Attractors for locally damped Bresse systems and a unique continuation property
aa r X i v : . [ m a t h . A P ] F e b Attractors for locally damped Bresse systems and a uniquecontinuation property
To Fu Ma
Department of Mathematics, University of Bras´ılia, 70910-900 Bras´ılia, DF, Brazil
Rodrigo N. Monteiro Department of Mathematics, State University of Londrina, 86057-970 Londrina, PR, Brazil
Paulo N. Seminario-Huertas
Department of Mathematics, University of Bras´ılia , 70910-900 Bras´ılia, DF, Brazil
Abstract
This paper is devoted to Bresse systems, a robust model for circular beams, given bya set of three coupled wave equations. The main objective is to establish the existence ofglobal attractors for dynamics of semilinear problems with localized damping. In order todeal with localized damping a unique continuation property (UCP) is needed. Thereforewe also provide a suitable UCP for Bresse systems. Our strategy is to set the problem ina Riemannian geometry framework and see the system as a single equation with differentRiemann metrics. Then we perform Carleman-type estimates to get our result.
Keywords : Bresse system, unique continuation, localized damping, Riemannian manifold,global attractor.
Contents Corresponding author. Dynamics of locally damped Bresse systems 14
The Bresse system is a model for circular beams given by three coupled wave equations,namely, $&% ρ ϕ tt ´ k p ϕ x ` ψ ` ℓw q x ´ k ℓ p w x ´ ℓϕ q “ p , L q ˆ p , ,ρ ψ tt ´ bψ xx ` k p ϕ x ` ψ ` ℓw q “ p , L q ˆ p , ,ρ w tt ´ k p w x ´ ℓϕ q x ` kℓ p ϕ x ` ψ ` ℓw q “ p , L q ˆ p , . The functions ϕ “ ϕ p x, t q , ψ “ ψ p x, t q , w “ w p x, t q correspond to the vertical displacement,shear angle and longitudinal displacement at a point x P p , L q and time t ě
0, respectively.The coefficients are all positive constants defined by ρ “ ρA , ρ “ ρI , k “ KAG , b “ EI and k “ AE , where the quantities ρ , A , I , K , G and E denote respectively, material density,cross-sectional area, second moment of the cross-section area, a shear factor, shear modulusand modulus of elasticity. In addition, ℓ ą ℓ “ ρ k “ ρ b and k “ k . (1.1)Such a condition was firstly observed for Timoshenko systems in [25]. Otherwise only poly-nomial stability can be obtained. See e.g. [1, 10, 12, 13, 17, 24, 26, 29].In a different direction, Charles et al [7] proved the exponential stability of Bresse systemsby adding a localized damping in each one of its three equations, without assuming the speedcondition (1.1). This is quite interesting since the equal speed assumption can not be realizedphysically, cf. [20].The main objective of this paper is to establish existence of global attractors for dynamicsof a semilinear Bresse system with locally defined damping (see problem 4.1), without assum-ing condition (1.1). Our approach is very different from the above one in [7]. Indeed, oneof ingredients for obtaining exponential stability of wave equations with localized dampingis a unique continuation property (UCP). To our purpose, the UCP says whether a waveequation that vanishes in a subdomain must be identically null. In [7] they have used a UCPderived from Holmgren uniqueness theorem, which is only valid for equations with analyticcoefficients. Because of nonlinear terms, our problem (4.1) has no longer analytic coefficients.To overcome this difficult we propose a new UCP for Bresse systems.2ore precisely, we discuss the unique continuation property for coupled wave equationsof the form $’’’&’’’% B t u ´ ∆ u “ f p u , u , ¨ ¨ ¨ , u n q in p , L q ˆ p , , B t u ´ ∆ u “ f p u , u , ¨ ¨ ¨ , u n q in p , L q ˆ p , , ... B t u n ´ ∆ n u n “ f n p u , u , ¨ ¨ ¨ , u n q in p , L q ˆ p , , (1.2)where for i “ , ¨ ¨ ¨ , n the following is assumed:1. We consider on the system (1.2) the Dirichlet boundary conditions u i p , t q “ u i p L, t q “ @ t P p , , (1.3)and initial data p u , B t u , ¨ ¨ ¨ , u n , B t u n q ˇˇ t “ “ p u , u , ¨ ¨ ¨ , u n , u n q in p , L q . (1.4)2. Given γ i ą
0, the operator ∆ i represents the one-dimensional Laplacian operator withwave propagation velocity ? γ i , defined by∆ i “ γ i B x . (1.5)3. The symbols f i denote the coupling functions with energy level terms such that f i P L p , T ; L p , L qq and f i p u , u , ¨ ¨ ¨ , u n q “ n ÿ j “ p ij B x u j ` n ÿ j “ q ij u j , (1.6)where p ij , q ij P L p , T ; L p , L qq . Additionally, given T ą C T ą ż T ż L | f i p u , u , ¨ ¨ ¨ , u n q| d x d t ď C T ż T F u p t q d t, (1.7)where F u p t q represents the energy of the system (1.2) defined by F u p t q “ n ÿ i “ F u,i p t q (1.8)and F u,i p t q is the energy of the i -th equation of the system given by F u,i p t q “ ż L ” | u i | ` γ i |B x u i | ` |B t u i | ı d x. (1.9)The UCP has been extensively used in the analysis of exact controllability, exponentialstability, and in the theory of attractors for locally damped wave equations. The stabilizationproblem for linear wave equations on a smooth compact Riemannian manifold was studied,e.g., in [2, 21, 22]. In these papers, to show the exponential decay rates for the energy the3uthors assume localized damping and their proof uses a UCP for the wave equation based onHolmgren Theorem. In [4, 5] the authors treated the nonlinear case exhibiting an exponentialdecay of the energy with sharp damping region, roughly speaking, a damping region witharbitrarily small measure. In this case, a new UCP is proven by means of energy estimatesand a escape vector field based on [16].The study of the existence of global attractors for the wave equation with external forcesof critical exponent and locally distributed damping has been established in [6, 8, 11, 19].Despite dealing with wave equations, the significant difference between these models is thedamping regions imposed on the damping parameter. Therefore different types of UCP areneeded. Reference [11] uses the UCP given in [23]. In [8, 19], the authors apply the UCPcorresponding to Carleman-type estimates for wave equations with linear potential and in [6]the authors introduced a new UCP using the techniques of [28].Here, the main idea for proving a new UCP for Bresse systems is to set Problem (1.2)in a Riemannian geometry framework and see the system as a single equation with differentRiemann metrics. Then we show how Carleman-type estimates obtained in [28] can be usedto obtain a UCP for our system p . q under above assumptions on the functions f i p¨q p i “ , , ¨ ¨ ¨ , n q .Our paper is organized in the following way. Our UCP - Theorem 3.2 - will follow aftera series of comparison results between reference [28]. p i q We begin in Section 2 with a Rie-mannian geometry background material. p ii q After, in Section 3, we introduce a preliminarymaterial that will lead the corresponding Carleman-type estimates for the Problem p . q . Fi-nally, as a consequence of Carleman-type estimates, we then achieve our goal, the proof ofthe Theorem 3.2. For completeness, to the best of our knowledge, this is the first UCP resultfor coupled wave equations.In the second part of the paper, we establish the existence of global attractors for theBresse system with a nonlinear foundation and nonlinear localized dissipation - see Problem(4.1). We note that in [18] the authors studied a Bresse system