EXISTENCE, UNIQUENESS OF WEAK SOLUTIONS

Manuscript submitted to AIMS’ Journals Volume 15, Number 3, July 2006

Website: http://AIMsciences.org pp. 777–809

EXISTENCE, UNIQUENESS OF WEAK SOLUTIONS AND GLOBAL ATTRACTORS FOR A CLASS OF NONLINEAR 2D KIRCHHOFF-BOUSSINESQ MODELS

Igor Chueshov Department of Mechanics and Mathematics Kharkov University Kharkov, 61077, Ukraine

Irena Lasiecka Department of Mathematics University of Virginia Charlottesville, VA 22901, USA

Abstract. We study dynamics of a class of nonlinear Kirchhoff-Boussinesq plate models. The main results of the paper are: (i) existence and uniqueness of weak (finite energy) solutions, (ii) existence of weakly compact attractors.

Introduction. The main object of the study in this paper is the following nonlinear plate equation referred to as Kirchhoff-Boussinesq model: wtt + kwt + ∆2 w = div [f0 (∇w)] + ∆ [f1 (w)] − f2 (w),

(1)

defined on a bounded domain Ω ⊂ R2 with a sufficiently smooth boundary Γ. With (1) we associate the initial data w(x, 0) = w0 (x),

wt (x, 0) = w1 (x),

x ∈ Ω.

(2)

Here k ≥ 0 is the damping parameter, the mapping f0 : R &→ R and the scalar, sufficiently smooth functions f1 and f2 represent (nonlinear) feedback forces acting upon the plate. Our main goal is to study the wellposedness along with long time behaviour of finite energy solutions associated with (1). In order to motivate the structure of nonlinear terms appearing in (1), we wish to point out that the model (1) arises naturally as the limit in Midlin–Timoshenko equations which describe the dynamics of a plate that accounts for transverse shear effects (see, e.g., [22] and [23, Chap.1], and the references therein). More specifically, the Midlin-Timoshenko (MT) system is given in the following canonical form: 2

2

α [vtt + kvt ] − Av + κ · (v + ∇w) + h0 (v) + v · h1 (w) = 0, wtt + k0 wt − κ · div(v + ∇w) + h2 (w) = 0,

(3) (4)

where v(x, t) = (v1 (x, t), v2 (x, t))T is a vector function and w(x, t) is a scalar function on Ω × R+ . The functions v1 (x, t) and v2 (x, t) represent the angles of deflection 2000 Mathematics Subject Classification. Primary: 60H25, 47H10; Secondary: 34D35. Key words and phrases. 2D Boussinesq models, weak well-posedness, global attractors. Research of the second author was partially supported by the NSF Grant DMS-0104305 and ARO Grant DAAD19-02-10179.

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IGOR CHUESHOV AND IRENA LASIECKA

of a filament (they are measures of transverse shear effects) and w(x, t) is the bending component (transverse displacement). The parameter α > 0 describes rotational inertia of filaments. The terms αkvt and k0 wt represent resistance forces (with the intensities k > 0 and k0 > 0). The factor κ > 0 is the so-called shear modulus. The second order elliptic differential operator A in (3) is defined by:   2 1+µ 2 2 ∂x1 + 1−µ 2 ∂x2 2 ∂x1 x2 , A= 1+µ 2 1−µ 2 2 2 ∂x1 x2 2 ∂x1 + ∂x2

where 0 < µ < 1 is the Poisson ratio. The mapping h0 : R2 &→ R2 and the scalar functions h1 and h2 represent feedback forces. Here we note that the presence of the term vh1 (w) destroys the conservative character of these forces and, as we will see later, the energy of the associated dynamical system will be no longer decreasing along trajectories. For details concerning the dynamics of Midlin–Timoshenko plates we refer to [22, 23] and also to our forthcoming paper [10]. We also note that the MT system (3) and (4) subjected to nonlinear damping has been discussed in [9, Chap.7]. What is of relevance to the present paper is the fact that equation (1) can be rigorously derived from the MT system (3) and (4) as a limit equation when the parameters α → 0 and κ → ∞ simultaneously and the following relations between feedback forces hold: % s h1 (ξ)dξ, f2 (s) = h2 (s). f0 (s1 , s2 ) = −h0 (−s1 , −s2 ), f1 (s) = 0

This limiting argument is given in the Appendix (see Proposition 4.3). From the mechanical point of view, the limit when α → 0 means that we neglect asymptotically rotational inertia of filaments and the limit when κ → +∞ corresponds to the absence of transverse shears. The absence of transverse shears is one of the Kirchhoff hypotheses in the shell theory (see, e.g., the discussion in [22]). We also note that in the case f0 (∇w) ≡ 0, f1 (w) = αw + βw2 and f2 (w) ≡ 0 problem (1) is referred to as the ”good” Boussinesq equation (see, e.g., [41] and the references therein). It is known that solutions to ”good” Boussinesq equations may blow-up in a finite time [19, 29, 30]. Various papers considered the additional dissipative effects, such as strong structural damping ∆wt , added to the right hand side of equation (1), see [42, 43] and references therein. Strong structural damping has regularizing effect on the dynamics which gives rise to analytic semigroup associated with linear part of the system. In such case the mathematical analysis is very different and the results obtained reflect strong regularity properties of solutions which are driven by finite energy initial conditions. In contrast, the model under consideration in this paper, does not account for any strong damping - it’s dynamics represent hyperbolic-like behavior. The control of long time behavior is achieved due to the presence of restoring forces f0 (∇w) and the viscous damping kwt which naturally arise in Kirchhoff plate models. This is why we refer to (1) as the Kirchhoff-Boussinesq equation. For the same reason we are mainly interested in the following particular choice of functions f0 and f1 that represent feedback forces acting upon the plate: f0 (∇w) = |∇w|2 ∇w, f1 (w) = w2 + w.

(5)

In order to provide a quantitative analysis of the model (1) it is necessary to impose boundary conditions. To make our analysis complete, we shall consider

2D KIRCHHOFF-BOUSSINESQ MODELS

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several types of physically relevant boundary conditions associated with plate theory ([5, 22]). These include clamped, simply supported and free boundary conditions. (C) Clamped B. C.: w(x, t) = 0, ∇w(x, t) = 0 on Γ × R+ .

(6)

(SS) Simply supported B. C.:

w(x, t) = 0, ∆w = −g

∂ wt ∂n

on Γ × R+ .

(7)

(F) Free B.C.: ∆w + (1 − µ)B1 w = −g1

∂ wt , ∂n

(8)

∂ ∆w + (1 − µ)B2 w = g2 wt + (f0 (∇w) + ∇[f1 (w)], n) on Γ × R+ . ∂n Here above the constants g, g1 , g2 ≥ 0 are given, n = (n1 , n2 ) is the outer normal to Γ. Parameters gi ≥ 0 represent the strength of boundary damping, often considered in the context of plate models [22]. The boundary operators B1 and B2 are standard ones [22] associated with boundary conditions representing moments and shear forces on the boundary Γ: 2n1 n2 ux1 x2 − n21 ux2 x2 − n22 ux1 x1 , (9) ( ) ∂ &' 2 n1 − n22 ux1 x2 + n1 n2 (ux2 x2 − ux1 x1 ) , B2 u = ∂τ where τ = (−n2 , n1 ) is the unit tangent vector along ∂Ω. The constant 0 < µ < 1 has a meaning of the Poisson modulus. We note that the presence of the nonlinear terms involving f0 and f1 in (8) is typical for the case of free (natural) boundary conditions. Indeed, differentiation of potential energy associated with the system and the resulting variational formulation based on higher order (free) boundary conditions lead to the boundary terms involving the nonlinearities f0 , f1 (see Chapter 2 in [22] where several models with free boundary conditions are derived). The dynamics defined above will be considered as an evolution defined on the finite energy space which is given below. Specific structure of the energy space H depends on the choice of boundary conditions. More specifically, we take  H ≡ H02 (Ω) × L2 (Ω) (clamped b.c. (6)),  H ≡ (H 2 ∩ H01 )(Ω) × L2 (Ω) (simply supported b.c. (7)), (10)  H = H 2 (Ω) × L2 (Ω) (free b.c. (8)). B1 u

=

Here H s (Ω) is the L2 based Sobolev space of order s and H0s (Ω) is the closure of C0∞ (Ω) in H s (Ω), s > 0. We will denote by , · ,s the norm in H s (Ω) and by , · , and (·, ·) the norm and the inner product in L2 (Ω). Below we also use the notation Cw ([0, T ]; X) for the space of weakly continuous functions on the interval [0, T ] with values in the Hilbert space X and Xw for the space X endowed with the weak topology. One of the key difficulty associated with the two-dimensional Kirchhoff–Boussinesq model is the fact that the nonlinear term corresponding to f0 is not bounded or locally Lipschitz on the space H of finite energy defined by (10). This poses a serious issue with the wellposedness (particularly uniqueness) of solutions. In fact, to the best of authors’ knowledge the uniqueness of finite energy solutions has been an open problem in the literature (see Remark 1).

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This brings us to the main goal of the present paper which is two-fold: (i) to prove wellposedness of finite energy solutions - including the uniqueness, and (ii) to establish existence of a weakly compact global attractor for the resulting semi-flow. Our main results read as follows: Main Theorem. Let f2 ∈ C 1 (R). With reference to the dynamics in (1), (5) subject to either set of boundary conditions specified in (C), or (SS) or ((F) the following hold: • Existence and uniqueness: For any initial data (w0 ; w1 ) ∈ H there exists a unique finite energy solution (w(t); wt (t)) ∈ Cw ([0, T ]; H) with T < T0 , where T0 = ∞ provided f2 satisfies the non-explosion condition specified in assumption (1.8) and (1.9). • Weak continuity of the flow: Solutions in the reference above depend continuously on initial data with respect to weak topology of H, i.e. problem (1) generates a continuous semi-flow St in the space Hw by the formula St W0 = W (t) ≡ (w(t); wt (t)),

W0 = (w0 ; w1 ).

• Weakly compact global attractors: Assuming, in addition, that the constant k > 0 is sufficiently large and the dissipativity condition imposed on function f2 in Assumption 3.1 is in force, the dynamical system (St ; Hw ) admits a (weak) global attractor A. We recall (see [1], [7], [39], for instance) that a bounded closed set A is said to be a (weak) global attractor for the dynamical system (St ; Hw ) if (i) A is strictly invariant, i.e. St A = A for every t ≥ 0, and (ii) A is uniformly attracting in Hw , i.e. for any bounded set B ⊂ H and any weakly open set O containing A there exists t0 = t0 (B, O) such that St B ⊂ O for any t ≥ t0 . Remark 1. We wish to point out that the existence of a ”global attractor in the energy space”, for the two-dimensional model (closely related to the model considered in this paper) & ) wtt + kwt + ∆2 w + ∆w − α|∇w|2 ∆w + γ∆ w2 = 0 (11)

with clamped boundary conditions is reported on page 357 in [37]. This result is attributed to the reference You (1997b) cited in [37]. However, upon inspection of that source reference, it turns out that the proof of this claim is based on the fact that the nonlinear term in the equation is bounded on the finite energy space. On the other hand, it is apparent via Sobolev’s embeddings that for two dimensional domains Ω the nonlinear term |∇w|2 ∆w is not bounded on L2 (Ω) for w ∈ H 2 (Ω). The same holds for the nonlinear term considered in (5). For this reason the question of wellposedness of solutions to (11) as well as to (1) and (5) appears to be an open problem. The contribution of this manuscript is to provide an affirmative answer to the wellposedness (particularly uniqueness) question along with a construction of a global attractor. Our method for uniqueness applies to nonlinear terms in both (5) as well as (11). It should be also noted that while uniqueness of solutions is now claimed for both models, it is not the case with continuous dependence of solutions with respect to the initial data. The strong continuity of solutions for both models is still an open question. The paper is organized as follows. Section 1 deals with existence and uniqueness of finite energy solutions. Theorem 1.1 provides existence of both local and


