ASYMPTOTIC ANALYSIS OF EVOLUTION EQUATIONS WITH NONCLASSICAL HEAT CONDUCTION

Transcription

1 POLITECNICO DI MILANO DEPARTMENT OF MATHEMATICS DOCTORAL PROGRAMME IN MATHEMATICAL MODELS AND METHODS IN ENGINEERING ASYMPTOTIC ANALYSIS OF EVOLUTION EQUATIONS WITH NONCLASSICAL HEAT CONDUCTION Doctoral Dissertation of: FILIPPO DELL ORO Avisor: Prof. VITTORINO PATA Tutor: Prof. DARIO PIEROTTI Chair of the PhD Program: Prof. ROBERTO LUCCHETTI CYCLE XXVI

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3 Summary The present octoral thesis eals with asymptotic behavior of evolution equations with nonclassical heat conuction. First, we consier the strongly ampe nonlinear wave equation on a boune smooth omain Ω R 3 u tt u t u + f(u t ) + g(u) = h which serves as a moel in the escription of type III thermal evolution within the theory of Green an Naghi. In particular, the nonlinearity f acting on u t is allowe to be nonmonotone an to exhibit a critical growth of polynomial orer 5. The main focus is the longterm analysis of the relate solution semigroup, which is shown to possess global an exponential attractors of optimal regularity in the natural weak energy space. Then, we analyze two evolution systems ruling the ynamics of type III thermoelastic extensible beams or Berger plates with memory. Specifically, we stuy the ecay properties of the solution semigroup generate by an abstract version of the linear system u tt + 2 u + α t = α tt α along with the limit situation without memory µ(s) [α(t) α(t s)] s u t = { utt + 2 u + α t = α tt α α t u t = an the existence of regular global attractors for an abstract version of the nonlinear moel u tt ω u tt + 2 u [ b + u 2 L (Ω)] u + 2 αt = g α tt α µ(s) [α(t) α(t s)] s u t =. I

4 Moreover, we iscuss the asymptotic behavior of the nonlinear type III Caginalp phasefiel system { ut u + ϕ(u) = α t α tt α t α + g(α) = u t on a boune smooth omain Ω R 3, with nonlinearities ϕ an g of polynomial critical growth 5, proving the existence of the regular global attractor. Finally, we analyze the linear ifferential system ρ 1 φ tt κ(φ x + ψ) x = ρ 2 ψ tt bψ xx + κ(φ x + ψ) + δθ x = ρ 3 θ t 1 β g(s)θ xx (t s) s + δψ tx = escribing a Timoshenko beam couple with a temperature evolution of Gurtin-Pipkin type. A necessary an sufficient conition for exponential stability is establishe in terms of the structural parameters of the equations. In particular, we generalize previously known results on the Fourier-Timoshenko an the Cattaneo-Timoshenko beam moels. In the first chapter of the thesis we introuce some preliminary results about infiniteimensional ynamical systems an linear semigroups neee in the course of the investigation. The remaining chapters correspon to the following papers, written uring the three years of PhD. F. Dell Oro an V. Pata, Long-term analysis of strongly ampe nonlinear wave equations, Nonlinearity 24 (211), , (Chapter 2 an Chapter 3). F. Dell Oro an V. Pata, Strongly ampe wave equations with critical nonlinearities, Nonlinear Anal. 75 (212), , (Chapter 4). F. Dell Oro, Global attractors for strongly ampe wave equations with subcritical-critical nonlinearities, Commun. Pure Appl. Anal. 12 (213), , (Chapter 5). M. Coti Zelati, F. Dell Oro an V. Pata, Energy ecay of type III linear thermoelastic plates with memory, J. Math. Anal. Appl. 41 (213), , (Chapter 6). F. Dell Oro an V. Pata, Memory relaxation of type III thermoelastic extensible beams an Berger plates, Evol. Equ. Control Theory 1 (212), , (Chapter 6). M. Conti, F. Dell Oro an A. Miranville, Asymptotic behavior of a generalization of the Caginalp phase-fiel system, Asymptot. Anal. 81 (213), , (Chapter 7). F. Dell Oro an V. Pata, On the stability of Timoshenko systems with Gurtin- Pipkin thermal law, submitte, (Chapter 8). II

5 Acknowlegments I thank my avisor Prof. Vittorino Pata for introucing me to the fascinating fiel of infinite-imensional ynamical systems, an for the competence, enthusiasm, professionality an constant guiance uring these three years of PhD. I also thank Prof. Monica Conti for the interest, help an several remarkable teachings, an Prof. Jaime Muñoz Rivera for the kin hospitality in Petrópolis an many stimulating iscussions. I express my sincere gratitue to my family for the support an the quotiian encouragement. I heartily thank Silvia for her love an her constant precious presence in my life. III

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7 Contents Introuction 1 1 Preliminaries Infinite-Dimensional Dynamical Systems Dissipative ynamical systems Global an exponential attractors Graient systems Linear Semigroups Strongly Dampe Nonlinear Wave Equations Introuction Preliminaries Notation General agreements A technical lemma An Equivalent Formulation Decompositions of the nonlinear terms The equation revisite The renorme spaces Formal estimates The Solution Semigroup Well-poseness Sketch of the proof The semigroup Dissipativity Partial Regularization Attractor for the SDNWE. The Critical-Subcritical Case Introuction The Global Attractor A Conitional Result V

8 Contents 3.4 Regularity of the Attractor Attractor for the SDNWE. The Fully-Critical Case Introuction The Dissipative Dynamical System Dissipative estimates The issipation integral Global an Exponential Attractors Statement of the result Proof of Theorem Proof of Lemma Proof of Lemma Proof of Lemma Attractor for the SDNWE. The Subcritical-Critical Case Introuction The Global Attractor Regularity of the Attractor Extensible Beams an Berger Plates Introuction Functional Setting an Notation The Solution Semigroup The past history formulation Well-poseness The Main Result Further Remarks The Lyapunov Functional An Auxiliary Functional Dissipation Integrals Proof of Theorem Conclusion of the proof of Theorem Proofs of Proposition an Corollary Proof of Proposition Proof of Corollary The Linear Case The Contraction Semigroups The first system The secon system Decay Properties of the Semigroup U 1 (t) Decay Properties of the Semigroup U 2 (t) Caginalp Phase-Fiel Systems Introuction Preliminaries Functional setting Technical lemmas VI

