Existence and exponential decay estimates for an N-dimensional nonlinear wave equation with a nonlocal boundary condition

Abstract

Motivated by the recent known results as regards the existence and exponential decay of solutions for wave equations, this paper is devoted to the study of an N-dimensional nonlinear wave equation with a nonlocal boundary condition. We first state two local existence theorems. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. The main tools are the Faedo-Galerkin method and the Lyapunov method.

Ngoc et al. Boundary Value Problems (2016) 2016:20 DOI 10.1186/s13661-016-0527-5

R E S E A R C H Open Access

http://crossmark.crossref.org/dialog/?doi=10.1186/s13661-016-0527-5&domain=pdf

Web End = Existence and exponential decay estimates for an N-dimensional nonlinear wave equation with a nonlocal boundary condition

Le Thi Phuong Ngoc1, Nguyen Anh Triet2 and Nguyen Thanh Long3*

*Correspondence: mailto:[email protected]

Web End [email protected]

3Department 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, VietnamFull list of author information is available at the end of the article

Abstract

Motivated by the recent known results as regards the existence and exponential decay of solutions for wave equations, this paper is devoted to the study of an N-dimensional nonlinear wave equation with a nonlocal boundary condition. We rst state two local existence theorems. Next, we give a sucient condition to guarantee the global existence and exponential decay of weak solutions. The main tools are the Faedo-Galerkin method and the Lyapunov method.

MSC: 35L05; 35L15; 35L70; 37B25

Keywords: Galerkin method; nonlinear wave equation; local existence; global existence; exponential decay

1 Introduction

In this paper, we consider the following initial-boundary value problem:

utt u + Ku + ut = a|u|pu + f (x, t), x , t > , (.)

u (x, t) = g(x, t) + h(x, y, t)u(y, t) dy, x , t , (.)

u(x, ) = u(x), ut(x, ) = u(x), (.)

where is a bounded domain in RN with a smooth boundary , is the unit outward normal on ; a = , K, , p are given constants, and u, u, f , g, h are given functions satisfying conditions specied later.

The wave equation

utt u = f (x, t, u, ut), (.)

with dierent boundary conditions, has been extensively studied by many authors, for example, we refer to [] and the references given therein. In these works, many interesting results about the existence, regularity and the asymptotic behavior of solutions

2016 Ngoc et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License

(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro

vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and

indicate if changes were made.

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 2 of 27

were obtained. In [], Beilin investigated the existence and uniqueness of a generalized solution for the following wave equation with an integral nonlocal condition:

(.)

where is a bounded domain in RN with a smooth boundary, is the unit outward normal on , f , u, u, k(x, , ) are given functions. Nonlocal conditions come up when values of the function on the boundary is connected to values inside the domain. There are various type of nonlocal boundary conditions of integral form for hyperbolic, parabolic or elliptic equations, introduced in []. In [], the following problem was considered:

(.)

where f (u) = b|u|pu, g(ut) = a( + |ut|m)ut, a, b > , m, p > , and is a bounded domain of RN, with a smooth boundary . Benaissa and Messaoudi showed that for suitably chosen initial data, (.) possesses a global weak solution, which decays exponentially even if m > . The proof of the global existence is based on the use of the potential well theory.

As [], Messaoudi [] also showed the problem (.), with f (u) = b|u|pu, b > has a unique global solution with energy decaying exponentially for any initial data (u, u)

H( ) L( ). So if f (u) = b|u|pu, and g(ut) = |ut|mut, Nakao [] showed that (.) has a unique global weak solution if p

N , N , and a global unique strong

N , N (of course if N = or N = then the only requirement is p ). On the other hand, in both cases it has been shown that the energy of the solution decays algebraically if m > and decays exponentially if m = . Also as [], Nakao and Ono [] extended this result to the Cauchy problem,

utt u + (x)u + g(ut) + f (u) = , x

RN, t > ,

u(x, ) = u(x), ut(x, ) = u(x), x

RN, (.)

where g(ut) behaves like |ut|mut, f (u) behaves like |u|pu and the initial data (u, u) is small enough in H( ) L( ). Later on, Ono [] studied the global existence and the decay properties of smooth solutions to the Cauchy problem related to (.), for f (u)

and gave sharp decay estimates of the solution without any restrictions on the data size (u, u).

In [], Munoz-Rivera and Andrade dealt with the global existence and exponential decay of solutions of the nonlinear one-dimensional wave equation with a viscoelastic boundary condition. In [], Santos also studied the asymptotic behavior of solutions to a coupled system of wave equations having integral convolutions as memory terms. The main results show that the solutions of that system decay uniformly in time, with rates depending on the rate of decay of the kernel of the convolutions.

In [], the global existence and regularity of weak solutions for the linear wave equation

utt uxx + Ku + ut = f (x, t), < x < , t > , (.)

utt u + c(x, t)u = f (x, t), (x, t) Q = (, T),

u

+

t

k(x, , )u(, ) d d = , (x, t) [, T), u(x, ) = u(x), ut(x, ) = u(x), x ,

utt u + g(ut) + f (u) = , x , t > ,u = , x , t ,u(x, ) = u(x), ut(x, ) = u(x), x ,

solution if p >

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 3 of 27

with the initial conditions as in (.) and two-point boundary conditions. The exponential decay of solutions was also given there by using Lyapunov method.

The works introduced as above lead to the study of the existence and exponential decay of solutions for the problem (.)-(.). This paper consists of three sections. The preliminaries are presented and two existence results with a = are done in Section . The decay of the solution with respect to a = , g = , K > , > , and < p NN, N is established in Section . The proofs of the existences are based on the Faedo-Galerkin method for strong solutions and standard arguments of density for weak solutions. Because this problem is solved in an N-dimensional domain, it causes technical diculties, so we need the relations between the norms as in Lemmas .-. below. To obtain the exponential decay, we use the multiplier technique combined with a suitable Lyapunov functional in the form L(t) = E(t) + (t), where

E(t) =

u (t)

+

u(t)

+ K

u(t)

p

u(t) p

Lp + h(x, t), u(t) u(x, t) dSx,

(t) =

u(t), u (t) +

u(t) , > is chosen suciently small, which allows us to show that if

u + K u u pLp + p h(x, y, )u(y) dy u(x) dSx >

and if the initial energy E(), f , h given are small enough, then the energy E(t) of the solution decays to zero exponentially when t goes to innity.

