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

R E S E A R C H

Open Access

Existence and exponential decay

estimates for an

N

-dimensional nonlinear

wave equation with a nonlocal boundary

condition

Le Thi Phuong Ngoc

_{, Nguyen Anh Triet}

_{and Nguyen Thanh Long}

*_{Correspondence:}

[email protected]

3_{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

Full 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 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.

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+λut=a|u|p–u+f(x,t), x∈,t> , (.)

–∂u

∂ν(x,t) =g(x,t) +

h(x,y,t)u(y,t)dy, x∈∂,t≥, (.) u(x, ) =u(x), ut(x, ) =u(x), (.)

whereis a bounded domain inRN _{with a smooth boundary∂,νis the unit outward}

normal on∂;a=±,K,λ,pare given constants, andu,u,f,g,hare given functions satisfying conditions specified later.

The wave equation

utt–u=f(x,t,u,ut), (.)

with different boundary conditions, has been extensively studied by many authors, for example, we refer to [–] and the references given therein. In these works, many in-teresting results about the existence, regularity and the asymptotic behavior of solutions

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

⎧ ⎪ ⎨ ⎪ ⎩

utt–u+c(x,t)u=f(x,t), (x,t)∈Q=×(,T), ∂u

∂ν+ t



k(x,ξ,τ)u(ξ,τ)dξdτ= , (x,t)∈∂×[,T),

u(x, ) =u(x), ut(x, ) =u(x), x∈,

(.)

whereis a bounded domain inRN_{with a smooth boundary,νis the unit outward normal}

on∂,f,u,u,k(x,ξ,τ) 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:

⎧ ⎪ ⎨ ⎪ ⎩

utt–u+g(ut) +f(u) = , x∈,t> ,

u= , x∈∂,t≥,

u(x, ) =u(x), ut(x, ) =u(x), x∈,

(.)

wheref(u) = –b|u|p–_u,g(u

t) =a( +|ut|m–)ut,a,b> ,m,p> , andis a bounded

do-main ofRN_{, with a smooth boundary∂. Benaissa and Messaoudi showed that for suitably}

chosen initial data, (.) possesses a global weak solution, which decays exponentially even ifm> . The proof of the global existence is based on the use of the potential well theory. As [], Messaoudi [] also showed the problem (.), withf(u) =b|u|p–_{u,b>  has a} unique global solution with energy decaying exponentially for any initial data (u,u)∈ H_()×L_{(). So iff(u) =b|u|}p–_{u, andg(u}

t) =|ut|m–ut, Nakao [] showed that (.)

has a unique global weak solution if ≤p– ≤_N_–,N≥, and a global unique strong solution ifp–  >_N_–,N≥ (of course ifN=  orN=  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 ifm>  and decays exponentially ifm= . Also as [], Nakao and Ono [] extended this result to the Cauchy problem,

utt–u+λ(x)u+g(ut) +f(u) = , x∈RN,t> ,

u(x, ) =u(x), ut(x, ) =u(x), x∈RN,

(.)

whereg(ut) behaves like|ut|m–ut,f(u) behaves like –|u|p–uand the initial data (u,u) is small enough inH()×L(). Later on, Ono [] studied the global existence and the decay properties of smooth solutions to the Cauchy problem related to (.), forf(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

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 prelimi-naries are presented and two existence results witha=  are done in Section . The decay of the solution with respect toa= ,g= ,K> ,λ> , and  <p≤N–

N–,N≥ is estab-lished 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 anN-dimensional domain, it causes technical difficulties, 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 formL(t) =E(t) +δψ(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,

ψ(t) =u(t),u(t)+λ  u(t)

 ,

δ>  is chosen sufficiently small, which allows us to show that if

∇u+Ku–up_Lp+p

∂

h(x,y, )u(y)dy

u(x)dSx> 

and if the initial energyE(),f,hgiven are small enough, then the energyE(t) of the solu-tion decays to zero exponentially whentgoes to infinity.

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

utt–u+Ku+λut+|u|p–u=f(x,t), x∈,t> . (.)