9 Contents 7.3 The Solution Semigroup A Priori Estimates an Dissipativity Further Dissipativity The Global Attractor Timoshenko Systems with Gurtin-Pipkin Thermal Law Introuction The Fourier thermal law The Cattaneo thermal law The Gurtin-Pipkin thermal law Comparison with Earlier Results The Fourier case The Cattaneo case The Coleman-Gurtin case Heat conuction of type III Functional Setting an Notation Assumptions on the memory kernel Functional spaces Basic facts on the memory space The Contraction Semigroup Some Auxiliary Functionals The functional I The functional J The functional K The functional L Proof of Theorem (Sufficiency) A further energy functional Conclusion of the proof of Theorem Proof of Theorem (Necessity) More on the Comparison with the Cattaneo Moel Appenix 123 Conclusions 129 Bibliography 13 VII

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11 Introuction The aim of the research containe in the present octoral thesis is the mathematical analysis of well-poseness an asymptotic behavior of linear an nonlinear issipative partial ifferential equations with nonclassical heat conuction, that is, thermal evolutions where the temperature may travel with finite spee propagation. In the linear case, we mainly focus on the stability properties of the associate semigroups, analyzing the ecay to zero of the solutions. In the nonlinear situation, we well on existence an regularity of finite-imensional global an exponential attractors, proviing a complete escription of the asymptotic ynamics by means of suitable small regions of the phase space. Hereafter is a etaile iscussion of the moels consiere in the thesis. In particular, the nonclassical character of the temperature is stresse an the physical meaning an relevance are explaine. Nonlinear Heat Conuction of Type III Let Ω R 3 be a boune omain with sufficiently smooth bounary Ω. The thermal evolution in a homogenous isotropic (rigi) heat conuctor occupying the space-time cyliner Ω T = Ω (, T ) is governe by the balance equation e t + iv q = F. Here, the internal energy e is a function of the relative temperature fiel ϑ = ϑ(x, t) : Ω T R, that is, the temperature variation from an equilibrium reference value, while q = q(x, t) : Ω T R 3 is the heat flux vector. Finally, F represents a source term. We also assume the Dirichlet bounary conition ϑ(x, t) x Ω =, 1

12 Introuction expressing the fact that the bounary Ω of the conuctor is kept at null (i.e. equilibrium) temperature for all times. Consiering only small variations of ϑ an ϑ, the internal energy fulfills with goo approximation the equality e(x, t) = e (x) + cϑ(x, t), where e is the internal energy at equilibrium an c > is the specific heat. Accoringly, the balance equation becomes For a general F of the form cϑ t + iv q = F. (1) F (x, t) = f(ϑ(x, t)) + h(x), (2) accounting for the simultaneous presence of a time-inepenent external heat supply an a nonlinearly temperature-epenent internal source, equation (1) reas cϑ t + iv q + f(ϑ) = h. (3) To complete the picture, a further relation is neee: the so-calle constitutive law for the heat flux, establishing a link between q an ϑ. In fact, the choice of the constitutive law is what really etermines the moel. At the same time, being a purely heuristic interpretation of the physical phenomenon, it may reflect ifferent iniviual perceptions of reality, or even philosophical beliefs. For instance, for the classical Fourier constitutive law q + κ ϑ =, κ >, (4) we euce from (3) the familiar reaction-iffusion equation cϑ t κ ϑ + f(ϑ) = h. Nevertheless, such an equation preicts instantaneous propagation of (thermal) signals, a typical sie-effect of parabolicity. This feature, sometimes calle the paraox of heat conuction (see e.g. [14, 32]), has often encountere strong criticism in the scientific community, up to be perceive as physically unrealistic by some authors. Therefore, several attempts have been mae through the years in orer to introuce some hyperbolicity in the mathematical moeling of heat conuction (see e.g. [8, 48]). A possible choice is aopting the Maxwell-Cattaneo law [8], namely, the ifferential perturbation of (4) q + εq t + κ ϑ =, κ ε >. (5) In which case, the sum (3)+ε t (3) entails the hyperbolic reaction-iffusion equation εcϑ tt κ ϑ + [c + εf (ϑ)]ϑ t + f(ϑ) = h, wiely employe in the escription of many interesting phenomena, such as chemical reacting systems, gene selection, population ynamics or forest fire propagation, to name a few (cf. [33, 59, 6]). Another strategy is relaxing (4) by means of a timeconvolution against a suitable (e.g. convex, ecreasing an summable) kernel µ. Precisely, omitting the epenence on x, q(t) = κ ϑ(t) t 2 µ(t s) ϑ(s) s. (6)

13 Introuction The constant κ can be either strictly positive or zero, accoring to the moels of Coleman-Gurtin [18] or Gurtin-Pipkin [48], respectively. Plugging (6) into (3) we en up with the integroifferential equation cϑ t κ ϑ µ(s) ϑ(t s) s + f(ϑ) = h. Quite interestingly, in the (fully hyperbolic) case κ =, we recover (5) as the particular instance of (6) corresponing to the kernel µ(s) = κ ε e s/ε. In a ifferent fashion, the theory of heat conuction of type III evise by Green an Naghi [43 47, 9] consiers instea a perturbation of the classical law (4) of integral kin. Inee, the Fourier law is moifie in the following manner: q + κ ϑ + ω u =, κ, ω >, (7) where an aitional inepenent variable appears: the thermal isplacement u : Ω T R, efine as hence satisfying the equality u(x, t) = u(x, ) + t ϑ(x, s) s, u t (x, t) = ϑ(x, t), (x, t) Ω T, an (for consistency) complying with the Dirichlet bounary conition u(x, t) x Ω =. Using (7) the balance equation (1) turns into Replacing for more generality (2) with cu tt κ u t ω u = F. F (x, t) = f(ϑ(x, t)) g(u(x, t)) + h(x), allowing the source term to contain a further contribution epening nonlinearly on the thermal isplacement, we finally arrive at the bounary-value problem cu tt κ u t ω u + f(u t ) + g(u) = h, u Ω =, u t Ω =, which will be analyze in Chapters 2-5. We refer the reaer to [8 87] for iscussions an other evelopments relate to type III heat conuction. 3

