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...

Research Article Received 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 Le Thi Phuong Ngoca and Nguyen Thanh Longb * † Communicated by M L Santos This 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 Introduction In this paper, we consider the initial-boundary value problem for the system of nonlinear wave equations ( utt uxx C jut jr1 ut D f1 u, v/ C F1 x, t/, < x < 1, < t < T, vtt ( vxx C jvt j r2 vt D f2 u, v/ C F2 x, t/, < x < 1, < t < T, u.0, t/ D u.1, t/ D 0, vx 0, t/ C K jv.0, t/jp ( v.0, t/ D jvt 0, t/jq vt 0, t/, v.1, t/ D 0, u.x, 0/ D uQ x/, ut x, 0/ D uQ x/, v.x, 0/ D vQ x/, vt x, 0/ D vQ x/, (1.1) (1.2) (1.3) i D 1, 2/ are given constants and f1 , f2 , F1 , F2 , uQ i , vQ i , i D 0, 1/ are given functions where p 2, q 2, K > 0, > 0, i > 0, ri 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 Dirichlet boundary 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 Copyright © 2013 John Wiley & Sons, Ltd Math Meth Appl Sci 2013 L T P NGOC AND N T LONG 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 authors proved 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 that system 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 arbitrary initial 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 associated with two-point boundary conditions have been established The proofs are based on the Galerkin method associated to a priori estimates, 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 nonlinear wave equations associated with initial and Dirichlet boundary conditions and with nonlinear boundary conditions (1.1)–(1.3) In this paper, we extend the results of [10,14] by combination of the methods used in [10,14] with some appropriate modifications for problem considered here Our main results are presented in three parts as follows Part is devoted to the presentation of the existence results based on Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions In this part, problems (1.1)–(1.3) are dealt with two cases of p, q/ : p 2, Ä q Ä or p 3, q > and r1 2, r2 In the cases q D and p 2, the solution obtained here is unique In parts and 3, problems (1.1)–(1.3) are considered with p > and q D r1 D r2 D Under some suitable conditions, by applying techniques 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 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 the norm in L2 On H1 , we shall use the following norm: jjvjjH1 D jjvjj2 C jjvx jj2 Á1=2 D 0, 1/, QT D 0, T/, T > 0, and k k is (2.1) We put V D fv H1 : v.1/ D 0g (2.2) V is a closed subspace of H1 , and the following lemma is known as a standard one Lemma 2.1 The imbedding H1 ,! C / is compact and jjvjjC0 / Ä p 2jjvjjH1 for all v H1 (2.3) Remark 2.1 On V, two norms v ! jjvjjH1 and v ! jjvx jj are equivalent Furthermore, jjvjjC0 / Ä jjvx jj for all v V (2.4) Remark 2.2 The weak formulation of the initial-boundary valued problems (1.1)–(1.3) can be given in the following manner: Find u, v/ « ˚ e D u, v/ L1 0, T; H1 \ H2 V \ H2 / : ut , vt / L1 0, T; H01 V , utt , vtt / L1 0, T; L2 L2 / , such that u, v/ satisfies the W following variational equation: ˆ < hutt t/, wi C hux t/, wx i C h‰r1 ut t//, wi D hf1 u, v/, wi C hF1 t/, wi , hvtt t/, i C hvx t/, x i C h‰r2 vt t//, i C ‰q vt 0, t// 0/ ˆ : D K‰p v.0, t// 0/ C hf2 u, v/, i C hF2 t/, i , for all w, / H01 V, together with the initial conditions u.0/, u0 0/ D Qu0 , uQ / , v.0/, v 0/ D Qv0 , vQ / , where ‰r z/ D jzjr (2.5) z, r fp, q, r1 , r2 g, with p, q Copyright © 2013 John Wiley & Sons, Ltd 2, ri (2.6) 2, i D 1, are given constants Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Now, we shall consider problems (1.1)–(1.3) with p assumptions: H1 / Qu0 , uQ / H01 \ H2 H2 / H3 / H01 , Qv0 , vQ / V \ H2 / 2, q 2, > 0, > 0, r1 2, r2 2, K > 0, > and make the following V; F1 , F2 L1 0, T; L2 / such that F10 , F20 L1 0, T; L2 /; there exists F C R2 ; R/ such that (i) @F u, v/ D f1 u, v/, @F u, v/ D f2 u, v/, @u @v (ii) there exist the constants ˛, ˇ > 2; C1 > 0, such that Á F u, v/ Ä C1 C juj˛ C jvjˇ , 8u, v R; H10 Qu0 , uQ / H01 H20 L2 Q F1 , F2 L2 , Qv0 , vQ / V L2 ; T / Remark 2.3 We present an example in which functions f1 and f2 satisfy assumption H3 / Consider the following functions: à  ˇ ˛ f1 u, v/ D ˛ juj˛ C juj 2 jvj u, à  ˇ ˛ f2 u, v/ D ˇ jvjˇ C juj jvj 2 v, where ˛, ˇ, , and are positive constants with a F C R2 ; R/ defined by < It is obvious that H3 , i// holds, and H3 , ii// is also valid, because there exists F u, v/ D Á ˇ ˛ juj˛ C jvjˇ C juj jvj such that @F u, v/ D ˛ @u  ˛ uC juj ˛ ˛ juj 2 ˇ jvj u D f1 u, v/, ˇ ˛ @F u, v/ D ˇ jvjˇ v C ˇ juj jvj 2 v D f2 u, v/, @v à à  Á Á 1 juj˛ C jvjˇ Ä F u, v/ Ä C juj˛ C jvjˇ , for all u, v/ R2 2 On the other hand, because of minf˛, ˇgF u, v/ Ä uf1 u, v/ C vf2 u, v/ Ä maxf˛, ˇgF u, v/, for all u, v/ R2 , functions f1 and f2 also satisfy hypothesis H30 in Sections and by choosing positive constants ˛, ˇ, , and We have the following theorem about the existence of a ‘strong solution’ suitably Theorem 2.2 Suppose that H1 /–.H3 / hold and the initial data satisfy the compatibility conditions vQ 0x 0/ C K jQv0 0/jp vQ 0/ D jQv1 0/jq vQ 0/ (2.7) If either of the following cases is valid (i) Ä q Ä and p 2; or (ii) q > and p f2g [ Œ3, C1/, then there exists a local weak solution u, v/ of problems (1.1)–(1.3) such that u, v/ L1 0, T ; H01 \ H2 V \ H2 / , ˆ ˆ ˆ ˆ ˆ < ut , vt / L1 0, T ; H01 V , ˆ utt , vtt / L1 0, T ; L2 L2 /, ˆ ˆ ˆ ˆ ˇ ˇq : ˇ ˇ r21 ˇ ˇ r22 ˇu ˇ u , ˇv ˇ v H1 QT /, ˇv 0, /ˇ for T > small