Decay of energy for second-order boundary hemivariational inequalities with coercive damping

Nonlinear Analysis 74 (2011) 1164–1181

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Decay of energy for second-order boundary hemivariational inequalities with coercive damping✩ Piotr Kalita ∗ Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Computer Science, ul. prof. S. Łojasiewicza 6, 30-348 Kraków, Poland

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Article history: Received 7 December 2009 Accepted 24 September 2010 MSC: 35L85 35B40 Keywords: Hemivariational inequality Asymptotic behavior Exponential decay of energy Coercive damping

abstract In this article we consider the asymptotic behavior of solutions to second-order evolution inclusions with the boundary multivalued term u′′ (t ) + A(t , u′ (t )) + Bu(t ) + γ¯ ∗ ∂ J (t , γ¯ u′ (t )) ∋ 0 and u′′ (t ) + A(t , u′ (t )) + Bu(t ) + γ¯ ∗ ∂ J (t , γ¯ u(t )) ∋ 0, where A is a (possibly) nonlinear coercive and pseudomonotone operator, B is linear, continuous, symmetric and coercive, γ¯ is the trace operator and J is a locally Lipschitz integral functional with ∂ denoting the Clarke generalized gradient taken with respect to the second variable. For both cases we provide conditions under which the appropriately defined energy decays exponentially to zero as time tends to infinity. We discuss assumptions and provide examples of multivalued laws that satisfy them. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Differential inclusions with the nonmonotone multivalued term given in the form of the Clarke subdifferential are known as hemivariational inequalities (HVIs). They are applied mainly to describe contact problems in the theory of (visco)elasticity where the multivalued and nonmonotone terms are used to represent various laws of normal contact, friction, adhesion, damage, or wear. It is well known that under appropriate conditions the solutions of the quasilinear hyperbolic PDE of second-order u′′ + Au′ + Bu = 0 and their time derivatives tend to zero in an appropriate sense as the time tends to infinity. This behavior is called the energy decay. In the case when the operator A is coercive (the prototype of such an operator is Au′ = −1u′ ) we speak about the structural damping, and energy decay results are relatively easy to obtain. The behavior of the model in such a case is parabolic. More interest in the last 40 years was devoted to the case of pure viscous damping (the prototype of a viscous damping operator is Au′ = u′ ) where the decay results are harder to obtain. It is well known that the type of decay (e.g. exponential, polynomial, logarithmic) and its rates are associated with the types of growth condition on the damping term at the origin and at infinity. In particular, the simplest case of linear growth both at the origin and at infinity leads to the exponential decay of energy. One method for proving such decay was developed by Nakao whose results concern the decay rates for the single-valued problems (see for instance [1,2]). The problem with multivalued and maximal monotone viscous damping defined in the problem domain was studied by Haraux [3]. Later, Baji et al. [4] analyzed the problem governed by the inclusion u′′ + ∂ Φ (u′ ) − 1u ∋ 0 with a continuous, convex function Φ such that int∂ Φ (0) ̸= ∅ and established conditions under which the velocity decays to zero in finite time.

✩ Supported in part by the Ministry of Science and Higher Education of Poland under grant No. N201 027 32/1449.

∗

Tel.: +48 12 664 7538; fax: +48 12 664 6673. E-mail address: [email protected].

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.09.053

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In the case of multivalued viscous damping defined on the boundary and given in the form of subdifferential of a convex and lower semicontinuous function, the asymptotic behavior of the energy was investigated by Lasiecka [5] (see also [6] for the existence, uniqueness and regularity analysis of the extended version of the problem). There, the result of Dafermos and Slemrod [7] was used to prove the existence of limits of trajectories, and then the method of multipliers was used to prove that under some additional conditions on the domain geometry this limit must be equal to zero. The results presented therein rely on the theory of maximal monotone operators. For the case of single-valued and monotone boundary damping the decay rates were studied by, among others, Martinez [8], who developed a variant of the multipliers method to derive the rates of decay of solutions depending on various growth conditions that the damping term satisfies at zero. Further results on the asymptotic behavior of solutions to initial and boundary value problems for second-order hyperbolic PDEs and their systems for various kinds of boundary conditions, nonlinear monotone damping and nonlinear sources can be found in [9–17]. In recent years the case of localized damping aroused much interest. Such problems yield significant difficulties since then the damping must propagate across the whole domain. The problem with a nonlinear source term and nonlinear monotone damping localized in some open set in the problem domain was considered in [18] (where a method was developed for dealing with the problem without any growth conditions on the damping term) and [19]. In turn, nonlinear monotone damping localized on a section of the domain boundary was considered in [20]. For the case of HVIs there are no reliable decay results known to the author. The aim of this paper is to provide assumptions under which the exponential decay occurs for the weak solutions of the inclusions u′′ (t ) + A(t , u′ (t )) + Bu(t ) + γ¯ ∗ ∂ J (t , γ¯ u(t )) ∋ 0 and u′′ (t ) + A(t , u′ (t )) + Bu(t ) + γ¯ ∗ ∂ J (t , γ¯ u′ (t )) ∋ 0, where J is a possibly nonconvex locally Lipschitz integral functional J (t , v) = j(x, t , v(x)) dx and the multivalued term has the form of the Clarke subdifferential with respect to v . The functional J is defined on the space of boundary functions which corresponds to the nonmonotone and multivalued boundary condition of Neumann–Robin type. This case is of particular interest, since such boundary conditions are useful in modeling various types of frictional and adhesive contact in the theory of (visco)elasticity (see [21,22]). The operator A is assumed to be (possibly) nonlinear, pseudomonotone and coercive, while B is assumed to be linear, continuous, symmetric and coercive. Various existence results for the weak solutions of second order in time hemivariational inequalities are known in literature. The case of a multivalued term dependent on u or u′ , or both, defined inside the problem domain was considered by Goeleven et al. [23], Ochal [24] and Gasiński [25]. The results of the last two articles were extended to the case where the multivalued term is defined on the part of the domain boundary in the articles [26] (where the two cases of dependence of the multivalued term on u and u′ are considered), [27] (where multivalued term depends on u′ and the relation between problem constants is replaced by the so called sign condition) and [28] (where the most general case of the arbitrary upper semicontinuous multifunction dependent on both u and u′ is considered). In all three papers mentioned we have the coercive operator A and linear growth conditions at infinity on A and ∂ J. In the case of noncoercive A the existence of solutions when the multivalued term is defined on the boundary is still an open problem. An existence result for the wave-like problem with a multivalued term defined inside the problem domain and dependent on u′ is available in [29]. For the case of dependence of the multivalued term on u there are existence results available in [30–32]. The methods used to prove the existence of solutions for second-order HVIs rely either on showing that the assumptions of an appropriate surjectivity theorem of Brezis type hold (see for instance [26]) or on the construction of auxiliary problems, either with added regularizing terms (see [28]) or with the smoothed multivalued term (see [29]). Unfortunately, if the multivalued term is defined on the boundary, without the coercive damping, neither of these approaches produces sufficient estimates that enable enough convergence to pass to the limit in the multivalued term. The other problem with HVIs is the lack of uniqueness and regularity results. There are uniqueness results only under the one-side Lipschitz condition (known also as the relaxed monotonicity condition; see [33]) on ∂ J which is equivalent to the semiconvexity of the superpotential J and relates the problems with nonconvex potential very close to the convex case. On the other hand, without this condition there is a recent multiplicity result of Carl for a parabolic problem (first order in time) with a periodic boundary condition (see [34]). Also regularity results, which are often used to prove the decay of energy, are still an open problem for HVIs. There are regularity results available for elliptic problems (see [29]) while there are no such results known to the author for the time dependent case. Now we briefly outline and discuss the results of this article. The paper takes the existence results of Migórski [26] for the boundary HVIs as the starting point. The existence result of [26] for the (easier) case of a multivalued term dependent on u′ is recalled in Section 3.1. Additional assumptions which imply the exponential decay of energy together with the decay result itself are provided in Section 3.2. The assumptions are the linear growth of A and ∂ J at zero (H (j)(iii) and H (A)(ii)) and coercivity of B. Since the structural damping A guarantees the exponential decay, the influence of the multivalued term does not have to be damping. In fact this term can represent an excitatory force, provided the ‘good’ damping properties of A are not ‘spoiled’, which is reflected by the assumption H (j)(iv). Key ideas behind the proof follow the aforementioned methodology of Nakao. In particular no multipliers method is needed. One standard estimate is obtained by taking the duality with u′ (Lemma 2) and, then, the other one by taking the duality with u (Lemma 3). The decay result (Theorem 4) is obtained by combining two estimates with each other. Section 4 is devoted to the problem with the multivalued term depending on u. Here, the straightforward approach cannot be used to obtain the exponential decay, so the problem is solved via the regularization of the multivalued term through the standard mollifier procedure. In Section 4.3 the existence of solutions to the underlying problem is established

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(Theorem 12) under the same conditions as in [26] using the Galerkin method which is an alternative to using the surjectivity result of Papageorgiou et al. [35]. Then, two estimates are obtained: the first one by taking the duality with velocity in the Galerkin problem (Lemma 14) and the other one by taking the duality with the function itself (Lemma 15). The estimates are combined with each other in Theorem 16 to obtain the decay result for the energy of the solutions of Galerkin problems and then, by passing to the limit, the decay result is obtained for the original problem (Theorem 17). It must be remarked that the decay is shown only for the solutions obtained by passing to the limit in the Galerkin problems and it is not known whether the solution is unique or whether all solutions can be obtained in this way. In addition to the assumptions required for the existence, the assumptions that ensure the energy decay are coercivity of B, linear growth at zero of A and ∂ J, the assumptions H (j)(v) on the direct time dependence of j and, in the case when the force given by the multivalued term is excitatory, its control by the elastic term B (the assumptions H (j)(vi)1 and the upper bound in H (j)(vi)2 ). The last part of the article, Section 5, contains a brief discussion of the assumptions used in the article and several constitutive laws used in mechanics. 2. Preliminaries 2.1. The lemma of Nakao We prove a simple generalization of Nakao lemma that will be useful in the sequel. Lemma 1. Let φ : [0, ∞) → R. If there exist constants δ > 0 and a ≥ 0 such that sup φ(s) ≤ δ(φ(t ) − φ(t + 1)) + a,

for all t ≥ 0,

(1)

t ≤s≤t +1

then there exist constants C > 0 and λ > 0 (both depending on δ ) such that

φ(t ) ≤ |φ(0) − φ(1)|C exp(−λt ) + a,

for all t ≥ 0.

