LONGTIME DYNAMICS FOR AN ELASTIC

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Supplement 2013

Website: www.aimSciences.org pp. 797–806

LONGTIME DYNAMICS FOR AN ELASTIC WAVEGUIDE MODEL

Zhijian Yang and Ke Li Department of Mathematics, Zhengzhou University No.100, Science Road, Zhengzhou 450001, China

Abstract. The paper studies the longtime dynamics for a nonlinear wave equation arising in elastic waveguide model utt − ∆u − ∆utt + ∆2 u − ∆ut − ∆g(u) = f (x). It proves that the equation possesses in trajectory phase space a global trajectory attractor Atr and the full trajectory of the equation in Atr is of backward regularity provided that the growth exponent of nonlinearity g(u) is supercritical.

1. Introduction. In this paper, we study the longtime dynamics for a nonlinear wave equation arising in elastic waveguide model utt − ∆u − ∆utt + ∆2 u − ∆ut − ∆g(u) = f (x)

in Ω × R+ ,

(1)

where Ω is a bounded domain in RN with smooth boundary ∂Ω, on which we consider either hinged boundary condition u|∂Ω = ∆u|∂Ω = 0,

(2)

or clamped boundary condition u|∂Ω = 0,

∂u |∂Ω = 0, ∂ν

(3)

and the initial condition u(x, 0) = u0 (x),

ut (x, 0) = u1 (x),

x ∈ Ω,

(4)

where ν is the unit outward normal on ∂Ω, and the assumptions on g(u) and f will be specified later. In the study of nonlinear wave propagation in elastic waveguide, taking account of energy exchange between the waveguide and the external medium through the lateral surfaces of the waveguide, Samsonov et al. [7, 8] established the model equations 1 utt − uxx = (cu3 + 6u2 + autt − buxx + dut )xx (5) 4 to describe the longitudinal displacement of the elastic rod. Obviously, equation (1) includes (5) as its special case when N = 1. There have been many researches on the global well-posedness of the equation of type (1) (Cf. [1, 2, 5, 6, 9, 10, 11, 12] 2010 Mathematics Subject Classification. Primary: 35B40, 35B41; Secondary: 35G31, 35L35, 37L30. Key words and phrases. Nonlinear wave equation, global solution, longtime dynamics, trajectory attractor, backward regularity. The first author is supported by NSF grant 11271336.

797

798

ZHIJIAN YANG AND KE LI

and references therein). For the study on the longtime dynamics of model (1.1), one can see [4, 13, 14] and references therein. When the space dimension N = 1, Dai and Guo [4] established in phase space E1 = V2 × V1 (see below for its definition) the finite dimensional global attractor for the dynamical system associated with equation (1), with initial boundary conditions (2), (4). For the multidimensional case, Yang [13] established in E1 the global attractor for the evolution semigroup S(t) related to problem (1)-(4) provided that the growth exponent of the nonlinearity g(u) is subcritical. And the author [14] also discussed the longtime dynamics to the Cauchy problem of equation (1). But when p is supercritical, that is, p > N/(N − 2)+ , we can not construct the evolution semigroup according to traditional manner because there is no the uniqueness of the weak energy solution yet. In this case, what happens to the longtime dynamics of the above mentioned problem? The question remains unanswered. The purpose of the present paper is to answer the question. we use the so-called trajectory dynamical system approach, which does not require the uniqueness and is developed by Chepyzhov and Vishik [3], and construct the so-called trajectory attractor associated with the weak energy solution which can be obtained by the Galerkin approximations of the above-mentioned problem and establish the backward regularity of the full trajectory in trajectory attractor provided that the growth exponent of nonlinearity g(u) is supercritical. The plan of the paper is as follows. In Section 2, the global existence of the weak energy solutions is studied. In Section 3, the existence of global trajectory attractor is established. In section 4, the backward regularity of the full trajectory is proved. 2. Global existence of the weak energy solutions. We use the same abbreviations as in [13]: Lp = Lp (Ω), H k = H k (Ω), H = L2 , k · kp = k · kLp , k · k = k · kL2 , with p ≥ 1, ξu = (u, ut ) and ( V2 =

H 2 ∩ H01 ,

for condition (2),

H02 ,

for condition (3).