with nonlinear foundationand dissipation acting on the whole domain. There, UCP and observability inequalities werenot necessary. In this sense, our application improves the previous results on the existenceof long-time dynamics of the Bresse system allowing the dissipation to be localized in anarbitrary subset of p , L q .The outline of the remainder of the paper is the following: p iii q In Section 4, we introducethe semilinear Bresse system with localized dissipation along with the well-posedness resultand energy estimates. The main result is the Theorem 4.3 and whose proof is based on thefollowing strategy: p a q we first show the existence of a strictly Lyapunov function for theassociated dynamical system by using the new UCP stated in Theorem 3.2 and p b q introduc-ing observability inequalities, we prove the asymptotic smoothness of the problem using theabstracts results on the recent theory of quasi-stable systems [9]. Here, we also mention theimportance of the UCP for the proof of a strictly Lyapunov function - see Definition 4.3. p iv q The Appendix is devoted to the well-posedness result for wave equations with over-determinedconditions. 4
A Riemannian geometry framework
Let p M, g q be an n -dimensional, compact Riemannian manifold, with smooth boundaryand smooth metric. The tangent space on M at p is denoted by T p M and fix a coordinatesystem p x , ¨ ¨ ¨ , x n q then pB x , ¨ ¨ ¨ , B x n q represents the associated coordinate vector fields. Inthis case g p X, Y q “ x
X, Y y “ n ÿ i,j “ g ij α i β j , | X | “ x X, X y , where X “ n ÿ i “ α i B x i , Y “ n ÿ i “ β i B x i in T p M for some p P M , (2.10)and g ij “ @ B x i , B x j D . Note that | ¨ | represents the norm with respect to the metric g p¨ , ¨q . In particular, we denotethe inner product g p¨ , ¨q by the matrix p g ij q n ˆ n and its inverse by p g ij q n ˆ n .The tangent and cotangent bundle of M are respectively detonate by T M and T ˚ M . Thesymbol D denotes the Levi-Civita connection of M such that for two vector fields X and Y on M given by (2.10) the following equality hods true D X Y “ n ÿ i,k “ « α i B x i β k B x k ` n ÿ j “ β j Γ kij B x k ff , where Γ kij represent the Christoffel symbols.Let f : M Ñ R and H P T p M for all p P M .1. If f P C p M q then the differential Df : T M Ñ R represents the gradient of theconnection D on f and Df p H q “ D H f “ H p f q “ x ∇ f, H y , where ∇ is the usual gradient defined in a coordinate system by ∇ f “ n ÿ i,j “ g ij B x i f B x j . (2.11)Thanks to the musical isomorphism we will identify Df with ∇ f . Here, we often denote Df by ∇ f . In particular, if t E , ¨ ¨ ¨ , E n u represents an orthonormal basis of T p M and H “ ř ni “ h i E i then Df p H q “ H p f q “ n ÿ i “ h i E i p f q .
2. If f P C p M q then D f represents the Hessian of f such that for all Y P T MD f p¨ , Y q “ D p Df qp¨ , Y q “ D Y p ∇ f p¨qq : T M Ñ R , D Y p ∇ f p X qq “ x D X p ∇ f q , Y y , @ X P T M.
In particular D f p X, X q “ x D X p ∇ f q , X y , @ X P T M.
3. Let f P C p M q . The function f is strictly convex in the metric g if and only if D f p X, X q ą X P T M .4. If t E , ¨ ¨ ¨ , E n u represents an orthonormal basis of T p M , then the divergent of H isdefined as div p H q “ n ÿ i “ x D E i H, E i y .
5. If f P C p M q then div p f H q “ f div p H q ` H p f q .
6. For a function f P C p M q we define the Laplace-Beltrami operator ∆ by∆ f “ div p ∇ f q “ n ÿ i “ x D E i p ∇ f q , E i y “ n ÿ i “ D f p E i , E i q .
7. The covariant derivate DH is the bilinear form given by DH p X, Y q “ x D X H, Y y , @ X, Y P T M.
In particular if f P C p M q D p ∇ f qp X, Y q “ D f p X, Y q , @ X, Y P T M.
8. If f P C p M q then x ∇ f, ∇ p H p f qqy “ DH p ∇ f, ∇ f q `
12 div p| ∇ f | H q ´ | ∇ f | div H. Remark 2.1.
Let Ω an open bounded, connected, compact subset of M with smooth boundary B Ω and f P C p M q a strictly convex function in the metric g . Then by translating andrescaling [28, Remark 1.2] , the function f satisfies the following conditions D f p X, X q ě | X | , @ p P Ω , @ X P T p M, min Ω f p x q ” m ą . (2.12)6 .2 Geometry on the wave system Let us consider R with usual topology and x the natural coordinate system. In particular,for each x P R the tangent space is T x R “ R x “ R .For a fixed i P t , ¨ ¨ ¨ , n u , we begin defining the metrics g i associated with ∆ i by g i p X, Y q “ x
X, Y y g i “ γ ´ i αβ, (2.13)with corresponding norm | X | g i “ x X, X y g i , where X “ α B x , Y “ β B x P R x , for each x P R .Recalling assumption γ i ą
0, one can see that the pairs p R , g i q are Riemannian manifolds.In this case, the Levi-Civita connection of p R , g i q will be denoted by D g i . Here, the symbol p R , g q denotes the R space with Euclidean metric and we use the following notations for themetric and norm g p X, Y q “ X ¨ Y and | X | g “ | X | , @ X, Y P R x , @ x P R . (2.14)Important properties of the above metrics are stated in the following lemma. Althoughmost of these results are followed straightforwardly from the known results, they are crucialfor what follows. So for the convenience of the reader, we give their proofs here. Lemma 2.1.
Let p , L q Ă R for some L ą . If x be the natural coordinate system in R , f, h P C pr , L sq and X “ α B x , Y “ β B x vector fields. Then . x X, γ i Y y g i “ X ¨ Y ; . ∇ g i f “ γ i B x f B x ; . X p f q “ x ∇ g i f, X y g i “ α B x f ; . x ∇ g i f, ∇ g i h y g i “ γ i B x f B x h ; . if ν represents the unit outward normal vector for pr , L s , g i q then x ∇ g i f p q , ν p qy g i “ ´? γ i B x f p q , x ∇ g i f p L q , ν p L qy g i “ ? γ i B x f p L q . Proof.
In light of (2.13) and (2.14), one can easily see that Item 1 holds true. In fact, bydefinition x X, γ i Y y g i “ γ ´ i γ i αβ “ X ¨ Y. (2.15)To prove Item 2, we recall the definition (2.11) to find ∇ g i f “ γ i B x f B x . Reasoning analogously to (2.15) and assuming h P C pr , L sq , we have the validity of Items2-4. To conclude, let us show Item 5. Firstly, note that if p P t , L u and tB x u represents theassociated coordinate vector field for T p r , L s , then x˘? γ i B x , ˘? γ i B x y g i “ .
7n particular E “ ˘? γ i B x represents an orthonormal basis for p P T p M . Let ν “ E , therefore ∇ g i f “ γ i B x f B x “ ˘? γ i B x f ν and x ∇ g i f, ν y g i “ ˘? γ i B x f. Note that ν p q “ ´? γ i B x and ν p L q “ ? γ i B x . We complete the proof of Lemma. Remark 2.2.
The Hessian of f P C p R q with respect to the metric g i is given by D g i f p X, X q “ x D g i ,X p ∇ g i f q , X y g i “ α γ ´ i γ i B x f “ α B x f, where X “ α B x . We observe that the Hessian of f is positive if and only if B x f is positive. As already observed Carleman estimates are an important tool for proving unique contin-uation property for solutions to partial differential equations [6, 15, 28]. In this section, weprove the Carleman estimates for the Problem p . q .In what follows we shall use the following notations.1. Let L a real number in r , L s . We define Ω and Ω the subsets of r , L s as followsΩ “ p , L q and Ω “ p L , L q . (3.16)Note that Ω Y Ω “ r , L s .For ε ą
0, we will also consider the following subsets of Ω and Ω V “ ´ , L ´ ε ¯ Xr , L s , V “ ´ L ` ε , L ¯ Xr , L s , ω “ ´ L ´ ε , L ` ε ¯ Xr , L s .
