Attractors for locally damped Bresse systems and a unique continuation property

arXiv:2102.12025v1 [math.AP] 24 Feb 2021 Attractors for locally damped Bresse systems and a unique continuation property To Fu Ma Department of Mathematics, University of Bras´ılia, 70910-900 Bras´ılia, DF, Brazil Rodrigo N. Monteiro 1 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 by a set of three coupled wave equations. The main objective is to establish the existence of global attractors for dynamics of semilinear problems with localized damping. In order to deal with localized damping a unique continuation property (UCP) is needed. Therefore we also provide a suitable UCP for Bresse systems. Our strategy is to set the problem in a Riemannian geometry framework and see the system as a single equation with different Riemann metrics. Then we perform Carleman-type estimates to get our result. Keywords: Bresse system, unique continuation, localized damping, Riemannian manifold, global attractor. Contents 1 Introduction 2 2 A Riemannian geometry framework 5 2.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Geometry on the wave system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Unique continuation property (UCP) 8 3.1 Analysis of the coupled system - Problem (1.2) . . . . . . . . . . . . . . . . . . 10 3.2 Carleman estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 AnewUCP...................................... 13 1Corresponding author. 1 4 Dynamics of locally damped Bresse systems 14 4.1 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Global attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2.1 Abstract existence theorems . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2.2 Gradient structure and quasi-stability . . . . . . . . . . . . . . . . . . . 18 4.2.3 Proof of the main result - Theorem 4.3 . . . . . . . . . . . . . . . . . . . 24 1 Introduction The Bresse system is a model for circular beams given by three coupled wave equations, namely, $ & % ρ1ϕtt ´kpϕx`ψ`ℓwqx´k0ℓpwx´ℓϕq “ 0 in p0, Lqˆp0,8q, ρ2ψtt ´bψxx `kpϕx`ψ`ℓwq “ 0 in p0, Lqˆp0,8q, ρ1wtt ´k0pwx´ℓϕqx`kℓpϕx`ψ`ℓwq “ 0 in p0, Lqˆp0,8q. The functions ϕ“ϕpx, tq,ψ“ψpx, tq,w“wpx, tqcorrespond to the vertical displacement, shear angle and longitudinal displacement at a point xP p0, Lqand time tě0, respectively. The coefficients are all positive constants defined by ρ1“ρA,ρ2“ρI,k“KAG,b“EI and k0“AE, where the quantities ρ,A,I,K,Gand Edenote respectively, material density, cross-sectional area, second moment of the cross-section area, a shear factor, shear modulus and modulus of elasticity. In addition, ℓą0 denotes the beam’s curvature. Its mathematical modeling can be found in [3, 14] and it is worth observing that when ℓ“0 the arched beam reduces to the Timoshenko beam [27]. The Bresse system and its viscoelastic and thermoelastic extensions were studied by many authors. Roughly speaking, most of results are concerned with a certain asymptotic stability dichotomy. Indeed, by analogy to the Timoshenko system, the Bresse system having damping terms in one or two of its equations is exponentially stable if and only if satisfies the equal wave speeds condition ρ1 k“ρ2 band k“k0.(1.1) Such a condition was firstly observed for Timoshenko systems in [25]. Otherwise only polynomial 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 systems by adding a localized damping in each one of its three equations, without assuming the speed condition (1.1). This is quite interesting since the equal speed assumption can not be realized physically, cf. [20]. The main objective of this paper is to establish existence of global attractors for dynamics of a semilinear Bresse system with locally defined damping (see problem 4.1), without assuming condition (1.1). Our approach is very different from the above one in [7]. Indeed, one of ingredients for obtaining exponential stability of wave equations with localized damping is a unique continuation property (UCP). To our purpose, the UCP says whether a wave equation that vanishes in a subdomain must be identically null. In [7] they have used a UCP derived from Holmgren uniqueness theorem, which is only valid for equations with analytic coefficients. 