Polynomial and non exponential stability of a weak dissipative Bresse system
aa r X i v : . [ m a t h . A P ] F e b POLYNOMIAL AND NON EXPONENTIAL STABILITY OF A WEAKDISSIPATIVE BRESSE SYSTEM
AISSA GUESMIA
Institut Elie Cartan de Lorraine, UMR 7502, Universit´e de Lorraine3 Rue Augustin Fresnel, BP 45112, 57073 Metz Cedex 03, France
Abstract.
In this paper, we study the Bresse system in a bounded domain with linear frictional dissi-pation working only on the veridical displacement. The longitudinal and shear angle displacements arefree. Our first main result is to prove that, independently from the velocities of waves propagations,this linear frictional dissipation does not stabilize exponentially the whole Bresse system. Our secondmain result is to show that the solution converges to zero at least polynomially. The proof of the well-posedness of our system is based on the semigroup theory. The stability results will be proved using acombination of the energy method and the frequency domain approach.
Keywords.
Bresse system, Frictional damping, Asymptotic behavior, Energy method,Frequency domain approach.
AMS Classification.
Introduction
The subject of this paper is studying the stability of Bresse system under linear frictional dampingeffective only on the the veridical displacement. This system is defined in (0 , × (0 , ∞ ) and takes theform(1.1) ρ ϕ tt − k ( ϕ x + ψ + l w ) x − lk ( w x − lϕ ) + δϕ t = 0 ,ρ ψ tt − bψ xx + k ( ϕ x + ψ + l w ) = 0 ,ρ w tt − k ( w x − lϕ ) x + lk ( ϕ x + ψ + l w ) = 0along with the initial data(1.2) ϕ ( x,
0) = ϕ ( x ) , ϕ t ( x,
0) = ϕ ( x ) in (0 , ,ψ ( x,
0) = ψ ( x ) , ψ t ( x,
0) = ψ ( x ) in (0 , ,w ( x,
0) = w ( x ) , w t ( x,
0) = w ( x ) in (0 , ( ϕ (0 , t ) = ψ x (0 , t ) = w x (0 , t ) = 0 in (0 , ∞ ) ,ϕ (1 , t ) = ψ x (1 , t ) = w x (1 , t ) = 0 in (0 , ∞ ) . The functions ϕ, ψ and w model, respectively, the vertical, shear angle and longitudinal displacementsof the filament. The coefficients ρ , ρ , b, k, k , l and δ are positive constants. The unique dissipationconsidered in (1 .
1) is played by the linear frictional damping δϕ t (it is well known that, when δ = 0, (1 . E-mail addresse: [email protected].
In case of bounded domain, it was proved that the exponential stabiliy holds under some restrictionson the velocities of waves propagations, and the plynomial stability is valid in general with a decay ratedepending on the regularity of initial data; see for example [2], [3], [5], [6], [7], [11], [13], [15], [16], [19],[20] and [22].However, when the domain is the whole line R , the situation is completely different in the sens thatno exponential stability result can be obtained and only polynomial stability results are proved on the L -norm of solutions (under some assumptions on the coefficients of the system) with a decay rate thatcan be improved by taking initial data more regular; see [8] (one frictional damping acting on the thirdequation) and [21] (two frictional dampings acting on the second and third equations). When the frictionaldamping is acting on the second equation, the authors of [8] proved that there is no decay of solutions atall.As far as we know, when only the veridical displacement (first equation) of Bresse system is dampedvia a frictional damping (the other two equations are totally free), the stability of Bresse system has neverbeen considered in the literature. Unlike the papers cited above concerned with the case of bounded do-main, we prove that, despite the presence of the linear frictional damping δϕ t , (1 .
1) is never exponentiallystable independently from the values of the coefficients of (1 . .
1) is at leastpolynomially stable with decay rates that can be improved by considering more smooth initial data.Our results show that the exponential stability of the overall Bresse system can not be guaranteedby a frictional dissipation working only in the veridical displacement. In comparaison with the knownresults cited above, this phenomenon means that Bresse system in a bounded domain is more dominatedby its longitudinal and shear angle displacements than by its veridical displacement.The paper is organized as follows: in section 2, we establish the existence, uniqueness and smoothnessof solutions of (1 . − (1 . . − (1 . . − (1 .
3) is given in section 4.2.
The semigroup setting
In this section, we study the existence, uniqueness and smoothness of solutions for (1 . − (1 .