We end the paper with a remark about a situation where a = , precisely we consider (.) in the form

utt u + Ku + ut + |u|pu = f (x, t), x , t > . (.)

With some suitable conditions for f , h, g, we obtain a unique global solution for (.)-(.) and (.), with energy decaying exponentially as t +, without any restrictions on the data size (u, u) as in [].

2 Preliminaries and existence results

In this paper,

RN is an open and bounded set with a smooth boundary and the usual function spaces Cm( ), Wm,p = Wm,p( ), Lp = W,p( ), Hm = Wm,( ), p , m = , , . . . are used. Let , be either the scalar product in L or the dual pairing of a continuous linear functional and an element of a function space. The notation stands for the norm in L and we denote by X the norm in the Banach space X. We call X

the dual space of X. We denote by Lp(, T; X), p , the Banach space of the real functions u : (, T) X measurable, such that

u L

p(,T;X) =

T

u(t) pX

dt

/p< for p <

and

u L(,T;X) = ess sup

<t<T

u(t)

X for p = .

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 4 of 27

Let u(t), u (t) = ut(t), u (t) = utt(t), u(t), u(t) denote u(x, t), ut(x, t), ut(x, t), ( ux (x, t), . . . , uxN (x, t)), N

i=

u

xi (x, t), respectively.

On H we shall use the following norm: v H

= ( v + v )/.

In cases N = or N = , by the continuity and compactness of the injections H( )

C( ) with N = or H( ) Lq( ) with N = , it is not dicult to study problem (.)-(.). On the other hand, it is obvious that the problem considered with a = is more dicult than the one with a = ,so in what follows we only consider problem (.)-(.) with N , a = . A remark in the end of this paper will give a note in the case a = .

First, we recall the following results, see [].

Lemma . Let

RN be an open and bounded set of class C. Then the embedding H Lq, is continuous if q and compact if q < , where =

N

N , N .

RN be an open and bounded set with a smooth boundary . Then

v(x) dSx / v H

Lemma . Let

for all v H, (.)

where is a positive constant depending only on the domain .

The proofs below also require the following lemma.

Lemma . Let

RN be an open and bounded set with a smooth boundary . Let p NN, N . Then there exists a constant Dp > depending on p, N and such that

(i)

|u|pu

|v|pv

Dp + u

H + v H

/N + u

H + v H

p

u v H ,

(.)

(ii)

|u|pv

Dp + u /NH + u pH v

H

for all u, v H.

Proof (i) We have

|u|pu

|v|pv

=

d d

v

+ (u v)

p v

+ (u v)

d

= (p )|u v|

v

+ (u v)

p

d (p )|u v||W|p, (.)

with W = |u| + |v|.

Hence, by Hlders inequality we have

|u|pu |v|pv

(p ) |u v||W|p dx /

(p ) |u v| dx / |W|(p) dx /

for all > . (.)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 5 of 27

Note that H Lq, q =

Cq v H

, v H, q .

N

N , N , and v L

q

Choose = =

N

N , we have =

=

NN

NN = N , and

|u v| dx / = u v L

C u v H

. (.)

By the condition p NN = +

N , N is equivalent to

(p ) =

NN , (.)

so we consider two cases as follows.Case . (p ) =

N

N :

|W|(p) dx /

= W pL(p) C

(p) W H

p

= Cp(p) W pH. (.)

Case . (p ) < =

N

N :

|W|(p) = |W| + |W|, (.)

|W|(p) dx /

+ |W| dx /

| | + |

|/ W /

| | + |

|/ W H

/

= | | + | |/ W H

/N

| |/N + | |/N W /NH. (.)

Consequently, in both cases we get

|W|(p) dx /

| |/N + | |/N W /NH + Cp(p)N W pH. (.)

Hence

|u|pu

|v|pv

(p )C u v H

| |/N + |

|/N W /NH + Cp(p)N W pH

Dp u v H

+ W /NH + W pH Dp + u

H + v H

N +

u H + v H

p

u v H

. (.)

Similarly (ii) is proved.

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 6 of 27

The proof of Lemma . is complete.

Next, we state two local existence theorems. We make the following assumptions:

(A) < p NN, N , (B) K,

(A) f , f L(, T; L),(A) h L(, T; L( )), h , h L(, T; L( )), (A) g L( ), g , g L( ),(A ) f L(QT),(A ) h L(, T; L( )), h L(, T; L( )), (A ) g L(, T; L( )), g L(, T; L( )).

Then we have the following theorem as regards the existence of a strong solution.

Theorem . Suppose that (A), (B), (A)-(A) hold and the initial data (u, u) H H satises the compatibility condition

(x) = g(x, ) + h(x, y, )u(y) dy. (.)

Then problem (.)-(.) has a unique local solution

u L , T; H , ut L , T; H , utt L , T; L (.)

for T > small enough.

Remark . The regularity obtained by (.) shows that problem (.)-(.) has a unique strong solution

(.)

With less regular initial data, we obtain the following theorem as regards the existence of a weak solution.

Theorem . Let (A), (B), (A )-(A ) hold and (u, u) H L.

Then problem (.)-(.) has a unique local solution

u C [, T]; H C [, T]; L (.)

for T > small enough.

Proof of Theorem . Let {wj} be a denumerable base of H. Under the assumptions of Theorem ., using the Faedo-Galerkin approximation and Lemmas .-., we nd the approximate solution of problem (.)-(.) in the form

um(t) =

m

j=

cmj(t)wj, (.)

R,

u

u L(, T; H) C(, T; H) C(, T; L), ut L(, T; H) C(, T; L),utt L(, T; L).

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 7 of 27

where the coecient functions cmj satisfy the system of ordinary dierential equations

u m(t), wj + um(t), wj + Kum(t) + u m(t), wj

+ ( h(x, t), um(t) + g(x, t))wj(x) dSx= |um(t)|pum(t), wj + f (t), wj , j m, um() = u, u m() = u.

(.)