The notation (·, ·) for the H-inner product will also be used for the notation of duality pairing between dual spaces. Define the operator A : V2 → V20 , (Au, v) = (∆u, ∆v)

for any u, v ∈ V2 . s

Then, the operators As (s ∈ R) are strictly positive and the spaces Vs = D(A 4 ) s s are Hilbert spaces with the scalar products and the norms (u, v)s = (A 4 u, A 4 v), s 1 kukVs = kA 4 uk, respectively. Obviously, kukV2 = kA 2 uk = k∆uk, |ukV1 = 1 kA 4 uk = k∇uk. We denote the Banach space E0 ≡ V1 ∩ Lp+1 × H, which is equipped with the norm k(u, v)k2E0 = kuk2V1 + kuk2p+1 + kvk2 . Especially, E0 = V1 × H as 1 ≤ p ≤ (N + 2)/(N − 2)+ for V1 ,→ Lp+1 , where a+ = max{a, 0}. 1 Rewriting equation (1) in the operator equation, and applying A− 2 to the resulting expression, we get the Cauchy problem which is equivalent to either problem

LONGTIME DYNAMICS FOR ELASTIC WAVEGUIDE MODEL

799

(1), (2), (4) or problem (1), (3), (4): 1

1

1

(A− 2 + I)utt + (I + A 2 )u + ut + g(u) = A− 2 f,

(6)

u(0) = u0 , ut (0) = u1 .

(7)

Theorem 2.1. Assume that (H1 ) √ lim inf g 0 (s) > − λ1 , |s|→∞

g ∈ C 1 (R). Either |g 0 (s| ≤ C(1 + |s|p−1 ),

s ∈ R,

where λ1 (> 0) is the first eigenvalue of the operator A, and 1 ≤ p < ∞ if N = 1, 2; 1 ≤ p ≤ p∗ ≡ (N + 2)/(N − 2) if N ≥ 3, or else, p > p∗ , N ≥ 3, and there exist K, C1 > 0 such that −K + C1 |s|p−1 ≤ g 0 (s) ≤ C(1 + |s|p−1 ),

s ∈ R.

(8)

(H2 ) (u0 , u1 ) ∈ E0 , f ∈ V−3 . Then problem (6), (7) admits a solution u, with ξu ∈ L∞ (R+ , E0 ) ∩ Cw (R+ , E0 ) and Z ∞ kξu (t)k2E0 + kut (τ )k2 dτ ≤ C(k(u0 , u1 )kE0 )e−κt + C(kf kV−3 ), (9) t

where and in the following κ > 0 denotes small constant. Especially, when 1 ≤ p ≤ N/(N − 2)+ , the solution is unique, ξu ∈ Cb (R+ , E0 ) and ξu depends continuously on initial data ξu0 in E0 . Proof. By the standard Galerkin approximation combined with the compactness arguments one can prove Theorem 2.1. We omit the process here. 3. Global trajectory attractor. When p is supercritical, that is, p > N/(N − 2), N ≥ 3, under the assumptions of Theorem 2.1, we can not get the uniqueness of the global weak solutions, so we can not define the evolution semigroup S(t) according to traditional manner. In this case, we define the trajectory phase space of problem (6)-(7): K+ ={ξu ∈ L∞ (R+ ; E0 )|ξu solves (6)-(7), there exists ξunk such that + ξunk → ξu weak∗ in L∞ loc (R ; E0 ) and ξunk (0) → ξu (0) weak* in E0

as k → ∞, where ξunk is the Galerkin approximation of ξu }.

(10)

We endow the trajectory phase space with the topology ∗

+ w J + : [L∞ , loc (R ; E0 )] + i.e. the weak∗ topology of the space L∞ loc (R ; E0 ). Then, one can easily shows that + + K is closed with respect to the J topology. Define on K+ the shift operator by the formula

T (s) : K+ → K+ , s ≥ 0, (T (s)u)(t) = u(t + s), t ≥ 0.

