Existence Of Solutions To Nonlinear th-Order Coupled Klein-Gordon Equations With Nonlinear

Existence Of Solutions To Nonlinear th-Order Coupled Klein-Gordon Equations With Nonlinear

Sources And Memory Terms

Khaled Zennir

, Amar Guesmia

Received 5 December 2014

Abstract

In this article, we consider a system of th-order derivatives of the dependent variables of coupled Klein-Gordon equations to improve recent results obtained in [10, 31, 33] using idea in [25]. Using the potential well method, we prove that the solutions of (1) exist globally, under some conditions on the initial datum.

1 Introduction

In this paper, we consider a system of th-order derivatives of the dependent variables of coupled Klein-Gordon equations

8>

>>

>>

><

>>

>>

>>

:

u⁰⁰₁+ ( 1) u₁+m²₁u₁+ ₁(t)Rt

0g₁(t s) u₁(x; s)ds+ju⁰₁j^{r 2}u⁰₁

=ju1j^{p 2}u1ju2j^p;

u⁰⁰₂+ ( 1) u2+m²₂u2+ 2(t)Rt

0g2(t s) u2(x; s)ds+ju⁰₂j^{r 2}u⁰₂

=ju2j^{p 2}u2ju1j^p

(1)

where m_i; i = 1;2 are non-negative constants, r; p 2; 1. In a bounded domain Rⁿ Yaojun Ye [33] introduced related problem to (1) with = 1; _i = 0; i = 1;2, supplemented with the initial and Dirichlet boundary conditions. By using the potential well method, global existence is discussed and asymptotic stability is also given, by using multiplied method.

Erhan Piskin and Necat Polat [10] considered a system of class of nonlinear higher- order wave equations (1) withmi=gi= 0; i= 1;2and strong nonlinearity in sources.

Under suitable conditions on the initial datum, theorems of global existence and decay rate are proved.

In (1),ui =ui(t; x); i= 1;2;wherex2 is a bounded domain ofRⁿ;(n 1)with a smooth boundary @ , t >0. Our system is supplemented with the following initial conditions

ui(x;0) =ui0(x)2H₀( ); i= 1;2; (2)

Mathematics Sub ject Classi…cations: 35L05, 35B40, 35G31, 58J45.

yLaboratory LAMAHIS,University 20 Août 1955- Skikda 21000, Algeria

zLaboratory LAMAHIS,University 20 Août 1955- Skikda 21000, Algeria

121

u⁰_i(x;0)) =u_i1(x)2L²( ); i= 1;2; (3) and boundary conditions

ui(x) = @ui

@ = = @ ¹ui

@ ¹ = 0forx2@ andi= 1;2; (4) where is the outward normal to the boundary.

We mention here that

jr uj²= ( ⁼²u)²for pair value of and

jr uj²=jr( ^{( 1)=2}u)j²for odd where

jruj²= Xn i=1

@u

@xi 2

and u= Xn i=1

@²u

@x²_i:

This kind of systems (gi6= 0; i= 1;2) appears in the models of nonlinear viscoelasticity.

Viscoelastic materials have properties between two types, elastic materials and viscous

‡uids. This two types of materials are usually considered in basic texts on continuum mechanics. At each material point of an elastic material the stress at the present time depends only on the present value of the strain. On the other hand, for an incompressible viscous ‡uid the stress at a given point is a function of the present value of the velocity gradient at that point. Such materials have memory: the stress depends not only on the present values of the strain and/or velocity gradient, but also on the entire temporal history of motion.

The systems of nonlinear wave equations go back to Reed [27] who proposed a system in three space dimensions, where this type of system was completely analyzed.

Existence and uniqueness of global weak solutions, asymptotic behavior for an anal- ogous hyperbolic-parabolic system of related problems have attracted a great deal of attention in the last decades, and many results have appeared. See in this directions [5, 6, 7, 8, 12, 17, 22, 21, 24] and references therein.

We mention the work of [2], where authors studied the following system:

( u_tt u+ju_tj^{m 1}u_t=f₁(u; v);

vtt v+jvtj^{r 1}vt=f2(u; v);

(5) in (0; T) with initial and boundary conditions and the nonlinear functions f1

and f₂ satisfying appropriate conditions. They proved under some restrictions on the parameters and the initial data many results on the existence of a weak solution. They also showed that any weak solution with negative initial energy blows up in …nite time using the same techniques as in [11].