2. There exist strictly convex functions d : r , L s Ñ R and d : r , L s Ñ R such that d p x q “ p x ` L q , and d p x q “ p x ´ L q . (3.17)In particular, if x denotes the natural coordinate system, Lemma 2.1 implies that ∇ g i d “ γ i p x ` L qB x , ∇ g i d “ γ i p x ´ L qB x p i “ , ¨ ¨ ¨ , n q , and D g i d j p X, X q “ α , for all x P r , L s and X “ α B x P T x r , L s p j “ , q . The functions d j p¨q p j “ , q have the following properties: Lemma 3.1.
Under the above definitions, the functions d j p¨q p j “ , q satisfy . d j P C pr , L sq ; . D g i d j p X, X q “ | X | g i , for all x P Ω j and X P T x r , L s ; . inf Ω j | ∇ g i d j | g i ą ; . min Ω j d j “ L ą .Moreover, if ν represents the unit outward normal vector then . x ∇ g i d j p x q , ν p x qy g i ă on t , L u X V j . The next section is devoted to proof the Carleman estimate compatible with the system(1.2). To this aim, we allocated the above notations in the same context as Section 1 in [28].First, without loss of generality (by rescaling), we can assume that k ij ” inf x P Ω j | ∇ g i d j | g i d j ą , @ i “ , ¨ ¨ ¨ , n, @ j “ , . (3.18)In what follows, for fixed i P t , ¨ ¨ ¨ , n u and j “ ,
2, we define T ,j ” x P Ω j d j p x q . (3.19)Let T ą max t T , , T , u and by p . q there exist δ ą c “ c δ P p , q satisfying cT ą x P Ω j d j p x q ` δ. (3.20)In above context, we define functions φ j : Ω j ˆ R Ñ R P C p Ω j q by φ j p x, t q ” d j p x q ´ c ˆ t ´ T ˙ , p x, t q P Ω j ˆ r , T s . The following properties are valid for φ j p¨q : p φ. q For the constant δ ą φ j p x, q “ φ j p x, T q “ d j p x q ´ c T ď max j “t , u ˜ max x P Ω j d j p x q ¸ ´ c T ď ´ δ, uniformly in Ω j . p φ. q There are t and t with 0 ă t ă T { ă t ă T , such thatmin j “ , ˜ min p x,t qP Ω j ˆr t ,t s φ j p x, t q ¸ ě σ, (3.21)for σ P ´ , min Ω j d j ¯ , where min Ω j d j “ L . i “ , ¨¨¨ ,n j “ , ˜ max p x,t qP Ω j ˆr ,T s ` |B t φ j | ` | ∇ g i φ j | g i ˘¸ ď p ` c q σ ˚ ε p ´ c q , (3.22)for any ε P p , min t ´ c, uq with σ ˚ P p , σ q and c P p , q .We end this section defining Q p σ q “ " p x, t q P r , L s ˆ r , T s | min j “ , φ j p x, t q ě σ * . (3.23)This set will play an important role in Carleman estimates by being able to separate the set r , L s ˆ r , T s from the level surface generated by the pseudo-convex function φ j at height of σ .Additionally, note thatΩ j ˆ r t , t s Ă Q p σ q Ă r , L s ˆ r , T s , @ j “ , . (1.2) In this section, we will study the Problem (1.2) under the new decomposition Ω and Ω (3.16). This decomposition will allow us to define the boundary terms for the solutions of thissystem. We begin with the following definition: Definition 3.1.
Let T ą . The vector function p u , ¨ ¨ ¨ , u n q is a weak solution to the Prob-lem (1.2) - (1.4) if the function u i solves the variational form of equation (1.2) i and possessesregularity u i P C p , T ; p H p M i qqq X C p , T ; p L p M i qqq , where M i denotes the Riemannian manifold pr , L s , g i q p i “ , ¨ ¨ ¨ , n q .If u i P C p M i ˆ R ` q the vector function p u , ¨ ¨ ¨ , u n q is a regular solution to the Problem (1.2) - (1.4) . Now, let p u , ¨ ¨ ¨ , u n q be a regular solution to Problem (1.2)-(1.4) and we define u i,j p x, t q “ χ j p x, t q u i p x, t q , i “ , ¨ ¨ ¨ , n and j “ , , (3.24)where χ j is a smooth cutoff function such that χ j “ " V j ˆ r , T s , pr , L sz Ω j q ˆ r T ` , , (3.25)where T is a positive constant satisfying (3.20).By definition, we can observe that u i,j P C p M i,j ˆ R ` q , where M i,j ” p Ω j , g i q representsa 1-dimensional compact connex smooth riemannian manifold with boundary B Ω j and withmetric g i .Recall that the objective of the present section is to show a unique continuation propertyfor the Problem (1.2). For this purpose we shall assume u i p x, t q “ , @p x, t q P ω ˆ r , , i “ , ¨ ¨ ¨ , n. (3.26)10nder the above notations, the function u i,j P C p M i,j ˆ R ` q solves the problem $’’&’’% B t u i,j ´ ∆ i u i,j “ f i p u ,j , u ,j , ¨ ¨ ¨ , u n,j q in M i,j ˆ p , T s ,u i,j p x, t q “ B M i,j ˆ p , T s ,u i,j p x, q “ χ j p x, q u i p x q in M i,j , B t u i,j p x, q “ χ j p x, q u i p x q in M i,j (3.27)In particular, the following decomposition is valid u i “ u i, ` u i, , p x, t q P r , L s ˆ r , T s p i “ , ¨ ¨ ¨ , n q . (3.28)On the other hand, note that the system $’’&’’% B t v i,j ´ ∆ i v i,j “ f i p v ,j , v ,j , ¨ ¨ ¨ , v n,j q in M i,j ˆ p , T s ,v i,j p x, t q “ B M i,j ˆ p , T s ,v i,j p x, q “ χ j p x, q u i p x q in M i,j , B t v i,j p x, q “ χ j p x, q u i p x q in M i,j (3.29)is well posed, for all i “ , ¨ ¨ ¨ , n and j “ , v i,j “ u i,j . In the context of the previous section, in regard to the study the boundary terms for thesystem (3.29), the following will be considered:1. For v i,j P C p , T ; p H p M i,j qqq X C p , T ; p L p M i,j qqq the weak solution of Problem (3.29)we have x ∇ g i v i,j , ∇ g i d j y g i “ x ∇ g i v i,j , ν y g i x ∇ g i d j , ν y g i , where ν denotes the outward unit normal field along the boundary B M i,j .2. Let τ be a positive parameter then we define BT τ v i,j ” τ ż T ż B Ω j e τφ j ´ x ∇ g i v i,j , ν y g i ¯ x ∇ g i d j , ν y g i d x d t, (3.30)with T ą f i p¨q p i “ , ¨ ¨ ¨ , n q in the contextof the new decomposition. We promptly have from (1.6)-(1.9) that there exists a positiveconstant C T such that ż T ż Ω j | f i p v ,j , ¨ ¨ ¨ , v n,j q| g i d x d t ď C T n ÿ i “ ż T ż L V i p x, t q d x d t, (3.31)where V i p x, t q ” | v i, ` v i, | ` γ i |B x p v i, ` v i, q| ` |B t p v i, ` v i, q| , (3.32)11ith v i, ` v i, “ " v i, in r , L s ˆ r , T s ,v i, in r L , L s ˆ r , T s . (3.33)Note that V i | Ω j ˆr ,T s “ | v i,j | ` γ i |B x v i,j | ` |B t v i,j | and ÿ j “ ż Ω j V i | Ω j ˆr ,T s d x “ ż L V i d x. Now, for any regular solution v i,j , we find from (3.31) that ż T ż Ω j e τφ j |B t v i,j ´ ∆ i v i,j | g i d x d t ď C T n ÿ i “ ż T ż L e τφ j V i p x, t q d x d t. In particular, by (3.21) and (3.23), we obtain ż T ż r Q p σ qs c e τφ j |B t v i,j ´ ∆ i v i,j | g i d x d t ď C T e τσ n ÿ i “ ż r Q p σ qs c V i p x, t q d x d t. Remark 3.1.
