The effect of the heat conduction of types I and III on the decay rate of the Bresse system via the longitudinal displacement

In this paper, we consider the thermoelastic Bresse system in one-dimensional bounded interval under mixed homogeneous Dirichlet–Neumann boundary conditions and two different kinds of dissipation working only on the longitudinal displacement and given by heat conduction of types I and III. We prove that the exponential stability of the two systems is equivalent to the equality of the three speeds of the wave propagations. Moreover, when at least two speeds of the wave propagations are different, we show the polynomial stability for each system with a decay rate depending on the smoothness of the initial data. The results of this paper complete the ones of Afilal et al. [On the uniform stability for a linear thermoelastic Bresse system with second sound (submitted), 2018], where the dissipation is given by a linear frictional damping or by the heat conduction of second sound. The proof of our results is based on the semigroup theory and a combination of the energy method and the frequency domain approach.

The authors of [13] considered the following system: and proved the exponential stability if and the polynomial stability in general. In [5], the authors proved that is exponentially stable if and only if (7) holds, and it is polynomially stable in general. The results of [5] were generalized in [15] to the case where δ and β are functions of x and vanish on some part of the domain. The authors of [9] proved that the following thermoelastic Bresse system is exponentially stable if = 0 and l is small, it is not exponentially stable if and it is polynomially stable in general. The author of [4] studied the stability of ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ where g : R + → R + is a given function satisfying some hypotheses. He provided a necessary and sufficient condition for exponential stability in terms of the structural parameters of the problem. For particular choices of g, the results of [4] cover the cases of Fourier, Cattaneo and Coleman-Gurtin heat conduction. For all the above stability results, at least the shear angle displacement ψ was damped via the heat conduction. The authors of [1] considered the Cattaneo heat conduction working only on the longitudinal displacement and proved that the exponential stability is equivalent to and Moreover, the polynomial stability of (10) in general was also proved in [1]. Similar stability results were proved in [1] when δθ x is replaced by δw t , the last two equations in (10) are neglected and (11) is replaced by (7). Our objective in this paper is to complete the results of [1] by considering the heat conduction of types I and III. We prove that, when l does not belong to two sequences of real numbers (conditions (15) and (24) below), the exponential stability of the two systems is equivalent to (7). Moreover, we show that the polynomial stability holds in general with two decay rates corresponding to the two cases, The proof of the well-posedness is based on the semigroup theory. However, the stability results are proved using the energy method combined with the frequency domain approach.
The paper is organized as follows. In Sect. 2, we give an idea on the proof of the well-posedness of (1)-(3) and (2)-(5). In Sects. 3 and 4, we prove, respectively, our exponential and polynomial stability results.

The semigroup setting
In this section, we give a brief idea on the proof of the well-posedness of (1)-( 3) and (2)- (5). We consider the energy space (1), 1) in case (4), We consider also (1), Systems (1)- (3) and (2)- (5) can be written as a first-order system given by where A is a linear operator defined by . The following well-posedness results for (14) hold: Then, for any m ∈ N and 0 ∈ D(A m ), System (14) admits a unique solution Proof First, from the definition of H 1 * (0, 1) and where c is a constant such that c = 0 or l ∈ π 2 + πN.
Then Condition (15) implies that ϕ = ψ = w = 0, and thus, H is a Hilbert space. Second, we prove that A is dissipative. Indeed, using the definition of A and ·, · H , and integrating by parts, we get 1) in case (4). (17) Hence, A is dissipative in H.
Third, we show that, for any F ∈ H, there exists Z ∈ D (A) satisfying . . , f j ) T and Z = (z 1 , . . . , z j ) T , where j = 7 in case (1), and j = 8 in case (4). The first, third and fifth equations in (18) are equivalent to and the seventh equation in case (4) becomes So, because F ∈ H, z 2 , z 4 , z 6 and z 8 have the required regularity in D (A). Then, the last equation in (18) is reduced to in case (4). By a direct integration, we see that each equation in (21) and (22) has a unique solution z 7 satisfying the needed regularity and Neumann boundary condition in D (A). Therefore, the second, fourth and sixth equations in (18) become To prove that (23) admits a solution (z 1 , z 3 , z 5 ) satisfying the required regularity and Neumann boundary condition in D (A), we consider the variational formulation of (23) and use the Lax-Milgram theorem and classical elliptic regularity arguments. So, this proves that (18) has a unique solution Z ∈ D (A). By the resolvent identity, we have λI − A is surjective, for any λ > 0 (see [14]). Consequently, the Lumer-Phillips theorem implies that A is the infinitesimal generator of a linear C 0 semigroup of contractions on H. Finally, Theorem 2.1 holds (see [16])