In [20], authors considered the nonlinear viscoelastic system 8>

><

>>

:

utt u+ Rt 0

g(t s) u(x; s)ds+jutj^{m 1}ut=f1(u; v);

v_tt v+ Rt 0

h(t s) v(x; s)ds+jv_tj^{r 1}v_t=f₂(u; v);

(6)

forx2 andt >0 where

f1(u; v) =aju+vj^{2( +1)}(u+v) +bjuj ujvj^{( +2)}; f2(u; v) =aju+vj^{2( +1)}(u+v) +bjuj^{( +2)}jvj v;

and they prove a global nonexistence theorem for certain solutions with positive initial energy, the main tool of the proof is a method used in [28].

The non-critical case of (1) where g_i = 0; m = 2; i = 1;2, has been studied re- cently in [26]. M. A. Rammaha and Sawanya Sakuntasathien focus on the global well-posedness of the system of nonlinear wave equations

8<

:

utt u+ djuj^k+ejvj^l jutj^{m 1}ut=f1(u; v);

vtt v+ d⁰jvj +e⁰juj jvtj^{r 1}vt=f2(u; v);

(7)

in a bounded domain Rⁿ, n= 1;2;3;and 0< r; m <1, with Dirichlet boundary conditions. The nonlinearitiesf1(u; v)andf2(u; v)act as strong sources in the system.

Under some restriction on the parameters in the system, they obtained several results on the existence and uniqueness of solutions. In addition, they proved that weak solutions blow up in …nite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms. This last result was extended by A. Benaissa et al. in [3] with positive initial energy, r; m >1 and forn >0.

The gender of our systems (m_i6= 0; i= 1;2) has been proposed …rst by Segal [30]

in the next coupled Klein-Gordon equations which is considered in the study of the quantum …eld theory and de…nes the motion of a charged meson in an electromagnetic

…eld (

u⁰⁰₁ u1+m²₁u1+h1u1u²₂= 0;

u⁰⁰₂ u₂+m²₂u₂+h₂u²₁u₂= 0;

(8) where m1; m2; h1 andh2 are non-negative constants.

Wheng_i= 0; i= 1;2;Yaojun Ye generalized the problem (8), where author studied coupled nonlinear Klein-Gordon equations with nonlinear damping and source terms, in a bounded domain with the initial and Dirichlet boundary conditions

( u⁰⁰₁ u₁+m²₁u₁+aju⁰₁j u⁰₁=bju₁j u₁ju₂j ⁺²; u⁰⁰₂ u₂+m²₂u₂+aju⁰₂j u⁰₂=bju₂j u₂ju₁j ⁺²;

(9)

where m1; m2; a; b are non-negative constants, > 0 and 0. The existence of global solutions is discussed by using the potential well method and the asymptotic stability is also given by applying a Lemma due to V. Komornik [14].

REMARK 1.1. Noting here that our contribution is: We investigate the same system in [33] with the presence of the viscoelastic terms and potential functions, under additional condition (18), we prove that the solutions stay in the stable set (13).

2 Preliminaries

From now on, we denote by c_i, i = 0;1;2; :::, used throughout this paper, various positive constants which may be di¤erent at di¤erent occurrences and in the sequel, for the sake of simplicity we will denote the tderivative valuedv=dtbyv⁰ andd²v=dt² byv⁰⁰.

We assume that, for i = 1;2; the relaxation functions g_i : R⁺ ! R⁺ and the potential _i:R⁺ !R⁺ are nonincreasing di¤erentiable satisfying:

gi(0)>0, +1>

+1

Z

0

gi(s)ds,1 i(t) Zt 0

gi(s)ds li>0, and i(t)>0: (10)

The following notation will be used throughout this paper ( )(t) =

Z t 0

(t )k (t) ( )k²2d : (11) The following technical Lemma will play an important role.