Note that the Problem (3.29) satisfies the following compatibility condition v i,j p L , t q “ in p , T s and v i p L q “ v i p L q “ . In particular, there exist positive constants k , k such that k n ÿ i “ ż L V i p x, t q d x ď n ÿ i “ ż L γ i |B x p v i, ` v i, q| `|B t p v i, ` v i, q| d x ď k n ÿ i “ ż L V i p x, t q d x. Collecting all the above ingredients and proceeding analogously to Theorem 6.1 in [28] wearrive at:
Theorem 3.1 (Carleman Estimates) . Let v i,j p i “ , ¨ ¨ ¨ , n and j “ , q be a regular solutionof the Problem (3.29) with initial data p χ j p x, q u i p x q , χ j p x, q u i p x qq . Then, for all τ ą sufficiently large and ε ą small, the following estimate holds true ÿ j “ n ÿ i “ BT τ v i,j ě „ k e τσ p t ´ t q p ετ p ´ c q ´ nC T q e ´ C T T n ÿ i “ ż L ” V i p x, q ` V i p x, T q ı d x ´ „ C ,T k e τσ k T e C T T ` C T τ e ´ τδ n ÿ i “ ż L ” V i p x, q ` V i p x, T q ı d x ě k T n ÿ i “ ”ż L V i p x, q ` V i p x, T q ı d x, where . k , k the positive constants from Remark . ; . σ defined in p φ. q ; . c P p , q given in p . q ; . C T , C ,T positive constants depending only on T, σ and Q p σ q ; . k T a positive constant depending only on σ, φ , φ , n and C ,T .Moreover, the above inequality may be extended to all weak solution of the system (3.29) withinitial data p χ j p x, q u i p x q , χ j p x, q u i p x qq P H p M i,j q ˆ L p M i,j q . Remark 3.2. p a q Note that the definition of σ and Q p σ q allow the right-hand term of previousinequality to be independent of j . p b q The inequality holds for weak solution of the system (3.29) because [28, Theorems 7.1 and 8.1] . Thanks to the Theorem 3.1 it is possible to state the main result of this part of paper.
Theorem 3.2.
Let I be an open interval such that ω ” I X r , L s ‰ H . Then, for T ą large enough, any weak solution u i P C p , T ; H p , L qq X C p , T ; L p , L qq of system (1.2) - (1.4) vanishing in ω ˆ r , T s must vanish all over r , L s ˆ r , T s .Proof. Without loss of generality we can assume ω “ ´ L ´ ε , L ` ε ¯ X r , L s . (3.34)where ε ą L P r , L s .The proof of the unique continuation property will be divided into four steps: Step 1. Equivalence of systems.
Firstly, we observe that if p u , ¨ ¨ ¨ , u n q is a weak so-lution of the Problem (1.2)-(1.4) with overdetermined condition p . q then u i,j “ χ j u i P C p , T ; H p M i,j qq X C p , T ; M i,j q is a solution of (3.27), where M i, “ pr , L s , g i q and M i, “ pr L , L s , g i q , for all i “ , ¨ ¨ ¨ , n and j “ , j “ ,
2, the system (3.29) with the following overdeterminedcondition p v ,j , ¨ ¨ ¨ , v n,j q “ p , ¨ ¨ ¨ , q in ω j ” Ω j X ω (3.35)is well-posed and generates a strongly continuous semigroup T M i,j : H ω j p M i,j q Ñ H ω j p M i,j q in the Hilbert space H ω j p M i,j q ” " p v ,j , ¨ ¨ ¨ , v n,j , w ,j , ¨ ¨ ¨ , w n,j q ˇˇˇˇ v i,j P H p M i,j q , w i,j P L p M i,j q ,v i,j “ w i,j “ ω j , i “ , ¨ ¨ ¨ , n * . In particular, T M i,j p χ j p q u , ¨ ¨ ¨ , χ j p q u n , χ j p q u , ¨ ¨ ¨ , χ j p q u n q “ p u ,j , ¨ ¨ ¨ , u n,j , B t u ,j , ¨ ¨ ¨ , B t u n,j q .
13s the weak solution of (3.29) satisfying (3.35) with initial data p χ j p q u , ¨ ¨ ¨ , χ j p q u n , χ j p q u , ¨ ¨ ¨ , χ j p q u n q P H ω j p M i,j q . Step 2. Carleman estimate.
From Step 1 and via Theorem 3.1 there exists a positiveconstant k T such that ÿ j “ n ÿ i “ BT τ u i,j ě k T n ÿ i “ „ż L V i p x, q ` V i p x, T q d x . Next, from (1.8), (3.28), (3.30), (3.32) and (3.33) we find that2 τ ÿ j “ n ÿ i “ ż T ż B Ω j e τφ j ´ x ∇ g i u i,j , ν y g i ¯ x ∇ g i d j , ν y g i d x d t ě k T p F u p q ` F u p T qq . (3.36) Step 3. Boundary estimates.
The fact that L P ω and ω is an open subset of r , L s wehave x ∇ g i u i,j p L q , ν p L qy g i “ , @ i “ , ¨ ¨ ¨ , n and j “ , . Now, recalling Item 5 from Lemma 3.1 we obtain x ∇ g i d j , ν y g i ă t , L u , @ i “ , ¨ ¨ ¨ , n and j “ , , Combining the above information with assumption that τ ą
0, we infer that2 τ ÿ j “ n ÿ i “ ż T ż B Ω j e τφ j ´ x ∇ g i u i,j , ν y g i ¯ x ∇ g i d j , ν y g i d x d t ď . (3.37) Step 4. Conclusion.
From inequalities p . q and p . q we find0 ě k T p F u p q ` F u p T qq ě . This last implies that F u p q “
0. Since (1.2)-(1.4) is well-posed, the result is followed.
Let us consider the semilinear Bresse system $&% ρ ϕ tt ´ k p ϕ x ` ψ ` lw q x ´ k l p w x ´ lϕ q ` a p x q g p ϕ t q ` f p ϕ, ψ, w q “ ,ρ ψ tt ´ bψ xx ` k p ϕ x ` ψ ` lw q ` a p x q g p ψ t q ` f p ϕ, ψ, w q “ ,ρ w tt ´ k p w x ´ lϕ q x ` kl p ϕ x ` ψ ` lw q ` a p x q g p w t q ` f p ϕ, ψ, w q “ , (4.1)with Dirichlet boundary conditions ϕ p , t q “ ϕ p L, t q “ ψ p , t q “ ψ p L, t q “ w p , t q “ w p L, t q “ , t P R ` , (4.2)and with initial condition ϕ p q “ ϕ , ϕ t p q “ ϕ , ψ p q “ ψ , ψ t p q “ ψ , w p q “ w , w t p q “ w . (4.3)14 .1 Well-posedness In this section, we summarize all the assumptions that will be used to prove the mainresult. We also introduce the well-posedness result along with some energy inequalities.
Notations.
Henceforth the symbols L p p , L q p p ě q and H m p , L q p m P N q denote theLebesgue and Sobolev spaces, respectively. The norms in L p p , L q are indicated by } ¨ } p and } ¨ } L p ,L q ” } ¨ } . We will also frequently use the inequality } u } ď Lπ } u x } , @ u P H p , L q . Assumptions.
