Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds

In this paper, we consider a one-dimensional linear Timoshenko system of thermoelasticity type III and prove a polynomial stability result for the non-equal wave-propagation speed case.

However, the stability in the case of only one damping in the second equation of (1.1) depends on the values of the constants ρ, I ρ , E, I and K . Precisely, if the wave-propagation speeds are equal K ρ = E I I ρ , then a uniform stability can be obtained for weak solutions. This has been showed by Soufyane and Wehbe [23], Ammar-Khodja et al. [1], Guesmia and Messaoudi [8,9], Messaoudi and Mustafa [12,14] and Messaoudi et al. [13], Fernández Sare and Rivera [4], Messaoudi and Said-Houari [16], Rivera and Racke [18,19] and Mustafa and Messaoudi [20]. Whereas, in the opposite case K ρ = E I I ρ , a weaker rate of decay is obtained for more regular solutions. In this regard, we quote, among others, the work of Fernández Sare and Rivera [4], Messaoudi and Said-Houari [16].
Regarding stabilization via heat effect, Rivera and Racke [17] considered the following system where ϕ, ψ, and θ are the transverse displacement of the beam, the rotation angle of the filament and the difference temperature, respectively. Under appropriate conditions of σ, ρ i , b, k, γ, they proved several exponential decay results for the linearized system and non-exponential stability result for the case of different wave speeds. In the above system, the heat flux is given by Fourier's law. As a result, this theory predicts an infinite speed of heat propagation. That is any thermal disturbance at one point has an instantaneous effect elsewhere in the body. Experiments showed that heat conduction in some dielectric crystals at low temperatures is free of this paradox and disturbances, which are almost entirely thermal, propagate in a finite speed. To overcome this physical paradox, many theories have merged such as thermoelasticity by second sound or thermoelasticity type III. For background related to this theory, we refer the reader to Green and Naghdi [5][6][7] and the review paper of Chandrasekharaiah [2].
Messaoudi and Said-Houari [15] considered the following Timoshenko-type system with thermoelasticity type III together with initial and boundary conditions, and showed, under the condition K ρ 1 = b ρ 2 , that weak solutions decay exponentially. In the present work, we consider (1.2) for the case K ρ 1 = b ρ 2 and prove a polynomial decay result for strong solutions.

Main result
In this section, we state and prove our main decay result. In order to exhibit the dissipative nature of System (1.2), we introduce the new variables φ = ϕ t and = ψ t , So, System (1.2) takes the form We supplement (2.1) with the following initial and boundary conditions From Eqs. (2.1) 1 , (2.1) 3 and (2.2), we easily verify that then simple substitution shows that (φ, ,θ) satisfies (2.1), the boundary conditions in (2.2), and more importantly In this case, Poincaré's inequality is applicable for θ and φ. In the sequel, we work with φ and θ but for convenience, we write φ and θ instead. Before stating and proving our stability result, we present a short discussion of the well-posedness and the semigroup formulation of (2.1) and (2.2). For this purpose, we introduce, as in [17], The energy space x L 2 (0,1) The proof can be carried similarly to [13].
To state our decay result, we introduce the first and second-order energy functionals: , then the strong solution (2.4) satisfies, for a positive constant k, independent of t and the initial data, the estimate The proof of our result will be established through several lemmas.

Lemma 2.4 Let
satisfies, for all ε 1 > 0, Proof By taking a derivative of I 1 and using (2.1) and (2.2), we conclude Using Young's inequality and ⎛ Proof By taking a derivative of I 2 and using (2.1) and (2.2), we get The assertion of the lemma then follows, using Young's and Poincaré's inequalities. (φ, , θ) be the strong solution of (2.1) and (2.2). Then, the functional
Proof of Theorem 2. 2 We define the Lyapunov functional L as follows A combination of (2.6)-(2.12) and use of 1 0 give where λ is a positive constant independent of N .
At this point, we choose our constants carefully. First, let us take N 1 large enough such that N 1 b 4 − 1 4 1 2 + b > 0, then pick ε 1 so small that 1 4 ρ 1 − N 1 ε 1 > 0. We then choose N 2 large enough so that Finally, we select ε 2 so small that Therefore (2.14) takes the form for some constant η > 0. Now, we handle the last term in the right-hand side of (2.15), using (2.1) 3 as follows: Multiplying by ρ 1 b K − ρ 2 , we get Therefore, recalling Young' s inequality and (2.13), we get, ∀ε 3 > 0, where C is a positive constant depending on δ, β, κ, ρ 3 only. We then define to get, from (2.15) and (2.16), . Using (2.5), choosing ε 3 small enough and taking N large enough so that L is positive and Nβκ − λ − C ε 3 > 0, (2.17) takes the form Simple integration, recalling that E 1 is non-increasing, leads to Consequently, This completes the proof.