BEAM EQUATION WITH NONLINEARITY OF KIRCHHOFF TYPE IN DOMAINS WITH MOVING BOUNDARY

BEAM EQUATION WITH NONLINEARITY OF KIRCHHOFF TYPE IN DOMAINS WITH MOVING BOUNDARY

M. L. SANTOS, J. FERREIRA, AND C. A. RAPOSO Received 2 October 2003

We prove the exponential decay in the casen >2, as time goes to infinity, of regular so- lutions for the nonlinear beam equation with memory and weak dampingutt+∆²u− M(∇u²_L²(Ωt))∆u+₀^tg(t−s)∆u(s)ds+αut =0 inQ^∧ in a noncylindrical domain of Rⁿ⁺¹ (n≥1) under suitable hypothesis on the scalar functionsM andg, and whereα is a positive constant. We establish existence and uniqueness of regular solutions for any n≥1.

1. Introduction

LetΩbe an open bounded domain ofRⁿcontaining the origin and havingC²boundary.

Letγ: [0,∞[→Rbe a continuously differentiable function. See hypotheses (1.24), (1.25), and (1.26) onγ. Consider the family of subdomains{Ωt}0≤t<∞ofRⁿgiven by

Ωt=T(Ω), T:y∈Ω −→x=γ(t)y, (1.1) whose boundaries are denoted byΓt, and letQ^∧ be the noncylindrical domain ofRⁿ⁺¹ given by

Q∧=

0≤t<∞Ωt× {t} (1.2)

with lateral boundary

∧

=

0_≤t<∞

Γt× {t_}. (1.3)

We consider the Hilbert spaceL²(Ω) endowed with the inner product (u,v)=

Ωu(x)v(x)dx (1.4)

Copyright©2005 Hindawi Publishing Corporation Abstract and Applied Analysis 2005:8 (2005) 901–919 DOI:10.1155/AAA.2005.901

and corresponding norm

u²_L²_(Ω)=(u,u). (1.5)

We also consider the Sobolev spaceH¹(Ω) endowed with the scalar product

(u,v)H¹(Ω)=(u,v) + (∇u,∇v). (1.6) We define the subspace ofH¹(Ω), denoted byH₀¹(Ω), as the closure of C₀^∞(Ω) in the strong topology ofH¹(Ω). ByH⁻¹(Ω), we denote the dual space ofH₀¹(Ω). This space endowed with the norm induced by the scalar product

(u,v)_H0¹_(Ω)=(∇u,∇v) (1.7) is, owing to the Poincar´e inequality

u²_L²_(Ω)≤C∇u²_L²_(Ω), (1.8) a Hilbert space. We define for all 1≤p <∞,

u_L^pp(Ω)=

Ω

u(x)^pdx, (1.9)

and ifp= ∞,

uL^∞(Ω)=sup

x∈Ωessu(x). (1.10)

In this work, we study the existence and uniqueness of strong solutions as well as the exponential decay of the energy to the nonlinear beam equation with memory given by

u_tt+∆²u−M∇u²_L²_(Ωt) ∆u+ _t

0g(t−s)∆u(s)ds+αu_t=0 inQ,^∧ (1.11) u=∂u

∂ν ⁼0 on ∧

, (1.12)

u(x, 0)=u0(x), ut(x, 0)=u1(x) inΩ0, (1.13) whereν=ν(σ,t) is the unit normal at (σ,t)_∈^∧ directed towards the exterior ofQ. If we^∧ denote byηthe outer unit normal to the boundaryΓofΩ, we have, using a parametriza- tion ofΓ,

ν(σ,t)=1 ν

η(ξ),−γ(t)ξ·η(ξ), ξ= σ

γ(t), (1.14)

where

ν=

1 +γ(t)ξ·η(ξ)^{2 1/2}. (1.15)

Indeed, fix (σ,t)∈^∧

. Let ϕ=0 be a parametrization of a part of Γ,containing ξ=σ/γ(t). The parametrization of a partof^∧ isψ(σ,t)=ϕ(σ/γ(t))=ϕ(ξ)=0. We have

∇ψ(σ,t)= 1 γ(t)

∇ϕ(ξ),−γ(t)ξ· ∇ϕ(ξ). (1.16)