14 Introuction Type III Thermoelastic Extensible Beams an Berger Plates For n = 1, 2, let Ω R n be a boune omain with sufficiently smooth bounary Ω. Given the parameters ω > an b R, we consier the evolution system of couple equations on the space-time cyliner Ω + = Ω R + in the unknown variables u tt ω u tt + 2 u [ b + u 2 L 2 (Ω)] u + ϑ = g, (8) ϑ t + iv q u t =, (9) u = u(x, t) : Ω + R, ϑ = ϑ(x, t) : Ω + R, q = q(x, t) : Ω + R n. Such a system, written here in normalize imensionless form, rules the ynamics of a thermoelastic extensible beam (for n = 1) or Berger plate (for n = 2) of shape Ω at rest (see [4, 93]). Accoringly, the variable u stans for the vertical isplacement from equilibrium, ϑ is the (relative) temperature an q is the heat flux vector obeying some constitutive law, epening on one s favorite choice of heat conuction moel. The term ω u tt appearing in the first equation witnesses the presence of rotational inertia, whereas the real parameter b accounts for the axial force acting in the reference configuration: b > if the beam (or plate) is stretche, b < if compresse. Finally, the function g : Ω R escribes a lateral loa istribution. We also assume that the ens of the beam (or plate) are hinge, which translates into the hinge bounary conitions for u u(x, t) x Ω = u(x, t) x Ω =, an we take the Dirichlet bounary conition for ϑ ϑ(x, t) x Ω =, expressing the fact that the bounary Ω is kept at null (i.e. equilibrium) temperature for all times. It is worth noting that ifferent bounary conitions for u are physically significant as well, such as the clampe bounary conitions u Ω = u =, ν Ω where ν is the outer normal vector. However, the mathematical analysis carrie out in this thesis (see Chapter 6) epens on the specific structure of the hinge bounary conitions (the so calle commutative case ). In the clampe case major moifications on the neee tools are require an the proofs become much more technical. We are left to specify the constitutive relation for the heat flux, establishing a link between q an ϑ. Aopting for instance the classical Fourier law (the physical constants are set to 1) q = ϑ, equation (9) turns into ϑ t ϑ u t =. 4

15 Introuction In a ifferent fashion, as alreay sai, the theory of heat conuction of type III evise by Green an Naghi consiers a perturbation of the classical law of integral kin, by means of the so-calle thermal isplacement satisfying the equality α t bounary conition α(x, t) = α (x) + Then, the Fourier law is moifie as so that (9) takes the form t ϑ(x, s) s, = ϑ an (for consistency) complying with the Dirichlet α(x, t) x Ω =. q = α t α, α tt α α t u t =. (1) Still, the equation preicts infinite spee propagation of (thermal) signals, ue to its partially parabolic character which provies an instantaneous regularization of α t. Such an effect is not expecte (nor observe) in real conuctors. Similarly to what one in [3] for the Fourier case, a possible answer is consiering a memory relaxations of the above constitutive law of the form q(t) = κ(s) α t (t s) s α(t), for some boune convex summable function κ (the memory kernel) of total mass κ(s) s = 1. Up to a rescaling, we may also suppose κ() = 1. Accoringly, (9) becomes α tt α κ(s) α t (t s) s u t =, (11) where the past history of the temperature is suppose to be known an regare as an initial atum of the problem. It is reaily seen that, when the function κ converges in the istributional sense to the Dirac mass at zero, equation (1) is formally recovere from (11). From the physical viewpoint, this means that (1) is close to (11) when the memory kernel is concentrate, i.e. when the system keeps a very short memory of the past effects. As a matter of fact, (11) can be given a more convenient form. Inee, efining the ifferentiate kernel a formal integration by parts yiels κ(s) α t (t s) s = µ(s) = κ (s), 5 µ(s) [α(t) α(t s)] s.

16 Introuction In summary, equation (8) an the particular concrete realization (11) of (9) give rise to the system u tt ω u tt + 2 u [ b + u 2 L (Ω)] u + 2 αt = g, α tt α µ(s) [α(t) α(t s)] s u t =, which will be analyze in Chapter 6 (actually, in a more general abstract form). As a matter of fact, from the physical viewpoint, it is also relevant to neglect the effect of the rotational inertia on the plate (see e.g. [4, 51, 55, 57]). This correspons to the limit situation when ω =. Hence, we will also consier the following linear an homogeneous version of the above moel u tt + 2 u + α t =, α tt α along with the system µ(s) [α(t) α(t s)] s u t =, { utt + 2 u + α t =, α tt α α t u t =, formally obtaine when the memory kernel collapses into the Dirac mass at zero. As we will see, the presence of the memory prouces a lack of exponential stability of the associate linear semigroups, preventing the analysis of the asymptotic properties in the nonlinear case. Type III Nonlinear Caginalp Phase-Fiel Systems Let Ω R 3 be a boune omain with smooth bounary Ω. The thermal evolution of a material occupying a volume Ω, with orer parameter u an (relative) temperature ϑ, is governe by the equations H t u t = Ψ u, (12) + iv q = F. (13) Here, Ψ enotes the total free energy of the system efine as ( 1 Ψ(u, ϑ) = 2 u 2 + Φ (u) uϑ 1 ) 2 ϑ2 x, where the potential Φ (s) = Ω s ϕ(y) y for some real ϕ : R R, 6

17 Introuction has a typical ouble-well shape (e.g. Φ (s) = (s 2 1) 2 ). Besies, H stans for the enthalpy of the material H = Ψ ϑ = ϑ + u, q is the heat flux vector an F represents a source term. Finally, we assume the Dirichlet bounary conition for u an ϑ Accoringly, equation (12) reas u(t) Ω = ϑ(t) Ω =. u t u + ϕ(u) = ϑ, (14) while the concrete form of (13) epens on the choice of the constitutive law for the heat flux. For example, within the classical Fourier law q = k ϑ, k >, we euce from (13) the reaction-iffusion equation ϑ t k ϑ = u t + F which, couple with (14), constitutes the original Caginalp phase-fiel system [6]. Instea, aopting the theory of heat conuction of type III, the heat flux takes the form where the variable q = k α k ϑ, k >, α(t) = t ϑ(τ) τ + α() represents the thermal isplacement. In turn, the balance equation (13) translates into α tt k α t k α = u t + F. In conclusion, for a general term F of the form F = g(α) accounting for the presence of a nonlinear internal source epening on the isplacement, an setting the physical constants to 1, we en up with the following nonlinear phase-fiel system of Caginalp type { ut u + ϕ(u) = α t, α tt α t α + g(α) = u t. This moel will be stuie in Chapter 7. We refer the reaer to [62 65] for further iscussions relate to phase-fiel systems with nonclassical heat conuction. 7

