On a nonlinear wave equation with a nonlocal boundary condition

345 ACTA MATHEMATICA VIETNAMICA Volume 36, Number 2, 2011, pp 345–374 ON A NONLINEAR WAVE EQUATION WITH A NONLOCAL BOUNDARY CONDITION LE THI PHUONG NGOC, TRAN MINH THUYET, PHAM THANH SON, NGUYEN THANH LONG Dedicated to Tran Duc Van on the occasion of his sixtieth birthday Abstract Consider the initial-boundary value problem for the nonlinear wave equation  utt − µ(t)uxx + K|u|p−2 u + λ|ut |q−2 ut = F (x, t), < x < 1, < t < T,    t   µ(t)ux (0, t) = K0 u(0, t) + k (t − s) u (0, s) ds + g(t),   −µ(t)ux (1, t) = K1 u(1, t) + λ1 |ut (1, t)|α−2 ut (1, t),    u(x, 0) = u0 (x), ut (x, 0) = u1 (x), where p, q, α ≥ 2; K0 , K1 , K ≥ 0; λ, λ1 > are given constants and µ, F, g, k, u0 , u1 , are given functions First, the existence and uniqueness of a weak solution are proved by using the Galerkin method Next, with α = 2, we obtain an asymptotic expansion of the solution up to order N in two small parameters λ, λ1 with error λ2 + λ21 N+ Introduction In this paper, we consider the following initial-boundary value problem (1.1) utt − µ(t)uxx + f (u, ut ) = F (x, t), < x < 1, < t < T, t (1.2) µ(t)ux (0, t) = K0 u(0, t) + k (t − s) u (0, s) ds + g(t), (1.3) −µ(t)ux (1, t) = K1 u(1, t) + λ1 |ut (1, t)|α−2 ut (1, t), (1.4) u(x, 0) = u0 (x), ut (x, 0) = u1 (x), where f (u, ut ) = K|u|p−2 u + λ|ut |q−2 ut , with K, K0 , K1 , λ, λ1 , p, q and α being given constants and u0 , u1 , g, k, µ, F being given functions satisfying conditions specified later Received October 8, 2010 2000 Mathematics Subject Classification 35L20, 35L70, 35Q72 Key words and phrases Galerkin method, a priori estimates, asymptotic expansion of the solution up to order N 346 L T P NGOC, T M THUYET, P T SON AND N T LONG It is well known that many various problems in the form (1.1)–(1.4) have been investigated For example, we refer to Cavalcanti et al [4], Long, Dinh and Diem [9], Ngoc, Hang and Long [10], Qin [11, 12], Rivera [13], Santos [14], and the references therein In these works, many interesting results about the unique existence, regularity, stability, asymptotic expansion or the decay of solutions are obtained Clearly, the boundary condition (1.2) is nonlocal and if we put t Y (t) = g(t) + K0 u(0, t) + k (t − s) u (0, s) ds, then (1.2) is written as follows (1.5) µ(t)ux (0, t) = Y (t), and problem (1.1)–(1.4) can be reduced to problem (1.1), (1.3)-(1.5), in which an unknown function u(x, t) and an unknown boundary value Y (t) satisfy the following Cauchy problem for the ordinary differential equation (1.6) Y ′′ (t) + γ1 Y ′ (t) + γ2 Y (t) = γ3 utt (0, t), < t < T, Y (0) = Y0 , Y ′ (0) = Y1 , where γ1 , γ2 , γ3 , Y0 , Y1 are certain constants such that γ12 − 4γ2 < In [1], An and Trieu studied a special case of problem (1.1), (1.4)-(1.6) associated with the following homogeneous boundary condition at x = : u(1, t) = 0, (1.7) with µ(t) ≡ 1, F = u0 = u1 = Y0 = γ1 = 0, and f (u, ut ) = Ku + λut , with γ3 , K ≥ 0, λ ≥ being given constants In the latter case, problem (1.1), (1.4)(1.7) is a mathematical model describing the shock of a rigid body and a linear viscoelastic bar resting on a rigid base [1] We note more that from (1.6), representing Y (t) in terms of γ1 , γ2 , γ3 , Y0 , Y1 , utt (0, t) and then integrating by parts, we shall obtain Y (t) as below t Y (t) = g(t) + K0 u(0, t) + k (t − s) u (0, s) ds, where g(t) = Y0 − γ3 u0 (0) e−γt cos ωt γ 1 Y1 + γ3 u0 (0) − γ3 u1 (0) e−γt sin ωt, γ ω ω sin ωt −γt e , −2γ cos ωt + (γ − ω ) ω + − Y0 − k (t) = γ3 with γ = 12 γ1 , ω = 21 4γ2 − γ12 , K0 = γ3 Therefore, problem (1.1), (1.3)-(1.5) leads to problem (1.1)-(1.4) ON A NONLINEAR WAVE EQUATION 347 The paper consists of three sections In Section 2, under the conditions (u0 , u1 ) ∈ H × H ; F, F ′ ∈ L1 (0, T ; L2 ); g, k, µ ∈ W 2,1 (0, T ) , µ(t) ≥ µ0 > 0; p, q, α ≥ 2; K, K0 , K1 ≥ 0; λ, λ1 > and some other conditions, we prove a theorem of