(11)

Obviously, {T (s)}s≥0 composes a semigroup acting on the topology space (K+ , J + ), and (T (s), (K+ , J + )) composes a trajectory dynamical system. Definition 3.1. Let ξu ∈ K+ . We define the M -energy functional + Mu (t) = inf{lim inf kξunk (t)kE0 |ξunk → ξu weak∗ in L∞ loc (R ; E0 ), k→∞

ξunk (0) → ξu (0) weak∗ in E0 },

(12)

800


where the external infimum in the right-hand side of (12) is taken over all the ∗ n sequence of the Galerkin approximations {ξunk }∞ k=1 , with ξu k → ξu weak in + ∞ Lloc (R ; E0 ). Definition 3.2. A set B ⊂ K+ is called M -bounded if kBkM = sup Mu (0) < ∞. ξu ∈B tr

Definition 3.3. A set A is called a global attractor of the trajectory dynamical system (T (s), (K+ , J + )) if (i) The set Atr is a compact M -bounded set in (K+ , J + ); (ii) T (s)Atr = Atr , s ≥ 0; (iii) Atr attracts M -bounded sets in K+ , that is, for every M -bounded subset B ⊂ K+ and every neighborhood O(Atr ) of Atr in (K+ , J + ), there exist s0 = s0 (B, O) such that T (s)B ⊂ O(Atr ) for s ≥ s0 . Theorem 3.4. Under the assumptions of Theorem 2.1, if p > N/(N − 2), N ≥ 3, then the trajectory dynamical system (T (s), (K+ , J + )) has a global attractor Atr and Atr = Π+ K, where K ={ξu ∈ L∞ (R; E0 )|ξu solves (6) and there exists ξunk such that J − lim ξunk = ξu },

(13)

k→∞

is the set of the full trajectory of (6), ξunk is the Galerkin approximation of ξu , that is, there exists a subsequence tk → −∞ such that ξunk solves the problem 1

1

((A− 2 + I)unttk , wj ) + ((I + A 2 )unk , wj ) + (unt k , wj ) + (g(unk ), wj ) 1

=(A− 2 f, wj ), t > tk , j = 1, · · · , nk , ξunk (tk ) =

ξ0nk

=

(14)

(u0nk , un1 k ),

kξ0nk kE0

w with ≤ C, where C is independent of k, the topology J : [L∞ loc (R; E0 )] + and Π+ ξu = ξu |t≥0 is the restriction of ξu ∈ K to the semi-axis R .

∗

Lemma 3.5. The shift operator T (s) defined in (11) is closed in (K+ , J + ) for every s ≥ 0, that is, for any sequence {ξul } ⊂ K+ , J + − lim ξul = ξu , l→∞

J + − lim T (s)ξul = z l→∞

imply z = T (s)ξu . Lemma 3.6. Let the assumptions of Theorem 3.4 hold, and ξu ∈ K+ . Then Mu (t) < ∞, kξu (t)k2 ≤ Mu (t), MT (s)u (t) ≤ Mu (t + s), Z ∞ Mu2 (t) + kut (τ )k2 dτ ≤ CMu2 (0)e−κt + C(kf kV−3 ), t > 0. t

Proof of Theorem 3.4. Let BR = {ξu ∈ K+ | sup Mu2 (t) ≤ R2 }. t≥0

Then BR is a precompact absorbing set of the trajectory dynamical system (T (s), (K+ , J + )) for R2 ≥ 2C(kf kV−3 ). Indeed, it follows from Lemma 3.6 that BR absorbs all M -bounded sets of K+ and T (s)BR ⊂ BR for s ≥ 0. Since BR is + + + bounded in L∞ loc (R ; E0 ), it is percompact in (K , J ). Let \ Atr = ωw∗ (BR ) = [T (h)BR ]J + , h≥0


