Time-dependent attractor of wave equations with nonlinear damping and linear memory

Qiaozhen Ma; Jing Wang; Tingting Liu

doi:10.1515/math-2019-0008

Article Open Access

Time-dependent attractor of wave equations with nonlinear damping and linear memory

Qiaozhen Ma , Jing Wang and Tingting Liu

Published/Copyright: March 10, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this article, we consider the long-time behavior of solutions for the wave equation with nonlinear damping and linear memory. Within the theory of process on time-dependent spaces, we verify the process is asymptotically compact by using the contractive functions method, and then obtain the existence of the time-dependent attractor in H01(Ω)×L2(Ω)×Lμ2(R+;H01(Ω)) .

Keywords: wave equation; linear memory; contractive functions; time-dependent attractor

MSC 2010: 35B25; 37L30; 45K05

1 Introduction

Let Ω be an open bounded set of ℝ³ with smooth boundary ∂Ω. For any τ ∈ ℝ, we consider the following equations

ε(t)utt+a(x)g(ut)−Δu−∫0∞μ(s)Δηt(s)ds+f(u)=h(x), in Ω×(τ, ∞),u(x,t)=0, x∈∂Ω, t∈R,u(x,t)=u0(x,t),ut(x,t)=∂tu0(x,t), x∈Ω, t≤τ, (1.1)

where u = u(x, t) : Ω ×[τ, ∞) → ℝ is an unknown function, and u₀ : Ω ×(−∞, τ] → ℝ is a given past history of u, h(⋅) ∈ L²(Ω) is independent of time, μ is a summable positive function. η = η^t(x, s) := u(x, t) − u(x, t − s), s ∈ ℝ⁺. ε ∈ C¹(ℝ) is a decreasing bounded function and satisfies

limt→+∞ε(t)=0. (1.2)

In particular, there exists L > 0, such that

supt∈R[|ε(t)|+|ε′(t)|]≤L. (1.3)

The function a(x) satisfies:

a(x)∈L∞(Ω), a(x)≥α0>0 in Ω, (1.4)

where α₀ is a constant.

Like in [1,2], the nonlinear damping g ∈ C¹(ℝ), g(0) = 0, g is strictly increasing, and satisfies

lim inf|s|→∞g′(s)>0, (1.5)

g(s)|≤C(1+|s|p), 1≤p<5. (1.6)

The nonlinear term f ∈ C¹(ℝ), f(0) = 0, and for some C₀ ≥ 0 satisfies

|f′(s)|≤C0(1+|s|2), ∀ s∈R, (1.7)

along with the dissipation condition

lim inf|s|→∞f(s)s>−λ1, ∀ s∈R, (1.8)

where λ₁ is the first eigenvalue of the strictly positive operator A = −Δ.

With respect to the memory component, as in [3,4,5, 6], we assume that

μ∈C1(R+)∩L1(R+), ∫0∞μ(s)ds=m0<∞, (1.9)

μ′(s)≤−ρμ(s)≤0, ∀s≥0, (1.10)

where ρ is a positive constant.

The problem (1.1) can be viewed as a description of viscoelastic solids with fading memory and dissipation due to the viscous resistance of the surrounding medium, as well as of composite materials, phase-fields, and wave phenomena [7,8,9].

When μ is a Dirac measure at some fixed time instant or when it vanishes, the equation (1.1) reduces to the nonlinear damped wave equation, which has been investigated extensively by many authors. For instance, in the case that ε is a positive constant independent of time, the long-time behavior of the solution can be well characterized by using the concept of global attractors in the framework of semigroup. The existence and regular properties of the global attractor have been studied in [2, 10,11,12]. When ε is a positive constant independent of time and the forcing term h depends on time, the system is a non-autonomous wave equation, the long-time behavior of the solution can be understood in the framework of process. We refer the reader to [11, 13,14,15,16] for some specific results involving the uniform attractor (or pullback attractors) about non-autonomous case.

When ε is still a positive constant and the nonlinear damping a(x)g(t) is either linear damping αu_t or strong damping Δu_t, Conti and Pata ([17]), Borni and Pata ([18]), Pata and Zucchi ([19]) investigated the existence of global attractors for (1.1). Sun, Cao and Duan ([20]) obtained the existence and asymptotic regularity of the uniform attractor about the non-autonomous system with strong damping, while the robust exponential attractors was scrutinized by Kloden, Real and Sun ([21])

However, provided that ε depends explicitly on time in (1.1), such as a positive decreasing function of time ε(t) vanishing at infinity, leading to time-dependent terms at functional level, these problems become more complex and interesting, because the corresponding dynamical system is still understood within the non-autonomous framework even the forcing term is independent of time, and the classical theory generally fails to capture the dissipation mechanism of the system, as mentioned in [22,23].