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. 2 More precisely, we discuss the unique continuation property for coupled wave equations of the form $ ’ ’ ’ & ’ ’ ’ % B2 tu1´∆1u1“f1pu1, u2,¨¨¨, unqin p0, Lqˆp0,8q, B2 tu2´∆2u2“f2pu1, u2,¨¨¨, unqin p0, Lqˆp0,8q, . . . B2 tun´∆nun“fnpu1, u2,¨¨¨, unqin p0, Lqˆp0,8q, (1.2) where for i“1,¨¨¨, n the following is assumed: 1. We consider on the system (1.2) the Dirichlet boundary conditions uip0, tq “ uipL, tq “ 0@tP p0,8q,(1.3) and initial data pu1,Btu1,¨¨¨, un,Btunqˇˇt“0“ pu0 1, u1 1,¨¨¨, u0 n, u1 nqin p0, Lq.(1.4) 2. Given γią0, the operator ∆irepresents the one-dimensional Laplacian operator with wave propagation velocity ?γi, defined by ∆i“γiB2 x.(1.5) 3. The symbols fidenote the coupling functions with energy level terms such that fiP L2p0, T ;L2p0, Lqq and fipu1, u2,¨¨¨, unq “ n ÿ j“1 pi jBxuj` n ÿ j“1 qi juj,(1.6) where pi j, qi jPL2p0, T ;L2p0, Lqq. Additionally, given Tą0 there exists a constant CTą0 such that żT 0żL 0|fipu1, u2,¨¨¨, unq|2dxdtďCTżT 0 Fuptqdt, (1.7) where Fuptqrepresents the energy of the system (1.2) defined by Fuptq “ n ÿ i“1 Fu,iptq(1.8) and Fu,iptqis the energy of the i-th equation of the system given by Fu,iptq “ żL 0”|ui|2`γi|Bxui|2`|Btui|2ıdx. (1.9) The UCP has been extensively used in the analysis of exact controllability, exponential stability, and in the theory of attractors for locally damped wave equations. The stabilization problem 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 the 3 authors assume localized damping and their proof uses a UCP for the wave equation based on Holmgren Theorem. In [4, 5] the authors treated the nonlinear case exhibiting an exponential decay of the energy with sharp damping region, roughly speaking, a damping region with arbitrarily small measure. In this case, a new UCP is proven by means of energy estimates and a escape vector field based on [16]. The study of the existence of global attractors for the wave equation with external forces of 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 the damping regions imposed on the damping parameter. Therefore different types of UCP are needed. Reference [11] uses the UCP given in [23]. In [8, 19], the authors apply the UCP corresponding 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 different Riemann metrics. Then we show how Carleman-type estimates obtained in [28] can be used to obtain a UCP for our system p1.2qunder above assumptions on the functions fip¨q pi“ 1,2,¨¨¨, nq. Our paper is organized in the following way. Our UCP - Theorem 3.2 - will follow after a series of comparison results between reference [28]. piqWe begin in Section 2 with a Riemannian geometry background material. piiqAfter, in Section 3, we introduce a preliminary material that will lead the corresponding Carleman-type estimates for the Problem p1.2q. Finally, as a consequence of Carleman-type estimates, we then achieve our goal, the proof of the Theorem 3.2. For completeness, to the best of our knowledge, this is the first UCP result for coupled wave equations. In the second part of the paper, we establish the existence of global attractors for the Bresse 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 foundation and dissipation acting on the whole domain. There, UCP and observability inequalities were not necessary. In this sense, our application improves the previous results on the existence of long-time dynamics of the Bresse system allowing the dissipation to be localized in an arbitrary subset of p0, Lq. The outline of the remainder of the paper is the following: piiiqIn Section 4, we introduce the semilinear Bresse system with localized dissipation along with the well-posedness result and energy estimates. The main result is the Theorem 4.3 and whose proof is based on the following strategy: paqwe first show the existence of a strictly Lyapunov function for the associated dynamical system by using the new UCP stated in Theorem 3.2 and pbqintroducing observability inequalities, we prove the asymptotic smoothness of the problem using the abstracts results on the recent theory of quasi-stable systems [9]. Here, we also mention the importance of the UCP for the proof of a strictly Lyapunov function - see Definition 4.3. pivq The Appendix is devoted to the well-posedness result for wave equations with over-determined conditions. 