Global Attractors for the Higher-Order Evolution Equation

Erhan Pişkin 1  and Hazal Yüksekkaya 1
  • 1 Dicle University, Diyarbakir, Turkey
Erhan Pişkin and Hazal Yüksekkaya

Abstract

In this paper, we obtain the existence of a global attractor for the higher-order evolution type equation. Moreover, we discuss the asymptotic behavior of global solution.

1 Introduction

We consider the following nonlinear evolution equation

{utt+(Δ)mu+(Δ)mut+(Δ)mutt+g(x,u)=f(x),(x,t)Ω×(0,),u(x,0)=u0(x),ut(x,0)=u1(x),xΩ,iu(x,t)ivi=0,i=1,2,,m1,(x,t)Ω×[0,),
where in a bounded domain Ω ⊂ Rn with smooth boundary Ω, the assumption on f, g, u0 and u1 will be made below.

When m = 1, the equation (1.1) is following form

uttΔuΔutΔutt+g(x,u)=f(x).
Chen and Wang [19] proved the existence of global attractor for the problem (1.2). Lately, Xie and Zhong in [8] studied the existence of global attractor of solution for the problem (1.1) with f = 0. Also, there are some authors studied the existence and nonexistence, asymptotic behavior of global solution for (1.2) (see [2, 3, 4, 5, 6, 7] for more details ). Nakao and Yang in [9] showed the global attractor of the Kirchhoff type wave equation.

In this paper, we improve our result by adopting and modifying the method of [19], we studied more general form of the equation.

This paper is organized as follows: In section 2, we give some assumptions and state the main results. In section 3, we prove the global existence of solution using the Faedo-Galerkin method. Also, we write some important estimates for the solution. In section 4, the existence of the global attractor is proved. In Section 5, the proof of decay property for solution is showed.

2 Preliminaries and main results

We write the Sobolev space Hk (Ω) = Wk,2 (Ω), H0k(Ω)=W0k,2(Ω) . Furthermore, we show by (.,.) the inner product of L2 (Ω), by ‖.‖p the norm of Lp (Ω), p ≥ 1 and by ‖.‖E the norm of any other Banach space E. As usual, we give u(t) instead of u(x,t), and u (t) for ut (t) and so on.

We write the following assumptions on f and g.

(A1) Assume f (x) ∈ L2 (Ω) and show F = ‖ f ‖2;

(A2) Suppose g(x,u) ∈ C1× R1) and ∃k1,k2> 0, h1 (x) ∈ L2 (Ω), h2 (x) ∈ L2 (Ω) ∩ Ln/2 (Ω) such that

g(x,u)u+h1(x)|u|k1(G(x,u)+h1(x)|u|)0,(x,u)Ω×R1
and the growth condition in u
|g(x,u)|k2(|u|α+h2(x)),|gu(x,u)|k2(|u|α1+h2(x)),(x,u)Ω×R1
with α ≥ 1, (n = 1,2), and 1αn+2n+2 , (n ≥ 3), G(x,u)=0ug(x,s)ds .

Later, we assume H1 = ‖h12, H2 = max {‖h22, ‖h2n/2}.

Clearly, the function g(x,u) = a(x)|u|α−1u − b(x)|u|β−1u(1 ≤ β < α) supplies (2.1) and (2.2) for some a(x), b(x).

Next, we show the definition and lemmas relating to the global attractor, (see [9, 11, 12]).

Definition 1

Suppose that E is Banach space and {S (t)}t≥0a semigroup on E. A set AE is said a (E,E)−global attractor if and only iff

  1. (1)A is never changing (invariant), namely, S (t)A = A for whole t ≥ 0;
  2. (2)A is compact in E;
  3. (3)A is a bounded set in E and absorbs all bounded subset B in E relating with E topology, that is, for whichever bounded subset B ⊂ E,
    distE(S(t)B,A*)=supinfyBxA*||S(t)yx||E0ast.

Lemma 2

Assume E is Banach space and {S (t)}t ≥0is a semigroup of continuous operators on E. Then, there exists (E,E)−global attractor A if the following conditions are supplied:

  1. (1)There exists a bounded absorbing set B0in E, that is, for whichever bounded subset B ⊂ E, there is a T = T (B) such that S (t)B ⊂ B0for any t ≥ T.
  2. (2){S (t)}t≥0as asymptotically compact in E, that is, for any bounded sequence {yn} in E and tnas n → ∞, {S(tn)yn}n=1has a convergent subsequence relating to E topology.

We show the basic results now.

