DSpace at VNU: Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions

DSpace at VNU: Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirch...

DOI 10.1007/s10440-009-9555-9

Linear Approximation and Asymptotic Expansion

of Solutions in Many Small Parameters for a Nonlinear

Kirchhoff Wave Equation with Mixed Nonhomogeneous

Conditions

Le Thi Phuong Ngoc · Nguyen Thanh Long

Received: 21 February 2009 / Accepted: 24 November 2009 / Published online: 1 December 2009

© Springer Science+Business Media B.V 2009

Abstract In this paper, we consider the following nonlinear Kirchhoff wave equation

In particular, motivated by the asymptotic expansion of a weak solution in only one, two

or three small parameters in the researches before now, an asymptotic expansion of a weaksolution in many small parameters appeared on both sides of (1)1is studied

Keywords Faedo–Galerkin method· Linear recurrent sequence · Asymptotic expansion of

whereu0,u1, μ, f, g are given functions satisfying conditions specified later In (1.1),

the nonlinear term μ(u, u x2) not only depends on u but also depends on the integral

One of the early classical studies dedicated to Kirchhoff equations was given by hozaev [27] After the work of Lions, for example see [11], (1.5) received much attentionwhere an abstract framework to the problem was proposed We refer the reader to, e.g.,J.J Bae, M Nakao [1], Cavalcanti et al [2,3], Ebihara, Medeiros and Miranda [5], Hosoyaand Yamada [6], Lasiecka and Ong [9], Miranda et al [22], Menzala [21], Park et al [25,

Po-26], Rabello et al [28], Santos et al [29], Yamada [31] for many interesting results andfurther references

In [1], Bae and Nakao proved the existence of global solutions to the initial-boundaryvalue problem for the Kirchhoff type quasilinear wave equations of the form

A survey of the results about the mathematical aspects of Kirchhoff model can be found

in the investigations by Medeiros, Limaco and Menezes [19,20] In these works, there aremany contributions about the mathematical aspects of the mixed problems associated to the

operator Kirchhoff or the operator Kirchhoff–Carrier, such as existence of local and globalsolutions, global regular solutions, the asymptotic behavior of the energy.

In [8], Larkin studied in a n+ 1-dimensional cylinder global solvability of the mixedproblem for the nonhomogeneous Carrier equation

u t t − μ(x, t, u2)u + g(x, t, u t ) = f (x, t). (1.8)Santos et al., [29], considered a nonlinear wave equation of Kirchhoff type

u t t − μ( u2)u − u t + f (u) = 0 (1.9)

in  × (0, ∞) with memory condition at the boundary and they studied the asymptotic

behavior of the corresponding solutions

In some other special cases, when the function μ = 1 or μ = μ(x, t) and the nonlinear term f has the simple forms, the problem (1.4), with various initial-boundary conditions,has been studied by many authors, for example, Long, Alain Pham, Diem [17], Ngoc, Hang,Long [23], and references therein

However, by the fact that it is difficult to consider the problem (1.4) with some

initial-boundary conditions in the case μ(x, t, u, u x2) depending on u and u x2,few workswere done as far as we know In order to solve this problem, the linearization method fornonlinear term is usually used Let us present this technique as follows

At first, we note that, for each v = v(x, t) belongs to X being a suitable space of function,

we can give some suitable assumptions to obtain a unique solution u ∈ X of the problem with respect to μ = μ(x, t, v(x, t), v x (t )2)= μ(x, t ) and f = f (x, t, v, v x , v t )= f (x, t ).