4 2 A Riemannian geometry framework 2.1 Basic notation Let pM, gqbe an n-dimensional, compact Riemannian manifold, with smooth boundary and smooth metric. The tangent space on Mat pis denoted by TpMand fix a coordinate system px1,¨¨¨, xnqthen pBx1,¨¨¨,Bxnqrepresents the associated coordinate vector fields. In this case gpX, Y q “ xX, Y y “ n ÿ i,j“1 gijαiβj,|X|2“ xX, Xy, where X“ n ÿ i“1 αiBxi, Y “ n ÿ i“1 βiBxiin TpMfor some pPM, (2.10) and gij “@Bxi,BxjD. Note that |¨| represents the norm with respect to the metric gp¨,¨q. In particular, we denote the inner product gp¨,¨q by the matrix pgijqnˆnand its inverse by pgij qnˆn. The tangent and cotangent bundle of Mare respectively detonate by T M and T˚M. The symbol Ddenotes the Levi-Civita connection of Msuch that for two vector fields Xand Y on Mgiven by (2.10) the following equality hods true DXY“ n ÿ i,k“1«αiBxiβkBxk` n ÿ j“1 βjΓk ijBxkff, where Γk ij represent the Christoffel symbols. Let f:MÑRand HPTpMfor all pPM. 1. If fPC1pMqthen the differential Df :T M ÑRrepresents the gradient of the connection Don fand DfpHq “ DHf“Hpfq “ x∇f, Hy, where ∇is the usual gradient defined in a coordinate system by ∇f“ n ÿ i,j“1 gijBxifBxj.(2.11) Thanks to the musical isomorphism we will identify Df with ∇f. Here, we often denote Df by ∇f. In particular, if tE1,¨¨¨, Enurepresents an orthonormal basis of TpMand H“řn i“1hiEithen DfpHq “ Hpfq “ n ÿ i“1 hiEipfq. 2. If fPC2pMqthen D2frepresents the Hessian of fsuch that for all YPT M D2fp¨, Y q “ DpDfqp¨, Y q “ DYp∇fp¨qq :T M ÑR, 5 with DYp∇fpXqq “ xDXp∇fq, Y y,@XPT M. In particular D2fpX, Xq “ xDXp∇fq, Xy,@XPT M. 3. Let fPC3pMq. The function fis strictly convex in the metric gif and only if D2fpX, Xq ą 0 for all XPT M. 4. If tE1,¨¨¨, Enurepresents an orthonormal basis of TpM, then the divergent of His defined as divpHq “ n ÿ i“1xDEiH, Eiy. 5. If fPC1pMqthen divpfHq “ fdivpHq`Hpfq. 6. For a function fPC2pMqwe define the Laplace-Beltrami operator ∆ by ∆f“divp∇fq “ n ÿ i“1xDEip∇fq, Eiy “ n ÿ i“1 D2fpEi, Eiq. 7. The covariant derivate DH is the bilinear form given by DHpX, Y q “ xDXH, Y y,@X, Y PT M. In particular if fPC2pMq Dp∇fqpX, Y q “ D2fpX, Y q,@X, Y PT M. 8. If fPC1pMqthen x∇f, ∇pHpfqqy “ DHp∇f, ∇fq` 1 2divp|∇f|2Hq´ 1 2|∇f|2divH. Remark 2.1. Let Ωan open bounded, connected, compact subset of Mwith smooth boundary BΩand fPC3pMqa strictly convex function in the metric g. Then by translating and rescaling [28, Remark 1.2], the function fsatisfies the following conditions D2fpX, Xq ě |X|2,@pPΩ,@XPTpM, min Ω fpxq ” mą0.(2.12) 6 2.2 Geometry on the wave system Let us consider Rwith usual topology and xthe natural coordinate system. In particular, for each xPRthe tangent space is TxR“Rx“R. For a fixed iP t1,¨¨¨, nu, we begin defining the metrics giassociated with ∆iby gipX, Y q “ xX, Y ygi“γ´1 iαβ, (2.13) with corresponding norm |X|2 gi“ xX, Xygi, where X“αBx, Y “βBxPRx,for each xPR. Recalling assumption γią0, one can see that the pairs pR, giqare Riemannian manifolds. In this case, the Levi-Civita connection of pR, giqwill be denoted by Dgi. Here, the symbol pR, g0qdenotes the Rspace with Euclidean metric and we use the following notations for the metric and norm g0pX, Y q “ X¨Yand |X|g0“ |X|,@X, Y PRx,@xPR.(2.14) Important properties of the above metrics are stated in the following lemma. Although most of these results are followed straightforwardly from the known results, they are crucial for what follows. So for the convenience of the reader, we give their proofs here. Lemma 2.1. Let p0, Lq Ă Rfor some Lą0. If xbe the natural coordinate system in R, f, h PC1pr0, Lsq and X“αBx, Y “βBxvector fields. Then 1.xX, γiYygi“X¨Y; 2.