Theorem 3

Suppose (A1)–(A2) satisfy and (u0,u1) ∈ X. Then, the problem (1.1) admits a unique weak solution u(t) in the class

C1([0,);H0mC([0,);H2mH0m)W2,([0,);H0mW1,(([0,);H2m)
holds.
P12u(t)22+P12ut(t)22C1eλ1t+C2,t0
utt(t)22+P12ut(t)22+P12utt(t)22C3eλ2t+C4,t0
and
P12ut(t)22+Pu(t)22+Put(t)22C5eλ3t+C4,t0
with some λ1, λ2,λ3> 0. In this theoremC1=C1(P12u02,P12u12) , C2 = C2(F,H1), C3=C3(P12u02,P12u12,F,H1,H2), C4 = C4 (F,H1,H2),C5 = C5 (‖Pu02 ,‖Pu12 ,F,H1,H2).

Show the solution in Theorem 1 by S (t)(u0,u1) = (u(t),ut (t)). We are now in a position to prove some continuity of S (t) relating to the initial data (u0,u1), which will be needed for the proof of the existence of global attractor.

Theorem 4

Suppose whole conditions in Theorem 3. Assume S (t)(u0k,u1k) and S (t)(u0,u1) are the solutions of the problem (1.1) with the initial data (u0k,u1k) and (u0,u1). If (u0k,u1k) (u0,u1) in X as k → ∞, then S (t)(u0k,u1k) → S (t)(u0,u1) in X as k → ∞.

Theorem 4 denotes that the semigroup S (t) : X → X is continuous on X.

Theorem 5

Assume every assumptions in Theorem 3 be provided. Then, the semigroup {S (t)}t≥0related with the solution of the problem (1.1) accepts a (X,X)−global attractor A.

For the decay property of solution u(t) for the problem (1.1), we get

Theorem 6

Suppose u is a weak solution in Theorem 3 with f = 0 and g(x,u) = g(u). Besides, suppose 0 2G(u) ≤ ug(u). Then, for whichever q > 0, there isC1=C1(P12u02,P12u12)such that

E(t)=12(u(t)22+P12u(t)22+P12ut(t)22)+ΩG(u(t))dxC1(1+t)1/q.

3 The Proof of Theorem 3

In this section, we suppose that all assumptions in Theorem 3 are supplied. Firstly, we establish the global existence of a solution to problem (1.1) with Fadeo-Galerkin method as in [16, 17].

Assume ωj (x) (j = 1,2,...) is the complete set of properly normalized eigenfunctions for the operator (Δ)m in H0m(Ω) . Then, the family 1,ω2...,ωk,...} holds an orthogonal basis for both H0m(Ω) and L2 (Ω), see [16, 17]. For each positive integer k, show Vk = span{ω1,ω2...,ωk,...}. We search for an approximation solution uk (t) to the problem (1.1) in the form

uk(t)=j=1kdjk(t)ωj
where djk (t) are the solution of the nonlinear ordinary differential equation (ODE) system in the variant t:
(uk,ωj)(Puk,ωj)(Puk',ωj)(Puk,ωj)+(g,ωj)=(f,ωj),j=1,2,,k,
with the initial conditions
djk(0)=(u0k,ωj),djk'(0)=(u1k,ωj)
where u0k and u1k are chosen in Vk so that
u0ku0,u1ku1inH2m(Ω)H0m(Ω)ask.
Here (.,.) shows the inner product in L2 (Ω). Then, Sobolev imbedding theorem means that ∃c0> 0, such that
uk(0)H0m2c0P12u022,uk'(0)H0m2c0P12u122k=1,2,,
and (3.1) shows that for any v ∈ Vk,
(uk,v)(Puk,v)(Puk',v)(Puk,v)+(g,v)=(f,v),vVk.
We know, the system (3.1) and (3.2) accept a unique solution uk (t) on the interval [0,T ] for any T > 0. Such a solution can be expanded to the overall interval [0,∞). We show by Ci (i = 1,2,...) the constants that are independent of k and t ≥ 0, by C0 the constant depending on k1, k2 in (A2) and Sobolev imbedding constant c0 in (3.4). These constants may be different from line to line.

Multiplying (3.1) by djk (t) and summing the resulting equations over j, we obtain

E1'(t)+P12uk'(t)22=0,t0
where
E1(t)=12(uk'(t)22+P12uk(t)22+P12uk'(t)22)+ΩG(x,uk(t))dxΩf(x)uk(t)dx.
Also, multiplying (3.1) by djk (t), we get
E21(t)+P12uk(t)22+Ωg(x,uk)uk(t)dx=uk'(t)22+P12uk'(t)22+Ωf(x)uk(t)dx
where
E2(t)=12P12uk(t)22+Ωuk(t)uk1(t)dx+ΩP12uk(t)P12uk'(t)dx.
If we take sufficient large k1> 0 and use the assumption (A2), we get
ψk'(t)+λ1ψk(t)C0(F2+H12m),ψk(t)=k1E1(t)+E2(t)
with some positive λ 1, relating to the indicated constants in (A2).