18 Introuction Timoshenko Systems with Gurtin-Pipkin Thermal Law Given l >, we consier the thermoelastic beam moel of Timoshenko type [92] ρ 1 φ tt κ(φ x + ψ) x =, ρ 2 ψ tt bψ xx + κ(φ x + ψ) + δϑ x =, (15) ρ 3 ϑ t + q x + δψ tx =, where the unknown variables φ, ψ, ϑ, q : (x, t) [, l] [, ) R represent the transverse isplacement of a beam with reference configuration [, l], the rotation angle of a filament, the relative temperature an the heat flux vector, respectively. Here, ρ 1, ρ 2, ρ 3 as well as κ, b, δ are strictly positive fixe constants. The system is complemente with the Dirichlet bounary conitions for φ an ϑ an the Neumann one for ψ φ(, t) = φ(l, t) = ϑ(, t) = ϑ(l, t) =, ψ x (, t) = ψ x (l, t) =. Such conitions, commonly aopte in the literature, seem to be the most feasible from a physical viewpoint. To complete the picture, we nee to establish a link between q an ϑ, through the constitutive law for the heat flux. We assume the Gurtin-Pipkin heat conuction law βq(t) + g(s)ϑ x (t s) s =, β >, (16) where the memory kernel g is a (boune) convex summable function on [, ) of total mass g(s) s = 1. As alreay sai, equation (16) can be viewe as a memory relaxation of the Fourier law, inucing (similarly to the Cattaneo law) a fully hyperbolic mechanism of heat transfer. In this perspective, it may be consiere a more realistic escription of physical reality. Accoringly, system (15) turns into ρ 1 φ tt κ(φ x + ψ) x =, ρ 2 ψ tt bψ xx + κ(φ x + ψ) + δϑ x =, ρ 3 ϑ t 1 β g(s)ϑ xx (t s) s + δψ tx =, an this moel will be stuie in the final Chapter 8. 8

19 CHAPTER1 Preliminaries In this first chapter, we recall some basic tools from the theory of infinite-imensional ynamical systems an linear semigroups. A more etaile presentation can be foun in the classical books [2, 13, 24, 49, 5, 53, 77, 88, 91]. 1.1 Infinite-Dimensional Dynamical Systems Nonlinear ynamical systems play a crucial role in the moern stuy of several physical phenomena where some kin of evolution is taken into account. In particular, many ynamics are characterize by the presence of some issipation mechanisms (e.g. friction or viscosity) which prouce a loss of energy in the system. Roughly speaking, from the mathematical viewpoint issipation is represente by the existence of a set in the phase space calle absorbing set (see Definition 1.1.3). Nevertheless, in orer to have a better unerstaning of the asymptotic behavior of the system, some aitional goo geometrical an topological properties (e.g. compactness or finite fractal/hausorff imension) are necessary. This leas to the moern concept of attractor (see Definition 1.1.7), that is, the minimal compact set which attracts uniformly all the boune sets of the phase space Dissipative ynamical systems We begin with some efinitions. Definition Let X be a real Banach space. A ynamical system (otherwise calle C -semigroup of operators) on X is a one-parameter family of functions S(t) : X X epening on t satisfying the following properties: 9

20 Chapter 1. Preliminaries (S.1) S() = I; 1 (S.2) S(t + τ) = S(t)S(τ) for all t, τ ; (S.3) t S(t)x C([, ), X) for all x X; (S.4) S(t) C(X, X) for all t. Remark In light of some recent evelopments (see [9] an [13, Chapter XI]), the notion of ynamical system can be actually given in a more general form, removing the continuity assumptions (S.3) an (S.4) from Definition (see the forthcoming Remark for a more etaile iscussion). Along this section, S(t) will always enote a ynamical system acting on a Banach space X. Definition A nonempty set B X is calle invariant for S(t) if S(t)B B, t. Definition A subset B X is calle absorbing set if it is boune 2 an for any boune set B X there exists an entering time t e = t e (B) such that S(t)B B, t t e. It is worth noting that, once we have prove the existence of an absorbing set B, an invariant absorbing set can be easily constructe through the formula S(t)B B, t e = t e (B). t t e As alreay mentione, a ynamical systems is calle issipative if it possesses an absorbing set. We also nee the notion of ω-limit set. Definition The ω-limit set of a nonempty set B X is efine as ω(b) = t S(τ)B. Thus, since ω(b) in some sense captures the ynamics of the orbits of B, if the ynamical system possesses an absorbing set B one might try to escribe the asymptotic behavior of the whole system through union ω(x), x B since any trajectory eventually enters into B. However, this set turns out to be too small in orer to provie the necessary information, as will be clear in the sequel. τ t 1 Here, I enotes the ientity on X. 2 Some authors o not require bouneness in the efinition of absorbing set. 1

21 1.1. Infinite-Dimensional Dynamical Systems Global an exponential attractors Due to the fact that the phase space X can be infinite-imensional, existence of absorbing set usually gives poor information on the longterm ynamics. Inee, for instance, balls are not compact in the infinite-imensional case. Therefore, one might think to investigate, for example, existence of compact absorbing set. However, when ealing with concrete ynamical systems generate by partial ifferential equations arising in Mathematical Physics, compact absorbing sets pop up when the equation exhibits regularizing effects on the solution, that is, when the ynamics is parabolic. Thus, in the hyperbolic case, compact absorbing set are out of reach. The central iea is then to consier compact sets that attract (rather than absorb) the orbits originating from boune sets. Definition If B 1 an B 2 are nonempty subsets of X, the Hausorff semiistance between B 1 an B 2 is efine as δ X (B 1, B 2 ) = sup z 1 B 1 inf z 1 z 2 X. z 2 B 2 Observe that the Hausorff semiistance is not symmetric. Moreover, it is easy to see that δ X (B 1, B 2 ) = if an only if B 1 B 2, where B 2 enotes the closure in X of the set B 2. Definition A set K X is calle attracting for S(t) if lim δ X(S(t)B, K) =, t for any boune set B X. The ynamical system S(t) is calle asymptotically compact if has a compact attracting set. Definition A compact set A X which is at the same time attracting an fully invariant (i.e. S(t)A = A for every t ) is calle the global attractor of S(t). It is well-known that the global attractor of a ynamical system, provie it exists, is unique an connecte (see e.g. [2, 49, 5, 91]). Besies, in several concrete situations arising in ynamical systems generate by partial ifferential equations, the attractor A has finite fractal imension im f (A) = lim sup ε ln N ε ln 1, ε where N ε is the smallest number of ε-balls of X covering A. In this situation, roughly speaking, the long-term ynamics becomes finite-imensional (see e.g. [91]). We now state one of the main abstract results concerning existence of global attractors. To this aim, we will lean on the notion of Kuratowski measure of noncompactness of a boune set B X. This is by efinition α(b) = inf { : B is covere by finitely many sets of iameter less than }. Accoringly, α(b) = if an only if B is totally boune, i.e. precompact in a Banach space framework. Further straightforwar properties are liste below (cf. [49]): 11