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global solutions. Theorem 1.6 formulated in Subsection 1.2 and proved in Section 2 provides uniqueness of finite energy solutions. Theorem 1.9 and Corollary 1.10 in Subsection 1.3 state weak Hadamard wellposedness of the system. Section 3 deals with global attractivity of the dynamics. Theorem 3.2 gives the result on existence of global weakly compact attractor. In the Appendix we discuss the relation between the Kirchhoff-Boussinesq equation (1) and the Mindlin-Timoshenko model (3) and (4). 1. Existence-uniqueness theorem for weak solutions. We recall that a function w(x, t) is said to be a weak solution to problem (1) and (2) (with one of the boundary conditions specified in (C), (SS) or (F)) on an interval [0, T ] , if W (t, x) ≡ (w(x, t); ∂t w(x, t)) ∈ L∞ (0, T ; H), where the space H is given by (10) depending on the choice of boundary conditions. Moreover, (i) the vector-valued function t &→ (w(t); ∂t w(t)) ∈ H is weakly continuous, (ii) w(0) = w0 and ∂t w(0) = w1 , and (iii) equality (1) is satisfied in the following sense • in the clamped case (6): d (wt (t), φ) + k(wt (t), φ) + (∆w(t), ∆φ) + F (w(t), φ) = 0 dt

(1.1)

for each φ ∈ H02 (Ω), where F (w, φ) = (f0 (∇w), ∇φ) + (∇[f1 (w)], ∇φ) + (f2 (w), φ);

(1.2)

• in the simply supported case (7): . % d ∂w(t) ∂φ · dΓ + k(wt (t), φ) (wt (t), φ) + g dt ∂n Γ ∂n +(∆w(t), ∆φ) + F (w(t), φ) = 0

(1.3)

for each φ ∈ (H 2 ∩ H01 )(Ω), where F (w, φ) is given by (1.2); • in the case of free boundary conditions (8): 0 . % / d ∂w(t) ∂φ · + g2 w(t) · φ dΓ + k(wt (t), φ) g1 (wt (t), φ) + dt ∂n ∂n Γ +a(w(t), φ) + F (w(t), φ) = 0

(1.4)

for each φ ∈ H 2 (Ω), where F (w, φ) is the same as above and the bilinear form a(w, u) is given by % a(w, u) ≡ [wx1 x1 ux1 x1 + wx2 x2 ux2 x2 ] dx (1.5) Ω % + [µ(wx1 x1 ux2 x2 + wx2 x2 ux1 x1 ) + 2(1 − µ)wx1 x2 ux1 x2 ] dx. Ω

We understand the time derivatives in relations (1.1), (1.3) and (1.4) in the sense of distributions.

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1.1. Existence of finite energy solutions. Theorem 1.1. Let f2 ∈ C 1 (R). With reference to (1), (5) under any set of boundary conditions specified in (C), (SS), (F) and for any initial conditions W0 = (w0 ; w1 ) ∈ H there exists a local (in time) weak solution W (t) = (w(t); wt (t)). More precisely, for any R > 0 and W0 ∈ H with |W0 |H ≤ R there exists T0 ≡ T0 (R) such that the problem possesses a weak solution w(t) on the interval [0, T0 ] with the property |W (t)|H ≤ C(R, T ) for all 0 ≤ t ≤ T < T0 . In addition, we have that • in the simply supported case (7), we have that the generalized time derivative √ of the function g ∂w ∂n belongs to L2 ([0, T ] × Γ), i.e. 1 % T% 1 1 ∂ ∂w 12 1 dΓdτ < ∞ for every 0 < T < T0 ; 1 g (1.6) 1 1 0 Γ ∂t ∂n • similarly, in the case of free boundary conditions (8), we have the relation 1 1 3 1 % T% 2 1 1 ∂ ∂w 12 1 ∂w 12 1 + g2 1 1 g1 11 (1.7) 1 ∂t 1 dΓdτ < ∞ for 0 < T < T0 . ∂t ∂n 1 0 Γ

Moreover, the solution is global, i.e. T0 = ∞, provided f2 satisfies the following non-explosion condition • in the clamped and simply supported cases: % s f2 (ξ)dξ ≥ −δs4 − β (1.8) F2 (s) ≡ 0

for some δ ≥ 0 small enough and β ∈ R; • in the free boundary conditions case: F2 (s) ≥ −αs2 − β

for some

α, β ∈ R.

(1.9)

Remark 1.2. It is easy to see that (1.8) holds with arbitrary small δ > 0 if ν(µ) ≡ lim inf |s|→∞

f2 (s) ≥0 s|s|µ−1

(1.10)

for µ = 3. Relation (1.9) is true when ν(1) > −∞. The sufficient condition (1.10) will be used later when studying dissipativity. Proof. Step 1 : preliminary energy estimates. In what follows we denote by E(w; w$ ) the energy of the system at the state (w; w$ ) ∈ H. The positive part of the energy is denoted by E0 (w; w$ ). This is given by the expression: for the clamped and simply supported boundary conditions one has 0 % / 1 1 $ $ 2 2 4 E0 (w; w ) ≡ |w (x)| + |∆w(x)| + |∇w(x)| dx. (1.11) 2 Ω 2 For the free case we define 0 % / 1 1 1 E0 (w; w$ ) ≡ |w$ (x)|2 + |∇w(x)|4 dx + a(w, w), 2 Ω 2 2

(1.12)

where the bilinear form a(w, u) is given by (1.5). With E0 (w; w$ ) given above we define next the energy of the system at the state W = (w; w$ ): % w(x)|∇w(x)|2 dx. (1.13) E(w; w$ ) ≡ E0 (w; w$ ) + Ω


783

On the strength of Sobolev’s embedding H 1 (Ω) ⊂ Lp (Ω), 1 ≤ p < ∞, we have that E0 and E are continuous functionals on H and ( ' E0 (w; w$ ) ≤ C · 1 + |W |4H , W = (w; w$ ) ∈ H. It is also clear that 1% 1 1 1 α 1 w|∇w|2 dx1 ≤ 1 |∇w|4 ,w,2 L4 (Ω) + 1 1 2α 2 Ω

for any α > 0.

(1.14)

For function w ∈ (H 2 ∩H01 )(Ω) one can use Poincare’s inequality in order to estimate the L2 component of the displacement w % ||w||2 ≤ C||∇w||2 ≤ + |∇w|4 dx + +−1 CΩ , w ∈ (H 2 ∩ H01 )(Ω). (1.15) Ω

Selecting in (1.15) suitably small + and combining with (1.14) yields the estimate 1% 1 1 1 1 w|∇w|2 dx1 ≤ 1 |∇w|4 for any w ∈ (H 2 ∩ H01 )(Ω). L4 (Ω) + CΩ 1 1 8 Ω

Hence, in the case of clamped and simply supported boundary conditions we have that 1 (1.16) E0 (w; w$ ) − CΩ ≤ E(w; w$ ) ≤ 2E0 (w; w$ ) + CΩ , (w; w$ ) ∈ H. 2 In the case of free boundary conditions (1.15) does not hold and superlinearity of the gradient term in potential energy can not be used in order to absorb the L2 component of the displacement appearing in (1.14). Therefore, (1.14) implies only 1 E0 (w; w$ ) − 2,w,2 ≤ E(w; w$ ) ≤ 2E0 (w; w$ ) + ,w,2 2

(1.17)

for any (w; w$ ) ∈ H 2 (Ω) × L2 (Ω).

Step 2: existence of solutions. We define next Faedo–Galerkin approximations of solutions. The algorithm depends on the structure of the boundary conditions. Step 2(i) : clamped case. We begin first with the clamped case. Let Hn ⊂ H02 (Ω) be a finite dimensional approximation of H02 (Ω). We denote by Wn (t) ≡ (wn (t); wnt (t)) ∈ Hn × Hn an appropriate Faedo–Galerkin approximation of the sought after solution. The following variational equation is satisfied for all test functions φn ∈ Hn : (wntt , φn ) + (∆wn , ∆φn ) + k(wnt , φn ) + (|∇wn |2 ∇wn , ∇φn )

with the initial conditions

= −(∇wn2 , ∇φn ) − (f2 (wn ), φn )

(wn (0), φn ) = (w0 , φn ),

(wnt (0), φn ) = (w1 , φn )

(1.18) (1.19)

such that Wn (0) ≡ (wn (0); wnt (0)) → (w0 ; w1 ) ≡ W0 in H as n → ∞. We assume that |W0 |H ≤ R and choose n0 such that |Wn (0)|H ≤ 2R for all n ≥ n0 . It is clear that the approximate solution wn (t) exists on some (maximal) interval [0, Tn ), where Tn > 0 also depends on Wn (0). Taking φn = wnt in the variational form (1.18) and noting that ∇wn2 ∇wnt =

d (wn |∇wn |2 ) − wnt |∇wn |2 dt

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gives En (t) + k

% t% 0

Ω

|wnt |2 dxdτ = En (0) +

% t% 0

Ω

& ) |∇wn |2 wnt − f2 (wn )wnt dxdτ.

(1.20) Here and below for shortness we denote En (t) = E(wn (t); wnt (t)) and E0,n (t) = E0 (wn (t); wnt (t)). Since E0,n (0) ≤ CR under the conditions |W0 |H ≤ R and n ≥ n0 , from (1.16) and (1.20) it follows that % t% ) & 2 E0,n (t) ≤ CR + + f22 (wn ) dxdτ |∇wn |4 + 2wnt 0 Ω 0 % t/ % f22 (wn )dx dτ. ≤ CR + 4E0,n (t) + 0

Ω

Applying Sobolev’s embedding H (Ω) ⊂ C(Ω) along with continuity of f2 , Gronwall’s inequality gives 5 6 40,n (t)) e4t 40,n (t) ≤ CR + t · Ψ(E (1.21) E 2

40,n (t) = for any t from the maximal interval [0, Tn ) of the existence, where E max[0,t] E0,n (τ ) and Ψ : R+ &→ R+ is an increasing continuous function. Since E0,n (t) is continuous in time, we obtain that for some 0 < T1n ≤ min{1, Tn }, there exists a constant C1n > 0 so that the following bound holds: 40n (t) ≤ C1n for 0 < t ≤ T1n . E

Therefore from (1.21) we have that Consequently

40,n (t) ≤ (CR + t · Ψ(C1n )) e4 for 0 < t ≤ T1n . E

40,n (t) ≤ 2e4 CR E (1.22) 7 8 −1 for any 0 ≤ t ≤ min T1n , CR Ψ(C1n ) . Let T0n > 0 be the maximal number such that (1.22) holds for all 0 ≤ t < T0n . Then for each n ≥ n0 we have that T0n ≤ Tn 40,n (T0n ) = 2e4 CR . We claim and either T0n = ∞ or T0n < ∞. In the latter case E that inf {T0n : n ≥ n0 } ≡ T0 > 0. (1.23) Indeed, if (1.23) is not true, then there exists a sequence {nk } such that T0nk → 0 as k → ∞. However (1.21) implies that ' ( 40,n (T0n ) ≤ CR + T0n · Ψ(2e4 CR ) e4T0nk 2e4 CR = E k k

for all k = 1, 2, . . .. After the limit transition k → ∞ we obtain a contradiction. Thus there exists T0 > 0 and C(R) > 0 such that E0,n (t) ≤ C(R) for all 0 ≤ t ≤ T0 , n ≥ n0 ,

(1.24)

,wntt ,−2 ≤ C(R), 0 ≤ t ≤ T0 , n ≥ n0 .