We note that

ψk(t)c0(k1+c0)(P12uk'(t)22+P12uk(t)22)+k12(F2+H12m)+k1Ω(G+h1|uk(t)|)dx
and
2ψk(t)(k11)(uk'(t)22+P12uk1(t)22)+(k15c0)P12uk(t)22+k1Ω(G+h1|uk(t)|)dxk12(F2+H12m)
with k1 max{3,2 + 5c0}.

The application of Gronwall lemma to (3.10) holds

P12uk(t)22+P12uk'(t)22+ΩG(x,uk(t))+h1(x)|uk(t)|)dxC1eλ1t+C2,t0
which shows
P12uk(t)22+P12uk'(t)22C1eλ1t+C2,t0
Where C1=C1(P12u02,P12u12) , C2 = C2 (F,H1).

Also, we differentiate (3.1) with respect to t and get

(uk',ωj)(Puk',ωj)(Puk,ωj)(Puk',ωj)+(guuk',ωj)=0,j=1,2,,k.
Multiplying (3.14) by djk(t) and summing the resulting equations over j, we obtain
E3'(t)+P12uk(t)22+Ωguuk'ukdx=0
with
E3(t)=12(uk(t)22+P12uk'(t)22+P12uk(t)22)C0(P12uk(t)22+P12uk'(t)22)2,t0
in which the Sobolev embedding theorem has been used.

Furthermore, the growth condition (2.2) and the Hölder inequality mean that

Ω|guuk'uk|dxk2Ω(|h2||uk'|+|uk|α1|uk'||uk|)dxC0(h2n/2+P12um2(α1))P12uk'2P12uk2.
Therefore, we get
Ω|guuk'uk|dx12P12uk(t)22+C0P12uk'(t)22(P12uk(t)22(α1)+H22m)
and
E3'(t)+12P12uk(t)22C0P12uk'(t)22(P12uk(t)22(α1)+H22m).
Then, the applications of the estimates (3.13) and (3.15)–(3.18) give that ∃λ 1 ≥ λ 2 > 0, depending on C0, such that
E3'(t)+λ2E3(t)C0P12uk'(t)22(1+P12uk(t)22(α1)+H22m)C3eλ1t+C4.
Here, assume C3=C3(P12u02,P12u12,F,H1,H2) , C4 = C4 (,F,H1,H2). Then (3.19) means that
E2(t)E3(0)eλ2t+C3eλ2t+λ21c4,t0.
We show that E3 (0) is uniformly bounded for k under the conditions in Theorem 3 now. It follows by (3.1) that
(uk(t)Puk(t)Puk'(t)Puk(t),uk(t))=(f,uk'(t))(g,uk(t)).
Especially, suppose t = 0, we get
uk(0)22+P12uk(0)22+ΩP12uk(0).(P12uk(0)+P12uk'(0))dx=Ωf(x)uk(0)dxΩ(g,uk(0))uk(0)dx.
By Young inequality with ɛ,
Ω|P12uk(0).P12uk(0)|dxεP12uk(0)22+CεP12uk(0)22,Ω|P12uk(0).P12uk'(0)|dxεP12uk(0)22+CεP12uk'(0)22,Ω|g(x,uk(0))uk(0)dx|uk(0)2nn2gμ1εP12uk(0)22+Cεgμ12,
and
Ω|f(x)uk(0)|dxεP12uk(0)22+Cεf22
with μ1 = 2n/(n + 2). Since μ1α = 2nα/(n + 2) 2n/(n − 2), we obtain by (2.2) that
Ω|g|μ1dxC0Ω(|uk(0)|μ1α+|h2|μ1)dxC0(P12u02μ1α+h22μ1).
Suppose 0 < ɛ ≤ 1/6. Then, from (3.22) to (3.24) that
E3(0)P12uk(0)22+uk(0)22+P12uk'(0)22C0(P12uk'(0)22+P12uk(0)22+F2+gμ12)C0(P12u122+P12u022+F2+P12u022α+h222)C3.
Therefore, the inequality (3.20) shows
P12uk(t)22+P12uk'(t)22+P12uk(t)22C3eλ2t+λ21C4,t0
and the estimates (3.13) and (3.26) give that
{{uk(t)}isboundedinL([0,);H0m(Ω)),{uk'(t)}isboundedinL([0,);H0m(Ω)),{uk(t)}isboundedinL([0,);H0m(Ω)).
So, there exists a subsequences in {uk} (still showed by {uk}) such that
{ukuweaklystarinL([0,);H0m(Ω)),uk'u'weaklystarinL([0,);L2(Ω)),ukuweaklystarinL2([0,);H0m(Ω)),.
From applying the fact that L ([0,∞); H0m(Ω))L2(|0,) ; H0m(Ω)) ) and the Lions-Aubin compactness Lemma in [20], we obtain from (3.27) and (3.28) that
uku,uk'stronglyinL2([0,);L2(Ω))
and then uk→ u a.e in Ω× [0,∞).

