Scholarly article on topic 'Global attractors for nonlinear wave equations with linear dissipative terms'

Global attractors for nonlinear wave equations with linear dissipative terms Academic research paper on "Mathematics"

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Academic research paper on topic "Global attractors for nonlinear wave equations with linear dissipative terms"

Pan et al. Boundary Value Problems (2015) 2015:16 DOI 10.1186/s13661-014-0276-2

0 Boundary Value Problems

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Global attractors for nonlinear wave equations with linear dissipative terms

Zhigang Pan1*, Dongming Yan2* and Qiang Zhang3

Correspondence: panzhigang@swjtu.edu.cn; 13547895541@126.com 'Schoolof Mathematics, Southwest Jiaotong University, Chengdu, 610031,China

2School ofMathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou, 310018, China

Fulllist of author information is available at the end of the article

Abstract

An initial boundary value problem of the semilinear wave equation of which the source term f(x, u) is without variational structure in a bounded domain is considered. Firstly, we prove that it has a unique globally weaksolution (u,ut) e C°([0,œ),Hjj(Q) x L2(Q)) by using our previous results (Pan etal. in Bound. Value Probl. 2012:42, 2012). Secondly, we obtain the existence of global attractors in Hjj(Q) x L2(Q) by using the «-limit compactness condition (Ma etal. in Indiana Univ. Math. J. 5(6):1542-1558, 2002), rather than the traditional method. MSC: 35B33; 35B41; 35L71

Keywords: dissipative terms; global attractor; «-limit compactness

ringer

1 Introduction

In this paper we are concerned with the existence of global attractors for nonlinear wave equations with linear dissipative terms in a bounded domain ^ in Rn:

ut + 2kut = Au - \u\p-1u + f (x, u) in ^ x (0, k),

u(x, t) = 0 on 3&, x (0, k), (1.1)

u(x, 0) = <p(x), ut(x, 0) = ^(x) in

where ut = ^, utt = djr, A = ^n=1 dx?, x = (x1,...,xn); the sourcing terms are -\u\p-1u + f (x, u), 1 < p < n-j, n > 3; 1 < p < k, n = 1,2; andf (x, u) satisfies

f (x,z)\< C\z\q + g(x), q < g e L2(Q). (1.2)

The attractor is an important concept describing the asymptotic properties of dynamical systems. A great deal of work has been devoted to the existence of global attractors of dynamical systems (see, e.g. [1-9] and references therein). The existence of a global attractor (1.1) with a source term only containingf was proved by Hale [7] forf satisfying for n > 3 the growth conditionf (u) < C0(\u\Y + 1), with 1 < y < n-j. For the case n = 2, Hale and Raugel [10] proved the existence of the attractor under an exponential growth condition of the type f (u)\ < exp Q (u) (such a condition previously appeared in the work of Gallouet [11]). The existence of the attractor in the critical case y = n-j was first proved by Babin and Vishik [1], and then more generally by Arrieta etal. [12]. For other treatments see Chepyzhov and Vishik [3], Ladyzhenskaya [13], Raugel [14] and Temam [9]. When ^

© 2015 Pan et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly credited.

is bounded and u is subjected to suitable boundary conditions, the general result is that the dynamical system associated with the problem possesses a global attractor in the natural energy space x L2(&) if nonlinear term f has a subcritical or critical exponent, because there exist typical parabolic-like flows with an inherent smoothing mechanism. By the traditional method (see [15] for examples), in order to obtain the existence of global attractors for semilinear wave equations, one needs to verify the uniform compactness of the semigroup by getting the boundedness in a more regular function space. However, in some cases it is difficult to obtain the uniform compactness of the semigroup. Fortunately, a new method for obtaining the global attractors has been developed in [16]. With this method, one only needs to verify a necessary compactness condition («-limit compactness) with the same type of energy estimates as those for establishing the absorbing sets. In this paper, we use this method to obtain the existence of global attractors for problem (1.1) with the general condition where the source term f (x, u) is without variational structure. This paper is organized as follows:

- in Section 2 we recall some preliminary tools, definitions and our previous results;

- in Section 3 we obtain the existence and uniqueness of weak solution by using our previous results [17] and the various conditions can also be found [18];

- in Section 4 we obtain our main results for problem (1.1) by using the new method («-compactness condition).