22 Chapter 1. Preliminaries α(b) = α(b); α(b) iam(b); α(b 1 B 2 ) = max{α(b 1 ), α(b 2 )}; B 1 B 2 α(b 1 ) α(b 2 ). The result reas as follows. Theorem Let S(t) : X X be a issipative ynamical system acting on a Banach space X, an let B an absorbing set. If there exists a sequence t n such that then ω(b) is the global attractor of S(t). lim α(s(t n)b) =, n We aress the reaer to the classical books [2,13,49,91] for a proof (but see also [75] an Theorem A.2 in the final appenix). Remark The basic objects of the theory introuce so far (absorbing an attracting sets, global attractors) can be in fact revisite only in terms of their attraction properties, without any continuity assumption on S(t). Within this approach, a slight ifferent notion of global attractor is necessary, the minimality with respect to attraction being the sole characterizing property. The invariance is iscusse only in a secon moment, as a consequence of some kin of continuity. A etaile iscussion can be foun in [9]. Nonetheless, the global attractor is usually affecte by an essential rawback. Inee, the attraction rate can be arbitrarily slow an, in general, cannot be explicitly estimate. As a consequence, the global attractor may be very sensitive to small perturbations. Although not crucial from the theoretical sie, this problem becomes significant for practical purposes (e.g. numerical simulations). In orer to overcome these ifficulties, a new object has been introuce in [26], namely, the so-calle exponential attractor. Definition An exponential attractor is a compact invariant set E X of finite fractal imension satisfying for all boune set B X the exponential attraction property δ X (S(t)B, E) I( B X )e ωt for some ω > an some increasing function I : R + R +. Contrary to the global attractor, an exponential attractor is not unique. With regar to sufficient conitions for the existence of exponential attractors in Hilbert spaces we refer to [1, 26]. In a Banach space setting, the first result was evise in [27] (see also [17] an the final appenix of this thesis). 12

23 1.1. Infinite-Dimensional Dynamical Systems Graient systems In this section we analyze a special class of ynamical systems, the so-calle graient systems, characterize by the existence of a Lyapunov functional. We begin with some efinitions. Definition A function Z C(R, X) is calle a complete boune trajectory (CBT) of S(t) if sup Z(τ) X < τ R an S(t)Z(τ) = Z(t + τ), t, τ R. We also introuce the set of stationary points of S(t) S = {z X : S(t)z = z, t }, an the unstable set of S, that is, { W (S) = Z() : Z CBT an } lim Z(τ) S X =. τ Definition A Lyapunov functional for the ynamical system S(t) is a function Λ C(X, R) such that (i) Λ(z) if an only if z X ; (ii) Λ(S(t)z) Λ(z) for every z X an every t ; (iii) Λ(S(t)z) = Λ(z) for all t implies that z S. If there exists a Lyapunov functional, then S(t) is calle a graient system. We report the following stanar abstract result on existence of global attractors for graient systems (see [49, 56]). Lemma Let S(t) : X X be a graient system acting on a Banach space X. Assume that (i) the set S of the stationary points of S(t) is boune in X; (ii) for every R there exist a positive function I R vanishing at infinity an a compact set K R X such that S(t) can be split into the sum S (t) + S 1 (t), where the one-parameter operators S (t) an S 1 (t) fulfill whenever z X R an t. S (t)z X I R (t) an S 1 (t)z K R, Then, S(t) possesses a connecte global attractor A, which consists of the unstable set W (S). Moreover, A is a subset of K R for some R >. In conclusion, roughly speaking, the asymptotic ynamics of graient systems can be fully escribe by means of complete boune trajectories eparting (at ) from the set of stationary points of S(t). 13

24 Chapter 1. Preliminaries 1.2 Linear Semigroups We now consier the particular situation where S(t) is a linear operator for every t. With stanar notation, we will enote by L(X) the space of boune linear operators from X into X. Definition Let X be a real Banach space. A linear ynamical system (otherwise calle C -semigroup of boune linear operators) acting on X is a family of maps S(t) L(X) epening on t satisfying the semigroup properties (S.1)-(S.2) together with (S.3 ) lim t S(t)x = x for all x X. Notice that assumption (S.3 ) an the semigroup properties imply the continuity t S(t)x C([, ), X) for every fixe x X. When ealing with linear ynamical systems, an important concept is the one of infinitesimal generator. Definition The linear operator A with omain { S(t)x x D(A) = x X : lim t t efine as } exists Ax = lim t S(t)x x t is the calle the infinitesimal generator of the linear ynamical system S(t). It is possible to show that A is a close, ensely efine operator which uniquely etermines the linear ynamical system (see e.g. [77]). Formally, one writes S(t) = e ta to inicate that the operator A is the infinitesimal generator of the semigroup S(t). One important an natural question is then to etermine whenever a close linear operator with ense omain in X is the infinitesimal generator of a linear ynamical system S(t). As a matter of fact, a necessary an sufficient conition is given by the Hille-Yosia Theorem (see e.g. [77]), so that the problem is nowaays completely solve from the theoretical viewpoint. However, the Hille-Yosia Theorem is somehow ifficult to apply in several concrete situations, as it involves the knowlege of the spectrum of the operator A (usually, not an easy task to achieve). Nevertheless, in a Hilbert space setting, there exists a more effective criterion. We nee two preliminary efinitions. Definition S(t) is calle a contraction semigroup if S(t) L(X) 1, t. 14

