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...

Acta Appl Math (2010) 112: 137–169 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: December 2009 © Springer Science+Business Media B.V 2009 Abstract In this paper, we consider the following nonlinear Kirchhoff wave equation ⎧ ∂ ⎪ ⎨ utt − ∂x (μ(u, ux )ux ) = f (x, t, u, ux , ut ), u(1, t) = 0, ux (0, t) = g(t), ⎪ ⎩ ut (x, 0) = u1 (x), u(x, 0) = u0 (x), < x < 1, < t < T , (1) where u0 , u1 , μ, f , g are given functions and ux = u2x (x, t)dx To the problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method 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 weak solution in many small parameters appeared on both sides of (1)1 is studied Keywords Faedo–Galerkin method · Linear recurrent sequence · Asymptotic expansion of order N + Mathematics Subject Classification (2000) 35L20 · 35L70 · 35Q72 L.T.P Ngoc Nhatrang Educational College, 01 Nguyen Chanh Str., Nhatrang City, Vietnam e-mail: phuongngoccdsp@dng.vnn.vn L.T.P Ngoc e-mail: ngoc1966@gmail.com N.T 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: longnt@hcmc.netnam.vn N.T Long e-mail: longnt2@gmail.com 138 L.T.P Ngoc, N.T Long Introduction In this paper, we consider the following initial and boundary value problem utt − ∂ (μ(u, ux )ux ) = f (x, t, u, ux , ut ), x ∈ ∂x u(1, t) = 0, ux (0, t) = g(t), = (0, 1), < t < T , (1.2) ut (x, 0) = u1 (x), u(x, 0) = u0 (x), (1.1) (1.3) where u0 , u1 , μ, f, g are given functions satisfying conditions specified later In (1.1), the nonlinear term μ(u, ux ) not only depends on u but also depends on the integral ux = u2x (x, t)dx Equation (1.1) constitutes a case, relatively simpler, of a more general equation, namely utt − ∂ (μ(x, t, u, ux )ux ) = f (x, t, u, ux , ut ), ∂x When f = 0, μ = μ( ux )ux , x∈ = (0, 1), t > (1.4) = (0, L), (1.4) is related to the Kirchhoff equation ρhutt = P0 + Eh 2L L ∂u (y, t) dy uxx , ∂y (1.5) presented by Kirchhoff in 1876 (see [7]) This equation is an extension of the classical D’Alembert’s wave equation by considering the effects of the changes in the length of the string during the vibrations The parameters in (1.5) have the following meanings: u is the lateral deflection, L is the length of the string, h is the area of the cross-section, E is the Young modulus of the material, ρ is the mass density, and P0 is the initial tension One of the early classical studies dedicated to Kirchhoff equations was given by Pohozaev [27] After the work of Lions, for example see [11], (1.5) received much attention where 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], Hosoya and Yamada [6], Lasiecka and Ong [9], Miranda et al [22], Menzala [21], Park et al [25, 26], Rabello et al [28], Santos et al [29], Yamada [31] for many interesting results and further references In [1], Bae and Nakao proved the existence of global solutions to the initial-boundary value problem for the Kirchhoff type quasilinear wave equations of the form utt − (1 + u ) u + ρ(ut ) = (1.6) is a compact in × R+ , where is an exterior domain in RN , N ≥ 1, such that RN set in RN In [3], Cavalcanti et al studied global existence and asymptotic stability for the following nonlinear and generalized damped extensible plate equation utt + u + αu − μ | u|2 dx u + f (u) + g(ut ) = (1.7) 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 are many contributions about the mathematical aspects of the mixed problems associated to the Linear Approximation and Asymptotic Expansion of Solutions 139 operator Kirchhoff or the operator Kirchhoff–Carrier, such as existence of local and global solutions, global regular solutions, the asymptotic behavior of the energy In [8], Larkin studied in a n + 1-dimensional cylinder global solvability of the mixed problem for the nonhomogeneous Carrier equation utt − μ(x, t, u ) u + g(x, t, ut ) = f (x, t) (1.8) Santos et al., [29], considered a nonlinear wave equation of Kirchhoff type utt − μ( u ) u − ut + f (u) = (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 = 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 