From the assumptions of Theorem ., system (.) has a solution um on an interval [, Tm] [, T]. The following estimates allow one to take Tm = T for all m, consisting of two key estimates.

In the rst key estimate, we put Sm(t) = u m(t) + um(t) , it implies from (.) that

Sm(t) = Sm() +

h(x, ), u + g(x, ) u

(x) dSx

t

Kum(s) + u m(s), u m(s) ds

t

+ f (s), u m(s) ds + t

u

m(s)

pu

m(s), u m(s) ds

g(x, t)um(x, t) dSx h(x, t), um(t) u

m(x, t) dSx

+

t ds h (x, s), um(s) + h(x, s), u m(s) + g (x, s) u

m(x, s) dSx

Ij. (.)

By Lemmas .-. and the following inequalities:

ab a + b for all a, b

R, > ,

(a + b + c)q q(aq + bq + cq) for all q , a, b, c

(.)

Sm() +

and

v v H

NN , N , (.)

with computing explicitly, all terms in the right-hand side of (.) are estimated, in which the following estimates are worthy of note:

Sm() + I = Sm() +

h(x, ), u + g(x, ) u

, v L

q

Cq v H

, v H, q =

(x) dSx

= u + u + h(x, ), u + g(x, ) u

(x) dSx

C; (.)

I =

t

Kum(s) + u m(s), u m(s) ds t

u

m(s)

ds +

K + || t

u

m(s)

ds

t

u +

s

u m(r)

dr

ds +

K + || t Sm(s) ds

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 8 of 27

T u + T

t

u

m(r)

dr +

K + || t Sm(s) ds

T u + T + K + |

|

t Sm(s) ds CT +

t Sm(s) ds ; (.)

I =

t

f (s), u m(s) ds

T

f

(s)

ds +

t

u

m(s)

ds CT +

t Sm(s) ds; (.)

I =

t

u

m(s)

pu

m(s), u m(s) ds

t

u

m(s)

p u

m(s)

ds

t

u

m(s)

p

ds +

t Sm(s) ds

=

t

u

m(s)

p

Lp ds +

t Sm(s) ds Cpp

t

u

m(s)

p

H ds +

t Sm(s) ds, (.)

since p , and H( ) Lp( ), we have

u

m(t)

p

H u + Sm(t) + t

t Sm(s) ds p

pp u p + p S

m(t)

p + pptp t

Sm(s)

p ds,

it leads to

I =

t

u

m(s)

pu

m(s), u m(s) ds CT + CT

t

Sm(s)

p ds + t Sm(s) ds; (.)

I =

g(x, t)um(x, t) dSx g L(,T;L

( ))

u

m(t)

H

g L(,T;L( )) + u

m(t)

H

g L(,T;L( )) + u + Sm(t) + t

t Sm(s) ds

CT + Sm(t) + CT

t Sm(s) ds for all <

< ; (.)

I =

h(x, t), um(t) u

m(x, t) dSx h L(,T;L

( ))

u

m(t)

u

m(t)

H

h L(,T;L( )) u

m(t)

+

u

m(t)

H

h L(,T;L( )) u + t

t Sm(s) ds

+

u + Sm(t) + t t Sm(s) ds

CT + Sm(t) +

CT

t Sm(s) ds for all

> , < ; (.)

I =

t ds h (x, s), um(s) + h(x, s), u m(s) + g (x, s) u

m(x, s) dSx

h

L(,T;L( ))

t

um(s)

H ds

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 9 of 27

+ h L(,T;L

( ))

t

u

m(s) u

m(s)

H ds

+

g

L(,T;L( ))

t

u

m(s)

H ds

t Sm(s) ds . (.)

Combining estimations of all terms and choosing = , we obtain after some rearrangements

Sm(t) CT +

t Sm(s) ds +

t

CT + CT

t

u

m(s)

H ds +

t

u

m(s)

ds CT +

p ds , t Tm, (.) where CT always indicates a constant depending on T.

Then, by solving a nonlinear Volterra integral inequality (.) (based on the methods in []), the following lemma is proved.

Lemma . There exists a constant T > depending on T (independent of m) such that

Sm(t) CT, m

N, t [, T], (.)

where CT is a constant depending only on T.

By Lemma ., we can take a constant Tm = T for all m.

In the second key estimate, we put Xm(t) = u m(t) + u m(t) ,and it follows from (.) that

Xm(t) = Xm() +

h (x, ), u + h(x, ), u + g (x, ) u

(x) dSx

Sm(s)

t

Ku m(s) + u m(s), u m(s) ds + t

f (s), u m(s) ds

+ (p )

t

um(s) pu

m(s), u m(s) ds

h (x, t), um(t) + h(x, t), u m(t) + g (x, t) u

m(x, t) dSx

+

t ds h (x, s), um(s) + h (x, s), u m(s)

+ h(x, s), u m(s) + g (x, s) u m(x, s) dSx

Xm() +

Ji. (.)

Letting t + in equation (.), multiplying the result by c mj(), and using the compatibility (.), we get

u m()

= u, u m() Ku +u, u m() + |u|pu, u m() + f (), u m() .

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 10 of 27

This implies that

u

m()

u + |K| u + || u + |u

|p

+

f

()

= X for all m, (.)

where X is a constant depending only on p, K, , u, u, f .Also note the following estimations:

Xm() + J = Xm() +

h (x, ), u + h(x, ), u + g (x, ) u

(x) dSx

X + u + h (x, ), u + h(x, ), u + g (x, ) u

(x) dSx

X; (.)

J =

t

Ku m(s) + u m(s), u m(s) ds

t

u

m(s)

ds +

K + || t

u

m(s)

ds

CT + K + |

|

t Xm(s) ds; (.)

J =

t

f (s), u m(s) ds t

f

(s)

ds +

t

f

(s) u

m(s)

ds

CT +

t

f

(s) X

m(s) ds. (.)