From this and observing thatη(ξ)= ∇ϕ(ξ)/|∇ϕ(ξ)|, expression (1.14) follows. Letν(·,t) be thex-component of unit normalν(·,·),|ν| ≤1. Then by relation (1.14), one has

ν(σ,t)=η σ

γ(t)

. (1.17)

In this paper, we deal with the nonlinear beam equation with memory in domains with moving boundary. We show the existence and uniqueness of strong solutions to the initial boundary value problem (1.11)–(1.13). The method we use to prove the result of exis- tence and uniqueness is based on transforming our problem into another initial bound- ary value problem defined over a cylindrical domain whose sections are not time depen- dent. This is done using a suitable change of variable. Then we show the existence and uniqueness for this new problem. Our existence result on domains with moving bound- ary will follow by using the inverse transformation, that is, by using the diffeomorfism

τ:Q^∧−→Q, (x,t)∈Ωt −→(y,t)= x

γ(t),t

(1.18)

andτ⁻¹:Q→Q^∧defined by

τ⁻¹(y,t)=(x,t)=

γ(t)y,t. (1.19)

Denoting byvthe function

v(y,t)=u◦τ⁻¹(y,t)=uγ(t)y,t, (1.20) the initial boundary value problem (1.11)–(1.13) becomes

v_tt+γ⁻⁴∆²v−γ⁻²Mγⁿ⁻²∇v²_L²_(Ω) ∆v+ _t

0g(t−s)γ⁻²(s)∆v(s)ds +αv_t−A(t)v+a1· ∇∂_tv+a2· ∇v=0 inQ,

v|Γ=∂v

∂ν

Γ=0,

v|t=0=v0, vt|t=0=v1 inΩ,

(1.21)

where

A(t)v= n i,j=1

∂_yia_{i j}∂_yjv, (1.22) ai j(y,t)= −

γγ⁻¹²yiyj (i,j=1,. . .,n) a1(y,t)= −γγ⁻¹y,

a2(y,t)= −γ⁻²yγγ+γαγ+ (n−1)γ.

(1.23)

To show the existence of strong solution, we will use the following hypotheses:

γ≤0 n >2, γ≥0 ifn≤2, (1.24) γ∈L^∞(0,∞), inf

0≤t<∞γ(t)=γ0>0, (1.25) γ∈W^2,∞(0,∞)∩W^2,1(0,∞). (1.26) Note that assumption (1.24) means thatQis decreasing ifn >2 and increasing ifn≤2 in the sense that whent > t andn >2, then the projection ofΩt on the subspacet= 0 contains the projection ofΩt on the same subspace and contrary in the casen≤2.

The above method was introduced by Dal Passo and Ughi [4] to study certain class of parabolic equations in noncylindrical domains. Concerning the functionM∈C¹[0,∞[, we assume that

M(τ)≥ −m0, M(τ)τ≥M(τ)^∧ ∀τ≥0, (1.27) whereM^∧(τ)=_τ

0M(s)dsand

0≤m0< λ1γ⁻_L∞², (1.28) whereλ1is the first eigenvalue of the spectral Dirichlet problem

∆²w=λ1w inΩ, w=∂w

∂η ⁼0 inΓ. (1.29)

We recall also the classical inequality

∆wL²(Ω)≥

λ1∇wL²(Ω). (1.30)

Remark 1.1. The hypotheses (1.27) and (1.30) are classic, as one can see, for instance, in [9,20,21] without the term of memory₀^tg(t−s)∆u(s)dsin fixed domain. In fact, the hypothesis (1.28) was introduced by the second author with some modifications, due to the complexity of working in noncylindrical domains in [1].

Unlike the existing papers on stability for hyperbolic equations in noncylindrical do- main, we do not use the penalty method introduced by Lions [16], but work directly in our noncylindrical domainQ. To see the dissipative properties of the system, we have to construct a suitable functional whose derivative is negative and is equivalent to the

first-order energy. This functional is obtained using the multiplicative technique follow- ing Komornik [10] or Rivera [18]. We only obtained the exponential decay of solution for our problem for the casen >2. The main difficult to obtain the decay forn≤2 is due to the geometry of the noncylindrical domain because it affects substantially the problem, since we work directly inQ. Therefore the case^∧ n≤2 is an important open problem. From the physics point of view, the system (1.11)–(1.13) describes the transverse deflection of a streched viscoelastic beam fixed in a moving boundary device. The viscoelasticity prop- erty of the material is characterized by the memory term

