On a nonlinear wave equation with a nonlocal boundary condition

Soriano, On existence and ymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions, J.. Dinh, A semilinear wave equation associated with a linea

ON A NONLINEAR WAVE EQUATIONWITH A NONLOCAL BOUNDARY CONDITION

LE THI PHUONG NGOC, TRAN MINH THUYET, PHAM THANH SON, NGUYEN THANH LONG Dedicated to Tran Duc Van on the occasion of his sixtieth birthday

Abstract Consider the initial-boundary value problem for the nonlinear

k (t − s) u (0, s) ds + g(t),

−µ(t)u x (1, t) = K 1 u(1, t) + λ 1 |u t (1, t)|α−2u t (1, t),

u (x, 0) = e u 0 (x), u t (x, 0) = e u 1 (x),

where p, q, α ≥ 2; K 0 , K 1 , K ≥ 0; λ, λ 1 > 0 are given constants and µ,

F, g, k, e u 0 , e u 1 , are given functions First, the existence and uniqueness of a

weak solution are proved by using the Galerkin method Next, with α = 2,

we obtain an asymptotic expansion of the solution up to order N in two small

parameters λ, λ 1 with error p

Received October 8, 2010.

2000 Mathematics Subject Classification 35L20, 35L70, 35Q72.

Key words and phrases Galerkin method, a priori estimates, asymptotic expansion of the solution up to order N

It is well known that many various problems in the form (1.1)–(1.4) have beeninvestigated For example, we refer to Cavalcanti et al [4], Long, Dinh andDiem [9], Ngoc, Hang and Long [10], Qin [11, 12], Rivera [13], Santos [14], andthe references therein In these works, many interesting results about the uniqueexistence, regularity, stability, asymptotic expansion or the decay of solutions areobtained.

Clearly, the boundary condition (1.2) is nonlocal and if we put

and problem (1.1)–(1.4) can be reduced to problem (1.1), (1.3)-(1.5), in which

an unknown function u(x, t) and an unknown boundary value Y (t) satisfy thefollowing Cauchy problem for the ordinary differential equation

(1.6)

 Y′′

(t) + γ1Y′(t) + γ2Y(t) = γ3utt(0, t), 0 < t < T,

Y(0) = eY0, Y′(0) = eY1,where γ1, γ2, γ3, eY0, eY1 are certain constants such that γ12− 4γ2 <0

In [1], An and Trieu studied a special case of problem (1.1), (1.4)-(1.6) ated with the following homogeneous boundary condition at x = 1 :

with µ(t) ≡ 1, F = eu0 = eu1 = eY0 = γ1 = 0, and f (u, ut) = Ku + λut, with γ3,

K ≥ 0, λ ≥ 0 being given constants In the latter case, problem (1.1), (1.7) is a mathematical model describing the shock of a rigid body and a linearviscoelastic bar resting on a rigid base [1]

(1.4)-We note more that from (1.6), representing Y (t) in terms of γ1, γ2, γ3, eY0, eY1,

utt(0, t) and then integrating by parts, we shall obtain Y (t) as below

bg(t) = Ye0− γ3ue0(0)

e−γtcos ωt+

4γ2− γ2

1, K0 = γ3 Therefore, problem (1.1), (1.3)-(1.5)leads to problem (1.1)-(1.4)

The paper consists of three sections In Section 2, under the conditions(eu0,ue1) ∈ H2× H1; F, F′ ∈ L1(0, T ; L2); g, k, µ ∈ W2,1(0, T ) , µ(t) ≥ µ0 >0;

p, q, α ≥ 2; K, K0, K1 ≥ 0; λ, λ1 > 0 and some other conditions, we prove

a theorem of global existence and uniqueness of a weak solution u of problem(1.1)-(1.4) The proof is based on the Galerkin method associated to a prioriestimates, the weak convergence and the compactness techniques Finally, inSection 3, with α = 2, we obtain an asymptotic expansion of the solution u up toorder N in two small parameters λ, λ1 with error p

λ2+ λ2 1

N + 1

.The resultsobtained here may be considered as the generalizations of those in An and Trieu[1] and in [2, 6, 7, 9, 10]

2 Existence and uniqueness of a weak solution

First, put Ω = (0, 1), QT = Ω × (0, T ), T > 0 and denote the usual functionspaces used in this paper by the notations Cm Ω