(2)

If φ(t ) ≥ 0 for t ∈ [0, ∞) and a = 0, then we recover the well known Nakao lemma (see for instance Lemma 2 in [2], case r = 0) as a simple consequence. Proof. By (1) for all t ≥ 0 we get a δ + . 1+δ 1+δ Now let us fix t ≥ 0. For some n ∈ N (possibly 0) we have t ∈ [n, n + 1). Using the above estimate n times yields   n  n−1  δ a δ δ φ(t ) ≤ φ(t − n) + 1+ + ··· + 1+δ 1+δ 1+δ 1+δ  n   n  δ δ = φ(t − n) +a 1− . 1+δ 1+δ

φ(t + 1) ≤ φ(t )

(3)

The assumption (1) yields φ(t − n) ≤ δ(φ(0) − φ(1)) + a, so we have

δ φ(t ) ≤ δ(φ(0) − φ(1)) 1+δ 

n

+ a.

(4)

If φ(t ) ≤ a then the thesis holds. In the opposite case, after subtracting a from both sides of the inequality, taking the logarithm, we obtain



ln(φ(t ) − a) ≤ ln δ(φ(0) − φ(1))



δ 1+δ

n 

= ln(δ(φ(0) − φ(1))) + n ln



δ 1+δ



.

(5)

Since for x > 0 we have ln(x) ≤ x − 1, we obtain ln(φ(t ) − a) ≤ ln(δ(φ(0) − φ(1))) −

n 1+δ

≤ ln(δ(φ(0) − φ(1))) −

t −1 1+δ

.

(6)

The last inequality implies

  t −1 φ(t ) ≤ δ(φ(0) − φ(1)) exp − + a. 1+δ

(7)

Summarizing, we get for t ≥ 0

φ(t ) ≤ |φ(0) − φ(1)|δ exp



1 1+δ



 exp −

t 1+δ



+ a. 

(8)

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2.2. Notation In the sequel all the problems are considered on the time interval (0, T ), where T is arbitrarily large. Let N ∈ N. The Euclidean norm and scalar product in RN are denoted respectively by | · | and (·, ·). Ω ⊂ RN is an open bounded set with Lipschitz boundary. The boundary ∂ Ω is divided into three mutually disjoint parts ∂ Ω = Γ¯ C ∪ Γ¯ D ∪ Γ¯ N , of which ΓD and ΓC have positive boundary measure. The trace operator is denoted by γ : V → L2 (ΓC ; RN ). We define the spaces ∗ V = {v ∈ H 1 (Ω ; RN ) : γ v = 0 on ΓD } and H = L2 (Ω ; RN ). The norm  onV is denoted by ‖·‖. Spaces V ⊂ H ⊂ V constitute an evolution triple. Furthermore Z = H δ (Ω ; RN ) ∩ V (where δ ∈ 21 , 1 ) and γ¯ : Z → L2 (ΓC ; RN ) is the trace operator on Z . The space V is compactly embedded in Z (with the embedding injection i : V → Z ) and for v ∈ V we have ‖i(v)‖Z ≤ β‖v‖ with β > 0. We have γ (v) = γ¯ (i(v)); however in the sequel we drop the embedding symbol i for simplicity and write γ (v) = γ¯ (i(v)) = γ¯ (v) for v ∈ V . We introduce the spaces of time dependent functions V = L2 (0, T ; V ) and W = {u ∈ V , u′ ∈ V ∗ }. We use the same symbol γ¯ to denote the Nemytskii trace operator γ¯ : L2 (0, T ; Z ) → L2 (0, T ; L2 (ΓC ; RN )) defined by (γ¯ u)(t ) = γ¯ (u(t )). We define ‖γ¯ ‖ := ‖γ¯ ‖L(Z ;L2 (ΓC ;RN )) . For 0 ≤ t1 < t2 ≤ T the space L2 (t1 , t2 ; V ) is embedded in L2 (t1 , t2 ; Z ) with the constant β (the same as V ⊂ Z ). By m we denote the Lebesgue measure in RN and by mN −1 the N − 1-dimensional boundary measure. Recall that if u ∈ V and u′ ∈ W then u ∈ C ([0, T ]; V ) and u′ ∈ C ([0, T ]; Z ), so we can talk about the pointwise values of u as elements of V , of u′ as elements of Z and of γ¯ u and γ¯ u′ as elements of L2 (ΓC ; RN ) for all t ∈ [0, T ]. All definitions appearing in the article are standard (see for instance [26,28]). We recall that the generalized directional derivative (in the sense of Clarke; see [36]) of a locally Lipschitz function h : E → R, where E is the Banach space, at x ∈ E in the direction v ∈ E is defined by h0 (x; v) = lim sup

h(y + λv) − h(y)

y→x,λ↘0

λ

,

and the generalized gradient (in the sense of Clarke) of h at x ∈ E is given by

∂ h(x) = {ζ ∈ E ∗ : h0 (x; v) ≥ ⟨ζ , v⟩E ∗ ×E for all v ∈ E }. Finally we recall (see [37]) that the single-valued operator A : X → X ∗ , where X is a reflexive Banach space, is pseudomonotone if for un → u weakly in X the convergence lim supn→∞ ⟨Aun , un − u⟩X ∗ ×X ≤ 0 implies ⟨Au, u − w⟩X ∗ ×X ≤ lim infn→∞ ⟨Aun , un − w⟩X ∗ ×X for all w ∈ X . Sometimes (see for instance [38]) for the sake of compatibility with the definition of pseudomonotonicity for multivalued operators the pseudomonotone operators are also assumed to be bounded. In this article, whenever we assume the pseudomonotonicity of an operator, we also assume the growth condition that implies boundedness, so the argument works well for both definitions. 3. A hemivariational inequality with a multivalued term depending on the velocity 3.1. The existence of solutions We consider the following HVI (see [26,28]):

 find u ∈ V with u′ ∈ W such that, u′′ (t ) + A(t , u′ (t )) + Bu(t ) + γ¯ ∗ (∂ J (t , γ¯ u′ (t ))) ∋ f (t ) for a.e. t ∈ (0, T ), u(0) = u , u′ (0) = u1 . 0

(9)

The assumptions are the following: H (A):

A : (0, T ) × V → V ∗ is an operator such that: (i) (ii) (iii) (iv)

H (B): H (j):

t → ⟨A(t , u), v⟩ is measurable on (0, T ) for all u, v ∈ V ; ‖A(t , v)‖V ∗ ≤ a(t ) + b‖v‖ for a.e. t ∈ (0, T ), for v ∈ V with a ∈ L2 (0, T ), a ≥ 0, b ≥ 0; ⟨A(t , v), v⟩ ≥ α‖v‖2 for a.e. t ∈ (0, T ), for all v ∈ V with α > 0; v → A(t , v) is pseudomonotone for every t ∈ (0, T ).

B : V → V ∗ is a bounded, linear, monotone and symmetric operator, i.e. B ∈ L(V , V ∗ ), ⟨Bv, v⟩ ≥ 0 for all v ∈ V , ⟨Bv, w⟩ = ⟨Bw, v⟩ for all v, w ∈ V . j : ΓC × (0, T ) × RN → R: (i) j(·, ·, ξ ) is measurable for all ξ ∈ RN and j(·, t , 0) ∈ L1 (ΓC ); (ii) j(x, t , ·) is locally Lipschitz for all x ∈ ΓC , t ∈ (0, T ); (iii) |η| ≤ c (1 + |ξ |) for all ξ ∈ RN , η ∈ ∂ j(x, t , ξ ), t ∈ (0, T ), x ∈ ΓC with c > 0.

H0 :

f ∈ V ∗ , u0 ∈ V , u1 ∈ H.

The symbol ∂ j denotes the Clarke subdifferential of j with respect to the variable ξ . Under assumption H (j) the functional J : (0, T ) × L2 (ΓC ; RN ) → R given by J (t , v) =

∫ ΓC

j(x, t , v(x)) dΓ ,

t ∈ (0, T ), v ∈ L2 (ΓC ; RN )

(10)

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P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

is well defined, is Lipschitz on bounded subsets of L2 (ΓC ; RN ) and its generalized gradient (with respect to the second variable) satisfies

ζ ∈ ∂ J (t , v) ⇒ ‖ζ ‖L2 (ΓC ;RN ) ≤ c1 (1 + ‖v‖L2 (ΓC ;RN ) ), (11) √ √ with c1 = 2c max{1, mN −1 (ΓC )} (see Lemma 2 in [26]). Migórski [26] proved that the above problem has a solution under assumptions H (A), H (B), H (j), H0 and the additional assumption : α2 > c1 β‖γ¯ ‖2 .

Hconst

In the sequel extra assumptions for A, B, j will be formulated. By H (A) we will always mean H (A)(i)–(iv) and by H (j) we will mean H (j)(i)–(iii). 3.2. The decay of energy Throughout this section, in addition to the previous assumptions, we will make use of the following ones: H(j)(iii)1 : |η| ≤ c |ξ | for all ξ ∈ RN and η ∈ ∂ j(x, t , ξ ), for a.e. t ∈ (0, T ) and for a.e. x ∈ ΓC with c > 0. H(j)(iv): There exists the real constant κ < β 2 ‖αγ¯ ‖2 such that for a.e. x ∈ ΓC , for a.e. t ∈ (0, T ) and for all ξ ∈ RN if

η ∈ ∂ j(x, t , ξ ) then 0 ≤ (η + κξ , ξ ). H(A)(ii)1 : ‖A(t , v)‖V ∗ ≤ b‖v‖ for a.e. t ∈ (0, T ) for v ∈ V with b > 0. H (B)1 : In addition to H (B) we assume that B is coercive, i.e. ⟨Bv, v⟩ ≥ d‖v‖2 for v ∈ V with d > 0.

Remark 1. Of course H(j)(iii)1 is stronger then H(j)(iii). In particular it implies that ∂ j(x, t , 0) = {0}. This assumption also implies that for v ∈ L2 (ΓC ; RN ), for a.e. t ∈ (0, T ) and η ∈ ∂ J (t , v), we have ‖η‖L2 (ΓC ;RN ) ≤ c ‖v‖L2 (ΓC ;RN ) . Remark 2. H(A)(ii)1 is H(A)(ii) with a = 0. Remark 3. Coercivity of A is the key assumption required for the velocity to tend to zero. In fact H(j)(iv) means that the boundary multivalued term does not spoil the damping effect of A. Let u be a solution to the problem considered. We define the energy E (t ) = is defined everywhere and continuous on (0, T ). We prove the following:

1 2

(‖u′ (t )‖2H + ⟨Bu(t ), u(t )⟩). The function E (t )

Lemma 2. Let f = 0. Under assumptions H(A)(i), (iii), H(B), H(j)(i)–(iv), if u is a solution to (9), then for all 0 ≤ t1 < t2 ≤ T the following estimate holds: E (t2 ) − E (t1 ) ≤ κβ 2 ‖γ¯ ‖2 − α ‖u′ ‖2L2 (t ,t ;V ) . 1 2





(12)

Proof. Without loss of generality we can assume that κ ≥ 0. Let u be a solution to (9). There exists η ∈ L2 (0, T ; L2 (ΓC ; RN )) such that η(t ) ∈ ∂ J (t , γ¯ u′ (t )) for a.e. t ∈ (0, T ) and u′′ (t ) + A(t , u′ (t )) + Bu(t ) + γ¯ ∗ η(t ) = 0

for a.e. t ∈ (0, T ).