Using the growth condition (2.2), for any T > 0, the integral

0TΩ|g(x,uk)(x,t)|α+1αdxdt
is bounded. Accordingly, by Lemma 2 in Chap. 1 [17], we conclude
g(x,uk)g(x,u)weaklyinLα+1α([0,T];Lα+1α(Ω))
with these convergences, by using the limit in the approximate equation (3.5), we get
(u(t),v)(Pu,v)(Pu,v)(Pu,v)+(g(x,u),v)=f(f,v),vH0m(Ω),
So, u(t) is a weak solution of (1.1) and supplies (2.5) and (2.6), and the proof of existence for the solution u(t) of (1.1) is completed.

We derive the estimates for ‖Pu(t)2 and ‖Put (t)2 now. Also, we write u instead of uk for convenience and view the estimates for u as a limit of uk. Supposing v = −Pu in (3.31), we obtain

E4'(t)+Pu(t)22P12ut(t)22+Put(t)22+C0(F2+g22)
with some C0> 0 and
E4(t)=12Pu(t)22+ΩP12ut(t)P12u(t)dx+ΩPut(t)Pu(t)dx.
Also, assuming v = −Put in (3.31), we get
ΩPut(utt+Pu+Putt)dx+Put22=ΩgP12utdxΩfPutdx12Put22+C0(F2+g22).
This means that
E5'+12Put(t)22C0(F2+g22)
with
E5(t)=12(P12ut(t)22+Put(t)22+Pu(t)22).
We note that
u2α2αC0P12u22αθ+Pu22α(1θ)ηPu22+CηP12u22β
with small η > 0 and 2αθ = (n − 2)α − n < 2, β = α (1 − θ )/(1 − αθ ) > 0. Then, (3.37) shows
g22C0(u2α2α+H22m)ηPu22CηP12u22β+C0H22m.
Then, by (2.5), (3.35) and (3.38) that
E5'(t)+12Pu(t)22Pu(t)22+CηP12u(t)22β+C0(F2+H22m)C1eλ1βt+ηPu(t)22+C2.
Assume φ (t) = k1E5 (t) + E4 (t). We get from (3.32) and (3.39) that
ϕ(t)+k112Put(t)22+(1(1+k1/2)η)Pu(t)22C1eλ1βt+C2.
Suppose k1 3 and η is small that 1 − η (1 + k1/2) 4/5. Then, (3.40) shows
ϕ(t)+Put22+12Pu(t)22C1eλ1βt+C2.
We note that
E4(t)35Pu22+3Put(t)22+12(P12u22+P12ut22)
and
ϕ(t)(35+k12)Pu22+(3+k12)Put22+12(P12u22+P12ut22)C0(Pu(t)22+Put(t)22+C1eλ1βt+C2).
Also (3.41) and (3.43) give that ∃λ 1β ≥ λ 3 > 0, depending on C0, such that
ϕ(t)+λ3ϕ(t)C1eλ1βt+C2,t0.
So,
ϕ(t)ϕ(0)eλ3t+C1eλ3t+C2λ31,t0.
Otherwise, we get
ϕ(t)=k1E4(t)+E3(t)k12(P12ut22+Put22+Pu22)12(P12ut22+Put22+P12u22)k112P12ut22+(k121)Put22+k1c02Pu22P12ut22+Put22+Pu22,
where the facts k1≥ {4,2 + c0} and Sobolev imbedding theorem (see [17])
P12u22c0Pu22uH2m(Ω)H0m(Ω)
have been used. So, by the estimates (3.45) and (3.46) that
P12ut22+Pu(t)22+Put(t)22C5eλ3t+C4λ31,t0
with C4 = C4 (F,H1,H2), C5 = C5 (‖Pu02, ‖Pu12 ,F,H1,H2).

To establish the uniqueness, we suppose that u(t) and v(t) are two solutions of (1.1), which supply the estimates (2.5)–(2.7) and u(0) = v(0), u (0) = v (0). Taking U (t) = ut (t), V (t) = vt (t) and W (t) = U (t) − V (t), then we see from (1.1) that