, Wm,p = Wm,p(Ω) , Lp =

W0,p(Ω) , Hm= Wm,2(Ω) , 1 ≤ p ≤ ∞, m = 0, 1,

Let h·, ·i be either the scalar product in L2 or the dual pairing of a continuouslinear functional and an element of a function space The notation || · || standsfor the norm in L2and we denote by || · ||X the norm in the Banach space X Wecall X′ the dual space of X We denote by Lp(0, T ; X), 1 ≤ p ≤ ∞ for the Banachspace of real functions u : (0, T ) → X measurable, such that ||u||L p (0,T ;X)<+∞,with

Let u(t), u′(t) = ut(t) = ˙u(t), u′′(t) = utt(t) = ¨u(t), ux(t) = ▽u(t), uxx(t) =

∆u(t), denote u(x, t), ∂ u∂ t(x, t), ∂∂ t2u2(x, t), ∂ u∂x(x, t), ∂∂x2u2(x, t), respectively

On H1 we shall use the following norm

kvk2+ kvxk21/2.Then we have the following lemma

Lemma 2.1 [The imbedding] H1 ֒→ C0([0, 1]) is compact and

(2.3) kvkC0 (Ω) ≤√2 kvkH 1 for all v ∈ H1

The proof of Lemma 2.1 is straightforward, and we omit it

Next, we make the following assumptions:

(H1) (eu0,eu1) ∈ H2× H1,

(H2) F, F′ ∈ L1(0, T ; L2),

(H3) µ ∈ W2,1(0, T ) , µ(t) ≥ µ0 >0,

(H4) g, k ∈ W2,1(0, T ) ,

(H5) p, q, α ≥ 2; K, K0, K1 ≥ 0; λ, λ1 >0

Finally, let us note more that the weak solution u of the initial and boundaryvalue problem (1.1)-(1.4) will be obtained in Theorem 2.2 in the following manner:Find u ∈ fW = {u ∈ L∞(0, T ; H2), ut ∈ L∞(0, T ; H1), utt ∈ L∞(0, T ; L2)},such that u satisfies the variational equation

u′′(t), v

+ µ(t) hux(t), vxi + Y (t)v(0) +K1u(1, t) + λ1Πα(u′(1, t))

v(1)+ KΠp(u(t)) + λΠq(u′(t)), v

= hF (t), vi , for all v ∈ H1, a.e., t ∈ (0, T ),and the initial conditions

u(0) = eu0, ut(0) = eu1,where Πr(z) = |z|r−2z, r∈ {p, q, α} and

Remark 2.1 If u ∈ L∞(0, T ; H2) and ut∈ L∞(0, T ; H1), then u : [0, T ] → H1

is continuous ([5], Lemma 1.2, p.7), so it is clear that u(0) is defined and u(0)belongs to H1 Similarly, with ut ∈ L∞(0, T ; H1) and utt ∈ L∞(0, T ; L2)}, itimplies that ut : [0, T ] → L2 is continuous, ut(0) ∈ L2 follows In order to getthe following result of existence, we need the assumptions (H1)-(H5), in whiche

u0 ∈ H2, ue1∈ H1.Then, u(0) = eu0 ∈ H2 and ut(0) = eu1∈ H1

Theorem 2.2 Let (H1) − (H5) hold For every T > 0, there exists a uniqueweak solution u of problem (1.1)-(1.4), such that

Step 1 (The Galerkin approximation) Let {wj} be a denumerable base of H2

We find the approximate solution of problem (1.1)-(1.4) in the form

where the coefficient functions cmj satisfy the system of ordinary differentialequations

From the assumptions of Theorem 2.2, the system (2.7)–(2.8) has a solution um

on an interval [0, Tm] ⊂ [0, T ] The following estimates allow one to take Tm = Tfor all m, (see [3])

Step 2 (A priori estimates I) Substituting (2.8) into (2.7), then multiplying the

jth equation of (2.7) by c′mj(t) and summing with respect to j, and afterwardsintegrating with respect to the time variable from 0 to t, we get after somerearrangements

where CT always indicates a bound depending on T,

Combining (2.10), (2.12) and (2.15)-(2.21), we obtain after some rearrangements

t

Z

0

q1T(ε, s)Sm(s)ds,for all ε > 0, where

2q1T(ε, t) = (2ε +

1

ε+ 4)CT + kF (t)k , q1T(ε, ·) ∈ L1(0, T ).Choosing ε > 0, with 2ε

µ0 ≤ 12,by Gronwall’s lemma, we deduce from (2.22) that(2.24) S m (t) ≤ k 1T (ε) exp

u′′′m(t), wj

+ µ(t) u′mx(t), wjx

+ µ′(t) humx(t), wjxi + Ym′ (t)wj(0)+ K1u′m(1, t)wj(1) + λ1Π′α(u′m(1, t))u′′m(1, t)wj(1)