With some suitable conditions forf,h,g, we obtain a unique global solution for (.)-(.) and (.), with energy decaying exponentially ast→+∞, 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 spacesCm_(),Wm,p_=Wm,p_(),Lp_=W,p_(),Hm_=Wm,_{(), ≤p≤ ∞,} m= , , . . . are used. Let·,·be either the scalar product inLor the dual pairing of a continuous linear functional and an element of a function space. The notation · stands for the norm inL and we denote by · Xthe norm in the Banach spaceX. We callX the dual space ofX. We denote byLp_{(,T;X), ≤p≤ ∞, the Banach space of the real}

functionsu: (,T)→Xmeasurable, such that

uLp_(,T;X)=

T



u(t) p_Xdt

/p

<∞ for ≤p<∞

and

uL∞_(,T;X)=ess sup

<t<T

Letu(t),u(t) =ut(t),u(t) =utt(t),∇u(t),u(t) denoteu(x,t),∂∂ut(x,t), ∂u ∂t(x,t), (

∂u ∂x(x,t), . . . ,_∂∂_xNu(x,t)),N_i=∂_∂u

x_i(x,t), respectively.

OnH_{we shall use the following norm:vH}

= (v+∇v)/.

In casesN=  orN= , by the continuity and compactness of the injectionsH_()→ C_{() withN=  orH}_()→Lq_{() withN= , it is not difficult to study problem}

(.)-(.). On the other hand, it is obvious that the problem considered with a=  is more difficult than the one witha= –,so in what follows we only consider problem (.)-(.) withN≥,a= . A remark in the end of this paper will give a note in the casea= –.

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≥.

Lemma . Let⊂RN _{be an open and bounded set with a smooth boundary∂.Then}

∂

v(x)dSx /

≤γvH for all v∈H, (.)

whereγis 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≤N–

N–,N≥.Then there exists a constant Dp> depending on p,N andsuch that

(i) |u|p–u–|v|p–v

≤Dp

 +uH+vH

/N

+uH+vH

p–

u–vH,

(ii) |u|p–v ≤Dp

 +u/_HN +u

p–

H

vH

(.)

for all u,v∈H.

Proof (i) We have

_|u|p–_u–|v|p–_v=



 d

dθv+θ(u–v)

p–

v+θ(u–v)dθ

= (p– )|u–v|





v+θ(u–v)p–dθ≤(p– )|u–v||W|p–_{, (.)}

withW=|u|+|v|.

Hence, by Hölder’s inequality we have

_|u|p–_u–|v|p–_{v ≤(p– )}

|u–v||W|p–dx

/

≤(p– )

|u–v|αdx

/α

|W|(p–)αdx

/α

Note thatH_→Lq_{, ≤q≤}∗₌ N

N–,N≥, andvLq≤CqvH,∀v∈H, ≤q≤∗. Chooseα=∗

 =

N

N–, we haveα=

α α–=

N N–

N N––

=N

, and

|u–v|αdx

/α

=u–v_L∗≤C∗u–vH. (.)

By the condition ≤p≤_NN_––=  +_N_–,N≥ is equivalent to

≤(p– )α≤∗= N

N– , (.)

so we consider two cases as follows. Case. ≤(p– )α≤∗=_NN_–:

|W|(p–)αdx

/α

=Wp–

L(p–)α≤

C(p–)αWH

p–

=C_(p–_p–)αWp_H–. (.)

Case. ≤β≡(p– )α< ≤∗= N N–:

|W|(p–)α=|W|β≤ +|W|, (.)

|W|(p–)αdx

/α ≤

 +|W|dx

/α

≤||+||/W/α

≤||+||/WH

/α

=||+||/WH

/N

≤ ||/N+||/NW/_HN. (.)

Consequently, in both cases we get

|W|(p–)αdx

/α

≤ ||/N+||/NW/_HN +C

p– (p–)NW

p–

H. (.)

Hence

_|u|p–u–|v|p–v ≤(p– )C∗u–vH

×||/N+||/NW/_HN +C

p– (p–)NW

p–

H

≤Dpu–vH

 +W/_HN +W

p–

H

≤Dp

 +uH+vH 

N _+uH_+vH_p–

× u–vH. (.)

The proof of Lemma . is complete.