25 1.2. Linear Semigroups Definition A linear operator A on a real Hilbert space X is issipative if Ax, x, x D(A). The result reas as follows. Lemma (Lumer-Phillips). Let A be a ensely efine linear operator on a real Hilbert space X. Then A is the infinitesimal generator of a contraction semigroup S(t) if an only if (i) A is issipative; an (ii) Range(I A) = X. We aress the reaer to [77] for the proof. The Lumer-Phillips Theorem turns out to be a very useful tool in the stuy of linear ynamical systems generate by partial ifferential equations, as we will see in Chapter 6. Another funamental problem is the one of asymptotic stability, that is, the stuy of the ecay properties of the trajectories. Definition The linear ynamical system S(t) is sai to be stable if lim S(t)x X =, x X; t exponentially stable if there are M 1 an ε > such that S(t) L(X) Me εt, t. Exploiting the semigroup properties, when lack of exponential stability occurs we can say that there is no ecay pattern vali for all x X (see [11, 77]). In orer to eal with concrete cases of linear ynamical systems generate by partial ifferential equations, we will also exploit an operative abstract criterion evelope in [79] (but see also [37] for the statement use here). First, we nee a efinition. Definition The complexification of a real Banach space X is the complex Banach space X C efine as an enowe with the norm X C = X ix = {z = x + iy with x, y X} x + iy XC = x 2 + y 2. Analogously, the complexification A C of a linear operator of A on X is the (linear) operator on X C with omain D(A C ) = {z = x + iy with x, y D(A)} efine by A C (x + iy) = Ax + iay. 15

26 Chapter 1. Preliminaries The result is the following. Lemma A linear ynamical system S(t) = e ta acting on a real Hilbert space X is exponentially stable if an only if there exists ε > such that inf λ R iλz A Cz XC ε z XC, z D(A C ), (1.2.1) where A C an X C are unerstoo to be the complexifications of the original infinitesimal generator A an space X, respectively. 16

27 CHAPTER2 Strongly Dampe Nonlinear Wave Equations 2.1 Introuction Let Ω R 3 be a boune omain with smooth bounary Ω. Calling H = L 2 (Ω), with inner prouct, an norm, an introucing the strictly positive Dirichlet operator A = with omain D(A) = H 2 (Ω) H 1 (Ω) H, we consier the evolution equation in the unknown u = u(x, t) : Ω R + R subject to the initial conitions u tt + Au t + Au + f(u t ) + g(u) = h (2.1.1) u(x, ) = a(x) an u t (x, ) = b(x), where a, b : Ω R are assigne ata. The time-inepenent external source h = h(x) is taken in H, while the nonlinearities comply with the following assumptions, λ 1 > being the first eigenvalue of A. Assumptions on f. Let f C 1 (R), with f() =, satisfy for every s R an some c the growth boun f (s) c + c s 4, (2.1.2) along with the issipativity conition lim inf s f (s) > λ 1. (2.1.3) 17

28 Chapter 2. Strongly Dampe Nonlinear Wave Equations Assumptions on g. Let g C 1 (R), with g() =, satisfy for every s R an some c the growth boun along with the issipativity conitions g (s) c + c s p 1 with p [1, 5], (2.1.4) lim inf s lim inf s g(s) s > λ 1, (2.1.5) sg(s) c 1 s g(y)y s 2 > λ 1 2, (2.1.6) for some c 1 >. In fact (2.1.5)-(2.1.6) are automatically verifie (with c 1 = 1) if g fulfills the same issipation conition (2.1.3) of f, slightly less general but still wiely use in the literature. As explaine in the introuction of the thesis, equation (2.1.1), here written in imensionless form, rules the thermal evolution in a rigi boy of shape Ω within the theory of heat conuction of type III evise by Green an Naghi. However, other physical interpretations are possible, for example viscoelasticity of Kelvin-Voigt type. After introucing the notation an the functional setting (see Section 2.2), in the successive Section 2.3 we consier an equivalent formulation of the problem, more suitable for our purposes. Well-poseness is prove in Section 2.4, yieling a solution semigroup S(t) (ynamical system) acting on the natural weak energy space. Finally, in Sections , we well on the issipative character of the semigroup, witnesse by the existence of (boune) regular absorbing sets. 2.2 Preliminaries Notation For σ R, we efine the hierarchy of (compactly) neste Hilbert spaces H σ = D(A σ 2 ), w, v σ = A σ 2 w, A σ 2 v, w σ = A σ 2 w. For σ >, it is unerstoo that H σ enotes the completion of the omain, so that H σ is the ual space of H σ. Moreover, the subscript σ is always omitte whenever zero. The symbol, also stans for uality prouct between H σ an its ual space H σ. In particular, H 2 = H 2 (Ω) H 1 (Ω) H 1 = H 1 (Ω) H = L 2 (Ω) H 1 = H 1 (Ω), an we have the Poincaré inequality Then we efine the natural energy spaces λ 1 w 2 w 2 1, w H 1. H σ = H σ+1 H σ 18

29 2.3. An Equivalent Formulation enowe with the stanar prouct norms {w 1, w 2 } 2 H σ = w 1 2 σ+1 + w 2 2 σ. We will also encounter the asymmetric energy spaces General agreements V σ = H σ H σ. Without loss of generality, we may (an o) suppose c 1 = 1 in (2.1.6). Along the chapter, we will perform a number of formal energy-type estimates, which are rigorously justifie in a Galerkin approximation scheme. In the proofs, we will always aopt the symbol Λ (or Λ ε ) to enote some energy functional, specifying its particular structure from case to case. Moreover, the Höler, Young an Poincaré inequalities will be tacitly use in several occasions, as well as the Sobolev embeing A technical lemma H 1 L 6 (Ω). We report a Gronwall-type lemma from the very recent paper [7]. Lemma Given k 1 an C, let Λ ε : [, ) [, ) be a family of absolutely continuous functions satisfying for every ε > small the inequalities 1 k Λ Λ ε kλ an t Λ ε + ελ ε Cε 6 Λ 3 ε + C for some continuous Λ : [, ) [, ). Then there are constants δ >, R an an increasing function I such that Λ (t) I(Λ ())e δt + R. 2.3 An Equivalent Formulation Equation (2.1.1) can be given an equivalent formulation, allowing to rener the calculations much simpler Decompositions of the nonlinear terms The first step is splitting f an g into the sums of suitable functions. Lemma For every fixe λ < λ 1 sufficiently close to λ 1, the ecomposition f(s) = ϕ(s) λs + ϕ c (s) hols for some ϕ, ϕ c C 1 (R) with the following properties: ϕ c is compactly supporte with ϕ c () = ; 19

