On existence of solutions for parabolic hemivariational inequalities

Journal of Computational and Applied Mathematics 129 (2001) 77–87 www.elsevier.nl/locate/cam

On existence of solutions for parabolic hemivariational inequalities  Stanis law Mig)orski Faculty of Mathematics and Physics, Institute of Computer Science, Jagiellonian University, ul. Nawojki 11, 30072 Cracow, Poland Received 23 February 1999

Abstract A class of parabolic hemivariational inequalities is considered in which the energy function is locally Lipschitz (not necessarily convex or di1erentiable) and the nonlinear evolution operator is multivalued and generalized pseudomonotone. Using techniques of multivalued analysis and a surjectivity result for L-generalized pseudomonotone operators, we prove the existence for a Cauchy problem and a periodic problem. Some applications to nonsmooth mechanics are also indicated. c 2001 Elsevier Science B.V. All rights reserved.
Keywords: Hemivariational inequality; Clarke subdi1erential; Evolution triple; L-generalized pseudomonotone operator; Nonconvex; Coercive operator; Compact embedding

1. Introduction In this paper, we study the problem of existence of solutions for the following evolution inclusion: du + Au + @J(u)
f; dt

u(0) = u0

(P)

driven by multivalued, coercive, generalized pseudomonotone operator A deAned within the framework of an evolution triple of spaces. The multivalued perturbation term is assumed to be in the form of the Clarke subdi1erential of a locally Lipschitz function J. The problem (P) can be considered as a multivalued version of a parabolic hemivariational inequality.  Expanded version of a talk presented at the International Conference on Nonlinear Programming and Variational Inequalities, Hong Kong, December 15 –18, 1998. E-mail address: [email protected] (S. Mig)orski).

c 2001 Elsevier Science B.V. All rights reserved. 0377-0427/01/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 0 0 ) 0 0 5 4 3 - 4

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Our e1orts here are prompted by the growing literature on the mathematical models for mechanical problems with nonconvex and nonsmooth energy functions. The lack of convexity and di1erentiability in boundary value problems in mechanics and engineering leads to new types of variational expressions called hemivariational inequalities. The latter have been introduced in (cf. [18,19]) and since that time they have proved to be an e1ective tool for the study of both practical and theoretical problems (cf. [7,8]). For the description of various mechanical problems which can be formulated as hemivariational inequalities as well as applications, we refer to [18,20,16]. It is worth mentioning here that in the context of elliptic systems, the problem of the existence of solutions to hemivariational inequalities was studied by many authors using di1erent methods. Namely, Rauch [24] used the molliAcation and regularization techniques, while Chang [4], and Costa and GonMcalves [6] based their approach on the critical point theory for nondi1erentiable functions. Moreover, the Galerkin method was applied by Panagiotopoulos [19,20], the Anite element approximation has been considered in [13], and the theory of pseudomonotone mappings of Browder and Hess [2] has been employed by Naniewicz and Panagiotopoulos [16]; see also references therein. The evolution hemivariational inequalities have been studied only recently. The existence results in the parabolic case have been obtained by Miettinen [12], who used a regularization technique and the Galerkin method, by Carl [3] (where Rauch’s method was adapted) and Papageorgiou [21], who combined the method of upper and lower solutions, the theory of pseudomonotone operators with truncation and penalization techniques. For the parabolic problem, an existence result based on a regularized approximating method can be also found in [14]. The goal of this note is to extend the above-mentioned works to the case of pseudomonotone multivalued operators considered within the setting of an evolution triple of spaces. We treat problem (P) with a general locally Lipschitz functional J which is deAned on inAnite-dimensional space and which satisAes a directional (unilateral) growth condition (cf. [16,15]). Our proof is based on a general surjectivity result for a sum of operators of monotone type and on a compactness argument. The outline of this paper is as follows. In Section 2, we present a boundary value problem which comes from the theory of semipermeable potentials and can be formulated in terms of hemivariational inequalities. In Section 3, after recalling basic notation, deAnitions and preliminary results, we formulate the problem under consideration. Section 4 contains the main existence result for problem (P). We will also show the weak compactness of the solution set of our problem. Finally, we elaborate and comment on the main hypotheses that we have used in our approach.