2 Preliminaries

Consider the abstract nonlinear evolution equation defined on X, given by

du + kft = G(u), k >0,

u(x,0)=^(x), (2.1)

_ ut(x, 0) = f (x),

where G: X2 x R+ ^ XL* is a mapping, X2 c XL, Xl, X2 are Banach spaces and X* is the dual space of Xl, R+ = [0, to), u = u(x, t) is an unknown function. First we introduce a sequence of function spaces:

Jx c H2 c X2 c Xi c H,

(x2 c Hi c H, (.)

where H, Hl, H2 are Hilbert spaces, X is a linear space, XL, X2 are Banach spaces and all inclusions are dense embeddings. Suppose that

IL: X ^ XL is a one to one dense linear operator, ^

(Lu, v)H = (u, v)Hi, Vu, v e X.

In addition, the operator L has an eigenvalue sequence

Lek = Xkek (k = 1,2,...) (2.4)

such that {ek} c X is the common orthogonal basis of H and H2.

Definition 2.1 [17] Set (p, f) e X2 x Hi, u e ^¿^((O, to),Hi) n L^((0, to),X2) is called a globally weak solution of (2.1), if Vv e XL, we have

(ut, v)H + k(u, v)H = I (Gu, v) dt + k(p, v)H + (f, v)H. (2.5)

Definition 2.2 [17] Let Yl, Y2 be Banach spaces, the solution u(t, p, f) of (2.1) is called uniformly bounded in Yl x Y2, if for any bounded domain x c Yl x Y2, there exists a constant C which only depends on the domain x such that

||uHYl + \\ut\\Y2 < C, V(p, f) e x and t > 0.

Suppose that G = A + B: X2 x R+ — XL*. Throughout this paper, we assume that:

(i) There exists a functional F e C1: X2 — R1 such that

(Au, Lv) = (-DF (u), v), Vu, v e X. (.6)

(ii) The functional F is coercive, i.e.

F(u) —^ to ^ ||u|X2 — to. (.7)

(iii) There exist constants Cl > 0 and C2 > 0 such that

|(Bu,Lv) | < CiF(u) +C2\v\\2Hl, Vu, v e X. (2.8)

Lemma 2.1 [17] Set G: X2 x R+ — XL* to be weakly continuous, (p, f) e X2 x Hl, then we obtain the following results:

(1) If G = A satisfies the assumptions (i) and (ii), then there exists a globally weak solution of (2.1),

u e Wl1o,cTO((0, to),Hi) nLTOc((0, to),X2),

and u is uniformly bounded in X2 x Hl .

(2) If G = A + B satisfies the assumptions (i), (ii) and (iii), then there exists a globally weak solution of (2.1),

u e W£CTO((0, to),Hi) n LTOTO ((0, to),X2).

(3) Furthermore, if G = A + B satisfies

|(Gu, v)|< 2 UvUH + CF(u) + g(t) (.9)

for someg e Lloc(0, to), then u e W^O, to),H).

A family of operators S(t): X — X (t > 0) is called a semigroup generated by (2.1) if it satisfies the following properties:

(1) S(t) : X ^ X is a continuous map for any t > 0,

(2) S(0) = id: X ^ X is the identity,

(3) S(t + 5) = S(t) • S(s), Vt, s > 0. Then the solution of (2.1) can be expressed as

u(t, u0) =S(t)uo.

Introducing the expression of the abstract semilinear wave equation: § + 2kj§ = Lu + T(u), k > 0,

u(x,0)=<p(x), (2.10)

ut (x, 0) = f (x),

where X1, X are Banach spaces, X1 c X is a dense inclusion, L : X1 ^ X is a sectorial linear operator, and T : X1 ^ X is a nonlinear bounded operator.

Lemma 2.2 [19] SetL : X1 ^ X, a sectorial linear operator and T : X1 ^ X, a nonlinear bounded operator, L = L + k2/, then the solution of (2.9) can be expressed as follows:

u = e-kt

cos t(-L)2y + k(-L)-3 sin (-L)2y + (-L)-2 sin t(-L)2 f

+ f e-k(t-T)(-£)-i sin(t - t)(-£)* T(u) dr io

-(-L) 2 sin t(-L) 2 y + kcos t(-L) 2 y + cos t(-L)2 f

ut = -ku + e kt

+ i e-k(t-r) cos(t - r)(-L)2 T(u) dr

Next, we introduce the concepts and definitions of invariant sets, global attractors, and «-limit compactness sets for the semigroup S(t).

