A new stability result for a thermoelastic Bresse system with viscoelastic damping

We investigate a thermoelastic Bresse system with viscoelastic damping acting on the shear force and heat conduction acting on the bending moment. We show that with weaker conditions on the relaxation function and physical parameters, the solution energy has general and optimal decay rates. Some examples are given to illustrate the findings.

In the present work, with the viscoelastic law applied to the shear force, the following constitutive laws are imminent: where k 1 = κGA, k 2 = EI, k 3 = EA, E, G, κ, and β are the modulus of elasticity, shear modulus, shear factor, and diffusivity, respectively. The viscoelastic term (represented by the convolution, see [24,25] for details) acts as a damping mechanism to decrease the effect of unwanted vibration from internal or external forces such as beam's weight and wind. Substituting (1.6) into (1.7), we arrive at the following Timoshenko-thermoelastic Bresse system: where x ∈ (0, L), t > 0, the physical parameters l, k 1 , k 2 , k 3 , ρ 1 , ρ 2 , ρ 3 , β, γ are all positive constants, and g is a given function to be specified later. Recently, Mukiawa [26] considered (1.8) with Maxwell-Cattaneo law and established a general decay estimate. Next, we supplement (1.8) with the Neumann-Dirichlet-Dirichlet-Neumann boundary conditions: (1.9) and the initial data (1.10) Our main focus is to show that the solution energy of (1.8)-(1.10) decays in the same way as the relaxation function g provided k 1 = k 3 . Thus, without additional conditions on the physical parameters k 1 , ρ 1 , k 2 , ρ 2 and without stronger regularity on the solution (u, v, w, θ ) of system (1.8)-(1.10), we obtain a general and optimal decay estimate for the solution energy. The result obtained in this paper is general and optimal in the sense that it agrees with the decay rate of g (see conditions on g in Sect. 2). This work is organized as follows: In Sect. 2, we present materials that will be helpful in establishing the main results. In Sect. 3, we state and prove some useful lemmas. In Sect. 4, we look at the stability rate of the energy functional associated with problem (1.8)-(1.10).

Assumptions and functional setting
We denote by ·, · and · 2 the usual inner product and the norm in L 2 (0, L), respectively. In addition, we assume that the relaxation function g satisfies (A 1 ) g : [0, +∞) − → (0, +∞) is a decreasing C 1 -function such that There exist a nonincreasing C 1 -function ξ : [0, +∞) → [0, +∞) and a C 1 function that is linear or is strictly convex C 2 -function on (0, r], r ≤ g(0), with H(0) = H (0) = 0 such that andH is a strictly increasing convex C 2 -function. For example, for any t > r, we can defineH bȳ 2. From (A 1 ), g is continuous, positive and g(0) > 0, hence for any t 0 > 0, we obtain 3. Again, condition (A 2 ) implies that ξ and g are continuous, nonincreasing, and positive. Moreover, H is continuous and positive. Hence, ∀t ∈ [0, t 0 ], we obtain where b 1 and b 2 are some positive constants. Hence, it follows that Integration of (1. Solving (2.8) and using the initial data u 0 , u 1 , and θ 0 yield where It follows that Therefore, we can use Poincaré on u and θ . Hence, due to (2.10), we have the Poincaré inequality Moreover, (ũ, v, w,θ) satisfies problem (1.8) with initial data forũ andθ given as From now onward, we work with (ũ, v, w,θ ); however, for convenience, we write (u, v, w, θ ) keeping in mind Remark (2.2). Let us define the following spaces: The well-posedness of problem (1.8)-(1.10) is the following.
then the weak unique solution of problem (1.8)-(1.10) has more regularity in the class The result in Theorem 2.1 can be established using the Galerkin approximation method.
The following notations and basic lemmas will be applied repeatedly throughout the paper. We set For any 0 < α < 1, we set as in [27] ds.
Proof Using the Cauchy-Schwarz inequality, we have

Lemma 2.3 Let F be a convex function on the close interval
and j be an integrable function on such that j(x) ≥ 0 and j(x) dx = β 1 > 0. Then we have the following Jensen inequality:

Technical lemmas
In this section, we establish some lemmas which will be used to prove the main stability result.
Proof Multiplying the equations in (1.8) by u t , v t , w t , and θ , respectively, then integrating by parts over (0, L) and using the boundary conditions, we get respectively and (3.7) We estimate J 1 as follows: , we obtain (3.1). Hence, E is decreasing and bounded above by satisfies, for any positive 0 , the estimate Proof Differentiation of F 1 , using (1.8) 3 , integration by parts, and the boundary conditions give Proof By direct differentiation, then using (1.8) 1 , integration by parts, boundary condition (1.9) and keeping in mind (2.13), we get (3.13) Using the Cauchy-Schwarz, Young inequalities and applying Lemmas 2.1-2.2, we have (3.14) Now, substituting (3.14) into (3.13), using the fact that (k 1 -t 0 g(s) ds) ≥ l 0 , and choosing δ 1 = l 0 2 , we obtain the result.

Lemma 3.8 The functional F 7 defined by
satisfies, along the solution of (1.8)-(1.10), the estimate Proof By differentiating F 7 and recalling that J (t) = -g(t) and Recalling that J (t) = -g(t) ≤ 0, then J is nonincreasing, hence The result follows.
Proof Applying the Cauchy-Schwarz, Young, and Poincaré inequalities, we get (3.32) from routine calculations. Recalling that h = αgg , it follows from Lemmas 3.1-3.7 that, for any t ≥ t 0 , Now, we select the constants in (3.36) carefully. First, we choose N 6 so large that then we select N 2 large enough such that Next, we select N 3 large enough such that Now, we choose N 5 large enough so that We then select l and 1 small enough so that Recalling that h = αgg , we have αg 2 (s) h(s) = αg 2 (s) αg(s)-g (s) < g(s); thus application of dominated convergence theorem gives ds → 0 as α → 0.

Main decay result
Now, we state the main stability result of this paper.
Then the solution energy (3.2) satisfies for some positive constants M andM.