2. Physical motivation The physical motivation of the paper comes from the so-called nonsmooth and nonconvex problems for semipermeable media. Such problems were considered in [9], where semipermeability relations were assumed to be monotone and they lead to variational inequalities. The case of nonmonotone semipermeability relations was Arst studied by Panagiotopoulos [17] in the stationary case under the name of hemivariational inequalities. The semipermeability conditions are realized by various types of membranes, natural and artiAcial ones. These conditions arise in hydraulics, electrostatics and Pow through porous media (cf. [18,20]), here we formulate an interior semipermeability problem in the language of heat conduction.

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79

Fig. 1. Semipermeablity relations.

Let  be an open bounded subset of R3 and 0 ¡ T ¡ + ∞. We consider the heat equation @u − u = f in  × (0; T ): @t We assume that f = f1 + f2 , where f2 is prescribed and − f1 (x; t) ∈ @j(x; u(x; t))

in  × (0; T ):

(1)

(2)

Moreover, suppose for simplicity that u=0

on @ × (0; T )

(3)

and u(x; 0) = u0 (x)

in :

(4)

Here j = j(x; ) is a locally Lipschitz energy function which is generally nonsmooth and nonconvex, and @j denotes its generalized (Clarke’s) gradient in the second variable (see [5]). It is known that (2) describes the behaviour of a semipermeable membrane of Anite thickness occupying  or the behaviour of temperature controller producing heat in  in order to regulate the temperature in the interior of . For instance, in Fig. 1(a), if the temperature u ¡ h1 the region  supplies constant heat per unit volume, say c. When u = h1 the heat is supplied (for constant temperature) until a value d is reached. Then the relation between the supplied heat and temperature follows the parabola until h2 is reached. At temperature u = h2 the heat is changed from value −a to −b, and then the heat supply remains constant whereas the temperature u may increase. Similarly, the graph of Fig. 1(b) corresponds to a temperature control problem when we regulate the temperature to deviate as little as possible from the interval [h1 ; h2 ] (see [17,16] for further details).

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Taking into account the deAnition of generalized gradient (see the next section), the weak formulation of (1) – (4) is the following hemivariational inequality: And u :  × (0; T ) → R such that u
(t); v − u(t)V ×V ∗ + − u(t); v − u(t)V ×V ∗ +






j 0 (x; u(x; t); v − u(x; t)) d x¿f2 (t); v − u(t)V ×V ∗

for a:e: t ∈ (0; T );

∀v ∈ V = H01 ();

u(0) = u0 : 3. Preliminaries and problem statement In this section, we Ax our notation and recall some of the basic deAnitions and facts from the multivalued analysis and the theory of monotone operators. For details, we refer to [11,26,10]. Let V be the rePexive Banach space and let  be an open bounded subset of RN with Lipschitz boundary @. We suppose that the embedding V ⊂ H = Lp (; RN ) is compact for some 26p ¡ ∞. Typically, V = W 1; p (; RN ) or V = W01; p (; RN ). Identifying X = L2 (; RN ) with its dual, we have the following Gelfand Avefold: V ⊂ H ⊂ X ⊂ H ∗ ⊂ V ∗; where all embeddings are dense and continuous. We denote by ·; · the duality of V and V ∗ as well as the pairing between H and H ∗ , by · ; | · | and · V ∗ the norms in V; X and V ∗ , respectively. Given a Axed real number 0 ¡ T ¡ + ∞, set Q =  × (0; T ). We introduce the spaces V = p L (0; T ; V ); H=Lp (0; T ; H ); X=L2 (0; T ; X ); H∗ =Lq (0; T ; H ∗ ); V∗ =Lq (0; T ; V ∗ ); (1=p+1=q=1) and W = {w ∈ V | w
∈ V∗ }, where the time derivative is understood in the sense of vector-valued distributions. Clearly, we have W ⊂ V ⊂ H ⊂ X ⊂ H ∗ ⊂ V∗ with continuous embeddings. Since we have assumed that V ⊂ H compactly, we have also that W ⊂ H compactly. Moreover, the embedding W ⊂ C(0; T ; X ) is continuous (see  T [11,26]). The pairing of V and V∗ and the duality between H and H∗ are denoted by f; v= 0 f(s); v(s) ds. With weak-V we denote the space V equipped with the weak topology. We recall the deAnitions of the generalized directional derivative and the generalized gradient of Clarke for a locally Lipschitz function g : E → R, where E is a Banach space (see [5]). The generalized directional derivative of g at x in the direction v, denoted by g0 (x; v), is deAned by g0 (x; v) = lim sup y→x; t↓0

g(y + tv) − g(y) : t

The generalized gradient of g at x, denoted by @g(x), is a subset of a dual space E ∗ given by @g(x) = {$ ∈ E ∗ : g0 (x; v)¿$; vE×E ∗ for all v ∈ E}: Let Z1 ; Z2 be the Hausdor1 topological spaces. A multifunction G : Z1 → 2Z2 \{∅} is said to be upper semicontinuous (usc), if for all C ⊆ Z1 closed, the set G − (C) = {z ∈ Z1 : G(z) ∩ C = ∅} is closed.