From

u

m(s)

pu

m(s)

Dp + u

m(s)

/N

H + u

m(s)

p

H

u

m(s)

H DpCT u

m(s)

H ,

by Lemma .(ii), it gives

J = (p )

t

u

m(s)

pu

m(s), u m(s) ds

(p )

t

u

m(s)

pu

m(s) u

m(s)

ds

(p )DpCT

t

u

m(s)

H

u

m(s)

ds

(p )DpCT

t

u

m(s)

H ds +

t

u

m(s)

ds

= (p )DpCT

t

u

m(s)

ds +

t

u

m(s)

ds

+

t

u

m(s)

ds

CT +

t Xm(s) ds ; (.)

J =

h (x, t), um(t) + h(x, t), u m(t) + g (x, t) u

m(x, t) dSx

um(t) h

L

(,T;L( )) + u

m(t) h L(,T;L( ))

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 11 of 27

+

g

L(,T;L( ))

u

m(t)

H

CT u

m(t)

H

CT + u

m(t)

H

CT + u + Xm(t) + t

t Xm(s) ds

CT + Xm(t) + CT +

t Xm(s) ds for all

(, ); (.)

J =

t ds h (x, s), um(s) + h (x, s), u m(s)

+ h(x, s), u m(s) + g (x, s) u m(x, s) dSx

CT

t

h (s)

L( )

u

m(s)

H ds + CT

t

u

m(s)

H ds

+ CT

t

u

m(s) u

m(s)

H ds +

t

g (s)

L( )

u

m(s)

H ds

CT h

L(,T;L( )) +

t

h (s)

L( )

u

m(s)

H ds

+ CTT +

t

u

m(s)

H ds + CT

t

u

m(s)

ds +

t

u

m(s)

H ds

+ g

L(,T;L( )) +

t

g (s)

L( )

u

m(s)

H ds

CT + CT

t Xm(s) ds +

t

(s)

u

m(s)

H ds

CT + CT

t Xm(s) ds +

t

(s)

u + Xm(s) + s s Xm(r) dr ds

CT + CT

t Xm(s) ds +

t

(s)Xm(s) ds, (.)

where

(s) = +

L( ), L(, T). (.)

Combining estimations and choosing = , we obtain after some rearrangements

Xm(t) CT +

t

h

(s)

L( ) + g

(s)

(s)Xm(s) ds, (.)

where CT always indicates a constant depending on T, and

(s) = CT

+

f (s)

+

h (s)

L( ) +

g (s)

L( )

, L(, T). (.)

By Gronwalls lemma, we deduce from (.) that

Xm(t) CT exp

T

(s) ds

CT for all t [, T]. (.)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 12 of 27

It veries the existence of a subsequence of {um}, denoted by the same symbol, such that

(.)

By means of the continuity of the function t |t|pt, we have

|um|pum |u|pu and a.e. in QT. (.)

On the other hand

|u

m|pum

p dt CppT um pL(,T;H) CT. (.)

Using the Lions lemma ([], Lemma ., p.), it follows from (.) and (.) that

|um|pum |u|pu in L(QT) weakly. (.)

Passing to the limit in (.) by (.), (.), and (.), we have u satisfying the problem

On the other hand, we have from (.), (.)

u = u + Ku + u |u|pu f L , T; L . (.) Thus u L(, T; H) and the proof of existence is complete. The uniqueness of a weak solution is proved as follows.

Let u, u be two weak solutions of problem (.)-(.), such that

ui L , T; H , u i L , T; H , u i L , T; L , i = , . (.)

um u in L(, T; H) weakly, u m u in L(, T; H) weakly, u m u in L(, T; L) weakly.

(.)

By the compactness lemma of Lions ([], p.), we can deduce from (.) the existence of a subsequence still denoted by {um}, such that

um u strongly in L(QT) and a.e. in QT, u m u strongly in L(QT) and a.e. in QT.

L(QT ) =

T ds u

m(x, t)

p

dx

=

T

T

u

m(t)

p

Lp dt

u (t), v + u(t), v + Ku(t) + u (t), v

+ ( h(x, t), u(t) + g(x, t))v(x) dSx= |u(t)|pu(t), v + f (t), v for all v H, u() = u, u () = u.

Cp

u

m(t)

H

(.)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 13 of 27

Then u = u u satisfy the variational problem

u (t), v + u(t), v + Ku(t) + u (t), v +

h(x, t), u(t) v(x) dSx = |u|pu |u|pu, v for all v H,u() = u () = .

(.)

We take v = u = u u in (.) and integrating with respect to t, we obtain

(t) =

t

Ku(s) + u (s), u (s) ds h(x, t), u(t) u(x, t) dSx

+

t ds h (x, s), u(s) + h(x, s), u (s) u(x, s) dSx +

t

|u|pu |u|pu, u (s) ds =

j, (.)

where

(t) =

u (t)

+ u(t) .

(.)

By (.) and the following inequalities:

ab a +

b for all a, b

R, > , (.)

u(t)

=

t

u (s)

ds

t

t

u (s)

ds t

t

(s) ds,

u(t)

H = u(t)

+

u(t)

(t) + t

t

(s) ds, (.)

t

u(s)

H ds

t

(s) + s

s

(r) dr

ds + t

t

(s) ds,

we estimate the following integrals in the right-hand side of (.):

=

t

Ku(s) + u (s), u (s) ds

t

u(s)

ds +

K + || t

u (s)

ds

t

s

s

(r) dr

ds +

K + || t(s) ds

T

t

(r) dr +

K + || t(s) ds CT t(s) ds; (.)

=

h(x, t), u(t) u(x, t) dSx h L(,T;L

( ))

u(t) u(t)

H

h L(,T;L( ))

u(t)

+

u(t)

H

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 14 of 27

h L(,T;L( ))t

t

(s) ds +

(t) + t

t

(s) ds

(t) +

CT

t

(s) ds; (.)

t ds h (x, s), u(s) + h(x, s), u (s) u(x, s) dSx h

L(,T;L( ))

=

t

u(s)

H ds

+ h L(,T;L

( ))

t

u (s) u(s)

H ds

CT

t

u(s)

H ds + CT

t

u (s)

ds CT

t

(s) ds. (.)

By Lemma .(i), we have

|u

|pu |u|pu

Dp + u

H

+ u H

/N + u H + u H

p

u(s)

H

Dp + M/N + Mp u(s)

H CT u(s)

H , (.)

where M = u L(,T;H

) + u L(,T;H

). Hence

=

t

|u|pu |u|pu, u (s) ds

CT

t

u(s)

H

u (s)

ds

CT

t

u(s)

H ds + CT

t

u (s)

ds

CT

t

(s) ds. (.)