+ K Π′p(um(t))u′m(t), wj

+ λ Π′q(u′m(t))u′′m(t), wj

= F′(t), wj

,(2.25)

for all 1 ≤ j ≤ m

Multiplying the jthequation of (2.25) by c′′

mj(t), summing up with respect to jand then integrating with respect to the time variable from 0 to t, we have aftersome rearrangements



|u′m(1, s)|α2 −1u′m(1, s) 2

ds(2.27)

where CT always indicates a bound depending on T

We again use the inequalities (2.3), (2.14), (2.29) and by (H1)-(H5), we shallestimate the terms Jj,1 ≤ j ≤ 8 on the right-hand side of (2.26) as follows(2.30)

µ0Xm(t) ≤ εXm(t) +1

εCT,(2.31)

k|um(1, ·)kW 1,∞ (0,T )≤ CT,

′

m(1, ·)|α2 −1u′m(1, ·)

H 1 (0,T )≤ CT

Step 3 (Limiting process) From (2.11), (2.24), (2.27), (2.40)-(2.42), we deducethe existence of a subsequence of {um}, denoted by the same symbol such that

By the Aubin-Lions lemma ([5, p.57]) and the imbeddings H1(QT) ֒→ L2(QT),

H1(0, T ) ֒→ C0([0, T ]) , W1,α(0, T ) ֒→ C0([0, T ]) , we can deduce from (2.43)1−8the existence of a subsequence still denoted by {um}, such that

um → u strongly in L2(QT) and a.e in QT,

u′m → u′ strongly in L2(QT) and a.e in QT,

Πp(um) → eΠp strongly in L2(QT) and a.e in QT,

Πq(u′m) → eΠq strongly in L2(QT) and a.e in QT,

We now continue the proof of Theorem 2.2.

Passing to the limit in (2.7)-(2.9) by (2.43)1,3, (2.44)3,4, (2.47)-(2.50), we obtainthat u satisfies the system

Similarly, it follows from

.The proof of the existence is complete.

Step 4 (Uniqueness of the solution) Let u1, u2be two weak solutions of problem(1.1)-(1.4) such that

1(t)) − Πq(u′

2(t)), vi = 0,for all v ∈ H1,

3 An asymptotic expansion of the weak solution

with respect to two small parameters

In this part, we assume that α = 2, and

(H6) K, K0, K1 ≥ 0, p > N + 2, q > N + 1, N ≥ 1

and (eu0,eu1, F, µ, g, k, K, K0, K1) satisfy the assumptions (H1)-(H4), (H6) Let

λ, λ1 > 0 By Theorem 2.2, problem (1.1)-(1.4) has a unique weak solution udepending on−→λ = (λ, λ

First, we note that if the small parameters λ, λ1 > 0 satisfy p

λ2+ λ21 < 1,then a priori estimates of the sequence {um} in the proof of Theorem 2.2 forproblem 

where CT is a constant depending only on T, eu0,ue1, F, µ, g, k (independent of

λ, λ1) Hence the limit u = u− →

λ = u(λ, λ1) of the sequence {um} as m → +∞

in suitable function spaces is a unique weak solution of problem

P− → λ

satisfying

u′′− →

λ(1, s)

2ds≤ CT,p

λ1u′− →

λ(1, ·) 2

H 1 (0,T )≤ CT.Let {−→λj}, −→λj = (λj, λ1j), be a sequence such that λj, λ1j > 0, −→λ

j → 0 as

j → ∞ We put uj = u− →

λ and deduce from (3.2), (3.3) that there exists a

subsequence of the sequence {uj} denoted again by {uj}, such that

By the Aubin-Lions lemma ([5, p 57]) and the imbeddings H1(QT) ֒→ L2(QT),

H1(0, T ) ֒→ C0([0, T ]) , we can deduce from (3.4)1−8 the existence of a quence denoted again by {uj} such that

uj → u0 strongly in L2(QT) and a.e in QT,

u′j → u′0 strongly in L2(QT) and a.e in QT,

Πp(uj) → Πp strongly in L2(QT) and a.e in QT,

Πq(u′j) → Πq strongly in L2(QT) and a.e in QT,

uj(0, ·) → u0(0, ·) strongly in C0([0, T ]) ,

uj(1, ·) → u0(1, ·) strongly in C0([0, T ]) ,

p

λ1ju′j(1, ·) → χ1 strongly in C0([0, T ]) Similarly, by (3.3)1 and (3.5)1−4, it is easy to prove that