3) usingsemigroup techniques. For this, let us considere the space H = H (0 , × L (0 , × H ∗ (0 , × L ∗ (0 , × H ∗ (0 , × L ∗ (0 , , where L ∗ (0 ,
1) = (cid:26) v ∈ L (0 , , Z v dx = 0 (cid:27) and H ∗ (0 ,
1) = H (0 , ∩ L ∗ (0 , , equipped with the inner product D ( ϕ , ˜ ϕ , ψ , ˜ ψ , w , ˜ w ) T , ( ϕ , ˜ ϕ , ψ , ˜ ψ , w , ˜ w ) T E H = k h ( ϕ x + ψ + l w ) , ( ϕ x + ψ + l w ) i L (0 , + k h ( w x − lϕ ) , ( w x − lϕ ) i L (0 , + b h ψ x , ψ x i L (0 , + ρ h ˜ ϕ , ˜ ϕ i L (0 , + ρ D ˜ ψ , ˜ ψ E L (0 , + ρ h ˜ w , ˜ w i L (0 , . Notice that, using the definition of H (0 ,
1) and H ∗ (0 , ϕ, ψ, w ) ∈ H (0 , × H ∗ (0 , × H ∗ (0 , k k ϕ x + ψ + l w k L (0 , + b k ψ x k L (0 , + k k w x − lϕ k L (0 , = 0 , then ψ = 0 , ϕ ( x ) = − c sin ( lx ) and w ( x ) = c cos ( lx ) , where c is a constant such that c = 0 or l = m π , for some m ∈ Z . Furthermore, by assuming that(2.1) l = mπ, ∀ m ∈ Z , TABILITY OF A WEAK DISSIPATIVE BRESSE SYSTEM 3 we get ϕ = ψ = w = 0, and so, H is a Hilbert space with respect to the generated norm (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ϕ, ˜ ϕ, ψ, ˜ ψ, w, ˜ w (cid:17) T (cid:13)(cid:13)(cid:13)(cid:13) H = k k ϕ x + ψ + l w k L (0 , + b k ψ x k L (0 , + k k w x − lϕ k L (0 , + ρ k ˜ ϕ k L (0 , + ρ k ˜ ψ k L (0 , + ρ k ˜ w k L (0 , . The definition of L ∗ (0 ,
1) allows to apply Poincar´e’s inequality in H ∗ (0 , Z v ( x ) dx = 0 , for v ∈ { ψ, w } can be assumed without lose of generality thanks to a classical change of variables; see,for example, Remark 2.1 of [9].Now, we consider the vectorsΦ = (cid:16) ϕ, ˜ ϕ, ψ, ˜ ψ, w, ˜ w (cid:17) T and Φ = ( ϕ , ϕ , ψ , ψ , w , w ) T , where ˜ ϕ = ϕ t , ˜ ψ = ψ t and ˜ w = w t . System (1 . − (1 .
3) can be formulated as the following first ordersystem:(2.2) ( Φ t = A Φ in (0 , ∞ ) , Φ (0) = Φ , where(2.3) A Φ = ˜ ϕkρ ( ϕ x + ψ + l w ) x + lk ρ ( w x − lϕ ) − δρ ˜ ϕ ˜ ψbρ ψ xx − kρ ( ϕ x + ψ + l w )˜ wk ρ ( w x − lϕ ) x − lkρ ( ϕ x + ψ + l w ) with domain D ( A ) = (cid:26) Φ ∈ H | ϕ ∈ H (0 , ∩ H (0 ,
1) ; ψ, w ∈ H (0 , ∩ H ∗ (0 ,
1) ;˜ ϕ ∈ H (0 ,
1) ; ˜ ψ, ˜ w ∈ H ∗ (0 ,
1) ; ψ x (0) = w x (0) = ψ x (1) = w x (1) = 0 (cid:27) . Theorem 2.1.
Assume that (2 . holds. Then, for any m ∈ N and Φ ∈ D ( A m ) , system (2 . admits aunique solution (2.4) Φ ∈ ∩ mj =0 C m − j (cid:0) R + ; D (cid:0) A j (cid:1)(cid:1) . Proof.
We remark that D ( A ) is dense in H . Now, direct calculation gives(2.5) hA Φ , Φ i H = − δ k ˜ ϕ k L (0 , ≤ . Hence, A is a dissipative operator.Next, we show that 0 ∈ ρ ( A ). Let F = ( f , · · · , f ) T ∈ H . We prove that there exists Z =( z , · · · , z ) T ∈ D ( A ) satisfying(2.6) A Z = F. Indeed, first, the first, third and fifth equations in (2 .
6) are equivalent to(2.7) z = f , z = f and z = f , and then(2.8) z ∈ H (0 ,
1) and z , z ∈ H ∗ (0 , . A. GUESMIA
Second, substitute z into the second equation in (2 . .
6) are reduced to(2.9) k ( z x + z + l z ) x + lk ( z x − lz ) = δf + ρ f ,bz xx − k ( z x + z + l z ) = ρ f ,k ( z x − lz ) x − lk ( z x + z + l z ) = ρ f . To prove that (2 .
9) admits a solution ( z , z , z ) satisfying(2.10) ( z ∈ H (0 , ∩ H (0 , , z , z ∈ H (0 , ∩ H ∗ (0 , ,z x (0) = z x (0) = z x (1) = z x (1) = 0 , we define the following bilinear form: a (( v , v , v ) , ( w , w , w )) = k h v x + v + lv , w x + w + lw i L (0 , + b h v x , w x i L (0 , + k h v x − lv , w x − lw i L (0 , , ∀ ( v , v , v ) T , ( w , w , w ) T ∈ H , and the following linear form: L ( v , v , v ) = h δf + ρ f , v i L (0 , + h ρ f , v i L (0 , + h ρ f , v i L (0 , , ∀ ( v , v , v ) T ∈ H , where H = H (0 , × H ∗ (0 , × H ∗ (0 , . Thus, the variational formulation of (2 .
9) is given by(2.11) a (( z , z , z ) , ( w , w , w )) = L ( w , w , w ) , ∀ ( w , w , w ) T ∈ H . From the Lax-Milgram theorem, it follows that (2 .
11) has a unique solution( z , z , z ) ∈ H . Therefore, using classical elliptic regularity arguments, we conclude that ( z , z , z ) solves (2 .
9) andsatisfies the regularity and boundary conditions (2 . .
6) has a unique solution Z ∈ D ( A ). By the resolvent identity, we have λI −A is surjective, for any λ > I denotesthe identity operator. Consequently, the Lumer-Phillips theorem implies that A is the infinitesimalgenerator of a linear C semigroup of contractions on H . So, Theorem 2.1 holds (see [17]). (cid:3) The proof of the non-exponential and polynomial stability for (2 .