Combining (.), (.)-(.), (.) and choosing = , we obtain

(t) CT

t

(s) ds. (.)

By Gronwalls lemma, it follows from (.) that , i.e., u u. Theorem . is proved completely.

Proof of Theorem . In order to prove this theorem, we use standard arguments of density.

First, we note that W(, T; L( )) = {g L(, T; L( )) : g L(, T; L( ))} is a Hilbert space with respect to the scalar product (see []):

f , g W

(,T;L( )) =

T

f (t), g(t)

L( ) + f (t), g (t)

L( )

dt. (.)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 15 of 27

Furthermore, we also have the embedding W(, T; L( )) C([, T]; L( )) is continuous and

g C

([,T];L( )) T g L(,T;L( )) + g

L(,T;L( ))

(,T;L( )) (.)

for all g W(, T; L( )), where T =

T g W

T + +

T (see the Appendix).

Similarly, W(, T; L( )) = {h L(, T; L( )) : h L(, T; L( ))} is a Hilbert space with respect to the scalar product

h, k W

(,T;L( )) =

T

h(t), k(t)

L( ) + h (t), k (t)

L( )

dt, (.)

and the embedding W(, T; L( )) C([, T]; L( )) is continuous and

h C

([,T];L( )) T h L(,T;L( )) + h

L(,T;L( ))

(,T;L( )) (.)

for all h W(, T; L( )), where T =

T h W

T + +

T (see the Appendix).

Consider (u, u, f , g, h) H L L(QT) W(, T; L( )) W(, T; L( )). Let the sequence {(um, um, fm, gm, hm)} H H C(QT)C( )C(

[, T]), such that

um u strongly in H, um u strongly in L, fm f strongly in L(QT),

(.)

gm g W(,T;L( )) gm g L(,T;L( )) + g

m g

L(,T;L( )) , (.)

hm h W(,T;L( )) hm h L(,T;L( )) + h

m h

L(,T;L( ))

. (.)

So {(um, um)} satisfy, for all m

N, the compatibility condition

(x) = gm(x, ) + hm(x, y, )um(y) dy. (.)

Then, for each m

N, there exists a unique function um under the conditions of Theorem .. Thus, we can verify

u m(t), v + um(t), v + Kum(t) + u m(t), v

+ ( hm(x, t), um(t) + gm(x, t))v(x) dSx = |um|pum, v + fm(t), v for all v H, um() = um, u m() = um

um

(.)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 16 of 27

and

um L(, T; H) C(, T; H) C(, T; L), u m L(, T; H) C(, T; L),u m L(, T; L).

(.)

By the same arguments used to obtain the above estimates, we get

u

m(t)

+

u

m(t)

H CT, (.)

t [, T], where CT always indicates a constant depending on T as above.On the other hand, we put wm,l = um ul, fm,l = fm fl, hm,l = hm hl, gm,l = gm gl,

hm,l(x, y, ) = h()m,l(x, y), gm,l(x, ) = g()m,l(x), from (.), it follows that

w m,l(t), v + wm,l(t), v + Kwm,l(t) + w m,l(t), v

+ ( hm(x, t), wm,l(t) + hm,l(x, t), ul(t) + gm,l(x, t))v(x) dSx = |um|pum |ul|pul, v + fm,l(t), v for all v H, wm,l() = um ul w()m,l, w m,l() = um ul w()m,l.

(.)

We take v = wm,l = um ul, in (.) and integrating with respect to t, we get

Sm,l(t) = Sm,l() +

h

m(x, ), w()m,l + h

m,l(x, ), ul

+ gm,l(x, ) w()m,l(x) dSx

t

+ fm,l(s), w m,l(s) ds t

Kwm,l(s) + w m,l(s), w m,l(s) ds

h

m(x, t), wm,l(t)

+ h

m,l(x, t), ul(t)

+ gm,l(x, t) w

m,l(x, t) dSx

+

t ds h

m(x, s), wm,l(s) + h

m(x, s), w m,l(s) w

m,l(x, s) dSx

+

t ds h

m,l(x, s), ul(s) + h

m,l(x, s), u l(s) w

m,l(x, s) dSx

t ds g m,l(x, s)wm,l(x, s) dSx +

t

+

|um|pum |ul|pul, w m,l(s) ds Sm,l() +

Zj, (.)

where

Sm,l(t) =

w

m,l(t)

+

w

m,l(t)

,

(.)

Sm,l() = um ul + um ul . (.)

After all terms of Sm,l(t) are estimated, in which we note the two main estimations Z, Z as follows:

Z =

h

m(x, ), w()m,l + h

m,l(x, ), ul

+ gm,l(x, ) w()m,l(x) dSx

w()m,l hm() L( )

+ ul

h()m,l L( )

+ g()m,l L( )

w()m,l H

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 17 of 27

T const.

w()m,l

H + hm,l W

(,T;L( )) + gm,l W

(,T;L( ))

w()m,l

H

, as m, l +; (.)

this result combined with (.)-(.) shows that

Sm,l() + Z = um ul + um ul + Z

R(m, l) , as m, l +. (.)

On the other hand

|u

m|pum |ul|pul

Dp + u

m H

+ ul H

/N + um H + ul H

p

w

m,l(s)

H

CT w

m,l(s)

H , (.) by Lemma .(i), we get

Z =

t

|um|pum |ul|pul, w m,l(s) ds CT t

w

m,l(s)

H

w

m,l(s)

ds

CT t

w()m,l

+

+ t t Sm,l(s) ds + CT t Sm,l(s) ds

CT

w()m,l

+

t Sm,l(s) ds . (.)

We obtain

Sm,l(t) R()T(m, l) + CT

t Sm,l(s) ds, (.)

with

R()T(m, l) = R(m, l) + fm,l L(QT)

+ CT

w()m,l

+ hm,l W(,T;L( )) + gm,l W(,T;L( )) , (.) as m, l +. By Gronwalls lemma, it follows from (.) that

Sm,l(t) R()T(m, l) exp(TCT) CTR()T(m, l), t [, T]. (.)