801

where [ ]J + denotes the J + closure. By virtue of Lemma 3.5, verifying Definition 3.3 step by step one easily shows that Atr is the desired trajectory attractor. Now, we prove Atr = Π+ K. Since T (s)Atr = Atr , s ≥ 0, for any semi-trajectory ξu0 (t) ∈ Atr starting from u0 , there exists a sequence of semi-trajectory ξzn (t) starting from z n such that T (1)ξzn (t) = ξzn−1 (t), t ∈ R+ . Let t ≥ 0, ξu0 (t), ξu (t) = ξzn (t + n), t ≥ −n. Obviously, ξu = J − lim ξzn = J − lim ξun , n→∞

n

n→−∞

−n

with z = u . where ξun is the Galerkin solution of equation (6), with t ≥ tn = −n and ξun (tn ) = un , that is, there exists the Galerkin approximation ξunk of ξun such that J − limk→∞ ξunk = ξun . Therefore, there is the subsequence nk (l) such that ξu = J − liml→∞ ξunk (l) , where ξunk (l) is the solution of problem (14). Theorem 3.4 is proved. 4. Backward regularity of the full trajectory. Theorem 4.1. In addition to the assumptions of Theorem 3.4, if also g ∈ C 2 (R), 2 ≤ p < (N − 2)/(N − 4) if N ≥ 5, and f ∈ V−2 , then for every full trajectory ξu ∈ K, there exists a time T = T (u) such that ξu ∈ Cb ((−∞, T ]; E1 ). Let ξu ∈ K be an arbitrary full trajectory of equation (6). Rewrite equation (6) in the form 1

1

1

(A− 2 + I)utt + (I + A 2 )u + ut + g(u) + LA− 2 u = h(t), with h(t) ≡ A

− 21

(15)

− 12

f + LA u. It follows from Theorem 3.4 and (9) that, Z +∞ 1 kh(t)k2 + kht (τ )k2V2 dτ ≤ C0 ≡ C(kA− 2 f k), t ∈ R, −∞

ht ∈ Cb (R; V2−δ ),

lim kht (t)kV2−δ = 0.

(16)

t→∞

Lemma 4.2. Under the assumptions of Theorem 4.1, for a sufficiently large L, there exists time T = T (u, L) such that the equation 1

1

1

(A− 2 + I)vtt + (I + A 2 )v + vt + g(v) + LA− 2 v = h(t), t ≤ T,

(17)

possesses a unique regular backward solution ξv ∈ C((−∞, T ]; E1 ), and kvt (t)k2V2 + kv(t)k2V2 ≤ C0 , t ≤ T,

lim kvt (t)kV2 = 0.

t→−∞

(18)

In order to prove Lemma 4.2, we first consider the parabolic equation 1

1

wt + (I + A 2 )w + g(w) + LA− 2 w = h(t), t ∈ R.

(19)

By the standard arguments, we have Lemma 4.3. Under the assumptions of Theorem 4.1, for sufficiently large L, equation (19) possesses a unique solution w, with ξw = (w, wt ) ∈ Cb (R; V3 × V2 ), and the following estimates hold kw(t)kV3 + kwt (t)kV2 ≤ C0 , wtt ∈ L2 (t, t + 1; V1 ), t ∈ R, lim (kwt (t)kV2 + kwtt kL2 (t,t+1;V1 ) = 0.

t→−∞

(20)

802


Proof of Lemma 4.2. Let v = w + z. The function z(t) solves the equation F (t, z) = 0,

(21)

where 1

1

1

˜ F (t, z) ≡ (A− 2 + I)ztt + (I + A 2 )z + zt + g(w + z) − g(w) + LA− 2 z + h(t), ˜ ≡ (A− 12 + I)wtt . We see from (20) that limT →−∞ khk ˜ L2 (T,T +1;V ) = and where h(t) 1 0. We applying the implicit function theorem to solve equation (21). Define the space L2b (−∞, T ; V1 ) = {f ∈ L2loc (−∞, T ; V1 )|

kf kL2 (t−1,t;V1 ) } < ∞}

sup t∈(−∞,T ]

equipped with the norm kf kL2b (−∞,T ;V1 ) ≡ supt∈(−∞,T ] kf kL2 (t−1,t;V1 ) . Obviously, F : (−∞, T ] × Cb ((−∞, T ]; E1 ) → L2b (−∞, T ; V1 ) and F (∞, 0) = 0. We verify that the variation equation at z = 0 1