To circumvent these issue, in [22], Conti, Pata and Temam presented a notion of time-dependent attractor by exploiting the minimality with respect to the pullback attraction property, and constructed a sufficient condition proving the existence of time-dependent attractor based on the theory established by Plinio, Duane and Temam([23]). Meanwhile, within the new framework, the authors studied the following weak damped wave equations with time-dependent speed of propagation

ε(t)utt+αut−Δu+f(u)=g(x). (1.11)

Besides, they proved that the time-dependent global attractor of (1.11) converged in a suitable sense to the attractor of the parabolic equation αu_t − Δ u + f(u) = g(x) when ε(t) → 0 as t → + ∞ ([24]). Successively, in [25], they continued to show the existence of an invariant time-dependent global attractor to the following specific one-dimensional wave equation ε(t)u_tt − u_xx + [1 + ε f′(u)]u_t + f(u) = h, which converges in suitable sense to the classical Fourier equation.

Recently, Meng et al. investigated the long-time behavior of the solution for the wave equation with nonlinear damping g(u_t) on the time-dependent space, in which they found a new technical method verifying compactness of the process via defining the contractive functions, see [1]. In [26], Meng and Liu also showed the necessary and sufficient conditions of the existence of time-dependent global attractor borrowed from the ideas in [10]. Liu and Ma ([27]) studied the existence of the pullback attractors for the plate equation with time-dependent forcing term on the strong time-dependent Hilbert space. Successively, exploiting the methods and framework of [22, 24], Liu and Ma obtained the existence and regularity of the time-dependent attractor for the plate equation with critical growth nonlinearity, as well as the asymptotic structure in [28].

As we know, in the study of the long-time behavior, especially for attractors, obtaining certain asymptotic compactness of the solution operator is a key step. However, if the equation contains the history memory, for instance, just for our problem (1.1), it makes impossible to utilize (I − P_m)u as the test function to capture the asymptotic compactness of the solution process, that is to say, the methods introduced in [10, 26] is out of action to our problem. On the other hand, because of the critical nonlinear damping, the technique of operator decomposition in [22] is not suitable to deal with (1.1) anymore. Thus, for our problem, we need to make a priori estimates to solution on a new triple solution space, and then verify the compactness of the solution process by exploiting the method of contractive function.

It is worth mentioning that we use the more weaker dissipative condition (1.8) than [1, 22]; indeed, for simplicity, in which the authors made use of the dissipative condition like lim inf|s|→∞f′(s)>−λ1 .

For convenience, hereafter, C (or c) denotes an arbitrary positive constant which may be different from line to line even in the same line.

The rest of this article consists of two Sections. In the next Section, we define some functions sets and iterate some useful lemmas. In Section 3, the existence of the time-dependent global attractor is obtained.

2 Preliminaries

As in Borini, Pata [18], Pata, Zucchi [19] and Dafermos [4], we introduce the past history of u, i.e. η = η^t(x, s), as a new variable of the system, which will be ruled by a supplementary equation: denoting

ηt=∂∂tη, ηs=∂∂sη, (2.1)

then we can rewrite (1.1) as

ε(t)utt+a(x)g(ut)−Δu−∫0∞μ(s)△ηt(s)ds+f(u)=h(x),ηtt=−ηst+ut, (2.2)

with initial boundary conditions

u(x,t)=0, x∈∂Ω, t≥τ,ηt(x,s)=0, (x,s)∈∂Ω×R+, t≥τ,u(x,τ)=u0(x), ut(x,τ)=u1(x), x∈Ω,ητ(x,s)=η0(x,s), (x,s)∈Ω×R+, (2.3)

where

u0(x)=u0(x,τ), u1(x)=∂tu0(x,t)|t=τ,η0=η0(x,s)=u0(x,τ)−u0(x,τ−s).

Without loss of generality, set H = L²(Ω) with the inner product 〈⋅, ⋅〉 and norm ∥⋅∥, respectively. For s ∈ ℝ⁺, we define the hierarchy of (compactly) nested Hilbert spaces

Hs=D(As2), 〈w,v〉s=〈As2w,As2v〉, ∥w∥s=∥As2w∥;

especially, we have the embeddings H^s+1 ↪ H^s.

For s ∈ ℝ⁺, let Lμ2 (ℝ⁺ ; H^s) be the family of Hilbert spaces of functions φ : ℝ⁺ → H^s, equipped with the inner product and norm, respectively,

〈φ1,φ2〉μ,s=〈φ1,φ2〉μ,Hs=∫0∞μ(s)〈φ1(s),φ2(s)〉Hsds,

∥φ∥μ,s2=∥φ∥μ,Hs2=∫0∞μ(s)∥φ(s)∥s2ds.

Now, for t ∈ ℝ and s ∈ ℝ⁺, we have the following time-dependent spaces

Hts=Hs+1×Hs×Lμ2(R+;Hs+1),

with the norm

∥z∥Hts2=∥(u,ut,ηt)∥Hts2=∥u∥s+12+ε(t)∥ut∥s2+∥ηt∥μ,s+12.

Especially, denote A = −Δ with domain D(A) = H²(Ω) ∩ H01 (Ω), we have

Ht1=H2×H1×(Lμ2(R+;H2)∩Hμ1(R+;H1))=D(A)×H01(Ω)×(Lμ2(R+;D(A))∩Hμ1(R+;H01(Ω))),

Ht=Ht0=H1×H×Lμ2(R+;H1)=H01(Ω)×L2(Ω)×Lμ2(R+;H01(Ω)).