(13)

By H(j)(iv) we have

∫ 0≤ ΓC

(η(x, t ) + κ(γ¯ u′ (t ))(x), (γ¯ u′ (t ))(x)) dΓ

for a.e. t ∈ (0, T ).

Thus we have for a.e. t ∈ (0, T )

−κ‖γ¯ u′ (t )‖2L2 (Γ

C ;R

N)

≤ (η(t ), γ¯ u′ (t ))L2 (ΓC ;RN ) = ⟨γ¯ ∗ η(t ), u′ (t )⟩.

By the continuity of the trace operator, we have

− κ‖γ¯ ‖2 ‖u′ (t )‖2Z ≤ ⟨γ¯ ∗ η(t ), u′ (t )⟩ for a.e. t ∈ (0, T ).

(14)

Taking the duality with u (t ) in (13), we get for a.e. t ∈ (0, T ) ′

⟨u′′ (t ), u′ (t )⟩ + ⟨A(t , u′ (t )), u′ (t )⟩ + ⟨Bu(t ), u′ (t )⟩ + ⟨γ¯ ∗ η(t ), u′ (t )⟩ = 0. Setting 0 ≤ t1 < t2 ≤ T , integrating the above equation over (t1 , t2 ) and using the identities and ⟨Bv(t ), v (t )⟩ = ′

1 2

∫

t2 t1

d  dt

1 d 2 dt

1 d 2 dt

‖v(t )‖2H = ⟨v ′ (t ), v(t )⟩

⟨Bv(t ), v(t )⟩ as well as assumption H(A)(iii) (coercivity of A) and (14), we get

 ‖u′ (t )‖2H + ⟨Bu(t ), u(t )⟩ dt + α‖u′ ‖2L2 (t

1 ,t2 ;V )

− κ‖γ¯ ‖2 ‖u′ ‖2L2 (t

1 ,t2 ;Z )

Using continuity of the embedding L2 (t1 , t2 ; V ) ⊂ L2 (t1 , t2 ; Z ) we obtain the thesis.

≤ 0. 

P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

1169

To prove the decay of the energy we need the following estimate. Lemma 3. Let f = 0. Assuming H(A)(i), (ii)1 , H(B)1 , H(j)(i), (ii), (iii)1 , if u is a solution to (9) then the following estimate holds for any 0 ≤ t1 < t2 ≤ T with the constants C1 , C2 > 0 independent of T :

‖u‖2L2 (t

1 ,t2 ;V )

≤ C1 (‖u′ (t1 )‖H ‖u(t1 )‖H + ‖u′ (t2 )‖H ‖u(t2 )‖H ) + C2 ‖u′ ‖2L2 (t

1 ,t2 ;V )

.

Proof. Let us take the duality in (13) with u(t ). We get

⟨u′′ (t ), u(t )⟩ + ⟨A(t , u′ (t )), u(t )⟩ + ⟨Bu(t ), u(t )⟩ + ⟨γ¯ ∗ η(t ), u(t )⟩ = 0. Using the identity ⟨u′′ (t ), u(t )⟩ = d dt

d dt

(u′ (t ), u(t ))H − ‖u′ (t )‖2H and H (B)1 we have

  (u′ (t ), u(t ))H − ‖u′ (t )‖2H + d‖u(t )‖2 ≤ ‖A(t , u′ (t ))‖V ∗ + ‖γ¯ ‖β‖η(t )‖L2 (ΓC ;RN ) ‖u(t )‖.

Taking advantage of H(A)(ii)1 and H(j)(iii)1 we get d‖u(t )‖2 ≤ −

d dt

  (u′ (t ), u(t ))H + ‖u′ (t )‖2H + b + c ‖γ¯ ‖2 β 2 ‖u′ (t )‖ ‖u(t )‖.

Now we integrate above inequality over interval (t1 , t2 ). We get

∫

t2

d

‖u(t )‖2 dt ≤ (u′ (t1 ), u(t1 ))H − (u′ (t2 ), u(t2 ))H +

t1

∫

t2

‖u′ (t )‖2H dt

t1

  + b + c ‖γ¯ ‖2 β 2

t2

∫

‖u′ (t )‖ ‖u(t )‖ dt ,

(15)

t1

which gives d‖u‖2L2 (t ,t ;V ) ≤ ‖u′ (t1 )‖H ‖u(t1 )‖H + ‖u′ (t2 )‖H ‖u(t2 )‖H + ‖u′ ‖2L2 (t ,t ;H ) 1 2 1 2

  + b + c ‖γ¯ ‖2 β 2 ‖u′ ‖L2 (t1 ,t2 ;V ) ‖u‖L2 (t1 ,t2 ;V ) .

(16)

Applying the Cauchy inequality in the last term, we obtain d‖u‖2L2 (t ,t ;V ) ≤ ‖u′ (t1 )‖H ‖u(t1 )‖H + ‖u′ (t2 )‖H ‖u(t2 )‖H + ‖u′ ‖2L2 (t ,t ;H ) 1 2 1 2

(b + c ‖γ¯ ‖2 β 2 )2

+ which gives the thesis.

2d

‖u′ ‖2L2 (t

d

1 ,t2 ;V )

+ ‖u‖2L2 (t 2

1 ,t2 ;V )

,

(17)



We formulate the following: Theorem 4. Assume H(A)(i), (ii)1 , (iii), H(B)1 , H(j)(i), (ii), (iii)1 , (iv). If u is a solution to (9) with f = 0 then there exist constants D > 0 and λ > 0 independent of T such that for all t ∈ (0, T ) we have E (t ) ≤ E (0)D exp(−λt ). Proof. Let us define F 2 (t ) = E (t ) − E (t + 1) for t ∈ (0, T ). By Lemma 2 the function F is well defined and for C > 0, t +1

∫

2

t +1

∫

‖u (s)‖ ds ≤ CF (t ) and ′

2

t

‖u′ (s)‖2H ds ≤ CF 2 (t ).

(18)

t

Since u′ ∈ C ([0, T ]; H ) there exist t1 ∈ t , t +



1 4

and t2 ∈ t + 34 , t + 1 such that we have







√ √ ‖u′ (t1 )‖H ≤ 2 C F (t ) and ‖u′ (t2 )‖H ≤ 2 C F (t ).

(19)

Now we are in a position to estimate for any t ∈ (0, T ) E (t ) = F 2 (t ) + E (t + 1) ≤ F 2 (t ) + E (t2 ) ≤ F 2 (t ) + 2(t2 − t1 )E (t2 )

≤ F 2 (t ) + 2

∫

t2

∫

E (s) ds ≤ F 2 (t ) +

t1

t2

‖u′ (s)‖2H ds + ‖B‖L(V ;V ∗ )

t1

∫

t2

‖u(s)‖2 ds.

t1

We use Lemma 3 as well as (18) to get for all t ∈ (0, T ) with C3 , C4 > 0 E (t ) ≤ C3 F 2 (t ) + C4 (‖u′ (t1 )‖H ‖u(t1 )‖ + ‖u′ (t2 )‖H ‖u(t2 )‖). Applying (19) we get E (t ) ≤ C3 F 2 (t ) + 2C4

‖u(s)‖ ≤

2 d

√

√

(20)

C F (t )(‖u(t1 )‖ + ‖u(t2 )‖) for all t ∈ (0, T ). By the coercivity of B we have √ √  √ 4 C C4 F (t ) E (t1 ) + E (t2 ) . By the fact that d

E (s) for all s ≥ 0. Thus for t ∈ (0, T ) we get E (t ) ≤ C3 F 2 (t ) +

1170

P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

√

√

E (t ) is nonincreasing (cf. Lemma 2) for all t ∈ (0, T ) we have E (t ) ≤ C3 F 2 (t ) + d 4 F (t ) E (t ). By the Cauchy inequality we get for all t ∈ (0, T ), E (t ) ≤ C5 F 2 (t ) with C5 > 0. Lemmas 1 and 2 yield the thesis.  8 CC

4. A hemivariational inequality with a multivalued term depending on the displacement 4.1. Assumptions Throughout this section, in addition to the previous assumptions, the following ones will be used: H(j)(v)1 : For all (t , ξ ) ∈ (0, T ) × RN and for a.e. x ∈ ΓC , there exists a partial derivative

   ∂ j(x,t ,ξ )  function of (t , ξ ) and  ∂ t  ≤ K (x), where K ∈ L1 (ΓC ).

∂ j(x,t ,ξ ) , ∂t

which is a continuous

∂ j(x,t ,ξ )

H(j)(v)2 : For all (t , ξ ) ∈ (0, T ) × RN and for a.e. x ∈ ΓC we have ∂ t ≤ 0. H(j)(vi)1 : There exists a real constant µ < β 2 ‖dγ¯ ‖2 (where d is the coercivity constant in H(B)(ii)) such that for a.e. x ∈ ΓC , for a.e. t ∈ (0, T ) and for all ξ ∈ RN if η ∈ ∂ j(x, t , ξ ), then 0 ≤ (η + µξ , ξ ). H(j)(vi)2 : There exist real constants ν > 0 and χ < 2β 2 ‖dγ¯ ‖2 such that for a.e. x ∈ ΓC , for a.e. t ∈ (0, T ) and for all ξ ∈ RN we have −χ|ξ |2 ≤ j(x, t , ξ ) ≤ ν|ξ |2 .