enough Furthermore, if q D and p (2.8) v 0, / H1 0, T /, 2, the solution is unique Copyright © 2013 John Wiley & Sons, Ltd Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Remark 2.3 The regularity obtained by (2.8) shows that problems (1.1)–(1.3) has a strong solution u, v/ L1 0, T ; H01 \ H2 V \ H2 / \ C Œ0, T ; H01 ˆ ˆ ˆ ˆ ˆ < ut , vt / L1 0, T ; H01 V \ C Œ0, T ; L2 L2 , ˆ utt , vtt / L1 0, T ; L2 L2 /, ˆ ˆ ˆ ˆ ˇ ˇq : ˇ ˇ r21 ˇ ˇ r22 ˇu ˇ u , ˇv ˇ v H1 QT /, ˇv 0, /ˇ v 0, V \ C Œ0, T ; L2 L2 , (2.9) / H1 0, T / With less regular initial data, we obtain the following theorem about the existence of a weak solution Theorem 2.3 Let q D 2, p Suppose that H10 , H20 , and H3 / hold Then problems (1.1)–(1.3) have a unique local solution ( u, v/ C Œ0, T ; H01 u0 , v / Lr1 Q T / V \ C Œ0, T ; L2 Lr2 Q T /, v.0, L2 , / H1 0, T (2.10) /, for T > small enough Proof of Theorem 2.3 The proof consists of four steps Step The Faedo–Galerkin approximation Let f.wi , j /g be a denumerable base of H01 \ H2 / V \ H2 / We find the approximate solution of problem (1.1)– (1.3) in the form Xm Xm cmj t/wj , vm t/ D dmj t/ j , (2.11) um t/ D jD1 jD1 where the coefficient functions cmj , dmj / satisfy the following system ˝ ˛ ˝ ˛ ˝ ˛ ˝ ˛ ˝ ˛ u00m t/, wj C umx t/, wjx C ‰r1 u0m t//, wj D f1 um , vm /, wj C F1 t/, wj ˆ ˆ ˆ ˆ ˝ ˛ ˝ ˛ ˝ ˛ ˆ < v 00 t/, j C vmx t/, jx C ‰r v t//, j C ‰q v 0, t// j 0/ m m m ˛ ˝ ˛ ˝ ˆ D K‰ C F , Ä j Ä m, v 0, t// 0/ C f u , v /, t/, ˆ p m m m j j j ˆ ˆ ˆ : 0 um 0/, um 0/ D Qu0 , uQ / , vm 0/, vm 0/ D Qv0 , vQ / (2.12) Under the assumptions of Theorem 2.2, system (2.12) has a solution um t/, vm t// on an interval Œ0, Tm  Œ0, T t/, d0 t// and summing with respect to j, and afterwards Step The first estimate Multiplying the jth equation of (2.12) by cmj mj integrating with respect to the time variable from to t, we obtain after some rearrangements Z t Sm t/ D Sm 0/ C K ‰p vm 0, s//vm 0, s/ds Z tÄ C2 Z t @F @F um s/, vm s//, u0m s/ C um s/, vm s//, vm s/ @u @v ds (2.13) ˝ ˛ ˝ ˛ F1 s/, u0m s/ C F2 s/, vm s/ ds D Sm 0/ C I1 C I2 C I3 , C2 where Sm t/ D jju0m t/jj2 C jjvm t/jj2 C jjumx t/jj2 C jjvmx t/jj2 C Z C2 t vm s/ r2 Lr2 Z t ds C Z t u0m s/ r1 Lr1 ds (2.14) ˇ ˇ ˇv 0, s/ˇq ds m By 2.12/3 and (2.14), we have Sm 0/ D jjQu1 jj2 C jjQv1 jj2 C jjQu0x jj2 C jjQv0x jj2 Á CN , for all m (2.15) We shall estimate respectively the following integrals in the right-hand side of (2.13) First integral Using the inequalities ab Ä Copyright © 2013 John Wiley & Sons, Ltd 1 q q ı a C 0ı q q q0 q0 b , for all ı > 0, a, b 0, q > 1, q0 D q q , (2.16) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG and s Ä C sN , 8s 0, 0, N, N D max 1, ˛ ˇ q.p , , 2 2.q 1/ , 1/ (2.17) we obtain Z t I1 D K 0 ‰p vm 0, s//vm 0, s/ds Ä K Z t ˇ ˇ 1ˇ vm 0, s/ˇ ds jvm 0, s/jp Z t Z t ˇ ˇ ˇv 0, s/ˇq ds C ı q0 K Ä ıq K jvm 0, s/j.p m q q0 0 Z t q.p 1/ 1 q q q1 Ä ıq K K Sm t/ C ı jvm 0, s/j q q q Z th i q q q q1 C SNm s/ ds Ä ı K K Sm t/ C ı q q 1/q0 ds (2.18) Choosing ı > such that q1 ı q K Ä 12 , it follows from (2.18) that I1 Ä CT C Sm t/ C CT Z th i C SNm s/ ds, (2.19) where we remark that, in what follows, CT always indicates a bound depending on T Second integral By (2.4), (2.14), and (2.17), we have kQu0 k˛L˛ kum t/k˛L˛ D˛ Z tD Z t IJ ˛ jum s/j kumx s/k˛ um s/, u0m s/ E Z u0m s/ ds Ä ˛ t ds Ä ˛ jj jum s/j˛ Z t ˛=2 Sm s/ds Ä ˛ jj u0m s/ ds Z th i C SNm s/ ds (2.20) Hence, by (2.12)3 , (2.14), (2.15), (2.17), and (2.20), it follows that ˇ ˇ kum t/k˛L˛ C kvm t/kLˇ Ä kQu0 k˛L˛ C kQv0 kLˇ C ˛ C ˇ/ Ä C0 C ˛ C ˇ/ Z th C SNm s/ Z th i i C SNm s/ ds (2.21) ds, where C0 always indicates a positive constant depending only on uQ , vQ , uQ , vQ , ˛, and ˇ Using H3 /, we deduce from (2.21) that Z tÄ @F @F um s/, vm s//, u0m s/ C um s/, vm s//, vm s/ @u @v Z tÄ Z d D2 F um x, s/, vm x, s//dx ds ds 0 Z Z D F Qu0 x/, vQ x//dx C F um x, t/, vm x, t//dx I2 D Ä2 Z supp jF y, z/j C 2C1 jyj,jzjÄ C0 D2 ds Á C jum x, t/j˛ C jvm x, t/jˇ dx (2.22) Ä Z th i C SNm s/ ds supp jF y, z/j C 2C1 C 2C1 C0 C ˛ C ˇ/ jyj,jzjÄ C0 Z th i C SNm s/ ds, Ä C C C0 where C0 always indicates a bound depending on uQ , vQ , uQ , vQ , ˛, and ˇ Third integral Using the assumption H2 /, we deduce from the Cauchy–Schwartz inequality that Z I3 D t ˝ ˛ ˝ ˛ F1 s/, u0m s/ C F2 s/, vm s/ ds Ä Ä kF1 k2L2 Q / T Copyright © 2013 John Wiley & Sons, Ltd C kF2 k2L2 Q / T Z t C Z t s/ kF1 s/k u0m s/ C kF2 s/k vm Z th i C SNm s/ ds Sm s/ds Ä CT C ds (2.23) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Hence, (2.13), (2.14), (2.15), (2.19), (2.22), and (2.23) lead to Sm t/ Ä CT C CT Z th i C SNm s/ ds, Ä t Ä Tm (2.24) Then by solving a nonlinear Volterra integral inequality (on the basis of the methods in [18]), we obtain the following lemma Lemma 2.4 There exists a constant T > depending on T (independent of m) such that Sm t/ Ä CT , 8m N, 8t Œ0, T , (2.25) where CT is a constant depending only on T as above Lemma 2.4 allows one to take constant Tm D T for all m The second estimate 00 0/ First of all, we are going to estimate u00m 0/ C vm 00 0/, we obtain Letting t ! 0C in (2.12)1 , multiplying the result by cmj u00m 0/ ˝ ˛ uQ 0xx , u00m 0/ C D jQu1 jr1 E ˝ ˛ ˝ ˛ uQ , u00m 0/ D f1 Qu0 , vQ /, u00m 0/ C F1 0/, u00m 0/ (2.26) jj C kf1 Qu0 , vQ /k C kF1 0/k Á C 01 for all m, (2.27) This implies that u00m 0/ Ä kQu0xx k C u1 j jj jQ r1 where C 01 is a constant depending only on r1 , , uQ , vQ , uQ , f1 , and F1 00 0/, and using the compatibility (2.7) to obtain Similarly, letting t ! 