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

(A)  <p≤_NN_––,N≥,

(B) K,λ∈R,

(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_(Q

T),

(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_{satisfies the compatibility condition}

–∂u

∂ν (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

⎧ ⎪ ⎨ ⎪ ⎩

u∈L∞(,T∗;H)∩C(,T∗;H)∩C(,T∗;L),

ut∈L∞(,T∗;H)∩C(,T∗;L),

utt∈L∞(,T∗;L).

(.)

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 ofH. Under the assumptions of Theorem ., using the Faedo-Galerkin approximation and Lemmas .-., we find the approximate solution of problem (.)-(.) in the form

um(t) = m

j=

where the coefficient functionscmjsatisfy the system of ordinary differential equations

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u_m(t),wj+∇um(t),∇wj+Kum(t) +λum(t),wj

+_∂(h(x,t),um(t)+g(x,t))wj(x)dSx

=|um(t)|p–um(t),wj+f(t),wj, ≤j≤m,

um() =u, um() =u.

(.)

From the assumptions of Theorem ., system (.) has a solutionum on an interval

[,Tm]⊂[,T]. The following estimates allow one to takeTm=T∗for allm, consisting of

two key estimates.

In the first key estimate, we putSm(t) =um(t)+∇um(t), it implies from (.) that

Sm(t) =Sm() + 

∂

h(x, ),u

+g(x, )u(x)dSx

– 

t



Kum(s) +λum(s),um(s)

ds

+ 

t



f(s),u_m(s)ds+ 

t



um(s)p–um(s),um(s)

ds

– 

∂

g(x,t)um(x,t)dSx– 

∂

h(x,t),um(t)

um(x,t)dSx

+ 

t

 ds

∂

h(x,s),um(s)

+h(x,s),u_m(s)+g(x,s)um(x,s)dSx

≡Sm() +



j=

Ij. (.)

By Lemmas .-. and the following inequalities:

ab≤βa₊

βb

 _{for alla,b∈R,β> ,}

(a+b+c)q_≤q–_(aq_+bq_+cq_{) for allq≥,a,b,c≥} (.)

and

v ≤ vH, vLq≤CqvH, ∀v∈H, ≤q≤∗= N

N– ,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(x)dSx

=u+∇u+ 

∂

h(x, ),u

+g(x, )u(x)dSx≡



C; (.)

I= –

t



Kum(s) +λum(s),um(s)

ds≤

t



um(s)



ds+K+ |λ|

t



u_m(s) ds

≤ t



u+

s



u_m(r) dr



ds+K+ |λ|

t



≤Tu+T

t



u_m(r) dr+K+ |λ|

t



Sm(s)ds

≤Tu+

T+K+ |λ|

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



um(s) p–

um(s),um(s)

ds≤

t



um(s) p–

u_m(s) ds

≤ t



um(s)p– ds+ t



Sm(s)ds

=

t



_um(s) p–

Lp–ds+

t



Sm(s)ds≤Cpp––

t



_um(s) p–

H ds+

t



Sm(s)ds, (.)

since ≤≤p– ≤∗, andH_()→Lp–_{(), we have}

_um(t) p–

H ≤

u+Sm(t) + t t



Sm(s)ds p–

≤p–p–up–+ p–

Sm(t) p–

+ p–p–tp–

t



Sm(s) p–

ds,

it leads to

I= 

t



um(s) p–

um(s),um(s)

ds≤CT+CT t



Sm(s) p–

ds+

t



Sm(s)ds; (.)

I= –

∂

g(x,t)um(x,t)dSx≤γgL∞(,T;L_(∂)) u_m(t)

H ≤  βγ  g 

L∞(,T;L_(∂))+β um(t)



H

≤ 

βγ



gL∞(,T;L_(∂))+β

u+Sm(t) + t t



Sm(s)ds

≤ 

βCT+βSm(t) +CT

t



Sm(s)ds for all  <β< ; (.)

I= –

∂

h(x,t),um(t)

um(x,t)dSx≤γhL∞(,T;L_(∂×)) u_m(t) u_m(t)

H ≤  βγ  h 

L∞(,T;L_(∂×)) um(t)



+β um(t)



H

≤ 

βγ



hL∞(,T;L_(∂×))

u+ t

t



Sm(s)ds

+β

u+Sm(t) + t t



Sm(s)ds

≤ 

βCT+βSm(t) +



βCT

t



Sm(s)ds for allβ> ,β< ; (.)