(1.25)

under the condition |W0 |H ≤ R. Using this estimate and also variational relation (1.18) it is easy to see that Therefore there exist a sequence {nk } and a function w(t, x) such that ( ' {w(t); wt (t); wtt (t)} ∈ L∞ 0, T0 ; H × H −2 (Ω)

and

{wnk (t); wnk t (t); wnk tt (t)} → {w(t); wt (t); wtt (t)},

k → ∞,

(1.26)


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' ( weakly star in L∞ 0, T0 ; H × H −2 (Ω) . By (1.26) we can assume that W (t) ≡ (w(t); wt (t)) ∈ Cw (0, T0 ; H) and by [38, Corollary 4] we have that ' ( (wnk (t); wnk t (t)) → (w(t); wt (t)) strongly in C 0, T0 ; H02−ε (Ω) × H −ε (Ω)

as k → ∞ for any ε > 0. Since the maps w &→ |∇w|2 ∇w, w &→ |∇w|2 and w &→ f2 (w) are continuous from H02−ε (Ω) into L2 (Ω) for ε small enough, we can pass with the limit on variational form (1.18) obtaining (1.1) for each φ ∈ H02 (Ω). By weak lower semicontinuity of E0 and (1.24) we also have E0 (w(t); wt (t)) ≤ lim inf E0,n (t) ≤ C(R), n→∞

t ∈ [0, T0 ], |W0 |H ≤ R.

Thus we obtain the local existence result stated in Theorem 1.1 for the clamped case. In order to obtain global existence it is convenient to introduce another energy function given by % E(w; w$ ) = E(w; w$ ) + F2 (w(x))dx, (1.27) Ω

where the energy E(w; w$ ) is defined by (1.13) and the function F2 is the antiderivative of f2 is given by (1.8). The energy relation (1.20) implies that % t% En (t) ≤ En (0) + |∇wn |2 wnt dxdτ, 0

Ω

where En (t) = E(wn (t); wnt (t)). Using continuity of F2 and Sobolev’s embedding H 2 (Ω) ⊂ C(Ω) one easily obtains from (1.16) that |E(w, w$ )| ≤ 2E0 (w, w$ ) + Ψ∗ (|w|2,Ω ),

(1.28)

where Ψ∗ (·) is an increasing function. In order to obtain the lower bound on E(w, w$ ) we notice that (1.16) implies % 1 1 |∇w|4 dx − CΩ . E(w, w$ ) ≥ E0 (w, w$ ) + 4 16 Ω

On the other hand, by (1.8) and Poincare’s inequality: % % 4 1 |w| dx ≤ CΩ · |∇w|4 dx, w ∈ H 2 (Ω) ∩ H01 (Ω) ⊂ W4,0 (Ω), Ω

(1.29)

Ω

we obtain %

Ω

F2 (w(x))dx ≥ −δ

%

Ω

w4 (x)dx − β ≥ −CΩ δ

%

Ω

|∇w|4 (x)dx − β

and using the definition of E yields the desired lower bound as long as we select δ in (1.8) small enough: 9 :% 1 1 1 $ $ − δCΩ |∇w|4 dx − C ≥ E0 (w, w$ ) − C. E(w; w ) ≥ E0 (w; w ) + 4 16 4 Ω Thus the upper bound for E given in (1.28) allows us to obtain the estimate % t E0,n (t) ≤ C1 (R) + C2 E0,n (τ )dτ, t > 0, 0

which implies the global existence result for the clamped case.

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Remark 1.3. Using energy relation (1.20) for approximate solutions we can also establish an energy inequality for the weak solutions constructed by Faedo–Galerkin approximations. Indeed, due to the above mentioned compactness and the convergence wnk t → wt weakly star in L∞ (0, T0 ; L2 (Ω)) we also have that % t% % t% |∇wnk |2 wnt dxdτ → |∇w|2 wt dxdτ 0

as well as

% t% 0

0

Ω

Ω

f2 (wn )wnt dxdτ →

Ω

% t% 0

f2 (w)wt dxdτ.

Ω

Weak lower semicontinuity of the energy E0 along with the convergence above allows us to conclude the energy inequality for the limit: % t% % t% |∇w|2 wt dxdτ − f2 (w)wt dxdτ. E(w(t); wt (t)) ≤ E(w0 ; w1 ) + 0

0

Ω

Ω

This inequality can be also written in the form: % t% E(w(t); wt (t)) ≤ E(w0 ; w1 ) + |∇w|2 wt dxdτ, 0

Ω

where E(w; w ) is given by (1.27). $

Step 2(ii)- simply supported boundary conditions. In the case of simply supported boundary conditions, Faedo–Galerkin approximation is defined as follows. Let Hn ⊂ H01 (Ω) ∩ H 2 (Ω) be a finite dimensional approximation of H 2 (Ω) ∩ H01 (Ω). We denote by Wn (t) ≡ (wn (t); wnt (t)) ∈ Hn × Hn a solution to the following variational equation 9 : ∂ ∂ wnt , φn (wntt , φn ) + (∆wn , ∆φn ) + k(wnt , φn ) + g · ∂n ∂n L2 (Γ) +(|∇wn |2 ∇wn , ∇φn ) = −(∇wn2 , ∇φn ) − (f2 (wn ), φn ),

(1.30)

where φn ∈ Hn is arbitrary, with the initial conditions (1.19). The corresponding energy equality then becomes 12 % t% % t% 1 1 ∂ 1 1 wnt 1 dΓdτ En (t) + k |wnt |2 dxdτ + g 1 ∂n 1 0 0 Ω Γ % t% & ) |∇wn |2 wnt − f2 (wn )wnt dxdτ. (1.31) = En (0) + 0

Ω

Therefore, using (1.16) we obtain that 12 0 % t% 1 % t/ % 1 ∂ 1 2 1 wnt 1 dΓdτ ≤ CR + C E0,n (t) + g (t) + f (w )dx dτ. E 0,n n 2 1 1 0 0 Γ ∂n Ω

Now, the rest of the argument is the same as in the clamped case, after noting that in the case g > 0 we have the estimate % T0 ,wntt ,2−2 dτ ≤ C (1.32) 0

instead of (1.25) and that the energy inequality implies also weak convergence g 1/2

∂ ∂ ∂ wnt → g 1/2 w weakly in L2 ([0, T0 ] × Γ) ∂n ∂t ∂n

(1.33)


787

along a subsequence. This allows to pass with the limit on Faedo–Galerkin approximation, thereby reconstructing the sought after variational form for the solution. The regularity of the normal derivatives in (1.33) establishes the additional boundary regularity required in (1.6). As for global solutions, the argument is the same as in the clamped case. This 1 (Ω) which is is due to the fact that (1.29) depends only on the fact that w ∈ W4,0 satisfied, as well, for simply supported boundary conditions. Step 2(iii): free boundary conditions. Finally, the free case. The arguments are as above after noticing that the Faedo–Galerkin approximation is now defined as follows: Let Hn ⊂ H 2 (Ω) be a finite dimensional approximation of H 2 (Ω) and Wn (t) ≡ (wn (t); wnt (t)) ∈ Hn × Hn be a solution to the following variational equation (wntt , φn ) + a(wn , φn ) + k(wnt , φn ) 0 % / ∂ ∂ + g1 wnt φn + g2 wnt φn dΓdτ + (|∇wn |2 ∇wn , ∇φn ) ∂n ∂n Γ = −(∇[w2 ], ∇φn ) − (f2 (wn ), φn ),

∀φn ∈ Hn ,

(1.34)

with the initial conditions (1.19). The energy identity for the approximation becomes: 3 12 % t% % t% 2 1 1 1 ∂ 2 |wnt |2 dxdτ + g1 11 wnt 11 + g2 |wnt | dΓdτ En (t) + k ∂n 0 0 Ω Γ % t% & ) = En (0) + |∇wn |2 wnt − f2 (wn )wnt dxdτ. (1.35) 0

Ω

Since E0,n (t) does not control the L2 component of the displacement (see (1.17)), it is natural to add to both sides of the original equation the term 2w(t). This leads to a new energy function denoted by Φn (t) ≡ E0,n (t) + ||wn (t)||2 . From (1.17) we have that ' ( Φn (t) ≤ C ,wn (t),2 + En (t) : 9 % t 2 2 ≤ C0 En (t) + C1 ,wn (0), + ,wnt , dτ . 0

Therefore (1.35) implies that 3 12 % t% 2 1 1 ∂ 1 2 1 1 Φn (t) + g1 1 wnt 1 + g2 |wnt | dΓdτ ∂n 0 Γ : % t% 9 % 2 ≤ CR + C [f2 (wn )] dx dτ. Φn (τ ) + 0

Ω

(1.36)

Ω

The further argument is the same as in the previous cases. We should also take into account that estimate (1.32) remains true for the free case and that (1.36) gives the bounds for the boundary terms such that 1/2 ∂ 1/2 ∂ ∂ wnt → g1 w weakly in L2 ((0, T0 ) × Γ) g1 ∂n ∂t ∂n 1/2 1/2 ∂ w weakly in L2 ((0, T0 ) × Γ) (1.37) g2 wnt → g2 ∂t ∂ along a subsequence (we understand time derivatives ∂t in the sense of distributions). Using the above convergence along with the arguments employed before

788


allows to pass with the limit on a variational form (1.34). Moreover, the convergence of traces in (1.37) establishes the boundary regularity (1.7). The proof existence of local weak solutions is thus completed. As for global solutions, we note that (1.35) implies % & ) F2 (wn (t)) + α|wn (t)|2 dx ≤ En (0) En (t) + Ω % % t% ; < & ) α 2 |∇wn |2 wnt + wn wnt dxdτ. + F2 (wn (0)) + α|wn (0)| dx + 2 0 Ω Ω Therefore property (1.9) and (1.17) allows us to obtain an estimate of the form % t Φn (t) ≤ C1 (R) + C2 Φn (τ )dτ, t > 0, 0

which implies the global existence result.

Remark 1.4. As in the clamped case (see Remark 1.3) we can also derive from (1.31) and from (1.35) the energy inequalities for the simply supported and free cases. Remark 1.5. It may be worth noting that the result of Theorem 1.1 remains valid for a model with nonlinear dissipation kg(wt ) where g is continuous and monotone. Similar generalizations can be applied to boundary damping as long as the nonlinearity is monotone. We can also consider more general (than (5)) functions f0 and f1 in (1) imposing on them some non-explosion and growth conditions. 1.2. Uniqueness of finite energy solutions. While existence of solutions relies on reasonably routine arguments (seen above), the uniqueness of weak solutions has been an open problem in the two dimensional case. This is due to the fact that in the two-dimensional case the nonlinear term div[f0 (∇w)] is neither bounded nor locally-Lipschitz as a mapping from H 2 (Ω) into L2 (Ω). The main contribution of the present paper is uniqueness of weak solutions. The corresponding result is formulated below. Since the uniqueness does not depend on structural properties of f0 and f1 , we shall prove this result (with no extra effort) for more general functions f0 and f1 than considered in the previous subsection. Theorem 1.6. Assume that • f0 (v) = (f01 (v1 , v2 ), f02 (v1 , v2 ))T is a continuously differentiable mapping from R2 into itself such that 1 1 1 ∂f0i (v1 , v2 ) 1 ' ( 1 ≤ C 1 + v12 + v22 , i, j = 1, 2; 1 (1.38) 1 1 ∂vj

• f1 (s) ∈ C 2 (R) and f2 (s) ∈ C 1 (R). Then the problem (1) and (2) with either set of boundary conditions (C), (SS) or (F) cannot possess more than one weak solution for any interval of existence [0, T ]. Remark 1.7. The result stated in the Theorem above does not depend on the presence of the damping in (1) or in (7) and (8). In particular, it applies to the undamped version of the problem. Moreover, as we will see below, the undamped case is much easier. Remark 1.8. An interesting problem that still remains unsolved is that of Hadamard wellposedness, i.e. continuous dependence of solutions with respect to initial data (in the strong topology of the finite energy space H). The above property


789

depends on strong continuity of solutions and related energy identity for solutions. It is known [20] that once the above properties are established, then Hadamard wellposedness follows via a soft type of argument based on uniqueness and lower weak semicontinuity of the energy (see also [32] for linear problems). In fact, in the case of full von Karman evolutions, the said properties - energy identity - has been proved in [20] by using an appropriate finite difference approximation. It is unfortunate that the method does not apply in the present context. Thus the issue of Hadamard wellposedness still remains open. On the other hand, Hadamard wellposedness in weak topology can be proved by using the uniqueness result just established (see Theorem 1.9 and Corollary 1.10 below). Another interesting and open question is whether one could consider nonlinear damping in the model. In the case of interior damping one could perhaps obtain results for nonlinear but linearly bounded damping. However, in the case of boundary damping the difficulty is very intrinsic. The boundary damping is not bounded with respect to finite energy space. The proof of Theorem 1.6 is given in Section 2. Here we just note that the proof is based on idea introduced by Sedenko [34, 35] in the study of uniqueness of solutions describing nonlinear oscillations of shallow shells. We shall apply variants of this method used in [6] for the von Karman evolution equations with homogenous boundary conditions and in [25] for full von Karman systems with boundary dissipation. We also refer to [11] where the same method was applied to 2D Zakharov system. 1.3. Hadamard wellposedness in weak topology. Equipped with the existence and uniqueness result we proceed to prove that the system is Hadamard well-posed with respect to weak topology. Theorem 1.9. With reference to (1) finite energy solutions depend continuously, with respect to weak topology, on the initial data. This is to say: W n (0) → W (0) weakly in H implies that max |(W n (t), Φ)H − (W (t), Φ)H | → 0,

[0,T0 ]

n → ∞.