WtPWPWtP(uv)=g(x,v)g(x,u),xΩ,t0.
Multiplying (3.48) by W, we obtain
12ddt(W(t)22+P12W(t)22)+P12W(t)22+ΩP12(uv)P12Wdx=Ω(g(x,v)g(x,u))Wdx
and
W(t)22+P12W(t)22+20tP12W(s)22ds+20tΩP12(u(s)v(s))P12W(s)dxds=20tΩ(g(x,v(s))g(x,u(s)))W(s)dxds.
Since
|P12(u(s)v(s))|0s|P12(uτ(τ)vτ(τ))|dτ=0s|P12W(τ)|dτ
then
P12(u(s)v(s))2s1/2(0sP12W(τ)22dτ)1/2
and
0tΩ|P12(u(s)v(s))P12W(s)|dxds0tΩ0s|P12W(s)||P12W(τ)|dxdτds0t0sP12W(s)2P12W(τ)2dτdst0tP12W(s)22ds.
Now, taking Uɛ (s) = ɛu(s) + (1 − ɛ)v(s), 0 ≤ ɛ ≤ 1, we get
G=0tΩ|g(x,u(s))g(x,v(s))||W(s)|dxds=0tΩ|01ddεg(x,Uε)dε||W(s)|dxds0tΩ01|gu(x,Uε)||u(s)v(s)||W(s)|dεdxdsk20tΩ(|u|α1+|v|α1+h2(x))|u(s)v(s)||W(s)|dxdsc00t(u(s)σ1σ1+v(s)σ1σ1+h2(s)σ2σ2)P12(u(s)v(s))2P12W(s)2ds
where σ1 = n(α − 1)/2 2n/(n − 2), σ2 = n/2.

From (2.5) and Sobolev imbedding theorem, there is C3> 0 such that

u(s)σ1σ1+v(s)σ1σ1+h2σ2σ2C0(P12u(s)2σ1+P12v(s)2σ1+h2σ2σ2)C3s0.
Then,
GC30ts1/2(0sP12W(τ)22dτ)1/2P12W(s)2dsC3t0tP12W(τ)22dτ.
Then, the estimates (3.50)–(3.52) indicate that
W(t)22+P12W(t)22+20tP12W(s)22(C3+1)t0tP12W(s)22ds.
The integral inequality (3.53) represents that there exists T1> 0, such that W (t) = 0 in [0,T1]. As a result, u(t) − v(t) = u(0) − v(0) = 0 in [0,T1].

Then, we conduce that u(t) = v(t) on [T1,2T1], [2T1,3T1],..., and u(t) = v(t) on [0,∞). This shows the proof of uniqueness.

Now, we establish u ∈ C ([0,∞); H0m(Ω)) ). Assume t > s ≥ 0. Then,

P12(u(t))u(s)22=Ω|stP12uτ(τ)dτ|2dx(ts)stP12uτ(τ)dτ22dτ0asts.
This indicates u(t) ∈ C ([0,∞); H0m(Ω)) . Also, we get
P(u(t)u(s))22=Ω|stPuτ(τ)dτ|2dx(ts)stPuτ(τ)22dτ0asts.
and u(t) ∈ C [0,∞); H2m(Ω)H0m(Ω)) .

Moreover, we get

P12(ut(t)ut)(s)22(ts)stP12utt(τ)22dτ0asts.
This shows that u(t) ∈ C1 ([0,∞); H0m ) and the proof of Theorem 3 is completed.

4 Global attractor for the problem (1)

By Theorem 3, we see that the solution operatör S (t)(u0,u1) = (u(t),ut (t)), t ≥ 0 of the problem (1.1) creates a semigroup on X=(H2m(Ω)H0m(Ω))×(H2m(Ω)H0m(Ω)) , which supplies these properties:

  1. (1)S (t) : X → X for all t ≥ 0;
  2. (2)S (t + s) = S (t)S (s) for t,s ≥ 0;
  3. (3)S (t)(u0,u1) → S (s)(u0,u1) in X as t → s for any (u0,u1) ∈ X.

For establishing the existence of the (X,X)-global attractor for the problem (1.1), firstly, we show the continuity of S (t) relating to the initial data (u0,u1).

The proof of Theorem 4

Suppose uk (t), u(t) is corresponding solution of the problem (1.1) with the initial data (u0k,u1k) and (u0,u1) respectively, k = 1,2,....

Since (u0k,u1k) (u0,u1) in X, {(u0k,u1k)} is bounded in X. Set wk (t) = uk (t) − u(t). Then, wk holds

{wkPwkPwk'Pwk=g(x,u)g(x,uk)=Gk,(x,t)Ω×(0,),wk(x,0)=u0k(x)u0(x),wk'(x,0)=u1k(x)u1(x),xΩ,wk(x,t)=0,(x,t)Ω×[0,).
Multiplying the equation in (4.1) by wk,−Pwk and −Pwk, we get
12ddt(wk'22+P12wk22+P12wk'22)+(1η)P12wk'22CηGk22
and
ddt(12Pwk22+Ω(Pwk'Pwk+P12wkP12wk')dx)+(1η)Pwk22Pwk'22+P12wk'22+CnGk22c0Pwk'22+CηGk22
and
12ddt(P12wk'22+Pwk22+Pwk'22)+(1η)Pwk'22CηGk22
with small η > 0. Then, by (4.2) and (4.4) we obtain
yk'(t)+(k(1η)c0)Pwk'22+(1η)P12wk'22+(1η)Pwk(t)22kCηGk22
where
yk(t)=k1+12(Pwk(t)22+P12wk'(t)22)+12(P12wk(t)22+wk'(t)22)+k12P12wk'(t)22+Ω(Pwk'(t)Pwk(t)+P12wk(t)P12wk'(t))dxk1+22(Pwk(t)22+P12wk'(t)22)+k1+12Pwk'(t)22+wk'(t)22+P12wk'(t)22C0(Pwk'(t)22+Pwk(t)22+P12wk'(t)22)
By taking k1 3
yk(t)=k1+12(Pwk(t)22+P12wk'(t)22)+12(P12wk(t)22+wk'(t)22)+k12P12wk'(t)2212(Pwk'22+Pwk'22)12(P12wk'22+P12wk'22)Pw(t)22+P12wk'(t)22+Pwk'(t)22,t0.
Otherwise, we obtain from assumption (A2),
Gk22=Ω|g(x,uk)g(x,u)|2dx=Ωgu2wk2dxc0Ω(|uk|2(α1)+|u|2(α1)+h22)wk2dx.
The application of Sobolev imbedding theorem and the estimate (2.7) gives
Ω|uk|2(α1)wk2dxwk2μ22uk2(α1)μ32(α1)C3wk2μ22C3wk22
with μ2 = n/(n − 4)+ and μ3 = μ2/(μ2 − 1). Similarly,
Ω|uk|2(α1)wk2dxwk2μ22u2(α1)μ32(α1)C3wk2μ22C3Pwk22
and
Ωh22wk2dxwk2μ22h2N/22C3wk2μ22C3Pwk22.
Then, we get from (4.5) to (4.11) that λ 4 > 0, such that
yk'(t)+λ4yk(t)C3Gk22C3wk2μ22C3Pwk2μ22C3yk(t)
where C3 is as in (2.6), independent of k. The differential inequality (4.12) means
yk(t)yk(0)e(C3λ4)t,t0.
Then, from (4.6) and (4.7), we obtain
yk(0)C0(P12(u1ku1)22+P(u0ku0)22+P(u1ku1)22)0ask
and
Pwk(t)22+P12wk'22+Pwk'22yk(t)yk(0)e(C3λ4)t0ask.
This indicates that S (t) : X → X is continuous. Now we show that {S (t)}t≥0 is asymptotically compact in X from the method in [9].

Assume {(u0k,u1k)} is a bounded sequence and {uk (t)} be the corresponding solutions of the problem (1.1) in C [0,∞); H2m(Ω)H0m(Ω)) . We suppose tk ∞ as k → ∞. For any T > 0, assume tn,tk> T. Then, the application of (4.12) to wkn (t) = un (t + tnT ) − uk (t + tnT ), we get

Ykn(t)Ykn(0)eλ4t+C30teλ4(ts)wkn(s)2μ22ds,t0
with
Ykn(t)=Pwkn(t)22+P12wkn'(t)22+Pwkn'(t)22.
Especially, we take t = T and obtain
P(un(tn))uk(tk)22+P12(un'(tn)uk'(tk))22+P(un'(tn)uk'(tk))22Ykn(0)eλ4T+C3sup0sTuk(tkT+s)un(tnT+s)2μ22.
Since the embedding (H2m(Ω)H0m(Ω))L2μ2(Ω) is compact, we can remove a subsequence {ukk1(tkk1T + s)} which converges in L2μ2 (Ω). Therefore, for any ɛ > 0, firstly we fix T > 0, such that
Ykn(0)eλ4T<ε2.
Supposing n0> 0 and k1, j > n0, we get
C3sup0sTukk1(tkk1T+s)ukj(tkjT+s)2μ22<ε2.
Then, it follows by (4.18) to (4.20) that {ukk1(tkk1)} is a Cauchy sequence in X and we finalize that {S (t)}t≥0 is asyptotically compact on X and now Theorem 4 is established.

Proof of Theorem 5

From Lemma 2, it is sufficient to indicate that there exists a continuous operator semigroup {S (t)} on X such that S (t)(u0,u1) = (u(t),ut (t)) for each t ≥ 0. By the estimates (2.7), we accomplish that

β0={(u,v)XP12v22+Pu22+Pv22C4}
is an absorbing set of {S (t)}t≥0 and for any (u0,u1) ∈ X,
distX(S(t)(u0,u1),β0)C5eλ3t,t0
where the constants C4, C5 are in (2.7). By Theorem 2, S (t) : X → X is continuous and asymptotically compact on X. From a general theory (see [1, 11]), we conclude that S (t) admits a global attractor A on X defined by
A=ω(β0)=τ0[S(t)tτβ0]X
where [D]X is the closure of the set D in X. Then we prove the Theorem 5.