global existence and uniqueness of a weak solution u of problem (1.1)-(1.4) The proof is based on the Galerkin method associated to a priori estimates, the weak convergence and the compactness techniques Finally, in Section 3, with α = 2, we obtain an asymptotic expansion of the solution u up to N + 21 The results λ2 + λ21 order N in two small parameters λ, λ1 with error obtained here may be considered as the generalizations of those in An and Trieu [1] and in [2, 6, 7, 9, 10] Existence and uniqueness of a weak solution First, put Ω = (0, 1), QT = Ω × (0, T ), T > and denote the usual function spaces used in this paper by the notations C m Ω , W m,p = W m,p (Ω) , Lp = W 0,p (Ω) , H m = W m,2 (Ω) , ≤ p ≤ ∞, m = 0, 1, Let ·, · be either the scalar product in L2 or the dual pairing of a continuous linear functional and an element of a function space The notation || · || stands for the norm in L2 and we denote by || · ||X the norm in the Banach space X We call X ′ the dual space of X We denote by Lp (0, T ; X), ≤ p ≤ ∞ for the Banach space of real functions u : (0, T ) → X measurable, such that ||u||Lp (0,T ;X) < +∞, with  1/p T      u(t) pX dt , if ≤ p < ∞, (2.1) u Lp (0,T ;X) =     if p = ∞  ess sup u(t) X , u′ (t) 0 Finally, let us note more that the weak solution u of the initial and boundary value problem (1.1)-(1.4) will be obtained in Theorem 2.2 in the following manner: Find u ∈ W = {u ∈ L∞ (0, T ; H ), ut ∈ L∞ (0, T ; H ), utt ∈ L∞ (0, T ; L2 )}, such that u satisfies the variational equation u′′ (t), v + µ(t) ux (t), vx + Y (t)v(0) + K1 u(1, t) + λ1 Πα (u′ (1, t)) v(1) + KΠp (u(t)) + λΠq (u′ (t)), v = F (t), v , for all v ∈ H , a.e., t ∈ (0, T ), and the initial conditions u(0) = u0 , ut (0) = u1 , where Πr (z) = |z|r−2 z, r ∈ {p, q, α} and t Y (t) = g(t) + K0 u(0, t) + k (t − s) u (0, s) ds Remark 2.1 If u ∈ L∞ (0, T ; H ) and ut ∈ L∞ (0, T ; H ), then u : [0, T ] → H is continuous ([5], Lemma 1.2, p.7), so it is clear that u(0) is defined and u(0) belongs to H Similarly, with ut ∈ L∞ (0, T ; H ) and utt ∈ L∞ (0, T ; L2 )}, it implies that ut : [0, T ] → L2 is continuous, ut (0) ∈ L2 follows In order to get the following result of existence, we need the assumptions (H1 )-(H5 ), in which u0 ∈ H , u1 ∈ H Then, u(0) = u0 ∈ H and ut (0) = u1 ∈ H Theorem 2.2 Let (H1 ) − (H5 ) hold For every T > 0, there exists a unique weak solution u of problem (1.1)-(1.4), such that (2.4) u ∈ L∞ (0, T ; H ), ut ∈ L∞ (0, T ; H ), utt ∈ L∞ (0, T ; L2 ), q α |ut | −1 ut ∈ H (QT ) , |ut (1, ·)| −1 ut (1, ·) ∈ H (0, T ) Remark 2.2 It follows from (2.4) that problem (1.1)-(1.4) has a unique weak solution u satisfying (2.5)   u ∈ H (QT ) ∩ C 0, T ; H ∩ C 0, T ; L2 ∩ L∞ 0, T ; H , ut ∈ L∞ (0, T ; H ) ∩ C 0, T ; L2 , q α  utt ∈ L∞ (0, T ; L2 ), |ut | −1 ut ∈ H (QT ) , |ut (1, ·)| −1 ut (1, ·) ∈ H (0, T ) Proof of Theorem 2.2 The proof consists of Steps 1–4 Step (The Galerkin approximation) Let {wj } be a denumerable base of H We find the approximate solution of problem (1.1)-(1.4) in the form m (2.6) cmj (t)wj , um (t) = j=1 ON A NONLINEAR WAVE EQUATION 349 where the coefficient functions cmj satisfy the system of ordinary differential equations  u′′m (t), wj + µ(t) umx (t), wjx + Ym (t)wj (0) + K1 um (1, t)wj (1)     +λ Π (u′ (1, t))w (1) + KΠ (u (t)) + λΠ (u′ (t)), w j j p m q m α m (2.7)  = F (t), wj , ≤ j ≤ m,    um (0) = u0m , u′m (0) = u1m , in which t (2.8) Ym (t) = g(t) + K0 um (0, t) + k(t − s)um (0, s)ds, and (2.9)    u0m   u1m m = j=1 m = j=1 αmj wj → u0 strongly in H , βmj wj → u1 strongly in H From the assumptions of Theorem 2.2, the system (2.7)–(2.8) has a solution um on an interval [0, Tm ] ⊂ [0, T ] The following estimates allow one to take Tm = T for all m, (see [3]) Step (A priori estimates I) Substituting (2.8) into (2.7), then multiplying the j th equation of (2.7) by c′mj (t) and summing with respect to j, and afterwards integrating with respect to the time