2) is based on the following twofrequency domain theorems:
Theorem 2.2. ( [10] and [18] ) A C semigroup of contractions on a Hilbert space H generated by anoperator A is exponentially stable if and only if (2.12) i R ⊂ ρ ( A ) and sup λ ∈ R (cid:13)(cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13)(cid:13) L ( H ) < ∞ . Theorem 2.3. ( [12] ) If a bounded C semigroup e t A on a Hilbert space H generated by an operator A satisfies, for some j ∈ N ∗ , (2.13) i R ⊂ ρ ( A ) and sup | λ |≥ λ j (cid:13)(cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13)(cid:13) L ( H ) < ∞ . Then, for any m ∈ N ∗ , there exists a positive constant c m such that (2.14) (cid:13)(cid:13) e t A z (cid:13)(cid:13) H ≤ c m k z k D ( A m ) (cid:18) ln tt (cid:19) mj ln t, ∀ z ∈ D ( A m ) , ∀ t > . TABILITY OF A WEAK DISSIPATIVE BRESSE SYSTEM 5 Lack of exponential stability of (1 . − (1 . Theorem 3.1.
Assume that (2 . holds. Then the semigroup associated with (2 . is not exponentiallystable.Proof. Using Theorem 2.2, it is enough to prove that the second condition in (2 .
12) is not satisfied. Todo so, we prove that there exists a sequence ( λ n ) n ⊂ R such thatlim n →∞ (cid:13)(cid:13)(cid:13) ( iλ n I − A ) − (cid:13)(cid:13)(cid:13) L ( H ) = ∞ , which is equivalent to find a sequence ( F n ) n ⊂ H satisfying(3.1) k F n k H ≤ , ∀ n ∈ N and(3.2) lim n →∞ k ( iλ n I − A ) − F n k H = ∞ . For this purpose, let Φ n = ( iλ n I − A ) − F n , ∀ n ∈ N , where Φ n = (cid:16) ϕ n , ˜ ϕ n , ψ n , ˜ ψ n , w n , ˜ w n (cid:17) T and F n = ( f n , · · · , f n ) T . Then we have to find ( λ n ) n ⊂ R , ( F n ) n ⊂ H and (Φ n ) n ⊂ D ( A ) satisfying (3 . n →∞ k Φ n k H = ∞ and iλ n Φ n − A Φ n = F n , ∀ n ∈ N . The equation in (3 .
3) is equivalent to(3.4) iλ n ϕ n − ˜ ϕ n = f n ,iρ λ n ˜ ϕ n − k ( ϕ nx + ψ n + l w n ) x − lk ( w nx − lϕ n ) + δ ˜ ϕ n = ρ f n ,iλ n ψ n − ˜ ψ n = f n ,iρ λ n ˜ ψ n − bψ nxx + k ( ϕ nx + ψ n + l w n ) = ρ f n ,iλ n w n − ˜ w n = f n ,iρ λ n ˜ w n − k ( w nx − lϕ n ) x + lk ( ϕ nx + ψ n + l w n ) = ρ f n . Choosing(3.5) f n = f n = f n = 0 . Then system (3 .
4) becomes(3.6) ˜ ϕ n = iλ n ϕ n , ˜ ψ n = iλ n ψ n , ˜ w n = iλ n w n , (cid:0) iδλ n − ρ λ n (cid:1) ϕ n − k ( ϕ nx + ψ n + l w n ) x − lk ( w nx − lϕ n ) = ρ f n , − ρ λ n ψ n − bψ nxx + k ( ϕ nx + ψ n + l w n ) = ρ f n , − ρ λ n w n − k ( w nx − lϕ n ) x + lk ( ϕ nx + ψ n + l w n ) = ρ f n . To simplify the calculations, we put N = nπ . Some of the computations below were done in [1]. Now,we consider three cases. Case 1 : bρ = k ρ . We choose(3.7) ϕ n = ˜ ϕ n = 0 ,ψ n ( x ) = α cos ( N x ) , ˜ ψ n ( x ) = iα λ n cos ( N x ) ,w n ( x ) = α cos ( N x ) , ˜ w n ( x ) = iα λ n cos ( N x ) , A. GUESMIA (3.8) f n = 0 , f n ( x ) = − lk ρ α cos ( N x ) , f n ( x ) = − l k ρ α cos ( N x )and(3.9) λ n = N s k ρ , where α , α ∈ R . We have Φ n ∈ D ( A ) and F n ∈ H . On the other hand, (3 .
6) is satisfied if and only if(3.10) kα + l ( k + k ) α = 0 , (cid:20) − λ n + bρ N + kρ (cid:21) α + lkρ α = − lk ρ α ,lkρ α + (cid:20) − λ n + k ρ N + l kρ (cid:21) α = − l k ρ α . According to (3 .
9) and because bρ = k ρ , we have − λ n + k ρ N = − λ n + bρ N = 0 , and therefore, the system (3 .
10) is equivalent to(3.11) α = − l (cid:18) k k (cid:19) α . Choosing α = ρ ρ lk p ρ + l ρ and using (3 .