∇gif“γiBxfBx; 3. Xpfq “ x∇gif, Xygi“αBxf; 4.x∇gif, ∇gihygi“γiBxfBxh; 5.if νrepresents the unit outward normal vector for pr0, Ls, giqthen x∇gifp0q, νp0qygi“ ´?γiBxfp0q,x∇gifpLq, νpLqygi“?γiBxfpLq. Proof. In light of (2.13) and (2.14), one can easily see that Item 1 holds true. In fact, by definition xX, γiYygi“γ´1 iγiαβ “X¨Y. (2.15) To prove Item 2, we recall the definition (2.11) to find ∇gif“γiBxfBx. Reasoning analogously to (2.15) and assuming hPC1pr0, Lsq, we have the validity of Items 2-4. To conclude, let us show Item 5. Firstly, note that if pP t0, Luand tBxurepresents the associated coordinate vector field for Tpr0, Ls, then x˘?γiBx,˘?γiBxygi“1. 7 In particular E“ ˘?γiBxrepresents an orthonormal basis for pPTpM. Let ν“E, therefore ∇gif“γiBxfBx“ ˘?γiBxfν and x∇gif, νygi“ ˘?γiBxf. Note that νp0q “ ´?γiBxand νpLq “ ?γiBx. We complete the proof of Lemma. Remark 2.2. The Hessian of fPC2pRqwith respect to the metric giis given by D2 gifpX, Xq “ xDgi,X p∇gifq, Xygi“α2γ´1 iγiB2 xf“α2B2 xf, where X“αBx.We observe that the Hessian of fis positive if and only if B2 xfis positive. 3 Unique continuation property (UCP) As already observed Carleman estimates are an important tool for proving unique continuation property for solutions to partial differential equations [6, 15, 28]. In this section, we prove the Carleman estimates for the Problem p1.2q. In what follows we shall use the following notations. 1. Let L0a real number in r0, Ls. We define Ω1and Ω2the subsets of r0, Lsas follows Ω1“ p0, L0qand Ω2“ pL0, Lq.(3.16) Note that Ω1YΩ2“ r0, Ls. For εą0, we will also consider the following subsets of Ω1and Ω2 V1“´0, L0´ε 4¯Xr0, Ls, V2“´L0`ε 4, L¯Xr0, Ls, ω “´L0´ε 2, L0`ε 2¯Xr0, Ls. 2. There exist strictly convex functions d1:r0, Ls Ñ Rand d2:r0, Ls Ñ Rsuch that d1pxq “ 1 2px`Lq2,and d2pxq “ 1 2px´2Lq2.(3.17) In particular, if xdenotes the natural coordinate system, Lemma 2.1 implies that ∇gid1“γipx`LqBx,∇gid2“γipx´2LqBxpi“1,¨¨¨, nq, and D2 gidjpX, Xq “ α2,for all xP r0, Lsand X“αBxPTxr0, Ls pj“1,2q. The functions djp¨q pj“1,2qhave the following properties: Lemma 3.1. Under the above definitions, the functions djp¨q pj“1,2qsatisfy 1. djPC8pr0, Lsq; 2. D2 gidjpX, Xq “ |X|gi,for all xPΩjand XPTxr0, Ls; 8 3.inf Ωj|∇gidj|gią0; 4.min Ωj dj“L2 2ą0. Moreover, if νrepresents the unit outward normal vector then 5.x∇gidjpxq, νpxqygiă0on t0, LuXVj. 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 kij ”inf xPΩj |∇gidj|2 gi dją4,@i“1,¨¨¨, n, @j“1,2.(3.18) In what follows, for fixed iP t1,¨¨¨, nuand j“1,2, we define T2 0,j ”4 max xPΩj djpxq.(3.19) Let TąmaxtT0,1, T0,2uand by p3.19qthere exist δą0 and c“cδP p0,1qsatisfying cT 2ą4 max xPΩj djpxq`4δ. (3.20) In above context, we define functions φj: ΩjˆRÑRPC3pΩjqby φjpx, tq ” djpxq´cˆt´T 2˙2 ,px, tq P Ωjˆr0, T s. The following properties are valid for φjp¨q: pφ.1qFor the constant δą0 φjpx, 0q “ φjpx, T q “ djpxq´cT2 4ďmax j“t1,2u˜max xPΩj djpxq¸´cT2 4ď ´δ, uniformly in Ωj. pφ.2qThere are t0and t1with 0 ăt0ăT{2ăt1ăT, such that min j“1,2˜min px,tqPΩjˆrt0,t1s φjpx, tq¸ěσ, (3.21) for σP´0,minΩjdj¯, where min Ωj dj“L2 2(Lemma 3.1). 9 Next, let A:DpAq Ă HÑHbe the differential operator AZ” » — — — — — — – ˜ϕ ˜ ψ ˜w kρ1´1pϕx`ψ`lwqx`k0lρ1´1pwx´lϕq bρ2´1ψxx ´kρ2´1pϕx`ψ`lwq k0ρ1´1pwx´lϕqx´klρ´1 1pϕx`ψ`lwq fi ffi ffi ffi ffi ffi ffi fl , with domain DpAq “ “H2p0, LqXH1 0p0, Lq‰3ˆH1 0p0, Lq3. Next, let be B:DpBq “ HÑHthe damping operator BZ” » — — — — — — – 0 0 0 ´α1pxqρ´1 1g1p˜ϕq ´α2pxqρ´1 2g2p˜ ψq ´α3pxqρ´1 1g3p˜wq fi ffi ffi ffi ffi ffi ffi fl . Finally, by F:HÑH, we represent the source terms operator FZ” » — — — — — — – 0 0 0 ´ρ´1 1f1pϕ, ψ, wq ´ρ´1 2f2pϕ, ψ, wq ´ρ´1 1f3pϕ, ψ, wq fi ffi ffi ffi ffi ffi ffi fl . Now, using the definitions of operators A, B, F, we can abstract represent the problem as follows d dtZptq´pA`BqZptq “ FpZptqq,Zp0q ” Z0“ pϕ0, ψ0, w0, ϕ1, ψ1, w1q,(4.5) where Zptq “ pϕptq, ψptq, wptq,˜ϕptq,˜ ψ, ˜wptqq with ˜ϕ“ϕt,˜ ψ“ψt,˜w“wt. 