variable from to t, we get after some rearrangements t Sm (t) = Sm (0) + 2g(0)u0m (0) − 2g(t)um (0, t) + t t g′ (s)um (0, s)ds + +2 µ′ (s)||umx (s)||2 ds F (s), u′m (s) ds t (2.10) − 2um (0, t) t k(t − r)um (0, r)dr + 2k(0) t +2 s um (0, s)ds 0 k′ (s − r)um (0, r)dr = Sm (0) + 2g(0)u0m (0) + Ij , j=1 u2m (0, s)ds 350 L T P NGOC, T M THUYET, P T SON AND N T LONG where Sm (t) = u′m (t) 2K + µ(t)||umx (t)|| + um (t) p t p Lp u′m (s) + 2λ q Lq ds t (2.11) u′m (1, s) + 2λ1 α ds + K0 u2m (0, t) + K1 u2m (1, t) By the assumptions (H1 ), (H3 )-(H5 ), and the imbedding H ֒→ C Ω , we deduce from (2.9) that (2.12) Sm (0) + 2g(0)u0m (0) + u0m H1 ≤ C0 , for all m, where C0 is a constant depending only on p, K, K0 , K1 , µ(0), g(0), u0 , u1 On the other hand, we deduce from (2.2), (2.11) and (2.12) that  2 ′   Sm (t) ≥ um (t) + µ0 umx (t) ,     t t     ′  um (t) = um (0) + um (s)ds ≤ 2C0 + 2t Sm (s)ds,      0   t (2.13)   Sm (t) + 2C0 + 2t Sm (s)ds, um (t) 2H ≤   µ       t t      um (s) H ds ≤ 2T C0 + + T2 Sm (s)ds    µ0 0 Using the assumptions (H1 )-(H5 ) and (2.3), (2.13) and the following inequality 2ab ≤ εa2 + b2 , for all ε > 0, a, b ≥ 0, ε (2.14) we shall estimate respectively the terms Ij , j = 1, , on the right-hand side of (2.10) as follows t ′ I1 = (2.15) µ (s) umx (s) µ′ ds ≤ µ0 t t ≤ CT Sm (s)ds, Sm (s)ds L∞ (0,T ) ON A NONLINEAR WAVE EQUATION 351 where CT always indicates a bound depending on T, g 2C ([0,T ]) I2 = −2g(t)um (0, t) ≤ ε um (t) 2H + ε t (2.16) ε Sm (t) + εCT Sm (s)ds + (ε + )CT , for all ε > 0, ≤ µ0 ε t t ′ I3 = (2.17) g (s)um (0, s)ds ≤ ds ≤ H1 ds  Sm (s)ds , t I4 = um (s) t ≤ CT  + F (s), u′m (s) +  t (2.18) g′ L∞ (0,T ) t F (s) ds + u′m (s) F (s) ds t ≤ CT + F (s) Sm (s)ds, t I5 = −2um (0, t) (2.19) ≤ ε um (t) ≤ k(t − r)um (0, r)dr H1 k + ε t L2 (0,T )  um (r) 0 t I6 = 2k(0) (2.20) I7 = (2.21) um (s) H1 ds  t Sm (s)ds , k′ (s − r)um (0, r)dr um (0, s)ds √ ≤ T k′ ≤ 4k(0) ≤ CT 1 + s Sm (s)ds , for all ε > 0, t  t dr  t ε Sm (t) + (ε + )CT 1 + µ0 ε u2m (0, s)ds H1 t um (s) L2 (0,T ) H1  ds ≤ CT 1 + t  Sm (s)ds 352 L T P NGOC, T M THUYET, P T SON AND N T LONG Combining (2.10), (2.12) and (2.15)-(2.21), we obtain after some rearrangements (2.22) 2ε 1 Sm (t) ≤ Sm (t) + k1T (ε) + µ0 2 t q1T (ε, s)Sm (s)ds, for all ε > 0, where  1  = C0 + 2(ε + + 2)CT ,  k1T (ε) ε (2.23) 1   q1T (ε, t) = (2ε + + 4)CT + F (t) , q1T (ε, ·) ∈ L1 (0, T ) ε 2ε ≤ , by Gronwall’s lemma, we deduce from (2.22) that Choosing ε > 0, with µ0 (2.24)  Sm (t) ≤ k1T (ε) exp   t q1T (ε, s)ds ≤ CT , ∀m ∈ N, ∀t ∈ [0, T ], ∀T > (A priori estimates II) Now differentiating (2.7)1 with respect to t, we have ′ ′ ′ u′′′ m (t), wj + µ(t) umx (t), wjx + µ (t) umx (t), wjx + Ym (t)wj (0) (2.25) + K1 u′m (1, t)wj (1) + λ1 Π′α (u′m (1, t))u′′m (1, t)wj (1) + K Π′p (um (t))u′m (t), wj + λ Π′q (u′m (t))u′′m (t), wj = F ′ (t), wj , for all ≤ j ≤ m Multiplying the j th equation of (2.25) by c′′mj (t), summing up with respect to j and then integrating with respect to the time variable from to t, we have after some rearrangements Xm (t) = Xm (0) + 2µ′ (0) u0mx , u1mx t µ′ (s)||u′mx (s)||2 ds − 2µ′ (t) umx (t), u′mx (t) +3 t t ′′ +2 µ (s) umx (s), u′mx (s) t (2.26) +2 ds − 2K Π′p (um (s))u′m (s), u′′m (s) ds t F ′ (s), u′′m (s) ds − g′ (s)u′′m (0, s)ds t − 2k(0) t um (0, s)u′′m (0, s)ds −2 s u′′m (0, s)ds 0 = Xm (0) + µ′ (0)u0mx , u1mx + Jj , j=1 k′ (s − r)um (0, r)dr ON A NONLINEAR WAVE EQUATION 353 where Xm (t) = u′′m (t) + µ(t) u′mx (t) t 8λ1 (α − 1) + α2 (2.27) t 8λ(q − 1) + q2 2 + K0 u′m (0, t) + K1 u′m (1, t) α ∂ |u′m (1, s)| −1 u′m (1, s) ∂s q ∂ |u′m (s)| −1 u′m (s) ∂s 2 ds ds By the assumptions (H1 )-(H3 ), (H5 ), and the imbedding H ֒→ C Ω , we deduce from (2.9) that Xm (0) + 2µ′ (0) u0mx , u1mx ≤ C0 , (2.28) for all m, where C0 is a constant depending only on p, q, K, K0 , K1 λ, µ, F (0), u0 , u1 On the other hand, (2.2), (2.3), (2.27) and (2.28) yield   Xm (t) ≥ u′′m (t) + µ0 u′mx (t) ,      