The following hypotheses will be used throughout the paper. p f. . q The sources functions f i P C p R q p i “ , , q are locally Lipschitz and there exists afunction F P C p R q such that ∇ F “ p f , f , f q . p f. . q There exists constants 0 ď α ă π βL and c F ą F p u, v, w q ě ´ α ” | u | ` | v | ` | w | ı ´ c F , @ u, v, w P R , ∇ F p u, v, w q¨p u, v, w q ě F p u, v, w q ´ α ” | u | ` | v | ` | w | ı ´ c F , @ u, v, w P R , where β ą } ϕ x } ` } ψ x } ` } w x } ď β ” b } ψ x } ` k } ϕ x ` ψ ` lw } ` k } w x ´ lϕ } ı . p f. . q There exists c f ą | ∇ f i p u, v, w q| ď c f ” ` | u | p ´ ` | v | p ´ ` | w | p ´ ı , i “ , , , p ě , @ u, v, w P R . p g. . q The damping functions g i P C p R q p i “ , , q are monotone increasing with g i p q “ m and M ą m ď g i p s q ď M, @ s P R . p a. . q The localizing functions a i P L p , L q p i “ , , q are non-negative and there existspositive constant a such that a i p x q ě a , x P I i , i “ , , , where I i Ă r , L s are open intervals with p L , L q ” Ş i I i ‰ H . Dynamical system generation.
Before introducing the well-posedness result, we start withthe necessary functional framework. First, the finite energy space H of the well-posedness isdefined as H “ H p , L q ˆ H p , L q ˆ H p , L q ˆ L p , L q ˆ L p , L q ˆ L p , L q . For Z “ p ϕ p t q , ψ p t q , w p t q , ˜ ϕ p t q , ˜ ψ, ˜ w p t qq P H , we define the H norm as } Z } H “ ρ } ˜ ϕ } ` ρ } ˜ ψ } ` ρ } ˜ w } ` b } ψ x } ` k } ϕ x ` ψ ` lw } ` k } w x ´ lϕ } . (4.4)15ext, let A : D p A q Ă H Ñ H be the differential operator A Z ” »——————– ˜ ϕ ˜ ψ ˜ wkρ ´ p ϕ x ` ψ ` lw q x ` k lρ ´ p w x ´ lϕ q bρ ´ ψ xx ´ kρ ´ p ϕ x ` ψ ` lw q k ρ ´ p w x ´ lϕ q x ´ klρ ´ p ϕ x ` ψ ` lw q fiffiffiffiffiffiffifl , with domain D p A q “ “ H p , L q X H p , L q ‰ ˆ H p , L q . Next, let be B : D p B q “ H Ñ H the damping operator B Z ” »——————– ´ α p x q ρ ´ g p ˜ ϕ q´ α p x q ρ ´ g p ˜ ψ q´ α p x q ρ ´ g p ˜ w q fiffiffiffiffiffiffifl . Finally, by F : H Ñ H , we represent the source terms operator F Z ” »——————– ´ ρ ´ f p ϕ, ψ, w q´ ρ ´ f p ϕ, ψ, w q´ ρ ´ f p ϕ, ψ, w q fiffiffiffiffiffiffifl . Now, using the definitions of operators
A, B, F , we can abstract represent the problem asfollows dd t Z p t q´p A ` B q Z p t q “ F p Z p t qq , Z p q ” Z “ p ϕ , ψ , w , ϕ , ψ , w q , (4.5)where Z p t q “ p ϕ p t q , ψ p t q , w p t q , ˜ ϕ p t q , ˜ ψ, ˜ w p t qq with ˜ ϕ “ ϕ t , ˜ ψ “ ψ t , ˜ w “ w t . We observe that the well-posedness of (4.5) induces the well-posedness for the Problem(4.1) -(4.3). In the following, we present the well-posedness for (4.5).
Theorem 4.1 (Well-posedness) . Assume the validness of Assumptions p f. q - p g. q . Then forany initial data Z P H and T ą , the Cauchy problem p . q admits a unique weak solution Z P C pr , T s ; H q that depends continuously on the initial data and is given by the variation ofparameters formula Z p t q “ e p A ` B q t Z ` ż t e p A ` B qp t ´ s q F p Z p s qq d s, t P r , T s . (4.6) Moreover, if Z P D p A q then the solution is strong. . H Q Z ÞÑ Z p t q “ p ϕ, ψ, w, ϕ t , ψ t , w t q , where Z p t q solves(4.5), defines a strongly continuous semigroup t S p t qu t ě on H . Energy.
Let Z p t q “ p ϕ p t q , ψ p t q , w p t q , ϕ t p t q , ψ t p t q , w t p t qq be a solution of p . q - p . q . Theenergy is defined by the following functional E Z p t q ” E Z p t q ` ż L F p ϕ, ψ, w q d x “ } Z p t q} H ` ż L F p ϕ, ψ, w q d x. (4.7)The weak solution Z p t q “ p ϕ p t q , ψ p t q , w p t q , ϕ t p t q , ψ t p t q , w t p t qq satisfies the energy identity E Z p t q ` ż ts ż L ” α p x q g p ϕ t q ϕ t ` α p x q g p ψ t q ψ t ` α p x q g p w t q w t ı d x d τ “ E Z p s q , (4.8)for all 0 ď s ă t .As in [18], the energy E Z p¨q (4.7) and the norm } ¨ } H (4.4) satisfy C E } Z p t q} H ´ Lc F ď E Z p t q ď } Z p t q} H ` c E p ` } Z p t q} p ` H q , @ t ě , (4.9)for some positive constants C E and c E . Some essential definitions and results from the theory of attractors for gradient systemsis collected
Definition 4.1.
A global attractor for a dynamical system p H, S p t qq , with evolution operator t S p t qu t ě on a complete metric space H is defined as a a compact set A Ă H that is fullyinvariant, that is S p t q A “ A for all t ě , and uniformly attracts all bounded subsets of H lim t Ñ8 sup ! dist H p S p t q Z , A q | Z P B ) “ , for any bounded set B Ă H. Definition 4.2.
The fractal dimension of a compact set A Ă H in a metric space H isdefined as dim Hf p A q “ lim sup ε Ñ ln N ε p A q ln p { ε q , where N ε p A q is the minimal number of closed balls of radius ε which cover the set A . To ascertain the existence of a global attractor, we use the concept of gradient and quasi-stable dynamical systems. The global attractor for this systems admits additional structureand properties: (i) the attractor for gradient systems has a regular structure, that is, theattractor is described by the unstable manifold emanating from the set of stationary points and(ii) quasi-stable systems provide several properties of attractors, such as finite dimensionality.
Definition 4.3.
Let Y Ă H be a forward invariant set of a dynamical system p H, S p t qq . p i q A continuous functional
Φ : Y Ñ R is said to be a Lyapunov function on Y for p H, S p t qq ifthe map t ÞÑ Φ p S p t q Z q is non-increasing for any Z P Y . p ii q The Lyapunov function is saidto be strict on Y if the equation if Φ p S p t q Z q “ Φ p Z q for all t ą for some Z P Y impliesthat y is a stationary point of p H, S p t qq . p iii q The dynamical system p H, S p t qq is said to begradient if there exists a strict Lyapunov function on the whole phase space H . efinition 4.4. Let
X, Y be reflexive Banach spaces, X compactly embedded in Y . Weconsider a dynamical system p H, S p t qq with H “ X ˆ Y and evolution operator defined by S p t q Z “ p u p t q , u t p t qq , Z “ p u p q , u t p qq P H, (4.10) where the function u possess the property u P C pr , ; X q X C pr , ; Y q . (4.11) A dynamical system of the the form p . q with regularity p . q is said to be quasi-stable on aset B Ă H , if there exist a compact semi-norm r ¨ s X on X and non-negative scalar functions a p t q , b p t q , c p t q , such that, p i q a p t q , b p t q are locally bounded on r , , p ii q b p t q P L p , with lim t Ñ8 b p t q “ and p iii q for any Z , Z P B the following estimates hold true } S p t q Z ´ S p t q Z } H ď a p t q} Z ´ Z } H , (4.12) and } S p t q Z ´ S p t q Z } H ď b p t q} Z ´ Z } H ` c p t q sup ă s ă t r u p s q ´ u p s qs X , (4.13) where S p t q Z i “ p u i p t q , u it p t qq , i “ , . Unifying the abstracts results from [9] we arrive at the following criteria for existence andproperties of global attractors.