Here

Hμ1(R+;Hs)={φ:φ(t),∂tφ(t)∈Lμ2(R+;Hs)}.

For every t ∈ ℝ, let X_t be a family of normed spaces, we introduce the R-ball of X_t

Bt(R)={z∈Xt:∥z∥Xt≤R}.

We denote the Hausdorff semi-distance of two (nonempty) sets B, C ⊂ X_t by:

δt(B,C)=supx∈BdistXt(x,C)=supx∈Binfy∈C∥x−y∥Xt.

For any given ϵ > 0, the ϵ-neighbourhood of a set B ⊂ X_t is defined as

Otϵ(B)=⋃x∈B{y∈Xt|∥y−x∥Xt<ϵ}=⋃x∈B{x+Bt(ϵ)}.

Finally, given any set B ⊂ X_t, the symbol B̄ stands for the closure of B in X_t.

Now we iterate some basic notations and abstract results, which are necessary for getting our main results.

Definition 2.1

^[22] Let X_t be a family of normed spaces. A process is a two-parameter family of mappings {U(t, τ) : X_τ → X_t, t ≥ τ ∈ ℝ} with properties

U(τ, τ) = Id is the identity on X_τ, τ ∈ ℝ;
U(t, s)U(s, τ) = U(t, τ), ∀, t ≥ s ≥ τ.

Definition 2.2

^[22] A family ℭ = {C_t}_t∈ℝ of bounded sets C_t ⊂ X_t is called uniformly bounded if there exist a constant R > 0 such that C_t ⊂ 𝔹_t(R), ∀ t ∈ ℝ.

Definition 2.3

^[22] A time-dependent absorbing set for the process U(t, τ) is a uniformly bounded family 𝔅 = {B_t}_t∈ℝ with the following property : for every R > 0 there exists a t₀ such that

τ≤t−t0⇒U(t,τ)Bτ(R)⊂Bt.

Definition 2.4

^[22] A (uniformly bounded) family 𝔎 = {K_t}_t∈ℝ is called pullback attracting if for all ϵ > 0, the family {Otϵ(Kt)}t∈R is pullback absorbing.

Definition 2.5

^[22] We call a time-dependent global attractor is the smallest element of 𝕂, i.e. the family 𝔄 = {A_t}_t∈ℝ ∈ 𝕂 such that A_t ⊂ K_t, ∀ t ∈ ℝ, for any element 𝔎 = {K_t}_t∈ℝ ∈ 𝕂.

Definition 2.6

^{[22, 24]} We say 𝔄 = {A_t}_t∈ℝ is invariant if

U(t,τ)Aτ=At, ∀ t ≥ τ.

Theorem 2.7

^[22]

If U(t, τ) is asymptotically compact, then there exists a unique time-dependent attractor 𝔄 = {A_t}_t∈ℝ;
If U(t, τ) is a T-closed process for some T > 0, which possesses a time-dependent global attractor 𝔄 = {A_t}_t∈ℝ, then 𝔄 is invariant.

Lemma 2.8

^{[1, 29]} Let g(⋅) satisfy condition (1.5). Then for any δ > 0, there exists a positive constant C_δ, such that |u − v|² ≤ δ + C_δ(g(u) − g(v))(u − v) for all u, v ∈ ℝ.

Theorem 2.9

^[1] Let U(⋅, ⋅) be a process in a family of Banach space {X_t}_t∈ℝ. Then U(⋅, ⋅) has a time-dependent global attractor 𝓤^* = {At∗}t∈R satisfying

At∗=⋂s≤t⋃τ≤sU(t,τ)Bτ¯,

if and only if

U(⋅, ⋅) has a pullback absorbing family 𝓑 = {B_t}_t∈ℝ;
U(⋅, ⋅) is pullback asymptotically compact.

Theorem 2.10

^[1] Let U(⋅, ⋅) be a process on {X_t}_t∈ℝ and has a pullback absorbing family 𝓑 = {B_t}_t∈ℝ. Moreover, assume that for any ε > 0 there exist T(ε) ≤ t, ϕTt ∈ 𝓒(B_T), such that

∥U(t,T)x−U(t,T)y∥≤ε+ϕTt(x,y), ∀x,y∈BT,

for ant fixed t ∈ ℝ. Then U(⋅, ⋅) is pullback asymptotically compact, where 𝓒(B_T) denotes the set of all contractive function on B_T × B_T.

For nonlinear function g, by condition (1.6) we have

|g(s)|p+1p=|g(s)|1p⋅|g(s)|≤C(1+|s|)|g(s)|≤C′+C′g(s)⋅s, (2.4)

furthermore, there holds

|g(s)|≤C+C(g(s)s)pp+1. (2.5)

Lemma 2.11

^[20] Let F(u) = ∫0uf(y)dy . From (1.8), we can get for 0 < ν < 1 and c_i > 0 (i = 1, 2), there hold

2〈F(u),1〉≥−(1−ν)∥u∥12−c1, (2.6)

〈f(u),u〉≥−(1−ν)∥u∥12−c2, ∀ u∈H1. (2.7)

3 Existence of the time-dependent global attractor

3.1 Well-posedness and time-dependent absorbing set

Now we state the results about the well-posedness of system (2.2)-(2.3) which can be found in [19, 20]. In fact, the existence of solution (u(t), u_t(t), η^t(s)) to (2.2)-(2.3) is obtained by using the standard Galerkin approximation method, which is based on Lemma 3.2 below.