It is obviously that u depends on v, so we can suppose that u = A(v) Therefore, the above problem can be reduced to a fixed point problem for operator A : X → X Based on these ideas, with the first term u0is chosen, the usual iteration u m = A(u m−1), m = 1, 2, , is

applied to establish a sequence{u m }, which converges to the solution of the problem, hence

the existence results follows

Without loss of generality, we need only consider the problem (1.1)–(1.3) instead ofProblem (1.2)–(1.4) in order to avoid making the treatment too complicated

The paper consists of four sections At first, some required preliminaries are done inSect.2 With the technique presented as above, we begin Sect.3by establishing a sequence

of approximate solutions of the problem (1.1)–(1.3) based on the Faedo-Galerkin method.Thanks to a priori estimates, this sequence is bounded in an appropriate space, from which,using compact imbedding theorems and Gronwall’s Lemma, one deduce the existence of aunique weak solution of the problem (1.1)–(1.3) In particular, an asymptotic expansion of a

weak solution u = u(ε1, , ε p ) of order N + 1 in p small parameters ε1, ε2, , ε pfor theequation

associated to (1.1)2,3 , with μ ∈ C N+2(R × R+), μ1∈ C N+1(R × R+), μ(y, z) ≥ μ0> 0,

μ1(y, z) ≥ 0, for all (y, z) ∈ R × R+, g ∈ C3(R+), f ∈ C N+1([0, 1] × R+× R3) and

f i ∈ C N ([0, 1] × R+× R3), i = 2, 3, , p, is established in Sect.4 This result is a relativegeneralization of [1,5,6,8,9,12–18,23–31]

Then V is a closed subspace of H1 and on V , v H1, v x

v V=√a(v, v) = ∇v are equivalent norms.

Then we have the following lemmas, the proofs of which are straightforward and areomitted

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

Lemma 2.2 The imbedding V → C0([0, 1]) is compact and

3 The Existence and Uniqueness Theorem of Solution

We make the following assumptions:

(H1) u0∈ V ∩ H2, u1∈ V,

(H2) g ∈ C3(R+),

(H3) μ ∈ C2(R × R+), μ(y, z) ≥ μ0> 0, ∀(y, z) ∈ R × R+,

(H4) f ∈ C1(× R+× R3).

Put ϕ(x, t) = (x − 1)g(t) By the transformation v(x, t) = u(x, t) − ϕ(x, t),

Prob-lem (1.1)–(1.3) reduces to the following problem with homogeneous boundary conditions

v0(x)= u0(x) − ϕ(x, 0) =  u0(x) − (x − 1)g(0),

v1(x)= u1(x) − ϕ t (x, 0)= u1(x) − (x − 1)g (0),

(3.2)

and g,u0satisfying the consistency condition g(0) = u x (0, 0)= u0(0).

Consider T∗> 0 fixed, let M > 0, we put

and associate with problem (3.1) the following problem:

Find v m ∈ W1(M, T ) (m ≥ 1) which satisfies the following linear variational problem

v m (t ), w + μ m (t )∇v m (t ), ∇w = F m (t ), w , ∀w ∈ V,

Then, we have the following theorem.

Theorem 3.1 Let (H1) –(H4) hold Then there exist positive constants M, T > 0 such that, for v0≡ v0, there exists a recurrent sequence {v m } ⊂ W1(M, T ) defined by (3.9), (3.10)

Proof The proof consists of several steps.

Step 1: The Faedo-Galerkin approximation (introduced by Lions [10]) Consider the

ba-sis in V

w j (x)=

2

m on interval[0, T ], so let us omit the details (see [4])

Step 2 A priori estimates Put

For all j = 1, 2, , k, multiplying (3.13) by˙c (k)

m (t ), summing on j, and integrating by parts with respect to the time variable from 0 to t, we have

By replacing w j in (3.13) by w j ,we obtain

a( ¨v (k)

m (t ), w j ) + a(μ m (t ) ∇v (k)

m (t ), ∇w j ) = −F m (t ), w j , 1 ≤ j ≤ k. (3.17)Similarly, by multiplying (3.17) by ˙c (k)

We shall estimate the integrals on the right hands of (3.16) and (3.18) as follows

First integral I1.From (3.4), (3.8), and (3.10) we have

We shall estimate the term I4∗(s)as follows We have

∂2μ m

∂x∂s (s) = D1D1μ(η m (x, s), z m (s))(v m−1(x, t ) + ϕ (x, t ))(∇v m−1(x, s) + g(s))