t t    2 ′    um (t) ≤ u1m + 2t Xm (s)ds ≤ 2C0 + 2t Xm (s)ds,     0   t (2.29)  Xm (t) + 2C0 + 2t Xm (s)ds, u′m (t) H ≤    µ0         t t      u′m (s) H ds ≤ CT 1 + Xm (s)ds ,    0 where CT always indicates a bound depending on T We again use the inequalities (2.3), (2.14), (2.29) and by (H1 )-(H5 ), we shall estimate the terms Jj , ≤ j ≤ on the right-hand side of (2.26) as follows (2.30) t J1 = µ ′ (s)||u′mx (s)||2 ds µ′ ≤ µ0 t L∞ (0,T ) t Xm (s)ds ≤ CT Xm (s)ds, where CT always indicates a bound depending on T, J2 = −2µ′ (t) umx (t), u′mx (t) ≤ µ′ (2.31) ≤ µ′ L∞ (0,T ) CT L∞ (0,T ) umx (t) 1 Xm (t) ≤ εXm (t) + CT , µ0 ε u′mx (t) 354 L T P NGOC, T M THUYET, P T SON AND N T LONG for all ε > 0, t t ′′ (2.32) J3 = umx (s), u′mx (s) µ (s) ds ≤ t ≤ 2CT Xm (s)ds ≤ CT + µ0 µ′′ (s) u′mx (s) ds µ′′ (s) umx (s) t µ′′ (s) Xm (s)ds, t (2.33) Π′p (um (s))u′m (s), u′′m (s) ds J4 = −2K √ ≤ 2K(p − 1) um t p−2 Sm (s) Xm (s)ds L∞ (0,T ;H ) t ≤ CT + Xm (s)ds, t (2.34) J5 = T F ′ (s), u′′m (s) ds ≤ t ′ F ′ (s) F (s) ds + u′′m (s) ds t ≤ CT + F ′ (s) Xm (s)ds, t J6 = − g′ (s)u′′m (0, s)ds t ′ = 2g (0)u1m (0) − 2g ′ (t)u′m (0, t) ′ (2.35) ≤ g′ (0)u1m (0) + g ε T g′′ (s) g (s) ds + C ([0,T ]) + ε u′m (t) H1 t ′′ +2 g′′ (s)u′m (0, s)ds +2 u′m (s) H1 ds ε ≤ C0 + (ε + )CT + Xm (t) + ε µ0 t CT + εCT + ′′ g (s) µ0 Xm (s)ds, 360 L T P NGOC, T M THUYET, P T SON AND N T LONG where (2.60) σ(t) = u′ (t) + µ(t) ux (t) + K0 u2 (0, t) + K1 u2 (1, t) t Πα (u′1 (1, s)) − Πα (u′2 (1, s)) u′ (1, s)ds + 2λ1 t Πq (u′1 (s)) − Πq (u′2 (s)), u′ (s) ds ≥ u′ (t) + 2λ + µ0 ux (t) We again use the inequalities (2.3) and (2.14), and the following inequalities  t t t   ′ (s)ds ′ (s) ds ≤ t σ(s)ds,  u(t) = u ≤ t u    0   t (2.61) u(t) H ≤ µ0 σ(t) + t σ(s)ds,   t  t    σ(s)ds,  u(s) H ds ≤ µ0 + 21 T 0 to respectively estimate the terms on the right-hand side of (2.59) as follows t (2.62) ′ σ1 = µ (s) ux (s) µ′ ds ≤ µ0 t L∞ (0,T ) t σ(s)ds ≡ σ1 σ(s)ds, t (2.63) σ2 = −2K Πp (u1 (s)) − Πp (u2 (s)), u′ (s) ds t ≤ 2K(p − 1)R where R = √ 2max ui i=1,2 t p−2 u(s) ′ u (s) ds ≤ σ2 L∞ (0,T ;H ) , t (2.64) σ(s)ds, t σ3 = 2k(0) u (0, s)ds ≤ σ3 σ(s)ds, t (2.65) σ4 = −2u(0, t) ≤ ε u(t) H1 k(t − r)u(0, r)dr k + ε t L2 (0,T ) u(r) H1 dr ON A NONLINEAR WAVE EQUATION ε ≤ σ(t) + σ4 (ε) µ0 t (2.66) σ5 = t σ(s)ds, s u(0, s)ds 0 k′ (s − r)u(0, r)dr t ≤4 s u(s) H1 ds |k′ (s − r)| u(r) t ≤4 361 H1 dr t u(s) H1 ds + T k′ L2 (0,T ) t u(r) H1 dr ≤ σ5 σ(s)ds Combining (2.59) and (2.62)-(2.66), we obtain (2.67) ε σ(t) + (σ1 + σ2 + σ3 + σ4 (ε) + σ5 ) σ(t) ≤ µ0 t σ(s)ds Choosing ε > 0, with µε0 ≤ , using Gronwall’s lemma, we obtain that σ(t) ≡ 0, i.e., u1 ≡ u2 Theorem 2.2 is completely proved An asymptotic expansion of the weak solution with respect to two small parameters In this part, we assume that α = 2, and (H6 ) K, K0 , K1 ≥ 0, p > N + 2, q > N + 1, N ≥ and (u0 , u1 , F, µ, g, k, K, K0 , K1 ) satisfy the assumptions (H1 )-(H4 ), (H6 ) Let λ, λ1 > By Theorem 2.2, problem (1.1)-(1.4) has a unique weak solution u − → depending on λ = (λ, λ1 ): − = u(λ, λ1 ) u = u→ λ We consider the following perturbed problem   utt − µ(t)uxx + KΠp (u) + λΠq (ut ) = F (x, t), < x < 1, < t < T,   t  µ(t)u (0, t) = K u(0, t) + k (t − s) u (0, s) ds + g(t), x − P→ λ    −µ(t)ux (1, t) = K1 u(1, t) + λ1 ut (1, t),    u(x, 0) = u0 (x), ut (x, 0) = u1 (x), where λ, λ1 are small parameters such that λ, λ1 > 0, λ2 + λ21 < 362 L T P NGOC, T M THUYET, P T SON AND N T LONG First, we note that if the small parameters λ, λ1 > satisfy λ2 + λ21 < 1, then a priori estimates of the sequence {um } in the proof of Theorem 2.2 for − satisfy problem P→ λ   u′m (t) + µ0 ||umx (t)||2 + K um (t) pLp + K0 u2m (0, t)     t t     q ′  u′m (1, s) ds ≤ CT , um (s) Lq ds + λ1  + K1 um (1, t) + λ    0 (3.1) 2 ′′ ′ ′ ′    um (t) + µ0 ||umx (t)|| + K0 um (0, t) + K1 um (1, t)    t t    q ∂  ′′ ′ −1 ′  + λ ds + λ u (1, s) |u (s)| u (s) ds ≤ CT ,  m m m   ∂s 0 where CT is a constant