(1.39)

for any Φ ∈ H and any (joint) interval [0, T0 ] of the existence the solutions W n (t) = (wn (t); wtn (t)) and W (t) = (w(t); wt (t)) with initial data W0n = (w0n ; w1n ) and W0 = (w0 ; w1 ) respectively. Proof. Since W0n → W0 weakly in H, we have that |W0n |H ≤ R is uniformly with respect to n. Hence the solution Wn (t) is defined locally on [0, T0 ] with some T0 > 0 independent of n. Moreover, by a priori estimate established in the proof of Theorem 1.1 we have that % t n 2 n 2 n ,wtt (τ ),2−2 dτ ≤ CR , t ∈ [0, T0 ], ,w (t),2 + ,wt (t), + 0

where T0 may depend only on R. By the same compactness based argument as used for the proof of the existence and by the uniqueness result in Theorem 1.6 we conclude that n {wn (t); wtn (t); wtt (t)} → {w(t); wt (t); wtt (t)},

n → ∞,

weakly star in L∞ (0, T0 ; H × V ). Here and below V = H (Ω) in the clamped & )$ & )$ case, V $ = H 2 (Ω) ∩ H01 (Ω) in simply supported case and V $ = H 2 (Ω) for the $

$

−2

790


free boundary conditions. By [38, Corollary 4] we also have that sup {,wn (t) − w(t),2−ε + ,wtn (t) − wt (t),−ε } → 0,

[0,T0 ]

n → ∞,

(1.40)

for any ε > 0. Let φ ∈ V $ . We obviously have that

max |(wn (t), φ) − (w(t), φ)| ≤ max |(wn (t), φN ) − (w(t), φN )| + C,φ − φN ,V !

[0,T0 ]

[0,T0 ]

for any sequence {φN } from L2 (Ω). We choose it such that ,φ − φN ,V ! → 0 as N → ∞. Thus by (1.40) we obtain that . lim sup max |(wn (t), φ) − (w(t), φ)| ≤ C,φ − φN ,V ! n→∞

[0,T0 ]

for every N = 1, 2, . . . and hence (wn (t), φ) → (w(t), φ) in C(0, T0 ) for every φ ∈ V $ . A similar argument can be applied to the time derivatives wtn . This implies (1.39). Corollary 1.10. Assume that (5), (1.8) and (1.9) hold. Then the mapping St : H &→ H defined by the formula St W0 = (w(t); wt (t)),

W0 = (w0 ; w1 ),

(1.41)

where w(t) is a weak solution to problem (1) and (2) with the corresponding boundary conditions, possesses the properties: • S0 = I and St ◦ Sτ = St+τ for all t, τ ≥ 0 (semigroup property); • (t, W0 ) &→ St W0 is a continuous mapping from R+ × Hw into Hw , where Hw is the space H endowed with the weak topology (weak continuity property). Thus problem (1) generates (topological) dynamical system (St ; Hw ) with the phase space Hw . Proof. The semigroup property follows from the uniqueness theorem. As for the continuity property, we obviously have that |(Stn W0n − St W0 , Φ)H | ≤ max |(Sτ W0n − Sτ W0 , Φ)H | + |(Stn W0 − St W0 , Φ)H | τ ∈[0,T0 ]

for any Φ ∈ H and tn , t ∈ [0, T ]. Therefore weak continuity of solutions with respect to t and relation (1.39) implies that (Stn W0n − St W0 , Φ)H → 0 as tn → t and W0n → W0 weakly in H. 2. Proof of Theorem 1.6. We start with some preliminary facts. We introduce the operator A in L2 (Ω) by the formula Au = ∆2 u with the domain D(A) given by • in the clamped case: D(A) = H 4 (Ω) ∩ H02 (Ω); • in the simply supported case: 7 8 D(A) = u ∈ H 4 (Ω) : u = ∆u = 0 on Γ ;

• in the free case: D(A) = u ∈ H 4 (Ω)

1 1 1 1

∆u + (1 − µ)B1 u = 0 ∂ ˜ ∂n ∆u + (1 − µ)B2 u = 0

(2.1) .

on Γ .

(2.2)


791

˜ 2 w ≡ B2 w − lw with some l > 0 and the boundary operators B1 and Here above B B2 are given by (9). The purpose of modifying the operator B2 is to define an invertible operator corresponding to the biharmonic problem with free boundary conditions. One can see that the operator A is a strictly positive self-adjoint operator with the compact resolvent. Let {ek } be the orthonormal basis in L2 (Ω) of eigenvectors of the operator A and {λk } be the corresponding eigenvalues: Aek = λk ek ,

k = 1, 2, ....;

W also note that for every s ∈ [0, 1/2] we have  4s H (Ω), s 1= 1/8, 3/8    0 4s (H ∩ H01 )(Ω), s ≥ 1/4 D(As ) = H 4s (Ω), s < 1/4, s 1= 1/8,    04s H (Ω)

0 ≤ λ1 ≤ λ2 ≤ . . .

(clamped case), (simply supported case), (simply supported case), (free case).

(2.3)

Moreover, the corresponding Sobolev norms are equivalent to the graph norms of the corresponding fractional powers of A, i.e. c1 ,As u, ≤ ,u,4s ≤ c2 ,As u,,

for all admissible s ∈ [0, 1/2]. The following assertion is critical for the proof.

u ∈ D(As ),

(2.4)

Lemma 2.1 ([6]). Let PN be the projector in L2 (Ω) onto the space spanned by {e1 , e2 , ..., eN } and f (x) ∈ D(A1/4 ). Then there exists N0 > 0 such that max |(PN f )(x)| ≤ C · {log(1 + λN )}1/2 , f ,1 . x∈Ω

for all N ≥ N0 . The constant C does not depend on N .

Proof. This assertion was proved in [6] for the case of clamped boundary conditions. For other boundary conditions the argument is basically the same and relies on (2.3) and (2.4) along with the relation max |φ(x)| ≤ Cσ −1/2 ,φ,H 1+σ (R2 ) ,

x∈R2

φ ∈ H 2 (R2 ),

for σ > 0 small enough (see the calculations in [6]). We note that one could also control the L∞ norm of f in Lemma 2.1 by sharp Sobolev embedding with logarithmic corrector involving second derivatives [4]. While this embedding turns out very useful in proving wellposedness of regular solutions to viscous Boussinesq equation [18], it is not appropriate in our Kirchhoff-Boussinesq model where the uniqueness of weak solutions is an issue. Below we will also use the following classical embedding (see, e.g., [40]) 2 (2.5) H s (Ω) ⊂ Lp (Ω) if s = 1 − , p ≥ 2. p If w(t) is a weak solution on the interval [0, T ], then (1) implies that ∂t2 w(x, t) ∈ L∞ (0, T ; H −2(Ω)).

Therefore by interpolation we can conclude that w(t) and ∂t w(t) are strongly continuous functions with values in H 1 (Ω) and H −1 (Ω) respectively. The main idea of the proof is based on the following principle: obtain energy estimates in ”negative” norms that depended on the large parameter N destined to tend to ∞. The role of the parameter N is to compensate the lack of embedding of H 1 into L∞ in the

792


dimension 2. By calibrating the estimates and letting N → ∞ one obtains the desired uniqueness statement. The first part of the program - energy estimates in ”negative” norms depend on the nature of boundary conditions. Indeed, the main tools are negative norm estimates generated by the linear part of the system. In the case of boundary damping, this step involves additional duality and interpolation arguments. The second part of the program - a large parameter - is based on certain sharp estimates derived for finite dimensional spectral projectors which give a ”quantitative” description of the lack of embedding of H 1 (Ω) into L∞ (Ω) (see Lemma 2.1 above). In what follows we shall present the proof of Theorem 1.6 treating each case of boundary conditions separately. 2.1. Clamped Case. Proof. Let w1 (t) and w2 (t) be weak solutions of the problem (1), (2) and (C). Let w(t) = w1 (t) − w2 (t). Then wN (t) = PN w(t) is a solution of the linear, but nonhomogenous problem wtt + kwt + ∆2 w = (PN M )(x, t),

x ∈ Ω, t > 0,

(2.6)

with the boundary and initial conditions ∂w w|∂Ω = |∂Ω = 0, w|t=0 = 0, ∂t w|t=0 = 0. ∂n Here as above PN is the orthoprojector in L2 (Ω) on the space spanned by the first N eigenvectors of the biharmonic operator A with the Dirichlet boundary conditions and M (x, t) ≡ M (t) = M1 (t) + M2 (t) + M3 (t), (2.7) where M1 (t) = M2 (t) = M3 (t) =

div [f0 (∇w1 (t)) − f0 (∇w2 (t))] ,

∆ [f1 (w1 (t)) − f1 (w2 (t))] , − [f2 (w1 (t)) − f2 (w2 (t))] .

Multiplying the equation (2.6) by A−1/2 wt and integrating by parts yield % t ||A−1/4 M (τ )||||A−1/4 wt (τ )||dτ ||A−1/4 PN wt (t)||2 + ||A1/4 PN w(t)||2 ≤ C 0

for all t ∈ [0, T ]. From here after accounting for (2.3) and (2.4) we obtain % t 2 2 , wt (t) ,−1 + , w(t) ,1 ≤ C · , M (τ ) ,−1 · , wt (τ ) ,−1 dτ. 0

In particular this implies that

, w(t) ,1 ≤ C ·

%

t 0

, M (τ ) ,−1 dτ.

(2.8)

The inequality above is the basis for further estimates. Below we also use the estimate which follows from the definition of weak solutions: sup {,w1 (t),2 + ,w2 (t),2 } ≤ R,

(2.9)

t∈[0,T ]

where R > 0 is a constant. Now we estimate the quantities ,Mi (t),−1 , i = 1, 2, 3. All constants appearing below may depend on R.


793

We start with ,M1 (t),−1 which is the most critical. It is clear that ,M1 (t),−1

C,f0 (∇w1 (t)) − f0 (∇w2 (t)), B B 2 % 1 A B ∂f0i B B C dλ B (λ∇w (t) + (1 − λ)∇w (t)) · w (t) 1 2 xj B ∂vj B i,j=1 0   2 2 A A B B Bwlx (t) · wmxi (t) · wxj (t)B , C · ,w(t),1 + k

≤

≤ ≤

l,m=1 k,i,j=1

where we have used the growth condition imposed on f0 in (1.38). We estimate every term in this sum. Before we proceed, let us note that the fact that prevents us from using standard estimates is that, in general, the last term on the RHS of the previous inequality is not bounded by C,w,1 under condition (2.9). To cope with the problem we shall use projectors PN that allow us splitting between large and small frequencies: wlxk · wmxi · wxj

=

(QN wlxk ) · wmxi · wxj + (PN wlxk ) · (QN wmxi ) · wxj +(PN wlxk ) · (PN wmxi ) · wxj ≡ z1 + z2 + z3 ,

where QN = I − PN . Let 0 < s < 1. Using the Hölder inequality and imbedding (2.5) we obtain that ,z1 , ≤ ≤ ≤

,QN wlxk ,L2/s(Ω) · ,wmxi · wxj ,L2/(1−s) (Ω)

C,QN wlxk ,1−s · ,wmxi ,L4/(1−s) (Ω) · ,wxj ,L4/(1−s) (Ω)

C,QN wlxk ,1−s · ,wmxi ,1 · ,wxj ,1 ≤ C,QN wlxk ,1−s .