5) and (3 . k F n k H = k f n k L (0 , + k f n k L (0 , = (cid:18) lk ρ (cid:19) " (cid:18) lρ ρ (cid:19) α Z cos ( N x ) dx ≤ (cid:18) lk ρ (cid:19) " (cid:18) lρ ρ (cid:19) α = 1 . On the other hand, we have k Φ n k H ≥ k k w nx − lϕ n k L (0 , = k k w nx k L (0 , = k α N Z [1 − cos (2 N x )] dx = k α N , hence(3.12) lim n →∞ k Φ n k H = ∞ . Case 2 : bρ = k ρ and k = k . We choose(3.13) f n = f n = 0 , f n ( x ) = cos ( N x ) , (3.14) ϕ n ( x ) = α sin ( N x ) , ˜ ϕ n ( x ) = iα λ n sin ( N x ) ,ψ n ( x ) = α cos ( N x ) , ˜ ψ n ( x ) = iα λ n cos ( N x ) ,w n ( x ) = α cos ( N x ) , ˜ w n ( x ) = iα λ n cos ( N x )and(3.15) λ n = s k ρ N + l kρ , TABILITY OF A WEAK DISSIPATIVE BRESSE SYSTEM 7 where α , α , α ∈ C depending on N . Notice that, according to these choices, Φ n ∈ D ( A ), F n ∈ H and(3.16) k F n k H = k f n k L (0 , = Z cos ( N x ) dx ≤ . On the other hand, thanks to (3 . .
13) and (3 . .
6) are satisfied, andthe last three ones are equivalent to(3.17) (cid:2) ( k − µ n ) N − ρ λ n + l k (cid:3) α + kN α + l ( k + k ) N α = 0 ,kN α + (cid:0) bN − ρ λ n + k (cid:1) α + klα = 0 ,l ( k + k ) N α + lkα + (cid:0) k N − ρ λ n + l k (cid:1) α = ρ , where we note(3.18) µ n = − iδλ n N . From the choice (3 . .
17) is equivalent to(3.19) α = − k + k k N α + ρ lk , so, substituting in the first two equations in (3 . α = a N α + a and(3.21) α = h l ( k + k ) a + ρ l i N [2 k + µ n − l ( k + k ) a ] N + l ( k − k ) , where a = k + k lk (cid:18) b − ρ k ρ (cid:19) N + k lk − lρ ( k + k ) ρ k ,a = ρ ( lk ) (cid:20)(cid:18) ρ k ρ − b (cid:19) N + l ρ kρ − k (cid:21) . To simplify the computations, we put a = ρ ( k + k ) lk (cid:18) ρ k ρ − b (cid:19) , a = ( k + k ) k (cid:18) ρ k ρ − b (cid:19) ,a = lρ ( k + k ) k − k ρ lk , a = l ρ ( k + k ) ρ k + k ( k − k ) k and d = k + k lk (cid:18) b − ρ k ρ (cid:19) , d = ρ ( lk ) (cid:18) ρ k ρ − b (cid:19) ,d = k lk − lρ ( k + k ) ρ k , d = ρ l k (cid:18) l ρ ρ − (cid:19) . Then
N α = a N + a N a N + ( µ n + a ) N + l ( k − k )and (notice that d a + d a = 0)(3.22) α = (cid:0) d N + d (cid:1) (cid:0) a N + a N (cid:1) a N + ( µ n + a ) N + l ( k − k ) + d N + d = ( d a + d a + d a + d a + d µ n ) N + (cid:0) d a + d a + l ( k − k ) d + d µ n (cid:1) N + l ( k − k ) d a N + ( µ n + a ) N + l ( k − k ) , A. GUESMIA
Because bρ = k ρ and k = k , then a = 0 and(3.23) d a + d a + d a + d a = ρ ( lk ) (cid:18) ρ k ρ − b (cid:19) ( k − k ) = 0 . On the other hand,(3.24) lim n →∞ µ n = lim n →∞ − iδλ n N = lim n →∞ − iδN s k ρ N + l kρ = 0 . Then we deduce from (3 . .
23) and (3 .
24) that(3.25) lim n →∞ α = d a + d a + d a + d a a = 0 , hence(3.26) lim n →∞ | α | λ n = ∞ . Now, we have k Φ n k H ≥ ρ k ˜ w n k L (0 , = ρ ( | α | λ n ) Z cos ( N x ) dx = ρ | α | λ n ) Z [1 + cos (2 N x )] dx = ρ | α | λ n ) , then by (3 .
26) we get (3 . Case 3 : bρ = k ρ and k = k . We consider the choices (3 . λ n = s bρ N + k ρ , (3.28) f n = 0 , f n ( x ) = α C n cos ( N x ) , f n ( x ) = α D n cos ( N x )and (3 .
14) with(3.29) α = (cid:18) ρ D n lk − (cid:19) α N and α = 0 , where C n = ρ lρ D n and D n = 2 lkρ − kk + l kN − µ n − ρ λ n N ! . According to (3 .
18) and (3 . n →∞ µ n = 0, and thenlim n →∞ D n = 2 lkρ − kk − ρ bρ ! and lim n →∞ C n = kρ − kk − ρ bρ ! (these limits exist since bρ = k ρ and k = k ), so, the sequence (cid:0) | C n | + | D n | (cid:1) n is bounded. Then wechoose(3.30) α = 1 p sup n ∈ N ( | C n | + | D n | ) . According to these choices, we see that Φ n ∈ D ( A ), F n ∈ H and, using (3 . .
28) and (3 . k F n k H = k f n k L (0 , + k f n k L (0 , = (cid:0) | C n | + | D n | (cid:1) α Z cos ( N x ) dx ≤ (cid:0) | C n | + | D n | (cid:1) α ≤ . On the other hand, thanks to (3 . .