depending only on T, u0 , u1 , F, µ, g, k (independent of − = u(λ, λ1 ) of the sequence {um } as m → +∞ λ, λ1 ) Hence the limit u = u→ λ − satisfying in suitable function spaces is a unique weak solution of problem P→ λ  p  ′  − (t)||2 + K u→ − (t) + K0 u→ u→ + µ0 ||u→ − (0, t) − (t)  λ x λ  λ λ Lp    t t    q  ′ ′   + K1 u→ ds + λ1 u→ u→ − (1, t) + λ − (s) − (1, s) ds ≤ CT ,  q λ λ λ  L  0 (3.2) 2  ′′ ′ ′ ′   u→ + µ0 ||u→ − (t) − (t)|| + K0 u→ − (0, t) + K1 u→ − (1, t)  λ λ x λ λ     t t    q ∂  ′′ ′ −1 ′  + λ ds + λ u (1, s) u (s) ds ≤ CT |u→ → − − (s)| → −   λ λ λ ∂s  0 It follows from (3.2) that  ′  −) ≤ CT , ≤ CT , Πq (u→ Πp (u→ −)  λ (Q )  λ H (QT ) H T     T   2   ′ ′  = λ1 λ1 u→ u→ − (1, ·) − (1, s) ds ≤ CT ,   λ λ  L (0,T )  (3.3) T   2   ′′ ′′  = λ λ1 u→ u − (1, ·) → − (1, s) ds ≤ CT ,   λ λ L2 (0,T )          ≤C λ u′ (1, ·) − → λ H (0,T ) T − → − → − → Let { λ j }, λ j = (λj , λ1j ), be a sequence such that λj , λ1j > 0, λ j → as − and deduce from (3.2), (3.3) that there exists a j → ∞ We put uj = u→ λ j ON A NONLINEAR WAVE EQUATION subsequence of the sequence {uj } denoted again by {uj }, such that  uj → u0 in L∞ (0, T ; H ) weakly*,   ′ ′   uj → u0 in L∞ (0, T ; H ) weakly*,    ′′ ′′  in L∞ (0, T ; L2 ) weakly*, uj → u0     in H (QT ) weakly,  Πp (uj ) → Πp ′ Πq (uj ) → Πq in H (QT ) weakly, (3.4)  ′   λ1j uj (1, ·) → χ1 in H (0, T ) weakly,    1,∞ (0, T )  u (0, ·) → u (0, ·) in W weakly*,  j   1,∞ (0, T )  u (1, ·) → u (1, ·) in W weakly*,   ′j uj (1, ·) → u′0 (1, ·) in L∞ (0, T ) weakly* 363 By the Aubin-Lions lemma ([5, p 57]) and the imbeddings H (QT ) ֒→ L2 (QT ), H (0, T ) ֒→ C ([0, T ]) , we can deduce from (3.4)1−8 the existence of a subsequence denoted again by {uj } such that  uj → u0 strongly in L2 (QT ) and a.e in QT ,    ′ ′  strongly in L2 (QT ) and a.e in QT , uj → u0     strongly in L2 (QT ) and a.e in QT ,  Πp (uj ) → Πp ′ Πq (uj ) → Πq strongly in L2 (QT ) and a.e in QT , (3.5)    uj (0, ·) → u0 (0, ·) strongly in C ([0, T ]) ,    u (1, ·) → u0 (1, ·) strongly in C ([0, T ]) ,    j λ1j u′j (1, ·) → χ1 strongly in C ([0, T ]) Similarly, by (3.3)1 and (3.5)1−4 , it is easy to prove that (3.6) Πp = Πp (u0 ), Πq = Πq (u′0 ), and t Yj (t) = g(t) + K0 uj (0, t) + k(t − s)uj (0, s)ds t (3.7) → g(t) + K0 u0 (0, t) + 0 k(t − s)u0 (0, s)ds ≡ Y0 (t) strongly in C ([0, T ]) Now, we shall prove that χ1 = It follows from (3.5)7 that (3.8) λ1j u′j (1, ·) → χ1 strongly inC ([0, T ]) On the other hand, it follows from (3.4)9 that (3.9) λ1j u′j (1, ·) → in L∞ (0, T ) weakly* Then, (3.8) and (3.9) imply (3.10) χ1 = 364 L T P NGOC, T M THUYET, P T SON AND N T LONG Hence we obtain from (3.8), (3.10) that λ1j u′j (1, ·) → strongly in C ([0, T ]) (3.11) Similarly, λj Πq (u′j ) → strongly in L2 (QT ) (3.12) By passing to the limit, as in the proof of Theorem 2.2, we conclude that u0 − → is a unique weak solution of problem (P0 ) corresponding to λ = satisfying (3.13) u0 ∈ H (QT ) ∩ C 0, T ; H ∩ C 0, T ; L2 ∩ L∞ 0, T ; H , u′0 ∈ L∞ (0, T ; H ), u′′0 ∈ L∞ (0, T ; L2 ) Note more that, u0 is a unique weak solution of problem (P0 ), i.e.,  ′′ u0 − µ(t)u0xx + KΠp (u0 ) = F (x, t), < x < 1, < t < T,     t      µ(t)u0x (0, t) = K0 u0 (0, t) + k (t − s) u0 (0, s) ds + g(t),      (P0 ) − µ(t)u0x (1, t) = K1 u0 (1, t),      u0 (x, 0) = u0 (x), u′0 (x, 0) = u1 (x),      u0 ∈ H (QT ) ∩ C 0, T ; H ∩ C 0, T ; L2 ∩ L∞ 0, T ; H ,     ′ u0 ∈ L∞ (0, T ; H ), u′′0 ∈ L∞ (0, T ; L2 ) − ) with We shall study the asymptotic expansion of the solution of problem (P→ λ respect to two small parameters λ and λ1 We use the following notations For a multi-index α = (α1 , α2 ) ∈ Z2+ and − → λ = (λ, λ1 ) ∈ R2 , we put  = α1 + α2 , α! = α1 !