DSpace at VNU: Existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions

DSpace at VNU: Existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations with nonline...

Received 7 March 2012 Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/mma.2803

MOS subject classification: 35L05; 35L15; 35L20; 35L55; 35L70

Existence, blow-up, and exponential decay

estimates for a system of nonlinear

wave equations with nonlinear

boundary conditions

Communicated by M L SantosThis paper is devoted to the study of a system of nonlinear equations with nonlinear boundary conditions First, on the basis of the Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we establish two local existence theorems of weak solutions Next, we prove that any weak solutions with negative initial energy will blow up in finite time Finally, the exponential decay property of the global solution via the construction of a suitable Lyapunov functional is presented Copyright © 2013 John Wiley & Sons, Ltd.

Keywords: system of nonlinear equations; Faedo–Galerkin method; local existence; global existence; blow up; exponential decay

where p  2, q  2, K > 0,  > 0,  i > 0, r i  2 i D 1, 2/ are given constants and f1, f2, F1, F2, Qu i , Qv i,.i D 0, 1/ are given functions

satisfying conditions specified later

Problems of this type arise in material science and physics, which have been studied by many authors For example, we refer to [1–16]and the references given therein In these works, many interesting results about the existence, regularity, and the asymptotic behavior

of solutions were obtained

In [12], Miao and Zhu proved the existence and regularity of global smooth solutions of a Cauchy problem for the nonlinear system

of wave equations with Hamilton structure Wu and Li [15] considered a system of nonlinear wave equations with initial and Dirichletboundary conditions, under some suitable conditions, the result on blow up of solutions and upper bound of blow-up time were given

a Nhatrang Educational College, 01 Nguyen Chanh Str., Nhatrang City, Vietnam

b 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

*Correspondence to: Nguyen Thanh Long, 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: longnt2@gmail.com

Cavalcanti et al [4] studied the existence of global solutions and the asymptotic behavior of the energy related to a degenerate

system of wave equations with boundary conditions of memory type By the construction of a suitable Lyapunov functional, the authorsproved that the energy decays exponentially The same method was also used in [13] to study the asymptotic behavior of the solutions

to a coupled system of wave equations having integral convolutions as memory terms The author showed that the solution of thatsystem decays uniformly in time, with rates depending on the rate of decay of the kernel of the convolutions

In [11], Messaoudi established a blow-up result for solutions with negative initial energy and a global existence result for arbitraryinitial data of a nonlinear viscoelastic wave equation associated with initial and Dirichlet boundary conditions

In [10, 14], the existence, regularity, blow-up, and exponential decay estimates of solutions for nonlinear wave equations associatedwith two-point boundary conditions have been established The proofs are based on the Galerkin method associated to a prioriestimates, weak convergence, and compactness techniques and also by the construction of a suitable Lyapunov functional The authors

in [14] proved that any weak solution with negative initial energy will blow up in finite time

The aforementioned works lead to the study of the existence, blow-up, and exponential decay estimates for a system of nonlinearwave equations associated with initial and Dirichlet boundary conditions and with nonlinear boundary conditions (1.1)–(1.3) In thispaper, we extend the results of [10,14] by combination of the methods used in [10,14] with some appropriate modifications for problemconsidered here Our main results are presented in three parts as follows

Part 1 is devoted to the presentation of the existence results based on Faedo–Galerkin method and standard arguments of densitycorresponding to the regularity of initial conditions In this part, problems (1.1)–(1.3) are dealt with two cases of.p, q/ : p  2, 2  q  4

or p  3, q > 4 and r1 2, r2 2 In the cases q D 2 and p  2, the solution obtained here is unique.

In parts 2 and 3, problems (1.1)–(1.3) are considered with p > 2 and q D r1D r2D 2 Under some suitable conditions, by applyingtechniques as in [14] with some necessary modifications and with some restrictions on the initial data, we prove that the solution of(1.1)–(1.3) blows up in finite time We also prove that the solution.u.t/, v.t// will exponential decay if the initial energy is positive and

small via the construction of a suitable Lyapunov functional

2 The existence and the uniqueness of a weak solution

The notation we use in this paper is standard and can be found in Lion’s book [17], with D 0, 1/, Q T D   0, T/, T > 0, and kk is the norm in L2

On H1, we shall use the following norm:

, such that.u, v/ satisfies the

following variational equation:

8ˆˆ

Now, we shall consider problems (1.1)–(1.3) with p  2, q  2, 1> 0, 2> 0, r1 2, r2 2, K > 0,  > 0 and make the following

, for all.u, v/ 2 R2

On the other hand, because of

minf˛, ˇgF.u, v/  uf1.u, v/ C vf2.u, v/  maxf˛, ˇg F.u, v/, for all u, v/ 2 R2,

functions f1and f2also satisfy hypothesis

H03

in Sections 3 and 4 by choosing positive constants˛, ˇ,  , and 1suitably

We have the following theorem about the existence of a ‘strong solution’

for T> 0 small enough

Furthermore, if q D 2 and p  2, the solution is unique.