I= 

t  ds ∂

h(x,s),um(s)

+h(x,s),u_m(s)+g(x,s)um(x,s)dSx

≤γ h _L∞_(,T;L_(∂×))

t



+ γhL∞(,T;L_(∂×))

t



u_m(s) um(s) _Hds

+ γ g _L∞_(,T;L_(∂))

t



_um(s) _Hds

≤CT+CT t



um(s) _Hds+

t



u_m(s) ds≤CT

 +

t



Sm(s)ds

. (.)

Combining estimations of all terms and choosingβ=_, we obtain after some rearrange-ments

Sm(t)≤CT

 +

t



Sm(s)ds+ t



Sm(s) p–

ds

, ≤t≤Tm, (.)

whereCTalways indicates a constant depending onT.

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 CTis a constant depending only on T.

By Lemma ., we can take a constantTm=T∗for allm.

In the second key estimate, we putXm(t) =um(t)+∇um(t),and it follows from

(.) that

Xm(t) =Xm() + 

∂

h(x, ),u

+h(x, ),u

+g(x, )u(x)dSx

– 

t



Ku_m(s) +λu_m(s),u_m(s)ds+ 

t



f(s),u_m(s)ds

+ (p– )

t



um(s)p–um(s),um(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() +



i=

Ji. (.)

Lettingt→+in equation (.), multiplying the result bycmj(), and using the

com-patibility (.), we get

u_m() =u,um()

–Ku+λu,um()

+|u|p–u,um()

This implies that

u_m() ≤ u+|K|u+|λ|u+ |u|p– + f() =X for allm, (.)

whereXis a constant depending only onp,K,λ,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) +λu_m(s),u_m(s)ds

≤ t



u_m(s) ds+K+ |λ|

t



u_m(s) ds

≤CT+

K+ |λ|

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) Xm(s)ds. (.)

From

_um(s)p–

u_m(s) ≤Dp

 + um(s)

/N

H + um(s) p–

H u_m(s) _H≤DpCT um(s) H,

by Lemma .(ii), it gives

J= (p– )

t



um(s)p–um(s),um(s)

ds

≤(p– )

t



um(s) p–

u_m(s) u_m(s) ds

≤(p– )DpCT t



u_m(s) _H um(s) ds

≤(p– )D_pC_T

t



u_m(s) _Hds+

t



u_m(s) ds

= (p– )D_pC_T

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

+ g _L∞_(,T;L_(∂)) um(t) H

≤CT um(t) H≤ 

βCT+β u

m(t)



H

≤ 

βCT+β

u+Xm(t) + t t



Xm(s)ds

≤ 

βCT+βXm(t) +CT

 +

t



Xm(s)ds

for allβ∈(, ); (.)

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

≤γCT t



h(s) _L_(∂×) um(s) Hds+ γCT t



u_m(s) _Hds

+ γCT t



u_m(s) u_m (s) _Hds+ γ t



g(s) _L_(∂) um(s) Hds

≤γC_T h _L_(,T;L_(∂×))+

t



h(s) _L_(∂×) um(s)



Hds

+ γC_TT+

t



u_m(s) _Hds+γCT t



u_m(s) ds+

t



u_m(s) _Hds

+γ 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) =  + h(s) _L_(∂×)+ g(s) _L_(∂), ∈L(,T). (.)

Combining estimations and choosingβ=_, we obtain after some rearrangements

Xm(t)≤CT+ t



(s)Xm(s)ds, (.)

whereCTalways indicates a constant depending onT, and

(s) =CT

 + f(s) + h(s) _L_(∂×)+ g(s) _L_(∂)

, ∈L(,T). (.)

By Gronwall’s lemma, we deduce from (.) that

Xm(t)≤CTexp

T



(s)ds

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

⎧ ⎪ ⎨ ⎪ ⎩

um→u inL∞(,T∗;H) weakly∗,

u_m→u inL∞(,T∗;H) weakly∗,

u_m→u inL∞(,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 inL(QT∗) and a.e. inQT∗,

u_m→u strongly inL_(Q

T∗) and a.e. inQT∗.