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81 ∗

Let Y be a rePexive Banach space. Recall (see [2,16]) that a mapping T from Y into 2Y is said to be the generalized pseudomonotone if the following holds: the conditions {yn } ⊂ Y; yn → y weakly in Y; yn∗ ∈ Tyn ; yn∗ → y∗ weakly in Y ∗ and lim supyn∗ ; yn Y ×Y ∗ 6y∗ ; yY ×Y ∗ imply y∗ ∈ Ty and yn∗ ; yn Y ×Y ∗ → y∗ ; yY ×Y ∗ . We say that T is coercive, if there exists a function c : R+ → R with limr→∞ c(r) = ∞ such that for all y ∈ Y and y∗ ∈ Ty, we have y; y∗ ¿c( y ) y or equivalently, inf {y; y∗ : y∗ ∈ Ty} → +∞ y

as y → ∞: ∗

Let L : D(L) ⊂ Y → Y ∗ be a linear densely deAned maximal monotone operator and let T : Y → 2Y . Also, we say that T is L-generalized pseudomonotone (or generalized pseudomonotone with respect to D(L)) if (a) for every y ∈ Y; Ty is a nonempty, convex and weakly compact set in Y ∗ , (b) T is usc from every Anite-dimensional subspace F of Y into weak-Y ∗ , (c) if {yn } ⊂ D(L); yn → y weakly in Y; y ∈ D(L); Lyn → Ly weakly in Y ∗ ; yn∗ ∈ Tyn ; yn∗ → y∗ weakly in Y ∗ and lim supyn∗ ; yn Y ×Y ∗ 6y∗ ; yY ×Y ∗ , then y∗ ∈ Ty and yn∗ ; yn Y ×Y ∗ → y∗ ; yY ×Y ∗ . The following surjectivity result for L-generalized pseudomonotone operators will be used in the sequel. Proposition 1. If Y is a re3exive; strictly convex Banach space; L : D(L) ⊂ Y → Y ∗ is a lin∗ ear densely de6ned maximal monotone operator and T : Y → 2Y is bounded; coercive and L-generalized pseudomonotone operator; then R(L + T ) = Y ∗ . The proof of Proposition 1 can be found in [11, Theorem 1:2, p. 319] or [25, Corollary 1, p. 610] for T single-valued and in [22] for T multivalued. We pass to the mathematical formulation of the problem. Let A be an operator between V and V∗ , let j : Q × RN → R; f ∈ V∗ and u0 ∈ V . We consider the following hemivariational inequality: And u ∈ W such that u
+ Au; v − u +




Q

j 0 (x; t; u(x; t); v(x; t) − u(x; t)) d x dt¿f; v − u for all v ∈ V;

u(0) = u0 :

(5)

Our goal is to prove the existence result for a problem which is more general than (5). The main result (Theorem 5) implies, in particular, that (5) admits a solution. In order to formulate a more general problem, we introduce the functional J: H → R deAned by J(v) =




Q

j(x; t; v(x; t)) d x dt:

We admit the following assumption. H (j): j : Q × RN → R satisAes the following conditions: (1) for all  ∈ RN , the function (x; t) → j(x; t; ) is measurable on Q, (2) for a.e. (x; t) ∈ Q, the function  → j(x; t; ) is locally Lipschitz on RN , (3) j(·; ·; ) ∈ L1 (Q),

(6)

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(4) Linear growth condition: there exists c¿0 such that for a.e. (x; t) ∈ Q and each  ∈ RN , for every element . of @ j(x; t; ), we have |.|6c(1 + ||p−1 ). Remark 2. It is well known (see [5]) that under the hypothesis H (j), the functional J given by (6) is well deAned, locally Lipschitz (and in fact, Lipschitz continuous on every bounded subset of H), and for u ∈ H and / ∈ H∗ such that / ∈ @J(u), we have /(x; t) ∈ @ j(x; t; u(x; t))

a:e: (x; t) ∈ Q:

Moreover, we observe that using Fatou’s lemma, we have J0 (u; v)6




Q

j 0 (x; t; u(x; t); v(x; t)) d x dt

for u; v ∈ H:

(7)

Now, we consider the following problem: And u ∈ W such that u
+ Au − f; v − u + J0 (u; v − u)¿0 u(0) = u0 ;

for all v ∈ V;

which can also be equivalently written in an abstract form as: And u ∈ W such that u
+ Au + @J(u)
f; u(0) = u0 :

(8)

If J is given by (6) with j : Q × RN → R satisfying H (j), then we deduce by (7) that every solution to (8) is also a solution of (5). Therefore, in the following we restrict ourselves to the study of the hemivariational inequality (8). Remark 3. It can be observed that if the function j satisAes H (j) and it is regular in the last variable (cf. [5]) for a.e. (x; t) ∈ Q, then the problems (5) and (8) are equivalent. This follows from the fact that in such a case we have equality in (7). 4. Main result In this section, we will prove the existence of solutions to the following abstract operator evolution inclusion: And u ∈ W such that u
+ Au + @J(u)
f; (9) u(0) = u0 ; ∗

where A: V → 2V is a multivalued map. Let the operator L : D(L) ⊂ V → V∗ be deAned by Lv = v


with D(L) = {v ∈ W: v(0) = 0}:

(10)

Recall that L is linear, densely deAned maximal monotone operator (see for example [26, Chapter 32] or [10, Chapter 3.9.2, p. 419]). The hypotheses on the data of (9) are the following. ∗ H (A): A : V → 2V is a multivalued operator with weakly compact and convex values, which is bounded, generalized pseudomonotone and coercive.

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H (J): J : H → R is Lipschitz continuous on each bounded subset of H and there exists k¿0 such that J0 (v; −v)6k(1 + v H )

for all v ∈ H:

(11)

Remark 4. It is well known (cf. e.g. [2, Proposition 4]) that the hypothesis H (A) is equivalent to the fact that A with weakly compact values is pseudomonotone. Theorem 5. If hypotheses H (A); H (J) hold; f ∈ V∗ and u0 ∈ V; then the problem (9) has a solution. ∗ Proof. Performing a translation by the initial condition, we deAne the operators Aˆ : V → 2V and  : H → 2H∗ by (Av)(·)  ˆ @J = A(v(·) + u0 ) and @J(v)(·) = @J(v(·) + u0 ), respectively. Consider the inclusion

Lz + Tz
f;

z ∈ D(L);

(12)

∗  ˆ + @J(z). We show the existence of solutions where the operator T : V → 2V is given by Tz =Az to (12). To this end we apply Proposition 1. Claim 1: T is a bounded operator (it maps bounded subsets of V into bounded subsets of V∗ ). This is an immediate consequence of the boundedness of Aˆ and of the fact that @J (and hence  is a bounded mapping from H into 2H∗ . also @J) Claim 2: T is coercive.  ⊂ H∗ . By the deAnition of generalized gradient, from (11) and from the Let v ∈ H and v∗ ∈ @J subadditivity of the function h → J0 (v; h) (see, e.g., [5]), we obtain

−v∗ ; v 6 J0 (v + u0 ; −v)6J0 (v + u0 ; −(v + u0 )) + J0 (v + u0 ; u0 ) 6 k(1 + v + u0 H ) + c u0  is subcoercive, i.e., with a suitable c ¿ 0. Hence, @J

v∗ ; v¿ − k1 (1 + v H + u0 ) − c u0 :  is coercive. By using the hypothesis H (A) one readily veriAes that A + @J Claim 3: T is L-generalized pseudomonotone operator.  are nonempty, weakly compact and convex subsets of H∗ , we easily get Since the values of @J that Tz is nonempty, weakly compact and convex in V∗ for all z ∈ V.  has a sequentially closed graph in H × (weak-H∗ ) topology (cf. [5]). Since The mapping @J it is also locally relatively weakly compact map in [23, Lemma 7:10] (or [10, Proposition 2:23,  is also usc in this topology. Hence, from the hypothesis H (A) it follows p. 43]), we know that @J that T is usc from every Anite-dimensional subspace of V into weak-V∗ .  ), ˆ n + @J(y Now let {y; yn } ⊂ D(L), yn → y weakly in V, Lyn → Ly weakly in V∗ , yn∗ ∈ Ay n yn∗ → y∗ weakly in V∗ and assume that lim supyn∗ ; yn 6y∗ ; y. Note that yn∗ = zn∗ + wn∗ with  ). Since W ⊂ H compactly, we suppose that ˆ n and wn∗ ∈ @J(y zn∗ ∈ Ay n

yn → y

in H:

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 is a bounded map, by passing to a subsequence, if necessary, we may assume that there Because @J ∗ exists w ∈ H∗ such that

wn∗ → w∗

weakly in H∗ :

 we have w∗ ∈ @J(y).  Observe next Again, by using the sequential closedness of the graph of @J, that

lim supzn∗ ; yn  + w∗ ; yH×H∗ = lim supzn∗ ; yn  n

n

+ limwn∗ ; yn H×H∗ = lim supzn∗ + wn∗ ; yn 6y∗ ; y; n

which gives

lim supzn∗ ; yn 6z ∗ ; y,

zn∗ = yn∗ − wn∗ → z ∗

n

∗

∗

∗

where z = y − w . We also have

weakly in V∗ :

ˆ and But from hypothesis, it follows that Aˆ is L-generalized pseudomonotone and so z ∗ ∈ Ay ∗ ∗ ∗ ∗ ∗ ∗  ˆ with w ∈ @J(y). Hence, y ∈ Ty and we zn ; yn  → z ; y. This means that y − w ∈ Ay have yn∗ ; yn  = zn∗ ; yn  + wn∗ ; yn H×H∗ → z ∗ ; y + w∗ ; yH×H∗ = y∗ ; y: Therefore, T is a generalized pseudomonotone operator with respect to D(L). Finally, by Proposition 1, we deduce that the problem (12) has at least one solution z ∈ D(L). It is clear now that u(t) = z(t) + u0 is the desired solution of the problem (9). The next proposition contains a priori estimates for solutions of the problem (9). Proposition 6. Under the hypotheses of Theorem 5; the solution set of (9) is a nonempty and weakly compact subset of W. Proof. Let S be a solution set of (9) and let {un } ⊂ S. We Arst show the a priori bounds for the solutions un . From the integration by parts formula concerning functions in W, it follows that Lun ; un  = 12 |un (T )|2X − 12 |un (0)|2X . As un
+ un∗ + gn∗ = f with un∗ ∈ Aun and gn ∈ @J(un ), exploiting the coercivity of A and the condition (11), we can write 



a ∗ u0 V + k + (k + f V ) un V ; c( un V ) un V 6 2 with c : R+ → R+ a function such that c(r) → ∞, as r → ∞ and a suitable constant a ¿ 0. Hence, we deduce that {un } is bounded in V uniformly with respect to n. Next since both A and @J are bounded operators, we infer from un
+ un∗ + gn = f that {un
} is bounded in V∗ . Thus, we have shown that {un } is bounded in W. Due to the weak compactness of a ball in the rePexive Banach space W, by passing to a subsequence if necessary, we assume that there exist u ∈ W such that un → u weakly in W. We will show that u ∈ S. From the proof of Theorem 5, we know that zn ∈ W, zn (t) = un (t) − u0 solves the problem  )
f; ˆ n + @J(z Lzn + Az n zn ∈ D(L);

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 are deAned as before. Moreover, we may assume that z → z weakly in W and where Aˆ and @J n also in H, and zn (t) → z(t) weakly in X , for every t ∈ [0; T ], where z(t) = u(t) − u0 (recall that W embeds compactly in H and continuously in C(0; T ; X )). We have

zn
; zn  + zn∗ ; zn  + wn∗ ; zn H×H∗ = f; zn ;

(13)

 ). Again, since Aˆ and @J  are bounded, we can suppose that z ∗ → z ∗ ˆ n and wn∗ ∈ @J(z with x∗n ∈ Az n n ∗ ∗ ∗ ∗ weakly in V and wn → w weakly in H . Because zn
+ zn∗ + wn∗ = f, in the limit, as n → ∞,  has a closed graph in H × (weak-H∗ ) topology, we have we obtain z
+ z ∗ + w∗ = f. Since @J  Note that by the Mazur’s lemma, the domain D(L) of L being closed and convex, is w∗ ∈ @J(z). a weakly closed subset of W. From this we see at once that z ∈ D(L). Subsequently, using once more the integration by parts formula and the weak lower semicontinuity of the norm, from (13), we have

lim supzn∗ ; zn 6f; z − w∗ ; zH×H∗ − 12 |z(T )|2X = z ∗ ; z: ˆ Thus, z ∈ W Since Aˆ is generalized pseudomonotone with respect to D(L), we deduce that z ∗ ∈ Az.
 ˆ is a solution of the problem z + Az + @J(z)
f, z(0) = 0, which implies that u solves (9), i.e., u ∈ S. Hence, the assertion of the proposition follows. Remark 7. If the operator A satisAes H (A) with the coercivity function c(r) ≈ r as r → ∞, then the condition (11) in H (J) can be replaced by a weaker one: 4