Definition 2.3 Let S(t) be a semigroup defined on X.Aset X c X is called an invariant set of S(t) if S(t)X = X, Vt > 0. An invariant set X is an attractor of S(t) if X is compact, and there exists a neighborhood U c X of X such that, for any u0 e U,

infl|S(t)u0 - v|L ^ 0, as t ^ 0.

In this case, we say that X attracts U. Especially, if X attracts any bounded set of X, X is called a global attractor of S(t) in X.

Definition 2.4 Let X be an infinite dimensional Banach space and A be a bounded subset of X. The measure of noncompactness y (A) of A is defined by

y (A) = inf{5 > 0 \ for A there exists a finite cover by sets whose diameter < 5}.

Lemma 2.3 [11] IfAn c X is a sequence bounded and closed sets, An = 0, An+i c An, and y (An) ^ 0 (n ^ k), then the set A = P|K=1 An is a nonempty compact set.

Definition 2.5 [16] A semigroup S(t) : X ^ X (t > 0) in X is called «-limit compact, if for any bounded set B c X and Ve > 0, there exists t0 such that

y(|J S(tB < e,

\>to '

where y is a noncompact measure in X. For a set D c X, we define the «-limit set of D as follows:

«(D) = RU S(t)D,

5>0 t>s

where the closure is taken in the X-norm.

Lemma 2.4 [19] Let S(t) be a semigroup in X, then S(t) has a global attractor A in X if and only if

(1) S(t) has «-limit compactness, and

(2) there is a bounded absorbing set B c X.

In addition, the «-limit set ofB is the attractor A = «(B).

Remark 2.1 Although the lemma has been proved partly in [19], we still give a proof here. Our proof is different from that in [20] but is similar to that in [16]. We adopt and present the proof also because we will use the same method to obtain the existence of the global attractor.

Proof Step 1. To prove the sufficiency of Lemma 2.4.

(a) S(t) has «-limit compactness, i.e., for any bounded set B c X and Ve > 0, there exists a t0, such that

y( U S(tB < e.

V>t0 '

So, we know that «(B) = P|^=0 (Jt>t S(t)B is a compact set from Lemma 2.3.

(b) «(B) is nonempty.

For B = 0, so Ut>sS(t)B = 0, Vs > 0, and

U S(t)B c \J S(t)B, Vs1 > s2,

t>s1 t>$2

we can obtain

«(B) = nu S(t)B=0.

s> 0 t> s

(c) «(B) is invariant.

For x e «(B) ^ there exist {xn} e B and tn ^ to, such that S(tn)xn ^ x. If y e S(t)«(B), then for some x e «(B), y = S(t)x.

Hence, there exist {xn} cB, tn — to, such that

S(t)S(tn)xn = S(t + t„)xn — S(t)x = y.

In conclusion, y e «(B), S(t)«(B) e «(B), Vt > 0. If x e «(B), fix {xn} c B and tn, such that

S(t)xn — x, as tn —^ to, n — to.

S(t) is «-limit compact, i.e., there exists ay e H, such that S(t) n U S(tn)xn ^ y, n ^TO.

tn >0 t>tn

Therefore y e «(B). For

p| |J S(tn)xn = f| U S(t)S(tn - t)xn ^ n U S(t)y

tn>0 t>tn tn>0 t>tn tn>0 t>tn

S(tn)xn ^ x e «(B),

which implies that

S(t)y ^ x, «(B) c S(t)«(B).

In conclusion, combining (a)-(c) and condition (2), Step 1 has been proved. Step 2. To prove the necessity of Lemma 2.4.

If A is a global attractor, then the e-neighborhood Us (A) c X is an absorbing set. So we need only to prove S(t) has «-limit compactness.

Since Ue (A) is an absorbing set, for any bounded set B c X and e >0, there exists a time te (B) > 0 such that

dist(x, A)< -4

(J S(t)B c Uj (A) = j x e X

t>tE (B)

On the other hand, A is a compact set, and there exist finite elements x1,x2,...,xn e X such that

A c Ù'U{xk< ,j) •

k=l V '

Uf(A) c y U(xk,0,

which implies that

y U S(t)B < y (Uf (A)) < e.

-t>tE (B)

Hence, Lemma 2.4 has been proved.

3 Existence and uniqueness of globally weak solution

Now, in this section, we begin to prove that problem (1.1) has a unique globally weak solution (u,ut) e C°([0,to),#0 x L2(Q)).