In the last inequality we have used the fact that the both w1 and w2 are bounded in H 2 (Ω) (see (2.9)), so the constant C depends on the constant R from (2.9). Since QN is the eigenprojector on the Span{ek : k ≥ N + 1}, we have from (2.4) and (2.9) that −s/4 −s/4 (2.10) ,z1 , ≤ C · λN +1 · ,wl ,2 ≤ C · λN +1 . In the same way we also obtain that −s/4

(2.11)

,z2 , ≤ C · λN +1 .

Now using Lemma 2.1 and estimate (2.9) we get that ,z3 , ≤ ≤

≤

sup |PN wlxk | · sup |PN wmxi | · ,wxj ,

x∈Ω

x∈Ω

C · log(1 + λN )· , wl ,2 · , wm ,2 · , w ,1

C · log(1 + λN )· , w ,1 .

(2.12)

Relations (2.10)–(2.12) imply that

−s/4

,M1 (t),−1 ≤ C1 · log(1 + λN )· , w(t) ,1 +C2 · λN +1 ,

(2.13)

for N large enough, where C1 and C2 depend on R from (2.9). The estimates for ,M2 (t),−1 and ,M3 (t),−1 is a simpler task. In fact, here we use straightforward calculations. Indeed, we have that A ,M2 (t),−1 ≤ C · ,∂xi [f1 (w1 (t)) − f1 (w2 (t))] , i=1,2

≤ C·

A %

i=1,2

0

1

d λ,∂xi [f1$ (λw1 (t) + (1 − λ)w2 (t)) · w(t)] ,.

794


Therefore, since H 2 (Ω) ⊂ C(Ω), we obtain that A % 1 d λ {,f1$ (λw1 (t) + (1 − λ)w2 (t)) · ∂xi w(t), ,M2 (t),−1 ≤ C · i=1,2 0 +,f1$$ (λw1 (t)

≤

C·

A

i=1,2

+ (1 − λ)w2 (t)) · ∂xi (λw1 (t) + (1 − λ)w2 (t)) · w(t),}

(,wxi (t), + ,w1xi · w(t), + ,w2xi (t) · w(t),) .

From the Hölder inequality and from (2.5) we have that ,wkxi · w, ≤ C,wkxi ,s · ,w,1−s ≤ C,wk ,1+s · ,w,1 ≤ C · ,w,1 ,

where k = 1, 2 and 0 < s < 1. Thus from the previous formula we obtain that ,M2 (t),−1 ≤ C · ,w,1 .

(2.14)

,M3 (t),−1 ≤ C,f2 (w1 ) − f2 (w2 ), ≤ C · ,w(t),1 .

(2.15)

Using the embedding H 2 (Ω) ⊂ C(Ω) and relation (2.9) it is also easy to see that Relations (2.13)–(2.15) give us that

−s/4

,M (t),−1 ≤ C1 · log(1 + λN )· , w(t) ,1 +C2 · λN +1

(2.16)

for each 0 < s < 1 and with the constants C1 and C2 depending on R from (2.9). Let ψ(t) = ,w(t),1 . It follows from (2.8) and (2.16) that % t −s/4 ψ(t) ≤ C1 · log(1 + λN ) ψ(τ )dτ + C2 · T · λN +1 , t ∈ [0, T ], 0

for 0 < s < 1. Therefore using Gronwall’s lemma we conclude that −s/4

ψ(t) ≤ C2 · T · λN +1 · (1 + λN )C1 t ,

t ∈ [0, T ].

If we let N → ∞, then for 0 ≤ t < t0 ≡ s · (4C1 )−1 we obtain ψ ≡ 0. Thus w1 (t) ≡ w2 (t) for 0 ≤ t < t0 . Now we can reiterate the procedure in order to conclude that w1 (t) ≡ w2 (t) for all 0 ≤ t ≤ T , where T is the time of existence. 2.2. Boundary dissipation- simply supported case. As it was seen in Subsection 2.1 one of the ingredients of the proof is relation (2.8). In the case of the zero boundary condition (g = 0 in (SS)) to obtain (2.8) we can use the same procedure as in the clamped case. However in the case of the presence of boundary dissipation we need a more sophisticated argument relying on the consideration of dual (negative) norms with respect to the generator that corresponds to a boundary dissipative evolution. To this end we introduce the following notation: • G : L2 (Γ) → L2 (Ω) with h &→ Gh given by ∆2 Gh = 0 in Ω, Gh = 0 on Γ, ∆[Gh] = h on Γ.

(2.17)

It is known from standard elliptic theory [32] that G : L2 (Γ) &→ H 5/2 (Ω) is bounded and hence G : L2 (Γ) &→ D(A1/2 ) ≡ H 2 (Ω) ∩ H01 (Ω). This implies that G∗ : [D(A1/2 )]$ &→ L2 (Γ) and, since A : D(A1/2 ) &→ [D(A1/2 )]$ , we can define the bounded operator G∗ A from D(A1/2 ) into L2 (Γ). Moreover, the divergence theorem yields ∂ 11 u1 , u ∈ H 2 (Ω) ∩ H01 (Ω) ≡ D(A1/2 ). G∗ Au = (2.18) ∂n Γ


795

We recall that in this subsection the operator A is defined by Au = ∆2 u on the domain given by (2.1). Let w1 (t) and w2 (t) be weak solutions of the problem (1), (2) and (SS). Using (1.3) and (2.18) one can see that w(t) = w1 (t) − w2 (t) satisfies the equation d [(wt (t), φ) + 2G∗ Aw(t), G∗ Aφ3] + k(wt (t), φ) + (A1/2 w(t), A1/2 φ) = (M (t), φ). dt (2.19) in the sense of distributions for each φ ∈ (H 2 ∩ H01 )(Ω) = D(A1/2 ), where M (t) is given by (2.7) and 2·, ·3 is the inner product in L2 (Γ). Relation (2.19) allow us to consider the function W (t) = (w(t); wt (t)) as a generalized (in the sense of distributions) solution to a first order equation. To this end we introduce a linear operator A in the space H = D(A1/2 ) × L2 (Ω) given by the operator matrix 9 : 0 −I A≡ (2.20) A gAGG∗ A + kI

with the domain

C D D(A) = (u; v) ∈ D(A1/2 ) : u + g · GG∗ Av ∈ D(A) .

(2.21)

The operator A is maximally accretive on H, hence by Lumer–Phillips Theorem A generates a semigroup e−At of contractions on H. Maximal accretivity of operators like A is well known and has been used many times in the past [26] and in particular Lemma 3.1 in [24] where m-accretivity property is proved for a more general nonlinear model (see also [27, 17], where dissipative simply supported boundary conditions are explicitly considered). It is also easy to see that the adjoint operator A∗ has the form : 9 0 I A∗ ≡ −A gAGG∗ A + kI with the domain

C D D(A∗ ) = (u; v) ∈ D(A1/2 ) : −u + g · GG∗ Av ∈ D(A) .

∗

By [33, Corollary 10.6] the operator A∗ generates a contraction semigroup e−A t (∗ ' −At −A∗ t and e =e . Since both A and A∗ generate semigroups of contractions and 0 is in the resolvent set of A and A∗ , fractional powers of A and A∗ are well defined -see Theorem 3 Chapter IX [45] and also many other standard references [33, 3, 21]. Lemma 2.2. For any Φ = (φ0 ; φ1 ) ∈ D(A∗ ) we have that d (W (t), Φ)H + (W (t), A∗ Φ)H = (F (t), Φ)H dt

(2.22)

in the sense of distributions, where W (t) = (w(t); wt (t)) and F (t) = (0; M (t)). Proof. By (2.19) we have that d (W (t), Φ)H dt

& ) d −(A1/2 w(t), A1/2 φ0 ) + 2G∗ Aw(t), G∗ Aφ1 3 + dt

+k(wt (t), φ1 ) + (A1/2 w(t), A1/2 φ1 ) = (M (t), φ1 ).

(2.23)

796


Since 2G∗ Aw(t), G∗ Aφ3 = (A1/2 w(t), A1/2 GG∗ Aφ) and −φ0 + gGG∗ Aφ1 lies in D(A), we have that −(A1/2 w, A1/2 φ0 ) + 2G∗ Aw, G∗ Aφ1 3

= (A1/2 w, −A1/2 φ0 + A1/2 GG∗ Aφ1 ) = (w, A[−φ0 + GG∗ Aφ1 ])

and hence we obtain that < d ; −(A1/2 w(t), A1/2 φ0 ) + 2G∗ Aw(t), G∗ Aφ1 3 = (wt (t), A[−φ0 + GG∗ Aφ1 ]) dt Therefore (2.23) and the definition of the operator A∗ imply (2.22). We also need the following properties of the operator A. Lemma 2.3. We have 1/4

,A

1 9 :1 1 −1/2 f 1 1 1 f , ≤ C 1A h 1H

for any (f ; h) ∈ H and 1 9 :1 1 −1/2 0 11 1A ≤ C,A−1/4 f ,, 1 f 1H

f ∈ L2 (Ω).

(2.24)

Proof. Since A is a generator of contraction semigroup and A is invertible with a bounded inverse, we have (see Proposition 6.1 [3] and also [44]): D(Aθ ) = [D(A), H]1−θ , θ ∈ [0, 1],

(2.25)

where [·, ·]θ denotes the complex interpolation functor (see [40], for example). On the other hand . ∂ (2.26) D(A) = (f ; h) : f, h ∈ D(A1/2 ), ∆2 f ∈ L2 (Ω), ∆f = −g h on Γ ∂n and C D D(A) ⊂ (f ; h) : f, h ∈ D(A1/2 ), f ∈ H 3 (Ω) ⊂ H 3 (Ω) × D(A1/2 ). (2.27) Indeed, by (2.21) and (2.3) we have that (f ; h) ∈ D(A) if and only if f, h ∈ (H 2 ∩ H01 )(Ω) and ∆2 (f + gGG∗ Ah) ∈ L2 (Ω),

∆(f + gGG∗ Ah)|Γ = 0.

Since by the definition (2.17) of the Green map we have that and

∆2 (f + gGG∗ Ah) = ∆2 f

∂ h, ∂n we obtain the desired characterization (2.26) of D(A). As for the inclusion in (2.27), this follows from elliptic regularity and from (2.26). To see this, we note that by ∂ the trace theorem ∂n h ∈ H 1/2 (Γ) for any h ∈ H 2 (Ω). Therefore by (2.26) for any (f ; h) ∈ D(A) we have that ∆(f + gGG∗ Ah)|Γ = ∆f |Γ + gG∗ Ah = ∆f |Γ + g

∆2 f ∈ L2 (Ω) and f = 0, ∆f ∈ H 1/2 (Ω) on Γ.

Hence the elliptic regularity (see [32], for example) yields that f ∈ H 3 (Ω), as desired in (2.27). Interpolating (2.27) with H = D(A1/2 ) × L2 (Ω) and applying (2.25) gives D(A1/2 ) ⊂ [H 3 (Ω), D(A1/2 )]1/2 × D(A1/4 )


797

This, in particular, yields 1/4

,A

1 9 :1 1 u 11 v, ≤ C 11A1/2 , v 1H

(u; v) ∈ D(A1/2 ).

