Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms

In this paper, we consider the following Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms: { ρ 1 ⁢ φ t ⁢ t - K ⁢ ( φ x + ψ ) x = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , ρ 2 ⁢ ψ t ⁢ t - b ⁢ ψ x ⁢ x + K ⁢ ( φ x + ψ ) + β ⁢ θ x = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , ρ 3 ⁢ θ t ⁢ t - δ ⁢ θ x ⁢ x + γ ⁢ ψ t ⁢ t ⁢ x + ∫ 0 t g ⁢ ( t - s ) ⁢ θ x ⁢ x ⁢ ( s ) ⁢ d s + μ 1 ⁢ θ t ⁢ ( x , t ) + μ 2 ⁢ θ t ⁢ ( x , t - τ ) = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , \left\{\begin{aligned} &\displaystyle\rho_{1}\varphi_{tt}-K(\varphi_{x}+\psi)_% {x}=0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\\ &\displaystyle\rho_{2}\psi_{tt}-b\psi_{xx}+K(\varphi_{x}+\psi)+\beta\theta_{x}% =0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\\ &\displaystyle\rho_{3}\theta_{tt}-\delta\theta_{xx}+\gamma\psi_{ttx}+\int_{0}^% {t}g(t-s)\theta_{xx}(s)\,\mathrm{d}s+\mu_{1}\theta_{t}(x,t)+\mu_{2}\theta_{t}(% x,t-\tau)=0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\end{aligned}\right. together with initial datum and boundary conditions of Dirichlet type, where g is a positive non-increasing relaxation function and μ 1 , μ 2 {\mu_{1},\mu_{2}} are positive constants. Under a hypothesis between the weight of the delay term and the weight of the friction damping term, we prove the global existence of solutions by using the Faedo–Galerkin approximations together with some energy estimates. Then, by introducing appropriate Lyapunov functionals, under the imposed constrain on the above two weights, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.