+ D2D1μ(η m (x, s), z m (s))( ∇v m−1(x, s) + g(s))z m (s) + D1μ(η m (x, s), z m (s))(∇v m−1(x, s) + g (s)), (3.26)

Fifth integral I5.By the Cauchy-Schwartz inequality, we have

D i f[v m−1] = D i f (x, t, v m−1(x, t ), ∇v m−1(x, t ), v m−1(x, t ), ∇v m−1(t ) + g(t)2

),

i = 1, 2, , 6 So, by (3.6), (3.8) and (3.35), we obtain

s m (k) (t )≤ C0(  f , μ, μ , v 0k , v 1k , g(0), β)+ C1(M, T , β)

+ 2 t

m ], for all m and k So we can take constant T (k)

m = T for all k and m.

Therefore, we have

v (k) m ∈ W(M, T ), for all m and k. (3.51)

Step 3 Limiting process From (3.51), we deduce the existence of a subsequence of{v (k)

m }still also so denoted, such that

Passing to limit in (3.13), we have v msatisfying (3.9), (3.10) in L2(0, T ), weak On the

other hand, it follows from (3.9), (3.10) and (3.52)4that v m= ∂

∂x (μ m (t )∇v m (t )) + F m∈

L∞(0, T ; L2), hence v m ∈ W1(M, T )and the proof of Theorem3.1is complete 

Theorem 3.2 Let assumptions (H1) –(H4) hold Then:

(i) There exist positive constants M and T satisfying (3.45), (3.47) and (3.48) such that the problem (3.1), (3.2) has a unique weak solution v ∈ W (M, T ).

(ii) The linear recurrent sequence {v m } defined by (3.9), (3.10) converges to the solution v

of the problem (3.1), (3.2) strongly in the space

Proof (i) Existence of the solution First, we note that W1(T )is a Banach space with respect

to the norm (see Lions [10])

w W1(T ) = w L∞(0,T ;V ) + wL∞(0,T ;L2) (3.55)

We shall prove that{v m } is a Cauchy sequence in W1(T ) Let w m = v m+1− v m Then w m

satisfies the variational problem

First integral J1 By (H3)and (3.19), we have

F m (t )→ f (·, t, v(t), v x (t ), v (t ), v x (t ) + g(t)2) strongly in L∞(0, T ; L2), (3.73)

μ m (t ) → μ(v(t) + ϕ(t), v x (t ) + g(t)2) strongly in L∞(Q T ). (3.74)Finally, passing to limit in (3.9)–(3.10) as m = m j → ∞, it implies from (3.69),(3.70)1,3 , 3.73) and (3.74) that there exists v ∈ W(M, T ) satisfying the equation

thus we have v ∈ W1(M, T ).The existence proof is completed

(ii) Uniqueness of the solution.

Let v1, v2∈ W1(M, T ) be two weak solutions of the problem (3.1), (3.2) Then v=

v1− v2satisfies the variational problem

Z(t ) = v (t )2+

μ1(t )∇v(t)2

≥ v (t )2+ μ0∇v(t)2. (3.81)

We now estimate the terms on the right hand of (3.80) as follows

First integral  J1 By (H2) and (H3),we have

This implies that

M K(M, μ). Using Gronwall’s lemma, it

follows that Z(t) ≡ 0, i.e., v1≡ v2.

Remark 1 (i) In the case of μ ≡ 1, f = f (x, t, u, u x , u t ) with f ∈ C1([0, 1]×R+×R3)andthe boundary condition in [12] standing for (1.2), we have obtained the results concerningthe ones in paper [12]

(ii) In the case of μ = μ(u x2) or μ = μ(u), f = f (x, t, u, u x , u t ) with f ∈ C1([0, 1]×

R+× R3),and some other boundary conditions standing for (1.2), we have also obtainedsome results in the papers [13,16,24]

4 Asymptotic Expansion of the Solution with Respect to Many Small Parameters

In this section, let (H1)–(H4)hold We also make the following assumptions:

We use the following notations For a multi-index α = (α1, , α p )∈ Zp

First, we shall need the following lemma

Lemma 4.1 Let m, N ∈ N and u α ∈ R, α ∈ Z p

The proof of Lemma4.1can be found in [15]

Now, we assume that

(H7) μ ∈ C N+2(R × R+), μ1∈ C N+1(R × R+), μ ≥ μ0> 0, μ1≥ 0,

(H8) f ∈ C N+1([0, 1] × R+× R3), f i ∈ C N ([0, 1] × R+× R3), i = 2, 3, , p.