(.)

By means of the continuity of the functiont−→ |t|p–t, we have

|um|p–um→ |u|p–u and a.e. inQT∗. (.)

On the other hand

_|um|p–um



L_(QT

∗) =

_T∗

 ds

um(x,t)

p– dx

=

T∗



um(t)

p–

Lp–dt

≤ _T∗



Cp– um(t) _H

p– dt

≤C__pp_––T∗um_Lp∞–_(,T∗;H₎≤CT. (.)

Using the Lions lemma ([], Lemma ., p.), it follows from (.) and (.) that

|um|p–um→ |u|p–u inL(QT∗) weakly. (.)

Passing to the limit in (.) by (.), (.), and (.), we haveusatisfying the problem

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u(t),v+∇u(t),∇v+Ku(t) +λu(t),v +_∂(h(x,t),u(t)+g(x,t))v(x)dSx

=|u(t)|p–_{u(t),v+f(t),v for allv∈H}_, u() =u, u() =u.

(.)

On the other hand, we have from (.), (.)

u=u+Ku+λu–|u|p–u–f ∈L∞,T∗;L. (.)

Thusu∈L∞(,T∗;H) and the proof of existence is complete. The uniqueness of a weak solution is proved as follows.

Letu,ube two weak solutions of problem (.)-(.), such that

ui∈L∞

,T∗;H, ui∈L∞

,T∗;H, ui ∈L∞

Thenu=u–usatisfy the variational problem

⎧ ⎪ ⎨ ⎪ ⎩

u(t),v+∇u(t),∇v+Ku(t) +λu(t),v+_∂h(x,t),u(t)v(x)dSx

=|u|p–u–|u|p–u,v for allv∈H, u() =u() = .

(.)

We takev=u=u_–u_in (.) and integrating with respect tot, we obtain

σ(t) = –

t



Ku(s) +λu(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|p–u–|u|p–u,u(s)

ds= 

j=

σj, (.)

where

σ(t) = u(t) + ∇u(t) . (.)

By (.) and the following inequalities:

ab≤βa+ 

βb

 _{for alla,b∈R,β> , (.)}

u(t) =

t



u(s) ds



≤t

t



u(s) ds≤t

t



σ(s)ds,

u(t) _H= ∇u(t) 

+ u(t) ≤σ(t) +t

t



σ(s)ds, (.)

t



u(s) _Hds≤

t



σ(s) +s

s



σ(r)dr

ds≤ +t

t



σ(s)ds,

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

σ= –

t



Ku(s) +λu(s),u(s)ds

≤ t



u(s) ds+K+ |λ|

t



u(s) ds

≤ t



s

s



σ(r)dr

ds+K+ |λ|

t



σ(s)ds

≤T

t



σ(r)dr+K+ |λ|

t



σ(s)ds≤CT t



σ(s)ds; (.)

σ= –

∂

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

≤γhL∞(,T;L_(∂×)) u(t) u(t)

H

≤ 

βγ



hL∞(,T;L_(∂×)) u(t) 

≤ 

βγ



hL∞(,T;L_(∂×))t

t



σ(s)ds+β

σ(t) +t

t



σ(s)ds

≤βσ(t) + 

βCT

t



σ(s)ds; (.)

σ= 

t

 ds

∂

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

≤γ h _L∞_(,T;L_(∂×))

t



u(s) _Hds

+ γhL∞(,T;L_(∂×))

t



u(s) u(s) _Hds

≤CT t



u(s) _Hds+CT t



u(s) ds≤CT t



σ(s)ds. (.)

By Lemma .(i), we have

_|u|p–u–|u|p–u

≤Dp

 +uH+u_H/N+u_H+u_Hp– u(s)

H

≤Dp

 +M/N

 +M

p–

 u(s) H≤CT u(s) _H, (.)

whereM=uL∞(,T∗;H₎+u_L∞_(,T∗;H₎. Hence

σ= 

t



|u|p–u–|u|p–u,u(s)

ds

≤CT t



u(s) _H u(s) ds

≤CT t



u(s) _Hds+CT t



u(s) ds

≤CT t



σ(s)ds. (.)