_t

0g(t−s)∆u(s)ds. (1.31)

The uniform stabilization of plates equations with linear or nonlinear boundary feed- back was investigated by several authors, see for example [8,9,11,13,14,15]. In a fixed domain, it is well known that if the relaxation functiong decays to zero, then the en- ergy of the system also decays to zero, see [3,12,19,22]. But in a moving domain, the transverse deflectionu(x,t) of a beam which changes its configuration at each instant of time increases its deformation, and hence increases its tension. Moreover, the horizon- tal movement of the boundary yields nonlinear terms involving derivatives in the space variable. To control these nonlinearities, we add in the system a frictional damping, char- acterized byut. This term will play an important role in the dissipative nature of the problem. In [1,6], a quite complete discussion about the model of transverse deflection and transverse vibrations can be found, respectively, for the nonlinear beam equation and elastic membranes. This model was proposed by Woinowsky-Krieger [23] for the case of cylindrical domains, without the dissipative term and₀^tg(t−s)∆u(s)ds. See also Eisley [5] and Burgreen [2] for physics justification and background of the model. Our results in this paper were more difficult to obtain than the results in [7], due to the introduction of the terms corresponding to the biharmonic operator∆²and to the nonlinear function of KirchhofftypeM(∇u²_L²_(Ω)), which generated nontrivial problems that were solved thanks to the hypotheses (1.27), (1.28), and (1.30) and to the hypothesis regarding the

“dilation function”. Besides, in [7], we made only two estimates, while here we had to make four estimates that introduce some technical ideas with regard to the existence, uniqueness, and regularity. Regarding the solution decay, we used a similar technical of [7] but we introduced Lemmas3.3 and3.4 to control the terms of energy and to use with success the technique of multipliers. We use the standard notations which can be found in Lion’s and Magenes’ books [16,17]. In the sequel byC(sometimesC1,C2,. . .), we denote various positive constants which do not depend ontor on the initial data. This paper is organized as follows. InSection 2, we prove a basic result on existence, regular- ity, and uniqueness of regular solutions. We use Galerkin approximation, Aubin-Lions theorem, energy method introduced by Lions [16], and some technical ideas to show ex- istence regularity and uniqueness of regular solution for problem (1.11)–(1.13). Finally, inSection 3, we establish a result on the exponential decay of the regular solution to the problem (1.11)–(1.13). We use the technique of the multipliers introduced by Komornik [10], Lions [16], and Rivera [18] coupled with some technical lemmas and some technical ideas.

2. Existence and regularity

In this section, we will study the existence and regularity of solutions for the system (1.11)–(1.13). For this, we assume that the kernelg:R+→R+is inW^2,1(0,∞), and satis- fies

g,−g≥0, m1

γ²_L∞ − _∞

0 g(s)γ⁻²(s)ds=β1>0, (2.1) where

m1= λ1

γ²_L∞ −m0

>0. (2.2)

To simplify our analysis, we define the binary operator gϕ(t)

γ(t)⁼

Ω

_t

0g(t−s)γ⁻²(s)ϕ(t)−ϕ(s)²ds dx. (2.3) With this notation, we have the following statement.

Lemma2.1. Forv∈C¹(0,T:H₀²(Ω)),

Ω

_t

0g(t−s)γ⁻²(s)∇v(s)· ∇vt(t)ds dx

= −1 2

g(t) γ²(0)

Ω|∇v|²dx+1 2g^∇v

γ ⁻ 1 2

d dt

g^∇v

γ ⁻ _t

0

g(s) γ²(s)ds

Ω|∇v|²dx

,

Ω

_t

0g(t−s)γ⁻²(s)∆v∆vtds dx

= −1 2

g(t) γ²(0)

Ω|∆v|²dx+1 2g∆v

γ ⁻ 1 2

d dt

g∆v

γ ⁻ _t

0

g(s) γ²(s)ds

Ω|∆v|²dx

. (2.4) The proof of this lemma follows by differentiating the terms g(∇u(t)/γ(t)) and g(∆u(t)/γ(t)). The well posedness of system (1.21) is given by the following theorem.