The assumptions H(j)(v)1 and H(j)(v)2 together will be denoted by H(j)(v) and similarly H(j)(vi)1 and H(j)(vi)2 will be denoted by H(j)(vi). H(j)(v) concerns the dependence of j on the time variable. H(j)(vi) is analogous to H(j)(iv) in the previous section. Under the technical assumption that j(x, t , 0) = 0, the upper bound in H(j)(iv)2 follows from the growth condition H(j)(iii)1 and the lower bound is implied by H(j)(iv)1 if µ < 2β 2 ‖dγ¯ ‖2 . 4.2. Regularization of the multivalued term Let us take j : ΓC × (0, T ) × RN → R which satisfies H (j). We introduce the standard mollifier p ∈ C0∞ (RN ) with the  properties 0 ≤ p(z ) on RN , supp(p) ⊂ [−1, 1]N and RN p(z ) dz = 1. We define pn (x) = nN p(nx) and we define supp(pn ) =

N

Kn , so Kn ⊂ − 1n , 1n for n ∈ N. The mapping z → pn (z )j(x, t , ξ − z ) is measurable and bounded for all (x, t , ξ ) ∈ ΓC × (0, T ) × RN , so we are in a position to define the functions jn : ΓC × (0, T ) × RN → R for n ∈ N as



jn (x, t , ξ ) =

∫

pn (z )j(x, t , ξ − z ) dz

for (x, t , ξ ) ∈ ΓC × (0, T ) × RN .

Kn

We observe that jn (x, t , ·) ∈ C ∞ (RN ) for all (x, t ) ∈ ΓC × (0, T ). Thus ∂ jn (x, t , ξ ) =



∂ jn (x,t ,ξ ) ∂ξ



.

We formulate the following: Lemma 5. If j satisfies H (j) then for every n ∈ N the function jn also satisfies H (j). The constant in H(j)(iii) for jn is independent of n. Proof. We already know that jn (x, t , ·) are C ∞ (RN ). Therefore it remains to show H(j)(i) and H(j)(iii). Proof of H (j)(i). Since j(·, ·, ξ ) is L(ΓC × (0, T )) measurable for all ξ ∈ RN and j(x, t , ·) is continuous for all (x, t ) ∈ ΓC × (0, T ), then j is L(ΓC × (0, T )) × B (RN ) measurable. Thus the mapping (x, t , ξ , z ) → pn (z )j(x, t , ξ − z ) is L(ΓC × (0, T )) × B (RN ) × B (RN ) measurable and, moreover, (x, t , ξ ) → jn (x, t , ξ ) is L(ΓC × (0, T )) × B (RN ) measurable. Thus also the mapping (x, t ) → jn (x, t , ξ ) is L(ΓC ×(0, T )) measurable for all ξ ∈ RN . Now we show that jn (·, t , 0) ∈ L1 (ΓC ) for all t ∈ (0, T ). For (x, t ) ∈ ΓC × (0, T ) we have

|jn (x, t , 0)| ≤

∫

pn (z )|j(x, t , −z )| dz .

Kn

By the Lebourg mean value theorem (see e.g. [39], Theorem 5.6.25) we get

|jn (x, t , 0)| ≤

∫

pn (z ) |j(x, t , 0)| + |(z ∗ , z )| dz ,





Kn

where z ∈ ∂ j(x, t , −rz ) for some r ∈ (0, 1). By H(j)(iii) we have |z ∗ | ≤ c (1 + |z |), so we have ∗

|jn (x, t , 0)| ≤ |j(x, t , 0)| + c

∫

pn (z )(1 + |z |)|z | dz K

n ≤ |j(x, t , 0)| + c 1 + which gives the thesis.

√ √ N

n

N

n

,

P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

1171

Proof of H (j)(iii). We follow the lines of proof of Lemma 5.2 in [22]. Let h ∈ RN . For any (x, t , ξ ) ∈ ΓC × (0, T ) × RN we have



∂ jn (x, t , ξ ) ,h ∂ξ

 = lim

jn (x, t , ξ + λh) − jn (x, t , ξ )

λ

λ→0

∫ = lim

λ→0 K n

pn ( z )

j(x, t , ξ + λh − z ) − j(x, t , ξ − z )

λ

dz . j(x,t ,ξ −z +λh)−j(x,t ,ξ −z )

By the Lebourg mean value theorem for some z ∗ ∈ ∂ j(x, t , ξ − z + r λh) where r ∈ (0, 1) we have = λ (z ∗ , h) ≤ c (1 + |ξ | + |z | + λ|h|)|h|, and therefore the integrand in the above formula can be estimated from above by the function cpn (z )(1 + |ξ | + |z | + |h|)|h|, which is bounded function of z ∈ RN for fixed h, ξ . Thus we are in a position to use the Fatou lemma to get



∂ jn (x, t , ξ ) ,h ∂ξ



∫

j(x, t , ξ + λh − z ) − j(x, t , ξ − z )

pn (z ) lim sup

≤ ∫

pn (z )j0 (x, t , ξ − z ; h) dz =

≤

∫

Kn

pn (z ) Kn

pn (z )c (1 + |z | + |ξ |)|h| dz ≤ c

≤

√ 1+

Kn

This gives the thesis with c¯ = c (1 +

sup

(z ∗ , h) dz

z ∗ ∈∂ j(x,t ,ξ −z )



∫

dz

λ

λ→0

Kn

N n

 + |ξ | |h|.

(21)

√

N ).



By Lemma 2 from [26] the functional Jn : (0, T ) × L2 (ΓC ; RN ) → R defined by Jn (t , u) =



j (x, t , u(x)) dΓ ΓC n

is well defined.

Furthermore, Jn (t , ·) is Lipschitz on bounded subsets of L2 (ΓC ; RN ) and its generalized gradient satisfies

ζ ∈ ∂ Jn (t , u) ⇒ ‖ζ ‖L2 (ΓC ;RN ) ≤ c¯1 (1 + ‖u‖L2 (ΓC ;RN ) ),

(22)

with some positive constant c¯1 dependent only on c , mN −1 (ΓC ) and N. Furthermore for (t , u) ∈ (0, T ) × L2 (ΓC ; RN ) we have

∂ jn (x, t , u(x)) for a.e. x ∈ ΓC , (23) ∂ξ so ∂ Jn (t , u) is a singleton for all (t , u). Therefore, since Jn (t , ·) is locally Lipschitz we deduce, by Proposition 5.6.15 (b) from [39], that it is strictly differentiable and we can write ∂ Jn (t , u) = {Du Jn (t , u)}. ζ ∈ ∂ Jn (t , u) ⇒ ζ (x) =

Lemma 6. Under assumptions H (j) and H(j)(v)1 if u ∈ C 1 (0, T ; L2 (ΓC ; RN )) then the following chain rule holds for n ∈ N and t ∈ (0, T ): d dt

∫

Jn (t , u(t )) =

ΓC

∂ jn (x, t , u(x, t )) dΓ + (Du Jn (t , u(t )), u′ (t ))L2 (ΓC ;RN ) . ∂t

Proof. Since the mapping (0, T ) ∋ t → (t , u(t )) ∈ (0, T ) × L2 (ΓC ; RN ) is Fréchet differentiable we need to show 2 N that  the mapping (0, T ) × L (ΓC ; R ) ∋ (t , v) → Jn (t , v) ∈ R is Fréchet differentiable and its derivative equals



ΓC

∂ jn (x,t ,v(x)) ∂t

dΓ , Du Jn (t , v) .

We observe that for all (t , ξ ) ∈ (0, T ) × Kn and for a.e. x ∈ ΓC we have

∂ j n ( x, t , ξ ) = ∂t

∫

pn (z )

Kn

∂ j(x, t , ξ − z ) dz . ∂t

Similarly for all (t , v) ∈ (0, T ) × L2 (ΓC ; RN ) we have Dt Jn (t , v) =

∂ ∂t

∫ ΓC

jn (x, t , v(x)) dΓ =

∫ ΓC

∂ jn (x, t , u(x)) dΓ . ∂t

Furthermore by the Lebesgue dominated convergence theorem Dt Jn (t , u) is a continuous function of (t , u). We are ready to prove the Fréchet differentiability of Jn . Let us fix (t , v) ∈ (0, T ) × L2 (ΓC ; RN ) and select λv ∈ L2 (ΓC ; RN ) and λt ∈ R such that t + λt ∈ (0, T ). Using the mean value theorem we have Jn (t + λt , v + λv ) − Jn (t , v) = Jn (t + λt , v + λv ) − Jn (t , v + λv ) + Jn (t , v + λv ) − Jn (t , v)

= Dt Jn (t + r λt , v + λv )λt + (Du Jn (t , v + sλv ), λv )L2 (ΓC ;RN ) = (Dt Jn (t + r λt , v + λv ) − Dt Jn (t , v)) λt + (Du Jn (t , v + sλv ) − Du Jn (t , v), λv )L2 (ΓC ;RN ) + Dt Jn (t , v)λt + (Du Jn (t , v), λv )L2 (ΓC ;RN ) ,

1172

P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

where r , s ∈ (0, 1) depend on t , v, λt , λv . We obtain the thesis by the fact that Dt Jn is continuous and Jn (t , ·) is strictly differentiable on L2 (ΓC ; RN ).  In the remainder of this section we prove three useful properties of Jn . First two lemmas will be useful for passing to the limit in a nonlinear regularized term. Lemma 7. Let j satisfy H (j). If un : (0, T ) → L2 (ΓC ; RN ) is a sequence such that un (t ) → u(t ) in L2 (ΓC ; RN ) for a.e. t ∈ (0, T ), Du Jn (·, un (·)) → η weakly in L2 (0, T ; L2 (ΓC ; RN )), then η(t ) ∈ ∂ J (t , u(t )) for a.e. t ∈ (0, T ). Proof. We use the idea of the proof of Theorem 5.5 in [22]. First we show that for t ∈ (0, T ) we have Graph(Limsupn→∞ Du Jn (t , ·)) ⊂ ∂ J (t , ·). Let us take vn → v in L2 (ΓC ; RN ) and let Du Jnk (t , vnk ) = ηnk → η in L2 (ΓC ; RN ) for {nk } ⊂ {n}. We need to show that η ∈ ∂ J (t , u). We have for h ∈ L2 (ΓC ; RN ) Jnk (t , vnk + λh) − Jnk (t , vnk )

(ηnk , h)L2 (ΓC ;RN ) = lim

λ

λ→0+

jnk (x, t , vnk (x) + λh(x)) − jnk (x, t , vnk (x))

∫ = lim

λ→0+

λ

ΓC

∫

pnk (z )

= lim

λ→0+

Knk

dΓ

j(x, t , vnk (x) − z + λh(x)) − j(x, t , vnk (x) − z )

∫

λ

ΓC

dΓ dz .

(24)

Now we fix ϵ > 0. There exists δ > 0 such that if ‖w − v‖L2 (ΓC ;RN ) ≤ δ and 0 < λ < δ , then for h ∈ L2 (ΓC ; RN ) j(x, t , w(x) + λh(x)) − j(x, t , w(x))

∫

λ

ΓC

dΓ ≤ J 0 (t , v; h) + ϵ.