1

1

¯ t≤T (A− 2 + I)vtt + (I + A 2 )v + vt + g 0 (w)v + LA− 2 v = h(t),

(22)

¯ ∈ L2 (−∞, T ; V1 ) if T is suffiis uniquely solvable in Cb ((−∞, T ]; E1 ) for every h b ciently small. Indeed, using the multiplier vt + v in (22) gives d ¯ vt + v) + (g 00 (w)wt , |v|2 ), E1 (t) + H1 (t) = 2(h, dt

(23)

where 1

1

E1 (t) = kA− 4 vt k2 + kvt k2 + (1 + )kvk2 + kA 4 vk2 + (g 0 (w)v, v) 1

1

+ LkA− 4 vk2 + 2((A− 2 + I)vt , v), 1

1

H1 (t) = 2kvt k2 + 2[kvk2 + kA 4 vk2 + LkA− 4 vk2 + (g 0 (w)v, v)] 1

− 2(kA− 4 vt k2 + kvt k2 ). By assumption (8) and the interpolation theorem, 1 (g 0 (w)v, v) ≥ − (kvk2V1 + Lkvk2V−1 ) + C1 2 Z E1 (t) ∼ kvt k2 + kvk2V1 + |w|p−1 |v|2 dx

Z

|w|p−1 |v|2 dx,

Ω

(24)

Ω

for > 0 suitably small. Since kwt (t)kV2 → 0 as t → −∞, we have |(g 00 (w)wt , v 2 )| ≤C(1 + kwkp−2 2N (p−2) )kwt k 8−N

2N N −4

kvk22N N −2

≤C(1 + ≤ kvk2V1 , 4 for T sufficiently small. Hence, when > 0 is sufficiently small, 2 kwkp−2 V2 )kwt kV2 kvkV1

t ≤ T,

d ¯ 2, E1 (t) + κE1 (t) ≤ Ckhk H1 (t) ≥ κE1 (t) + (kvt k2 + kvk2V1 ), 2 dt Z Z t 2 2 p−1 2 2 −κ(t−s) ¯ kvt (t)k + kv(t)kV1 + |w| |v| dx ≤ C kh(s)k e ds, Ω

−∞

(25)


803

with t ≤ T. So, the solution of equation (22) is unique. Similarly, using the multiplier 1 1 A 2 vt + A 2 v in (22), and noticing that 1 1 1 1 p−1 0 2 2 |(g (w)v, A v)| ≤ CkA vk kvk + kvk 2N kwk N (p−1) ≤ kA 2 vk2 + Ckvk2 , N −2(2−δ) 4 2−δ 1 1 2N |(g 0 (w)vt + g 00 (w)wt v, A 2 v)| ≤ CkA 2 vk (1 + kwkp−1 N (p−1) )kvt k N −2(1−δ) 1−δ + (1 + kwkp−2 2N (p−2) )kwt k 2N kvk 2N N −4 N −4 8−N 1 p−2 ≤CkA 2 vk (1 + kwkp−1 V2 )kvt kV1−δ + (1 + kwkV2 )kwt kV2 kvkV2 1 1 ≤ (kA 2 vk2 + kA 4 vt k2 ) + Ckvt k2 , t ≤ T, 2 for T sufficiently small, we get Z t 2 2 2 −κ(t−s) ¯ kvt (t)kV1 + kv(t)kV2 ≤ C kh(s)k ds, t ≤ T. (26) V1 e −∞