5 Decay property of solution for (1)

In this section, we search the decay property of solution to (1.1) with f ≣ 0. Firstly, we present a well-known Lemma that will be needed.

Lemma 7

([18])

Assume E : [0,∞) [0,∞) is a non-increasing function and suppose that there are constants q ≥ 0 and γ > 0 such that

SEq+1(t)dtγ1E(0)qE(s),S0.
Then, we get
E(t)E(0)(1+q1+qγt)1/qt0ifq>0
and
E(t)E(0)e1γtt0ifq=0.

Proof of Theorem 7

Suppose u(t) is a weak solution in Theorem 3 with f = 0. Show

E(t)=12(u(t)22+P12u(t)22+P12ut(t)22)+ΩG(u(t))dx,t0.
Then, we obtain by (1.1) that
E(t)+P12ut2=0,0.
This indicates that E (t) is non-increasing in [0,∞).

Multiplying the equation in (1.1) by Eq (t)u(t), q > 0, we obtain

STEqΩu(uttPuPutt+g(u))dxdt=0,T>S0.
We note that
STEq(t)(u,utt)dt=Eq(t)(u,ut)|STST(qE(t)q1E(t)(u,ut)+Eq(t)ut(t)22)STEq(t)(u,Pu)dt=STEq(t)P12u22STEq(t)(u,Put)dt=STEq(t)(P12u,P12ut)dt
and
STEq(t)(u,Putt)dt=ST(qE(t)q1E(t)(P12u,P12ut)+Eq(t)P12ut(t)22)dt+Eq(t)(P12u,P12ut)|ST.
Then, we get by (5.6) that
2STEq+1(t)dt=Eq(t)[(u,ut)+(P12u,P12ut)|ST]+qSTE(t)q1E(t)[(u,ut)+(P12u,P12ut)]dt+2STEq(t)(ut(t)22+P12ut(t)22)dt+STEq(t)(P12u,P12ut)dt+STEq(t)(2G(u)ug(u))dt.
Since G(u) 0, E (t) 0. Moreover, we get the following estimates from (5.5):
P12ut(t)2(E(t))1/2,P12u(t)222(E(t))1/2,P12ut(t)22(E(t))1/2,0,
|Eq(t)((u,ut)+(P12u,P12ut))|C0Eq(t)P12u2P12ut(t)2C0Eq+1(t),
ST|E(t)q1E(t)|[(u,ut)+(P12u,P12ut)]dtC0STE(t)q1|E(t)|P12u2P12ut2dtC0Eq+1(S),
2STEq(t)(ut(t)22+P12ut(t)22)dtC0STEq(t)(E(t))1/2C0Eq+1(s),
STEq(t)(P12u,P12ut)dtSTEq(t)P12u2P12ut2STEq+1(t)dt+C1Eq+1(S).
Then we obtain from (5.8) to (5.12) that
STEq+1(t)dtC0Eq+1(S)C0Eq(0)E(S)γ1Eq(0)E(S).
From Lemma 10, we get
E(t)=12(u(t)22+P12u(t)22+P12ut(t)22)+ΩG(u(t))dxE(0)(1+q1+qγt)1/qC1(1+t)1/q.
This is the estimates (2.8) and the proof of Theorem 7 is completed.

Conclusion 8.In this paper, we obtained the global attractor and the asymptotic behavior of global solution for the higher-order evolution equation with damping term. This improves and extends many results in the literature such as (Xie and Zhong (2007); Chen et al. (2011)).

References

  • [1]

    C.E. Seyler, D.L. Fanstermacher, A symmetric regularized long wave equation, Phys. Fluids, 27(1) (1984) 58–66.

  • [2]

    W. Q. Zhu, Nonlinear waves in elastic rods, Acta Mech. Solida Sin., 1(2) (1980) 247–253.

  • [3]

    B. L. Guo, The Vanishing Viscosity Method and the Viscosity of the Difference Scheme, Beijing Science Press, 1993.

  • [4]

    Y. D. Shang, Initial boundary value problem of equation u tt − Δu − Δu t − Δu tt = f (u), Acta Mathematicae Applicatae Sinica, 23(3) (2000) 385–393.

  • [5]

    R. Z. Xu, X. R. Zhao, J. H. Shen, Asymptotic behavior of solution for fourth order wave equation with dispersive and dissipative terms, J. Appl. Math. Mech. Engl. Ed., 29(2) (2008) 259–262.

  • [6]

    Z. J. Yang, C.M. Song, On the problem of the existence of global solutions for a class of nonlinear evolution equations, Acta Mathematicae Applicatae Sinica, 20(3) (1997) 321–331.

  • [7]

    Z. J. Yang, Existence and nonexistence of global solutions to a generalized modification of the improved Boussinesq equation, Math. Methods Appl. Sci., 21(16) (1998) 1467–1477.