LEMMA 2.1. For anyv2C¹(0; T; H ( ))we have Z

(t) Z t

0

g(t s) v(s)v⁰(t)dsdx

= 1 2

d

dt (t) (g r v) (t) 1 2

d dt (t)

Z t 0

g(s) Z

jr v(t)j²dxds 1

2 (t) (g⁰ r v) (t) +1

2 (t)g(t) Z

jr v(t)j²dxds 1

2

0(t) (g r v) (t) +1 2

0(t) Z t

0

g(s)ds Z

jr v(t)j²dxds:

PROOF. It’s not hard to see Z

(t) Z t

0

g(t s) v(s)v⁰(t)dsdx

= (t)

Z t 0

g(t s) Z

r v⁰(t)r v(s)dxds

= (t)

Z t 0

g(t s) Z

r v⁰(t) [r v(s) r v(t)]dxds (t)

Z t 0

g(t s) Z

r v⁰(t)r v(t)dxds:

Consequently,

Z (t)

Z t 0

g(t s) v(s)v⁰(t)dsdx

= 1 2 (t)

Z t 0

g(t s)d dt

Z

jr v(s) r v(t)j²dxds (t)

Z t 0

g(s) d dt

1 2

Z

jr v(t)j²dx ds;

which implies, Z

(t) Z t

0

g(t s) v(s)v⁰(t)dsdx

= 1

2 d dt (t)

Z t 0

g(t s) Z

jr v(s) r v(t)j²dxds 1

2 d dt (t)

Z t 0

g(s) Z

jr v(t)j²dxds 1

2 (t) Z t

0

g⁰(t s) Z

jr v(s) r v(t)j²dxds +1

2 (t)g(t) Z

jr v(t)j²dxds 1 2

0(t) Z t

0

g(t s) Z

jr v(s) r v(t)j²dxds

+1 2

0(t) Z s

0

g(s)ds Z

jr v(t)j²dxds:

This completes the proof.

The energy functionalE(t)associated with our system is given by

E(t) =1 2

X2 i=1

ku⁰_ik²2+J(t) (12)

where

J(t) = 1 2

X2 i=1

1 i(t) Z t

0

gi(s)ds kr uik²2

+1 2

X2 i=1

i(t)(gi r ui) +1 2

X2 i=1

m²_ikuik²2

1

pku1u2k^pp: Now, we introduce the stable set as follows:

W = (u1; u2)2(H₀( ))²:I(t)>0and J(t)< d [ f(0;0)g (13)

where

I(t) = X2 i=1

1 _i(t) Z t

0

g_i(s)ds kr u_ik²2

+ X2 i=1

i(t)(g_i r u_i) + X2 i=1

m²_iku_ik²2 ku₁u₂k^pp:

REMARK 2.2. We notice that the mountain pass leveldgiven in (13) de…ned by d= inf

8<

: sup

(u1;u2)2(^H0( ))²nf(0;0)g 0

J( (u₁; u₂)) 9=

;: Also, by introducing the so called "Nehari manifold"

N = n

(u1; u2)2(H₀( ))²n f(0;0)g:I(t) = 0 o

: It is readily seen that the potential depthdis also characterized by

d= inf

(u1;u2)2NJ(t):

This characterization of dshows that dist((0;0);N) = min

(u1;u2)2Nk(u₁; u₂)k(^H0( ))²:

The notation k:k stands for the norm in L² and we denote by k:kX the norm in the space X. Also, the following imbedding will be used frequently without mention kukp Ckr uk2 foru2H₀( )where

( 2 p <+1 if n= ;2 ;

2 p _n²ⁿ₂ if; n 3 : (14)

We introduce the following de…nition of weak solution to (1)-(4)

DEFINITION 2.3. A pair of functions (u₁; u₂) is said to be a weak solution of (1)-(4) on [0; T] ifu₁; u₂ 2 C_w([0; T]; H₀( )), u⁰₁; u⁰₂ 2C_w([0; T]; L²( )), (u₁₀; u₂₀)2 H₀( ) H₀( ),(u₁₁; u₂₁)2L²( ) L²( )and(u₁; u₂)satis…es