Theorem 3.1 (Existence) /fV(p, f ) e x L2(^),f satisfies condition (1.2) and 1 <p <

, n > 3; 1 < p < to, n = 1,2, then (1.1) has a globally weak solution

Remark 3.1 Divide the operator G(u) in Lemma 2.1 into two parts: A and B, where A has a variational structure and B has a non-variational structure. Then we obtain the globally weak solution by applying our result (2) in Lemma 2.1.

Proof Fix spaces as follows:

u e Wt00((0,to),l2(a)) nL£((0,to),H^)).

X2= Xi = fl0(n) n Lp+1(iï), X = C°0°(ß), H1= H = L2(Q).

In problem (1.1), set G(u) = Au - |u|p 1u + f (x, u). Define the map G(u)= A + B : Xi ^ Xf as

Note the functional I : X1 ^ R1,

Obviously, we obtain

{Au, v) = --DI[u], v), Vu, v e X

I [u] ^TO ^

u^Xi ^ o,

which implies that conditions (1) and (2) in Lemma 2.1 hold.

From the growth restriction condition (1.2), we get \(Bu, v)| =

< I \f(x, u)\|v| dx Jq

< 1 i |v|2dx +1 i \f(x,u)\2dx "2 q ' 2Jq » ( , ^

< 1 i V2 dx + C f [|u|2q + g2(x) J dx 2 Jq Jq

< 1 Mh + Cif |u|p+1 dx + C2 2 Jq

< 2 ||vyH1 + CiI[u] + C2,

where C, C1, C2 > 0. It implies that condition (3) in Lemma 2.1 holds. In conclusion, we see that problem (1.1) has a globally weak solution

u e Wf ((0,to),L2(Q)) nL£((0,to),h0(Q))

from the second result in Lemma 2.1. □

Next, we prove the uniqueness of the globally weak solution to problem (1.1).

Theorem 3.2 Ifu e Wof ((0, f), L2(Q)) n Lfc((0, to), Hj(Q)) is a weak solution of problem (1.1), then the solution u is unique.

Remark 3.2 From the formula of the wave equation in Lemma 2.2 and using the Gronwall inequality, we obtain the uniqueness of the globally weak solution.

Proof Set u1, u2 e Wl1o,c00((0, f), L2(Q)) n Lfc((0, f ), Hi(Q)) as the solutions of problem (1.1), then from Lemma 2.2, we get u e C0([0, f), h0(Q)), i = 1,2, and

||u1- u2|h0 = IK-A 2)(u1- u2)|L2

< c i I^r1^-ur^ + [f(x, u) -f (x, u2)J|L2 dT

< C1 f [( 11 u |p11 + |Df(x, u)|) • ||u1- u2Ih1 J dT ;

by using the Gronwall inequality, we easily obtain

f (x, u)vdx

||Ml-K2||tfi < 0,

where ii is the mean value between ul and u2. It implies that

llui - m2 ll^i < 0 ^ mi = u2. □

4 Existence of global attractor

In this section, we proved the existence of global attractor to problem (1.1).

Theorem 4.1 For any (<p, ^) e (Hj(^) x L2(^)), the sourcing termf satisfies the growth restriction (1.2) and the exponent ofp satisfies 1 <p < ^, n > 3 or 1 <p < k, n = 1,2; then problem (1.1) has a global attractor A in (Hj(^) x L2(^)).

Remark 4.1 Comparing Remark 3.1, we divide the operator G(u) of (2.1) into two parts: L and T, where L is a linear operator, while T is a nonlinear operator. We obtain the global attractor of problem (1.1) by using Lemma 2.4.

Proof According to Lemma 2.4, we prove Theorem 4.1 in the following three steps. Step 1. Problem (1.1) has a globally unique weak solution. Step 2. S(t) has a bounded absorbing set in x L2(^).

From Theorems 3.1 and 3.2, we see that problem (1.1) has a globally unique weak solution (u, ut) e C°([0, k),Hj x L2). Equation (1.1) generates a semigroup:

S(t) :H 1 x H ^ H1 x H.

Fix the spaces as follows:

H = L2(Q), h1= h2(q) n H0(Œ), L : H1 ^ H, T : H1 ^ H.

Note that

Lu = Au,

Tu = -|u|p 1u + f (x, u),

(.1) (4.2)

and L generates the fractional space, H1 = H^(^). Obviously, there exists a C1 functional F: H1 ^ R1 such that

and we easily get

T (u) = -DF (u), Vu e H1

then we get

F(u) > -C1

(DF (u), u)H - k{u, v)h > --\\ v\\H - C2) C2 > 0.