30 Chapter 2. Strongly Dampe Nonlinear Wave Equations ϕ vanish insie [ 1, 1] an fulfills for some c an every s R the bouns ϕ (s) c s 4. Proof. In light of (2.1.3), fix any λ subject to the bouns lim inf s f (s) > λ > λ 1. Hence, f (s) λ, s k, for k 1 large enough. Choosing then any smooth function ϱ : R [, 1] satisfying sϱ (s) an { if s k, ϱ(s) = 1 if s k + 1, it is immeiate to check that ϕ(s) = ϱ(s)[f(s) + λs] an ϕ c (s) = [1 ϱ(s)][f(s) + λs] comply with the requirements. Lemma For every fixe λ < λ 1 sufficiently close to λ 1, the ecomposition g(s) = γ(s) λs + γ c (s) hols for some γ, γ c C 1 (R) with the following properties: γ c is compactly supporte with γ c () = ; γ vanish insie [ 1, 1] an fulfills for some c an every s R the bouns s γ(y) y sγ(s) an γ (s) c s p 1. Proof. Using this time (2.1.5)-(2.1.6), where we put c 1 = 1, for any fixe λ < λ 1 close to λ 1 an every s k 1 large enough we get s[g(s) + λs] an Similarly to the previous proof, we efine s g(y)y sg(s) λs2. γ(s) = ϱ(s)[g(s) + λs] an γ c (s) = [1 ϱ(s)][g(s) + λs]. Then, the first inequality tells that s s [ s γ(y) y ϱ(s) [g(y) + λy] y = ϱ(s) g(y) y + 1 ], 2 λs2 an by applying the secon one we establish the esire integral estimate, whereas the growth boun on γ is straightforwar. 2

31 2.3. An Equivalent Formulation Due to Lemma an Lemma 2.3.2, the functionals on H 1 given by w(x) w(x) Φ (w) = 2 ϕ(y) yx, Γ (w) = 2 γ(y) yx, an Ω Φ 1 (w) = ϕ(w), w, fulfill for every w H 1 the inequalities Moreover, since Ω Γ 1 (w) = γ(w), w, Φ (w) 2Φ 1 (w), (2.3.1) Γ (w) 2Γ 1 (w). (2.3.2) ϕ(s) 6 5 = ϕ(s) 1 5 ϕ(s) c s ϕ(s) = csϕ(s), we euce that for all C > sufficiently large The equation revisite ϕ(w) L 6/5 C[Φ 1 (w)] 5 6, w H1. (2.3.3) For a fixe λ < λ 1 complying with Lemma an Lemma 2.3.2, we rewrite (2.1.1) in the equivalent form where an u tt + Bu t + Bu + ϕ(u t ) + γ(u) = q, (2.3.4) q = h ϕ c (u t ) γ c (u) L (R + ; H) B = A λi with omain D(B) = D(A) is a positive operator commuting with A. In particular, the bilinear form (w, v) σ = w, A 1 Bv σ = w, v σ λ w, v σ 1 efines an equivalent inner prouct on the space H σ whose inuce norm σ, in light of the Poincaré inequality, satisfies The renorme spaces λ 1 λ λ 1 w 2 σ w 2 σ = w 2 σ λ w 2 σ 1 w 2 σ. (2.3.5) Aiming to eal with the reformulate version (2.3.4) of the original equation, it is convenient to reefine the norms of the energy spaces. Accoringly, we agree to consier H σ an V σ enowe with the equivalent norms {w, v} 2 H σ = w 2 σ+1 + v 2 σ an {w, v} 2 V σ = w 2 σ + v 2 σ. Whenever neee, the norm inequalities (2.3.5) will be applie without explicit mention. 21

32 Chapter 2. Strongly Dampe Nonlinear Wave Equations Formal estimates We conclue by establishing some general ifferential relations, wiely use in the forthcoming proofs. Given a vector-value function w, as regular as require, we set w = 2w tt + 2Bw t + 2Bw. The following ientities are verifie by irect calculations: w, w = [ w 2 1 t + 2 w, w t ] + 2 w w t 2, (2.3.6) w, w t = [ w 2 1 t + w t 2] + 2 w t 2 1, (2.3.7) w, w tt = t[ wt (w, w t) 1 ] + 2 wtt 2 2 w t 2 1. (2.3.8) Next, we efine the family of energy functionals epening on ε Exploiting (2.3.6)-(2.3.7), the inequalities Π ε (w) = (1 + ε) w w t 2 + 2ε w t, w. (2.3.9) Π (w) 2Π ε (w) 4Π (w) (2.3.1) an t Π ε(w) + επ ε (w) + 1 [ ε w w t 2 1] w, wt + εw (2.3.11) are easily seen to hol for every ε > sufficiently small. 2.4 The Solution Semigroup Well-poseness First, we stipulate the efinition of solution. Definition Given T >, we call weak solution to (2.3.4) on [, T ] a function u C([, T ], H 1 ) C 1 ([, T ], H) W 1,2 (, T ; H 1 ) satisfying for almost every t [, T ] an every test θ H 1 the equality u tt, θ + (u t, θ) 1 + (u, θ) 1 + ϕ(u t ), θ + γ(u), θ = q, θ. Theorem For every T > an every z = {a, b} H there is a unique weak solution u to (2.3.4) on [, T ] subject to the initial conitions {u(), u t ()} = z. Moreover, given any pair of initial ata z 1, z 2 H, there exists a constant C epening (increasingly) on the norms of z ı such that the ifference ū = u 1 u 2 of the corresponing solutions satisfies the continuous epenence estimate {ū, ū t } L (,T ;H) + ū t L 2 (,T ;H 1 ) Ce CT z 1 z 2 H. 22

33 2.4. The Solution Semigroup Sketch of the proof The continuous epenence is essentially containe in the proof of Theorem in the next chapter (by setting ε = ). Concerning existence, we follow the usual Galerkin proceure, consiering the solutions u n to the corresponing n-imensional approximating problems. Arguing as in the forthcoming Theorem an Corollary 2.5.1, with the ai of (2.3.3), we euce the uniform bouneness of u n in L (, T ; H 1 ), an, calling Ω T = Ω (, T ), those of t u n in L (, T ; H) L 2 (, T ; H 1 ), γ(u n ) an ϕ( t u n ) in L 6 5 (ΩT ). Hence, we can extract weakly or weakly- convergent subsequences u n u, t u n u t, γ(u n ) γ, ϕ( t u n ) ϕ, in the respective spaces. Proving the claime continuity in time of u is stanar matter. The only ifficulty is ientifying the limits of the nonlinearities, i.e. showing the equalities γ = γ(u) an ϕ = ϕ(u t ). The first is a consequence of the so-calle weak ominate convergence theorem. Inee, from the Sobolev compact embeing we learn that, up to a subsequence, W 1,2 (, T ; H 1 ) C([, T ], H), u n u a.e. in Ω T γ(u n ) γ(u) a.e. in Ω T. For every fixe β L 6 (Ω T ), the latter convergence together with the L 6 5 -boun of γ(u n ) entail the limit T This provies the equality in turn implying T γ(u n (t)), β(t) t γ (t), β(t) t = T T γ = γ(u) a.e. in Ω T. γ(u(t)), β(t) t. γ(u(t)), β(t) t, Instea, the ientification of ϕ requires an aitional argument. For τ < T arbitrarily chosen, proceeing as in the proof of the forthcoming Theorem 2.6.1, we obtain the uniform bouneness of tt u n in L 2 (τ, T ; H 1 ), 23