Theorem 4.2.
Let p H, S p t qq be a gradient quasi-stable dynamical system. Assume its Lya-punov function Φ p¨q is bounded from above on any bounded subset of H and the set Φ p R q “ Z P H | Φ p Z q ď R ( is bounded for every R . If the set N of stationary points of p H, S p t qq isbounded, then p H, S p t qq possesses a finite dimensional global attractor A defined by the un-stable manifold emanating from set of stationary solution. Moreover, any trajectory stabilizesto the set N of stationary points, that is, lim t Ñ`8 dist H p S p t q Z , N q “ , @ Z P H. We now state the main result of the present chapter.
Theorem 4.3.
Under the Assumptions p f. q - p a. q the dynamical system p H, S p t qq generatedby the problem p . q - p . q has a global attractor A characterized by A “ M ` p N q , where M ` p N q is the unstable manifold emanating from N , the set of stationary points of t S p t qu t ě . Our strategy centers on establishing the conditions from the Theorem 4.2. Starting ex-hibiting the gradient structure for p H, S p t qq and focusing our attention on the strictness ofthe Lyapunov function where the new observability result stated in Theorem 3.2 plays anessential role in the proof. 18 roposition 4.1. Let the assumptions of Theorem . be satisfied. Then, p H, S p t qq is agradient dynamical system.Proof. The dynamical system p H, S p t qq is gradient with full energy E Z p¨q - defined in (4.7) -being the strict Lyapunov function Φ p¨q . In fact, from identity (4.8), we find that t Ñ Φ p S p t q Z q is a non-increasing function for any Z P H .Next, we suppose that Φ p S p t q Z q “ Φ p Z q , for all t ą
0. Then, from identity (4.8), weobtain ż ts ż L ” α p x q g p ϕ t q ϕ t ` α p x q g p ψ t q ψ t ` α p x q g p w t q w t ı d x d τ “ . This shows that ϕ t “ ψ t “ w t “ p L , L q ˆ r , T s , where p L , L q “ Ş i I i . Thus, p ϕ, ψ, w q satisfies the problem $&% ρ ϕ tt ´ k p ϕ x ` ψ ` lw q x ´ k l p w x ´ lϕ q ` f p ϕ, ψ, w q “ p , L q ˆ r , T s ,ρ ψ tt ´ bψ xx ` k p ϕ x ` ψ ` lw q ` f p ϕ, ψ, w q “ p , L q ˆ r , T s ,ρ w tt ´ k p w x ´ lϕ q x ` kl p ϕ x ` ψ ` lw q ` f p ϕ, ψ, w q “ p , L q ˆ r , T s . (4.14)Using the notation u “ ϕ t , u “ ψ t , u “ w t and taking the derivative in the distributionalsense of p . q , we find that p u , u , u q solves the problem $&% u tt ´ γ u xx “ F p u , u , u q in p , L q ˆ r , T s ,u tt ´ γ u xx “ F p u , u , u q in p , L q ˆ r , T s ,u tt ´ γ u xx “ F p u , u , u q in p , L q ˆ r , T s , with γ “ kρ , γ “ bρ , γ “ k ρ and with forcing F i p¨q defined by F p u , u , u q “ γ u x ` γ u x ` k ρ ´ l p u x ´ lu q ´ ρ ´ B t r f p ϕ, ψ, w qs ,F p u , u , u q “ ´ kρ ´ p u x ´ u ` lu q ´ ρ ´ B t r f p ϕ, ψ, w qs ,F p u , u , u q “ ´ γ lu x ´ kρ ´ l p u x ` u ` lu q ´ ρ ´ B t r f p ϕ, ψ, w qs . Now, we apply the UCP - Theorem 3.2 - to conclude that p ϕ t , ψ t , w t q “ p u , u , u q “ p , , q .Therefore, the solution Z P H must be stationary. This implies that the energy E Z p¨q is stricton H .Our next aim is to show the quasi-stability of p H, S p t qq . According to the Definition4.4, the difference of two trajectories should obeys estimates p . q and p . q . Taking theadvantage of the locally Lipschitz property of f i p¨q and the variation of parameter formula(4.6), one can easily show the validity of p . q . Next, by means of multiplier technique, weprove the stabilization inequality p . q . Proposition 4.2.
Let the assumptions of Theorem . be satisfied. Then, p H, S p t qq is aquasi-stable dynamical system.Proof. The proof is carried out through several energy estimates. In the text that follows, weuse the notations˜ v “ v ´ v , G p v q “ g i p v q ´ g i p v q and F i p v q “ f i p v q ´ f i p v q , i “ , , . Z , Z P B ,where B is a bounded subset of H . The corresponding solution S p t qp Z ´ Z q ” Z ´ Z “p ˜ ϕ, ˜ ψ, ˜ w, ˜ ϕ t , ˜ ψ t , ˜ w t q verifies the following problem $&% ρ ˜ ϕ tt ´ k p ˜ ϕ x ` ˜ ψ ` l ˜ w q x ´ k l p ˜ w x ´ l ˜ ϕ q “ ´ a p x q G p ˜ ϕ t q ´ F p ˜ ϕ, ˜ ψ, ˜ w q ,ρ ˜ ψ tt ´ b ˜ ψ xx ` k p ˜ ϕ x ` ˜ ψ ` l ˜ w q “ ´ a p x q G p ˜ ψ t q ´ F p ˜ ϕ, ˜ ψ, ˜ w q ,ρ ˜ w tt ´ k p ˜ w x ´ l ˜ ϕ q x ` kl p ˜ ϕ x ` ˜ ψ ` l ˜ w q “ ´ a p x q G p ˜ w t q ´ F p ˜ ϕ, ˜ ψ, ˜ w q , (4.15)with zero Dirichlet boundary conditions and initial conditions Z ´ Z .Second, we let ǫ be a positive real number, such that, ǫ ď p L ´ L q , where p L , L q “ Ş i I i . We consider the following real-function ξ p¨q defined as follows ξ p x q “ $&% p λ ´ q x, if x P r , L ` ǫ q ,λ p x ´ L ´ ǫ q ` p L ´ L ` ǫ q{p L ` ǫ q , if x P p L ` ǫ , L ´ ǫ s , p λ ´ qp x ´ L q , if x P r L ´ ǫ , L s . . Now, we take ˜ ϕ x ξ, ˜ ψ x ξ, ˜ w x ξ as multipliers for p . q . Thus, we find12 ż T ż L p λ ´ q E p t q d x d t “ ´ ż L ” ˜ ϕ t ˜ ϕ x ` ˜ ψ t ˜ ψ x ` ˜ w t ˜ w x ı ξ d x ˇˇˇ T ´ ż T ż L ´ ǫ L ` ǫ E p t q d x d t ´ ż T ż L ” k p ˜ ϕ x ´ ˜ ψ ` l ˜ w qp ˜ ψ ` l ˜ w q ` k l p ˜ w x ´ l ˜ ϕ q ˜ ϕ ı ξ d x d t ´ ż T ż L ” a G p ˜ ϕ t q ˜ ϕ x ` a G p ˜ ψ t q ˜ ψ x ` a G p ˜ w t q ˜ w x ı ξ d x d t ´ ż T ż L ” F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ϕ x ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ψ x ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ w x ı ξ d x d t, (4.16)where E p t q “ ρ | ˜ ϕ t | ` ρ | ˜ ψ t | ` ρ | ˜ w t | ` b | ˜ ψ x | ` k | ˜ ϕ x ` ˜ ψ ` l ˜ w | ` k | ˜ w x ´ l ˜ ϕ | . Let us estimate the left-hand side of p . q . Note that, from the definition of energy, we find E Z p t q “ ş L E p t q d t “ } Z } H . Then, we can show that there exists c ą ˇˇˇˇż L ” ˜ ϕ t ˜ ϕ x ` ˜ ψ t ˜ ψ x ` ˜ w t ˜ w x ı ξ d x ˇˇˇˇ ď c