14) and (3 . .
6) are satisfied, andbecause k = k and α = 0, the last three equations in (3 .
6) are equivalent to(3.31) (cid:2) ( k − µ n ) N − ρ λ n + l k (cid:3) α + kN α = 0 ,kN α + (cid:0) bN − ρ λ n + k (cid:1) α = ρ α C n , lkN α + lkα = ρ α D n . TABILITY OF A WEAK DISSIPATIVE BRESSE SYSTEM 9
The first equation in (3 .
31) is satisfied thanks to the definition of α and D n , the second equation in(3 .
31) holds according to the definition of λ n , α and C n , and the last equation in (3 .
31) is satisfied fromthe definition of α .Now, we have k Φ n k H ≥ ρ (cid:13)(cid:13)(cid:13) ˜ ψ n (cid:13)(cid:13)(cid:13) L (0 , = ρ ( α λ n ) Z cos ( N x ) dx = ρ α λ n ) Z [1 + cos (2 N x )] dx = ρ α λ n ) , consequently, (3 .
12) holds.Finally, there exist sequences ( F n ) n ⊂ H , (Φ n ) n ⊂ D ( A ) and ( λ n ) n ⊂ R satisfying (3 .
1) and (3 . .
2) is not exponentially stable. (cid:3) Polynomial stability of (1 . − (1 . .
2) is polynomially stable. Oursecond main result is stated as follow:
Theorem 4.1.
Assume that l satisfies (2 . and (4.1) l = k ρ − bρ k ρ ( mπ ) − kρ ρ ( k + k ) , ∀ m ∈ Z . Then, for any m ∈ N ∗ , there exists a constant c m > such that (4.2) ∀ Φ ∈ D ( A m ) , ∀ t ≥ , (cid:13)(cid:13) e t A Φ (cid:13)(cid:13) H ≤ c m k Φ k D ( A m ) (cid:18) ln tt (cid:19) m t. Proof.
Using Theorem 2.3, it is sufficient to show that(4.3) i IR ⊂ ρ ( A )and(4.4) sup | λ | ≥ λ (cid:13)(cid:13)(cid:13) ( iλI − A ) − (cid:13)(cid:13)(cid:13) H < ∞ . We start by proving (4 . ∈ ρ ( A ) (section 2), A − is boundedand it is a bijection between H and D ( A ). Since D ( A ) has a compact embedding into H , so, it followsthat A − is a compact operator, which implies that the spectrum of A is discrete.Let λ ∈ R ∗ and Φ = (cid:16) ϕ, ˜ ϕ, ψ, ˜ ψ, w, ˜ w (cid:17) T ∈ D ( A ) . We prove that iλ is not an eigenvalue of A by proving that the equation(4.5) A Φ = i λ
Φhas only Φ = 0 as a solution. Assume that (4 .
5) is true, then we have(4.6) ˜ ϕ = iλϕ, ˜ ψ = iλψ, ˜ w = iλw,kρ ( ϕ x + ψ + l w ) x + lk ρ ( w x − lϕ ) − δρ ˜ ϕ = iλ ˜ ϕ,bρ ψ xx − kρ ( ϕ x + ψ + l w ) = iλ ˜ ψ,k ρ ( w x − lϕ ) x − lkρ ( ϕ x + ψ + l w ) = iλ ˜ w. Using (2 . Re iλ k Φ k H = Re h iλ Φ , Φ i H = Re hA Φ , Φ i H = − δ k ˜ ϕ k L (0 , . So, by the first equation in (4 . ϕ = ˜ ϕ = 0 . Using (4 . .
6) leads to(4.8) ˜ ψ = iλψ, ˜ w = iλw,kψ x + l ( k + k ) w x = 0 ,bψ xx − k ( ψ + l w ) = − ρ λ ψ,k w xx − lk ( ψ + l w ) = − ρ λ w. The third equation in (4 .
8) implies that kψ + l ( k + k ) w is a constant, then, thanks to the definition of L ∗ (0 , ψ = − l (cid:18) k k (cid:19) w. Using the last two equations in (4 . lbψ xx − k w xx = − ρ lλ ψ + ρ λ w. Then, combining with (4 . w xx + α λ w = 0 , where(4.11) α = s ρ l ( k + k ) + kρ bl ( k + k ) + kk . This implies that, for c , c ∈ C , w ( x ) = c cos ( αλx ) + c sin ( αλx ) . The boundary condition w x (0) = 0 leads to c = 0, and then, using (4 . ψ ( x ) = − l (cid:18) k k (cid:19) c cos ( αλx ) and w ( x ) = c cos ( αλx ) . Because ψ x (1) = w x (1) = 0, we have c = 0 or ∃ m ∈ Z : αλ = mπ. Assume by contradiction that(4.13) ∃ m ∈ Z : αλ = mπ. Therefore, using (4 .
11) and (4 . .
8) are equivalent to(4.14) ( k ρ − bρ ) λ = k k + k (cid:2) bl ( k + k ) + kk (cid:3) . So, combining (4 . .
13) and (4 . ∃ m ∈ Z : l = k ρ − bρ k ρ ( mπ ) − kρ ρ ( k + k ) , which is a contraduction with (4 . c = 0 and hence(4.15) ψ = w = 0 . Using (4 .
15) and the first two equations in (4 . ψ = ˜ w = 0 . Finally, Φ = 0 and thus(4.16) iλ ∈ ρ ( A ) . This ends the proof of (4 . .