That is, the variational equation (22) is indeed solvable in space Cb ((−∞, T ]; E1 ). Applying the implicit function theorem to equation (21) we deduce that it has a unique solution ξz ∈ Cb ((−∞, T ]; E1 ) if T ∈ R is sufficiently small. Repeating the similar arguments to equation (21) one easily gets Z t 2 −κ(t−s) ˜ ˜ L2 (−∞,T ;V ) → 0 (27) kzt (t)k2 + kz(t)k2 ≤ C kh(s)k e ds ≤ Ckhk V1

V2

−∞

V1

b

1

as T → −∞. The combination of (25) with (20) yields kvt (t)k2V1 + kv(t)k2V2 ≤ C,

lim kvt (t)kV1 = 0.

t→−∞

(28)

Differentiating equation (17) with respect to t and setting y˜ = vt , we get 1

1

1

(A− 2 + I)˜ ytt + (I + A 2 )˜ y + y˜t + g 0 (v)˜ y + LA− 2 y˜ = ht (t), t ≤ T. Repeating the similar arguments used above one easily gets Z t k˜ yt (t)k2V1 + k˜ y (t)k2V2 ≤ C kht (s)k2V1 e−κ(t−s) ds → 0 −∞

as t → −∞. That is, (18) holds. Lemma 4.2 is proved. Lemma 4.4. ([15]). Assume that g satisfies condition (8). Then for any v, U ∈ R, G(v + U ) − G(v) − g(v)U ≥ −K|U |2 + δp |U |2 (|v|p−1 + |U |p−1 ), K G(v + U ) − G(v) − g(v + U )U ≤ |U |2 − δp0 |U |2 (|v|p−1 + |U |p−1 ), 2 |g(v + U ) − g(v) − g 0 (v)U | ≤ C|U |2 (1 + |v|p−2 + |U |p−2 ), Rs where G(s) = 0 g(τ )dτ, δp , δp0 are positive constants depending only on p. Proof of Theorem 4.1. By Lemma 4.2, it is sufficient to prove u(t) = v(t) for t ≤ T . Let ξunk (t), t ≥ tk , be a sequence of the Galerkin approximation of ξu ∈ K, that is tk → −∞, ξu = K − lim ξunk , k→∞

804


and the sequence ξunk (tk ) = ξk0 is uniformly bounded in E0 with respect to k. Let v nk (t) = Pnk v(t), t ≤ T, where v is constructed in Lemma 4.2, and ξv ∈ Cb ((−∞, T ]; V2 × V2 ), lim kξvnk − ξv kCb ((−∞,T ];V2 ×V2 ) = 0.

k→∞

We set U (t) = u(t) − v(t), U nk (t) = unk (t) − v nk (t). Then 1

1

1

(A− 2 + I)Uttnk + (I + A 2 )U nk + Utnk + Pnk (g(ζ nk ) − g(v nk )) + LA− 2 U nk =hnk (t) ≡ Pnk (g(v) − g(v nk )),

t > tk ,

(29)

with ξU nk (tk ) = ξk0 − Pnk ξv (tk ), kξU nk (tk )kE0 ≤ C and where ζ nk = v nk + U nk , p−1 khnk (t)k ≤ C 1 + kv nk kp−1 + kvk kv nk − vk 2N ≤ Ckv nk − vkV2 → 0 N (p−1) N (p−1) 2

N −4

2

uniformly on (−∞, T ] as k → ∞. Using the multiplier Utnk + U nk in (29) yields d E2 (t) + H2 (t) = 2(hnk , Utnk + U nk ), dt

(30)

where 1

1

1

E2 (t) = kA− 4 Utnk k2 + kUtnk k2 + (1 + )kU nk k2 + kA 4 U nk k2 + LkA− 4 U nk k2 1

1

+ 2[(A− 4 Utnk , A− 4 U nk ) + (Utnk , U nk )] + 2 G(ζ nk ) − G(v nk ) − g(v nk )U nk , 1 , 1 H2 (t) = 2 (g(ζ nk ) − g(v nk ))U nk , 1 + (1 − 2)kUtnk k2 − 2kA− 4 Utnk k2 − 2 g(ζ nk ) − g(v nk ) − g 0 (v nk )U nk , vtnk 1