We also use the notations f [u] = f (x, t, u, u x , u t ), μ[u] = μ(u, u x2).

Let u0be a unique weak solution of the problem (P0)(as in Theorem 3.2) corresponding

Then, we have the following lemma.

Lemma 4.2 Let ρ ν [μ], π ν [f ], |ν| ≤ N, be the functions are defined by the formulas (4.5)

and (4.8) Put h=|γ |≤N u γ−→ε γ , then we have

whereR N (1) [μ, −→ε ]L∞(0,T ;L2) + R (1)

N [f, −→ε]L∞(0,T ;L2) ≤ C, with C is a constant ing only on N, T , f, μ, u γ , |γ | ≤ N.

depend-Proof (i) In the case of N = 1, the proof of (4.10) is easy, hence we omit the details, which

we only prove with N ≥ 2 Put h = u0+1≤|α|≤N u α−→ε α ≡ u0+ h1,we rewritten

μ[h] = μ(h, ∇h2) = μ(u0+ h1, ∇u0+ ∇h12) = μ(u0+ h1, ∇u02+ ξ), (4.12)

where ξ = ∇u0+ ∇h12− ∇u02.

By using Taylor’s expansion of the function μ(u0+ h1,∇u02+ ξ) around the point (u0, ∇u02) up to order N + 1, we obtain

On the other hand, we have also

ξ = ∇u0+ ∇h12− ∇u02= 2∇u0,∇h1 + ∇h12≡

1≤|α|≤2N

σ α−→ε α , (4.16)

with σ α ,1≤ |α| ≤ 2N are defined by (4.6)

We again use formula (4.2), it follows from (4.16), that

where σ = (σ α ), α∈ Zp

+,1≤ |α| ≤ 2N.

Hence, it follows from (4.14) and (4.15), that

By the boundedness of the functions u γ , ∇u γ , u γ , |γ | ≤ N in the function space

L∞(0, T ; H1), we obtain from (4.14)–(4.17), (4.19), (4.21), that R N (1) (μ, u0,

h1, ξ ) L∞(0,T ;L2) ≤ C, with and C is a constant depending only on N, T , μ, u γ , |γ | ≤ N.

Hence, the part 1 of Lemma4.2is proved

(ii) We only prove (4.11) with N ≥ 2 By using Taylor’s expansion of the function

f [u0+ h1] around the point u0up to order N + 1, we obtain from (4.2), that

m! D m f [u0]

Remark 2 Lemma4.2is a generalization of a formula contained in ([13], p 262, formula(4.38)) and it is useful to obtain the following Lemma4.3 These Lemmas are the key to

the asymptotic expansion of a weak solution u = u(ε1, , ε p ) of order N + 1 in p small parameters ε1, , ε pas it will be said below

Let u− →ε = u(ε1, , ε p ) ∈ W1(M, T ) be a unique weak solution of the problem (P− →ε ) Then v = u− →ε −|γ |≤N u γ−→ε γ ≡ u− →ε − h satisfies the problem

Lemma 4.3 Let (H1), (H2), (H7), and (H8) hold Then there exists a constant  K such that

E− →εL∞(0,T ;L2)≤ K−→ε N+1, (4.28)

where  K is a constant depending only on N, T , f, f1, μ, μ1, u γ , |γ | ≤ N.

Proof In the case of N = 1, the proof of Lemma4.3is easy, hence we omit the details,

which we only prove with N ≥ 2.