4) by contradiction. Assume that (4 .
4) is false, then there exist sequences(Φ n ) n ⊂ D ( A ) and ( λ n ) n ⊂ R satisfying(4.17) k Φ n k H = 1 , ∀ n ∈ N , TABILITY OF A WEAK DISSIPATIVE BRESSE SYSTEM 11 (4.18) lim n →∞ | λ n | = ∞ and(4.19) lim n →∞ λ n k ( iλ n I − A ) Φ n k H = 0 . Let Φ n = (cid:18) ϕ n , ∼ ϕ n , ψ n , ∼ ψ n , w n , ∼ w n (cid:19) T . The limit (4 .
19) implies that(4.20) λ n h iλ n ϕ n − ∼ ϕ n i → H (0 , ,λ n h iλ n ρ ∼ ϕ n − k ( ϕ nx + ψ n + lw n ) x − lk ( w nx − lϕ n ) + δ ˜ ϕ n i → L (0 , ,λ n (cid:20) iλ n ψ n − ∼ ψ n (cid:21) → H ∗ (0 , ,λ n (cid:20) iλ n ρ ∼ ψ n − bψ nxx + k ( ϕ nx + ψ n + lw n ) (cid:21) → L ∗ (0 , ,λ n h iλ n w n − ∼ w n i → H ∗ (0 , ,λ n h iλ n ρ ∼ w n − k ( w nx − lϕ n ) x + lk ( ϕ nx + ψ n + lw n ) i → L ∗ (0 , . We will prove that k Φ n k H →
0, which gives a contradiction with (4 . Step 1.
Using (2 . Re (cid:10) λ n ( i λ n I − A ) Φ n , Φ n (cid:11) H = Re (cid:16) iλ n k Φ n k H − λ n hA Φ n , Φ n i H (cid:17) = δλ n k ˜ ϕ n k L (0 , . So, (4 . .
18) and (4 .
19) imply that(4.21) λ n ˜ ϕ n −→ L (0 , . Step 2.
Multiplying (4 . by 1 λ n , and using (4 .
18) and (4 . λ n ϕ n −→ L (0 , . Step 3.
Using an integration by parts, (4 .
18) and (4 . , we see that Dh iλ n ρ ∼ ϕ n − kψ nx − l ( k + k ) w nx + l k ϕ n + δ ˜ ϕ n i , λ n ϕ n E L (0 , + kλ n k ϕ nx k L (0 , −→ , so, using (4 . .
21) and (4 . λ n ϕ nx −→ L (0 , . Moreover, because ϕ n ∈ H (0 , λ n ϕ n −→ L (0 , , and by (4 . and (4 . ∼ ϕ nx → L (0 , . Step 4.
Multiplying (4 . and (4 . by 1 λ n , and using (4 .
17) and (4 . ψ n −→ L (0 ,
1) and w n −→ L (0 , . Step 5.
Taking the inner product of (4 . with 1 λ n [ kψ nx + l ( k + k ) w nx ] in L (0 ,
1) and using(4 . ρ D ( iρ λ n + δ ) ∼ ϕ n , [ kψ nx + l ( k + k ) w nx ] E L (0 , − k h ϕ nxx , [ kψ nx + l ( k + k ) w nx ] i L (0 , − k kψ nx + l ( k + k ) w nx k L (0 , + l k h ϕ n , [ kψ nx + l ( k + k ) w nx ] i L (0 , → . Integrating by parts and using the boundary conditions, we get(4.28) h ϕ nxx , [ kψ nx + l ( k + k ) w nx ] i L (0 , = − (cid:28) λ n ϕ nx , (cid:20) k ψ nxx λ n + l ( k + k ) w nxx λ n (cid:21)(cid:29) L (0 , . On the other hand, multiplying (4 . and (4 . by 1 λ n and using (4 . iρ ∼ ψ n − b ψ nxx λ n + kλ n ( ϕ nx + ψ n + lw n ) → L (0 , ,iρ ∼ w n − k w nxx λ n + lk ϕ nx λ n + lkλ n ( ϕ nx + ψ n + lw n ) → L (0 , . Exploiting (4 . (cid:18) λ n ψ nxx (cid:19) n and (cid:18) λ n w nxx (cid:19) n are bounded in L (0 , , then, using (4 . . .
28) and (4 . h ϕ nxx , [ kψ nx + l ( k + k ) w nx ] i L (0 , → , so, exploiting (4 . . .
21) and (4 . kψ nx + l ( k + k ) w nx → L (0 , . Step 6.
Taking the inner product of (4 . with ψ n λ n in L (0 , . − ρ (cid:28) ∼ ψ n , (cid:18) iλ n ψ n − ∼ ψ n (cid:19)(cid:29) L (0 , − ρ (cid:13)(cid:13)(cid:13)(cid:13) ∼ ψ n (cid:13)(cid:13)(cid:13)(cid:13) L (0 , + b k ψ nx k L (0 , + k h ( ϕ nx + ψ n + lw n ) , ψ n i L (0 , → , then, using (4 . . . and (4 . b k ψ nx k L (0 , − ρ (cid:13)(cid:13)(cid:13)(cid:13) ∼ ψ n (cid:13)(cid:13)(cid:13)(cid:13) L (0 , → . On the other hand, taking the inner product of (4 . with w n λ n in L (0 , . − ρ D ∼ w n , (cid:16) iλ n w n − ∼ w n (cid:17)E L (0 , − ρ (cid:13)(cid:13)(cid:13) ∼ w n (cid:13)(cid:13)(cid:13) L (0 , + k k w nx k L (0 , + lk h ϕ nx , w n i L (0 , + lk h ( ϕ nx + ψ n + lw n ) , w n i L (0 , → . By (4 . . . and (4 . k k w nx k L (0 , − ρ (cid:13)(cid:13)(cid:13) ∼ w n (cid:13)(cid:13)(cid:13) L (0 , → . TABILITY OF A WEAK DISSIPATIVE BRESSE SYSTEM 13
Step 7.