initialboundary conditions in the case μ(x, t, u, ux ) depending on u and ux , few works were done as far as we know In order to solve this problem, the linearization method for nonlinear 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), vx (t) ) = μ(x, t) and f = f (x, t, v, vx , vt ) = 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 u0 is chosen, the usual iteration um = A(um−1 ), m = 1, 2, , is applied to establish a sequence {um }, 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 of Problem (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 in Sect With the technique presented as above, we begin Sect by 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 a unique weak solution of the problem (1.1)–(1.3) In particular, an asymptotic expansion of a weak solution u = u(ε1 , , εp ) of order N + in p small parameters ε1 , ε2 , , εp for the equation utt − ∂ ([μ(u, ux ) + ε1 μ1 (u, ux )]ux ) ∂x p = f (x, t, u, ux , ut ) + εi fi (x, t, u, ux , ut ), (1.10) i=2 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 ∈ C (R+ ), f ∈ C N+1 ([0, 1] × R+ × R3 ) and fi ∈ C N ([0, 1] × R+ × R3 ), i = 2, 3, , p, is established in Sect This result is a relative generalization of [1, 5, 6, 8, 9, 12–18, 23–31] 140 L.T.P Ngoc, N.T Long Preliminaries Put = (0, 1) Let us omit the definitions of usual function spaces that will be used in the sequel such as Lp = Lp ( ), H m = H m ( ) The norm in L2 is denote by · We denote by ·, · the scalar product in L2 or a pair of dual products of continuous linear functional with an element of a function space We denote by · X the norm of a Banach space X and by X the dual space of X We denote Lp (0, T ; X), ≤ p ≤ ∞ the Banach space of real functions u : (0, T ) → X measurable, such that u Lp (0,T ;X) < +∞, with ⎧ ⎨ ( 0T u(t) pX dt)1/p , if ≤ p < ∞, u Lp (0,T ;X) = (2.1) if p = ∞ ⎩ ess sup u(t) X , 0 fixed, let M > 0, we put M∗ = g C ([0,T ∗ ]) = sup (|g(t)| + |g (t)|), (3.3) 0≤t≤T ∗ K = K(M, μ) = sup |D β μ(y, z)|, (3.4) (y,z)∈A∗ (M) |β|≤2 K0 (M, f ) = sup{|f (x, t, u, v, w, z)| : (x, t, u, v, w, z) ∈ A∗ (M)}, (3.5) K1 = K1 (M, f ) = (3.6) K0 (M, Di f ), i=1 where ⎧ ∗ ⎪ ⎨ A∗ (M) = {(y, z) ∈ R × R+ : |y| ≤ M, ≤ z ≤ (M + M ) }, ∗ A∗ (M) = {(x, t, u, v, w, z) ∈ [0, 1] × [0, T ] × R : ⎪ ⎩ |u|, |v|, |w| ≤ M, ≤ z ≤ (M + M ∗ )2 } For each T ∈ (0, T ∗ ] and M > 0, we get ⎧ ∞ ∞ ⎪ ⎨ W (M, T ) = {v ∈ L (0, T ; V ∩ H ) : vt ∈ L (0, T ; V ), vtt ∈ L (QT ), with v L∞ (0,T ;V ∩H ) , vt L∞ (0,T ;V ) , vtt L2 (QT ) ≤ M}, ⎪ ⎩ W1 (M, T ) = {v ∈ W (M, T ) : vtt ∈ L∞ (0, T ; L2 )}, (3.7) where QT = × (0, T ) We shall choose the first initial term v0 ≡ v0 Suppose that vm−1 ∈ W1 (M, T ), (3.8) and associate with problem (3.1) the following problem: Find vm ∈ W1 (M, T ) (m ≥ 1) which satisfies the following linear variational problem vm (t), w + μm (t)∇vm (t), ∇w = Fm (t), w , vm (0) = v0 , vm (0) = v1 , ∀w ∈ V , (3.9) 142 L.T.P Ngoc, N.T Long where ⎧ ⎪ ⎨ μm (t) = μ(ηm (x, t), zm (t)), ηm (x, t) = vm−1 (x, t) + ϕ(x, t), zm (t) = ∇vm−1 (t) + g(t) , ⎪ ⎩ Fm (x, t) = f (x, t, vm−1 (x, t), ∇vm−1 (x, t), vm−1 (x, t), ∇vm−1 (t) + g(t) ) (3.10) Then, we have the following theorem Theorem 3.1 Let (H1 )–(H4 ) hold Then there exist positive constants M, T > such that, for v0 ≡ v0 , there exists a recurrent sequence {vm } ⊂ 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 basis in V cos(λj x), + λ2j wj (x) = π λj = (2j − 1) , j ∈ N, constructed by the eigenfunctions of the Laplace operator − (3.11) ∂ = − ∂x Put k (k) cmj (t)wj , vm(k) (t) = (3.12) j =1 (k) satisfy the system of linear differential equations where the coefficients cmj ·· (k) v m (t), wj + μm (t)∇vm(k) (t), ∇wj = Fm (t), wj , vm(k) (0) v˙m(k) (0) = v0k , ≤ j ≤ k, = v1k , (3.13) where k (k) j =1 αj wj (k) k j =1 βj wj v0k = v1k = → v0 strongly in V ∩ H , → v1 strongly in V (3.14) Note that by (3.8), it is not difficult to prove that the system (3.13) has a unique solution vm(k) on interval [0, T ], so let us omit the details (see [4]) Step A priori estimates Put ⎧ √ ⎪ μm (t)∇vm(k) (t) , p (k) (t) = v˙m(k) (t) + ⎪ ⎨ m √ μm (t) vm(k) (t) , qm(k) (t) = ∇ v˙m(k) (t) + (3.15) ⎪ ⎪ ⎩ s (k) (t) = p (k) (t) + q (k) (t) + t v·· (k) (s) ds m m m m (k) (t), summing on j, and integrating by For all j = 1, 2, , k, multiplying (3.13) by c˙m parts with respect to the time variable from to t, we have t (k) (k) (t) = pm (0) + pm ds (k) = pm (0) + I1 + I2 t μm (x, s)|∇vm(k) (x, s)|2 dx + Fm (s), v˙m(k) (s) ds (3.16) Linear Approximation and Asymptotic Expansion of Solutions By replacing wj in (3.13) by 143 wj , we obtain a(văm(k) (t), wj ) + a(μm (t)∇vm(k) (t), ∇wj ) = − Fm (t), wj , ≤ j ≤ k (3.17) (k) Similarly, by multiplying (3.17) by c˙m (t), we have qm(k) (t) = qm(k) (0) + t + ∂μm (0)∇v0k , v0k + Fm (0), v0k ∂x μm (x, s)| vm(k) (x, s)|2 dx ds 0 t ∂μm ∂ ∂μm (s)∇vm(k) (s) , vm(k) (s) ds − (t)∇vm(k) (t), vm(k) (t) ∂s ∂x ∂x +2 t − Fm (t), vm(k) (t) + ∂Fm (s), vm(k) (s) ds ∂t = qm(k) (0) + ∂μm (0)∇v0k , v0k + Fm (0), v0k + Ij ∂x j =3 (3.18) 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 K(M, μ), |μm (x, t)| ≤ 2ηM (3.19) where ηM = + M + M ∗ Hence, by (3.15) t I1 = ds μm (x, s)|∇vm(k) (x, s)|2 dx ≤ 2 η K(M, μ) μ0 M t (k) pm (s)ds (3.20) Second integral I2 From (3.5), (3.8), (3.10) and (3.15), we have t I2 = t Fm (s), v˙m(k) (s) ds ≤ 2K0 (M, f ) 0 t (k) pm (s)ds ≤ T K02 (M, f ) + (k) pm (s)ds (3.21) Third integral I3 By using and (3.15) and (3.19), we obtain t I3 = ds μm (x, s)| vm(k) (x, s)|2 dx ≤ 2 η K(M, μ) μ0 M t qm(k) (s)ds (3.22) Fourth integral I4 By the Cauchy-Schwartz inequality, we have t |I4 | = t ≤2 where I4∗ (s) = ∂ ∂μm (s)∇vm(k) (s) , vm(k) (s) ds ∂s ∂x ∂ ∂μm (s)∇vm(k) (s) ∂s ∂x ∂ ∂μm ( (s)∇vm(k) (s)) ∂s ∂x vm(k) (s) ds ≤ √ μ0 t I4∗ (s) qm(k) (s)ds, (3.23) 144 L.T.P Ngoc, N.T Long We shall estimate the term I4∗ (s) as follows We have I4∗ (s) = ∂ ∂μm (s)∇vm(k) (s) ∂s ∂x ≤ ∂μm (s) ∂x ≤ ∂μm (s) ∂x ≤ C0( ) C0( ) ∂μm (s) ∂x On the other hand, by plies that C0( ∂ μm ∂μm (s)∇ v˙m(k) (s) + (s)∇vm(k) (s) ∂x ∂x∂s ∇ v˙m(k) (s) + ∂ μm (s) ∂x∂s ∇vm(k) (s) ∇ v˙m(k) (s) + ∂ μm (s) ∂x∂s vm(k) (s) ∂ μm (s) +√ μ0 ∂x∂s ) ∂μm (x, s) ∂x ∂μm (s) ∂x = C0( ) qm(k) (s) (3.24) = D1 μ(ηm (x, s), zm (s))(∇vm−1 (x, s) + g(s)), we im- ≤ K(M, μ)( ∇vm−1 (s) C0( ) + M ∗) C0( ) ≤ K(M, μ)( vm−1 (s) + M ∗ ) ≤ K(M, μ)(M + M ∗ ) ≤ ηM K(M, μ) (3.25) Similarly, from the following equality ∂ μm (s) = D1 D1 μ(ηm (x, s), zm (s))(vm−1 (x, t) + ϕ (x, t))(∇vm−1 (x, s) + g(s)) ∂x∂s + D2 D1 μ(ηm (x, s), zm (s))(∇vm−1 (x, s) + g(s))zm (s) + D1 μ(ηm (x, s), zm (s))(∇vm−1 (x, s) + g (s)), (3.26) we obtain that ∂ μm (s) ≤ K(M, μ) vm−1 (s) + ϕ (s) ∂x∂s C0( ) ∇vm−1 (s) + g(s) + K(M, μ) ∇vm−1 (s) + g(s) |zm (s)| + K(M, μ) ∇vm−1 (s) + g (s) ≤ [1 + M + M ∗ + 2(M + M ∗ )2 ](M + M ∗ )K(M, μ) ≤ 2ηM K(M, μ) (3.27) It follows from (3.24), (3.25), (3.27), that 2 I4∗ (s) ≤ ηM + √ ηM K(M, μ) qm(k) (s) μ0 (3.28) Hence we obtain from (3.23) and (3.28), that 2 |I4 | ≤ √ ηM + √ ηM K(M, μ) μ0 μ0 t qm(k) (s)ds (3.29) Linear Approximation and Asymptotic Expansion of Solutions 145 Fifth integral I5 By the Cauchy-Schwartz inequality, we have |I5 | = − ≤ ∂μm ∂μm (t)∇vm(k) (t), vm(k) (t) ≤ (t)∇vm(k) (t) ∂x ∂x ∂μm (t)∇vm(k) (t) β ∂x vm(k) (t) +β vm(k) (t) , (3.30) for all β > On the other hand t ∂μm ∂μm (t)∇vm(k) (t) = (0)∇v0k + ∂x ∂x ∂μm (0) ∂x ≤ ∂ ∂μm (s)∇vm(k) (s) ds ∂s ∂x t ∇v0k + C0( ) I4∗ (s)ds (3.31) It follows from (3.28) and (3.31), that ∂μm ∂μm (t)∇vm(k) (t) ≤ (0) ∂x ∂x C0( ) 2 ∇v0k +ηM 1+ √ ηM K(M, μ) μ0 t qm(k) (s)ds (3.32) Hence we obtain from (3.30), (3.32), that |I5 | ≤ β (k) ∂μm (0) qm (t) + μ0 β ∂x + ∇v0k C0( ) 2 2 T K (M, μ)ηM + √ ηM β μ0 t qm(k) (s)ds, for all β > (3.33) for all β > (3.34) Sixth integral I6 By (3.10) and (3.15), we obtain |I6 | = | − Fm (t), vm(k) (t) | ≤ ≤ ≤ Fm (0) + β Fm (0) β t + Fm (t) β ∂Fm (s)ds ∂s T β T + +β vm(k) (t) β (k) q (t) μ0 m β (k) ∂Fm (s) ds + q (t), ∂s μ0 m It is known that ∂Fm (t) = D2 f [vm−1 ] + D3 f [vm−1 ]vm−1 (t) + D4 f [vm−1 ]∇vm−1 (t) + D5 f [vm−1 ]vm−1 (t) ∂t + 2D6 f [vm−1 ] ∇vm−1 (t) + g(t), ∇vm−1 (t) + g (t) , (3.35) where we have used the notations Di f [vm−1 ] = Di f (x, t, vm−1 (x, t), ∇vm−1 (x, t), vm−1 (x, t), ∇vm−1 (t) + g(t) ), i = 1, 