Thus, convergence of the sequence {(um, um, fm, gm, hm)} implies the convergence to zero as m, l + of the term on the right-hand side of (.). Therefore, we get

um u strongly in C [, T]; H C [, T]; L . (.)

On the other hand, from (.), we get the existence of a subsequence of {um}, still also so denoted, such that

um u in L(, T; H) weakly, u m u in L(, T; L) weakly.

(.)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 18 of 27

By the compactness lemma of Lions ([], p.), we can deduce from (.) the existence of a subsequence, still denoted by {um}, such that

um u strongly in L(QT) and a.e. in QT. (.)

Similarly, by (.), it follows from (.) that

|um|pum |u|pu in L(QT) weakly. (.)

Passing to the limit in (.) by (.)-(.), we have u satisfying the problem

Next, the uniqueness of a weak solution is obtained by using the well-known regularization procedure due to Lions. Theorem . is proved completely.

Remark . In the case < p , f L(QT), g W(, T; L( )), h W(, T; L(

)), and (u, u) H L, the integral inequality (.) leads to the following global estimation:

Sm(t) CT, m

N, t [, T], T > . (.)

Then, by applying a similar argument to the proof of Theorem ., we can obtain a global weak solution u of problem (.)-(.) satisfying

u L , T; H , ut L , T; L . (.)

However, in the case < p < , we do not imply that a weak solution obtained here belongs to C([, T]; H) C([, T]; L). Furthermore, the uniqueness of a weak solution is also not asserted.

3 Exponential decay

In this section, we study the exponentially decay of solutions of problem (.)-(.) corresponding to a = , g = , K > , > , and < p NN. For this purpose, we make the following assumptions:

(A ) f L(, ; L) = L(Q), Q =

R+, such that f (t) Cet, for all t , with C > , > are given constants,(A ) h L(, ; L( )) L(

R+ ), h L(, ; L( ))

Let K > , on H we shall use the following norm:

v = K v + v /.

d

dt u (t), v + u(t), v + Ku(t) + u (t), v

+ ( h(x, t), u(t) + g(x, t))v(x) dSx = |u|pu, v + f (t), v for all v H, u() = u, u () = u.

(.)

L(, ; L( )), (A ) g = .

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 19 of 27

Then we have the following lemma.

Lemma . On H, two norms v , v H

are equivalent and

max{, K}

v v H

min{, K}

v C v for all v H,

where C =

min{,K} .

The proof of this lemma is simple, we omit the details.

We construct the following Lyapunov functional:

L(t) = E(t) + (t), (.)

where > is chosen later and

E(t) =

u (t)

+

u(t)

p

u(t) p

Lp + h(x, t), u(t) u(x, t) dSx, (.)

(t) =

u(t), u (t) +

u(t) .

(.)

Put

I(t) = I

u(t) = u(t) u(t) pLp + p h(x, t), u(t) u(x, t) dSx, (.)

J(t) = J

u(t) =

u(t)

p

u(t) p

Lp + h(x, t), u(t) u(x, t) dSx

=

p

u(t)

+

pI(t), (.)

we rewrite

E(t) =

u (t)

+ J(t) =

u (t)

+

p

u(t)

+

pI(t). (.)

Then we have the following theorem.

Theorem . Assume that (A )-(A ) hold. Let I() > and the initial energy E() satisfy

= (CpC)p

pp E

p

+ pC

h L(,;L

( )) < , (.)

where

E = (E() + f (t) dt) exp( pp h(t) dt),

h(t) =

(C h (t) L

( ) +

h(t) L( )),

(.)

= C and Cp is a constant satisfying the inequality v L p

Cp v H

, for all v H.

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 20 of 27

Then, for E(), f L(,;L

), h L(,;L

( )), h L(,;L

( )) suciently small,

there exist positive constants C, such that

E(t) C exp( t) for all t . (.)

Proof of Theorem . At rst, we state and prove Lemmas .-. as follows.

Lemma . The energy functional E(t) satises

E (t)

u (t)

+

f

(t)

+

C

h (t)

L( ) +

h(t)

L( )

u(t)

. (.)

Proof Multiplying (.) by u (x, t) and integrating over [, ], we get

E (t) = u (t)

+ f (t), u (t) + h (x, t), u(t) + h(x, t), u (t) u(x, t) dSx. (.)

We have

f (t), u (t)

u (t)

+

f

(t)

.

(.)

By Lemmas ., ., ., we obtain

h (x, t), u(t) u(x, t) dSx u(t)

h (x,

t)

u(x,

t)

dSx

C u(t)

h (x,

t)

dSx

/

u(x, t) dSx /

C

h (t)

L( )

u(t)

, (.)

h(x, t), u (t) u(x, t) dSx u (t)

h(x,

t)

u(x,

t)

dSx

u (t)

h(t)

L( )

u(t)

u (t)

+

h(t)

L( )

u(t)

. (.)

Combining (.)-(.), (.) follows. Lemma . is proved completely.

Lemma . Suppose that (A )-(A ) hold. Then, if we have I() > and

= (CpC)p

pp E

p

+ pC

h L(,;L

( )) < , (.)

then I(t) > , t .

Proof By the continuity of I(t) and I() > , there exists T > such that

I(t) = I

u(t)

, t [, T], (.)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 21 of 27

this implies

J(t) =

p

u(t)

+

pI(t)

p

u(t)

p p

u(t)

, t [, T]. (.) It follows from (.), (.) that

u(t)

pp J(t)

pp E(t), t [, T]. (.)

Equation (.) leads to

E (t)

u (t)

+

f

(t)

+

C

h (t)

L( ) +

h(t)

L( )

u(t)

f

(t)

+ p

p

C

h (t)

L( ) +

h(t)

L( )

E(t)

p h(t)E(t). (.)

Integrating with respect to t, we obtain

E(t) E() +

=

f

(t)

+ p

f

(t)

dt + p

p

t h(s)E(s) ds, (.)

where h(t) is as in (.).

Combining (.), (.), and using the Gronwall lemma, we have

E(t) E() +

f

(t)

dt

exp p p

h(s) ds = E (.)

and

u(t)

pp E, t [, T]. (.)