We take k sufficiently large such that ‖vnk − v‖L2 (ΓC ;RN ) ≤

‖vnk − v − z ‖L2 (ΓC ;RN ) ≤ (ηnk , v)L2 (ΓC ;RN )

δ 2

and supp(pnk ) ⊆ B 0, √

√

δ



2

δ mN −1 (ΓC )

 . We observe that

+ |z | mN −1 (ΓC ) ≤ δ . We obtain ∫ ≤ lim pnk (z )(J 0 (t , u; v) + ϵ) dz = J 0 (t , u; v) + ϵ. 2

λ→0+

Knk

Moreover (η, v)L2 (ΓC ;RN ) ≤ J 0 (t , u; v) + ϵ and, since ϵ is arbitrary, (η, v)L2 (ΓC ;RN ) ≤ J 0 (t , u; v). Thus η ∈ ∂ J (t , u).

Now we prove that n≥1 Du Jn (t , B(v, 1)) is bounded for every v ∈ L2 (ΓC ; RN ). Indeed let w ∈ B(v, 1). From (22) it follows that ‖Du Jn (t , w)‖L2 (ΓC ;RN ) ≤ c¯1 (1 + ‖w‖L2 (ΓC ;RN ) ) ≤ c¯1 (2 + ‖v‖L2 (ΓC ;RN ) ). We are in a position to use the convergence theorem of Aubin and Cellina (see Theorem 7.2.1 in [40]) to get η(t ) ∈ convGraph(Limsupn→∞ Du Jn (t , u(t ))) for a.e. t ∈ (0, T ). Thus η(t ) ∈ conv∂ J (t , u(t )) = ∂ J (t , u(t )) and we have the thesis. 



Lemma 8. Let j satisfy H (j). If un → u in L2 (ΓC ; RN ), then for all t ∈ (0, T ) we have J (t , u) = limn→∞ Jn (t , un ). Proof. Let us estimate

∫     |J (t , u) − Jn (t , un )| =  j(x, t , u(x)) − jn (x, t , un (x)) dΓ  Γ ∫ C ≤ |j(x, t , u(x)) − jn (x, t , u(x))| dΓ ΓC

∫ + ΓC

|jn (x, t , u(x)) − jn (x, t , un (x))| dΓ := A + B.

(25)

We estimate each term separately:

∫ A = ΓC

∫ ≤ ΓC

∫     pn (z )(j(x, t , u(x)) − j(x, t , u(x) − z )) dz  dΓ   Kn ∫ pn (z )|j(x, t , u(x)) − j(x, t , u(x) − z )| dz dΓ . Kn

(26)

P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

1173

The Lebourg mean value theorem implies

∫

∫

pn (z )|(η∗ , z )| dz dΓ ,

A≤ ΓC

(27)

Kn

where η∗ ∈ ∂ j(x, t , η) with η ∈ {u(x) − θ z : 0 < θ < 1}. By H(j)(iii) we have

∫

∫

pn (z )c (1 + |u(x)| + |z |)|z | dz dΓ

A ≤ ΓC

≤

≤

c

Kn

√ ∫ n

c

√ 



N

ΓC

√  N

N

1 + |u(x)| +

√ 



N

mN −1 (ΓC ) 1 +

n

dΓ

n

n

 +



mN −1 (ΓC )‖u‖L2 (ΓC ;RN )

n→∞

−→ 0.

(28)

Estimating B, we get

∫ B = ΓC

∫ = ΓC

|jn (x, t , u(x)) − jn (x, t , un (x))| dΓ    ∂ jn (x, t , (1 − θ (x, t ))u(x) + θ (x, t )un (x))   , u(x) − un (x)  dΓ ,  ∂ξ

(29)

where 0 < θ < 1. Since jn satisfies H(j)(iii) we further get

∫ B ≤ ΓC

c (1 + |u(x)| + θ (x, t )|u(x) − un (x)|)|u(x) − un (x)| dΓ

 n→∞ ≤ c ( mN −1 (ΓC ) + ‖u‖L2 (ΓC ) + ‖u − un ‖L2 (ΓC ) )‖u − un ‖L2 (ΓC ) −→ 0. 

(30)

Lemma 9. Under assumptions H (j) and H(j)(vi)1 we have for a.e. t ∈ (0, T ) and for all u ∈ L2 (ΓC ; RN )

−(µ + an )‖u‖2L2 (Γ

N C ;R )

− bn ≤ (Du Jn (t , u), u),

where the an , bn → 0 are positive decreasing sequences which do not depend on the choice of t and u. Proof. Let us fix n ∈ N and take (t , ξ ) ∈ (0, T ) × RN and x ∈ ΓC such that H(j)(vi)1 holds. We have

∂ j n ( x, t , ξ ) · ξ = lim λ→0 ∂ξ

∫

pn (z )

j(x, t , ξ + λξ − z ) − j(x, t , ξ − z )

λ

Kn

dz .

The Lebourg mean value theorem gives

∂ j n ( x, t , ξ ) · ξ = lim λ→0 ∂ξ

∫

pn (z )(w ∗ , ξ ) dz ,

Kn

where w ∗ ∈ ∂ j(x, t , w) with w = ξ − z + ϵλξ and ϵ ∈ (0, 1) depends on x, t , ξ , z , λ. Now since pn (z )(w ∗ , ξ ) ≥ −pn (z ) c (1 + |ξ | + |z | + λ|ξ |)|ξ |, by the Fatou lemma and the fact that without loss of generality the constant µ in H(j)(vi)1 can be assumed to be nonnegative, we get

∂ j n ( x, t , ξ ) ·ξ ≥ ∂ξ

∫

pn (z ) lim inf(w ∗ , ξ ) dz λ→0

Kn

∫

pn (z ) lim inf(−µ|w|2 − (w ∗ , z − ϵλξ )) dz

≥

λ→0

Kn

∫

pn (z )(−µ(|ξ | + |z |)2 − c (1 + |ξ | + |z |)|z |) dz .

≥

(31)

Kn

√

Now, since |z | ≤

N n

and

 Kn

pn (z ) dz = 1, we get

 √ √  ∂ j n ( x, t , ξ ) N N N 2 · ξ ≥ −µ|ξ | − |ξ | (2µ + c ) − (µ + c ) 2 + c ∂ξ n n n   √  √ ≥ − µ+

N

2n

|ξ | − (µ + c ) 2

N

n2

+c

N

n

√ 

+ (2µ + c )

2

N

2n

.

(32)

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P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

 √ √  √ We set bn = mN −1 (ΓC ) (µ + c ) nN2 + c nN + (2µ + c )2 2nN and an = 2nN . Taking u ∈ L2 (ΓC ; RN ) we have for a.e. t ∈ (0, T ) ∫ ∂ jn (x, t , u(x)) (Du Jn (t , u), u) = · u(x) dΓ ≥ −(µ + an )‖u‖2L2 (Γ ;RN ) − bn , C ∂ξ ΓC which yields the thesis.



4.3. The existence of solutions We consider the following evolution inclusion:

 find u ∈ V with u′ ∈ W such that u′′ (t ) + A(t , u′ (t )) + Bu(t ) + γ¯ ∗ (∂ J (t , γ¯ u(t ))) ∋ f (t ) for a.e. t ∈ (0, T ), u(0) = u , u′ (0) = u1 , 0

(33)

where H (A), H (B), H (j) and H0 are assumed to hold. For (33) we also consider the Galerkin problem

 find un ∈ C 1 (0, T ; Vn ) with u′n ∈ AC (0, T ; Vn ) such that (34) u′′ (t ) + A(t , u′ (t )) + Bu (t ) + γ¯ ∗ (Du Jn (t , γ¯ un (t ))) = f (t ) for a.e. t ∈ (0, T ), un(0) = u , n u′ (0) = nu , 0n 1n  where Vn are finite dimensional spaces such that n≥1 Vn = V and each Vn is a subspace of Vn+1 . For simplicity we assume that dim(Vn ) = n. Moreover let {u0n , u1n } ∈ Vn be the sequences such that u0n → u0 in V and u1n → u1 in H. Eq. (34) is understood in Vn∗ , i.e.

⟨u′′n (t ) + A(t , u′n (t )) + Bun (t ) + γ¯ ∗ (Du Jn (t , γ¯ un (t ))), v⟩ = ⟨f (t ), v⟩ (35) is assumed to hold for a.e. t ∈ (0, T ) and all v ∈ Vn . ∑n Assuming that Vn = span(w1 , w2 , . . . , wn ) we can represent un (t ) in this basis as un (t ) = k=1 ckn (t )wk . Taking (35) with wk as v we obtain the system of nonlinear ODEs equivalent to (35). Lemma 10. Assume H (A), H (B), H (j), H0 . If un is a solution to (34) then the following bound holds with the constant C > 0 independent of n and t ∈ (0, T ):

‖un (t )‖ + ‖u′n (t )‖H + ‖u′n ‖V ≤ C (1 + ‖u0n ‖ + ‖u1n ‖ + ‖f ‖V ∗ ).

(36)

Proof. Let us first assume that T <

α c¯1 β 2 ‖γ¯ ‖2

,

(37)

∑n ′ ′ where c¯1 is the constant in (22). Since u′n (t ) = k=1 ckn (t )wk we can take the duality in (35) with un (t ) to get, for a.e. t ∈ (0, T ), ⟨u′′n (t ), u′n (t )⟩ + ⟨A(t , u′n (t )), u′n (t )⟩ + ⟨Bun (t ), u′n (t )⟩ + (Du Jn (t , γ¯ un (t )), γ¯ u′n (t ))L2 (ΓC ;RN ) = ⟨f (t ), u′n (t )⟩. Since ⟨u′′ (t ), u′ (t )⟩ = t ∈ (0, T ) 1 d 2 dt

1 d 2 dt

‖u′ (t )‖2H and ⟨Bu(t ), u′ (t )⟩ =

‖u′n (t )‖2H + α‖u′n (t )‖2 +

1 d 2 dt

By the inequality ‖un (t )‖ ≤ ‖un0 ‖ + d dt

‖u′n (t )‖2H + 2α‖u′n (t )‖2 +

≤

t

d dt

0

1 d 2 dt

(38)

⟨Bu(t ), u(t )⟩, by assumption H(A)(iii) we infer that for a.e.

⟨Bun (t ), un (t )⟩ ≤ ‖f (t )‖V ∗ ‖u′n (t )‖ + c¯1 ‖γ¯ ‖β(1 + ‖γ¯ ‖β‖un (t )‖)‖u′n (t )‖.