sup x Pr ,L s t ξ p x qu E Z p t q . This last implies that ˇˇˇˇż L ” ˜ ϕ t ˜ ϕ x ` ˜ ψ t ˜ ψ x ` ˜ w t ˜ w x ı ξ d x ˇˇˇ T ˇˇˇˇ ď c sup x Pr ,L s t ξ p x qup E Z p T q ` E Z p qq . (4.17)Also using the definition of E Z p t q , one obtains ˇˇˇˇż T ż L ” k p ˜ ϕ x ´ ˜ ψ ` l ˜ w qp ˜ ψ ` l ˜ w q ` k l p ˜ w x ´ l ˜ ϕ q ˜ ϕ ı ξ d x d t ˇˇˇˇ ď ǫ ż T E Z p t q d t ` c ǫ l.o.t p ˜ ϕ, ˜ ψ, ˜ w q , (4.18)20ith lower order terms defined byl.o.t p ˜ ϕ, ˜ ψ, ˜ w q ” sup σ Pr ,T s ” } ˜ ϕ p σ q} p ` } ˜ ψ p σ q} p ` } ˜ w p σ q} p ı . To estimate the damping terms, we use Assumption p g. q to obtain ˇˇˇˇż T ż L a G p ˜ ϕ t q ˜ ϕ x ξ d x d t ˇˇˇˇ ď sup x Pr ,L s t ξ p x qu ż T ż L a M | ϕ t ´ ϕ t || ˜ ϕ x | d x d t ď ǫ ż T E Z p t q d t ` c ǫ ż T ż L a ˜ ϕ t d x d t. This allows us to conclude the following estimate ˇˇˇˇż T ż L ” a G p ˜ ϕ t q ˜ ϕ x ` a G p ˜ ψ t q ˜ ψ x ` a G p ˜ w t q ˜ w x ı ξ d x d t ˇˇˇˇ ď ǫ ż T E Z p t q d t ` c ǫ ż T ż L ” a ˜ ϕ t ` a ˜ ψ t ` a ˜ w t ı d x d t. (4.19)Let us estimate the kinetic energy in (4.19). Assumption p g. . q implies that ż T ż L ” a ˜ ϕ t ` a ˜ ψ t ` a ˜ w t ı d x d t ď c ż T ż L ” a G p ˜ ϕ t q ˜ ϕ t ` a G p ˜ ψ t q ˜ ψ t ` a G p ˜ w t q ˜ w t ı d x d t. (4.20)Next, we estimate the source terms. Invoking Assumption p f. q , we find a positive constant c such that ˇˇˇˇż T ż L F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ϕ x ξ d x d t ˇˇˇˇ ď c ż T ż L c p ∇ f qp| ϕ | ` | ψ | ` | w |q| ϕ x | d x d t ď ǫ ż T E Z p t q d t ` c ǫ, B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q , where c p ∇ f q “ ` | ϕ | p ´ ` | ϕ | p ´ ` | ψ | p ´ ` | ψ | p ´ ` | w | p ´ ` | w | p ´ . The above implies that ˇˇˇˇż T ż L ” F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ϕ x ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ψ x ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ w x ı ξ d x d t ˇˇˇˇ ď ǫ ż T E Z p t q d t ` c ǫ, B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q . (4.21)Next, we combine (4.17)-(4.21) with p . q . For sufficiently small ǫ ą
0, we obtain ż T E Z p t q d t ď c “ E Z p T q ` E Z p q ‰ ` ż T ż L ´ ǫ L ` ǫ E p t q d x d t ` c B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q` c B ż T ż L ” a G p ˜ ϕ t q ˜ ϕ t ` a G p ˜ ψ t q ˜ ψ t ` a G p ˜ w t q ˜ w t ı d x d t. (4.22)21he next step is to estimate the integral of E p¨q over the interval r L ` ǫ, L ` ǫ s . To this end,we consider the function r , s Q η P C p , L q defined as follows η p x q “ " η p x q “ , if x P p , L q Y p L , L q ,η p x q “ , if x P p L ` ǫ , L ´ ǫ q . . We start multiplying the equations p . q by ˜ ϕη , ˜ ψη and ˜ wη , respectively, and after integrateover r , T s ˆ r , L s , we add the kinetic energy ş T ş L “ ρ ˜ ϕ ` ρ ˜ ψ t ` ρ ˜ w t ‰ η d x d t to obtain ż T ż L E p t q η d x d t “ ´ ż L ” ρ ˜ ϕ t ˜ ϕ ` ρ ˜ ψ t ˜ ψ ` ρ ˜ w t ˜ w ı η d x ˇˇˇ T ` ż T ż L ” ρ ˜ ϕ ` ρ ˜ ψ t ` ρ ˜ w t ı η d x d t ´ ż T ż L ” k p ˜ ϕ x ` ˜ ψ ` l ˜ w q ˜ ϕ ` b ˜ ψ x ˜ ψ ` k l p ˜ w x ´ l ˜ ϕ q ˜ w ı η d x d t ´ ż T ż L ” a G p ˜ ϕ t q ˜ ϕ ` a G p ˜ ψ t q ˜ ψ ` a G p ˜ w t q ˜ w ı η d x d t ´ ż T ż L ” F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ϕ ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ψ ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ w ı η d x d t. (4.23)We shall estimate the right-hand side of (4.23). To this end, we repeat the pattern of estimates(4.17)-(4.21) to find ż T ż L E p t q η d x d t ď c “ E Z p T q ` E Z p q ‰ ` c B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q` c B ż T ż L ” a G p ˜ ϕ t q ˜ ϕ t ` a G p ˜ ψ t q ˜ ψ t ` a G p ˜ w t q ˜ w t ı d x d t. (4.24)Applying the estimate p . q above in p . q , we obtain ż T E Z p t q d t ď c “ E Z p T q ` E Z p q ‰ ` c B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q` c B ż T ż L ” a G p ˜ ϕ t q ˜ ϕ t ` a G p ˜ ψ t q ˜ ψ t ` a G p ˜ w t q ˜ w t ı d x d t. (4.25)Next, we estimate damping terms on the right-hand side of p . q . Multiply the equations(4.15) by ϕ t , ψ t , w t , respectively. Then we find that ż T ż L ” a G p ˜ ϕ t q ˜ ϕ t ` a G p ˜ ψ t q ˜ ψ t ` a G p ˜ w t q ˜ w t ı d x d t ´ E Z p q ` E Z p T q“ ´ ż T ż L ” F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ϕ t ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ψ t ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ w t ı d x d t. (4.26)Based on estimate (4.21), we obtain ˇˇˇˇż T ż L ” F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ϕ t ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ψ t ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ w t ı d x d t ˇˇˇˇ ď ǫ ż T E Z p t q d t ` c ǫ, B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q . (4.27)22sing both p . q and p . q , we find ż T ż L ” a G p ˜ ϕ t q ˜ ϕ t ` a G p ˜ ψ t q ˜ ψ t ` a G p ˜ w t q ˜ w t ı d x d t ď E Z p q ´ E Z p T q ` ǫ ż T E Z p t q d t ` c ǫ, B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q . (4.28)We return to (4.25) and obtain by use of (4.28), with ǫ ą ż T E Z p t q d t ď p c ´ c B q E Z p T q ` p c ` c B q E Z p q ` c B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q . (4.29)The next step is to estimate the energy E Z p¨q . To this end, we use the multipliers ϕ t , ψ t , ˜ w t for p . q . Then, after integration, we find T E Z p T q “ ż T E Z p t q d t ´ ż T ż Ts ż L ” a G p ˜ ϕ t q ˜ ϕ t ` a G p ˜ ψ t q ˜ ψ t ` a G p ˜ w t q ˜ w t ı d x d t d s ´ ż T ż Ts ż L ” F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ϕ t ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ψ t ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ w t ı d x d t d s. (4.30)Now, the forcing assumptions give ż L ” F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ϕ t ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ψ t ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ w t ı d x ď T E Z p t q ` c T, B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q . The above implies ż T ż Ts ż L ” F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ϕ t ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ ψ t ` F p ˜ ϕ, ˜ ψ, ˜ w q ˜ w t ı d x d t d s ď ż T E Z p t q d t ` c T, B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q . We combine the above estimates with (4.30)