Remark 2.3

The regularity obtained by (2.8) shows that problems (1.1)–(1.3) has a strong solution

8ˆˆˆˆ

H02, and.H3/ hold

Then problems (1.1)–(1.3) have a unique local solution

The proof consists of four steps

Step 1 The Faedo–Galerkin approximation Let f.w i,j /g be a denumerable base of H1\ H2/ .V \ H2/ We find the approximatesolution of problem (1.1)– (1.3) in the form

Under the assumptions of Theorem 2.2, system (2.12) has a solution.u m t/, v m t// on an interval Œ0, T m   Œ0, T.

Step 2 The first estimate Multiplying the jth equation of (2.12) by c0mj t/, d mj0 .t// and summing with respect to j, and afterwards integrating with respect to the time variable from 0 to t, we obtain after some rearrangements

We shall estimate respectively the following integrals in the right-hand side of (2.13)

First integral Using the inequalities

ab1

qıq a qC 1

q0ıq0b q0, for allı > 0, a, b  0, q > 1, q0D q

where we remark that, in what follows, C T always indicates a bound depending on T.

Second integral By (2.4), (2.14), and (2.17), we have

where C0always indicates a positive constant depending only on Qu0, Qv0, Qu1, Qv1,˛, and ˇ

Using.H3/, we deduce from (2.21) that

F.Qu0.x/, Qv0.x//dx C 2

Z 1 0

where C0always indicates a bound depending on Qu0, Qv0, Qu1, Qv1,˛, and ˇ

Third integral Using the assumption H2/, we deduce from the Cauchy–Schwartz inequality that

Hence, (2.13), (2.14), (2.15), (2.19), (2.22), and (2.23) lead to

Lemma 2.4 allows one to take constant T m D Tfor all m.

The second estimate.

First of all, we are going to estimate

Letting t! 0Cin (2.12)1, multiplying the result by c00mj.0/, we obtain

m.0/  kQu 0xxk C 1jj j Qu1jr1 1jj C kf1 Qu0, Qv0/k C kF1.0/k  C01for all m, (2.27)

where C01is a constant depending only on r1,1, Qu0, Qv0, Qu1, f1, and F1

Similarly, letting t! 0Cin (2.12)2, multiplying the result by d00mj.0/, and using the compatibility (2.7) to obtain

where C02is a constant depending only on r2,2, Qu0, Qv0, Qv1, f2, and F2

Now differentiating (2.12) with respect to t, the results are

8ˆˆˆˆˆˆˆˆ

X m 0/ D jju00m.0/jj2C jjv00m.0/jj2C jj Qu 1xjj2C jjQv 1xjj2 C0, for all m, (2.33)

where we note that, in the sequel, C0always indicates a positive constant depending only on Qu0, Qv0, Qu1, Qv1, f1, f2, F1, F2, r1, r2,1, and2.Put

K2.T, F/ D sup

jyj, jzjpC T, j˛jD2

ˇ

We shall estimate all integrals in the right-hand side of (2.31)

First integral From (2.14), (2.25), (2.34), and the Hölder inequality

2=q X m t// 2=q

 C0Cq 2

q ı

q q2

Chooseı > 0, with2qıq 12, from (2.44), (2.45) and (2.48), (2.38) follows

and also using (2.48), we obtain (2.38)

Combining (2.31), (2.32), (2.33), and (2.35)–(2.38) leads to

By the compactness lemma of Lions [17, p 57] and the imbeddings H2.0, T/ ,! C1.Œ0, T/ , H1.0, T/ ,! C0.Œ0, T/ , W 1,q 0, T/ ,!

C0.Œ0, T/, we can deduce from (2.52) the existence of a subsequence still denoted by f.u m , v m/g such that

8ˆˆˆˆ

.u m , v m / ! u, v/ strongly in L2.Q T/  L2.Q T/ and a.e in Q T,

.u0m , v0m / ! u0, v0/ strongly in L2.Q T/  L2.Q T/ and a.e in Q T,

v m 0, / ! v.0, / strongly in C0.Œ0, T/,ˇ

for all x, y 2 ŒR, R, R > 0, r  2, it follows from (2.32), (2.51), and (2.53)2that

8ˆˆ

Step 4 Uniqueness of the solution Assume now that q D 2 and p  2 Let u i , v i /, i D 1, 2 be two weak solutions of problems (1.1)–(1.3)

such that

8ˆˆˆˆ

We take.w, / D u0, v0/ in (2.68)1,2, and integrating with respect to t, we obtain

We estimate all terms on the right-hand side of (2.69) as follows:

Integral Z1 t/ Applying the Cauchy–Schwartz inequalities (2.72)–(2.74) give

jZ1.t/j D 2

ˇ

ˇZ t0

˝

f1.u1, v1/  f1.u2, v2/, u0.s/˛

ds

ˇˇ

˝

f2.u1, v1/  f2.u2, v2/, v0.s/˛

ds

ˇˇ

Now, we consider two cases for p.