Z t 0

Z

ju1j^{p 2}u1ju2j^p dxds = Z t

0

Z

u⁰⁰₁ dxds+m²₁ Z t

0

Z

u1 dxds +

Z t 0

Z

r u₁r dxds+ Z t

0

Z

ju⁰₁j^{r 2}u⁰₁ dxds Z t

0

Z

1(t) Z s

0

g1(t )r u1(x; )r d dxds

and Z t

0

Z

ju2j^{p 2}u2ju1j^p dxds = Z t

0

Z

u⁰⁰₂ dxds+m²₂ Z t

0

Z

u2 dxds +

Z t 0

Z

r u2r dxds+ Z t

0

Z

ju⁰₂j^{r 2}u⁰₂ dxds Z t

0

Z

2(t) Z s

0

g₂(t )r u₂(x; )r d dxds for all test functions ; 2H₀( )\L²( )and almost allt2[0; T]whereCw([0; T]; X) denotes the space of weakly continuous functions from[0; T]into Banach spaceX.

In order to state the local existence result, we introduce the following complete metric space (the proof is similar to that in [29, 32, 31])

YT = f(u; v) :u; v2C([0; T];H₀( ) H₀( )); u⁰; v⁰2C [0; T];L²( ) L²( )

THEOREM 2.4. Let(u₁₀; u₂₀)2 (H₀( ))² and (u₁₁; u₂₁)2(L²( ))² fori= 1;2 be given. Suppose thatr >2andpsatis…es

( 1 p <+1 if n= ;2 ;

1 p ⁴_{n 2}ⁿ if n 3 : (15)

Then, under assumptions on two functionsg_i,i= 1;2, the problem (1)-(4) has a unique local solution(u₁(t; x); u₂(t; x))2Y_T forT small enough.

3 Global Existence Result

LEMMA 3.1. Suppose that (10) and (15) hold. Let (u1; u2)be the solution of the system (1)-(4). Then the energy functional is a non-increasing function, that is for all t 0,

E⁰(t) = 1 2

X2 i=1

i(t)(g_i⁰ r ui) 1 2

X2 i=1

i(t)gi(t)kr uik²2

+1 2

X2 i=1

0(t)(g_i r u_i) 1 2

X2 i=1

0(t) Z t

0

g_i(s)ds kr u_ik²2

1 2

X2 i=1

0(t)(g_i r u_i) 1 2

X2 i=1

0(t) Z t

0

g_i(s)ds kr u_ik²2: (16)

We will prove the invariance of the setW:That is for somet₀>0if(u₁(t₀); u₂(t₀))2 W; then(u₁(t); u₂(t))2W fort t₀ andi= 1;2. We begin with by the existence of the potential depth in the next Lemma.

LEMMA 3.2. dis a positive constant.

PROOF. We have

J( (u₁; u₂)) =

2

2 X2 i=1

1 _i(t) Z t

0

g_i(s)ds kr u_ik²2+

2

2 X2 i=1

i(t)(g_i r u_i)

+

2

2 X2

i=1

m²_iku_ik²2 2p

p ku₁u₂k^pp:

Using (10) to get

J( (u₁; u₂)) K( );

where

K( ) =

2

2 X2 i=1

likr uik²2+

2

2 X2 i=1

m²_ikuik²2 2p

p ku1u2k^pp: By di¤erentiating the second term in the last equality with respect to ;to get

d

d K( ) = X2 i=1

l_ikr u_ik²2+ X2 i=1

m²_iku_ik²2 2 ^{2p 1}ku₁u₂k^pp:

For ₁= 0and

2= 2 ^2(p11) P2

i=1likr uik²2+P2

i=1m²_ikuik²2

ku₁u₂k^pp

!_2(p¹₁₎

;

then we have

d

d K( ) = 0:

As

d

d K( ₂) = 0; K( ₁) = 0;

and since

d² d ²K( )

= 2 <0;

we see that

sup

0

j( ) sup

0

K( ) =K( ₂)

= 2^2(p2p1) P2

i=1likr uik²2+P2

i=1m²_ikuik²2

ku₁u₂k^pp

!_2(p²₁₎

X2 i=1

l_ikr u_ik²2+ X2 i=1

m²_iku_ik²2

!