(2.28)

Direct computations of A−1 give 9 : 9 −1 : f A h + gGG∗ Af + kA−1 f A−1 = . h −f

Thus

1 9 :1 1 −1/2 f 1 1A 1 1 h 1H

1 9 :1 1 1/2 −1 f 1 1 1 = 1A A h 1H 1 :1 9 1 1/2 A−1 h + gGG∗ Af + kA−1 f 1 1 . 1 = 1A 1 −f H

(2.29)

Applying (2.28) with u = A−1 h+gGG∗ Af +kA−1 f and v = −f and evoking (2.29) yields the desired conclusion in the first part of the Lemma. As for the second part, the argument is simpler. From (2.29) we have 1 1 9 :1 9 −1 :1 1 −1/2 0 1 1 1 1A 1 = 1A1/2 A v 1 . (2.30) 1 1 1 1 v 0 H H

On the other hand, since D(A) × 0 ⊂ D(A) and D(A1/2 ) × 0 ⊂ H, by interpolation we have that D(A(1+θ)/2 ) × 0 ⊂ D(Aθ ) for any θ ∈ [0, 1]. Hence 1 9 :1 1 θ f 1 1A 1 ≤ C,A1/2+θ/2 f ,. 1 0 1H Applying the above inequality with θ = 1/2 yields 1 9 :1 1 1/2 A−1 v 1 1A 1 ≤ C,A3/4 A−1 v, = C,A−1/4 v,. 1 1 0 H

Therefore by (2.30) we obtain the inequality in (2.24). The proof of the Lemma is thus completed. By the same argument as in the clamped case from Lemma 2.1 we have that −s/4

,M (t),−1 ≤ C1 log (1 + λN ) ,w(t),1 + C2 λN +1 .

(2.31)

Therefore to prove the uniqueness it remains to prove relation (2.8) for the case considered. Since D(A1/4 ) = H01 (Ω) (see (2.3)), from (2.31) and Lemma 2.3 we have that |A−1/2 F (s)|H ≤ C,M (t),−1 ≤ C,

t ∈ [0, T ].

(2.32)

By Lemma 2.2 we have that Y (t) = A−1/2 W (t) ∈ L∞ (0, T ; D(A1/2 ) satisfies the relation d (Y (t), Φ)H + (Y (t), A∗ Φ)H = (G(t), Φ)H , Φ ∈ D(A∗ ), dt in the sense of distributions, where G(t) ≡ A−1/2 F (t) ∈ L∞ (0, T ; H). Therefore we can conclude that % t e−(t−τ )A A−1/2 F (τ )dτ in H. (2.33) A−1/2 W (t) = 0

798


Hence it follows from Lemma 2.3 and from (2.32) that % t ,M (s),−1 ds, ,A1/4 w(t), ≤ C1 |A−1/2 W (t)|H ≤ C2 0

which implies relation (2.8) for the case considered. Relying on (2.31) and (2.8) by the same argument as in the clamped case we can obtain that w ≡ 0 as desired.

2.3. Boundary dissipation- free boundary conditions. The main difference with respect to the previous case is functional setup for the problem. The following spaces and operators are now relevant. • Au = ∆2 u with the domain D(A) given by (2.2). The corresponding bilinear form is % a ˜(u, v) = a(u, v) + (1 − µ)l uvdΓ, u, v ∈ H 2 (Ω), Γ

where a(u, v) is given by (1.5). • G1 : L2 (Γ) → L2 (Ω) with h &→ G1 h given by

∆2 G1 h = 0 in Ω,

∆G1 h + (1 − µ)B1 G1 h = h on Γ,

∂ ˜2 G1 h = 0 on Γ ∆G1 h + (1 − µ)B ∂n • G2 : L2 (Γ) → L2 (Ω) with h &→ G2 h given by

∆2 G2 h = 0 in Ω,

∆G2 h + (1 − µ)B1 G2 h = 0 on Γ,

∂ ˜2 G2 h = h on Γ ∆G2 h + (1 − µ)B ∂n It is known from elliptic theory and characterization of fractional powers corresponding to elliptic operators that [28, 14] G1 : L2 (Γ) G2 : L2 (Γ)

&→ H 5/2 (Ω), G1 : H 1/2 (Γ) &→ H 3 (Ω); (2.34) C D 7 &→ u ∈ H 7/2 (Ω) : ∆u + (1 − µ)B1 u = 0 on Γ ⊂ D(A 8 −% )

are bounded, where + can be taken arbitrarily small. Moreover, the divergence theorem implies that ∂ G∗1 Au = u and G∗2 Au = −u on Γ for all u ∈ D(A1/2 ) = H 2 (Ω). ∂n Let w1 (t) and w2 (t) be weak solutions of the problem (1), (2) and (F). We denote w(t) = w1 (t) − w2 (t). According to the definition of weak solution (1.4) we infer that w(t) satisfies the following variational form d [(wt (t), φ) + g1 2G∗1 Aw, G∗1 Aφ3 + g2 2G∗2 Aw, G∗2 Aφ3] + k(wt (t), φ) dt ˜ (t), φ) +(A1/2 w(t), A1/2 φ) = (M & ) ˜ (t) is an element in H 2 (Ω) $ given by with any φ ∈ H 2 (Ω) and where M ˜ (t), φ) (M

=

(2.35)

−(f0 (∇w1 ) − f0 (∇w2 ), ∇φ) − (∇[f1 (w1 ) − f1 (w2 )], ∇φ) % −(f2 (w1 ) − f2 (w2 ), φ) + (1 − µ)l wφdΓ (2.36) Γ


799

for any φ ∈ H 2 (Ω). Following the same idea as in the simply supported case we introduce the linear operator A in H = D(A1/2 ) × L2 (Ω) by the formula : 9 0 −I (2.37) A≡ A g1 AG1 G∗1 A + g2 AG2 G∗2 A + kI on the domain

C D D(A) = (u; v) ∈ D(A1/2 ) : u + g1 G1 G∗1 Av + g2 G2 G∗2 Av ∈ D(A) .

It is well known [26], in particular Lemma 3.1 in [24], that A generates a semigroup e−At of contractions on H. The corresponding analogs of Lemma 2.2 and relation (2.33) makes it possible to prove that % t |A−1/2 W (t)|H ≤ |A−1/2 F (s)|H ds, W (t) = (w(t); wt (t)), 0

˜ (t)). Similar to the simply supported case we claim that where F (t) = (0; M % t 1/4 ˜ (s),ds. (2.38) ,A−1/4 M ,A w(t), ≤ C 0

To prove it we use the following lemma.

Lemma 2.4. We have the relations 1 9 :1 1 f 11 ,A1/4 f , ≤ C 11A−1/2 , g 1H and

1 9 :1 1 −1/2 0 11 1A ≤ C,A−1/4 f ,, 1 f 1H

9

f g

:

∈ H,

(2.39)

f ∈ L2 (Ω).

Proof. Since A−1 is bounded, and A is m-accretive, by Proposition 6.1 in [3, p.113] we have that D(Aθ ) = [D(A), H]1−θ , 0 ≤ θ ≤ 1, (2.40) where, as in the proof of Lemma 2.3, [·, ·]θ denotes the complex interpolation functor. By the definition of A given in (2.37) and arguing as in the simply supported case we have   1 1 f, h ∈ D(A1/2 ), ∆2 f ∈ L2 (Ω)   1 ∂ h on Γ D(A) = (f ; h) 11 ∆f + (1 − µ)B1 f = −g1 ∂n .  1 ∂ ∆f + (1 − µ)B ˜2 f = g2 h on Γ  ∂n Moreover D C (2.41) D(A) ⊂ (u; v) : u, v ∈ D(A1/2 ), u ∈ H 3 (Ω) = H 3 (Ω) × D(A1/2 ). Indeed, since h ∈ D(A1/2 ) = H 2 (Ω) and by the trace theorem 1 ∂ 11 h1 ∈ H 1/2 (Γ), h|Γ ∈ H 3/2 (Γ), ∂n Γ

the last assertion follows from elliptic regularity (2.34) applied to ∆2 f ∈ ∂ ∆f + (1 − µ)B1 f = −g1 h ∈ ∂n ∂ ˜ 2 f = g2 h ∈ ∆f + (1 − µ)B ∂n ∂ h + g2 G2 h + A−1 ∆2 f which then gives f = −g1 G1 ∂n

L2 (Ω),

H 1/2 (Γ), H 3/2 (Γ), ∈ H 3 (Ω). Thus (2.41) holds.

800


Interpolating (2.41) with H = D(A1/2 ) × L2 (Ω) and applying (2.40) gives D(A1/2 ) ⊂ [H 3 (Ω), D(A1/2 )]1/2 × D(A1/4 ).

This, in particular, yields

1/4

,A

1 9 :1 1 1/2 u 1 1 1 . v, ≤ C 1A v 1H

(2.42)

Direct computations of A−1 give : 9 : 9 −1 f A h + g1 G1 G∗1 Af + g2 G2 G∗2 Af + kA−1 f −1 A = −f h

and 1 1 9 :1 9 −1 :1 ∗ ∗ −1 1 −1/2 f 1 1 1 1A 1 = 1A1/2 A h + g1 G1 G1 Af + g2 G2 G2 Af + kA f 1 . 1 1 h 1H 1 −f H

Therefore applying (2.42) with u = A−1 h + g1 G1 G∗1 Af + g2 G2 G∗2 Af + kA−1 f and v = −f yields (2.39). The second part of the Lemma is proved in a manner identical to the simply supported case. Since by (2.3) D(A1/4 ) = H 1 (Ω), using (2.38) we obtain that % t ˜ (τ ), dτ ,A−1/4 M ,w(t),1 ≤ C

(2.43)

0

and by (2.36) we have that ˜ (t), ,A−1/4 M

≤

C1 (,f0 (∇w1 ) − f0 (∇w2 ), + ,∇[f1 (w1 ) − f1 (w2 )],) +C2 (,f2 (w1 ) − f2 (w2 ), + ,w,1 ) .

Therefore by the same argument as above from Lemma 2.1 we have that ˜ (t), ≤ C1 log (1 + λN ) ,w(t),1 + C2 λ ,A−1/4 M N +1 . −s/4

and hence using (2.43) we can conclude that w(t) ≡ 0.

3. Weakly compact global attractor. The main step in proving existence of weakly compact global attractor is showing that the flow is ultimately dissipative. Our main result requires additional structure from the term f2 . In fact, we shall assume that f2 is dissipative in the following sense: Assumption 3.1. Assume that ν(µ) ≡ lim inf |s|→∞

f2 (s) ≥0 s|s|µ−1

(3.1)

for µ = 3 in the clamped and simply supported case and ν(η) > 0 for some η > 1 in the free case. The dissipativity condition assumed on f2 implies (see Remark 1.2) that problem (1)–(5) generates a dynamical system (see Corollary 1.10). Theorem 3.2. Let f2 ∈ C 1 (R). Under Assumption 3.1 the dynamical system (St ; Hw ) generated by equations (1), (2) and (5), under any set of the boundary conditions considered possesses a (weak) global attractor.


801

Remark 3.3. We conjecture that for the models with boundary dissipation one should be able to replace the condition that k is large by the condition that the parameters g and gi are sufficiently large. However, the analysis of ultimate dissipativity is more involved in that case. Techniques developed in [8] should be helpful in deriving this type of the result. The proof of Theorem 3.2 is given below. 3.1. Existence of a bounded absorbing set. Lemma 3.4. Under Assumption 3.1 the semi-flow St given by (1.41) is ultimately dissipative provided that k > 0 is large enough, i.e. there is k∗ > 0 such that for any k ≥ k∗ there exists R > 0 such that for any bounded set B in H we have that |St W |H ≤ R for all t ≥ tB .