0C in (2.12)2 , multiplying the result by dmj 00 vm 0/ ˝ ˛ 00 vQ 0xx , vm 0/ C D jQv1 jr1 E ˝ ˛ ˝ ˛ 00 00 00 0/ D f2 Qu0 , vQ /, vm 0/ C F2 0/, vm 0/ vQ , vm (2.28) jj C kf2 Qu0 , vQ /k C kF2 0/k Á C 02 for all m, (2.29) This implies that 00 vm 0/ Ä kQv0xx k C v1 j jj jQ r2 where C 02 is a constant depending only on r2 , , uQ , vQ , vQ , f2 , and F2 Now differentiating (2.12) with respect to t, the results are ˛ ˝ ˛ ˝ ˛ ˝ 000 um t/, wj C u0mx t/, wjx C ‰r01 u0m t//u00m t/, wj ˆ ˆ ˆ ˆ ˆ ˛ ˝ @2 F @2 F ˆ 0 ˆ ˆ um , vm /vm u , v /u C , wj C F10 t/, wj , D m m ˆ m ˆ @u @u@v ˆ 4, p f2g [ Œ3, C1/, and we have Z J4 D 2K Copyright © 2013 John Wiley & Sons, Ltd t 00 ‰p0 vm 0, s//vm 0, s/vm 0, s/ds Ä CT C Xm t/ (2.38) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Proof of Lemma 2.5 We consider two cases for p, q/ : Case Ä q Ä 4, p : Case 1.a Ä q Ä 4, p D : We have Z t 00 vm 0, s/vm 0, s/ds D 2K J4 D 2K D K q Ä K q Using the inequality a4 q Z ˇ ˇ ˇv 0, s/ˇ1 m q ˇ ˇ ˇv 0, s/ˇ2 m q t Z t Ä C aq , 8a vm 0, s/ Z t ˇ ˇ ˇv 0, s/ˇ1 m  ˇq @ ˇˇ vm 0, s/ˇ @s  ˇq @ ˇˇ vm 0, s/ˇ j @s ˇ ˇq vm 0, s/ ˇvm 0, s/ˇ q vm 0, s/ vm 0, s/ 00 vm 0, s/ds à (2.39) ds à jds 0, Ä q Ä and the inequality 2ab Ä ı1 a2 C b , 8a, b ı1 0, ı1 > 0, (2.40) we deduce from (2.14), (2.25), and (2.32) that J4 Ä Ä Z tÄ 2K q Z 2K qı1 ˇ ˇ ˇv 0, s/ˇ4 m t  à ˇq @ ˇˇ vm 0, s/ˇ vm 0, s/ j2 ds @s Ãˇ2 Z ˇ  ˇq ˇ 2K t ˇˇ @ ˇˇ q ˇ v 0, s/ ˇ ds v ds C ı1 0, s/ m m ˇ q ˇ @s ˇ4 ˇˇ vm 0, s/ˇ ı1 q C ı1 j Z th ˇ ˇq i C ˇvm 0, s/ˇ ds C ı1 2K Ä qı1 (2.41) Kq Xm t/ q 1/ Ä Ä because of Xm t/ q 1/ R t @ j @s q2 Kq Kq 2K Sm t/ C ı1 Xm t/ Ä CT C ı1 Xm t/, TC qı1 q 1/ ı1 q 1/  ˇ ˇq ˇv 0, s/ˇ m Kq q 1/ Choosing ı1 > 0, with ı1 vm 0, s/ à j2 ds, Sm t/ ˇq Rtˇ ˇ ˇ vm 0, s/ ds Ä 12 , we obtain (2.38) Case 1.b Ä q Ä 4, p > : We have Z t J4 D 2K 0 00 ‰p0 vm 0, s//vm 0, s/vm 0, s/ds Z D 2K.p t 1/ 00 vm 0, s/vm 0, s/ds jvm 0, s/jp Z D 2K.p t 1/ D By jvm 0, s/j Ä jjvmx t/jj Ä K.p q p Z t 1/ Sm t/ Ä J4 Ä ˇ1 ˇ 0, s/ˇ jvm 0, s/jp ˇvm K.p q jvm 0, s/jp ˇ1 ˇ 2ˇ vm 0, s/ˇ q q (2.42) ˇ ˇ 00 vm 0, s/ ˇvm 0, s/ˇ vm 0, s/ds q vm 0, s/  ˇq @ ˇˇ vm 0, s/ˇ @s vm 0, s/ à ds p CT , we deduce from (2.42) that 1/ p CT Áp Z t ˇ ˇ ˇv 0, s/ˇ2 m q j  ˇq @ ˇˇ vm 0, s/ˇ @s vm 0, s/ à jds, (2.43) so (2.38) holds Copyright © 2013 John Wiley & Sons, Ltd Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Case q > 4, p f2g [ Œ3, C1/ : Case 2.a q > 4, p : By using integration by parts, we obtain Z t 00 ‰p0 vm 0, s//vm 0, s/vm 0, s/ds D K.p J4 D 2K Z D K.p ˇ2 ˇ 0, t/ˇ 1/ jvm 0, t/jp ˇvm Z K.p 1/.p t 2/ jvm 0, s/jp Ä K.p Ä K.p Ä K.p ˇ2 ˇ 2ˇ vm 0, t/ˇ 1/ jvm 0, t/jp 1/ 1/ p Áp Sm t/ p Áp CT Using the inequality a3 Ä C aq , 8a Z t 1/.p ˇ ˇv 0, t/ˇ2 C K.p m ˇ ˇv 0, t/ˇ2 C K.p m Z 0, 8q t ds t 2/ jvm 0, s/jp 2ˇ Z 1/.p ˇ2 d ˇˇ v 0, s/ˇ ds ds m Q2 v1 0/ Z C K.p 0 vm 0, s/ vm 0, s/ 2ˇ ˇ2 ˇ Ä CT ˇvm 