Theorem 2.2. Take v0∈H₀²(Ω)∩H⁴(Ω), v1∈H₀²(Ω), and suppose that assumptions (1.24), (1.25), (1.26), (1.27), (1.28), (1.30), and (2.1) hold. Then there exists a unique solu- tionvof the problem (1.21) satisfying

v∈L^∞0,∞:H₀²(Ω)∩H⁴(Ω), v_t∈L^∞0,∞:H₀¹(Ω), vtt∈L^∞0,∞:L²(Ω).

(2.5)

Proof. We denote byBthe operator

Bw=∆²w, D(B)=H₀²(Ω)∩H⁴(Ω). (2.6) It is well known thatBis a positive selfadjoint operator in the Hilbert spaceL²(Ω) for which there exist sequences{w_n}n∈Nand{λ_n}n∈Nof eigenfunctions and eigenvalues ofB such that the set of linear combinations of{w_n}n∈Nis dense inD(B) andλ1< λ2≤ ··· ≤ λn→ ∞asn→ ∞. We denote by

v^m₀ = m j=1

v0,wj

wj, v^m₁ = m j=1

v1,wj

wj. (2.7)

Note that for any (v0,v1)∈D(B)×H₀²(Ω), we havev₀^m→v0strong inD(B) andv₁^m→v1

strong inH0²(Ω).

We denote byV_mthe space generated byw1,. . .,w_m. Standard results on ordinary dif- ferential equations imply the existence of a local solutionv^mof the form

v^m(t)= m j=1

g_jm(t)w_j, (2.8)

to the system

Ωv_tt^mwjd y+α

Ωv^m_t wjd y+

Ωγ⁻⁴∆²v^mwjd y

−γ⁻²Mγⁿ⁻²∇v^m2_L2(Ω) Ω∆v^mw_jd y +

Ω

_t

0g(t−s)γ⁻²(s)∇v^m(s)ds· ∇w_jd y+

ΩA(t)v^mw_jd y +

Ωa1· ∇v^m_t wjd y+

Ωa2· ∇v^mwjd y=0, (j=1,. . .,m)

(2.9)

v^m(x, 0)=v^m₀, v^m_t (x, 0)=v^m₁. (2.10) The extension of these solutions to the interval [0,∞[ is a consequence of the first estimate which we are going to prove below.

A priori estimate I. Multiplying (2.9) by g_jm(t), summing up the resulting product in j=1, 2,. . .,m, and after some calculations usingLemma 2.1, we get

1 2

d

dt£^m₁t,v^m+αv_t^m2_L2(Ω)−(n−2)γ 2γⁿ⁺¹

×

γⁿ⁻²∇v^m2₂Mγⁿ⁻²∇v^m2_L2(Ω) −M^∧γⁿ⁻²∇v^m2_L2(Ω)

+

ΩA(t)v^mv^m_t d y+

Ωa1· ∇v^m_t v_t^md y+

Ωa2· ∇v^mv_t^md y

= −1 2

g(t)

γ²(0)^∇v^m2_L2(Ω)+1

2g^∇v^m γ ⁻4γ

γ⁵∆v^m2_L2(Ω),

(2.11)

where

£^m₁t,v^m=v_t^m2_L2(Ω)+

− _t

0g(s)γ⁻²(s)ds∇v^m2_L2(Ω)

+γ⁻⁴∆v^m2_L2(Ω)+γ⁻ⁿM^∧γⁿ⁻²∇v^m2_L2(Ω) +g^∇v^m γ .

(2.12)

From (1.27), (1.28), and (1.30), it follows that

γ⁻⁴∆v^m2_L2(Ω)+γ⁻ⁿM^∧γⁿ⁻²∇v^m2_L2(Ω) ≥ m1

γ²L^∞

∇v^m2_L2(Ω). (2.13)

Taking into account (1.24), (1.27), the last inequality, and (2.1), it follows that the equality (2.11) can be written as

1 2

d dt£^m1

t,v^m+αv^m_t ²_L2(Ω)≤Cγ+γ£^m1(t). (2.14)

Integrating the inequality (2.14), using Gronwall’s lemma, and taking into account (1.26), we get

£^m₁t,v^m+ _t

0

v^m_s (s)²_L2(Ω)ds≤C, ∀m∈N,∀t∈[0,T]. (2.15)

A priori estimate II. Now, if we multiply (2.9) byλjg_jm(t) and summing up in j= 1,. . .,m, we get after some calculations

1 2

d

dt^∇v_t^m2_L2(Ω)+α∇v_t^m2_L2(Ω)+γ⁻²

2 Mγⁿ⁻²∇v^m2_L2(Ω) d

dt∆v^m2_L²_(Ω) +γ⁻⁴

2 d

dt^∇∆v^m2_L²(Ω)−

Ω

_t

0g(t−s)γ⁻²(s)∆v^m(s)∆v^mt ds d y +

ΩA(t)v^m∆v^m_t d y+

Ωa1· ∇v^m_t ∆v^m_t d y+

Ωa2· ∇v^m∆v_t^md y=0.