Proof. We first assume that g = g1 = g2 = 0 and consider the energy E defined by (1.27). It is clear that : % 9 ( 1 1' $ 2 $ 4 2 ,w , + 4 |∇w| − |w||∇w| + F2 (w) dx a(w, w) + E(w, w ) ≥ 2 4 Ω : 0 % 9/ ( 1 1' $ 2 1 α 2 4 ≥ ,w , + 4 − a(w, w) + |∇w| − |w| + F2 (w) dx, 2 4 2α 2 Ω

where 4 a(w, w) is either ,∆w,2 or a(w, w) given by (1.5) depending on boundary conditions. We claim that ( ' (3.2) E(w, w$ ) ≥ β E0 (w$ ; w) + ,w,2 − C

for some positive constants β and C. In the clamped and simply supported case the argument is as in the proof of global existence and relies on relation (1.15). In the free case we note that the property ν(η) > 0 implies that F2 (s) ≥ ν0 |s|1+η − C,

η > 1,

for some ν0 > 0. Thus we can choose appropriate α > 0 to obtain (3.2). By the Sobolev embedding H 2 (Ω) ⊂ C(Ω) we also obtain: |E(w, w$ )| ≤ C(|W0 |H ).

(3.3)

In addition, the following energy identity holds for regular solutions. Since we are dealing with weak solutions, we shall consider Faedo–Galerkin approximations for which the energy equality holds true. The passage with the limit on finite quantities will reconstruct the estimates for all weak solutions. Thus, in what follows we consider Faedo–Galerkin approximations defined by wn (t) given in (1.18), or (1.30), or (1.34) depending on the choice of boundary conditions. For simplicity of notation we are suppressing index n. Thus by (1.20), or (1.31), or (1.35) we have that % % d E(w(t), wt (t)) = −k |wt (t, x)|2 dx + |∇w|2 wt dx. dt Ω Ω We note that the energy is not necessarily decreasing. In order to prove ultimate dissipativity we introduce the Lyapunov function: % V (t) = E(t) + + w(t)wt (t)dx, Ω

where we denote E(t) = E(w(t), wt (t)). The parameter + > 0 will be chosen later. We have & ) |V (t) − E(t)| ≤ + E0 (t) + ,w(t),2 ,

802


where E0 (t) = E0 (w(t); wt (t)) is given by (1.11) (or (1.12)). Hence by (3.2) there exists +0 > 0 such that E(t) ≤ C1 V (t) + M1 ,

V (t) ≤ C2 E(t) + M2

for all 0 < + < +0 . As a consequence, V (t) is bounded from below: & ) E(t) − M1 ≥ c0 E0 (t) + ,w(t),2 − c1 , c0 , c1 > 0. V (t) ≥ C1 We also have Vt

(3.4)

(3.5)

= Et + +,wt ,2 + +(w, −kwt − ∆2 w

+div(|∇w|2 ∇w) + ∆(w2 ) − f2 (w))

a(w, w) = Et + +,wt ,2 − +k(w, wt ) − +4 % % % 4 2 −+ |∇w| dx − 2+ w|∇w| dx − + wf2 (w)dx Ω : 9 Ω %Ω 1 1 ≤ − k + + ,wt ,2 + |∇w|2 wt dx + +2 k,w,2 2 2 Ω / 0 % ' ( 4 2 |∇w| + wf2 (w) + 2|∇w| w dx . −+ 4 a(w, w) + Ω

Estimating we obtain Vt

%

Ω

|∇w|2 wt dx ≤

k 1 ,wt ,2 + 4 k

%

Ω

|∇w|4 dx,

: 9 1 ≤ Ψ(t) ≡ − k + + ,wt ,2 − +4 a(w, w) (3.6) 4 9 : : : % /9 1 k+ 1 −+ + 1−δ− |∇w|4 − |w|2 + wf2 (w) dx. k+ δ 2 Ω

Now we split the cases. Case 1 (clamped and simply supported b.c.): If k > max{2, +−1 0 }, then we can choose + ∈ (0, +0 ) and δ > 0 such that 1 1 − k + + < 0, 1 − δ − > 0. (3.7) 4 k+ Using (3.1) we easily obtain that sf2 (s) ≥ −νs4 − Cν for every ν > 0. Therefore from (1.29) and (3.6) we obtain that Vt (t) + c0 E0 (t) ≤ c1

(3.8)

for some positive constants c0 and c1 . The above inequality along with a lower bound for V (t) (3.5) and inequalities in (3.3) and (3.4), which are valid for + ∈ (0, +0 ), yields the desired ultimate dissipativity in the clamped and simply supported cases with g = 0. In the simply supported case with g > 0 by the same calculations as above we have that Vt ≤ Ψ(t) − g ,∂n wt ,2L2 (Γ) − +g (∂n wt , ∂n w)L2 (Γ) , where Ψ(t) is given in (3.6) and ∂n denotes the normal derivative. It is clear that 1 1 < g; 1 1 2 2 ,∂n wt ,L2 (Γ) + +2 ,∂n w,L2 (Γ) +g 1(∂n wt , ∂n w)L2 (Γ) 1 ≤ 2 & ) g 2 ≤ ,∂n wt ,L2 (Γ) + C+2 ,w,2 + ,∆w,2 . 2


803

Therefore it is easy to see that under condition (3.7) we obtain (3.8) for the case g > 0. Case 2 (free b.c.): From Assumption 3.1 we have that wf2 (w) > ν1 |w|1+η − c,

η > 1,

for some ν1 > 0. Therefore one can see that & ) Vt (t) + c0 ,w(t),2 + E0 (t) ≤ c1

for some positive constants c0 and c1 provided relations (3.7) hold. Thus in the same way we can obtain dissipativity for the free boundary conditions with g1 = g2 = 0. In the case of positive gi we obviously have that Vt

≤ Ψ(t) − g1 ,∂n wt ,2L2 (Γ) − g2 ,wt ,2L2 (Γ) 5 6 −+ g1 (∂n wt , ∂n w)L2 (Γ) + g2 (wt , w)L2 (Γ) ,

where Ψ(t) is given in (3.6). Therefore the argument similar to the simply supported case leads to the conclusion desired. 3.2. Existence of an attractor. In order to complete the proof of the existence of a weak global attractor, hence of Theorem 3.2, it suffices to appeal to general results in dynamical systems. Indeed, the flow is weakly continuous with absorbing set that is weakly compact. Thus [1, Theorem 1, Chap.2, Sect.2] (see also Theorem 1.5.3 in [7]) provides the final result. 4. Appendix. In this Appendix we establish the relation between the KirchhoffBoussinesq equation (1) and the Mindlin-Timoshenko models (3) and (4). Our aim is to show that the Kirchhoff-Boussinesq model is a limit, when κ → ∞, of the Mindlin-Timoshenko models. Asymptotic analysis of the long time behavior, in particular upper semi-continuity of a family of attractors corresponding to the two models, is given in [10]. (See [12] for related asymptotic analysis of inertial sets associated with a damped wave equation). For the sake of some simplification of exposition we consider problem (3) and (4) with h0 (v) = v|v|2 ,

h1 (w) = 2w,

h2 (w) = w3 ,

(4.1)

and restrict ourselves to the Dirichlet boundary conditions v1 (x, t) = v2 (x, t) = 0, w(x, t) = 0 on Γ × R+ .

(4.2)

Unlike the wellposedness, the limiting argument does not depend on the type of boundary conditions considered. It is also immediate to see that the nonlinear functions in (4.1) correspond to nonlinear vector functions f0 and f1 given in (5). Our goal here is to show that dynamics of problem (3), (4), (4.1), (4.2) is close to (1), (C) in the limit κ → ∞ and α → 0 when f2 (w) = w3 and (5) holds. We consider the abstract Cauchy problem associated with (3), (4), (4.1) and (4.2). To this end the following functional framework is introduced. Let H ≡ L2 (Ω)×L2 (Ω)×L2 (Ω). On H we define the following abstract second order equation: Mα u tt (t) + Aκ u (t) + Dα u t (t) = F (u(t)), (4.3) 1/2 u |t=0 = u 0 ∈ D(Aκ ), u t |t=0 = u 1 ∈ H. Here u(t) = (v1 (t), v2 (t), w(t))T , and / −A + κI Aκ = −κdiv

κ∇ −κ∆

0

804


with the domain 7 8 D(Aκ ) = u ≡ (v1 , v2 , w) ∈ (H 2 ∩ H01 )(Ω) × (H 2 ∩ H01 )(Ω) × (H 2 ∩ H01 )(Ω) . It is clear that Aκ is a positive self-adjoint operator and

1 1 1 D(A1/2 κ ) = H0 (Ω) × H0 (Ω) × H0 (Ω).

We define the inertia operator Mα and the damping operator Dα by the formulas / 0 / 0 αI 0 αkI 0 Mα = , Dα = . 0 1 0 k0

Here I is the unit operator in L2 (Ω) × L2 (Ω). It is clear that Mα and Dα are bounded operators in H. The nonlinear term F is given by (  '    −v1 · '2w + |v1 |2 + |v2 |2 ( v1 (4.4) F (u ) =  −v2 · 2w + |v1 |2 + |v2 |2  , u =  v2  . w −w3 We have the following result concerning well-posedness for the problem (4.3).

Proposition 4.1. For any time interval [0, T ] problem (4.3) possesses a unique weak solution u(t) ≡ (v1 (t), v2 (t), w(t)) from the class 1 C([0, T ]; D(A1/2 κ )) ∩ C ([0, T ]; H)

provided that initial data (u0 , u1 ) ≡ (v10 , v20 , w0 , v11 , v21 , w1 ) belong to the space 1/2 D(Aκ ) × H. For this solution u(t) = (v1 (t), v2 (t), w(t)) the energy relation % t % t% E(u(t), ut (t)) + (Dα ut (τ ), ut (τ ))dτ = E(u0 , u1 ) + |v(x, τ )|2 wt (x, τ )dxdτ 0

0

holds, where E(u, ut ) = E0 (u, ut ) + with E0 (u, ut ) =

%

Ω

Ω

|v(x, t)|2 · w(x, t)dx

< 1 1; 2 + (Mα ut , ut ) + ,A1/2 u, κ 2 4

%

Ω

' ( |v(x, t)|4 + |w(x, t)|4 dx.

(4.5) (4.6)

(4.7)

Here and below we denote by , · , and (·, ·) the L2 -based norm and scalar products in the corresponding spaces. We note that the energy E(u, ut ) of the system is not necessarily positive. Moreover, the energy identity (4.5) indicates that the energy is not decreasing along trajectories. Proof. Proof of the wellposedness of solutions is rather routine. It relies on two fundamental facts: (i) the nonlinear term is locally Lipschitz with respect to the finite energy space, (ii) there is a natural a-priori bound for weak solutions. In order to exhibit these properties, appropriate functional analytic framework should be put in place. A simple calculation shows that the quadratic part E00 (u , u t ) of the energy E(u (t), u t (t)) has the form < 1; 2 (Mα u t , u t ) + ,A1/2 E00 (u , u t ) ≡ κ u, 2 ) 1& = α,vt ,2 + ,wt ,2 + a(v, v) + κ,v + ∇w,2 , (4.8) 2


805

where u ≡ (v1 , v2 , w) and the bilinear form a(v, vˆ) on V = H01 (Ω) × H01 (Ω) is given by % [(1 − µ) (∂x1 v1 · ∂x1 vˆ1 + ∂x2 v2 · ∂x2 vˆ2 ) + µdiv v · div vˆ] dx a(v, vˆ) = Ω % 1−µ + (∂x2 v1 + ∂x1 v2 ) (∂x2 vˆ1 + ∂x1 vˆ2 ) dx (4.9) 2 Ω

for any v = (v1 , v2 ) and vˆ = (ˆ v1 , vˆ2 ) from V . It is easy to see that the form a(v, vˆ) is elliptic on V , i.e. there exists γ0 > 0 such that ' ( a(v, v) ≥ γ0 ,v,2 + ,∇v,2 , v ∈ V = H01 (Ω) × H01 (Ω). This property implies that ' ( γ1 α,vt ,2 + ,wt ,2 + ,v,2H 1 + ,w,2H 1 ' ( ≤ E00 (u , u t ) ≤ γ2 α,vt ,2 + ,wt ,2 + ,v,2H 1 + ,w,2H 1 1/2

for any (u, u t ) ∈ D(Aκ ) × H, where γ1 and γ2 are positive constants depending on κ. We also note that the force F (u) can be written in the form F (u ) = −Π$ (u ) + F ∗ (u),

Π(u ) = Π0 (u) + Π1 (u).

