of system (1.6) and established an exponential decay result under the condition \(\frac{K}{\rho_{1}}=\frac {b}{\rho_{2}}\) and a polynomial decay result under the condition \(\frac{K}{\rho_{1}}\neq \frac{b}{\rho_{2}}\). For a Timoshenko system with time delay and forcing term at the same time Feng and Pelicer [16] obtained an exponential stability under equal wave speeds. In the present paper, when \(\frac{K}{\rho_{1}}=\frac{b}{\rho_{2}}\), we extend their result to nonlinear Timoshenko system of thermoelasticity of type III by using the perturbed energy functional technique as well, and when \(\frac{K}{\rho_{1}}\neq\frac{b}{\rho_{2}}\), we achieve a polynomial decay estimate.

In order to deal with the delay term, we define the following new variable: Thus we have Then system (1.1)–(1.2) is transformed to In order to use the Poincaré inequality for θ, as in [15], we introduce Then by (2.4)₃ we have After a simple substitution, we see that \((\phi,\psi,\bar{\theta},z)\) satisfies (2.4). From now on, we work with θ̄ but write θ for convenience.

In this section, we shall investigate the well-posedness of problem (2.1) with semigroup theory, we start with the vector function \(\mathcal{U}(t)=(\phi,\varphi,\psi,u,\theta ,v,z)^{T}\), where \(\varphi=\phi_{t}\), \(u=\psi_{t}\), and \(v=\theta_{t}\). We introduce as in [15] Then we define the energy space by equipped with the following inner product: in which ξ is a positive constant satisfying Thus system (2.4) can be re-written as where the operators \(\mathcal{A}\) and \(\mathcal{F}\) are defined by with and the initial value \((\theta_{0},\theta_{1})\) satisfies By using the same methods as those in [15] and in [16], we can obtain the following Lemmas 3.1 and 3.2, respectively. We omit the proof.

We introduce the first order energy of problem (2.4) as and the second order energy of problem (2.4) as (if \(\mathcal {U}_{0}\in\mathcal{D(A)}\))

Next we deal with the boundary term in (4.18). We introduce the function

Now we define the Lyapunov functional \(\mathscr{L}(t)\) as follows: where N, \(N_{2}\) are positive constants to be chosen properly later. For N large enough, it is not difficult to prove that there exist two positive constants \(\gamma_{1}\) and \(\gamma_{2}\) such that, for any \(t>0\),