(2.16)

UsingLemma 2.1, we obtain

Ω

_t

0g(t−s)γ⁻²(s)∆v^m(s)∆v^m_t d y

= −1

2g(t)γ⁻²(0)∆v^m2_L²_(Ω)+1

2g∆v^m γ

−1 2

d dt

g∆v^m γ ⁻

_t

0g(s)γ⁻²(s)ds∆v^m2_L2(Ω)

.

(2.17)

Substituting (2.17) into (2.16), we get 1

2 d

dt£^m₂(t) +α∇v^m_t ²_L2(Ω)

= −1 2

g(t)

γ²(0)∆v^m2_L²(Ω)+1

2g∆v^m γ ⁻

2γ

γ^5∇∆v^m2_L²(Ω)

+A(t)v^m,∆v^m_t +a1· ∇v_t^m,∆v^m_t +a2· ∇v^m,∆v_t^m +1

2 d dt

γ⁻²Mγⁿ⁻²∇v^m2_L2(Ω)

∆v^m2_L2(Ω),

(2.18)

where

£^m₂(t)=∇v_t^m2_L2(Ω)+g∆v^m γ ⁻

_t

0g(s)γ⁻²(s)ds∆v^m2_L²(Ω)

+γ⁻⁴∇∆v^m2_L2(Ω)+γ⁻²Mγⁿ⁻²∇v^m2_L2(Ω) ∆v^m2_L2(Ω).

(2.19)

From (1.27), (1.28), and (1.30), we have

γ⁻⁴∇∆v^m2_L²(Ω)+γ⁻²Mγⁿ⁻²∇v^m2_L2(Ω) ∆v^m2_L²(Ω)≥ m1

γ²_L^∞∆v^m2_L²(Ω). (2.20) Using relation (2.18) and taking into account (2.20), we get

£^m₂(t) +α _t

0

∇v^m_s(s)²_L2(Ω)ds≤C1+C2

_t

0

γ+γ£^m₂(s)ds. (2.21)

Using Gronwall’s and taking into account (1.24), we get

£^m2(t) +α _t

0

∇v^m_s(s)²_L2(Ω)ds≤C ∀t∈[0,T],∀m∈N. (2.22) A priori estimate III. Differentiating (2.9) with respect to the time, multiplying byg_jm(t), and using similar arguments as (2.22), we obtain, after some calculations and taking into account (2.22),

1 2

d

dt£^m₃(t) +αv^m_tt(t)²_L2(Ω)≤Cγ+γv_t^m(t)²_L2(Ω)

+Cγ+γ£^m3(t), ∀t∈[0,T],∀m∈N,

(2.23) where

£^m₃(t)=v^m_tt²_L2(Ω)+γ⁻²Mγⁿ⁻²∇v^m2_L2(Ω) ∇v^m_t ²_L2(Ω)

+g^∇v_t^m γ ⁻

t

0g(s)γ⁻²(s)ds∇v_t^m2_L2(Ω). (2.24) Using Gronwall’s lemma and relations (2.15), (2.22), we get

£^m₃(t) +α _t

0

∆v^m(s)²_L2(Ω)ds≤C, ∀t∈[0,T], ∀m∈N. (2.25)

It easy to see from (2.9) that

v_tt^m(0)²_L²_(Ω)≤C ∀m∈N. (2.26)

A priori estimate IV. Settingw=v^m(t) in (2.9), we deduce that 1

2 d

dt£^(m)4 (t)−v^m_t ²_L2(Ω)+γ⁻⁴∆v^m2_L²(Ω)

+γ⁻²Mγⁿ⁻²∇v^m2_L2(Ω) ∇v^m2_L2(Ω)