Taking the inner product of (4 . with w n λ n and of (4 . with ψ n λ n in L (0 , . (cid:28)(cid:20) iλ n ρ ∼ ψ n − bψ nxx + k ( ϕ nx + ψ n + lw n ) (cid:21) , w n (cid:29) L (0 , → , Dh iλ n ρ ∼ w n − k ( w nx − lϕ n ) x + lk ( ϕ nx + ψ n + lw n ) i , ψ n E L (0 , → . Integrating by parts and using the boundary conditions, we obtain − ρ (cid:28) ∼ ψ n , (cid:16) iλ n w n − ∼ w n (cid:17)(cid:29) L (0 , − ρ (cid:28) ∼ ψ n , ∼ w n (cid:29) L (0 , + b h ψ nx , w nx i L (0 , + k h ( ϕ nx + ψ n + lw n ) , w n i L (0 , → − ρ (cid:28) ∼ w n , (cid:18) iλ n ψ n − ∼ ψ n (cid:19)(cid:29) L (0 , − ρ (cid:28) ∼ w n , ∼ ψ n (cid:29) L (0 , − lk h ϕ n , ψ nx i L (0 , + k h w nx , ψ nx i L (0 , + lk h ( ϕ nx + ψ n + lw n ) , ψ n i L (0 , → , then, using (4 . . . , (4 . , (4 .
22) and (4 . − ρ (cid:28) ∼ ψ n , ∼ w n (cid:29) L (0 , + b h ψ nx , w nx i L (0 , → , − ρ (cid:28) ∼ ψ n , ∼ w n (cid:29) L (0 , + k h ψ nx , w nx i L (0 , → , which implies that(4.34) (cid:18) ρ b − ρ k (cid:19) (cid:28) ∼ ψ n , ∼ w n (cid:29) L (0 , → (cid:18) bρ − k ρ (cid:19) h ψ nx , w nx i L (0 , → . Step 8.
We distinguish in this step two cases.
Case 1: bρ = k ρ . From (4 .
34) and (4 . (cid:28) ∼ ψ n , ∼ w n (cid:29) L (0 , → h ψ nx , w nx i L (0 , → . Therefore, taking the inner product in L (0 ,
1) of (4 . ψ nx , and second, with w nx , we obtain(4.37) ψ nx → w nx → L (0 , , and then, by (4 . .
33) and (4 . ∼ ψ n → ∼ w n → L (0 , . Finally, combining (4 . . . . .
37) and (4 . k Φ n k H −→ , which is a contradiction with (4 . .
4) holds. Consequentely, (4 .
2) is satisfied.
Case 2: bρ = k ρ . Using (4 . and (4 . , we obtain λ n (cid:20) − iρ b λ n (cid:16) iλ n ψ n − ˜ ψ n (cid:17) − ρ b λ n ψ n − ψ nxx + kb ( ϕ nx + ψ n + lw n ) (cid:21) → L (0 , ,λ n (cid:20) − iρ b λ n ( iλ n w n − ˜ w n ) − ρ b λ n w n − ( w nx − lϕ n ) x + lkk ( ϕ nx + ψ n + lw n ) (cid:21) → L (0 , , so, using (4 . and (4 . , we find(4.40) λ n (cid:20) − ρ b λ n ψ n − ψ nxx + kb ( ϕ nx + ψ n + lw n ) (cid:21) → L (0 , ,λ n (cid:20) − ρ b λ n w n − ( w nx − lϕ n ) x + lkk ( ϕ nx + ψ n + lw n ) (cid:21) → L (0 , . Then, using (4 . .
23) and (4 . ρ b λ n ψ n + ψ nxx → L (0 ,
1) and ρ b λ n w n + w nxx → L (0 , . Multiplying (4 . by k and (4 . by l ( k + k ) and adding the obtained limits, and multiplying (4 . by k and (4 . by − l ( k + k ) and adding the limits, we obtain(4.42) ρ b λ n [ kψ n + l ( k + k ) w n ] + [ kψ nxx + l ( k + k ) w nxx ] → L (0 , ,ρ b λ n [ kψ n − l ( k + k ) w n ] + [ kψ nxx − l ( k + k ) w nxx ] → L (0 , . Taking the inner product in L (0 ,
1) of (4 . and (4 . with [ kψ n + l ( k + k ) w n ], integrating by partsand using the boundary conditions, we get ρ b k kλ n ψ n + l ( k + k ) λ n w n k L (0 , − k kψ nx + l ( k + k ) w nx k L (0 , → ρ b (cid:10) λ n [ kψ n − l ( k + k ) w n ] , [ kψ n + l ( k + k ) w n ] (cid:11) L (0 , − h [ kψ nx − l ( k + k ) w nx ] , [ kψ nx + l ( k + k ) w nx ] i L (0 , → , then, using (4 .