1 p2^2(p2p1)

P2

i=1l_ikr u_ik²2+P2

i=1m²_iku_ik²2

ku1u2k^pp

!_2(p^2p₁₎

ku1u2k^pp

= 2^2(p2p1) p 1 p

P2

i=1likr uik²2+P2

i=1m²_ikuik²2

ku1u2k^p

!_2(p^2p₁₎ :

It follows from the Holder inequality for someC >0and assumptions (10)

ku1u2k^p ku1k^2pku2k^2p C²kr u1k²kr u2k² 1

2C² X2 i=1

kr uik²2

! 1 2C²

X2 i=1

likr uik²2+ X2 i=1

m²_ikuik²2

!

;

which implies that

ku₁u₂kp

P2

i=1likr uik²2+P2

i=1m²_ikuik²2

1 2C²:

Sincep >1, we obatin that

sup

0

j( ) 2^2(p2p1) p 1 p

" P2

i=1l_ikr u_ik²2+P2

i=1m²_iku_ik²2

ku1u2k^p

#_2(p^2p₁₎

(p 1)

p C^(p2p1) =d >0:

Then, by the de…nition ofd;we conclude thatd >0forp >1:

LEMMA 3.3. W is a bounded neighborhood of0 inH₀( ).

PROOF. Foru2W;andu6= 0;we have J(t) = 1

2 X2 i=1

1 _i(t) Z t

0

g_i(s)ds kr u_ik²2+1 2

X2 i=1

i(t)(g_i r u_i)

+1 2

X2 i=1

m²_iku_ik²2

1

pku₁u₂k^pp

= p 2

2p

" ₂ X

i=1

1 _i(t) Z t

0

g_i(s)ds kr u_ik²2

+ X2 i=1

i(t)(g_i r u_i) + X2

i=1

m²_iku_ik²2

# +1

pI(t) p 2

2p

" ₂ X

i=1

1 _i(t) Z t

0

g_i(s)ds kr u_ik²2

+ X2 i=1

i(t)(g_i r u_i) + X2

i=1

m²_iku_ik²2

#

: (17)

By using (10), (17) becomes

J(t) p 2

2p X2 i=1

1 i(t) Z t

0

gi(s)ds kr uik²2

p 2 2p

X2 i=1

likr uik²2

p 2

2p min(l1; l2) X2 i=1

kr uik²2:

It follows that X2 i=1

kr u_ik²2

1 min(l₁; l₂)

2p

p 2 J(t)< 1 min(l₁; l₂)

2p

p 2 d=R:

Consequently,8(u1; u2)2W;we have (u1; u2)2B;where

B= (

(u1; u2)2(H₀( ))²: X2 i=1

kr uik²2< R )

:

This completes the proof.

In the following Lemma, we will see that if the initial data (or for somet₀>0) is in the setW, then the solution stays there forever.

LEMMA 3.4. Suppose that (10), (15) and C²

(2 min(l₁; l₂))

p 2pE(0) p 2

(p 1)

<1: (18)

hold, where C is the best Poincare’s constant. If (u10; u20) 2 W and (u11; u21) 2 L²( ) ², then the solution(u1(t); u2(t))2W fort 0.

PROOF. Since(u₁₀; u₂₀)2W, we see that I(t) =

X2 i=1

kr u_i0k²2+ X2 i=1

m²_iku_i0k²2 ku₁₀u₂₀k^pp >0:

Consequently, by continuity, there exists T_m T such that I(u(t)) =

X2 i=1

1 _i(t) Z t

0

g_i(s)ds kr u_ik²2+ X2 i=1

i(t)(g_i r u_i)

+ X2 i=1

m²_iku_ik²2 ku₁u₂k^pp 0 fort2[0; T_m]: This gives

J(t) = 1 2

X2 i=1

1 _i(t) Z t

0

g_i(s)ds kr u_ik²2+1 2

X2 i=1

i(t)(g_i r u_i) (19) +1

2 X2 i=1

m²_iku_ik²2

1

pku₁u₂k^pp

= p 2

2p

" ₂ X

i=1

1 _i(t) Z t

0

g_i(s)ds kr u_ik²2+ X2 i=1

i(t)(g_i r u_i)

+ X2 i=1

m²_iku_ik²2

# +1

pI(u(t)) p 2

2p

" ₂ X

i=1

1 _i(t) Z t

0

g_i(s)ds kr u_ik²2+ X2 i=1

i(t)(g_i r u_i)