Lemma 3.1

Under the assumptions (1.2)-(1.10), for every initial data z_τ = (u₀, u₁, η⁰) ∈ 𝓗_τ, there exists a unique solution z(t) = (u(t), u_t(t), η^t(s)) of problem (2.2)-(2.3) in space 𝓗_t, and for any τ ∈ ℝ, t ≥ τ, it satisfies

u∈C([τ,t];H1), ut∈C([τ,t];H), ηt∈C([τ,t];Lμ2(R+;H1)).

Furthermore, let z_i(τ) ∈ 𝓗_τ be the initial data such that ∥z_i(τ)∥_{𝓗_τ} ≤ R (i = 1, 2), and z_i(t) be the solution of problem (2.2)-(2.3). Then there exists C = C(R) > 0, such that

∥z1(t)−z2(t)∥Ht≤eC(t−τ)∥z1(τ)−z2(τ)∥Hτ, ∀ t≥τ. (3.1)

Thus, the system (2.2)-(2.3) generates a strongly continuous process U(t, τ), where

U(t,τ):Hτ→Ht acting as U(t,τ)z(τ)={u(t),ut(t),ηt(s)},

with the initial data z_τ = z(τ) = {u₀, u₁, η⁰} ∈ 𝓗_τ.

To prove Lemma 3.1, we first need the following estimate:

Lemma 3.2

Under the assumptions (1.2)-(1.10), for any initial data z(τ) ∈ 𝔹_τ(R) ⊂ 𝓗_τ, there exists R₀ > 0, such that

∥U(t,τ)z(τ)∥Ht≤R0, ∀ τ≤t.

Proof

Denote

E0(t)=12∥U(t,τ)z∥Ht2+∫ΩF(u)dx−∫Ωh⋅udx.

Multiplying (2.2)₁ with u_t in L² and exploiting (2.2)₂, we achieve

ddtE0(t)+∫Ωa(x)g(ut)⋅utdx+〈ηt(s),ηst(s)〉μ,1−ε′(t)2∥ut∥2=0. (3.2)

By Hölder, Young inequalities, and combining with (1.10) we obtain

〈ηt,ηst〉μ,1=12∫0∞μ(s)dds∥ηt(s)∥12ds=−12∫0∞μ′(s)∥ηt(s)∥12ds≥ρ2∫0∞μ(s)∥ηt(s)∥12ds=ρ2∥ηt(s)∥μ,12; (3.3)

Using that g is strictly increasing, ε(t) is decreasing, and (3.3), we have

∫Ωa(x)g(ut)⋅utdx−ε′(t)2∥ut∥2+〈ηt(s),ηst(s)〉μ,1≥α0∫Ωg(ut)⋅utdx−ε′(t)2∥ut∥2+ρ2∥ηt(s)∥μ,12>0.

Integrating (3.2) over [τ, t] we get

E0(t)≤E0(τ), ∀t≥τ. (3.4)

From (2.6) and Sobolev’s embeddings we deduce

E0(t)≥12ε(t)∥ut∥2+ν2∥u∥12+12∥ηt∥μ,12−ν4∥u∥12−1λ1ν∥h∥2−c12≥ν4[ε(t)∥ut∥2+∥u∥12+∥ηt∥μ,12]−(1λ1ν∥h∥2+c12)≥−(1λ1ν∥h∥2+c12). (3.5)

Consequently, there exist some proper positive constant C₁, C₂ and C₃, such that

C1∥U(t,τ)z∥Ht2−C2≤E0(t)≤C3∥U(t,τ)z∥Ht2.

And from (3.4), (3.5) yields

∫τt∫Ωa(x)g(ut)⋅utdx−12∫τtε′(s)∥ut(s)∥2ds+ρ2∫τt∥ηt(s)∥μ,12ds≤E0(τ)−E0(t)≤E0(τ)+1λ1ν∥h∥2+c12. (3.6)

On the other hand, (1.6), (2.5) along with the Hölder and Young inequality imply

|∫Ωa(x)g(ut)⋅udx|≤C∫Ωa(x)|u|dx+C∫Ωa(x)(g(ut)⋅ut)pp+1|u|dx ≤C∫Ωa(x)|u|dx+C(∫Ωa(x)g(ut)⋅utdx)pp+1(∫Ωa(x)|u|p+1)1p+1 ≤C∫Ωa(x)|u|dx+η∥u∥12+Cη∥u∥1p−1p∫Ωa(x)g(ut)⋅utdx, (3.7)

where η > 0 is a small enough constant, which will be determined later.