Equation (1.1) is equivalent to the equations that follow:

ff = -ku + v, k > 0,

dV = Lu + k2u - kv - |u|-p-1u + f (x, u).

Multiply (4.7) by (-Lu, v) and take the inner product in H:

Id u \dt / d v

, -Lu) = -k{u, -Lu)H + {-Lu, v)h , Ih

, v ) = {Lu, v)h + (k2u, VH - k{v, v)h + (T (u), v)h.

Summing (4.8) and (4.9), it follows that

I du \ I dV —, -Lu) + —, V

t Ih \ d t

= -k{u, -Lu)H - k{v, v)H + k2 (u, v>H + {Tu, v)H. Furthermore,

{-Lu,a>)H = ((-L2)u, (-L2)«jH, Vu,rn e Hi. From (4.4) and (4.7), we get

{Tu, v)h = (Tu, + ku

= (-DF(u), ^ + ku

= --DF(u), ^ - k[DF(u), u) dF (u)

- k(DF (u), uH.

(.8) (4.9)

(4.10)

Integrating (4.10) over [0, t] with respect to time t and combining the two formulas, we get

K ||2 1

u \ \ rr 2 + - \\v\2 - ■

2..--"h| +2',n,H-2\\H2

Idu,-L

\—,-Lu) + ( —, V L\3t H \dt ,hj

-k I [{u, -Lu)H + {v, v)h - k{u, v)h] dr + / {Tu, v)h dr Jo Jo

= -k i [{(-LI )u, (-L 2 )u)H + \\v\\H - k{u, v)h] dT

i [-T-kDF<u).4

= -k i [\\u\H + \\ v\H - k{u, v)h] dT - F(u(t)) + F(u(0))

- k J [DF(u), uHdT

= -k [\\u\\H2 + (DF(u), u)H - k{u, v)h] dx - F(u)+F(p);

combining with (4.6), it follows that

\\u\\h2 + \\V\\H < -M [M H2 + \\v\\H]dr + /(p, f ) + Ct, C >0.

1 Jo 2

Applying the Gronwall inequality, we get

\\u\\Hi + \\V\H </(p, f )e-kt + Ci(l - e-t). (.12)

It implies that 5(t) has a bounded absorbing set in Hi x H. Step 3.5(t) has «-limit compactness.

From the formula in Lemma 2.2, the solution of problem (l.l) can be expressed as follows:

u = e-kt[cos t(-A)2p + k(-A)-2 sint(-A)2p + (-A)-2 sint(-A)2f]

+ jQt[e-k(t-T)(-A)-i sin(t- r)(-A)i(-|u|p-1 u + /)] dr, (4.13)

ut = -ku + e-kt [-(-A)2 sin t(-A)22 p + k cos t(-A)22 p + cos t(-A)22 f ]

+ / [e k(t T)cos(t- t)(-A)2(-|u|p-1u + /)]dr. (4.14)

Since the linear operator

L = A :H2(Q) x H0(^) ^ L2(^) is a symmetrical sector operator, it has the eigenvalue sequence:

0>X1 > k2 > ■, kk ^ -to, k ^ to. Then

sin t(-A)2 v = ^ Vj sin^J-kjtej, (4.15)

cos t(-A)2 v = ^ Vjcos-J-^jte j. (.16)

For any v = vjej e L2(ft) and -Xj > 0 (j > l), the operator

sin t(-A) 2, cos t(-A) 2 : L2 (ft) ^ L2 (ft)

is uniformly bounded, i.e.

I sin t(-A)22 ||l2, I cos t(-A)2 ||l2 < l, Vt > 0. (4.17)

Furthermore, (u, ut) contains two parts: degenerative term

_ -kt /cos(-A)2 + k(-A)-2 sint(-A)2 (-A)-2 sint(-A)2\ / \kcost(-A)2-(-A)2 sint(-A)2 cost(-A)2 j\tj '

integral term

A //0te-k(t-T)(-A)-2 sin(t- r)(-A)2(-|M|p-1M +f)dr '2 ' V /0e-k(t-r) cos(t- r)(-A)2(-|M|p-1M + f)dr