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 pf.1q-pg.1q. Then for any initial data Z0PHand Tą0, the Cauchy problem p4.5qadmits a unique weak solution ZPCpr0, T s;Hqthat depends continuously on the initial data and is given by the variation of parameters formula Zptq “ epA`BqtZ0`żt 0 epA`Bqpt´sqFpZpsqqds, t P r0, T s.(4.6) Moreover, if Z0PDpAqthen the solution is strong. 16 The well-posedness result stated above is known and can be found in [18]. Theorem 4.1, in particular, implies that the map HQZ0ÞÑ Zptq “ pϕ, ψ, w, ϕt, ψt, wtq, where Zptqsolves (4.5), defines a strongly continuous semigroup tSptqutě0on H. Energy. Let Zptq “ pϕptq, ψptq, wptq, ϕtptq, ψtptq, wtptqq be a solution of p4.1q-p4.3q. The energy is defined by the following functional EZptq ” EZptq`żL 0 Fpϕ, ψ, wqdx“ }Zptq}2 H`żL 0 Fpϕ, ψ, wqdx. (4.7) The weak solution Zptq “ pϕptq, ψptq, wptq, ϕtptq, ψtptq, wtptqq satisfies the energy identity EZptq`żt sżL 0”α1pxqg1pϕtqϕt`α2pxqg2pψtqψt`α3pxqg3pwtqwtıdxdτ“EZpsq,(4.8) for all 0 ďsăt. As in [18], the energy EZp¨q (4.7) and the norm }¨}H(4.4) satisfy CE}Zptq}2 H´LcFďEZptq ď }Zptq}2 H`cEp1`}Zptq}p`1 Hq,@tě0,(4.9) for some positive constants CEand cE. 4.2 Global attractors 4.2.1 Abstract existence theorems Some essential definitions and results from the theory of attractors for gradient systems is collected Definition 4.1. A global attractor for a dynamical system pH, Sptqq, with evolution operator tSptqutě0on a complete metric space His defined as a a compact set AĂHthat is fully invariant, that is SptqA“Afor all tě0, and uniformly attracts all bounded subsets of H lim tÑ8 sup !distHpSptqZ,Aq|ZPB)“0,for any bounded set BĂH. Definition 4.2. The fractal dimension of a compact set AĂHin a metric space His defined as dimH fpAq “ lim sup εÑ0 ln NεpAq lnp1{εq, where NεpAqis the minimal number of closed balls of radius εwhich cover the set A. To ascertain the existence of a global attractor, we use the concept of gradient and quasistable dynamical systems. The global attractor for this systems admits additional structure and properties: (i) the attractor for gradient systems has a regular structure, that is, the attractor 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 YĂHbe a forward invariant set of a dynamical system pH, Sptqq.piq A continuous functional Φ : YÑRis said to be a Lyapunov function on Yfor pH, Sptqq if the map tÞÑ ΦpSptqZqis non-increasing for any ZPY.piiqThe Lyapunov function is said to be strict on Yif the equation if ΦpSptqZq “ ΦpZqfor all tą0for some ZPYimplies that yis a stationary point of pH, Sptqq.piiiqThe dynamical system pH, Sptqq is said to be gradient if there exists a strict Lyapunov function on the whole phase space H. 17 Definition 4.4. Let X, Y be reflexive Banach spaces, Xcompactly embedded in Y. We consider a dynamical system pH, Sptqq with H“XˆYand evolution operator defined by SptqZ“ puptq, utptqq,Z“ pup0q, utp0qq P H, (4.10) where the function upossess the property uPCpr0,8q;XqXC1pr0,8q;Yq.(4.11) A dynamical system of the the form p4.10qwith regularity p4.11qis said to be quasi-stable on a set BĂH, if there exist a compact semi-norm r¨sXon Xand non-negative scalar functions aptq, bptq, cptq, such that, piqaptq, bptqare locally bounded on r0,8q,piiqbptq P L1p0,8q with limtÑ8 bptq “ 0and piiiqfor any Z1,Z2PBthe following estimates hold true }SptqZ1´SptqZ2}2 Hďaptq}Z1´Z2}2 H,(4.12) and }SptqZ1´SptqZ2}2 Hďbptq}Z1´Z2}2 H`cptqsup 0ăsătru1psq´u2psqs2 X,(4.13) where SptqZi“ puiptq, ui tptqq,i“1,2. Unifying the abstracts results from [9] we arrive at the following criteria for existence and properties of global attractors. Theorem 4.2. Let pH, Sptqq be a gradient quasi-stable dynamical system. Assume its Lyapunov function Φp¨q is bounded from above on any bounded subset of Hand the set ΦpRq “ ZPH|ΦpZq ď R(is bounded for every R. If the set Nof stationary points of pH, Sptqq is bounded, then pH, Sptqq possesses a finite dimensional global attractor Adefined by the unstable manifold emanating from set of stationary solution. Moreover, any trajectory stabilizes to the set Nof stationary points, that is, lim tÑ`8 distHpSptqZ,Nq “ 0,@ZPH. We now state the main result of the present chapter. Theorem 4.3. Under the Assumptions pf.1q-pa.1qthe dynamical system pH, Sptqq generated by the problem p4.1q-p4.3qhas a global attractor Acharacterized by A“M`pNq, where M`pNqis the unstable manifold emanating from N, the set of stationary points of tSptqutě0. 