Multiplying (2.2)₁ by u_t + δu and integrating over Ω, we get

ddtE(t)+I(t)=0, (3.8)

where

E(t)=E0(t)+δε(t)〈ut,u〉,

I(t)=δ∥u∥12+(−ε′(t)2−δε(t))∥ut∥2+〈ηt(s),ηst(s)〉μ,1−δε′(t)〈ut,u〉+δ∫0∞μ(s)〈∇ηt(s),∇u(t)〉ds+δ〈f(u),u〉−δ〈h,u〉+∫Ωa(x)g(ut)⋅(ut+δu)dx,

therefore, we get

E(t)=E(τ)−∫τtI(s)ds. (3.9)

Together with Hölder, Young, Poincaré inequalities, it follows that

E(t)≥12ε(t)∥ut∥2+ν4∥u∥22+12∥ηt∥μ,12−δε(t)∥ut∥⋅∥u∥≥(ν4−δ2Lλ1)∥u∥12+14ε(t)∥ut∥2+12∥ηt∥μ,12−(1λ1ν∥h∥2+c12)≥ν8(∥u∥12+ε(t)∥ut∥2+∥ηt∥μ,12)−(1λ1ν∥h∥2+c12), (3.10)

here we use ν4−δ2Lλ1>ν8 for δ < ν small enough.

Thanks to (1.5)-(1.6), for any δ > 0, there exist C_δ > 0 such that

∫Ωa(x)g(ut)⋅utdx≥2δ∥ut∥2−Cδ|Ω|;

moreover,

δ|∫0∞μ(s)〈∇ηt(s),∇u(t)〉ds|≤δ∫0∞μ(s)∥∇ηt(s)∥⋅∥∇u(t)∥ds≤ρ4∥ηt(s)∥μ,12+δ2m0ρ∥u∥12. (3.11)

Collecting all the above estimates and due to (2.7), Hölder, Young inequalities, it leads to

I(t)≥2δ∥ut∥2−Cδ|Ω|−δ[C∫Ωa(x)|u|dx+η∥u∥12+Cη∥u∥1p−1p∫Ωa(x)g(ut)⋅utdx]+ρ4∥ηt(s)∥μ,12+(|ε′(t)|2−δε(t))∥ut∥2−12|ε′(t)|∥ut∥2−δ2L2λ1∥u∥12+δν∥u∥12−δc2−δ∥h∥⋅∥u∥−δ2m0ρ∥u∥12≥(2δ−δε(t))∥ut∥2+(δν2−δη−δ2L2λ1−δ2m0ρ)∥u∥12+ρ4∥ηt(s)∥μ,12−δCη∥u∥1p−1p∫Ωa(x)g(ut)⋅utdx−Cδ|Ω|−δ2ν∥h∥2−(C∥a∥L∞)2≥δν4(ε(t)∥ut∥2+∥u∥12+∥ηt(s)∥μ,12)−Cδ(E0(τ)∫Ωa(x)g(ut)⋅utdx)−Cδ(|Ω|+∥h∥2)−δc2−(C∥a∥L∞)2, (3.12)

where for δ < min{ν, ρ} small enough. Then, from (3.9)-(3.12), (3.6) yields

ν8(∥u∥12+ε(t)∥ut∥2+∥ηt(s)∥μ,12)−m1≤−δ∫τt[δν4(∥u(r)∥12+ε(r)∥ut(r)∥2+∥ηr(s)∥μ,12)−m2]dr+E(τ),

where m₁ = 1λ1ν∥h∥2+c12 + C_δE₀(τ)(E₀(τ) + 1λ1ν∥h∥2+c12 ), m₂ = C_δ(|Ω| + ∥h∥²) + δ c₂ + (C∥a∥_L^∞)². So, for any K₀ > 4m2δν , there exists t₀ > τ such that

∥u(t0)∥12+ε(t0)∥ut(t0)∥2+∥ηt0(s)∥μ,12≤K0.

As a result, let B_t = ⋃_t≥τ U(t, τ)B₀, where

B0={(u0,u1,η0)∈Hτ:∥u0∥12+ε(τ)∥u1∥2+∥η0(s)∥μ,12≤K0},

then B_t is a bounded absorbing set for process U(t, τ).

In addition, from the above discussion, there exists a positive constant R₀ such that

∥u(t)∥12+ε(t)∥ut(t)∥2+∥ηt(s)∥μ,12≤R0, ∀ t ≥t0>τ.

The proof for Lemma 3.1

Let z₁(τ), z₂(τ) ∈ 𝓗_τ such that ∥z_i(τ)∥_{𝓗_τ} ≤ R, i = 1, 2, and denote by C a generic positive constant depending on R but independent of z_i. We first observe that the energy estimates in Lemma 3.2 above ensure:

∥U(t,τ)zi(τ)∥Ht≤C. (3.13)

We set {u_i(t), ∂_t u_i(t), ηit (s)} = U(t, τ)z_i(τ) and denote z̄(t) = {ū(t), ū_t(t), η̄^t(s)} = U(t, τ)z₁(τ) − U(t, τ)z₂(τ).

Then the difference between the two solutions with initial data z̄(τ) = z₁(τ) − z₂(τ) fulfills

ε(t)u¯tt−Δu¯+a(x)(g(u1t)−g(u2t))−∫0∞μ(s)Δη¯t(s)ds+f(u1)−f(u2)=0.