Combining (.), (.)-(.), (.) and choosingβ=

, we obtain

σ(t)≤CT t



σ(s)ds. (.)

By Gronwall’s lemma, it follows from (.) thatσ ≡,i.e.,u≡u. Theorem . is

proved completely.

Proof of Theorem. In order to prove this theorem, we use standard arguments of den-sity.

First, we note thatW_(,T;L_{(∂)) ={g∈L}_(,T;L_{(∂)) :g∈L}_(,T;L_{(∂))}is a} Hilbert space with respect to the scalar product (see []):

f,gW_(,T;L_(∂))=

T



f(t),g(t)_L_(∂)+

f(t),g(t)_L_(∂)

Furthermore, we also have the embedding W_(,T;L_(∂))→C_([,T];L_{(∂)) is} continuous and

gC_([,T];L_(∂))≤γ_T

g

L_(,T;L_(∂))+ g 

L_(,T;L_(∂))

≡γTgW_(,T;L_(∂)) (.)

for allg∈W_(,T;L_{(∂)), whereγ}

T=

 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,kW_(,T;L_(∂×))=

T



h(t),k(t)_L_(∂×)+

h(t),k(t)_L_(∂×)

dt, (.)

and the embeddingW_(,T;L_(∂×))→C_([,T];L_{(∂×)) is continuous and}

hC_([,T];L_(∂×))≤γ_T

h_L_(,T;L_(∂×))+ h 

L_(,T;L_(∂×))

≡γThW_(,T;L_(∂×)) (.)

for allh∈W_(,T;L_{(∂×)), whereγ}

T=

 T+

 +_T (see the Appendix).

Consider (u,u,f,g,h)∈H×L×L(QT)×W(,T;L(∂))×W(,T;L(∂×)).

Let the sequence{(um,um,fm,gm,hm)} ⊂H×H×C∞(QT)×C∞(∂×)×C∞(∂×

×[,T]), such that

⎧ ⎪ ⎨ ⎪ ⎩

um→u strongly inH, um→u strongly inL, fm→f strongly inL(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{(um,um)}satisfy, for allm∈N, the compatibility condition

–∂um

∂ν (x) =gm(x, ) +

hm(x,y, )um(y)dy. (.)

Then, for eachm∈N, there exists a unique functionumunder the conditions of

Theo-rem .. Thus, we can verify

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u_m(t),v+∇um(t),∇v+Kum(t) +λum(t),v

+_∂(hm(x,t),um(t)+gm(x,t))v(x)dSx

=|um|p–um,v+fm(t),v for allv∈H,

um() =um, um() =um

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)



+ um(t)



H≤CT, (.)

∀t∈[,T∗], whereCTalways indicates a constant depending onTas above.

On the other hand, we putwm,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) +λw_m,l(t),v

+_∂(hm(x,t),wm,l(t)+hm,l(x,t),ul(t)+gm,l(x,t))v(x)dSx

=|um|p–um–|ul|p–ul,v+fm,l(t),v for allv∈H,

wm,l() =um–ul≡ ¯w()m,l, wm,l() =um–ul≡ ¯w()m,l.

(.)

We takev=wm,l=um–ul, in (.) and integrating with respect tot, we get

Sm,l(t) =Sm,l() + 

∂

hm(x, ),w¯()m,l

+hm,l(x, ),ul

+gm,l(x, )

¯

w()_m,l(x)dSx

+ 

t



fm,l(s),wm,l(s)

ds– 

t



Kwm,l(s) +λwm,l(s),wm,l(s)

ds

– 

∂

hm(x,t),wm,l(t)

+hm,l(x,t),ul(t)

+gm,l(x,t)

wm,l(x,t)dSx

+ 

t

 ds

∂

h_m(x,s),wm,l(s)

+hm(x,s),wm,l(s)

wm,l(x,s)dSx

+ 

t

 ds

∂

h_m,l(x,s),ul(s)

+hm,l(x,s),ul(s)

wm,l(x,s)dSx

+ 

t

 ds

∂

g_m,l(x,s)wm,l(x,s)dSx

+ 

t



|um|p–um–|ul|p–ul,wm,l(s)

ds≡Sm,l() +



j=

Zj, (.)

where

Sm,l(t) = wm,l(t)



+ ∇wm,l(t) , (.)