4.2.2 Gradient structure and quasi-stability Our strategy centers on establishing the conditions from the Theorem 4.2. Starting exhibiting the gradient structure for pH, Sptqq and focusing our attention on the strictness of the Lyapunov function where the new observability result stated in Theorem 3.2 plays an essential role in the proof. 18 Proposition 4.1. Let the assumptions of Theorem 4.3be satisfied. Then, pH, Sptqq is a gradient dynamical system. Proof. The dynamical system pH, Sptqq is gradient with full energy EZp¨q - defined in (4.7) - being the strict Lyapunov function Φp¨q. In fact, from identity (4.8), we find that tÑΦpSptqZq is a non-increasing function for any ZPH. Next, we suppose that ΦpSptqZq “ ΦpZq, for all tą0. Then, from identity (4.8), we obtain żt sżL 0”α1pxqg1pϕtqϕt`α2pxqg2pψtqψt`α3pxqg3pwtqwtıdxdτ“0. This shows that ϕt“ψt“wt“0 a.e. in pL1, L2q ˆ r0, T s, where pL1, L2q “ ŞiIi. Thus, pϕ, ψ, wqsatisfies the problem $ & % ρ1ϕtt ´kpϕx`ψ`lwqx´k0lpwx´lϕq`f1pϕ, ψ, wq “ 0 in p0, Lqˆr0, T s, ρ2ψtt ´bψxx `kpϕx`ψ`lwq`f2pϕ, ψ, wq “ 0 in p0, Lqˆr0, T s, ρ1wtt ´k0pwx´lϕqx`klpϕx`ψ`lwq`f3pϕ, ψ, wq “ 0 in p0, Lqˆr0, T s. (4.14) Using the notation u1“ϕt,u2“ψt,u3“wtand taking the derivative in the distributional sense of p4.14q, we find that pu1, u2, u3qsolves the problem $ & % u1 tt ´γ1u1 xx “F1pu1, u2, u3qin p0, Lqˆr0, T s, u2 tt ´γ2u2 xx “F2pu1, u2, u3qin p0, Lqˆr0, T s, u3 tt ´γ2u3 xx “F3pu1, u2, u3qin p0, Lqˆr0, T s, with γ1“k ρ1,γ2“b ρ2,γ3“k0 ρ1and with forcing Fip¨q defined by F1pu1, u2, u3q “ γ1u2 x`γ1u3 x`k0ρ´1 1lpu3 x´lu1q´ρ´1 1Btrf1pϕ, ψ, wqs, F2pu1, u2, u3q “ ´kρ´1 2pu1 x´u2`lu3q´ρ´1 2Btrf2pϕ, ψ, wqs, F3pu1, u2, u3q “ ´γ3lu1 x´kρ´1 1lpu1 x`u2`lu3q´ρ´1 1Btrf3pϕ, ψ, wqs. Now, we apply the UCP - Theorem 3.2 - to conclude that pϕt, ψt, wtq“pu1, u2, u3q“p0,0,0q. Therefore, the solution ZPHmust be stationary. This implies that the energy EZp¨q is strict on H. Our next aim is to show the quasi-stability of pH, Sptqq. According to the Definition 4.4, the difference of two trajectories should obeys estimates p4.12qand p4.13q. Taking the advantage of the locally Lipschitz property of fip¨q and the variation of parameter formula (4.6), one can easily show the validity of p4.12q. Next, by means of multiplier technique, we prove the stabilization inequality p4.13q. Proposition 4.2. Let the assumptions of Theorem 4.3be satisfied. Then, pH, Sptqq is a quasi-stable dynamical system. Proof. The proof is carried out through several energy estimates. In the text that follows, we use the notations ˜v“v1´v2, Gpvq “ gipv1q´gipv2qand Fipvq “ fipv1q´fipv2q, i “1,2,3. 19 First, we shall consider the difference of two trajectories with initial data Z1 0,Z2 0PB, where Bis a bounded subset of H. The corresponding solution SptqpZ1 0´Z2 0q ” Z1´Z2“ p˜ϕ, ˜ ψ, ˜w, ˜ϕt,˜ ψt,˜wtqverifies the following problem $ & % ρ1˜ϕtt ´kp˜ϕx`˜ ψ`l˜wqx´k0lp˜wx´l˜ϕq “ ´a1pxqG1p˜ϕtq´F1p˜ϕ, ˜ ψ, ˜wq, ρ2˜ ψtt ´b˜ ψxx `kp˜ϕx`˜ ψ`l˜wq “ ´a2pxqG2p˜ ψtq´F2p˜ϕ, ˜ ψ, ˜wq, ρ1˜wtt ´k0p˜wx´l˜ϕqx`klp˜ϕx`˜ ψ`l˜wq “ ´a3pxqG3p˜wtq´F3p˜ϕ, ˜ ψ, ˜wq, (4.15) with zero Dirichlet boundary conditions and initial conditions Z1 0´Z2 0. Second, we let ǫ0be a positive real number, such that, ǫ0ď pL2´L1q, where pL1, L2q “ ŞiIi. We consider the following real-function ξp¨q defined as follows ξpxq “ $ & % pλ´1qx, if xP r0, L1`ǫ0q, λpx´L1´ǫ0q`pL1´L2`2ǫ0q{pL1`ǫ0q,if xP pL1`ǫ0, L2´ǫ0s, pλ´1qpx´Lq,if xP rL2´ǫ0, Ls. . Now, we take ˜ϕxξ, ˜ ψxξ, ˜wxξas multipliers for p4.15q. Thus, we find 1 2żT 0żL 0pλ´1qEptqdxdt “ ´żL 0”˜ϕt˜ϕx`˜ ψt˜ ψx`˜wt˜wxıξdxˇˇˇ T 0´1 2żT 0żL2´ǫ0 L1`ǫ0 Eptqdxdt ´żT 0żL 0”kp˜ϕx´˜ ψ`l˜wqp˜ ψ`l˜wq`k0lp˜wx´l˜ϕq˜ϕıξ1dxdt ´żT 0żL 0”a1G1p˜ϕtq˜ϕx`a2G2p˜ ψtq˜ ψx`a3G3p˜wtq˜wxıξdxdt ´żT 0żL 0”F1p˜ϕ, ˜ ψ, ˜wq˜ϕx`F2p˜ϕ, ˜ ψ, ˜wq˜ ψx`F3p˜ϕ, ˜ ψ, ˜wq˜wxıξdxdt, (4.16) where Eptq “ ρ1|˜ϕt|2`ρ2|˜ ψt|2`ρ1|˜wt|2`b|˜ ψx|2`k|˜ϕx`˜ ψ`l˜w|2`k0|˜wx´l˜ϕ|2. Let us estimate the left-hand side of p4.16q. Note that, from the definition of energy, we find EZptq“şL 0Eptqdt“}Z}2 H. Then, we can show that there exists cą0 satisfying ˇˇˇˇżL 0”˜ϕt˜ϕx`˜ ψt˜ ψx`˜wt˜wxıξdxˇˇˇˇďcsup xPr0,LstξpxquEZptq. This last implies that ˇˇˇˇżL 0”˜ϕt˜ϕx`˜ ψt˜ ψx`˜wt˜wxıξdxˇˇˇ T 0ˇˇˇˇďcsup xPr0,LstξpxqupEZpTq`EZp0qq.(4.17) Also using the definition of EZptq, one obtains ˇˇˇˇżT 0żL 0”kp˜ϕx´˜ ψ`l˜wqp˜ ψ`l˜wq`k0lp˜wx´l˜ϕq˜ϕıξ1dxdtˇˇˇˇ ďǫżT 0 EZptqdt`cǫl.o.tp˜ϕ, ˜ ψ, ˜wq, (4.18) 20 with lower order terms defined by l.o.tp˜ϕ, ˜ ψ, ˜wq ” sup σPr0,T s”}˜ϕpσq}2 2p`}˜ ψpσq}2 2p`}˜wpσq}2 2pı. To estimate the damping terms, we use Assumption pg.1qto obtain ˇˇˇˇżT 0żL 0 a1G1p˜ϕtq˜ϕxξdxdtˇˇˇˇďsup xPr0,LstξpxqużT 0żL 0 a1M|ϕ1 t´ϕ2 t|| ˜ϕx|dxdt ďǫ 3żT 0 EZptqdt`cǫżT 0żL 0 a1˜ϕ2 tdxdt. This allows us to conclude the following estimate ˇˇˇˇżT 0żL 0”a1G1p˜ϕtq˜ϕx`a2G2p˜ ψtq˜ ψx`a3G3p˜wtq˜wxıξdxdtˇˇˇˇ ďǫżT 0 EZptqdt`cǫżT 0żL 0”a1˜ϕ2 t`a2˜ ψ2 t`a3˜w2 tıdxdt. (4.19) Let us estimate the kinetic energy in (4.19). Assumption pg.1.qimplies that żT 0żL 0”a1˜ϕ2 t`a2˜ ψ2 t`a3˜w2 tıdxdt ďcżT 0żL 0”a1G1p˜ϕtq˜ϕt`a2G2p˜ ψtq˜ ψt`a3G3p˜wtq˜wtıdxdt. (4.20) Next, we estimate the source terms. Invoking Assumption pf.3q, we find a positive constant csuch that ˇˇˇˇżT 0żL 0 F1p˜ϕ, ˜ ψ, ˜wq˜ϕxξdxdtˇˇˇˇďcżT 0żL 0 cp∇f1qp|ϕ|`|ψ|`|w|q|ϕx|dxdt ďǫżT 0 EZptqdt`cǫ,Bl.o.tp˜ϕ, ˜ ψ, ˜wq, where cp∇f1q “ 1`|ϕ1|p´1`|ϕ2|p´1`|ψ1|p´1`|ψ2|p´1`|w1|p´1`|w2|p´1. The above implies that ˇˇˇˇżT 0żL 0”F1p˜ϕ, ˜ ψ, ˜wq˜ϕx`F2p˜ϕ, ˜ ψ, ˜wq˜ ψx`F3p˜ϕ, ˜ ψ, ˜wq˜wxıξdxdtˇˇˇˇ ďǫżT 0 EZptqdt`cǫ,Bl.o.tp˜ϕ, ˜ ψ, ˜wq. (4.21) Next, we combine (4.17)-(4.21) with p4.16q. For sufficiently small ǫą0, we obtain żT 0 EZptqdtďc“EZpTq`EZp0q‰`1 2żT 0żL2´ǫ0 L1`ǫ0 Eptqdxdt`cBl.o.tp˜ϕ, ˜ ψ, ˜wq `cBżT 0żL 0”a1G1p˜ϕtq˜ϕt`a2G2p˜ ψtq˜ ψt`a3G3p˜wtq˜wtıdxdt. (4.22) 21 The next step is to estimate the integral of Ep¨q over the interval rL1`ǫ, L2`ǫs. To this end, we consider the function r0,1s Q ηPC8p0, Lqdefined as follows ηpxq “ "ηpxq “ 0,if xP p0, L1qYpL2, Lq, ηpxq “ 1,if xP pL1`ǫ0, L2´ǫ0q.. We start multiplying the equations p4.15qby ˜ϕη,˜ ψη and ˜wη, respectively, and after integrate over r0, T sˆr0, Ls, we add the kinetic energy şT 0şL 0“ρ1˜ϕ2`ρ2˜ ψ2 t`ρ1˜w2 t‰ηdxdtto obtain żT 0żL 0 Eptqηdxdt “ ´żL 0”ρ1˜ϕt˜ϕ`ρ2˜ ψt˜ ψ`ρ1˜wt˜wıηdxˇˇˇ T 0`2żT 0żL 0”ρ1˜ϕ2`ρ2˜ ψ2 t`ρ1˜w2 tıηdxdt ´żT 0żL 0”kp˜ϕx`˜ ψ`l˜wq˜ϕ`b˜ ψx˜ ψ`k0lp˜wx´l˜ϕq˜wıη1dxdt ´żT 0żL 0”a1G1p˜ϕtq˜ϕ`a2G2p˜ ψtq˜ ψ`a3G3p˜wtq˜wıηdxdt ´żT 0żL 0”F1p˜ϕ, ˜ ψ, ˜wq˜ϕ`F2p˜ϕ, ˜ ψ, ˜wq˜ ψ`F3p˜ϕ, ˜ ψ, ˜wq˜wıηdxdt. (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 żT 0żL 0 Eptqηdxdtďc“EZpTq`EZp0q‰`cBl.o.tp˜ϕ, ˜ ψ, ˜wq `cBżT 0żL 0”a1G1p˜ϕtq˜ϕt`a2G2p˜ ψtq˜ ψt`a3G3p˜wtq˜wtıdxdt. (4.24) Applying the estimate p4.24qabove in p4.22q, we obtain żT 0 EZptqdtďc“EZpTq`EZp0q‰`cBl.o.tp˜ϕ, ˜ ψ, ˜wq `cBżT 0żL 0”a1G1p˜ϕtq˜ϕt`a2G2p˜ ψtq˜ ψt`a3G3p˜wtq˜wtıdxdt. (4.25) Next, we estimate damping terms on the right-hand side of p4.25q. Multiply the equations (4.15) by ϕt,ψt,wt, respectively. Then we find that żT 0żL 0”a1G1p˜ϕtq˜ϕt`a2G2p˜ ψtq˜ ψt`a3G3p˜wtq˜wtıdxdt´EZp0q`EZpTq “ ´żT 0żL 0”F1p˜ϕ, ˜ ψ, ˜wq˜ϕt`F2p˜ϕ, ˜ ψ, ˜wq˜ ψt`F3p˜ϕ, ˜ ψ, ˜wq˜wtıdxdt. (4.26) Based on estimate (4.21), we obtain ˇˇˇˇżT 0żL 0”F1p˜ϕ, ˜ ψ, ˜wq˜ϕt`F2p˜ϕ, ˜ ψ, ˜wq˜ ψt`F3p˜ϕ, ˜ ψ, ˜wq˜wtıdxdtˇˇˇˇ ďǫżT 0 EZptqdt`cǫ,Bl.o.tp˜ϕ, ˜ ψ, ˜wq. (4.27) 22 Using both p4.26qand p4.27q, we find żT 0żL 0”a1G1p˜ϕtq˜ϕt`a2G2p˜ ψtq˜ ψt`a3G3p˜wtq˜wtıdxdt ďEZp0q´EZpTq`ǫżT 0 EZptqdt`cǫ,Bl.o.tp˜ϕ, ˜ ψ, ˜wq. (4.28) We return to (4.25) and obtain by use of (4.28), with ǫą0 small enough, the following estimate żT 0 EZptqdtď pc´cBqEZpTq`pc`cBqEZp0q`cBl.o.tp˜ϕ, ˜ ψ, ˜wq.