Multiplying the above equation with 2ū_t and integrating over Ω, we obtain

ddt∥z¯∥Ht2−ε′(t)∥u¯t∥2+2∫Ωa(x)((g(u1t)−g(u2t)))u¯tdx+2〈η¯t(s),η¯st(s)〉μ,1=−2〈f(u1)−f(u2),u¯t〉.

From (1.4) and the strict increase of g, we have

2∫Ωa(x)(g(u1t)−g(u2t))u¯tdx≥0.

By exploiting (1.7), (3.13), and Hölder, Young inequality, and the embedding H¹ ↪ L⁶, it yields

−2〈f(u1)−f(u2),u¯t〉≤C∫Ω(1+|u1|2+|u2|2)⋅|u¯|⋅|u¯t|dx ≤C[1+∥u1∥L62+∥u2∥L62]⋅∥u¯∥1⋅∥u¯t∥ ≤C[1+∥u1∥12+∥u2∥12]⋅∥u¯∥1⋅∥u¯t∥ ≤C∥u¯∥1∥u¯t∥ ≤2∥u¯t∥2+C∥u¯∥12,

from (3.3) yields

〈η¯t,η¯st〉μ,1≥ρ2∥η¯t∥μ,12.

Consequently, we end up with the differential inequality

ddt∥z¯(t)∥Ht2≤C(L+1)1ε(t)∥z¯(t)∥Ht2,

then applying the Gronwall’s Lemma on [τ, t], we obtain

∥z¯(t)∥Ht2≤eC(L+1)∫τt1ε(s)ds∥z¯(τ)∥Hτ2.

3.2 A priori estimates

The main purpose of this part is to establish (3.25)-(3.27), which will be used to obtain the asymptotic compactness of the process.

Let (u_i(t), u_{i_t}(t), ηit (⋅)) (i = 1, 2) be the corresponding solution of (2.2)-(2.3) with initial datum (u0i(τ),v0i(τ),ηiτ) ∈ {B_τ}_τ∈ℝ. For convenience, we introduce notations

gi(t)=g(uit(t)), fi(t)=f(ui(t)), i=1,2,

and

w=u1(t)−u2(t), ζt=η1t−η2t,

then w(t) satisfies

ε(t)wtt−Δw+a(x)(g1(t)−g2(t))−∫0∞μ(s)Δζt(s)ds+f1(t)−f2(t)=0, t>T,w(x,T)=u01(T)−u02(T), wt(x,T)=v01(T)−v02(T), ζT=η1T−η2T,w∣∂Ω=ζt∣∂Ω=0. (3.14)

Denote

Ew(t)=12(∥w∥12+ε(t)∥wt∥2+∥ζt∥μ,12).

Taking the inner product (3.14)₁ with w_t in L²(Ω), and by (3.14)₂, we find

ddtEw(t)+〈a(x)(g1(t)−g2(t)),wt〉−ε′(t)2∥wt∥2+〈ζt,ζst〉μ,1+〈f1−f2,wt(ξ)〉=0. (3.15)

Integrating (3.15) over [s, t], we have

Ew(t)−Ew(s)+∫st〈a(x)(g1(ξ)−g2(ξ)),wt(ξ)〉dξ−12∫stε′(ξ)∥wt(ξ)∥2dξ+ρ2∫st∥ζt(ξ)∥μ,12dξ+∫st〈f1−f2,wt(ξ)〉dξ≤0, (3.16)

where T ≤ s ≤ t. Thanks to ε′(t) < 0, then from (3.16) yields

∫st〈a(x)(g1(ξ)−g2(ξ)),wt(ξ)〉dξ+ρ2∫st∥ζt(ξ)∥μ,12dξ≤Ew(s)−∫st〈f1−f2,wt(ξ)〉dξ,

thus, for a(x) ≥ α₀ > 0, g′ > 0, there holds

∫st∥ζt(ξ)∥μ,12dξ≤2ρEw(s)−2ρ∫st〈f1−f2,wt(ξ)〉dξ. (3.17)

Combining with (1.3) and Lemma 2.8 we get that, for any δ > 0, there exists C_δ > 0, such that

ε(ξ)|wt|2≤L|wt|2≤δL+LCδ(g1(ξ)−g2(ξ))wt(ξ),

namely

∫Ttε(ξ)∥wt(ξ)∥2dξ≤δL|Ω|(t−T)+CδLα0Ew(T)−LCδα0∫Tt〈f1−f2,wt(ξ)〉ds. (3.18)

Multiplying (3.14)₁ by w, and integrating over Ω × [T, t], we obtain

∫Tt∥w(s)∥12ds+ε(t)〈wt(t),w(t)〉=〈ε(T)wt(T),w(T)〉+∫Ttε′(s)〈wt(s),w(s)〉ds+∫Ttε(s)∥wt(s)∥2ds−∫Tt∫0∞μ(s)〈∇ζt(s),∇w(t)〉ds−∫Tt〈f1−f2,w(s)〉ds−∫Tt〈a(x)(g1−g2),w(s)〉ds.