Hence, it follows from (.), (.) that

u(t) p

pp E(t)

Lp p h(x, t), u(t) u(x, t) dSx

(CpC)p u(t) p

+ pC

h(t)

L( )

u(t)

(CpC)p

pp E

u(t)

p

+ pC

h L(,;L

( ))

= u(t)

<

u(t) ,

t [, T]. (.)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 22 of 27

Therefore, I(t) > , t [, T].

Now, we put T = sup{T > : I(u(t)) > , t [, T]}. If T < + then, by the continuity of I(t), we have I(T) . By the same arguments as in the above part, we can deduce that there exists T > T such that I(t) > , t [, T]. Hence, we conclude that I(t) > , t .

Lemma . is proved completely.

Lemma . Let I() > and (.) hold. Then there exist the positive constants , such

that

E(t) L(t) E(t), t (.)

for is suciently small.

Proof A simple computation gives

L(t) =

u (t)

+

p

u(t)

+

pI(t) +

u(t), u (t) +

u(t) ,

(.)

E(t) =

u (t)

+ J(t) =

u (t)

+

p

u(t)

+

pI(t).

From the following inequalities:

u(t), u (t) C

u(t)

u (t)

u (t)

+

C u(t)

,

C u(t)

,

(.)

u(t)

we deduce from (.) that

L(t)

u (t)

+

p

u(t)

+

pI(t)

u (t)

C u(t)

C u(t)

=

u

(t)

+

p

C +

u(t)

+

pI(t)

= ( )

u (t)

+

C( + )

p

p

u(t)

+

pI(t)

E(t), (.)

where we choose

= min

, , C( + )

p

, (.)

p

with being small enough, < < min{ ,

C

(+

) }.

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 23 of 27

Similarly, we can prove that

L(t)

+

u

(t)

+

p +

C +

u(t)

+

pI(t)

= ( + )

u (t)

+

+ C( + )

p

p

u(t)

+

pI(t)

E(t), (.)

where

= max

+ , + C( + )

p

. (.)

Lemma . is proved completely.

Lemma . Let I() > and (.) hold. Then the functional (t) dened by (.) satises

(t) u (t)

+

C f

(t)

I(t)

(

) (p )C

h L(,;L

( ))

u(t)

(.)

for all > .

Proof By multiplying (.) by u(x, t) and integrating over [, ], we obtain

(t) = u (t)

u(t)

+ u(t) p

Lp + f (t), u(t) h(x, t), u(t) u(x, t) dSx

=

u (t)

I(t)

I(t) + (p ) h(x, t), u(t) u(x, t) dSx + f (t), u(t) . (.)

Note that

I(t) ( ) u(t) ,

h(x, t), u(t) u(x, t) dSx C

h L(,;L

( ))

u(t)

, (.)

f (t), u(t)

u(t)

+

C f

(t)

,

hence, Lemma . is proved by using some estimates.

Now, we prove Theorem ..

It follows from (.), (.), and (.) that

L (t)

u

(t)

I(t)

(

) (p )C

h L(,;L

( ))

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 24 of 27

C

h

L(,;L

( )) +

h L(,;L( )) u(t)

+

+

C f

(t)

=

u

(t)

I(t)

(

) [h]

u(t)

+

+

C f

(t)

(.)

for all , > , where

[h]

(p )C

h L(,;L

( ))

+

C

h

L(,;L

( )) +

h L(,;L( )) . (.)

Let , satisfy

< <

, <

). (.)

Then, for small enough such that < < and if h satisfy

[h]

<

<

(

) , (.)

we deduce from (.), (.), and (.) that there exists a constant > such that

L (t) ( )

(

u (t)

p

pI(t)

{

[ ( ) ] [h] }

p

p

u(t)

+ (t)

E(t) + (t)

L(t) +

(t) L(t) + (t), (.)

where

= min

,

p, {

[ ( ) ] [h] }

p

> ,

(t) =

+

C f

(t)

Cet, (.)

< < min{, }.

Combining (.) and (.), we get (.). Theorem . is proved completely.

Remark We consider the following problem:

utt u + Ku + ut + |u|pu = f (x, t), x , t > , u(x, t) = g(x, t) +

h(x, y, t)u(y, t) dy, x , t , u(x, ) = u(x), ut(x, ) = u(x).

(.)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 25 of 27

With the suitable conditions for K, , p, u, u, f , g, h, we prove that problem (.) has a unique global solution u(t) with energy decaying exponentially as t +, without the initial data (u, u) being small enough. The results obtained are as follows and their proofs are not dicult with a procedure analogous to the ones in Theorems ., ..

Theorem . Suppose that < p NN, K > , > , g , (u, u) H L and (A ), (A ) hold. Then problem (.) has a unique global solution u L(, ; H)

C([, ); H) C([, ); L) such that ut L(, ; L).Furthermore, if h L(,;L

( )), h L(,;L

( )) are suciently small then there

exist positive constants C, such that

u (t)

+

u(t)

+ u(t) p

Lp C exp( t) for all t .

Appendix

Lemma A. Let H be Hilbert space with respect to the scalar product , . Then the embedding W(, T; H) = {F L(, T; H) : F L(, T; H)} C([, T]; H) is continuous and

F C

([,T];H) T F L(,T;H) + F

L(,T;H)

T F W

(,T;H)

for all F W(, T; H), where T =

T + +

T .

Proof Let F W(, T; H), for all t, s [, T], we have

F(t)

F(s)

=

ts F (r) dr

t s

F (r)

dr

|t s| F

L(,T;H). (A.)

Hence F C([, T]; H). On the other hand

F(t)

=

F(s)

+

t s

d dt

F(r)

dr =

F(s)

t s

+ F(r), F (r) dr. (A.)

Integrating with respect to s, we obtain

T

F(t)

=

T

F(s)

ds +

T ds

t s

F(r), F (r) dr

= F L(,T;H) +

T ds

t

F(r), F (r) dr

T ds s

F(r), F (r) dr

= F L(,T;H) + T

t

F(r), F (r) dr

T ds s

F(r), F (r) dr. (A.)