(4.29) The next step is to estimate the energy EZp¨q. To this end, we use the multipliers ϕt,ψt, ˜wt for p4.15q. Then, after integration, we find T EZpTq “ żT 0 EZptqdt´żT 0żT sżL 0”a1G1p˜ϕtq˜ϕt`a2G2p˜ ψtq˜ ψt`a3G3p˜wtq˜wtıdxdtds ´żT 0żT sżL 0”F1p˜ϕ, ˜ ψ, ˜wq˜ϕt`F2p˜ϕ, ˜ ψ, ˜wq˜ ψt`F3p˜ϕ, ˜ ψ, ˜wq˜wtıdxdtds. (4.30) Now, the forcing assumptions give żL 0”F1p˜ϕ, ˜ ψ, ˜wq˜ϕt`F2p˜ϕ, ˜ ψ, ˜wq˜ ψt`F3p˜ϕ, ˜ ψ, ˜wq˜wtıdxď1 TEZptq`cT,Bl.o.tp˜ϕ, ˜ ψ, ˜wq. The above implies żT 0żT sżL 0”F1p˜ϕ, ˜ ψ, ˜wq˜ϕt`F2p˜ϕ, ˜ ψ, ˜wq˜ ψt`F3p˜ϕ, ˜ ψ, ˜wq˜wtıdxdtds ďżT 0 EZptqdt`cT,Bl.o.tp˜ϕ, ˜ ψ, ˜wq. We combine the above estimates with (4.30) T EZpTq ď 2żT 0 EZptqdt`cT,Bl.o.tp˜ϕ, ˜ ψ, ˜wq.(4.31) Next, we using estimate (4.29) in (4.31) we arrive at T EZpTq ď 2pc´cBqEZpTq`2pc`cBqEZp0q`cT,Bl.o.tp˜ϕ, ˜ ψ, ˜wq. Taking Tą4c, we find EZpTq ď 2pc`cBq T´2pc´cBqEZp0q`cT,Bl.o.tp˜ϕ, ˜ ψ, ˜wq. Using standard stabilization arguments, we obtain the existence of positive constants c1“cB and ω“ωBsuch that }Zptq}2 Hďc1}Zp0q}2 He´ωt `c1sup σPr0,ts”}ϕpσq}2 2p`}ψpσq}2 2p`}wpσq}2 2pı. Therefore, the inequality p4.13qholds with X“ rH1 0p0, Lqs3,Y“ rL2p0, Lqs3,bptq “ c1e´ωt, cptq “ c1and with compact semi-norm rp˜ϕ, ˜ ψ, ˜wqs2 X“ }˜ϕ}2 2p`}˜ ψ}2 2p`}˜w}2 2p. 23 4.2.3 Proof of the main result - Theorem 4.3 Proof of Theorem 4.3: From Proposition 4.1 and Proposition 4.2, we have that pH, Sptqq is a gradient quasi-stable system. Moreover, by inequality (4.9) one can see that the Lyapunov function defined as the energy EZp¨q satisfies the following: (i) Φp¨q is bounded from above on any bounded set and (ii) the ΦpRq “ ZPH|ΦpZq ď R(is bounded for every R. To conclude the proof, we note that if ZPN, then Z“ pϕ, ψ, w, 0,0,0qsolves the stationary problem $ & % ´kpϕx`ψ`lwqx´k0lpwx´lϕq`f1pϕ, ψ, wq “ 0, ´bψxx `kpϕx`ψ`lwq`f2pϕ, ψ, wq “ 0, ´k0pwx´lϕqx`klpϕx`ψ`lwq`f3pϕ, ψ, wq “ 0. (4.32) Multiplying in L2p0, Lqthe equations in p4.32qby pϕ, ψ, wq, we find b}ψx}2`k}ϕx`ψ`lw}2`k0}wx`lϕ}2“ ´żL 0 ∇Fpu, v, wq¨pu, v, wqdx. Now, we use Assumption (f.1) to show „1´2αβL2 π2”}ϕx}2`}ψx}2`}wx}2ıď2βcFL. Therefore, the set of stationary solutions Nis 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 Lą0and Tą0large enough. If the Problem (1.2)-(1.4) satisfies (1.5)-(1.9) with supplementary condition pu1,¨¨¨, unq “ p0,¨¨¨,0qin ωˆr0, T s,(A.33) with ωĂ r0, L0sas in (3.34). Then, the overdetermined problem is well-posed and generates a strongly continuous semigroup over the Hilbert space HωjpMiq ” "pu1,¨¨¨, un, v1,¨¨¨, vnqˇˇˇˇ uiPH1 0pMiq, viPL2pMiq, ui“vi“0in ω, i “1,¨¨¨, n *, where Mi“ pr0, Ls, giq. Proof. First, for the state vector Zptq ” pu1,¨¨¨, un,Btu1,¨¨¨,BtunqJ, the Problem (1.2)-(1.4) is equivalent to the following vectorial Cauchy problem BtZptq`AZptq “ FpZptqq,Zp0q “ pu0 1,¨¨¨, u0 n, u1 1,¨¨¨ , u1 nqJ,(A.34) 24 with operators defined by A“„0´I|V2pMiq B|V1pMiq0, B “» — — — – ´∆10¨¨¨ 0 0´∆2¨¨¨ 0 . . .. . .¨¨¨ . . . 0 0 0 ´∆n fi ffi ffi ffi fl , I “ pδijqnˆn and FpZq “ » — — — — — — — — – 0 . . . 0 f1pu1,¨¨¨, unq . . . fnpu1,¨¨¨, unq fi ffi ffi ffi ffi ffi ffi ffi ffi fl . The domain of operator Ais defined by DpAq “ rV1pMiqsnˆrV2pMiqsnwhere V1pMiq ” pu1,¨¨¨, unq P DpBq | pu1,¨¨¨, unq “ p0,¨¨¨,0qin ω(, V2pMiq ” pv1,¨¨¨, vnq P DpB1 2q | pv1,¨¨¨, vnq “ p0,¨¨¨,0qin ω( and DpBq “ Dp´∆1qˆ¨¨¨ˆDp´∆nq “ H2pM1qXH1 0pM1qˆ¨¨¨H2pMnqXH1 0pMnq. The finite energy space for (A.34) is the Hilbert space defined by HωpMiq ” pu1,¨¨¨, un, v1,¨¨¨, vnq P HpMiq | ui“vi“0 in ω, @i“1,¨¨¨, n(, where HpMiq ” rDpB1 2qsnˆLpMiqand LpMiq ” L2pM1qˆ¨¨¨ˆL2pMnq. Using classical semigroup theory, one can establish existence and uniqueness of a solution to the Cauchy problem (A.34). Moreover, the solution operator generates a strongly continuous semigroup TMiptq:HwpMiq Ñ HwpMiq, defined by pu0 1,¨¨¨, u0 n, u1 1,¨¨¨, u1 nq ÞÑ pu1ptq,¨¨¨, unptq,Btu1ptq,¨¨¨,Btunptqq, t ě0, where pu1,¨¨¨, un,Btu1,¨¨¨ ,Btunqis the weak solution corresponding to the initial data pu0 1,¨¨¨, u0 n, u1 1,¨¨¨, u1 nq. In addition, tTMiptqutě0is also strongly continuous semigroup on HωpMiqsatisfying the compatibility condition (A.33). Remark A.1. It is not difficult to show that if (A.33) is fulfilled, then we also have pBtu1,¨¨¨,Btunq “ p0,¨¨¨,0qin ωˆr0, T s. 25