By using the formulas (4.10), (4.11) for the functions μ1[h] and f i [h], we obtain

Similarly, with f i [h], we also obtain

function space L∞(0, T ; L2) by a constant depending only on N, T , f, f1, u γ , |γ | ≤ N.

On the other hand, we deduce from (4.10) and (4.29)1, that

We decompose the sum 

1≤|γ |≤2N into the sum of two the sums

By the boundedness of the functions u γ , ∇u γ , u γ , |γ | ≤ N in the function space

L∞(0, T ; H1),we obtain from (4.32) and (4.36), that

E− →εL∞(0,T ;L2)≤ K−→ε N+1, (4.39)where K is a constant depending only on N, T , f, f1, μ, μ1, u γ , |γ | ≤ N.

Now, we consider the sequence of functions{v m} defined by

By multiplying the two sides of (4.41) by v1,we find without difficulty from (4.28) that

2+ ζ0

2μ0



We shall prove that there exists a constant C T , independent of m and −→ε ,such that

v mW1(T ) ≤ C T−→ε N+1, with−→ε  ≤ ε∗ for all m. (4.47)

By multiplying the two sides of (4.40) with v m and after integration in t, we obtain

without difficulty from (4.28) that

0

F− →ε [v m−1+ h] − F− →ε [h]v m (s)ds

+ 2 t

With I3(1) (s)we have



I3(1) (s) = μ[v m−1+ h] − μ[h] C0() ≤ (1 + 2M∗)  K(M∗, μ)v m−1W1(T ) (4.55)With I3(2) (s)we also obtain



T

3

2+ ζ1

2μ

√

T

We assume that

σ T < 1, with the suitable constant T > 0. (4.64)

We shall now require the following lemma whose proof is immediate

Lemma 4.4 Let the sequence {ψ m } satisfy

ψ m ≤ σψ m−1+ δ for all m ≥ 1, ψ0= 0, (4.65)

where 0 ≤ σ < 1, δ ≥ 0 are the given constants Then

Applying Lemma 4.4 with ψ m = v mW1(T ) , σ = σ T = ζ2+ ζ2η T < 1, δ =

η T K−→εN+1,it follows from (4.66), that

On the other hand, the linear recurrent sequence {v m} defined by (4.40) converges

strongly in the space W1(T ) to the solution v of problem (4.26) Hence, letting m→ +∞

in (4.67) gives

v W1(T ) ≤ C T−→εN+1,or

Theorem 4.5 Let (H1), (H2), (H7), and (H8) hold Then there exist constants M > 0 and

T > 0 such that, for every −→ε , with−→ε  ≤ ε

∗ the problem (P−→ε ) has a unique weak solution

u→ε ∈ W1(M, T ) satisfying an asymptotic estimation up to order N + 1 as in (4.68), the functions u γ , |γ | ≤ N being the weak solutions of the problems ( P γ ), |γ | ≤ N, respectively Remark 3 Typical examples about asymptotic expansion of the solution in a small parameter

can be found in the researches of many authors, such as [12–14,16,23] However, to ourknowledge, in the case of asymptotic expansion in many small parameters, there is onlypartial results, for example, [15,17,18,24] concerning asymptotic expansion of thesolution in two or three small parameters

Acknowledgements The authors wish to express their sincere thanks to the referees for the suggestions and valuable comments.

References

1 Bae, J.J., Nakao, M.: Existence problem for the Kirchhoff type wave equation with a localized weakly

nonlinear dissipation in exterior domains Discrete Contin Dyn Syst 11(2–3), 731–743 (2004)

1 Bae, J.J., Nakao, M.: Existence problem for the Kirchhoff type wave equation with a localized weakly< /small>

nonlinear dissipation in exterior domains Discrete Contin... class="text_page_counter">Trang 25

Remark Lemma4.2is a generalization of a formula contained in ([13], p 262, formula(4.38)) and it is... ourknowledge, in the case of asymptotic expansion in many small parameters, there is onlypartial results, for example, [15,17,18,24] concerning asymptotic expansion of thesolution in two or three small