34 Chapter 2. Strongly Dampe Nonlinear Wave Equations with a boun epening on τ (an blowing up when τ ). Still, this is enough to infer the uniform bouneness of t u n in W 1,2 (τ, T ; H 1 ) C([τ, T ], H), an conclue that, calling Ω τ,t = Ω (τ, T ), t u n u t a.e. in Ω τ,t ϕ( t u n ) ϕ(u t ) a.e. in Ω τ,t. Then, repeating the previous argument with Ω τ,t in place of Ω T, we establish the equality ϕ = ϕ(u t ) a.e. in Ω τ,t, which extens on the whole cyliner Ω T by letting τ The semigroup The main consequence of Theorem is that the family of maps S(t) : H H acting as S(t)z = {u(t), u t (t)}, (2.4.1) where u is the solution on any interval [, T ] containing t with initial ata z = {a, b} H, efines a ynamical system on H (norme by H ). In what follows, we will often refer more correctly to (2.4.1) when speaking of solution with initial ata z, whose corresponing (ouble) energy is by efinition E(t) = S(t)z 2 H = u(t) u t(t) Dissipativity In this section, we consier a nonlinearity g(u) of critical orer p = 5. The issipative character of S(t) is witnesses by the existence of absorbing sets, capturing the trajectories originating from boune sets of initial ata uniformly in time. The existence of an invariant absorbing set for S(t) is an immeiate corollary of the next result. Theorem The issipative estimate E(t) I(E())e δt + R hols for some structural quantities δ >, R an I : [, ) [, ) increasing. Proof. Along the proof, C will enote a generic constant, possibly epening on ϕ, γ, q, but inepenent of the initial energy E(). Due to (2.3.9)-(2.3.1) an the positivity of Γ, the family of functionals satisfies for every ε > small Λ ε = Π ε (u) + Γ (u) E = Π (u) Λ 2Λ ε 4Λ. (2.5.1) 24

35 2.5. Dissipativity The prouct in H of (2.3.4) an 2u t + 2εu reas u, u t + εu + 2Φ 1 (u t ) = 2 γ(u), u t + εu 2ε ϕ(u t ), u + 2 q, u t + εu, an an application of (2.3.11) entails Recalling (2.3.2), we have t Π ε(u) + επ ε (u) + 1 [ ε u u t 2 1] + 2Φ1 (u t ) (2.5.2) 2 γ(u), u t + εu 2ε ϕ(u t ), u + 2 q, u t + εu. 2 γ(u), u t + εu = t Γ (u) 2εΓ 1 (u) t Γ (u) εγ (u), whereas (2.3.3) an (2.5.1) yiel Finally, 2ε ϕ(u t ), u 2ε ϕ(u t ) L 6/5 u L 6 Cε[Φ 1 (u t )] 5 6 u 1 Φ 1 (u t ) + Cε 6 Λ 3 ε. 2 q, u t + εu 2ε q u + 2 q u t 1 2 Plugging the three inequalities in (2.5.2), we en up with [ ε u u t 2 1] + C. t Λ ε + ελ ε + u t Φ 1(u t ) Cε 6 Λ 3 ε + C. (2.5.3) Within (2.5.1) an (2.5.3), we meet the hypotheses of Lemma Thus, E(t) Λ (t) I(Λ ())e δt + R, for some δ >, R an I increasing. On the other han, from the growth boun on γ we infer the control Λ = E + Γ (u) ce [ 1 + E 2], c 1. Accoringly, upon reefining I in the obvious way, the theorem is proven. For initial ata z B invariant absorbing, an integration in time of (2.5.3) with ε = provies a corollary. Corollary For any invariant absorbing set B there is C = C(B) such that sup z B T t [ ut (τ) Φ 1(u t (τ)) ] τ C + C(T t), T > t. 25

36 Chapter 2. Strongly Dampe Nonlinear Wave Equations 2.6 Partial Regularization In the same spirit of [73], a full exploitation of the partially parabolic features of the equation allows us to gain aitional regularity on the velocity component of the solution. Theorem There exists an invariant absorbing set B satisfying [ t+1 ] u t (t) 1 + u tt (t) + u tt (τ) 2 1 τ <. sup t sup z B In particular, B is a boune subset of V 1. The proof will require some passages. We begin with a simple observation, state as a lemma. Lemma If B is an invariant absorbing set, then B 1 = S(1)B B remains invariant an absorbing, an any (boune) function Λ : B 1 R satisfies sup t sup Λ(S(t)z) = sup z B 1 t t sup Λ(S(t + 1)z) sup Λ(S(1)z). z B z B Lemma There exists an invariant absorbing set B 1, together with a constant C = C(B 1 ), such that for all initial ata in B 1 sup t u t (t) 1 C an 1 u tt (t) 2 t C. Proof. Fix an arbitrary invariant absorbing set B, an consier initial ata z B. In what follows, C is a generic constant epening only on B. By (2.3.1) an the growth boun on γ, the functional Λ = Λ(S(t)z) = u t Φ (u t ) + 2(u, u t ) γ(u), u t + K fulfills for K = K(B ) > large enough the uniform controls u t 2 1 2Λ C[ 1 + u t Φ 1(u t ) ]. In particular, we see from Corollary that 1 Λ(S(t)z) t C. Recalling (2.3.8), the prouct in H of (2.3.4) an 2u tt yiels Estimating the right-han sie as t Λ + 2 u tt 2 = 2 u t γ (u)u t, u t + 2 q, u tt. 2 u t γ (u)u t, u t 2 u t γ (u) L 3/2 u t 2 L 6 C( 1 + u 4 1) ut 2 1 CΛ, 26