17) and (4 . ( kλ n ψ n + l ( k + k ) λ n w n → L (0 , ,k k λ n ψ n k L (0 , − l ( k + k ) k λ n w n k L (0 , → . Taking the inner product in L (0 ,
1) of (4 . with w n , and (4 . with ψ n , integrating by parts andusing the boundary conditions, we get(4.44) − ρ b λ n h ψ n , w n i L (0 , + λ n h ψ nx , w nx i L (0 , − kb (cid:10) λ n ϕ n , w nx (cid:11) L (0 , + kb h λ n ψ n , λ n w n i L (0 , + lkb k λ n w n k L (0 , → − ρ b λ n h ψ n , w n i L (0 , + λ n h ψ nx , w nx i L (0 , − l (cid:18) kk (cid:19) (cid:10) ψ nx , λ n ϕ n (cid:11) L (0 , + lkk k λ n ψ n k L (0 , + l kk h λ n ψ n , λ n w n i L (0 , → , then, multiplying (4 .
44) by bk k , and (4 .
45) by − bk k , adding the obtained limits and using (4 .
17) and(4 . lk k λ n w n k L (0 , − lb k λ n ψ n k L (0 , + (cid:0) k − l b (cid:1) h λ n ψ n , λ n w n i L (0 , → . Multiplying (4 . and (4 . by 1 λ n , and using (4 .
17) and (4 . λ n ψ n ) n and ( λ n w n ) n are bounded in L (0 , . So, by taking the inner product in L (0 ,
1) of (4 . with λ n ψ n , and using (4 . k k λ n ψ n k L (0 , + l ( k + k ) h λ n w n , λ n ψ n i L (0 , → . TABILITY OF A WEAK DISSIPATIVE BRESSE SYSTEM 15
Combining (4 . and (4 . l ( k + k ) (cid:2) k k − bl ( k + k ) (cid:3) k λ n ψ n k L (0 , + (cid:0) k − l b (cid:1) h λ n w n , λ n ψ n i L (0 , → , so, multiplying (4 .
48) by ( k + k ) (cid:0) k − l b (cid:1) k , and (4 .
49) by − l ( k + k ) k , adding the obtained limits andnoting that bρ = k ρ , we obtain (cid:2) kk + bl ( k + k ) (cid:3) k λ n ψ n k L (0 , → . Then(4.50) λ n ψ n → L (0 , . ,(4.51) λ n w n → L (0 , . Using (4 . . , (4 . , (4 .
50) and (4 . ∼ ψ n → L (0 ,
1) and ∼ w n → L (0 , . Taking the inner product in L (0 ,
1) of (4 . with ψ n , and (4 . with w n , integrating by parts andusing the boundary conditions, we get ρ b k λ n ψ n k L (0 , − k ψ nx k L (0 , → ρ b k λ n w n k L (0 , − k w nx k L (0 , → , then, from (4 .
50) and (4 . ψ nx → L (0 ,
1) and w nx → L (0 , . Finally, (4 . . . . .
52) and (4 .
53) imply (4 . . bρ = k ρ and bρ = k ρ , (4 .
4) holds, and so (4 .
2) is satisfied. Hence, theproof of Theorem 4.1 is completed. (cid:3)
References [1] M. Afilal, A. Guesmia and A. Soufyane, On the exponential and polynomial stability for a linear Bresse system, Math.Meth. Appl. Scie., 43 (2020), 2626-2645.[2] F. Alabau-Boussouira, J. E. Mu˜noz Rivera and D. S. Almeida J´unior, Stability to weak dissipative Bresse system, J.Math. Anal. Appl., 374 (2011), 481-498.[3] M. O. Alves, L. H. Fatori, M. A. Jorge Silva and R. N. Monteiro, Stability and optimality of decay rate for weaklydissipative Bresse system, Math. Meth. Appl. Sci., 38 (2015), 898-908.[4] J. A. C. Bresse, Cours de M´ecanique Appliqu´ee, Mallet Bachelier, Paris, 1859.[5] W. Charles, J. A. Soriano, F. A. Nascimentoc and J. H. Rodrigues, Decay rates for Bresse system with arbitrarynonlinear localized damping, J. Diff. Equa., 255 (2013), 2267-2290.[6] L. H. Fatori and R. N. Monteiro, The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25(2012), 600-604.[7] L. H. Fatori and J. M. Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010),881-904.[8] T. E. Ghoula, M. Khenissi and B. Said-Houari, On the stability of the Bresse system with frictional damping, J. Math.Anal. Appl., 455 (2017), 1870-1898.[9] A. Guesmia and M. Kirane, Uniform and weak stability of Bresse system with two infinite memories, ZAMP, 67 (2016),1-39.[10] F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff.Equa., 1 (1985), 43-56.[11] A. Keddi, T. Apalara and S. Messaoudi, Exponential and polynomial decay in a thermoelastic-Bresse system withsecond sound, Appl. Math. Optim., 77 (2018), 315-341.[12] Z. Liu and B. Rao, Characterization of polymomial decay rate for the solution of linear evolution equation, Z. Angew.Math. Phys., 56 (2005), 630-644.[13] Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69.[14] Z. Liu and S. Zheng, Semigroups associated with dissipative systems, 398 Research Notes in Mathematics, Chapmanand Hall/CRC, 1999.[15] N. Najdi and A. Wehbe, Weakly locally thermal stabilization of Bresse systems, Elec. J. Diff. Equa., 2014 (2014), 1-19. [16] N. Noun and A. Wehbe, Weakly locally internal stabilization of elastic Bresse system, C. R. Acad. Scie. Paris, S´er. I,350 (2012), 493-498.[17] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York,1983.[18] J. Pruss, On the spectrum of C0