(39)

‖u′n (τ )‖ dτ and the Cauchy inequality with ϵ , we have for every ϵ > 0

⟨Bun (t ), un (t )⟩

‖f (t )‖2V ∗ + 2c¯12 ‖γ¯ ‖2 β 2 + 2c¯12 ‖γ¯ ‖4 β 4 ‖u0n ‖2 + 2ϵ‖u′n (t )‖2 + 2c¯1 ‖γ¯ ‖2 β 2 ϵ

t

∫

‖u′n (τ )‖ dτ ‖u′n (t )‖.

(40)

0

Integrating the above inequality over the interval (0, s) for s ∈ (0, T ) we get

‖u′n (s)‖2H + 2α‖u′n ‖2L2 (0,s;V ) + ⟨Bun (s), un (s)⟩ ≤ ‖u1n ‖2H + ⟨Bu0n , u0n ⟩ + 2ϵ‖u′n ‖2L2 (0,s;V ) + + 2c¯1 ‖γ¯ ‖2 β 2

s

∫

‖u′n (t )‖ dt 0

2

.

‖f ‖2L2 (0,s;V ∗ ) + 2c¯12 ‖γ¯ ‖2 β 2 s + 2c¯12 ‖γ¯ ‖4 β 4 ‖u0n ‖2 s ϵ (41)

P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

1175

Now using the Jensen inequality to the last term as well as the hypothesis H (B) we get

‖u′n (s)‖2H + 2α‖u′n ‖2L2 (0,s;V ) ≤ ‖u1n ‖2H + ‖B‖L(V ;V ∗ ) ‖u0n ‖2 + 2(ϵ + T c¯1 ‖γ¯ ‖2 β 2 )‖u′n ‖2L2 (0,s;V ) + We take ϵ =

‖f ‖2V ∗ + 2T c¯12 ‖γ¯ ‖2 β 2 + 2T c¯12 ‖γ¯ ‖4 β 4 ‖u0n ‖2 . ϵ α−T c¯1 ‖γ¯ ‖2 β 2 2

(42)

to get

‖u′n (s)‖H + ‖u′n ‖L2 (0,s;V ) ≤ D1 (T )(1 + ‖u0n ‖ + ‖u1n ‖H + ‖f ‖V ∗ ),

(43)

where D1 (T ) is a finite positive number depending on T . Furthermore we note that

‖un (s)‖ ≤ ‖u0n ‖ +

s

∫

‖u′ (t )‖ dt ≤ ‖u0n ‖ +

√

T D1 (T )(1 + ‖u0n ‖ + ‖u1n ‖H + ‖f ‖V ∗ ).

0

Thus we obtain for t ∈ (0, T )

‖un (t )‖ + ‖u′n (t )‖H + ‖u′n ‖L2 (0,T ;V ) ≤ D2 (T )(1 + ‖u0n ‖ + ‖u1n ‖H + ‖f ‖V ∗ ), where 0 < D2 (T ) < ∞ (without loss of generality we can make the technical assumption that D2 (T ) ≥ 1). K −1 Now we take arbitrary T > 0. For some K ∈ N we have (0, T ) = i=0 (ai , bi ), where ai = i · bi = min{ai + τ , T }, where τ =

α 3 . 4 c1 β 2 ‖γ¯ ‖2

(44) 1 α 2 c1 β 2 ‖γ¯ ‖2

and

We observe that the operators A(i) : (0, bi − ai ) × V → V ∗ defined as A(i) (t , u) :=

A(t + ai , u) satisfy H (A) and the functions j(i) : ΓC × (0, bi − ai ) × RN → R defined as j(i) (x, t , ξ ) := j(x, t + ai , ξ ) satisfy H (j). Furthermore the f(i) : (0, bi − ai ) → V ∗ defined as f(i) (t ) := f (t + ai ) belong to L2 (0, bi − ai ; V ∗ ). For i ∈ {0, . . . , K − 1} the functions u(i)n : (0, bi − ai ) → Vn defined as u(i)n (t ) := un (t + ai ) (where un is the solution to (34)) solve the following shifted Galerkin problems:

 find vn ∈ C 1 (0, bi − ai ; Vn ) with vn′ ∈ AC (0, bi − ai ; Vn ) such that, v ′′ (t ) + A (t , v ′ (t )) + Bv (t ) + γ¯ ∗ (D J (t , γ¯ vn (t ))) = f(i) (t ), vn (0) = u(i)(a ), n v ′ (0) n= u′ (a ), u (i)n n n i n n i

(45)

where J(i)n is defined analogously to Jn and the equality holds for a.e. t ∈ (0, bi − ai ). By (44) we have for s ∈ (ai , bi )

‖un (s)‖ + ‖u′n (s)‖H + ‖u′n ‖L2 (ai ,bi ;V ) ≤ D2 (τ )(1 + ‖un (ai )‖ + ‖u′n (ai )‖H + ‖f ‖V ∗ ), and therefore, for any s ∈ (0, T ) we have

‖un (s)‖ + ‖u′n (s)‖H ≤ D2 (τ )K (‖un0 ‖ + ‖un1 ‖H ) + (1 + ‖f ‖V ∗ )D2 (τ )

D2 (τ )K − 1 D2 (τ ) − 1

≤ D3 (τ , K )(1 + ‖un0 ‖ + ‖un1 ‖H + ‖f ‖V ∗ ),

(46)

where 0 < D3 (τ , K ) < ∞. Furthermore we have

‖u′n ‖V ≤

K −1 −

‖u′n ‖L2 (ai ,bi ;V )

i =0

≤ KD2 (τ )(1 + ‖f ‖V ∗ ) + D2 (τ )

K −1 −

(‖un (ai )‖ + ‖u′n (ai )‖H )

i=0

≤ KD2 (τ )(1 + ‖f ‖V ∗ ) + D2 (τ )KD3 (τ , K )(1 + ‖un0 ‖ + ‖un1 ‖H + ‖f ‖V ∗ ), which gives the thesis.

(47)



On the basis of the derived bounds we establish the existence of solutions to Galerkin problems: Lemma 11. Under assumptions H (A), H (B), H (j) and H0 the Galerkin problem (34) has a solution. Proof. We observe that Vn ∋ v → ⟨A(t , v) + Bv + γ¯ ∗ (Du Jn (t , γ¯ v)), w⟩ is continuous for w ∈ Vn and t ∈ (0, T ). Indeed since A(t , ·) is pseudomonotone (cf. H(A)(iv)) and locally bounded (H(A)(ii)) then it is also demicontinuous (see Proposition 27.7(b) in [37]) and ⟨A(t , ·), w⟩ is continuous. Moreover ⟨B(·), w⟩ is continuous by H (B), and ⟨γ¯ ∗ (Du Jn (t , γ¯ (·))), w⟩ is continuous since Jn (t , ·) is strictly differentiable on L2 (ΓC ; RN ). By H (A)(i) the function ⟨A(·, v), w⟩ is measurable. Moreover, since jn (·, ·, ξ ) is measurable and jn (x, t , ·) ∈ C ∞ (RN ), then ∂ j (·,·,ξ ) ∂ j (x,t ,·) ,t ,u(x)) also n ∂ξ is measurable and n ∂ξ ∈ C ∞ (RN ; RN ). Thus for (x, t , u) ∈ ΓC × (0, T ) × L2 (ΓC ; RN ) the function ∂ jn (x∂ξ ∗ is measurable with respect to (x, t ) and ⟨γ¯ Du Jn (t , γ¯ v), w⟩ is measurable with respect to t.

1176

P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

Finally we have |⟨−A(t , v)− Bv − γ¯ ∗ (Jn′ (t , γ¯ v))+ f (t ), wi ⟩| ≤ (a(t )+ b‖v‖+‖B‖L(V ;V ∗ ) ‖v‖+β‖γ¯ ∗ ‖¯c1 (1 +β‖γ¯ ‖ ‖v‖)+ ‖f (t )‖V ∗ )‖wi ‖, which can be estimated from above by a function that belongs to L1 (0, T ) for v in some bounded set in Vn . The above observations together with the bounds ‖un (t )‖ ≤ C and ‖u′n (t )‖H ≤ C which hold for solutions of (34), for all n ∈ N and t ∈ (0, T ) (see Lemma 10), by the Caratheodory theorem, yield the existence of solutions to (34) on (0, T ).  Theorem 6 in [26] establishes the existence of a solution to (33) on the interval (0, T ) for T <

α

2c1 β 2 ‖γ¯ ‖2

using the

surjectivity result of [35]. The result in [26] can be easily extended to the arbitrary interval (0, T ) since it can be covered by a finite number of subintervals of length satisfying the above bound analogously to in the proof of Lemma 10. The following theorem establishes the solution on (0, T ) for arbitrary T by means of the Galerkin procedure. Theorem 12. Under assumptions H (A), H (B), H (j), H0 the problem (33) has a solution. Proof. Step 1: Weak convergence. By A : V → V ∗ and B : V → V ∗ we denote the Nemytskii operators for A and B defined respectively as (Av)(t ) = A(t , v(t )) and (B v)(t ) = B(v(t )). Let us take the sequence {un }∞ n=1 of solutions of Galerkin problems. By Lemma 10 the sequences ‖un ‖V , ‖u′n ‖V , ‖un (t )‖, ‖u′n (t )‖H are bounded for t ∈ (0, T ) by a constant independent of n. By the standard argument (see for instance the proof of Lemma 6 in [26]) the sequences ‖u′′n ‖V ∗ , ‖Au′n ‖V ∗ and ‖Du Jn (·, γ¯ un (·))‖L2 (0,T ;L2 (ΓC ;RN )) are also bounded by some constants independent of n. Thus for some u ∈ V such that ∞ u′ ∈ W we get (for some subsequence {unk }∞ k=1 ⊂ {un }n=1 which we still denote by un ) weakly in V ,

un → u ′

′

′′

′′

un → u

un → u

(48)

weakly in V ,

(49)

weakly in V ,

(50)

∗

and furthermore for some ζ ∈ V ∗ and η ∈ L2 (0, T ; L2 (ΓC ; RN )) we get

Au′n → ζ

weakly in V ∗ ,

Du Jn (·, γ¯ un (·)) → η

(51)

weakly in L (0, T ; L (ΓC ; R )). 2

2

N

(52)

By continuous embeddings H 1 (0, T ; V ) ⊂ C ([0, T ]; V ) and W ⊂ C ([0, T ]; H ) we also have un (t ) → u(t ) weakly in V for t ∈ [0, T ],

(53)

un (t ) → u (t ) weakly in H for t ∈ [0, T ].

(54)

′

′

By the compact embedding V ⊂ Z we get un (t ) → u(t ) in Z for t ∈ [0, T ],

(55)

and so furthermore

γ¯ un (t ) → γ¯ u(t ) in L2 (ΓC ; RN ) for t ∈ [0, T ].