J0 (v; −v)6k(1 + v H )

for all v ∈ H;

(14)

 still holds and the problem (9) has at least with 164 ¡ p. In this case the coercivity of Aˆ + @J one solution.

If we deAne L1 : D(L1 ) ⊂ V → V∗ by L1 v = v
, where D(L1 ) = {v ∈ W: v(0) = v(T )}, then by Proposition 32:10 of Zeidler [26], L1 is linear densely deAned maximal monotone mapping. Using the same method as developed above, we have the following result. Theorem 8. If hypotheses H (A); H (J) hold and f ∈ V∗ ; then the problem u
+ Au + @J(u)
f; u(0) = u(T ) has at least one solution. We will consider now the problem (9) with the mapping A which belongs to a class of single-valued operators. The hypothesis H (A) given below is satisAed for divergence operators of classical Leray–Lions type, see [11,10,26]. H (A): A : (0; T ) × V → V ∗ is an operator such that (i) t → A(t; v) is weakly measurable, (ii) v → A(t; v) is demicontinuous and pseudomonotone, p−1 (iii) A(t; v) V ∗ 66(t) + c1 v , a.e. t ∈ (0; T ) with 26p ¡ ∞, 6 ∈ Lq (0; T ); 1=p + 1=q = 1; p

r

(iv) A(t; v); v¿c v − a v − 7(t), 16r6p − 1. Let A : V → V∗ be the Nemitsky operator corresponding to A(·; ·), i.e., (Av)(t) = A(t; v(t)).

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Corollary 9. Assume that the Nemitsky operator A is generated by A(·; ·) which satis6es H (A). If H (J) holds; f ∈ V∗ and u0 ∈ V; then the problem (9) admits a solution. Proof. The result may be proved in almost the same way as Theorem 5. We remark only two points.  with (Av)(t) ˆ = A(t; v(t) + u0 ) is usc from every Anite-dimensional In order to show that T =Aˆ + @J ∗ subspace of V into weak-V , we use the fact that under our hypothesis Aˆ : V → V∗ is demicontinuous (cf. [1, Lemma 1]). Moreover, as it was observed in [1] the operator Aˆ inherits all needed properties of A(·; ·). Therefore, in particular, from Theorem 2(b) of [1] (or [21, Proposition 1]), we infer that Aˆ is L-generalized pseudomonotone. Remark 10. The functional J deAned by (6) satisAes the hypothesis H (J), if for example, j : Q × RN → R satisAes H (j) and the following directional growth condition holds: there exists a nonnegative function 8 ∈ Lq (Q) such that for a.e. (x; t) ∈ Q j 0 (x; t; ; −)68(x; t)(1 + ||);

∀  ∈ RN :

(15)

In fact, condition (15) implies (11) since by (7) and the HSolder inequality, we have J0 (v; −v)6


Q

8(x; t)(1 + |v(x; t)|) d x dt6 8 Lq (1 + v H )

for each v ∈ H. Condition (15) was introduced in the elliptic case in [16, Chapter 4]. Remark 11. If the operator A satisAes H (A) with the coercivity function c(r) ≈ r as r → ∞, then condition (14) holds for the functional J of the form (6) even when (15) is replaced by a weaker condition j 0 (x; t; ; −)681 (x; t)(1 + ||4 );

(16)

with 164 ¡ p and a nonnegative function 81 ∈ Lp=(p−4) (Q). Example 12. The simple example of a function j which satisAes H (j), (15) and (16) is j() = min(h1 (); h2 ()); where hk : RN → R, k = 1; 2, are convex quadratic functions. It is easy to see that in this case for all  ∈ RN , we have j 0 (; −)60

and

|.|6const(1 + ||)

for . ∈ @j():

Acknowledgements This work was supported by the State Committee for ScientiAc Research of the Republic of Poland (KBN) under Research Grant No. 2 P03A 040 15. The author would like to thank Prof. Jason Zhang from City University of Hong Kong for the very nice organization of the conference.

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