Then, using the following inequality

∫0∞μ(s)〈∇ζt(s),∇w(t)〉ds≤12σ∥ζt∥μ,12+σm02∥w∥12, (3.19)

yields

(1−σm02)∫Tt∥w(s)∥12ds+ε(t)〈wt(t),w(t)〉≤〈ε(T)wt(T),w(T)〉+∫Ttε′(s)〈wt(s),w(s)〉ds+∫Ttε(s)∥wt(s)∥2ds+12σ∫Tt∥ζξ(s)∥μ,12dξ−∫Tt〈f1−f2,w(s)〉ds−∫Tt〈a(x)(g1−g2),w(s)〉ds. (3.20)

Therefore, let 0 < σ < 2m0 , from (3.17)-(3.18), (3.20) yields

2∫TtEw(s)ds≤(1+22−σm0)LCδα0+2σ(2−σm0)ρ+1Ew(T)+(1+22−σm0)δL|Ω|(t−T)−(1+22−σm0)LCδα0+2σ(2−σm0)ρ+1∫Tt〈f1−f2,wt(s)〉ds+22−σm0∫Ttε′(s)〈wt(s),w(s)〉ds−22−σm0ε(t)〈wt(t),w(t)〉+22−σm0ε(T)〈wt(T),w(T)〉−22−σm0∫Tt〈a(x)(g1−g2),w(s)〉ds−22−σm0∫Tt〈f1−f2,w(s)〉ds. (3.21)

Integrating (3.16) over [T, t], we have

(t−T)Ew(t)+∫Tt∫sta(x)〈g1(ξ)−g2(ξ),wt(ξ)〉dξds−12∫Tt∫stε′(ξ)∥wt(ξ)∥2dξds+ρ2∫Tt∫st∥ζt∥μ,12dξds≤−∫Tt∫st〈f1(ξ)−f2(ξ),wt(ξ)〉dξds+∫TtEw(s)ds. (3.22)

Since 〈g₁(ξ) − g₂(ξ), w_t(ξ)〉 − 12 ε′(ξ)∥w_t∥² + ∥ζt∥μ,22 > 0, together with (3.21)-(3.22), there holds

(t−T)Ew(t)≤(12+12−σm0)LCδα0+1σ(2−σm0)ρ+12Ew(T)+(12+12−σm0)δL|Ω|(t−T)+12−σm0〈ε(T)wt(T),w(T)〉−12−σm0〈ε(t)wt(t),w(t)〉+12−σm0∫Ttε′(s)〈wt(s),w(s)〉ds−12−σm0∫Tt〈f1(s)−f2(s),w(s)〉ds−12−σm0∫Tta(x)〈(g1(s)−g2(s)),w(s)〉ds−[(12+12−σm0)LCδα0+1σ(2−σm0)ρ+12]∫Tt〈f1−f2,wt(s)〉ds−∫Tt∫st〈f1−f2,wt(ξ)〉dξds. (3.23)

Next, we will deal with ∫Tt∫Ω a(x)(g₁(s) − g₂(s))w(s) dx ds.

Multiplying (2.2) by u_{i_t} and integrating over Ω, we achieve

12ddt[ε(t)∥uit∥2+∥ui∥12+∥ηi∥μ,12+2∫ΩF(ui)dx−2∫Ωh(x)uidx]+〈ηi,ηsi〉μ,1+〈a(x)g(uit),uit〉=0,

which combines with (3.4)-(3.6) and the existence of time-dependent absorbing set, means that

∫Tt〈a(x)g(uit),uit〉ds≤E0(T)−E0(t)≤CT,

where the constant C_T depends on T. Then, similar to the computation in [1, 16] it follows that

Now, combining with (3.23), (3.24) we have

(t−T)Ew(t)≤(12+12−σm0)LCδα0+1σ(2−σm0)ρ+12Ew(T)+(12+12−σm0)δL|Ω|(t−T)+12−σm0〈ε(T)wt(T),w(T)〉−12−σm0〈ε(t)wt(t),w(t)〉+12−σm0∫Ttε′(s)〈wt(t),w(t)〉ds−∫Tt∫st〈f1−f2,wt(ξ)〉dξds−(12+12−σm0)LCδα0+1σ(2−σm0)ρ+12∫Tt〈f1−f2,wt(s)〉ds+(C′|Ω|∥a∥L∞(t−T))pp+1+(C′CT)pp+12−σm0∫Tt∫Ωa(x)|w(x,s)|p+1dxds1p+1−12−σm0∫Tt〈f1(s)−f2(s),w(s)〉ds