α2 !,   |α| − → − → λ = λ2 + λ21 , λ α = λα1 λα1 ,   α, β ∈ Z2+ , α ≤ β ⇐⇒ αi ≤ βi ∀i = 1, First, we shall need the following lemma Lemma 3.1 Let m, N ∈ N and uα ∈ R, α ∈ Z2+ , ≤ |α| ≤ N Then (3.14)   1≤|α|≤N m − →α  uα λ = − → (m) TN [u]α λ α , m≤|α|≤mN ON A NONLINEAR WAVE EQUATION 365 (m) where the coefficients TN [u]α , m ≤ |α| ≤ mN depend on u = (uα ), α ∈ Z2+ , ≤ |α| ≤ N defined by the recurrent formulas (3.15)  (1)  TN [u]α = uα , ≤ |α| ≤ N,      (m) (m−1) [u]β , m ≤ |α| ≤ mN, m ≥ 2, uα−β TN TN [u]α = (m)   β∈Aα (N )     A(m) (N ) = {β ∈ Z2 : β ≤ α, ≤ |α − β| ≤ N, m − ≤ |β| ≤ (m − 1)N } α + The proof of Lemma 3.1 can be found in [8] Let us consider the sequence of the weak solutions uγ , γ ∈ Z2+ , ≤ |γ| ≤ N, defined by the following problems:  ′′ uγ − µ(t)uγxx = Fγ (x, t), < x < 1, < t < T,     t      µ(t)uγx (0, t) = K0 uγ (0, t) + k (t − s) uγ (0, s) ds,      (Pγ ) − µ(t)uγx (1, t) = K1 uγ (1, t) + G1γ (t),      uγ (x, 0) = u′γ (x, 0) = 0,      uγ ∈ H (QT ) ∩ C 0, T ; H ∩ C 0, T ; L2 ∩ L∞ 0, T ; H ,     ′ uγ ∈ L∞ (0, T ; H ), u′′γ ∈ L∞ (0, T ; L2 ), where Fγ , G1γ , γ ∈ Z2+ , ≤ |γ| ≤ N, are defined by the recurrent formulas  ′ ′  −KΠp (u0 ) u1,0 − Πq (u0 ) , γ1 = 1, γ2 = 0, −KΠ′p (u0 ) u0,1 , γ1 = 0, γ2 = 1, Fγ = Fγ (x, t) =  (3.16) [1] ′ −KΦγ [Πp , u] − Φγ [Πq , u ], ≤ |γ| ≤ N, [2] G1γ = G1γ (t) = u˙ γ (1, t), ≤ |γ| ≤ N, [1] [2] with Φγ [Πp , u], u˙ γ (1, t), ≤ |γ| ≤ N, defined by the formulas Φγ1 −1,γ2 [Πq , u′ ], 0, ′ Φ[1] γ [Πq , u ] = u′γ1 ,γ2 −1 (1, t), 0, u˙ [2] γ (1, t) = (3.17) and     Φγ [Πp , u] (3.18) u′ |γ| =    Φγ [Πq , u′ ] = u′γ m=1 |γ| m=1 γ1 ≥ 1, γ1 = 0, γ2 ≥ 1, γ2 = 0, (m) (m) m! Πp (u0 )TN [u]γ , (m) ′ (m) ′ m! Πq (u0 )TN [u ]γ Here u = (uγ ) , = , |γ| ≤ N Then, we have the following theorem 366 L T P NGOC, T M THUYET, P T SON AND N T LONG − → − → Theorem 3.2 Let (H1 )-(H4 ), (H6 ) hold Then, for every λ , with λ < 1, − satisfying the asymptotic − has a unique weak solution u = u→ problem P→ λ λ estimation up to order N as follows ′ − u→ − λ (3.19) |γ|≤N − → u′γ λ γ L∞ (0,T ;L2 ) √ ′ (1, ·) − + λ1 u→ − λ |γ|≤N − − + u→ λ − → u′γ (1, ·) λ γ − → uγ λ γ |γ|≤N L2 (0,T ) ≤ CT L∞ (0,T ;H ) − → N + 21 λ , where CT is a constant depending only on T, the functions uγ , |γ| ≤ N, are the weak solutions of problems (P0 ), (Pγ ), ≤ |γ| ≤ N, respectively − Then v, with − be the unique weak solution of problem P→ Proof Let u = u→ λ λ (3.20) v =u− satisfies (3.21) |γ|≤N − → uγ λ γ = u − h,  ′′ v − µ(t)vxx = −K [Πp (v + h) − Πp (h)] − λ Πq v ′ + h′ − Πq h′     − →   + EN ( λ ), < x < 1, < t < T,      t        µ(t)vx (0, t) = K0 v(0, t) + k (t − s) v (0, s) ds,  − →   − µ(t)vx (1, t) = K1 v(1, t) + λ1 v ′ (1, t) + E1N ( λ ),      v(x, 0) = v ′ (x, 0) = 0,       v ∈ H (QT ) ∩ C 0, T ; H ∩ C 0, T ; L2 ∩ L∞ 0, T ; H ,     ′ v ∈ L∞ (0, T ; H ), v ′′ ∈ L∞ (0, T ; L2 ), where (3.22)  − →  EN ( λ ) − → ≡ EN ( λ ; x, t) = −K [Πp (h) − Πp (u0 )] − λΠq (h′ ) − − → − →  E1N ( λ ) ≡ E1N ( λ , t) = |γ|≤N − → λ1 u′γ (1, t) λ γ − − → Fγ λ γ , 1≤|γ|≤N − → G1γ λ γ 1≤|γ|≤N Then, we have the following lemma Lemma 3.3 Let (H1 )-(H4 ), (H6 ) hold Then − → N +1 − → , (i) EN ( λ ) ∞ ≤ CN λ L (0,T ;L ) t (3.23) (ii) − → E1N ( λ , s)v ′ (1, s) ds ≤ λ1 t v ′ (1, s) ds + C1N − → λ 2N +1 , ON A NONLINEAR WAVE EQUATION 367 − → − → for all λ , λ < 1, CN , C1N , are the constants depending only on N, T, p, q and the constants uγ L∞ (0,T ;H ) , − → Proof (i) Estimate EN ( λ ) u′γ L∞ (0,T ;L2 ) L∞ (0,T ;H ) , |γ| ≤ N In case N = 1, the proof of Lemma 3.3 is easy, hence we omit the details, therefore we only prove for N ≥ Put − → − → (3.24) h= uγ λ γ = u0 + h1 and h1 = uγ λ γ |γ|≤N 1≤|γ|≤N By using Taylor’s expansion of the function Πp (h) = Πp (u0 + h1 ) around the point u0 up to order N, we obtain N (3.25) Πp (h) = Πp (u0 ) + m=1 (m) +1 Πp (u0 )hm Π(N +1) (u0 + θ1 h1 )hN , + k! (N + 1)! p where < θ1 < By Lemma 3.1, we obtain from (3.18)1 , (3.24), after some − → rearrangements in order to of λ , that   |γ| − → (m) (m)  Πp (h) − Πp (u0 ) = Πp (u0 )TN [u]γ  λ γ m! m=1 1≤|γ|≤N   N − → (m) (m) TN [u]γ λ γ  Πp (u0 )  + m! m=1 + = (3.26) N +1≤|γ|≤m(N +1) +1 Π(N +1) (u0 + θ1 h1 )hN (N + 1)! p − → − → N +1 − → RN (Πp , u, λ ), Φγ [Πp , u] λ γ + λ 1≤|γ|≤N where − → λ N +1 − → RN (Πp , u, λ ) = N  (m) Πp (u0 )  m! m=1 N +1≤|γ|≤m(N +1)  − → (m) TN [u]γ λ γ  +1 + Π(N +1) (u0 + θ1 h1 )hN (N + 1)! p (3.27) Similarly, using Taylor’s expansion of the function Πq (h′ ) = Πq (u′0 + h′1 ) up to order N − 1, we obtain Πq h′ = Πq u′0 + h′1 (3.28) − → − → Φγ [Πq , u′ ] λ γ + λ = Πq u′0 + 1≤|γ|≤N −1 N − → RN −1 (Πq , u′ , λ ), 368 L T P NGOC, T M THUYET, P T SON AND N T LONG where − → λ N − → RN −1 (Πq , u′ , λ ) = N −1 m=1 + (3.29)  (m) ′  Π (u0 ) m! q N ≤|γ|≤mN (N ) ′ Π (u0 + θ2 h′1 ) h′1 N! q  − → (m) TN [u′ ]γ λ γ  N , and < θ2 < Combining (3.16), (3.22)1 , (3.26), and (3.28), we obtain − → − → EN ( λ ) ≡ EN ( λ ; x, t)   − → − → − → N +1  λ ′  (3.30) KRN (Πp , u, λ ) + − =− λ → RN −1 (Πq , u , λ ) λ We shall respectively estimate the following terms on the right-hand side of (3.30) − → N +1 − → Estimate K λ RN (Πp , u, λ ) √ uγ L∞ (0,T ;H ) + u′γ L∞ (0,T ;H ) , First, we note that if we put DT = |γ|≤N then by the boundedness of the functions uγ , u′γ , |γ| ≤ N, in the function space L∞ (0, T ; H ), we obtain from (3.24) that (3.31)  √ √ − → |γ| − →   |h1 | ≤ uγ H λ γ ≤ uγ L∞ (0,T ;H ) λ     1≤|γ|≤N 1≤|γ|≤N   √ − → − → uγ L∞ (0,T ;H ) ≤ DT λ ≤ DT , ≤ λ  1≤|γ|≤N   √  − → |γ|   |u0 + θ1 h1 | ≤ |u0 | + |h1 | ≤ uγ L∞ (0,T ;H ) λ ≤ DT   |γ|≤N Hence (3.32) +1 Π(N +1) (u0 + θ1 h1 )hN (N + 1)! p − → − → N +1 +1) (z) ≡ D1T (p) λ sup Π(N ≤ DT λ p (N + 1)! |z|≤DT N +1 , and (3.33) − → λ ≤ −N −1 max  N (m) Πp (u0 )  m! m=1 sup 1≤m≤N |z|≤DT Π(m) p (z) N +1≤|γ|≤m(N +1) N m! m=1  − → (m) TN [u]γ λ γ  (m) TN [u]γ N +1≤|γ|≤m(N +1) L∞ (0,T ;L2 ) − → λ |γ|−N −1 ON A NONLINEAR WAVE EQUATION 369 N ≤ max sup 1≤m≤N |z|≤DT m! m=1 Π(m) p (z) (m) TN [u]γ N +1≤|γ|≤m(N +1) ≡ D2T (p) L∞ (0,T ;L2 ) This implies (3.34) − → K λ N +1  (m) Π (u0 )  m! p m=1 N → − RN (Πp , u, λ ) ≤ K N +1≤|γ|≤m(N +1)  − → (m) TN [u]γ λ γ  +1 Π(N +1) (u0 + θ1 h1 )hN (N + 1)! p − → − → N +1 ≡ D∗T (p) λ ≤ (D2T (p) + D1T (p)) λ +K N +1 − → − → N Estimate λ λRN −1 (Πq , u′ , λ ) We also obtain from (3.29) in a similar manner corresponding to the above part that   N −1 − → − → N − → (m) TN [u′ ]γ λ γ  λ λ RN −1 (Πq , u′ , λ ) = λ Π(m) (u′0 )  m! q m=1 N ≤|γ|≤mN (N ) ′ N Π (u0 + θ2 h′1 ) h′1 N! q − → − → N +1 ≡ D∗T (q) λ ≤ (D2T (q) + D1T (q)) λ +λ (3.35) where (3.36)   D1T (q) =      D2T (q) = N DT N! , (N ) Πq (z) , sup |z|≤DT max sup 1≤m≤N −1 |z|≤D T (m) N −1 Πq (z) m=1 m! (m) N ≤|γ|≤mN TN [u′ ]γ L∞ (0,T ;L2 ) Therefore, it follows from (3.30), (3.34) and (3.35) that − → − → EN ( λ ) ≤ [D∗T (p) + D∗T (q)] λ (3.37) N +1 ≡ CN − → λ N +1 , where CN = D∗T (p) + D∗T (q) and the proof of (i) is complete t − → (ii) Estimate E1N ( λ , s)v ′ (1, s) ds We note that (3.38) N +1 − → − → E1N ( λ ) ≡ E1N ( λ , t) = = λ1 |γ|=N − → λ1 u′γ (1, t) λ γ − |γ|≤N − → ′ uγ (1, t) λ γ − → G1γ λ γ 1≤|γ|≤N 370 L T P NGOC, T M THUYET, P T SON AND N T LONG Using Cauchy’s inequality, we obtain t − → E1N ( λ , s)v ′ (1, s) ds t ≤ 2λ1 (3.39) − → λ ≤ u′γ (1, s) v ′ (1, s) ds |γ|=N √ − → ≤ 2λ1 λ ≤ |γ| t λ1 t N u′γ L∞ (0,T ;H ) |γ|=N − → v ′ (1, s) ds + 4λ1 λ 2N t λ1 v ′ (1, s) ds v ′ (1, s) ds + C1N − → λ  T 2N +1 u′γ L∞ (0,T ;H ) |γ|=N 2  , where  C1N = 4T  (3.40) u′γ L∞ (0,T ;H ) |γ|=N The proof of (ii) is complete The proof of Lemma 3.3 is complete 2  We now continue the proof of Theorem 3.2 Next, by multiplying the two sides of (3.21)1 with v ′ , after integration with respect to t, we find without difficulty from Lemma 3.3 that Z(t) ≤ T CN − → λ t −2 2N +2 µ′ + 2+ µ0 Z(s)ds L∞ (0,T ) − → E1N ( λ , s)v ′ (1, s) ds − 2v(0, t) t +2 t t t k(t − r)v(0, r)dr + 2k(0) s v(0, s)ds 0 ≡ − → λ 2N +2 t k′ (s − r)v(0, r)dr + K Πp (v + h) − Πp (h) (3.41) T CN v (0, s)ds µ′ + 2+ µ0 t Z(s)ds + L∞ (0,T ) Zi , i=1 ds ON A NONLINEAR WAVE EQUATION 371 where Z(t) = v ′ (t) 2 + µ(t) vx (t) + K0 v (0, t) + K1 v (1, t) t + 2λ (3.42) t ′ ′ ′ ′ Πq v + h − Πq h , v ds + 2λ1 v ′ (1, s) ds Using