Inverting the variables s and r in the last integral of (A.), we rewrite it as follows:

T ds

s

F(r), F (r) dr =

T (T r) F(r), F (r) dr. (A.)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 26 of 27

By the inequality ab a + b, for all a, b

R, > , we deduce from (A.), (A.)

that

T

F(t)

F L(,T;H) + T

t

F(r) F (r)

dr +

T (T r) F(r) F (r)

dr

F L(,T;H) + T

T

F(r) F (r)

dr

F L(,T;H) + T F L

(,T;H)

L(,T;H)

F L(,T;H) + T F L(,T;H) +

F

F

L(,T;H)

( + T) F L(,T;H) +

T

F

L(,T;H). (A.)

Choose > such that + T = T , or =

T + +

T . Hence

F C([,T];L)

T + +

T

F L(,T;H) + F

L(,T;H)

T F W(,T;H). (A.)

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally in this article. They read and approved the nal manuscript.

Author details

1University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam. 2Department of Mathematics, University of Architecture of Ho Chi Minh City, 196 Pasteur Str., Dist. 3, Ho Chi Minh City, Vietnam. 3Department 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.

Acknowledgements

The authors wish to express their sincere thanks to the referees for their valuable comments. The authors are also extremely grateful to Vietnam National University Ho Chi Minh City for the encouragement.

Received: 19 August 2015 Accepted: 11 January 2016

References

1. Guezane-Lakoud, A, Dabas, J, Bahuguna, D: Existence and uniqueness of generalized solutions to a telegraph equation with an integral boundary condition via Galerkins method. Int. J. Math. Math. Sci. 2011, Article ID 451492 (2011). doi:10.1155/2011/451492

2. Bergounioux, M, Long, NT, Dinh, APN: Mathematical model for a shock problem involving a linear viscoelastic bar. Nonlinear Anal. TMA 43, 547-561 (2001)

3. Beilin, SA: On a mixed nonlocal problem for a wave equation. Electron. J. Dier. Equ. 2006, 103 (2006)4. Benaissa, A, Messaoudi, SA: Exponential decay of solutions of a nonlinearly damped wave equation. Nonlinear Dier. Equ. Appl. 12, 391-399 (2005)

5. Clark, HR: Global classical solutions to the Cauchy problem for a nonlinear wave equation. Int. J. Math. Math. Sci. 21(3), 533-548 (1998)

6. Long, NT, Dinh, APN: On the quasilinear wave equation: utt u + f(u, ut) = 0 associated with a mixed nonhomogeneous condition. Nonlinear Anal. TMA 19, 613-623 (1992)

7. Long, NT, Ngoc, LTP: On a nonlinear wave equation with boundary conditions of two-point type. J. Math. Anal. Appl. 385(2), 1070-1093 (2012)

8. Long, NT, Dinh, APN, Diem, TN: On a shock problem involving a nonlinear viscoelastic bar. Bound. Value Probl. 2005(3), 337-358 (2005)

9. Long, NT, Truong, LX: Existence and asymptotic expansion for a viscoelastic problem with a mixed nonhomogeneous condition. Nonlinear Anal. TMA 67(3), 842-864 (2007)

10. Messaoudi, SA: Decay of the solution energy for a nonlinearly damped wave equation. Arab. J. Sci. Eng. 26, 63-68 (2001)

Ngoc et al. Boundary Value Problems (2016) 2016:20 Page 27 of 27

11. Medeiros, LA, Limaco, J, Menezes, SB: Vibrations of elastic strings: mathematical aspects, part one. J. Comput. Anal. Appl. 4(2), 91-127 (2002)

12. Medeiros, LA, Limaco, J, Menezes, SB: Vibrations of elastic strings: mathematical aspects, part two. J. Comput. Anal. Appl. 4(3), 211-263 (2002)

13. Menzala, GP: On global classical solutions of a nonlinear wave equation. Appl. Anal. 10, 179-195 (1980)14. Messaoudi, SA: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. 260, 58-66 (2003)

15. Nakao, M: Decay of solutions of some nonlinear evolution equations. J. Math. Anal. Appl. 60, 542-549 (1977)16. Nakao, M: Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equations. Math. Z. 206, 265-275 (1991)

17. Nakao, M, Ono, K: Global existence to the Cauchy problem of the semilinear wave equation with a nonlinear dissipation. Funkc. Ekvacioj 38, 417-431 (1995)

18. Ngoc, LTP, Hang, LNK, Long, NT: On a nonlinear wave equation associated with the boundary conditions involving convolution. Nonlinear Anal. TMA 70(11), 3943-3965 (2009)

19. Ono, K: On the global existence and decay of solutions for semilinear telegraph equations. Int. J. Appl. Math. 2(9), 1121-1136 (2000)

20. Pulkina, LS: A nonlocal problem with an integral condition of the rst kind for a multidimensional hyperbolic equation. Ross. Akad. Nauk 416(5), 597-599 (2007)

21. Munoz-Rivera, JE, Andrade, D: Exponential decay of non-linear wave equation with a viscoelastic boundary condition. Math. Methods Appl. Sci. 23, 41-61 (2000)

22. Santos, ML: Asymptotic behavior of solutions to wave equations with a memory condition at the boundary. Electron.J. Dier. Equ. 2001, 73 (2001)23. Santos, ML: Decay rates for solutions of a system of wave equations with memory. Electron. J. Dier. Equ. 2002, 38 (2002)

24. Santos, ML, Ferreira, J, Pereira, DC, Raposo, CA: Global existence and stability for wave equation of Kirchho type with memory condition at the boundary. Nonlinear Anal. TMA 54, 959-976 (2003)

25. Truong, LX, Ngoc, LTP, Dinh, APN, Long, NT: The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions. Nonlinear Anal., Real World Appl. 11(3), 1289-1303 (2010)

26. Brezis, H: Functional Analysis, Sobolev Spaces and Partial Dierential Equations. Springer, New York (2010)27. Lakshmikantham, V, Leela, S: Dierential and Integral Inequalities, vol. 1. Academic Press, New York (1969)28. Lions, JL: Quelques Mthodes de Rsolution des Problmes aux Limites Non Linaires. Dunod/Gauthier-Villars, Paris (1969)