Set

ϕTt((u01(T),v01(T)),(u02(T),v02(T)))=−1(t−T)(2−σm0)〈ε(t)wt,w〉+1(t−T)(2−σm0)∫Ttε′(s)〈wt(s),w(s)〉ds−1(t−T)(2−σm0)∫Tt〈f1(s)−f2(s),w(s)〉ds−1(t−T)∫Tt∫st〈f1−f2,wt(ξ)〉dξds+(C′|Ω|∥a∥L∞(t−T))pp+1+(C′CT)pp+1(t−T)(2−σm0)∫Tt∫Ωa(x)|w(x,s)|p+1dxds1p+1−[(12+12−σm0)LCδα0+1σ(2−σm0)ρ+12]t−T∫Tt〈f1−f2,wt(s)〉ds, (3.25)

and

CM=[(12+12−σm0)LCδα0+1σ(2−σm0)ρ+12]Ew(T)+(12+12−σm0)δL|Ω|(t−T) +12−σm0〈ε(T)wt(T),w(T)〉. (3.26)

Then we have

Ew(t)≤CMt−T+ϕTt((u01(T),v01(T)),(u02(T),v02(T))). (3.27)

3.3 Asymptotically compact

Theorem 3.1

Under the assumption (1.2)-(1.10), for any fixed t ∈ ℝ, bounded sequence {xn}n=1∞ ⊂ X_{τ_n} and any {τn}n=1∞ ⊂ ℝ^−t, with τ_n → −∞ as n → ∞, sequence {U(t,τn)xn}n=1∞ has a convergent subsequence.

Proof

For any fixed ε > 0, we first choose some proper δ such that (12+12−σm0)δL|Ω|≤ε4 , then for some fixed t, let T < t such that t − T so large that

CMt−T<ε2.

Hence, thanks to Theorem 2.10, we only need to verify that ϕTt ∈ 𝓒(B_t), for each fixed T.

Let (u_n, u_{n_t}, ηnt ) be the solution corresponding to initial data (u0n,v0n,η0n) ∈ B_T for the problem (2.2). From (3.4), ∥un∥12+ε(ξ)∥unt∥2+∥ηnt∥μ,12 is bounded, where the bound depends on the T, furthermore, ∥un∥12 is bounded. Moreover, by (1.2), (1.3), for fixed T, ξ ∈ [T, t], ε(ξ) is bounded, hence ∥u_{n_t}∥² is bounded.

According to Alaoglu Theorem, without loss of generality, we assume that (at most by passing to subsequence)

u_n → u *-weakly in L^∞(T, t; H01 (Ω)) ;
u_{n_t} → u_t *-weakly in L^∞(T, t ; L²(Ω)) ;
u_n → u strong in L^p+1(T, t; L^p+1(Ω)) ;
u_n(T) → u(T) and u_n(t) → u(t) strong in L²(Ω).

Here we use the compact emdedding H01 ↪ L^p+1(p < 5).

Now, we will deal with each term in (3.25) one by one.

Firstly, from Lemma 3.2, (i)-(iv) we get

limn→∞limm→∞∫Ωε(t)(unt(t)−umt(t))(un(t)−um(t))dx=0, (3.28)

limn→∞limm→∞∫Tt∫ΩL(unt(s)−umt(s))(un(s)−um(s))dxds=0, (3.29)

limn→∞limm→∞C(∫Tt∫Ωa(x)|un(s)−um(s)|p+1dxds)1p+1=0, (3.30)

where C = (C′|Ω|∥a∥L∞(t−T))pp+1+(C′CT)pp+1(t−T)(2−σm0) , and

limn→∞limm→∞∫Tt∫Ω(f(un)−f(um))(un(s)−um(s))dxds=0. (3.31)

Similar to the proof of the Theorem 5.4 in [1], we have

limn→∞limm→∞∫Tt∫Ω(f(un)−f(um))(unt(s)−umt(s))dxds=0. (3.32)

At the same time, for each fixed t, | ∫st∫Ω (u_{n_t}(ξ) − u_{m_t} (ξ))(f(u_n(ξ) − f(u_m(ξ)))dxd ξ| is bounded, then by the Lebesgue dominated convergence theorem we have

limn→∞limm→∞∫Tt∫st∫Ω(unt(ξ)−umt(ξ))(f(un(ξ)−f(um(ξ)))dxdξds=∫Tt(limn→∞limm→∞∫st∫Ω(unt(ξ)−umt(ξ))(f(un(ξ)−f(um(ξ)))dxdξ)ds=0. (3.33)

Hence, collecting all (3.28)-(3.33), we get that ϕTt ∈ 𝓒(B_t), so the proof is completed. □

3.4 Existence of the time-dependent global attractor

Theorem 3.2

Under the conditions (1.2)-(1.10), the process U(t, τ) : 𝓗_τ → 𝓗_t generated by problem (1.1) has an invariant time-dependent global attractor 𝓤 = {A_t}_t∈ℝ.

Proof

From Lemma 3.2, Theorem 3.1 and Theorem 2.9, we know that there exists a unique time-dependent global attractor 𝓤 = {A_t}_t∈ℝ. Furthermore, by virtue of the strong continuity of the process stated in Lemma 3.1, we can obtain that 𝓤 is invariant. □

Acknowledgement

This work was partly supported by the NSFC (11561064, 11761062), and partly supported by NWNU-LKQN-14-6.

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Received: 2018-06-08

Accepted: 2018-12-22

Published Online: 2019-03-10

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0008

Keywords for this article

wave equation; linear memory; contractive functions; time-dependent attractor

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