Sm,l() =um–ul+∇um–∇ul. (.)

After all terms ofSm,l(t) are estimated, in which we note the two main estimationsZ, Zas follows:

Z = 

∂

hm(x, ),w¯()m,l

+hm,l(x, ),ul

+gm,l(x, )

¯

w()_m,l(x)dSx

≤ γ w¯()_m,l hm() _L_(∂×)+ul h¯()m,l L_(∂×)+ g¯ ()

m,l L_(∂) w¯ ()

≤ γγT·const. w¯()m,l H+hm,lW_(,T;L_(∂×))+g_m,lW_(,T;L_(∂)) w¯()_m,l

H

→, asm,l→+∞; (.)

this result combined with (.)-(.) shows that

Sm,l() +Z =um–ul+∇um–∇ul+Z

≡R(m,l)→, asm,l→+∞. (.)

On the other hand

_|um|p–um–|ul|p–ul ≤Dp

 +umH+ulH

/N

+umH+ulH

p–

wm,l(s) _H

≤CT wm,l(s) _H, (.)

by Lemma .(i), we get

Z= 

t



|um|p–um–|ul|p–ul,wm,l(s)

ds≤CT t



wm,l(s) _H wm,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_(Q

T)

+CT w¯()m,l



+hm,l_W_(,T;L_(∂×))+gm,l_W_(,T;L_(∂))

→, (.)

asm,l→+∞. By Gronwall’s 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{(um,um,fm,gm,hm)}implies the convergence to

zero asm,l→+∞of the term on the right-hand side of (.). Therefore, we get

um→u strongly inC

[,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 inL∞(,T∗;H) 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 inL(QT∗) and a.e. inQT∗. (.)

Similarly, by (.), it follows from (.) that

|um|p–um→ |u|p–u inL(QT∗) weakly. (.)

Passing to the limit in (.) by (.)-(.), we haveusatisfying the problem

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ d

dtu(t),v+∇u(t),∇v+Ku(t) +λu(t),v

+_∂(h(x,t),u(t)+g(x,t))v(x)dSx

=|u|p–_{u,v+f(t),v for allv∈H}_, u() =u, u() =u.

(.)

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

Remark . In the case  <p≤,f∈L_(Q

T),g∈W(,T;L(∂)),h∈W(,T;L(∂×

)), and (u,u)∈H×L, the integral inequality (.) leads to the following global es-timation:

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 solutionuof 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 be-longs toC([,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 (.)-(.) cor-responding toa= ,g= ,K> ,λ> , and  <p≤_NN_––. For this purpose, we make the following assumptions:

(A_) f ∈L_(,∞;L_{) =L}_(Q

∞),Q∞=×R+, such thatf(t) ≤Ce–γt, for allt≥, with C> ,γ> are given constants,

(A_) h ∈ L∞(,∞;L(∂ × ))∩L(R+ × ∂× ), h ∈ L∞(,∞;L(∂ × ))∩ L_(,∞;L_(∂×)),

(A_) g= .

LetK> , onH_{we shall use the following norm:}

v=

Then we have the following lemma.

Lemma . On H_{,two normsv}

,vHare equivalent and



√

max{,K}v≤ vH≤ 

√

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) +δψ(t), (.)

whereδ>  is chosen later and

E(t) =   u

_(t)  +

 u(t)  –

 p u(t)

p Lp+

∂

h(x,t),u(t)u(x,t)dSx, (.)

ψ(t) =u(t),u(t)+λ  u(t)



. (.)

Put

I(t) =Iu(t)= u(t) _– u(t) p_Lp+p

∂

h(x,t),u(t)u(x,t)dSx, (.)

J(t) =Ju(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

η_∗= (CpC)p

p p– E∗

p– 

+pCγ¯hL∞(,∞;L_(∂×))< , (.)

where

E∗= (E() +λ ∞

 f(t)dt)exp( p p–

∞

 h(t)¯ dt),

¯

h(t) =γ¯(Ch(t)L_(∂×)+_λγ¯h(t)_L_(∂×)),

(.)

¯