2, , So, by (3.6), (3.8) and (3.35), we obtain 146 L.T.P Ngoc, N.T Long ∂Fm (t) ≤ K1 (M, f ) + vm−1 (t) + ∇vm−1 (t) + vm−1 (t) ∂t + ∇vm−1 (t) + g(t) ∇vm−1 (t) + g (t) ≤ K1 (M, f )(1 + 2M + 2(M + M ∗ )2 + vm−1 (t) ) + vm−1 (t) ) ≤ K1 (M, f )(2ηM (3.36) Hence, we deduce from (3.8), (3.34) and (3.36), that |I6 | ≤ ≤ Fm (0) β 2 Fm (0) β + + T K12 (M, f ) β T [4ηM + vm−1 (s) ]ds + β (k) T K12 (M, f )(4T ηM + M 2) + q (t), β μ0 m β (k) q (t) μ0 m for all β > (3.37) Seventh integral I7 By (3.6), (3.8), (3.10), (3.15) and (3.36), we obtain t ∂Fm (s), vm(k) (s) ds ∂t |I7 | = t ≤ t ∂Fm (s) ds + ∂t ∂Fm (s) ∂t √ ≤ K1 (M, f ) 2T ηM + vm(k) (s) ds 1/2 T vm−1 (s) ds T + K1 (M, f ) μ0 t (2ηM + vm−1 (s) )qm(k) (s)ds ≤ K1 (M, f )(2T ηM + √ T M) + K1 (M, f ) μ0 t (2ηM + vm−1 (s) )qm(k) (s)ds (3.38) t It remains to estimate văm(k) (s) ds Equation (3.13) can be rewritten as follows văm(k) (t), wj (m (t)vm(k) (t)), wj = Fm (t), wj , ∂x ≤ j ≤ k (3.39) ·· (k) Hence, It follows after replacing wj with v m (t) and integrating that t t văm(k) (s) ds 0 t ≤2 We estimate the term ∂ (μm (s)∇vm(k) (s)) ds + ∂x t Fm (s) ds ∂ (μm (s)∇vm(k) (s)) ds + 2T K02 (M, f ) ∂x ∂ (μm (s)∇vm(k) (s)) ∂x (3.40) Linear Approximation and Asymptotic Expansion of Solutions 155 p → We use the following notations For a multi-index α = (α1 , , αp ) ∈ Z+ , and − ε = p−1 (ε1 , , εp ) ∈ R+ × R , we put ⎧ α! = α1 ! αp !, |α| = α1 + · · · + αp , ⎪ ⎨ α α − → − → ε = ε12 + · · · + εp2 , ε α = ε1 εpp , ⎪ ⎩ p α, β ∈ Z+ , α ≤ β ⇐⇒ αi ≤ βi ∀i = 1, , p (4.1) First, we shall need the following lemma p Lemma 4.1 Let m, N ∈ N and uα ∈ R, α ∈ Z+ , ≤ |α| ≤ N Then → εα uα − 1≤|α|≤N m → ε α, TN(m) [u]α − = (4.2) m≤|α|≤mN p where the coefficients TN(m) [u]α , m ≤ |α| ≤ mN depending on u = (uα ), α ∈ Z+ , ≤ |α| ≤ N defined by the recurrence formulas ⎧ (1) T [u]α = uα , ≤ |α| ≤ N, ⎪ ⎪ ⎨ N(m) TN [u]α = β∈A(m) (N) uα−β TN(m−1) [u]β , m ≤ |α| ≤ mN, m ≥ 2, α ⎪ ⎪ ⎩ (m) p Aα (N ) = {β ∈ Z+ : β ≤ α, ≤ |α − β| ≤ N, m − ≤ |β| ≤ (m − 1)N } (4.3) The proof of Lemma 4.1 can 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 ), fi ∈ C N ([0, 1] × R+ × R3 ), i = 2, 3, , p We also use the notations f [u] = f (x, t, u, ux , ut ), μ[u] = μ(u, ux ) Let u0 be a unique weak solution of the problem (P0 ) (as in Theorem 3.2) corresponding → to − ε = (0, , 0), i.e., (P0 ) ⎧ ∂ (μ(u0 , u0x )u0x ) = f (x, t, u0 , u0x , u0 ) ≡ f [u0 ], u0 − ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ (0, 1), < t < T , ⎨ u0 (1, t) = 0, u0x (0, t) = g(t), ⎪ ⎪ ⎪ u0 (x, 0) = u1 (x), u0 (x, 0) = u0 (x), ⎪ ⎪ ⎪ ⎩ u0 − ϕ ∈ W1 (M, T ) p Let us consider the sequence of weak solutions uγ , γ ∈ Z+ , ≤ |γ | ≤ N, defined by the following problems: (Pγ ) ⎧ u − ∂ (μ(u0 , u0x )uγ x ) = Fγ , ⎪ ⎪ γ ∂x ⎪ ⎨ u (0, t) = u (1, t) = 0, γx γ ⎪ uγ (x, 0) = uγ (x, 0) = 0, ⎪ ⎪ ⎩ uγ ∈ W1 (M, T ), x ∈ (0, 1), < t < T , 156 L.T.P Ngoc, N.T Long p where Fγ , γ ∈ Z+ , ≤ |γ | ≤ N, are defined by the recurrence formulas ⎧ ⎪ ⎪ f [u0 ] ≡ f (x, t, u0 , u0x , u0 ), |γ | = 0, ⎨ p (i) Fγ = πγ [f ] + i=2 πγ [fi ] + 1≤|ν|≤|γ |, ν≤γ ⎪ ⎪ ⎩ ≤ |γ | ≤ N, ∂ [(ρν [μ] + ρν(1) [μ1 ])∇uγ −ν ], ∂x (4.4) with ρδ [μ] = ρδ [μ; {uγ }γ ≤δ ], ρδ(1) [μ] = ρδ(1) [μ; {uγ }γ ≤δ ], πδ [f ] = πδ [f ; {uγ }γ ≤δ ], πδ(i) [f ] = πδ(i) [f ; {uγ }γ ≤δ ], |δ| ≤ N, defined by the recurrence formulas ρδ [μ] = ⎧ ⎨ μ[u0 ], |δ| = 0, γ |γ |≤|δ| γ ! D μ[u0 ] ⎩ γ1 ≤|α|≤γ1 N, γ2 ≤|δ−α|≤2γ2 N (γ ) (γ ) TN [u]α T2N2 [σ ]δ−α , ≤ |δ| ≤ N, (4.5) p where σ = (σα ), α ∈ Z+ , ≤ |α| ≤ 2N, defined by ⎧ |α| = 1, ∇u0 , ∇uα , ⎪ ⎪ ⎨ σα = ∇u0 , ∇uα + β≤α ∇uβ , ∇uα−β , ≤ |α| ≤ N, ⎪ ⎪ ⎩ N + ≤ |α| ≤ 2N, β≤α ∇uβ , ∇uα−β , ⎧ p δ (1−) = (δ1 − 1, δ2 , , δp ), ⎪ ⎨ δ = (δ1 , δ2 , , δp ) ∈ Z+ , (1) ρδ [μ] = ρδ(1−) [μ] = ρδ1 −1,δ2 , ,δp [μ], ⎪ ⎩ (1) ρδ [μ] = ρ−1,δ2 , ,δp [μ] = 0, if δ1 = 0, πδ [f ] = ⎧ f [u0 ], ⎪ ⎨ ⎪ ⎩ (4.6) (4.7) |δ| = 0, 1≤|m|≤|δ| (m ) (m1 ) (m ) m [u]α TN [∇u]β TN [u ]γ , (α,β,γ )∈A(m,N) m! D f [u0 ]TN α+β+γ =δ ≤ |δ| ≤ N, (4.8) where m = (m1 , m2 , m3 ) ∈ Z3+ , |m| = m1 + m2 + m3 , m! = m1 !m2 !m3 !, D m f = m m m p D3 D4 D5 f, A(m, N ) = {(α, β, γ ) ∈ (Z+ )3 : m1 ≤ |α| ≤ m1 N, m2 ≤ |β| ≤ m2 N, m3 ≤ |γ | ≤ m3 N, }, ⎧ (i) ⎪ ⎨ πδ [f ] = πδ(i−) [f ] = πδ1 ,δ2 , ,δi−1 ,δi −1,δi+1 , ,δp [f ], i = 2, 3, , p, πδ(i) [f ] = πδ1 ,δ2 , ,δi−1 ,−1,δi+1 , ,δp [f ] = 0, if δi = 0, ⎪ ⎩ p δ = (δ1 , δ2 , , δp ) ∈ Z+ , δ (i−) = (δ1 , , δi−1 , δi − 1, δi+1 , , δp ) (4.9) Then, we have the following lemma Lemma 4.2 Let ρν [μ], πν [f ], |ν| ≤ N, be the functions are defined by the formulas (4.5) → ε γ , then we have and (4.8) Put h = |γ |≤N uγ − μ[h] = → → ρν [μ]− εν+ − ε N+1 → RN(1) [μ, − ε ], (4.10) → → πν [f ]− εν+ − ε N+1 → RN(1) [f, − ε ], (4.11) |ν|≤N f [h] = |ν|≤N Linear Approximation and Asymptotic Expansion of Solutions → → where RN(1) [μ, − ε ] L∞ (0,T ;L2 ) + RN(1) [f, − ε] ing only on N, T , f, μ, uγ , |γ | ≤ N L∞ (0,T ;L2 ) 157 ≤ C, with C is a constant depend- Proof (i) In the case of N = 1, the proof of (4.10) is easy, hence we omit the details, which → ε α ≡ u0 + h1 , we rewritten we only prove with N ≥ Put h = u0 + 1≤|α|≤N uα − μ[h] = μ(h, ∇h ) = μ(u0 + h1 , ∇u0 + ∇h1 ) = μ(u0 + h1 , ∇u0 where ξ = ∇u0 + ∇h1 − ∇u0 By using Taylor’s expansion of the function μ(u0 + h1 , ∇u0 (u0 , ∇u0 ) up to order N + 1, we obtain μ(u0 + h1 , ∇u0 2 + ξ ), (4.12) + ξ ) around the point + ξ) = μ(u0 , ∇u0 ) + 1≤|γ |≤N = μ[u0 ] + 1≤|γ |≤N γ γ D μ(u0 , ∇u0 )h11 ξ γ2 + RN (μ, u0 , h1 , ξ ) γ! γ γ D μ[u0 ]h11 ξ γ2 + RN [μ, u0 , h1 , ξ ], γ! (4.13) where RN [μ, u0 , h1 , ξ ] = N +1 (1 − θ )N D γ μ(u0 + θ h1 , ∇u0 γ ! |γ |=N+1 → = − ε N+1 γ + θ ξ )h11 ξ γ2 dθ RN(1) [μ, u0 , h1 , ξ ] (4.14) By the formula (4.2), we obtain → εα uα − γ h1 = γ1 → ε α, TN [u]α − (γ ) = (4.15) γ1 ≤|α|≤γ1 N 1≤|α|≤N p where u = (uα ), α ∈ Z+ , ≤ |α| ≤ N On the other hand, we have also ξ = ∇u0 + ∇h1 − ∇u0 = ∇u0 , ∇h1 + ∇h1 → ε α, σα − ≡ (4.16) 1≤|α|≤2N with σα , ≤ |α| ≤ 2N are defined by (4.6) We again use formula (4.2), it follows from (4.16), that → εα σα − ξ γ2 = 1≤|α|≤2N p where σ = (σα ), α ∈ Z+ , ≤ |α| ≤ 2N γ2 → ε α, T2N2 [σ ]α − (γ ) = γ2 ≤|α|≤2γ2 N (4.17) 158 L.T.P Ngoc, N.T Long Hence, it follows from (4.14) and (4.15), that → εα TN [u]α − γ → εα T2N2 [σ ]α − (γ ) h1 ξ γ = γ1 ≤|α|≤γ1 N (γ ) γ2 ≤|α|≤2γ2 N → ε α+β TN [u]α T2N2 [σ ]β − (γ ) = (γ ) γ1 ≤|α|≤γ1 N γ2 ≤|α|≤2γ2 N → ε α+β TN [u]α T2N2 [σ ]β − (γ ) = (γ ) γ1 ≤|α|≤γ1 N, γ2 ≤|β|≤2γ2 N → εδ TN [u]α T2N2 [σ ]β − (γ ) = (γ ) γ1 +γ2 ≤|δ|≤(γ1 +2γ2 )N γ1 ≤|α|≤γ1 N, γ2 ≤|β|≤2γ2 N, α+β=δ (γ ) (γ ) → εδ TN [u]α T2N2 [σ ]δ−α − = γ1 +γ2 ≤|δ|≤(γ1 +2γ2 )N γ1 ≤|α|≤γ1 N, γ2 ≤|δ−α|≤2γ2 N − → εδ = δ [α, γ1 , γ2 , N, u, σ ] γ1 +γ2 ≤|δ|≤(γ1 +2γ2 )N − → εδ+ = − → εδ δ [α, γ1 , γ2 , N, u, σ ] γ1 +γ2 ≤|δ|≤N δ [α, γ1 , γ2 , N, u, σ ] N+1≤|δ|≤(γ1 +2γ2 )N − → → εδ+ − ε = δ [α, γ1 , γ2 , N, u, σ ] N+1 → RN [α, γ1 , γ2 , N, u, σ, − ε ], (4.18) |γ |≤|δ|≤N where ⎧ ⎨ δ [α, γ1 , γ2 , N, u, σ ] = ⎩ − → ε N+1 (γ ) γ1 ≤|α|≤γ1 N, γ2 ≤|δ−α|≤2γ2 N → RN [α, γ1 , γ2 , N, u, σ, − ε ]= (γ ) TN [u]α T2N2 [σ ]δ−α , N+1≤|δ|≤(γ1 +2γ2 )N − → ε δ (4.19) δ [α, γ1 , γ2 , N, u, σ ] Hence, we deduce from (4.13), (4.14), (4.18), (4.19), that μ[h] ≡ μ(h, ∇h ) = μ(u0 + h1 , ∇u0 = μ[u0 ] + 1≤|γ |≤N = μ[u0 ] + 1≤|γ |≤N → + − ε γ D μ[u0 ] γ! γ 1≤|γ |≤N 1≤|γ |≤N → + − ε N+1 + ξ) γ γ → D μ[u0 ]h11 ξ γ2 + − ε γ! N+1 = μ[u0 ] + N+1 RN(1) [μ, u0 , h1 , ξ ] − → εδ δ [α, γ1 , γ2 , N, u, σ ] +γ2 ≤|δ|≤N γ → D μ[u0 ]RN [α, γ1 , γ2 , N, u, σ, − ε ] + RN(1) [μ, u0 , h1 , ξ ] γ! γ D μ[u0 ] γ! |γ |≤|δ|≤N RN(1) (μ, u0 , h1 , ξ ) − → εδ δ [α, γ1 , γ2 , N, u, σ ] Linear Approximation and Asymptotic Expansion of Solutions = μ[u0 ] + → + − ε γ D μ[u0 ] γ! 1≤|δ|≤N |γ |≤|δ| N+1 − → εδ δ [α, γ1 , γ2 , N, u, σ ] RN(1) (μ, u0 , h1 , ξ ) → → ρδ [μ]− εδ+ − ε = 159 N+1 RN(1) (μ, u0 , h1 , ξ ), (4.20) |δ|≤N p where ρδ [μ], δ ∈ Z+ , |δ| ≤ N, are defined by (4.5), (4.6) and RN(1) (μ, u0 , h1 , ξ ) = 1≤|γ |≤N γ → D μ[u0 ]RN [α, γ1 , γ2 , N, u, σ, − ε ] + RN(1) [μ, u0 , h1 , ξ ] γ! (4.21) By the boundedness of the functions uγ , ∇uγ , uγ , |γ | ≤ N in the function space L∞ (0, T ; H ), we obtain from (4.14)–(4.17), (4.19), (4.21), that RN(1) (μ, u0 , h1 , ξ ) L∞ (0,T ;L2 ) ≤ C, with and C is a constant depending only on N, T , μ, uγ , |γ | ≤ N Hence, the part of Lemma 4.2 is proved (ii) We only prove (4.11) with N ≥ By using Taylor’s expansion of the function f [u0 + h1 ] around the point u0 up to order N + 1, we obtain from (4.2), that f [u0 + h1 ] = f [u0 ] + D3 f [u0 ]h1 + D4 f [u0 ]∇h1 + D5 f [u0 ]h1 + 2≤|m|≤N m=(m1 ,m2 ,m3 )∈Z3+ m m D f [u0 ]h1 (∇h1 )m2 (h1 )m3 + RN(1) [f, h1 ] m! = f [u0 ] + D3 f [u0 ]h1 + D4 f [u0 ]∇h1 + D5 f [u0 ]h1 + 2≤|m|≤N m=(m1 ,m2 ,m3 )∈Z3+ m D f [u0 ] m! (m1 ) × TN (m2 ) [u]α TN (m3 ) [∇u]β TN → εν [u ]γ − |m|≤|ν|≤|m|N (α,β,γ )∈A(m,N) α+β+γ =ν + RN(1) [f, h1 ] = f [u0 ] + D3 f [u0 ]h1 + D4 f [u0 ]∇h1 + D5 f [u0 ]h1 + 2≤|m|≤N m=(m1 ,m2 ,m3 )∈Z3+ m D f [u0 ] m! (m1 ) × TN |m|≤|ν|≤N (α,β,γ )∈A(m,N) α+β+γ =ν + 2≤|m|≤N m=(m1 ,m2 ,m3 )∈Z3+ (m2 ) [u]α TN m D f [u0 ] m! (m3 ) [∇u]β TN → εν [u ]γ − 160 L.T.P Ngoc, N.T Long (m1 ) × TN (m2 ) [u]α TN (m3 ) [∇u]β TN → εν [u ]γ − N+1≤|ν|≤|m|N (α,β,γ )∈A(m,N) α+β+γ =ν + RN(1) [f, h1 ], (4.22) where RN(1) [f, h1 ] = |m|=N+1 m=(m1 ,m2 ,m3 )∈Z3+ N +1 m! m (1 − θ )N D m f [u0 + θ h1 ]h1 (∇h1 )m2 (h1 )m3 dθ (4.23) We also note that f [u0 ] + D3 f [u0 ]h1 + D4 f [u0 ]∇h1 + D5 f [u0 ]h1 + 2≤|m|≤N m=(m1 ,m2 ,m3 )∈Z3+ m D f [u0 ] m! |m|≤|ν|≤N (m1 ) × TN (m2 ) [u]α TN (m3 ) [∇u]β TN → εν [u ]γ − (α,β,γ )∈A(m,N) α+β+γ =ν = f [u0 ] + 1≤|m|≤N m=(m1 ,m2 ,m3 )∈Z3+ m D f [u0 ] m! (m1 ) × TN (m2 ) [u]α TN (m3 ) [∇u]β TN → εν [u ]γ − |m|≤|ν|≤N (α,β,γ )∈A(m,N) α+β+γ =ν = f [u0 ] + 1≤|ν|≤N 1≤|m|≤|ν| m=(m1 ,m2 ,m3 )∈Z3+ (m1 ) × TN (m2 ) [u]α TN m D f [u0 ] m! (m3 ) [∇u]β TN → εν [u ]γ − (α,β,γ )∈A(m,N) α+β+γ =ν → πν [μ]− ε ν = (4.24) |ν|≤N Similarly, we have also 2≤|m|≤N m=(m1 ,m2 ,m3 )∈Z3+ m (m ) (m ) (m ) → εν D f [u0 ] TN [u]α TN [∇u]β TN [u ]γ − m! N+1≤|ν|≤|m|N (α,β,γ )∈A(m,N) → + RN(1) [f, h1 ] = − ε α+β+γ =ν N+1 → RN(1) [f, − ε ], (4.25) → where RN(1) [f, − ε ] L∞ (0,T ;L2 ) ≤ C, with C is a constant depending only on N, T , f, uγ , |γ | ≤ N Then (4.11) holds The Lemma 4.2 is proved Linear Approximation and Asymptotic Expansion of Solutions 161 Remark Lemma 4.2 is a generalization of a formula contained in ([13], p 262, formula (4.38)) and it is useful to obtain the following Lemma 4.3 These Lemmas are the key to the asymptotic expansion of a weak solution u = u(ε1 , , εp ) of order N + in p small parameters ε1 , , εp as it will be said below → → Let u− ε = u(ε1 , , εp ) ∈ W1 (M, T ) be a unique weak solution of the problem (P− ε ) − →γ ≡ u − → → Then v = u− ε − ε − h satisfies the problem |γ |≤N uγ ε ⎧ ∂ (με1 [v + h]vx ) v − ∂x ⎪ ⎪ ⎪ ⎪ ∂ ⎪ → → = F− ⎪ ε [v + h] − F− ε [h] + ∂x [(με1 [v + h] − με1 [h])hx ] ⎪ ⎪ ⎪ ⎪ → x ∈ (0, 1), < t < T , + E− ⎪ ε (x, t), ⎨ vx (0, t) = v(1, t) = 0, ⎪ ⎪ ⎪ v(x, 0) = v (x, 0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ με1 [v] = μ[v] + ε1 μ1 [v] = μ(v, ∇v ) + ε1 μ1 (v, ∇v ), ⎪ ⎪ ⎩ p p → F− ε [v] = f [v] + i=2 εi fi [v] = f (x, t, v, vx , vt ) + i=2 εi fi (x, t, v, vx , vt ), (4.26) where p → E− ε (x, t) = f [h] − f [u0 ] + εi fi [h] + i=2 ∂ → ε γ [(μ[h] − μ[u0 ] + ε1 μ1 [h])hx ] − Fγ − ∂x 1≤|γ |≤N (4.27) Then, we have the following lemma Lemma 4.3 Let (H1 ), (H2 ), (H7 ), and (H8 ) hold Then there exists a constant K such that → E− ε L∞ (0,T ;L2 ) → ≤K − ε N+1 (4.28) , where K is a constant depending only on N, T , f, f1 , μ, μ1 , uγ , |γ | ≤ N Proof In the case of N = 1, the proof of Lemma 4.3 is easy, hence we omit the details, which we only prove with N ≥ By using the formulas (4.10), (4.11) for the functions μ1 [h] and fi [h], we obtain μ1 [h] = fi [h] = (1) − → → → εν+ − ε N RN−1 [μ1 , − ε ], − → − → − → ν N (1) RN−1 [fi , ε ], ≤ i ≤ p |ν|≤N−1 πν [fi ] ε + ε |ν|≤N−1 ρν [μ1 ] (4.29) By (4.7), (4.29)1 , we rewrite ε1 μ1 (h) as follows → → ρν1 −1,ν2 , ,νp [μ1 ]− ε ν + ε1 − ε ε1 μ1 [h] = N (1) → RN−1 [μ1 , − ε] 1≤|ν|≤N, ν1 ≥1 → → ρν(1) [μ1 ]− ε ν + ε1 − ε = N (1) → RN−1 [μ1 , − ε] 1≤|ν|≤N, ν1 ≥1 → → ρν(1) [μ1 ]− ε ν + ε1 − ε = 1≤|ν|≤N N (1) → RN−1 [μ1 , − ε ] (4.30) 162 L.T.P Ngoc, N.T Long Similarly, with fi [h], we also obtain → → ε ν + εi − πν [fi ]εi − ε εi fi [h] = N (1) → RN−1 [fi , − ε] |ν|≤N−1 → → πν1 ,ν2 , ,νi−1 ,νi −1,νi+1 , ,νp [fi ]− ε ν + εi − ε = N (1) → RN−1 [fi , − ε] 1≤|ν|≤N, νi ≥1 → → πν(i) [fi ]− ε ν + εi − ε = N (1) → RN−1 [fi , − ε] 1≤|ν|≤N, νi ≥1 → → πν(i) [fi ]− ε ν + εi − ε = N (1) → RN−1 [fi , − ε ] (4.31) 1≤|ν|≤N First, we deduce from (4.11) and (4.31), that p p f [h] − f [u0 ] + εi fi [h] = 1≤|ν|≤N i=2 → εν πν(i) [fi ] − πν [f ] + → + − ε i=2 N+1 → RN(1) [f, f2 , , fp , − ε ], (4.32) p (1) εi → → → ε ] = RN(1) [f, − ε ] + i=2 − RN−1 [fi , − ε ] is bounded in the where RN(1) [f, f2 , , fp , − → ε ∞ function space L (0, T ; L ) by a constant depending only on N, T , f, f1 , uγ , |γ | ≤ N On the other hand, we deduce from (4.10) and (4.29)1 , that (μ[h] − μ[u0 ] + ε1 μ1 [h])hx → εγ (ρν [μ] + ρν(1) [μ1 ])∇uγ −ν − = 1≤|γ |≤2N 1≤|ν|≤N, |γ −ν|≤N, ν≤γ → + − ε N+1 → RN(1) [μ, μ1 , − ε ], (4.33) where ε1 (1) → → → RN(1) [μ, μ1 , − ε ] = RN(1) [μ, − ε ]+ − [μ1 , − ε] RN−1 → ε → ε α ∇uα − (4.34) |α|≤N We decompose the sum 1≤|γ |≤2N into the sum of two the sums Therefore, we deduce from (4.33), (4.34), that N+1≤|γ |≤2N 1≤|γ |≤2N and → εγ (ρν [μ] + ρν(1) [μ1 ])∇uγ −ν − (μ[h] − μ[u0 ] + ε1 μ1 [h])hx = 1≤|γ |≤N 1≤|ν|≤|γ |, ν≤γ → + − ε N+1 → RN(2) [μ, μ1 , − ε ], (4.35) where − → ε N+1 → = − ε → ε] RN(2) [μ, μ1 , − N+1 → RN(1) [μ, μ1 , − ε] → ε γ (ρν [μ] + ρν(1) [μ1 ])∇uγ −ν − + N+1≤|γ |≤2N 1≤|ν|≤N, |γ −ν|≤N, ν≤γ (4.36) Linear Approximation and Asymptotic Expansion of Solutions 163 Hence ∂ [(μ[h] − μ[u0 ] + ε1 μ1 [h])hx ] ∂x ∂ → = [(ρν [μ] + ρν(1) [μ1 ])∇uγ −ν ]− εγ ∂x 1≤|γ |≤N 1≤|ν|≤|γ |, ν≤γ → + − ε N+1 ∂ (2) → R [μ, μ1 , − ε ] ∂x N (4.37) Combining (4.4), (4.5), (4.8), (4.27), (4.32) and (4.37), we then obtain → E− ε (x, t) p = f [h] − f [u0 ] + εi fi [h] + i=2 ∂ → εγ [(μ[h] − μ[u0 ] + ε1 μ1 [h])hx ] − Fγ − ∂x 1≤|γ |≤N p = πγ [f ] + 1≤|γ |≤N πγ(i) [fi ] + 1≤|ν|≤|γ |, ν≤γ i=2 → → εγ+ − Fγ − ε − N+1 ∂ → [(ρν [μ] + ρν(1) [μ1 ])∇uγ −ν ] − εγ ∂x → ε ]+ RN(1) [f, f2 , , fp , − 1≤|γ |≤N → = − ε N+1 → ε ]+ RN(1) [f, f2 , , fp , − ∂ (2) → R [μ, μ1 , − ε] ∂x N ∂ (2) → R [μ, μ1 , − ε] ∂x N (4.38) By the boundedness of the functions uγ , ∇uγ , uγ , |γ | ≤ N in the function space L∞ (0, T ; H ), we obtain from (4.32) and (4.36), that → E− ε L∞ (0,T ;L2 ) → ≤K − ε N+1 , (4.39) where K is a constant depending only on N, T , f, f1 , μ, μ1 , uγ , |γ | ≤ N The proof of Lemma 4.3 is complete Now, we consider the sequence of functions {vm } defined by ⎧ v0 ≡ 0, ⎪ ⎪ ⎪ ⎪ ∂ ⎪ (με1 [vm−1 + h]vmx ) vm − ∂x ⎪ ⎪ ⎪ ⎪ ⎪ − → → = F ⎪ ε [vm−1 + h] − F− ε [h] ⎪ ⎨ ∂ + ∂x [(με1 [vm−1 + h] − με1 [h])hx ] ⎪ ⎪ ⎪ ⎪ → x ∈ (0, 1), < t < T , + E− ⎪ ε (x, t), ⎪ ⎪ ⎪ ⎪ ⎪ vmx (0, t) = vm (1, t) = 0, ⎪ ⎪ ⎩ vm (x, 0) = vm (x, 0) = 0, m ≥ With m = 1, we have the problem ⎧ ∂ → ⎪ ε (x, t), ⎨ v1 − ∂x (με1 [h]v1x ) = E− v1x (0, t) = v1 (1, t) = 0, ⎪ ⎩ v1 (x, 0) = v1 (x, 0) = (4.40) x ∈ (0, 1), < t < T , (4.41) 164 L.T.P Ngoc, N.T Long By multiplying the two sides of (4.41) by v1 , we find without difficulty from (4.28) that v1 (t) + μ1,ε1 (t)v1x (t) t =2 t → E− ε (s), v1 (s) ds + μ1,ε1 (x, s)v1x (x, s)dx ds 0 t → ≤ T K2 − ε 2N+2 t + v1 (s) ds + ds 0 |μ1,ε1 (x, s)|v1x (x, s)dx, (4.42) where μ1,ε1 (x, t) = με1 [h(x, t)] By μ1,ε1 (x, t) = D1 με1 (h(x, t), ∇h(t) )h (x, t) + 2D2 με1 (h(x, t), ∇h(t) ) ∇h(t), ∇h (t) , (4.43) we have |μ1,ε1 (x, t)| ≤ (N + 1)M[1 + 2(N + 1)M](K(M∗ , μ) + K1 (M∗ , μ1 )) ≡ ζ0 , (4.44) with M∗ = (N + 2)M It follows from (4.42), (4.44), that v1 (t) + μ0 v1x (t) → ≤ T K2 − ε t 2N+2 + t v1 (s) ds + ζ0 v1x (s) ds (4.45) Using Gronwall’s lemma we obtain from (4.45), that v1 W1 (T ) = v1 L∞ (0,T ;L2 ) + v1 L∞ (0,T ;V ) √ → ≤ 1+ √ TK − ε μ0 N+1 exp T ζ0 + 2μ0 (4.46) → ε , such that We shall prove that there exists a constant CT , independent of m and − vm W1 (T ) → ≤ CT − ε N+1 , → with − ε ≤ ε ∗ , for all m (4.47) By multiplying the two sides of (4.40) with vm and after integration in t, we obtain without difficulty from (4.28) that vm (t) + μ0 vmx (t) → ≤ T K2 − ε t 2N+2 + vm (s) ds t + ds 0 |μm,ε1 (x, s)|vmx (x, s)dx t +2 → → F− ε [vm−1 + h] − F− ε [h] vm (s) ds t +2 → ε = T K2 − ∂ [(με1 [vm−1 + h] − με1 [h])hx ] ∂x vm (s) ds t 2N+2 + vm (s) ds + I1 (t) + I2 (t) + I3 (t), (4.48) Linear Approximation and Asymptotic