0, t/ˇ C CT jvm 0, s/jp 1/ jQv0 0/jp K.p t 1/ p t 2/ ˇ3 ˇ 3ˇ vm 0, s/ˇ ds Áp Sm s/ ˇ ˇv 0, s/ˇ3 ds m 3ˇ 1/.p p 2/ CT Áp Z t (2.44) ˇ ˇ ˇv 0, s/ˇ3 ds m ˇ ˇ ˇv 0, s/ˇ3 ds m and because ˇ ˇ ˇv 0, s/ˇ3 ds Ä T C m Z t ˇq Rtˇ ˇ ˇ vm 0, s/ ds Ä Sm t/, we obtain ˇ ˇ ˇv 0, s/ˇq ds Ä T C Sm t/ Ä CT m (2.45) On the other hand, ˇq ˇ ˇv 0, t/ˇ m q vm 0, t/ D jv1m 0/j v1m 0/ C ˇ Hence, because of Xm t/ q 1/ R t ˇˇ @ ˇ @s q2  ˇ ˇq ˇv 0, s/ˇ m vm 0, s/ ˇZ ˇ ˇ ˇ ˇv 0, t/ˇq Ä jv1m 0/jq C ˇ m ˇ Z t  ˇq @ ˇˇ vm 0, s/ˇ @s à ds (2.46) Ãˇ2 ˇ ˇ ds, it follows from (2.46) that ˇ  ˇq @ ˇˇ vm 0, s/ˇ @s Z tˇ  ˇ@ ˇ ˇq ˇ ˇ ˇ2 Ä jv1m 0/jq C 2t ˇ @s vm 0, s/ Ä jv1m 0/jq C vm 0, s/ t vm 0, s/ à vm 0, s/ ˇ2 ˇ dsˇˇ Ãˇ2 ˇ ˇ ds ˇ (2.47) 2Tq2 Xm t/ q 1/ Using the inequalities a C b/2=q Ä a2=q C b2=q , 8a, b ab Ä q q ı q q q aq 2 q q C ı b , 8a, b q 0, 8q 2, 0, 8q > 2, 8ı > 0, we obtain from (2.12)3 and (2.47) that ˇ2 ˇ 0, t/ˇ Ä CT 22=q jv1m 0/j2 C CT CT ˇvm Ä C0 C q q q ı q "   CT 2Tq2 q 1/ 2Tq2 q 1/ Ã2=q Xm t//2=q Ã2=q # q 2 iq qh C ı Xm t//2=q q (2.48) q Ä CT ı/ C ı Xm t// q q Choose ı > 0, with q2 ı Ä 12 , from (2.44), (2.45) and (2.48), (2.38) follows Copyright © 2013 John Wiley & Sons, Ltd Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Case 2.b q > 4, p D : We have Z J4 D 2K t 00 ‰p0 vm 0, s//vm 0, s/vm 0, s/ds D K ˇ2 ˇ 0, t/ˇ D K ˇvm Z t ˇ2 d ˇˇ vm 0, s/ˇ ds ds (2.49) ˇ ˇ2 K vQ 12 0/ Ä K ˇvm 0, t/ˇ , and also using (2.48), we obtain (2.38) Lemma 2.5 is proved completely Combining (2.31), (2.32), (2.33), and (2.35)–(2.38) leads to Z t Xm t/ Ä 2C0 C 8CT C 2 C F10 s/ C F20 s/ Xm s/ds (2.50) By Gronwall’s lemma, (2.50) gives Ä Z Xm t/ Ä 2C0 C 8CT / exp t C F10 s/ C F20 s/ ds Ä CT , 8m N, 8t Œ0, T  (2.51) Step Limiting process From (2.14), (2.25), (2.32), and (2.51), we deduce the existence of a subsequence of f.um , vm /g still also so denoted, such that ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < um , vm / ! u, v/ in L1 0, T ; H01 V weak*, V weak*, / ! u0 , v / u0m , vm 00 / ! u00 , v 00 / u00m , vm / ! u0 , v / u0m , vm in L1 in L1 0, T in Lr1 QT / vm 0, / ! v.0, / in W 1,q 0, T / weakly, in Lq 0, T / weakly, in H1 0, T / weakly, 0, vm / ! v 0, ˆ ˆ ˆ ˆ ˇ ˇq ˆ ˆ ˇv 0, /ˇ ˆ m ˆ ˆ  ˆ ˆ ˇ ˇ r1 ˆ ˆ @ ˇ ˇ2 ˆ um ˆ @t ˆ ˆ ˆ ˆ  ˆ ˇ ˇ r1 ˆ ˆ : @ ˇv ˇ m @t / vm 0, um vm à /! ; H01 ; L2 0, T L2 / Lr2 QT / weak*, weakly, ! in L2 QT / weakly, ! in L2 QT / weakly à (2.52) By the compactness lemma of Lions [17, p 57] and the imbeddings H2 0, T / ,! C Œ0, T / , H1 0, T / ,! C Œ0, T / , W 1,q 0, T / ,! C Œ0, T /, we can deduce from (2.52) the existence of a subsequence still denoted by f.um , vm /g such that um , vm / ! u, v/ strongly in L2 QT / L2 QT / and a.e in QT , ˆ ˆ ˆ ˆ ˆ / ! u0 , v / < u0m , vm strongly in L2 QT / L2 QT / and a.e in QT , (2.53) ˆ strongly