1

+ 2[kU nk k2 + kA 4 U nk k2 + LkA− 4 U nk k2 ], H2 (t) − E2 (t) 1

1

1

=(1 − 3)kUtnk k2 − 3kA− 4 Utnk k2 + [kA 4 U nk k2 + LkA− 4 U nk k2 + (1 − )kU nk k2 ] 1 1 − 22 [(A− 4 Utnk , A− 4 U nk ) + (Utnk , U nk )] + 2 g(ζ nk )U nk − G(ζ nk ) + G(v nk ), 1 − 2 g(ζ nk ) − g(v nk ) − g 0 (v nk )U nk , vtnk . (31) By Lemma 4.4, for L suitably large and suitably small, Z C1 kUtnk k2 + kU nk k2V1 + |U nk |2 (|U nk |p−1 + |v nk |p−1 )dx Ω nk 2 nk 2 nk p+1 + kv k ≤E2 (t) ≤ C2 kUt k + kU kV1 + kU nk kp+1 p+1 p+1 , 2 g(ζ nk )U nk − G(ζ nk ) + G(v nk ), 1 Z nk 2 nk 2 0 ≥ − kU kV1 − CkU kV−1 + δp |U nk |2 (|U nk |p−1 + |v nk |p−1 )dx, 4 Ω n −2 g(ζ nk ) − g(v nk ) − g 0 (v nk )U nk , vt k

(32)


805

≤C |U nk |2 (1 + |v nk |p−2 + |U nk |p−2 ), |vtnk | Z δp0 nk ≤ |U nk |2 (|U nk |p−1 + |v nk |p−1 )dx + CkU nk k2V1 (kvtnk kp−1 V2 + kvt kV2 ). (33) 2 Ω Substituting (32)-(33) into (31), taking L suitably large and suitably small and noticing limt→−∞ kvtnk (t)kV2 = 0, we get H2 (t) − E2 (t) ≥κ kUtnk (t)k2 + kU nk (t)k2V1 + kU nk (t)kp+1 p+1 Z + |U nk |2 |v nk |p−1 dx (34) Ω

for T sufficiently small. Substituting (34) into (30) gives d E2 (t) + E2 (t) ≤ Ckhnk (t)k2 , t > tk , dt kUtnk (t)k2 + kU nk (t)k2V1 + kU nk (t)kp+1 p+1 Z t ≤C(kξU nk (tk )kE0 )e−(t−tk ) + C khnk (s)k2 e−(t−s) ds.

(35)

tk

Letting k → ∞ we get kUt (t)k2 + kU (t)k2V1 + kU (t)kp+1 p+1 = 0,

t ∈ (−∞, T ].

(36)

Theorem 4.1 is proved. Corollary 4.5. Under the assumptions of Theorem 4.1, let ξu ∈ K be a bounded full trajectory of equation (6). Then kξu kCb ((−∞,T ];V2 ×V2 ) ≤ C0 .

(37)

Theorem 4.6. Let the assumptions of Theorem 4.1 hold, ξv ∈ K be a bounded full trajectory of equation (6) satisfying (37), and ξu ∈ K be another full trajectory of equation (6) which satisfies ξu (t) = ξv (t) for t ≤ T 0 < T. Then ξu (t) = ξv (t) for t ≤ T. Proof. Let ξvnk = Pnk ξv , U (t) = u(t) − v(t), U nk (t) = unk (t) − v nk (t). Then U nk solves the equation 1

1

1

(A− 2 + I)Uttnk + (I + A 2 )U nk + Utnk + Pnk (g(ζ nk ) − g(v nk )) + LA− 2 U nk 1

=hnk (t) ≡ Pnk (g(v) − g(v nk )) + LA− 2 U nk ,

t > tk ,

(38)

with ξU nk (tk ) = ξk0 − Pnk ξv (tk ). Obviously, 1

hnk → LA− 2 U

in Cb ((−∞, T ]; H), khnk kCb ((−∞,T ];H) ≤ C0 .