Exponential stability
Our objective in this section is to show the following exponential stability result: Theorem 3. 1 We assume that (15) holds. Then the semigroup associated with (14) is exponentially stable if and only if The proof is based on the following theorem:

contractions on a Hilbert space H generated by an operator A is exponentially stable if and only if
and sup Proof We prove that (24) is equivalent to (26), and (25) is equivalent to (27). So Theorem 3.2 implies Theorem 3.1.

Conditions (24) and (26) are equivalent
Note that, according to the fact that 0 ∈ ρ (A) (see Sect. 2), A −1 is bounded and it is a bijection between H and D(A). Since D(A) has a compact embedding into H, so it follows that A −1 is a compact operator, which implies that the spectrum of A is discrete. Then iλ ∈ ρ (A) if and only if λ is not an eigenvalue of A. Let λ ∈ R * . We prove that iλ is not an eigenvalue of A by proving that the unique solution ∈ D (A) of the equation is = 0. Let be given by (13). The Eq. (28) means that in case (4). Using (17) and (28), we find 1) in case (4).
Then (1), , then, using the Poincaré's inequality, (31) and the fourth equation in (30), we deduce that θ = 0 in case (1), Therefore, from (32) and the third and last equations in (29) and (30), we find As w,w ∈ ∼ H 1 * (0, 1) and according to Poincaré's inequality, we have Using (32) and (34), we see that (29) and (30) are reduced to Now, we follow the proof given in [1]. By deriving the fifth equation in (35) and combining the third one, we see that . We distinguish three cases.
for c 1 , c 2 ∈ C. Using the boundary conditions we find which implies that, using the first two equations and the last one in (35), Consequently, we get = 0. (41) Using again the boundary conditions (37), we find (38), and similarly to case 1, we arrive at (41).
Using the boundary conditions (37), we deduce that c 1 = 0, and If c 2 = 0, then (38) holds, and as before, we find (41). If c 2 = 0, then, by (42), we have Therefore, the fifth equation in (35) is equivalent to and then the third and fourth equations in (35) are reduced to We see that (43) and (45) lead to that is (24) does not hold. So, if (24) holds, we get a contradiction, and hence, c 2 = 0 and, as before, we find (41). If (24) does not hold, then, for λ ∈ R satisfying ( 45), the function (27) We assume that (25) holds and prove (27). Let us proceed by contradiction. So, we assume that (27) is false, then there exist sequences ( n ) n ⊂ D (A) and (λ n ) n ⊂ R satisfying and lim

Estimates on
∼ w n and λ n w n As in case (1) (Sect. 3.2.1), taking the inner product of (74) 6 with i ∼ w n λ n in L 2 (0, 1), integrating by parts and using the boundary conditions, we find (59) and (60). Estimate on ∼ ϕ n and conclusion The same computations as in case (1)

Condition (27) implies (25)
We prove this implication by contradiction. So, we assume that (25) does not hold and prove that (27) is not satisfied; that is we prove that there exists a sequence (λ n ) n ⊂ R such that which is equivalent to prove that there exists a sequence (F n ) n ⊂ H satisfying Then, from the second equality in (82), we have the following systems: in case (4). Choosing where N = (2n+1)π 2 and c is a constant satisfying 0 < |c| ≤ 1 √ ρ 2 , so On the other hand, the systems (83) and (84) become, respectively, Let us consider the choices where α 1 , . . . , α 4 are constants depending on N (will be fixed later). Then the last equation in (86) and the last one in (87) are equivalent to α 4 = μ n N α 3 , where where Because (25) is assumed to be not satisfied, then so we distinguish these two cases.

Polynomial stability
In this section, we prove the following polynomial stability independently from (25): Theorem 4.1 Assume that (15) and (24) hold. Then, for any m ∈ N * , there exists a constant c m > 0 such that, for any 0 ∈ D (A m ) and t > 0, The key of the proof of Theorem 4.1 is the following known theorem: Theorem 4.2 [12] If a bounded C 0 semigroup e tA on a Hilbert space H generated by an operator A satisfies (26) and, for some j ∈ N * , Then, for any m ∈ N * , there exists a positive constant c m such that
Hence, the proof of our Theorem 4.1 is completed.