Case pD 2 Note that, by (2.72),

Z t

0

12

By Gronwall’s lemma, (2.83) leads to 1 u2 0, v D v1 v2 0

Theorem 2.2 is proved completely

Proof of Theorem 2.3

Let Qu0, Qu1/ 2 H1 L2,.Qv0, Qv1/ 2 V  L2,.F1, F2/ 2 L2.Q T /, and q D 2, p  2.

In order to obtain the existence of a weak solution, we use standard arguments of density

Let us consider Qu0, Qu1/ 2 H1 L2,.Qv0, Qv1/ 2 V  L2,.F1, F2/ 2 L2.Q T / and let sequences f.u 0m , u 1m /g  C10 



 C01

,

f.v 0m , v 1m /g  C10 ./  C10 

, andf.F 1m , F 2m /g  C01

Q T

 C10 

Q T, such that8

ˆˆ

By the same arguments used to obtain the aforementioned estimates, we obtain

8t 2 Œ0, T, where C T is a positive constant independent of m and t.

On the other hand, we put U m,l D u m  u l , V m,l D v m  v l, from (2.86), it follows that

8ˆˆˆˆˆˆˆˆˆˆ

Convergences of the sequencesf.u 0m , u 1m /g and f.v 0m , v 1m /g imply the convergence to zero (when m, l ! 1) of terms on the

right-hand side of (2.100) Therefore, we obtain

8ˆˆ

Next, the uniqueness of a weak solution is obtained by using the well-known regularization procedure due to Lions Theorem 2.3 is

Furthermore, the uniqueness of a weak solution is also not asserted

3 Finite time blow up

In this section, we consider problems (1.1)–(1.3) corresponding to F1D F2D 0, q D r1D r2D 2, p > 2, 1> 0, 2> 0,  > 0 We shallshow that the solution of this problem blows up in finite time if

.u, v/ of problems (1.1)–(1.3) blows up in finite time.

F.u.x, t/, v.x, t//dx, (3.2)and we put

Hence, we deduce from (3.2), (3.5), and (3.6) that

ˇ  22ˇ

As (3.4), (3.5), (3.8), (3.14), and the following inequality

hf1.u.t/, v.t//, u.t/i C hf2.u.t/, v.t//, v.t/i  d1

Z 1 0

Now we continue with proof of Theorem 3.1.

Using the inequality

X6

iD1 x i

r

 6r1X6

iD1 x i r , for all r > 1, and x1,: : : , x6 0, (3.20)

we deduce from (3.9) and (3.10) that

kuk s L˛ kuk2L˛ ku xk2 ku xk2C kuk˛L˛ (3.26)

Case 2.kuk L˛ 1 : By 2  s  ˛, we have

We consider two cases forkuk :

Case 1.kuk  1 :

By 2 2=.1  /  ˛, we have

Case 2.kuk  1 : By 2  2=.1  /  ˛, we have

Therefore,

Combining (3.28) and (3.31), we obtain

whereı is a positive constant and

 2p

p 2E

ˇ2 23

where 1and 2are two positive constants Then there exist positive constants C and such that

Combining (4.11) and (4.13), it is easy to see that (4.10)iholds.

Similarly, we have also

(4.14)

Combining (4.11), (4.14), it is easy to see (4.10)iiholds

ku x t/k2C 2p

p2 E

ˇ2 2

C 2p

p2 E

ˇ2 2

Now, we put TD sup fT > 0 : I.t/ > 0, 8t 2 Œ0, Tg If T< C1; then, because of the continuity of I.t/, we have I.T/  0 By the

same arguments as above, we can deduce that there exists T2> Tsuch that I.t/ > 0, 8t 2 Œ0, T2 This leads to I.t/ > 0, 8t  0.

1

21

pı2

1

On the other hand,

hf1.u.t/, v.t//, u.t/i C hf2.u.t/, v.t//, v.t/i  d2

Z 1 0

and by I.t/ D ku x t/k2C kv x t/k2 K jv.0, t/j p  pR1

0 F.u.x, t/, v.x, t//dx > 0, for all t  0, we have

hf1.u.t/, v.t//, u.t/i C hf2.u.t/, v.t//, v.t/i  d2

Now we continue with the proof of Theorem 4.1

It follows from (4.1), (4.10)ii, and (4.27) that

L0.t/  

"12

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