(4.10)

Here, for u = (v1 , v2 , w)T ≡ (v, w)T , we denote the positive part of the energy % & ) 1 Π0 (u) = |v(x)|4 + |w(x)|4 dx (4.11) 4 Ω and

Π1 (u ) =

%

Ω

|v(x)|2 · w(x)dx,

The non-conservative part of the force F ∗ (u) is given by (T ' F ∗ (u ) = 0, 0, |v(x)|2 .

(4.12)

One can see that % & 2 ) 1 Π0 (u) − C1 ≤ Π1 (u ) ≤ C2 |w| + |v(x)|4 dx + C3 , 2 Ω

which implies that

c1 E0 (u , u t ) − c2 ≤ E(u , u t ) ≤ c3 E0 (u , u t ) + c4 ,

(4.13)

where ci are positive constants. We also have that % % & ) (F ∗ (u), u t ) = |wt |2 + |v(x)|4 dx + C2 . |v|2 · wt dx ≤ C1 Ω

Ω

EN Consequently, by considering Galerkin approximation u N (t) = i=1 gi (t)ei , where {ei } are eigenvectors of the operator Aκ , one can easily obtain a priori estimate of the form 2 1/2 N 2 ,u N t ∈ [0, T ]. (4.14) t (t), + ,Aκ u (t), ≤ CT ,

It is also easy to see that by the standard nonlinear analysis argument along with the Sobolev’s embedding (2.5) we obtain ( ' ,F (u) − F (u ∗ ), ≤ C(ρ) ,v − v ∗ ,H 1−δ (Ω) + ,w − w∗ ,H 1−δ (Ω) , δ > 0,

806

IGOR CHUESHOV AND IRENA LASIECKA T

1/2

T

for any u = (v1 , v2 , w) and u ∗ = (v1∗ , v2∗ , w∗ ) from the domain D(Aκ ) such that 1/2 1/2 ,Aκ u,, ,Aκ u ∗ , ≤ ρ. Therefore estimate (4.14) makes it possible to prove the existence of weak solutions. Further arguments are standard (see, e.g., [13, 9]). Remark 4.2. One can prove that the dynamical system generated by equation (3) and (4) subjected to conditions (4.1) and (4.2) possesses a compact global attractor (in the strong topology of the phase space). To obtain this result one can use either the methods developed in [9] or the approach suggested by J. Ball [2]. The following assertion justifies the relation between problems (3), (4) and (1) mentioned above. Proposition 4.3. Assume that the initial data (u0 , u1 ) ≡ (v10 , v20 , w0 , v11 , v21 , w1 )

to problem (4.3) satisfy the relations and

v10 + ∂x1 w0 = 0, v20 + ∂x2 w0 = 0, w0 ∈ H02 (Ω)

(4.15)

v11 ∈ L2 (Ω), v22 ∈ L2 (Ω), w1 ∈ H01 (Ω).

(4.16)

(v1α,κ (t), v2α,κ (t), wα,κ (t))

denote a weak solution to (4.3) with initial Let u (t) ≡ data (u0 , u1 ). Then for any interval [0, T ] we have that, as α + κ−1 → 0, 5 & )3 6 • uα,κ (t) → (−∂x1 u(t), −∂x2 u(t), u(t)) ∗ -weakly in L∞ 0, T ; H01 (Ω) ; α,κ

• wtα,κ (t) → ∂t u(t) ∗ -weakly in L∞ (0, T ; L2(Ω)), where u = u(x, t) is a weak solution to problem (1)and (C) with the initial data u(x, 0) = w0 (x) and ∂t u(x, 0) = w1 (x) in the case when f0 (w) = w3 and (5) holds. Proof. We adapt the method used in [22] in the case of the fixed α > 0. It follows from energy relation (4.5) that α,vtα,κ (t),2 + ,wtα,κ (t),2 + a(v α,κ (t), v α,κ (t)) + κ,v α,κ (t) + ∇wα,κ (t),2 ≤ C1 E0 (u 0 , u 1 )eC2 t + C3

for all t ≥ 0, where the constants Ci does not depend on κ ≥ 1 and on α ≤ 1. By (4.15) the positive part of the energy E0 (u 0 , u 1 ) is also independent of κ. Therefore sup t∈[0,T ]

and

7 8 α,vtα,κ (t),2 + ,wtα,κ (t),2 + ,v α,κ (t),2H 1 + ,wα,κ (t),2H 1 ≤ CT CT sup ,v α,κ (t) + ∇wα,κ (t), ≤ √ κ t∈[0,T ]

(4.17)

(4.18)

for any interval [0, T ] and for all κ ≥ 1 and α ≤ 1. Therefore there exists sequences 4 (t) = (4 {κn } and {αn } such that αn + κ−1 v1 (t), v42 (t), w(t)) 4 n → 0 and a function u with properties 5 & )3 6 4 (t) ∈ L∞ 0, T ; H01 (Ω) , w 4t (t) ∈ L∞ (0, T ; L2(Ω)) , u (4.19) 5 6 & )3 4 (t) ∗ -weakly in L∞ 0, T ; H01 (Ω) u αn ,κn (t) → u , (4.20) wtαn ,κn (t) → w 4t (t) ∗ -weakly in L∞ (0, T ; L2(Ω)) .

(4.21)


807

Moreover, by Aubin’s compactness theorem (see, [38, Corollary 4]) we have that 4 sup ,wαn ,κn (t) − w(t), H 1−ε (Ω) → 0 as n → ∞

(4.22)

0

t∈[0,T ]

for every ε > 0, and by (4.18) we can conclude that there exists a function u(t) such that ' ' ( ( u(t) ∈ L∞ 0, T ; H02 (Ω) , ut (t) ∈ L∞ 0, T ; H01 (Ω) and 4 (t) = (−∂x1 u(t), −∂x2 u(t), u(t)). u (4.23) By (4.18) and (4.22) we have that sup ,v αn ,κn (t) + ∇u(t),[H −ε (Ω)]2 → 0,

(4.24)

n → ∞,

t∈[0,T ]

for every ε > 0. By interpolation, using estimate (4.17) we obtain that ,v αn ,κn (t) + ∇u(t),[H 1−ε ]2 0

αn ,κn

(t) + ∇u(t),

1/(1+ε)

≤

C,v

≤

C,v αn ,κn (t) + ∇u(t),[H −ε ]2

[H01 ]2

ε/(1+ε)

,v αn ,κn (t) + ∇u(t),[H −ε ]2

ε/(1+ε)

for ε > 0 small enough. Therefore (4.24) implies sup ,v αn ,κn (t) + ∇u(t),[H 1−ε (Ω)]2 → 0,

(4.25)

n → ∞,

t∈[0,T ]

for every ε > 0 small enough. To conclude the proof of we only need to show that the function u(t) is a weak (variational) solution to (1) with the initial data (w0 , w1 ). Any weak solution u α,κ (t) to (4.3) satisfies the variational relation: % T % T α,κ − (Mα u α,κ (τ ) + D u (τ ), ψ (τ ))dτ + (Aκ u α,κ (t), ψ(τ ))dτ α t t 0

= (Mα u 1 + Dα u 0 , ψ(0)) +

%

0

T

(F (u α,κ (t)), ψ(τ ))dτ

(4.26)

0

for any function ψ(t) such that 5 & )3 6 , ψ(t) ∈ L∞ 0, T ; H01 (Ω)

5 6 ψt (t) ∈ L∞ 0, T ; [L2 (Ω)]3 ,

and ψ(T ) = 0. We choose ψ(t) = (−∂x1 φ, −∂x2 φ, φ) · f (t) ≡ ψ · f (t), where φ ∈ H02 (Ω) and f (t) is a scalar C 1 -function such that f (T ) = 0. By (4.22), (4.25) and (4.23) we have that % T lim (F (u αn ,κn (t)), ψ(τ ))dτ (4.27) n→∞

0

= −

%

T

f (τ )

0

%

Ω

' ( |∇u|2 ∇u + 2u ∇u∇φdxdτ −

%

T

f (τ )

0

%

√ By (4.21) and (4.23), since αn · ,vtαn ,κn (t), ≤ C, we obtain that % T n ,κn (Mαn u α (τ ) + Dαn u αn ,κn (τ ), ψt (τ ))dτ lim t n→∞

0

=

%

0

T

ft (τ )

%

(ut + k0 u) φdxdτ. Ω

u3 φdxdτ. Ω

(4.28)

808


At last, by the choice of the test function ψ, relation (4.9) implies that % div v αn ,κn · ∆φdx (Aκn u αn ,κn (t), ψ) = −a(v αn ,κn , ∇φ) = −µ Ω % & ∂x1 v1αn ,κn · ∂x21 φ + ∂x2 v2αn ,κn · ∂x22 φ − (1 − µ) Ω

+ (∂x2 v1αn ,κn + ∂x1 v2αn ,κn ) ∂x1 x2 φdx.

Thus from (4.20) and (4.23) we obtain that % % T lim (Aκn u αn ,κn (t), ψ(τ ))dτ = n→∞

0

0

T

f (τ )

%

∆u∆φdxdτ

(4.29)

Ω

Thus it follows from (4.26)–(4.29) that the function u(x, t) in representation (4.23) is a weak solution to problem (1) and (C) with initial data (w0 , w1 ). By the uniqueness theorem for weak solutions to (1) the function u(x, t) in (4.23) does not depend on sequences {αn } and {κn }. The proof is completed. REFERENCES [1] A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. [2] J. Ball, Global attractors for semilinear wave equations, Discr. Cont. Dyn. Sys. 10 (2004), 31–52. [3] A. Bensoussan and G. Daprato and M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, 1, Birkhauser, 1992. [4] H. Brezis and S. Waigner, A note on the limiting cases of Sobolev embedding and convolution inequalities, Comm. PDE 5 (1980), 773–789. [5] M. Berger and P. Fife, Von Karman equations and the buckling of a thin elastic plate, II. Plate with general edge conditions, Commun. Pure Appl. Math., 21 (1968), 227–241. [6] A. Boutet de Monvel and I. Chueshov, Uniqueness theorem for weak solutions of von Karman evolution equations, J. Math. Anal. Appl., 221 (1998), 419–429. [7] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/ [8] I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with nonlinear boundary dissipation, J. Diff. Equations, 198 (2004), 196–231. [9] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Scuola Normale Superiore (SNS) Preprint no.14, (2004); http://math.sns.it /papers/chulas04/ [10] I. Chueshov and I. Lasiecka, Global attractors for Mindlin–Timoshenko plates and for their Kirchhoff limits, in preparation. [11] I. Chueshov and A. Shcherbina, On 2D Zakharov system in a bounded domain, Diff. Int. Equations, 18 (2005), 781?-812. [12] P. Fabrie and C. Galusinskiand and A. Miranville. Uniform inertial sets for damped wave equations, Discr. Cont. Dyn. Sys., 6 (2000),393-419. [13] J.M. Ghidaglia and R. Temam, Regularity of the solutions of second order evolution equations and their attractors, Ann. della Scuola Norm. Sup. Pisa, 14 (1987), 485–511. [14] P. Grisvard, Characterization de quelques espaces d’interpolation, Arch. Rat. Mech. Anal., 25 (1967), 40–63. [15] J.K. Hale Asymptotic Behavior of Dissipative Systems, AMS, Providence, 1989 [16] A. Haraux Nonlinear Evolution Equations-Global Behaviour of Solutions, Springer-Verlag, New York, 1981. [17] M.A. Horn and I. Lasiecka, Asymptotic behavior with respect to thickness of boundary stabilizing feedback for the Kirchoff plate, J. Diff. Equations, 114 (1994), 396–433. [18] T.Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discr. Cont. Dyn. Sys., 12 (2005), 1-13


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Received May 2005; revised December 2005. E-mail address: [email protected] E-mail address: [email protected]