+ _t

0g(t−s)∆v^m(s)v^m(s)ds≤Cγ+γ

×v^m2_L2(Ω)+∆v^m2_L2(Ω)+v_t^m2_L2(Ω) ,

(2.27)

where

£^m₄(t)=2

Ωv^mv^m_t d y+αv^m2_L2(Ω). (2.28) From (1.27), (1.28), and (1.30), we have

γ⁻⁴∆v^m2_L2(Ω)+γ⁻ⁿM^∧γⁿ⁻²∇v^m2_L2(Ω) ≥ m1

γ²_L∞

∇v^m2_L2(Ω), (2.29)

wherem1=(λ1/γ²_L^∞−m0)>0. Moreover, it is easy to see that choosingk >2/α(see also (2.29)), we obtain

k£^m₁(t) + £^m₄(t)≥

k−2 α

v_t^m2_L2(Ω)+v^m2_L2(Ω)

+

k−2 α

γ⁻⁴∆v^m2_L²_(Ω)+γ⁻ⁿM^∧γⁿ∇v^m2_L2(Ω)

>0.

(2.30)

Now, multiplying (2.11) byk and combining with (2.27), we get, taking into account (2.29),

1 2

d dt

k£^m₁(t) + £^m₄(t)+ (kα−1)v^m_t ²_L2(Ω)+γ⁻⁴∆v^m2_L2(Ω)

≤Cγ+γk£^m₁(t) + £^m₄(t).

(2.31)

From (2.31), using Gronwall’s lemma, we obtain the following estimate:

k£ⁿ₁(t) + £^m₄(t) + _t

0

v^m_t ²_L2(Ω)+∆v^m2_L2(Ω) ds

≤Cv1²

L²(Ω)+∆v0²

L²(Ω) exp

C _∞

0

γ(t) +γ(t)dt

.

(2.32)

In virtue of (2.29) and (1.26), it follows from (2.32) that v^m_t ²_L2(Ω)+∆v^m2_L2(Ω)+

_t

0

v_t^m2_L2(Ω)+∆v^m2_L2(Ω) ds

≤Cv1²

L²(Ω)+∆v0²

L²(Ω) .

(2.33)

From estimates (2.15), (2.22), (2.25), and (2.33), it follows thatv^mconverges strongly in L²(0,∞:H¹(Ω)) to somev∈L²(0,∞:H¹(Ω)). Moreover, sinceM∈C¹[0,∞) and∇v^m is bounded inL^∞(0,∞:L²(Ω))∩L²(0,∞:L²(Ω)), we have

_t

0

Mγⁿ⁻²∇v^m2_L2(Ω) −Mγⁿ⁻²∇v²_L²_(Ω) ds≤C _t

0

v^m−v²_H1(Ω)ds, (2.34)

whereCis a positive constant independent ofmandtso that Mγⁿ⁻²∇v^m2_L2(Ω) ∆v^m,wj

−→Mγⁿ⁻²∇v²_L²(Ω) ∆v,wj

. (2.35) Therefore, we have thatvsatisfies

v∈L^∞0,∞:H₀¹(Ω)∩L²0,∞:H₀²(Ω), vt∈L^∞0,∞:H₀¹(Ω),

vtt∈L^∞0,∞:L²(Ω).

(2.36)

Lettingm→ ∞in (2.9), we conclude that

vtt+γ⁻⁴∆²v−γ⁻²Mγⁿ⁻²∇v²_L²_(Ω) ∆v+ _t

0g(t−s)γ⁻²(s)∆v(s)ds +αvt−A(t)v+a1· ∇∂tv+a2· ∇v=0

(2.37)

inL^∞(0,∞:L²(Ω)). Therefore, we have

v∈L^∞0,∞:H₀²(Ω)∩H⁴(Ω). (2.38) To prove the uniqueness of solutions of the problem (1.21), we use the method of the energy introduced by Lions [16], coupled with Gronwall’s inequality and the hypotheses introduced in the paper about the functionsM,g, and the obtained estimates.

To show the existence in noncylindrical domains, we return to our original problem in the noncylindrical domains by using the change variable given in (1.18) by (y,t)=τ(x,t), (x,t)∈Q. Let vbe the solution obtained fromTheorem 2.2andudefined by (1.20), then ubelongs to the classes

u∈L^∞0,∞:H₀²Ωt

∩H⁴Ωt

, ut∈L^∞0,∞:H₀¹Ωt

, utt∈L^∞0,∞:L²Ωt

.

(2.39)