Expansion of Solutions 165 where μm,ε1 (t) = με1 [vm−1 + h] We now estimate the integrals on the right hand of (4.48) as follows Estimating I1 (t) We have μm,ε1 (x, t) = D1 με1 [vm−1 + h](vm−1 + h ) + 2D2 με1 [vm−1 + h] ∇vm−1 + ∇h(t), ∇vm−1 + ∇h (t) , (4.49) hence |μm,ε1 (x, t)| ≤ M∗ (1 + 2M∗ )(K(M∗ , μ) + K1 (M∗ , μ1 )) ≡ ζ1 , with M∗ = (N + 2)M (4.50) It follows from (4.50), that t I1 (t) = ds 0 t |μm,ε1 (x, s)|vmx (x, s)dx ≤ ζ1 vmx (s) ds (4.51) Estimating I2 (t) We also note that f [vm−1 + h] − f [h] ≤ 2K1 (M∗ , f ) vm−1 W1 (T ) , and fi [vm−1 + h] − f1 [h] ≤ 2K1 (M∗ , fi ) vm−1 W1 (T ) , hence, we have → → F− ε [vm−1 + h] − F− ε [h] ≤ ζ2 vm−1 where ζ2 = ζ2 (M∗ , f, f1 ) = 2K1 (M∗ , f ) + from (4.52), that (4.52) W1 (T ) , p i=2 Ki (M∗ , f1 ) Therefore, we deduce t t I2 (t) = 2 → → F− ε [vm−1 + h] − F− ε [h] vm (s) ds ≤ T ζ2 vm−1 Estimating I3 (t) First, we need an estimation From the equation ∂ [(μ[vm−1 ∂x W1 (T ) + vm (s) ds (4.53) + h] − μ[h])hx ] ∂ ∂ [(μ[vm−1 + h] − μ[h])hx ] = (μ[vm−1 + h] − μ[h])hxx + (μ[vm−1 + h] − μ[h])hx , ∂x ∂x it follows that ∂ [(μ[vm−1 + h] − μ[h])hx ] ∂x ≤ μ[vm−1 + h] − μ[h] + ≤ ≡ √ hxx (s) ∂ (μ[vm−1 + h] − μ[h]) ∂x h(s) √ C0( ) h(s) H2 hx (s) μ[vm−1 + h] − μ[h] (1) (2) H (I3 (s) + I3 (s)) C0( ) C0( ) + ∂ (μ[vm−1 + h] − μ[h]) ∂x (4.54) 166 L.T.P Ngoc, N.T Long With I3(1) (s) we have I3(1) (s) = μ[vm−1 + h] − μ[h] C0( ) ≤ (1 + 2M∗ )K(M∗ , μ) vm−1 W1 (T ) (4.55) With I3(2) (s) we also obtain I3(2) (s) = ∂ (μ[vm−1 + h] − μ[h]) ∂x ≤ D1 μ[vm−1 + h]∇vm−1 + (D1 μ[vm−1 + h] − D1 μ[h])∇h ≤ (1 + M∗ + 2M∗2 )K(M∗ , μ) vm−1 (4.56) W1 (T ) We deduce from (4.54), (4.55), and (4.56), that √ ∂ ([μ(vm−1 + h) − μ(h)]hx ) ≤ 2(2 + 3M∗ + 2M∗2 )M∗ K(M∗ , μ) vm−1 ∂x W1 (T ) (4.57) Next, by με1 = μ + ε1 μ1 , it follows that ∂ [(με1 [vm−1 + h] − με1 [h])hx ] ≤ ζ3 vm−1 ∂x (4.58) W1 (T ) , where ζ3 = ζ3 (M, N, T , μ, μ1 ) = √ 2(2 + 3M∗ + 2M∗2 )M∗ (K(M∗ , μ) + K(M∗ , μ1 )) (4.59) By (4.58), we obtain t I3 (t) = ∂ [(με1 [vm−1 + h] − με1 [h])hx ] ∂x vm (s) ds t ≤ T ζ32 vm−1 W1 (T ) + vm (s) ds (4.60) Combining (4.48), (4.51), (4.53), (4.60), we then obtain vm (t) + μ0 vmx (t) → ≤ T K2 − ε 2N+2 + T (ζ22 + ζ32 ) vm−1 t +3 W1 (T ) t vm (s) ds + ζ1 vmx (s) ds (4.61) By using Gronwall’s lemma we deduce from (4.61) that vm W1 (T ) ≤ σT vm−1 W1 (T ) + δ, for all m ≥ 1, (4.62) where σT = ζ22 + ζ32 ηT , → ε δ = ηT K − ζ1 + exp T ηT = + √ μ0 2μ0 N+1 √ , (4.63) T Linear Approximation and Asymptotic Expansion of Solutions 167 We assume that σT < 1, with the suitable constant T > (4.64) We shall now require the following lemma whose proof is immediate Lemma 4.4 Let the sequence {ψm } satisfy ψm ≤ σ ψm−1 + δ for all m ≥ 1, ψ0 = 0, (4.65) where ≤ σ < 1, δ ≥ are the given constants Then ψm ≤ δ/(1 − σ ) Applying Lemma 4.4 with ψm = vm → ε N+1 , it follows from (4.66), that ηT K − vm where CT = 1− √ηT2K ζ2 +ζ32 ηT W1 (T ) for all m ≥ W1 (T ) , (4.66) σ = σT = → ≤ δ/(1 − σT ) = CT − ε N+1 ζ22 + ζ32 ηT < 1, δ = (4.67) , On the other hand, the linear recurrent sequence {vm } 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 → ≤C − ε N+1 , v W1 (T ) T or → εγ uγ − → u− ε − |γ |≤N → ≤ CT − ε N+1 (4.68) W1 (T ) Thus, we have the following theorem Theorem 4.5 Let (H1 ), (H2 ), (H7 ), and (H8 ) hold Then there exist constants M > and → → → T > such that, for every − ε , with − ε ≤ ε∗ , the problem (P− ε ) has a unique weak solution → ∈ W (M, T ) satisfying an asymptotic estimation up to 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Cavalcanti et al [2, 3], Ebihara, Medeiros and Miranda [5], Hosoya and Yamada [6], Lasiecka and Ong [9], Miranda et al [22], Menzala [21], Park et al [25, 26], Rabello et al [28], Santos et al... mathematical aspects of the mixed problems associated to the Linear Approximation and Asymptotic Expansion of Solutions 139 operator Kirchhoff or the operator Kirchhoff Carrier, such as existence of. .. case of asymptotic expansion in many small parameters, there is only partial results, for example, [15, 17, 18, 24] concerning asymptotic expansion of the solution in two or three small parameters