the following inequality ∀α ≥ 2, ∃Cα > : (Πα (x) − Πα (y)) (x − y) ≥ Cα |x − y|α , ∀x, y ∈ R, (3.43) we deduce from (3.42) that t t ′ (3.44) Z(t) ≥ v (t) + µ0 vx (t) q Lq ′ v (s) + 2λCq v ′ (1, s) ds ds + 2λ1 0 Using (3.44) and the inequalities  2  t t t    2  ′ ′  v (s) ds ≤ t v (s) ds ≤ t Z(s)ds, v(t) =       0    t      v(t) ≤ Z(t) + t Z(s)ds, H µ0 (3.45)     t t    1   v(s) H ds ≤ + T Z(s)ds,   µ0    0   √   |v(x, t)| ≤ v(t) v(t) H , ∀x ∈ [0, 1], C (Ω) ≤ we shall respectively estimate the terms Zi , i = 1, , on the right-hand side of (3.41) as follows t Z1 = −2 (3.46) − → E1N ( λ , s)v ′ (1, s) ds ≤ λ1 − → ≤ Z(t) + C1N λ 2N +1 t v ′ (1, s) ds + C1N ≤ v(r) H1 , k(t − r)v(0, r)dr ≤ β v(t) β Z(t) + βT + k µ0 β 2N +1 t Z2 = −2v(0, t) − → λ L2 (0,T ) H1 1 + T2 µ0 + k β t L2 (0,T ) t Z(s)ds dr 372 L T P NGOC, T M THUYET, P T SON AND N T LONG (3.47) β Z(t) + Z2T (β) ≡ µ0 t Z(s)ds, ∀β > 0, (3.48) t Z3 = 2k(0) v (0, s)ds ≤ 4|k(0)| t Z4 = t Z(s)ds ≡ Z3T Z(s)ds, s k′ (s − r)v(0, r)dr v(0, s)ds 0 √ ≤ T k′ (3.49) t 1 + T2 µ0 2 t t v(s) L2 (0,T ) H1 ds ≤ Z4T Z(s)ds With Z5 , first, by using the same arguments as in the above part, we can show − − of problem P→ satisfies that the weak solution u→ λ λ − u→ λ (3.50) L∞ (0,T ;H ) ≤ CT , − → λ < 1, − → where CT is a constant independent of λ On the other hand, we have   h L∞ (0,T ;H ) ≤ |γ|≤N uγ L∞ (0,T ;H ) ≡ C1T , (3.51)  v + h L∞ (0,T ;H ) = u→ − ≤ CT λ ∞ L (0,T ;H ) Next, with RT = max{C1T , CT }, it follows from (3.51) that t Z5 = K (3.52) Πp (v + h) − Πp (h) ≤ T (p − 1)2 RT2p−4 2 ds ≤ K (p − 1) t t RT2p−4 v(s) ds t Z(s)ds ≡ Z5T Z(s)ds Combining (3.41), (3.46)-(3.49) and (3.52), we then obtain (3.53) − → Z(t) ≤ Z1T λ 2N +1 + β + µ0 t Z(t) + Z2T (β) Z(s)ds, for all β > 0, where (3.54) 2 Z1T = T CN + C1N , 1 ′ Z2T (β) = + µ0 µ L∞ (0,T ) + Z2T (β) + Z3T + Z4T + Z5T ON A NONLINEAR WAVE EQUATION Choosing β > 0, with (3.55) + β µ0 Z(t) ≤ Z1T 373 ≤ 12 , it follows from (3.53), that − → λ 2N +1 t + Z2T (β) Z(s)ds Using Gronwall’s lemma, it follows that − → − → 2N +1 exp(T Z2T (β)) ≡ Z3T (β) λ (3.56) Z(t) ≤ Z1T λ 2N +1 This implies (3.19) Theorem 3.2 is proved completely Acknowledgements The authors wish to express their sincere thanks to the referee and Professor Dinh Nho Hao for the valuable comments and important remarks The authors thank the Editors of Acta Mathematica Vietnamica, the Institute of Mathematics for support in writing this paper in the special volume dedicated to Professor Tran Duc Van on the occasion of his 60th birthday The authors are also extremely grateful to support from the National Foundation for Science and Technology Development (NAFOSTED) References [1] N T An and N D Trieu, Shock between absolutely solid body and elastic bar with the elastic viscous frictional resistance at the side, J Mech NCSR.Vietnam 13 (2) (1991), 1–7 [2] M Bergounioux, N T Long and A P N Dinh, Mathematical model for a shock problem involving a linear viscoelastic bar, Nonlinear Anal 43 (2001), 547–561 [3] E L A Coddington and N Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955 [4] M M Cavalcanti, V N Domingos Cavalcanti and J A Soriano, On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions, J Math Anal Appl 281 (1) (2003), 108–124 [5] J L Lions, Quelques m´ethodes de R´esolution des Problems aux Limites Non-lin´eares, Dunod; 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L N K Hang and N T Long, On a nonlinear wave equation associated with the boundary conditions involving convolution, Nonlinear Analysis, Theory, Methods & Applications, Series A: Theory and Methods... Breakdown of solutions to nonlinear wave equation with a viscoelastic boundary condition, Arabian J Sci Engrg 19 ( 2A) (1994), 195–202 [13] J E M Rivera and D Andrade, A boundary condition with. .. Cavalcanti, V N Domingos Cavalcanti and J A Soriano, On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions, J Math Anal Appl 281 (1) (2003),