  • [8]

    Y.Q. Xie, C. K. Zhong, The existence of global attractors for a class nonlinear evolution equation, J. Math. Anal. Appl., 336 (2007) 54–69.

  • [9]

    M. Nakao, Z. J. Yang, Global attractors for some quasi-linear wave equations with a strong dissipation, Adv. Math. Sci. Appl, 1(17) (2007) 89–105.

  • [10]

    P. A. Martinez, A new method to obtain decay rate estimates for dissipative system, ESAIM Control, Optimization and Calculus of Variations, 4(1999) 419–444.

  • [11]

    A. V. Babin, M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

  • [12]

    R. Temam, Infinite-Dimensional Dynamical in Mechanics and Physics, Spring, New York, 1997.

  • [13]

    O. Goubet, R. Rosa, Asymptotic smoothing and the global attractor for a weakly damped kdv equation on the real line, Journal of Differential Equations, 185(1) (2002) 25–53.

  • [14]

    X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications to their attractors, Physica D, 88 (1995) 167–175.

  • [15]

    S. Frigeri, Attractors for semilinear damped wave equations with an acoustic boundary condition, J. Evol. Equ., 10 (2010) 29–58.

  • [16]

    L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics. AMS: Providence, RI, 1998.

  • [17]

    J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non linearies, Dunod-Gauthier Villars: Paris, 1969.

  • [18]

    V. Komornik, Exact controllability and stabilization, The Multiplier Method. Masson-Wiley: Paris, 1994.

  • [19]

    C. Chen, H. Wang, S. Zhu, Global attractor and decay estimates of solutions to a class of nonlinear evolution equations, Math. Meth. Appl. Sci., 34 (2011) 497–508.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    C.E. Seyler, D.L. Fanstermacher, A symmetric regularized long wave equation, Phys. Fluids, 27(1) (1984) 58–66.

  • [2]

    W. Q. Zhu, Nonlinear waves in elastic rods, Acta Mech. Solida Sin., 1(2) (1980) 247–253.

  • [3]

    B. L. Guo, The Vanishing Viscosity Method and the Viscosity of the Difference Scheme, Beijing Science Press, 1993.

  • [4]

    Y. D. Shang, Initial boundary value problem of equation u tt − Δu − Δu t − Δu tt = f (u), Acta Mathematicae Applicatae Sinica, 23(3) (2000) 385–393.

  • [5]

    R. Z. Xu, X. R. Zhao, J. H. Shen, Asymptotic behavior of solution for fourth order wave equation with dispersive and dissipative terms, J. Appl. Math. Mech. Engl. Ed., 29(2) (2008) 259–262.

  • [6]

    Z. J. Yang, C.M. Song, On the problem of the existence of global solutions for a class of nonlinear evolution equations, Acta Mathematicae Applicatae Sinica, 20(3) (1997) 321–331.

  • [7]

    Z. J. Yang, Existence and nonexistence of global solutions to a generalized modification of the improved Boussinesq equation, Math. Methods Appl. Sci., 21(16) (1998) 1467–1477.

  • [8]

    Y.Q. Xie, C. K. Zhong, The existence of global attractors for a class nonlinear evolution equation, J. Math. Anal. Appl., 336 (2007) 54–69.

  • [9]

    M. Nakao, Z. J. Yang, Global attractors for some quasi-linear wave equations with a strong dissipation, Adv. Math. Sci. Appl, 1(17) (2007) 89–105.

  • [10]

    P. A. Martinez, A new method to obtain decay rate estimates for dissipative system, ESAIM Control, Optimization and Calculus of Variations, 4(1999) 419–444.

  • [11]

    A. V. Babin, M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

  • [12]

    R. Temam, Infinite-Dimensional Dynamical in Mechanics and Physics, Spring, New York, 1997.

  • [13]

    O. Goubet, R. Rosa, Asymptotic smoothing and the global attractor for a weakly damped kdv equation on the real line, Journal of Differential Equations, 185(1) (2002) 25–53.

  • [14]

    X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications to their attractors, Physica D, 88 (1995) 167–175.

  • [15]

    S. Frigeri, Attractors for semilinear damped wave equations with an acoustic boundary condition, J. Evol. Equ., 10 (2010) 29–58.

  • [16]

    L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics. AMS: Providence, RI, 1998.

  • [17]

    J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non linearies, Dunod-Gauthier Villars: Paris, 1969.

  • [18]

    V. Komornik, Exact controllability and stabilization, The Multiplier Method. Masson-Wiley: Paris, 1994.

  • [19]

    C. Chen, H. Wang, S. Zhu, Global attractor and decay estimates of solutions to a class of nonlinear evolution equations, Math. Meth. Appl. Sci., 34 (2011) 497–508.

OPEN ACCESS

Journal + Issues

Search