+ X2 i=1

m²_iku_ik²2

# :

By using (10) and the fact that Rt

0gi(s)ds R₁

0 gi(s)ds; we easily see that, for t 2 [0; Tm],

X2 i=1

kr uik²2

1 min(l1; l2)

2p

p 2 J(t) 1

min(l1; l2) 2p

p 2 E(t) 1

min(l1; l2) 2p

p 2 E(0):

We then exploit (10), (15) and from the Holder inequality for someC >0. So we have ku1u2k^p ku1k^2pku2k^2p C²kr u1k²kr u2k² 1

2C² X2 i=1

kr uik²2

! : forC=C(n; p; ):

Consequently, we have ku₁u₂k^pp 2 ^pC^2p

X2 i=1

kr u_ik²2

!p

2 ^pC^2p X2 i=1

kr u_ik²2

!p 1 X2 i=1

kr u_ik²2

!

C^2p(2 min(l₁; l₂)) ^p 2p p 2

(p 1)

E(0)^{(p 1)} X2 i=1

l_ikr u_ik²2

!

X2 i=1

likr uik²2

!

; where

=C^2p(2 min(l1; l2)) ^p 2p p 2

(p 1)

E(0)^{(p 1)}: Which means, by the de…nition ofli; i= 1;2;

ku1u2k^pp

X2 i=1

likr uik²2

!

X2 i=1

1 i(t) Z t

0

gi(s)ds kr uik²2

X2 i=1

1 i(t) Z t

0

gi(s)ds kr uik²2+ X2 i=1

i(t)(gi r ui) + X2 i=1

m²_ikuik²2: Therefore,I(t)>0for allt2[0; T_m], by taking the fact that

t7!limTm

C^2p(2 min(l1; l2)) ^p 2p p 2

(p 1)

E(0)^{(p 1)} <1:

This shows that the solution (u1(t); u2(t))2W; for all t2[0; Tm]: By repeating this procedure Tmextends toT:

THEOREM 3.5. Suppose that (10), (15) and (18) hold. If(u₁₀; u₂₀)2W;(u₁₁; u₂₁)2 L²( ) ². Then the local solution (u₁; u₂) is global in time such that (u₁; u₂)2 G_T where

GT = (u; v) :u; v2L¹ R⁺;H₀( ) H₀( ) andu⁰; v⁰2L¹ R⁺;L²( ) L²( ) :

PROOF. In order to prove Theorem 3.5, it su¢ ces to show that the following norm X2

i=1

ku⁰_ik²2+ X2 i=1

kr u_ik²2+ X2 i=1

m²_iku_ik²2

is bounded independently oft. To achieve this, we use (12), (16) and (19) to get E(0) E(t) =J(t) +1

2 X2 i=1

ku⁰_i(t)k²2

p 2 2p

" ₂ X

i=1

1 _i(t) Z t

0

g_i(s)ds kr u_ik²2+ X2 i=1

i(t)(g_i r u_i)

+ X2

i=1

m²_iku_ik²2

# +1

2 X2 i=1

ku⁰_i(t)k²2+1 pI(t) p 2

2p

" ₂ X

i=1

l_ikr u_ik²2+ X2 i=1

i(t)(g_i r u_i) + X2 i=1

m²_iku_ik²2

#

+1 2

X2 i=1

ku⁰_i(t)k²2+1 pI(t) p 2

2p

" ₂ X

i=1

likr uik²2+ X2 i=1

m²_ikuik²2

# +1

2 X2 i=1

ku⁰_i(t)k²2: SinceI(t)and i(t)(g ru)(t)are positive, hence

X2 i=1

ku⁰_i(t)k²2+ X2 i=1

kr uik²2+ X2 i=1

m²_ikuik²2 CE(0);

where Cis a positive constant depending only on pandli. This completes the proof.

Open problem Let us mention here that, it will be interesting to discuss the as- ymptotic stability of this problem where also, one can establish a general decay rate estimate for the energy, which will depend on the behavior of both and g under following assumption

There exists a non-increasing di¤erentiable function _i; i= 1;2 :R⁺ !R⁺ satis- fying _i(t)>0; g_i⁰(t) _i(t)g_i(t);8t 0;and perhaps other conditions imposed by the nature of our system.

Acknowledgments. The authors want to thank the referee for his/her careful reading of the proofs.

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