(56)

By Lemma 7 we get η(t ) ∈ ∂ J (t , γ¯ u(t )) for a.e. t ∈ (0, T ), so, moreover, γ¯ η(t ) ∈ γ¯ ∂ J (t , γ¯ u(t )) for a.e. t ∈ (0, T ). T We observe that by H (B) the operator B is linear and ‖B v‖2V ∗ = 0 ‖B(v(t ))‖2V ∗ dt ≤ ‖B‖2L(V ;V ∗ ) ‖v‖2V , so B is bounded. Thus B is weakly continuous and by (48) we get ∗

∗

B un → B u weakly in V ∗ .

(57)

Step 2: Initial conditions. Remembering that un (0) = u0n → u0 in V , and moreover that we have un (0) → u0 weakly in V , by (53) we have u(0) = u0 . Furthermore since u′n (0) = u1n → u1 in H, moreover u′n (0) = u1n → u1 weakly in H and by (54) we get u′ (0) = u1 . Thus u satisfies the initial conditions of (33). Step 3: Passing to the limit in Galerkin problems. If we take v ∈ n=1 Vn then there exists m ∈ N such that (35) holds for all n ≥ m. We multiply (35) by h ∈ C ∞ ([0, T ]) and integrate over (0, T ) to get

∞

T

∫

⟨u′′n (t ), h(t )v⟩ dt + 0

T

∫ 0

∫ + 0

T

(Du Jn (t , γ¯ un (t )), h(t )γ¯ v)L2 (ΓC ;RN ) dt =

T

∫

⟨A(t , u′n (t )), h(t )v⟩ dt +

⟨Bun (t ), h(t )v⟩ dt 0

∫

T

⟨f (t ), h(t )v⟩ dt .

0

If we define V ∋ w := h(·)v then the above equation can be equivalently written as

⟨u′′n , w⟩V ∗ ×V + ⟨Au′n , w⟩V ∗ ×V + ⟨B un , w⟩V ∗ ×V + (Du Jn (·, γ¯ un (·)), γ¯ w)L2 (0,T ;L2 (ΓC ;RN )) = ⟨f , w⟩V ∗ ×V . Passing to the limit in the above equation we get

⟨u′′ , w⟩V ∗ ×V + ⟨ζ , w⟩V ∗ ×V + ⟨B u, w⟩V ∗ ×V + (η, γ¯ w)L2 (0,T ;L2 (ΓC ;RN )) = ⟨f , w⟩V ∗ ×V .

(58)

P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

Now since {h(t )v : h ∈ C ∞ ([0, T ]), v ∈ u′′ + ξ + B u + γ¯ ∗ η = f

∞

n=1

1177

Vn } is dense in V we get

in V ∗ ,

(59)

or, equivalently, u′′ (t ) + ζ (t ) + Bu(t ) + γ¯ ∗ η(t ) = f (t )

in V ∗ for a.e. t ∈ (0, T ).

(60)

Step 4: Passing to the limit in the nonlinear term. In order to prove that u solves (33) it suffices to show that ζ (t ) = A(t , u (t )) for a.e. t ∈ (0, T ) or, equivalently, that ζ = Au′ in V ∗ . By Lemma 7(e) from [26] and (49)–(51) to establish the above fact it is enough to show that ′

lim sup⟨Au′n , u′n − u′ ⟩V ∗ ×V = lim sup⟨Au′n , u′n ⟩V ∗ ×V − ⟨ξ , u′ ⟩V ∗ ×V ≤ 0. n→∞

n→∞

The required thesis is obtained by the standard argument analogous to the argument in the proof of Theorem 6 in [26].



4.4. The decay of energy We define the following energy functional suitable for the problem (33): 1 1 ⟨Bu(t ), u(t )⟩ + ‖u′ (t )‖2H + J (t , γ¯ u(t )). 2 2 Similarly, for the solutions of the Galerkin problem (34), we define the approximate energy E (t ) =

En (t ) =

1

(61)

1

⟨Bun (t ), un (t )⟩ + ‖u′n (t )‖2H + Jn (t , γ¯ un (t )).

2 2 Note that both E (t ) and En (t ) are defined for t ∈ [0, T ]. First we prove a simple lemma that relates En (t ) to E (t ).

(62)

Lemma 13. Let H (A), H (B), H (j) and H0 hold. If un solve (34) then there exists u which is a solution of (33) and the increasing sequence of natural numbers {nk }∞ k=1 such that E (t ) ≤ lim infk→∞ Enk (t ) for every t ∈ (0, T ). Remark. Observe that we cannot expect the uniqueness of the solution of (33). It is only proved that the energy of the solution obtained during the limiting process satisfies the desired property. Proof. Let u be the solution to (33) constructed in the proof of Theorem 12. There exists the subsequence unk (we denote it by un ) such that (53)–(56) hold. We estimate lim inf En (t ) = lim inf n→∞



2

n→∞

≥

1 2

1

1

⟨Bun (t ), un (t )⟩ + ‖u′n (t )‖2H + Jn (t , γ¯ un (t ))



2

lim inf⟨Bun (t ), un (t )⟩ + n→∞

1 2

lim inf ‖u′n (t )‖2H + lim Jn (t , γ¯ un (t )). n→∞

(63)

n→∞

By the monotonicity of B we have lim infn→∞ ⟨Bun (t ), un (t )⟩ ≥ ⟨Bu(t ), u(t )⟩. By Lemma 8 we get the thesis lim inf Enk (t ) ≥ k→∞

1 2

1

⟨Bu(t ), u(t )⟩ + ‖u′ (t )‖2H + J (t , γ¯ u(t )) = E (t ). 

(64)

2

In the following lemma we establish the first required estimate. Lemma 14. Let f = 0. Under assumptions H(A), H(B), H(j)(i)–(iii), (v) and H0 , if un is the solution to (34), then for all 0 ≤ t1 < t2 ≤ T the following estimate holds: En (t2 ) − En (t1 ) ≤ −α‖u′n ‖2L2 (t ,t ;V ) . 1 2

(65)

Proof. Taking the duality in (34) with u′n (t ) we get for a.e. t ∈ (0, T )

⟨u′′n (t ), u′n (t )⟩ + ⟨A(t , u′n (t )), u′n (t )⟩ + ⟨Bun (t ), u′n (t )⟩ + (Du Jn (t , γ¯ un (t )), γ¯ u′n (t ))L2 (ΓC ;RN ) = 0. Integrating the above equation over (t1 , t2 ) and using the identities ⟨Bv(t ), v(t )⟩ as well as the coercivity of A and Lemma 6 we get

∫

t2 t1

d dt



1 2

1

‖un (t )‖ + ⟨Bun (t ), un (t )⟩ + Jn (t , γ¯ un (t )) ′

2 H

2



1 d 2 dt

‖v(t )‖2H = ⟨v ′ (t ), v(t )⟩ and ⟨Bv(t ), v ′ (t )⟩ =

dt + α‖u′n ‖2L2 (t ,t ;V ) ≤ 1 2

∫

t2 t1

∫ ΓC

1 d 2 dt

∂ jn (x, t , γ¯ un (x, t )) dΓ dt . ∂t

Observing that, by H(j)(v)2 , the right hand side of the above inequality is nonpositive, we get the thesis.



1178

P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

Lemma 15. Let f = 0. Under assumptions H(A)(i), (ii)1 , (iii), H(B)1 , H(j)(i)–(iii), (vi)1 and H0 , there exists n0 ∈ N, constants C1 , C2 > 0 and the positive decreasing sequence bn → 0 independent of T such that if n ≥ n0 and un is the solution to (34) then the following estimate holds for any 0 ≤ t1 < t2 ≤ T : C1 ‖un ‖2L2 (t ,t ;V ) ≤ bn (t2 − t1 ) + ‖u′n (t1 )‖H ‖un (t1 )‖H + ‖u′n (t2 )‖H ‖un (t2 )‖H + C2 ‖u′n ‖2L2 (t ,t ;V ) . 1 2 1 2 Proof. Let us take the duality in (34) with un (t ). Using the identity ⟨u′′ (t ), u(t )⟩ = we have d dt

d dt

(66)

(u′ (t ), u(t ))H − ‖u′ (t )‖2H and H (B)1

(u′n (t ), un (t ))H − ‖u′n (t )‖2H + d‖un (t )‖2 + (Du Jn (t , γ¯ un (t )), γ¯ un (t ))L2 (ΓC ;RN ) ≤ ‖A(t , u′n (t ))‖V ∗ ‖u(t )‖.

Assuming without loss of generality that in H(j)(vi)1 we have µ ≥ 0 and taking advantage of H(A)(ii)1 and Lemma 9 we get with an and bn as in Lemma 9

(d − (µ + an )β 2 ‖γ¯ ‖2 )‖un (t )‖2 − bn ≤ −

d dt

(u′n (t ), un (t ))H + ‖u′n (t )‖2H + b‖u′n (t )‖ ‖un (t )‖.

(67)

Now we integrate the above inequality over the interval (t1 , t2 ). We get

(d − (µ + an )β 2 ‖γ¯ ‖2 )‖un ‖2L2 (t

1 ,t2 ;V )

≤ bn (t2 − t1 ) + ‖u′n (t1 )‖H ‖un (t1 )‖H + ‖u′n (t2 )‖H ‖un (t2 )‖H + ‖u′n ‖2L2 (t

1 ,t2 ;H )

+ b‖u′n ‖L2 (t1 ,t2 ;V ) ‖un ‖L2 (t1 ,t2 ;V ) .   Let us choose n0 such that for n ≥ n0 we have an < 21 β 2 ‖dγ¯ ‖2 − µ . Then, for n ≥ n0 , using the Cauchy inequality, we have 1 2

(d − µβ 2 ‖γ¯ ‖2 )‖un ‖2L2 (t

1 ,t2 ;V )

≤ bn (t2 − t1 ) + ‖u′n (t1 )‖H ‖un (t1 )‖H + ‖u′n (t2 )‖H ‖un (t2 )‖H + ‖u′n ‖2L2 (t +

which gives the thesis.

b2 d − µβ ‖γ¯ ‖ 2

2

‖u′n ‖2L2 (t

1

1 ,t2 ;V )

+ (d − µβ 2 ‖γ¯ ‖2 )‖un ‖2L2 (t 4

1 ,t2 ;V )

1 ,t2 ;H )

,

(68)



We are in a position to establish the theorem on the decay of the approximate energy. Theorem 16. Let f = 0 and un solve (34). Under assumptions H(A)(i), (ii)1 , (iii), H(B)1 , H(j)(i)–(iii), (v), (vi) and H0 , there exist n0 ∈ N, the positive and decreasing sequence gn → 0 and constants C > 0, λ > 0 independent of T such that for all natural n > n0 and t ∈ (0, T ) we have En (t ) ≤ En (0)C exp(−λt ) + gn . Proof. First we observe that H (B)1 together with the lower bound in H(j)(vi)2 gives us a lower bound for the approximate energy. Without loss of generality we assume that χ ≥ 0 in H(j)(vi)2 . We have for a.e. x ∈ ΓC and for a.e. t ∈ (0, T ) jn (x, t , ξ ) =

∫

pn (z )j(x, t , ξ − z ) dz ≥ −χ Kn

   N 1 (1 + ϵ)|ξ |2 + 2 1 + , n ϵ

where ϵ is an arbitrary positive constant. Thus for a.e. t ∈ (0, T ) En ( t ) =

≥ Taking ϵ =

1 2 1 2

(⟨Bun (t ), un (t )⟩ + ‖u′n (t )‖2H ) + Jn (t , γ¯ un (t )) (d − 2χ β 2 ‖γ¯ ‖2 (1 + ϵ))‖un (t )‖2 − χ mN −1 (ΓC )

d−2χβ 2 ‖γ¯ ‖2 4χ 2 β 2 ‖γ¯ ‖2

N n2

 1+

1

ϵ



.