T E Z p T q ď ż T E Z p t q d t ` c T, B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q . (4.31)Next, we using estimate (4.29) in (4.31) we arrive at T E Z p T q ď p c ´ c B q E Z p T q ` p c ` c B q E Z p q ` c T, B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q . Taking T ą c , we find E Z p T q ď p c ` c B q T ´ p c ´ c B q E Z p q ` c T, B l.o.t p ˜ ϕ, ˜ ψ, ˜ w q . Using standard stabilization arguments, we obtain the existence of positive constants c “ c B and ω “ ω B such that } Z p t q} H ď c } Z p q} H e ´ ωt ` c sup σ Pr ,t s ” } ϕ p σ q} p ` } ψ p σ q} p ` } w p σ q} p ı . Therefore, the inequality p . q holds with X “ r H p , L qs , Y “ r L p , L qs , b p t q “ c e ´ ωt , c p t q “ c and with compact semi-norm rp ˜ ϕ, ˜ ψ, ˜ w qs X “ } ˜ ϕ } p ` } ˜ ψ } p ` } ˜ w } p . .2.3 Proof of the main result - Theorem 4.3 Proof of Theorem p H, S p t qq isa gradient quasi-stable system. Moreover, by inequality (4.9) one can see that the Lyapunovfunction defined as the energy E Z p¨q satisfies the following: (i) Φ p¨q is bounded from aboveon any bounded set and (ii) the Φ p R q “ Z P H | Φ p Z q ď R ( is bounded for every R . Toconclude the proof, we note that if Z P N , then Z “ p ϕ, ψ, w, , , q solves the stationaryproblem $&% ´ k p ϕ x ` ψ ` lw q x ´ k l p w x ´ lϕ q ` f p ϕ, ψ, w q “ , ´ bψ xx ` k p ϕ x ` ψ ` lw q ` f p ϕ, ψ, w q “ , ´ k p w x ´ lϕ q x ` kl p ϕ x ` ψ ` lw q ` f p ϕ, ψ, w q “ . (4.32)Multiplying in L p , L q the equations in p . q by p ϕ, ψ, w q , we find b } ψ x } ` k } ϕ x ` ψ ` lw } ` k } w x ` lϕ } “ ´ ż L ∇ F p u, v, w q¨p u, v, w q d x. Now, we use Assumption (f.1) to show „ ´ αβL π ” } ϕ x } ` } ψ x } ` } w x } ı ď βc F L. Therefore, the set of stationary solutions N is bounded. This completes the proof. Appendix: Well-possednes for overdetermined wave equations
In this appendix we will guarantee the well-posedness for the system presented in (1.2)with overdetermined condition.
Theorem A.4.
Let L ą and T ą large enough. If the Problem (1.2) - (1.4) satisfies (1.5) - (1.9) with supplementary condition p u , ¨ ¨ ¨ , u n q “ p , ¨ ¨ ¨ , q in ω ˆ r , T s , (A.33) with ω Ă r , L s as in (3.34) . Then, the overdetermined problem is well-posed and generatesa strongly continuous semigroup over the Hilbert space H ω j p M i q ” " p u , ¨ ¨ ¨ , u n , v , ¨ ¨ ¨ , v n q ˇˇˇˇ u i P H p M i q , v i P L p M i q ,u i “ v i “ in ω, i “ , ¨ ¨ ¨ , n * , where M i “ pr , L s , g i q .Proof. First, for the state vector Z p t q ” p u , ¨ ¨ ¨ , u n , B t u , ¨ ¨ ¨ , B t u n q J , the Problem (1.2)-(1.4)is equivalent to the following vectorial Cauchy problem B t Z p t q ` A Z p t q “ F p Z p t qq , Z p q “ p u , ¨ ¨ ¨ , u n , u , ¨ ¨ ¨ , u n q J , (A.34)24ith operators defined by A “ „ ´ I | V p M i q B | V p M i q , B “ »———– ´ ∆ ¨ ¨ ¨ ´ ∆ ¨ ¨ ¨ ¨ ¨ ¨ ...0 0 0 ´ ∆ n fiffiffiffifl , I “ p δ ij q n ˆ n and F p Z q “ »————————– f p u , ¨ ¨ ¨ , u n q ... f n p u , ¨ ¨ ¨ , u n q fiffiffiffiffiffiffiffiffifl . The domain of operator A is defined by D p A q “ r V p M i qs n ˆ r V p M i qs n where V p M i q ” p u , ¨ ¨ ¨ , u n q P D p B q | p u , ¨ ¨ ¨ , u n q “ p , ¨ ¨ ¨ , q in ω ( ,V p M i q ” p v , ¨ ¨ ¨ , v n q P D p B q | p v , ¨ ¨ ¨ , v n q “ p , ¨ ¨ ¨ , q in ω ( and D p B q “ D p´ ∆ q ˆ ¨ ¨ ¨ ˆ D p´ ∆ n q “ H p M q X H p M q ˆ ¨ ¨ ¨ H p M n q X H p M n q . The finite energy space for (A.34) is the Hilbert space defined by H ω p M i q ” p u , ¨ ¨ ¨ , u n , v , ¨ ¨ ¨ , v n q P H p M i q | u i “ v i “ ω, @ i “ , ¨ ¨ ¨ , n ( , where H p M i q ” r D p B qs n ˆ L p M i q and L p M i q ” L p M q ˆ ¨ ¨ ¨ ˆ L p M n q . Using classical semigroup theory, one can establish existence and uniqueness of a solu-tion to the Cauchy problem (A.34). Moreover, the solution operator generates a stronglycontinuous semigroup T M i p t q : H w p M i q Ñ H w p M i q , defined by p u , ¨ ¨ ¨ , u n , u , ¨ ¨ ¨ , u n q ÞÑ p u p t q , ¨ ¨ ¨ , u n p t q , B t u p t q , ¨ ¨ ¨ , B t u n p t qq , t ě , where p u , ¨ ¨ ¨ , u n , B t u , ¨ ¨ ¨ , B t u n q is the weak solution corresponding to the initial data p u , ¨ ¨ ¨ , u n , u , ¨ ¨ ¨ , u n q . In addition, t T M i p t qu t ě is also strongly continuous semigroup on H ω p M i q satisfying the com-patibility condition (A.33). Remark A.1.
It is not difficult to show that if (A.33) is fulfilled, then we also have pB t u , ¨ ¨ ¨ , B t u n q “ p , ¨ ¨ ¨ , q in ω ˆ r , T s . unding: The first author is partially supported by CNPq grant 312529/2018-0. The thirdauthor is supported by INCTMat-CAPES grant 88887.507829/2020-00.
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Special issuededicated to the memory of Jacques-Louis Lions. Appl. Math. Optim. (2002), no. 2-3,331–375.[29] A. Wehbe and W. Youssef, Exponential and polynomial stability of an elastic Bressesystem with two locally distributed feedbacks , J. Math. Phys. 51 (2010), article 103523, 17pp. 27 mail addresses • T. F. Ma: [email protected] • R. N. Monteiro: [email protected] ••