Repeating the proof of Theorem 4.1 one can easily sees that (35) still holds for L sufficiently large and suitably small, and hence Z T0 Z t E2 (t) ≤ E2 (tk )e−(t−tk ) + C + e−(t−s) khnk (s)k2 ds. (39) tk

T0

Letting k → ∞, we get kut (t) − vt (t)k2 + ku(t) − v(t)k2V1 + ku(t) − v(t)kp+1 p+1 Z T ≤CL2 ku(s) − v(s)k2V−2 ds, t ∈ [T 0 , T ]. T0

(40)

806


Applying the Gronwall inequality to (40) yields u(t) = v(t) for t ∈ (−∞, T ]. Theorem 4.6 is proved. Remark 4.7. When p ≤ N/(N − 2)+ , we have known that problem (6), (7) possesses a unique solution ξu ∈ Cb (R+ ; E0 ). So we can define the evolution semigroup according to the standard manner: S(t) : E0 → E0 , S(t)ξu (0) = ξu (t). Define the operator Π0 : K+ → E0 , Π0 ξu = ξu (0). Obviously, Π0 is one to one and it is a tr homeomorphism between K+ and E0 , T (t) = Π−1 = A is the 0 S(t)Π0 , and Π0 A + −1 tr global attractor of S(t) in E0 . Since A ⊂ E1 , A = Π0 A ⊂ Cb (R ; E1 ) and thus the set of the full trajectory K ⊂ Cb (R; E1 ). REFERENCES [1] G. W. Chen, Y. P. Wang and S. B. Wang, Initial boundary value problem of the generalized cubic double dispersion equation, J. Math. Anal. Appl., 299 (2004), 563–577. [2] G. W. Chen and H. X. Xue, Periodic boundary value problem and Cauchy problem of the generalized cubic double dispersion equation, Acta Mathematica Scientia, 28B(3) (2008), 573–587. [3] V. Chepyzhov and M. Vishik, “Attractors for Equations of Mathematical Physics”, American Mathematical Society Colloquium Publications, 49 (Providence, RI: American Mathematical Society), 2002. [4] Z. D. Dai and B. L. Guo, Global attractor of nonlinear strain waves in elastic wave guides, Acta Math. Sci., 20(B) (2000), 322–334. [5] Y. C. Liu and R.Z. Xu, Potential well method for Cauchy problem of generalized double dispersion equations, J. Math. Anal. Appl., 338 (2008), 1169–1187. [6] Y. C. Liu and R. Z. Xu, Potential well method for initial boundary value problem of the generalized double dispersion equations, Communications on Pure and Applied Analysis, 7 (2008), 63–81. [7] A. M. Samsonov and E. V. Sokurinskaya, Energy exchange between nonlinear waves in elastic wave guides in external media, in “Nonlinear Waves in Active Media”, Springer, Berlin, (1989),99–104. [8] A. M. Samsonov, Nonlinear strain waves in elastic waveguide, “Nonlinear Waves in Solids, in Cism Courses and Lecture”, (eds. A. Jeffery and J. Engelbrechet), vol. 341, Springer, Wien, 1994. [9] A. M. Samsonov, On Some Exact Travelling Wave Solutions for Nonlinear Hyperbolic Equation, in “Pitman Research Notes in Mathematics Series”, vol. 227, Longman, (1993), 123–132. [10] S. B. Wang and G. W. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal. TMA., 64 (2006), 159–173. [11] R. Z. Xu and Y. C. Liu, Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations, J. Math. Anal. Appl., 359 (2009), 739–751. [12] R. Z. Xu, Y. C. Liu and T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations, Nonlinear Anal. TMA., 71 (2009), 4977–4983. [13] Z. J. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model, Nonlinear Anal. TMA., 70 (2009), 2132–2142. [14] Z. J. Yang, A global attractor for the elastic waveguide model in RN , Nonlinear Anal. TMA., 74 (2011), 6640–6661. [15] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Disc. Cont. Dyn. System-A, 11 (2004), 351–392.

Received for publication September 2012. E-mail address: [email protected] E-mail address: [email protected]