(69)

and en = χ mN −1 (ΓC ) nN2 1 + 1ϵ we obtain for a.e. t ∈ (0, T ) with the constant C3 > 0





En (t ) ≥ C3 ‖un (t )‖2 − en .

(70)

(t ) = En (t ) − En (t + 1) for t ∈ (0, T ). By Lemma 14 the functions Fn are well defined and for C4 > 0 we have ∫ t +1 t +1 ‖u′n (s)‖2 ds ≤ C4 Fn2 (t ) and ‖u′n (s)‖2H ds ≤ C4 Fn2 (t ). (71)

Now we set

∫

Fn2

t

t

Due to the fact that un ∈ C ([0, T ]; H ) there exist t1 ∈ t , t + ′



1 4

  ‖u′n (t1 )‖H ≤ 2 C4 Fn (t ) and ‖u′n (t2 )‖H ≤ 2 C4 Fn (t ).



and t2 ∈ t + 34 , t + 1 such that (70) holds in t1 , t2 and





(72)

P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

1179

Now we estimate the energy from above. For t ∈ (0, T ) we have En (t ) = Fn2 (t ) + En (t + 1) ≤ Fn2 (t ) + En (t2 ) ≤ Fn2 (t ) + 2(t2 − t1 )En (t2 )

≤ Fn2 (t ) + 2

∫

t2

t2

∫

En (s) ds ≤ Fn2 (t ) +

‖u′n (s)‖2H ds

t1

t1

∫

t2

+ ‖B‖L(V ;V ∗ )

t2

∫

‖un (s)‖ ds + 2 2

Jn (s, γ¯ un (s)) ds.

(73)

t1

t1

Using H(j)(vi)(b) we get for all ξ ∈ RN , for a.e. x ∈ ΓC and for a.e. t ∈ (0, T ), j n ( x, t , ξ ) =

∫

pn (z )j(x, t , ξ − z ) dz ≤

Kn

≤ 2ν

∫

pn (z )ν|ξ − z |2 dz Kn

∫

pn (z )(|ξ |2 + |z |2 ) dz ≤ 2ν



|ξ |2 +

Kn

N n2



.

(74)

Thus for u ∈ L2 (ΓC ; RN ) and for a.e. t ∈ (0, T ), Jn (t , u) ≤ 2ν‖u‖2L2 (Γ ;RN ) + C

2ν NmN −1 (ΓC ) n2

,

and

∫

t2

Jn (s, γ¯ un (s)) ds ≤ 2νβ ‖γ¯ ‖ 2

t1

2

∫

t2

‖un (s)‖2 ds +

2ν NmN −1 (ΓC ) n2

t1

(t2 − t1 ).

We take n0 as in Lemma 15 and n > n0 . The last inequality and (73), by Lemma 15 and (71), gives for all t ∈ (0, T ) En (t ) ≤ C5 Fn2 (t ) + dn + C6 (‖u′n (t1 )‖H ‖un (t1 )‖ + ‖u′n (t2 )‖H ‖un (t2 )‖)

(75)

with some constants C5 , C6 > 0 and the positive decreasing √ sequence dn → 0 independent of T , un , t1 , t2 . Applying (72) we get for all t ∈ (0, T ) the inequality En (t ) ≤ C5 Fn2 (t ) + 2 C4 C6 Fn (t )(‖un (t1 )‖ + ‖un (t2 )‖) +√dn .

√ (t ) En (t ) + en + dn . By the Cauchy inequality we get En (t ) ≤ C7 Fn2 (t ) + fn with C7 > 0 and fn = en + 2dn . Lemma 1 yields En (t ) ≤ |En (0) − En (1)|C exp(−λt ) + fn with C , λ > 0 independent of T , n, t. By Lemma 14, we get En (t ) ≤ (En (0) − En (1))C exp(−λt ) + fn , and by (70), En (t ) ≤ En (0)C exp(−λt ) + gn with gn → 0 positive and decreasing. This completes the proof.  By (70) for t ∈ (0, T ) and Lemma 14 for all t ∈ (0, T ) we have En (t ) ≤ C5 Fn2 (t ) +

4 C4 C6 √ Fn C3

As a consequence of Lemma 13 and Theorem 16 we formulate: Theorem 17. If f = 0 and assumptions H(A)(i), (ii)1 , (iii), (iv), H(B)1 , H(j)(i)–(iii), (v), (vi) and H0 hold then there exist constants C > 0, λ > 0 independent of T such that if u is the solution to the problem (33) obtained by Theorem 12 (as a limit of Galerkin approximations) then for t ∈ (0, T ) we have E (t ) ≤ E (0)C exp(−λt ). Proof. By Lemma 13 we have E (t ) ≤ lim infk→∞ Enk (t ). By Theorem 16 this yields E (t ) ≤ lim inf(Enk (0)C exp(−λt ) + gnk ) = C exp(−λt ) lim inf Enk (0). k→∞

Since Enk (0) → E (0), the thesis holds.

k→∞



Remark. In Theorem 17 only the asymptotic properties of the solutions obtained from the Galerkin procedure are proved. It is not known whether there are solutions which are not limits of Galerkin approximations, and no decay for such solutions is obtained. 5. Discussion and examples We show and discuss examples of one-dimensional multivalued laws that satisfy our assumptions. For the sake of simplicity we assume that j does not depend on x and t directly and we set j(0) = 0. If this is not the case, then the superpotential can be shifted to zero without effects on its subgradients, i.e. one can take ¯j(ξ ) := j(ξ ) − j(0) in place of j. The assumption H(j)(iii) holds if the graph of the subdifferential ∂ j does not intersect the interior of the marked area in the first column (labelled Existence) in Fig. 1. The slope of the lines and the location of the vertex are arbitrary. Then the solution exists on an arbitrarily long time interval for the cases of a multivalued term dependent on u and of one dependent on u′ . The second column in Fig. 1 summarizes the assumptions needed for asymptotic decay of energy. For the cases of the multivalued term depending on u′ and on u the elasticity operator must be coercive (H (B)1 ) and the viscosity term must satisfy the assumption H(A)(ii)1 . As for the multivalued term, the assumption H(j)(iii)1 in the case of dependence on u′ (and the upper bound in H(j)(vi)2 in the case of dependence on u) is required to hold. This is the case if the graph of ∂ j does not

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P. Kalita / Nonlinear Analysis 74 (2011) 1164–1181

Fig. 1. Graphs summarizing the assumptions required for the existence/decay of energy. The solution exists (first column) or the energy decays to zero exponentially (second column) provided the graph of ∂ j does not intersect the interior of the marked area.

intersect the area marked with horizontal lines. The slope that bounds this area is arbitrary (it is given by the constant c in the case of H(j)(iii)1 and ν in the case of H(j)(vi)2 ). Assumptions H(j)(iv) (the case of HVI depending on u′ ) or H(v)(vi)1 and the lower bound in H(v)(vi)2 (the case of HVI depending on u) hold if the graph of ∂ j does not intersect the area marked by vertical lines. In the case of dependence on the velocity the slope of the line that bounds this area is controlled by the coercivity constant of A. In the case of the multivalued term dependent on u the slope of the line depends on the coercivity constant of B. The assumptions have physical significance. If the boundary force represented by the multivalued law (this force belongs to −∂ j) is directed opposite to the velocity (resp., displacement) or, more generally, the angle between force and velocity (resp., displacement) is obtuse (which means that the boundary force is actually the resistance force) then the graph of ∂ j must be separated from the vertical axis ξ = 0 by the straight lines g (ξ ) = ±c ξ (where the constant c is arbitrary). The decay of energy can still hold. However, suppose the boundary force is directed in the same way as the velocity (resp., displacement) or, more generally, the respective angle is acute. In that case the projection of the boundary force on the direction of the velocity (resp., displacement) must not be too large, which is controlled by the coercivity constant of the viscosity (resp., elasticity) operator and geometrical properties of the domain. Roughly speaking, the boundary force can be possibly enforcing as long as it does not spoil the ‘good’ properties of the viscosity (resp., elasticity) force. Now we look at some typical one-dimensional examples of multivalued laws used in contact models to check whether they satisfy the proposed assumptions. (1) The (simplified) Coulomb law of friction and the Tresca law of friction (see Examples 2 and 4 in [21]). Here ∂ j(0) is a ball which has some positive radius and is centered around 0. Since for the decay of energy it is required that ∂ j(0) = {0} the assumptions are not satisfied. This is the case even if we have the simplest convex law j(u) = |u| since there is the adhesion area where nonzero force is allowed where there is no displacement (or velocity). For friction laws without an adhesion area (see Fig. 4.6.5 [22]) the assumptions are satisfied and we have the decay of energy. The argument presented in this article does not show what happens if int ∂ j(0) ̸= ∅. (2) The Winkler contact law (see Example 1 in [21]) and the zigzag laws describing interlaminar forces in sandwich plates (see [22], Section 3.5.2). The laws are nonconvex and the assumptions are satisfied since the graph of the subdifferential is contained in the first and third quadrants and separated from the vertical axis. Acknowledgement The author would like to thank the referee for valuable comments. References [1] M. Nakao, Decay of solutions of some nonlinear evolution equations, Journal of Mathematical Analysis and Applications 60 (1977) 542–549. [2] M. Nakao, Energy decay for the quasilinear wave equation with viscosity, Mathematische Zeitschrift 219 (1995) 289–299.

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