Now, we shall prove that χ1 = 0

It follows from (3.5)7 that

Hence we obtain from (3.8), (3.10) that

λ1ju′j(1, ·) → 0 strongly in C0([0, T ]) Similarly,

(3.12) λjΠq(u′j) → 0 strongly in L2(QT)

By passing to the limit, as in the proof of Theorem 2.2, we conclude that u0

is a unique weak solution of problem (P0) corresponding to−→λ

= 0 satisfying(3.13)

We use the following notations For a multi-index α = (α1, α2) ∈ Z2+ and

First, we shall need the following lemma

Lemma 3.1 Let m, N ∈ N and uα ∈ R, α ∈ Z2+, 1 ≤ |α| ≤ N Then

where the coefficients TN(m)[u]α, m ≤ |α| ≤ mN depend on u = (uα), α ∈ Z2+,

1 ≤ |α| ≤ N defined by the recurrent formulas

Let us consider the sequence of the weak solutions uγ, γ ∈ Z2

+, 1 ≤ |γ| ≤ N,defined by the following problems:

0)TN(m)[u′]γ.Here u = (uγ) , u′ = u′γ

,|γ| ≤ N

Then, we have the following theorem

Theorem 3.2 Let (H1)-(H4), (H6) hold Then, for every −→λ , with −→λproblem 

P− →

λ



has a unique weak solution u = u− →

λ satisfying the asymptoticestimation up to order N as follows

.Then v, with

|γ|≤N

uγ−→λγ = u − h,satisfies

), 0 < x < 1, 0 < t < T,µ(t)vx(0, t) = K0v(0, t) +

Lemma 3.3 Let (H1)-(H4), (H6) hold Then

is easy, hence we omit the details, therefore we only prove for N ≥ 2 Put

Similarly, using Taylor’s expansion of the function Πq(h′) = Πq(u′

and 0 < θ2<1 Combining (3.16), (3.22)1, (3.26), and (3.28), we obtain

We shall respectively estimate the following terms on the right-hand side of(3.30)

then by the boundedness of the functions uγ, u′γ,|γ| ≤ N, in the function space

L∞(0, T ; H1), we obtain from (3.24) that

N

X

m=1

1 m!Π

N −1X

m=1

1m!Π

Using Cauchy’s inequality, we obtain





2

The proof of (ii) is complete

We now continue the proof of Theorem 3.2

Next, by multiplying the two sides of (3.21)1 with v′, after integration withrespect to t, we find without difficulty from Lemma 3.3 that

Using the following inequality

µ0 +1

2T

2Zt 0

Z(s)ds

λ L∞ (0,T ;H 1 )≤ CT, −→λwhere CT is a constant independent of −→λ On the other hand, we have

(0,T ;H 1 ) ≡ C1T,

kv + hkL ∞ (0,T ;H 1 )= − →

λ L∞ (0,T ;H 1 )≤ CT.Next, with RT = max{C1T, CT}, it follows from (3.51) that

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[9] N T Long and L X Truong, Existence and asymptotic expansion of solutions to a linear wave equation with a memory condition at the boundary, Electron J Differential Equations 2007 (48) (2007), 1–19.

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& Applications, Series A: Theory and Methods 70 (11) (2009), 3943–3965.

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Nha Trang Educational College

01 Nguyen Chanh Str., Nha Trang City, Vietnam

E-mail address: ngoc1966@gmail.com, ngocltp@gmail.com

Department of Mathematics, University of Economics of Ho Chi Minh City 59C Nguyen Dinh Chieu Str., Dist 3, Ho Chi Minh City, Vietnam

E-mail address: tmthuyet@ueh.edu.vn

Department of Mathematics, University of Economics of Ho Chi Minh City 59C Nguyen Dinh Chieu Str., Dist 3, Ho Chi Minh City, Vietnam

E-mail address: thanhsonpham27@gmail.com

Department of Mathematics and Computer Science

University of Natural Science

Vietnam National University Ho Chi Minh City

227 Nguyen Van Cu Str., Dist.5, Ho Chi Minh City, Vietnam

E-mail address: longnt@hcmc.netnam.vn, longnt2@gmail.com

[6] N T Long and A P N Dinh, A semilinear wave equation associated with a linear differential equation with Cauchy data, Nonlinear Anal 24 (1995),... 842–864.

[9] N T Long and L X Truong, Existence and asymptotic expansion of solutions to a linear wave equation with a memory condition at the boundary, Electron J Differential Equations... Equations 2007 (48) (2007), 1–19.

non-[10] L T P Ngoc, L N K Hang and N T Long, On a nonlinear wave equation associated with the boundary conditions involving convolution, Nonlinear