From the uniformly bounded condition (4.17), we get

lim (u1, u1) =0 inH0(Q) x L2(Q); (4.18)

and for any (<p, ty ) e B,

(J(u2, a?) is a compact set in hJ(Q) x L2(Q), (4.19)

where B c Ho (ft) x L2(ft) is a bounded set. From (1.2) and H0(ft) ^ L2p(ft) (p < n^), we get

T: H0(^) ^ L2(ft) is a compact map,

Hence, combining (4.18) and (4.19), for the noncompact measure y we get

y( y S(t)B

V>to ^

= y(U (u(t, B), ut (t, B))

V>to y

- Y ( U (u1, -ku1 + u1)l + y ( y (u2, -ku2 + u2) ) V>to ' V>to '

= Y ( U (u1, -ku1 + uj) I

^ 0 (to ^ to), (4.20)

it implies that

S(t) = (u(t, •), ut(t, •)) has «-limit compactness.

Finally, combining Step 2 and Step 3, applying Lemma 2.4, problem (1.1) has a global attractor A in H^(Q) x L2(Q). □

Competing interests

The authors declare that they have no competing interests. Authors' contributions

DY and QZ discussed with ZP the paper who helped to check and prove the whole paper. All authors read and approved the final manuscript.

Author details

1 School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China. 2School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou, 310018, China. 3School of Computer Science, Civil Aviation Flight University of China, Guanghan, 618307, China.

Acknowledgements

The authors are grateful to professor Tian Ma for his helpful comments. This work is supported by the Fundamental Research Funds for the Central Universities (No. 2682014BR036).

Received: 26 September 2014 Accepted: 26 December 2014 Published online: 30 January 2015 References

1. Babin, AV, Vishik, MI: Attractors of Evolution Equations. Nauka, Moscow (1989); English translation: North-Holland, Amsterdam (1992)

2. Ball, JM: Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10,31-52 (2004)

3. Chepyzhov, VV, Vishik, MI: Attractors for Equations of Mathematical Physics. American Mathematical Society Colloquium Publications, vol. 49. Am. Math. Soc., Providence (2002)

4. Ghidaglia, JM,Temam, R: Attractors for damping nonlinear hyperbolic equations. J. Math. Pures Appl. 66, 273-319 (1987)

5. Haraux, A: Two remarks on dissipative hyperbolic problems. In: Lions, J-L (ed.) Séminaires de Collège de France (1984)

6. Haraux, A: Systèmes Dynamiques Dissipatifs et Applications. RMA, vol. 17. Masson, Paris (1991)

7. Hale, JK: Asymptotic Behavior and Dynamics in Infinite Dimensions. Pitman, London (1984)

8. Sell, GR, You, Y: Dynamics of Evolution Equations. Springer, New York (2002)

9. Temam, R: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin (1997)

10. Hale, JK, Raugel, G: Attractors for dissipative evolutionary equations. In: Perelló, C, Simó, C, Solà-Morales, J (eds.) Proceedings of the Conference EQUADIFF 91, Universitat de Barcelona, Barcelona, 26-31 August 1991, vol. 1, pp. 3-22. World Scientific, Singapore (1993)

11. Gallouët, T: Sur les injections entre espaces de Sobolev et espaces d'Orlicz et application au comportement à l'infini pour des équations des ondes semi-linéaires. Port. Math. 42,97-112 (1983/84)

12. Arrieta, JM, Carvalho, AN, Hale, JK: A damped hyperbolic equation with critical exponent. Commun. Partial Differ. Equ. 17, 841-866(1992)

13. Ladyzhenskaya, O: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)

14. Raugel, G: Global attractors in partial differential equations. In: Handbook of Dynamical Systems, vol. 2, pp. 885-982. North-Holland, Amsterdam (2002)

15. Robinson, C: Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge University Press, Cambridge (2001)

16. Ma, Q, Wang, S, Zhong, C: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana Univ. Math. J. 5(6), 1542-1558 (2002)

17. Pan, ZG, Pu, ZL, Ma, T: Global solutions to a class of nonlinear damped wave operator equations. Bound. Value Probl. 2012,42 (2012)

18. Li, XS, Huang, NJ, O'Regan, D: Differential mixed variational inequalities in finite dimensional spaces. Nonlinear Anal. 72,3875-3886 (2010)

19. Ma, T: Theories and Methods in Partial Differential Equations. Science Press, Beijing (2011)

20. Zhang, YH, Zhong, CK: Existence of global attractors for a nonlinear wave equation. Appl. Math. Lett. 18,77-84(2005)