in C Œ0, T /, v 0, / ! v.0, / ˆ ˆ m ˆ ˆ ˇq : ˇˇ vm 0, /ˇ vm 0, / ! strongly in C Œ0, T / By means of the continuity of f1 , we have f1 um , vm / ! f1 u, v/ a.e .x, t/ in QT By kf1 um , vm /kL2 QT / Ä p T sup p jyj, jzjÄ CT (2.54) jf1 y, z/j < 1, we deduce from (2.54) and Lions’s lemma [17, Lemma 1.3, p 12] that f1 um , vm / ! f1 u, v/ in L2 QT / weakly (2.55) f2 um , vm / ! f1 u, v/ in L2 QT / weakly (2.56) Similarly, By means of the following inequality j‰r x/ Copyright © 2013 John Wiley & Sons, Ltd ‰r y/j Ä r 1/Rr jx yj , (2.57) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG for all x, y Œ R, R, R > 0, r 2, it follows from (2.32), (2.51), and (2.53)2 that ( ‰r1 u0m / ! ‰r1 u0 / strongly in L2 QT /, (2.58) / ! ‰ v / strongly in L2 Q / ‰r2 vm r2 T By means of the continuity of ‰p , it follows from (2.53)3 that ‰p vm 0, t// ! ‰p v.0, t// strongly in C Œ0, T / ˇ ˇq Put Ám D ˇvm 0, /ˇ vm 0, /! (2.59) , from (2.53)4 , and we have Ám ! strongly in C Œ0, T / (2.60) It follows from (2.60) that vm 0, / D jÁm j q Ám ! j 0j q strongly in C Œ0, T / (2.61) We deduce from (2.52)6 and (2.61) that j 0j q D v 0, / (2.62) Because of the continuity of ‰q , it follows from (2.61) and (2.62) that 0, // ! ‰q v 0, // strongly in C Œ0, T / ‰q vm (2.63) Passing to the limit in (2.12) by (2.52)1,2,3 , (2.53), (2.55), (2.56), (2.58), (2.59), and (2.63), we have u, v/ satisfying the problem ˝ ˛ hu00 t/, wi C hux t/, wx i C ‰r1 u0 t//, w D hf1 u, v/, wi C hF1 t/, wi , ˆ ˆ < ˝ ˛ hv 00 t/, i C hvx t/, x i C ‰r2 v t//, C ‰q v 0, t// 0/ ˆ ˆ : D K‰p v.0, t// 0/ C hf2 u, v/, i C hF2 t/, i , for all w, / H01 (2.64) V, together with the initial conditions u.0/, u0 0/ D Qu0 , uQ / , v.0/, v 0/ D Qv0 , vQ / (2.65) On the other hand, we have from (2.51)1,2,3 , (2.63), and H2 / that ( uxx D u00 C vxx D v 00 C ‰r1 u / ‰r2 v / f1 u, v/ F1 L1 0, T ; L2 /, f2 u, v/ F2 L1 0, T ; L2 / (2.66) Thus u, v/ L1 0, T; H01 \ H2 / V \ H2 // and the existence is proved completely Step Uniqueness of the solution Assume now that q D and p Let ui , vi /, i D 1, be two weak solutions of problems (1.1)–(1.3) such that ui , vi / L1 0, T ; H01 \ H2 V \ H2 / , ˆ ˆ ˆ ˆ ˆ 0 1 < ui , vi L 0, T ; H0 V \ Lr1 QT / Lr2 QT / , (2.67) u00i , vi00 L1 0, T ; L2 L2 /, ˆ ˆ ˆ ˆ ˆ : ˇˇ ˇˇ r21 ˇˇ ˇˇ r22 ui ui , vi vi H1 QT /, vi0 0, / H1 0, T / , i D 1, Then u, v/ D u1 u2 , v1 v2 / satisfies the variational problem ˝ ˛ hu00 t/, wi C hux t/, wx i C ‰r1 u01 t/ ‰r1 u01 t//, w ˆ ˆ ˆ ˆ ˆ ˆ D hf1 u1 , v1 / f1 u2 , v2 /, wi , ˆ ˆ ˆ ˛ ˝ ˆ < hv 00 t/, i C hvx t/, x i C ‰r v t// ‰r v t/ , C v 0, t/ 0/ 2 ˆ D K ‰p v1 0, t// ‰p v2 0, t// 0/ ˆ ˆ ˆ ˆ ˆ ˆ C hf2 u1 , v1 / f2 u2 , v2 /, i , for all w, / H01 V, ˆ ˆ ˆ : u.0/ D v.0/ D u0 0/ D v 0/ D Copyright © 2013 John Wiley & Sons, Ltd (2.68) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG We take w, / D u0 , v / in (2.68)1,2 , and integrating with respect to t, we obtain Z t t/ D ˝ ˛ f1 u2 , v2 /, u0 s/ ds C f1 u1 , v1 / Z t ˝ ˛ f2 u2 , v2 /, v s/ ds f2 u1 , v1 / Z t (2.69) ‰p v2 0, s// v 0, s/ds Á Z1 t/ C Z2 t/ C Z3 t/, ‰p v1 0, s// C 2K where t/ D u0 t/ C v t/ Z C2 t C2 t C kux t/k2 C kvx t/k2 ˝ ‰r1 u01 s// Z ˝ ‰r2 v10 s// ˛ ‰r1 u01 s//, u0 s/ ds (2.70) ˛ ‰r2 v20 s//, v s/ ds C Z t ˇ ˇ ˇv 0, s/ˇ2 ds Using the following inequality 8r 2, 9Cr > : ‰r x/ u0 t/ ‰r y//.x Cr jx y/ yjr , 8x, y R, (2.71) Z (2.72) it follows from (2.70) that t/ C v t/ C kux t/k2 C kvx t/k2 Z t Z t r u0 s/ L1r1 ds C 2Cr2 v s/ C 2Cr1 0 r2 Lr2 t ds C ˇ ˇ ˇv 0, s/ˇ2 ds Put R D max kuix kL1 0,T;H1 / C kvix kL1 0,T;H1 / , (2.73) ˇ ˇ ˇÃ ¡ ˇ ˇ @fi ˇ ˇ @fi ˇ ˇ ˇ ˇ Li R/ D sup ˇ @y y, z/ˇ C ˇ @z y, z/ˇ , i D 1, jyj, jzjÄR (2.74) iD1,2 and 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 ˇZ t ˇ ˝ f1 u1 , v1 / jZ1 t/j D ˇˇ Z t Ä 2L1 R/ Z ˇ Z t ˛ ˇ f1 u2 , v2 /, u0 s/ dsˇˇ Ä kf1 u1 , v1 / Œku.s/k C kv.s/k u0 s/ ds (2.75) t Ä 2L1 R/ f1 u2 , v2 /k u0 s/ ds Œkux s/k C kvx s/k u0 s/ ds Ä 2L1 R/ Z t s/ds Integral Z2 t/ Similarly, ˇZ t ˇ ˝ f2 u1 , v1 / jZ2 t/j D ˇˇ ˇ Z ˛ ˇ f2 u2 , v2 /, v s/ dsˇˇ Ä 2L2 R/ t s/ds (2.76) Integral Z3 t/ We have Z t ‰p v2 0, s// v 0, s/ds ‰p v1 0, s// Z3 t/ D 2K (2.77) Now, we consider two cases for p Case p D Note that, by (2.72), t/ v 0, t/ C Z Copyright © 2013 John Wiley & Sons, Ltd t ˇ ˇ ˇv 0, s/ˇ2 ds (2.78) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Thus Z t Z3 t/ D 2K v.0, s/v 0, s/ds Ä K2 Z Ä K2 Z t v 0, s/ds C Z t t ˇ ˇ ˇv 0, s/ˇ2 ds (2.79) s/ds C t/ Case p > By using the inequality (2.57), it follows from (2.73) that Z t ‰p v1 0, s// ‰p v2 0, s// v 0, s/ds Z3 D 2K Ä 2K.p 1/Rp Z t ˇ ˇ K2 p jv.0, s/j ˇv 0, s/ˇ ds Ä 1/2 R2p Z t s/ds C (2.80) t/ Put ( p D K p K2 1/2 R2p 4, p 3, (2.81) p D , Hence, it follows from (2.79)–(2.81) that Z Z3 t/ Ä t s/ds C p t/, if p f2g [ Œ3, C1/ (2.82) Combining (2.69), (2.75), (2.76), and (2.82), we obtain Z t/ Ä 2L1 R/ C 2L2 R/ C t s/ds p (2.83) By Gronwall’s lemma, (2.83) leads to t/ Á 0, that is, u D u1 Theorem 2.2 is proved completely u2 Á 0, v D v1 v2 Á Proof of Theorem 2.3 Let Qu0 , uQ / H01 L2 , Qv0 , vQ / V L2 , F1 , F2 / L2 QT /, and q D 2, p In order to obtain the existence of a weak solution, we use standard arguments of density Let us consider Qu0 , uQ / H01 L2 , Qv0 , vQ / V L2 , F1 , F2 / L2 QT / and let sequences f.u0m , u1m /g f.v0m , v1m /g C01 / C01 C01 , and f.F1m , F2m /g QT ˆ < u0m , u1m / ! Qu0 , uQ / v0m , v1m / ! Qv0 , vQ / ˆ : F1m , F2m / ! F1 , F2 / C01 C01 C01 , QT , such that strongly in H01 strongly in V strongly in L2 Q L2 , L2 , T/ (2.84) L2 Q T / So f.v0m , v1m /g satisfy, for all m N, the compatibility conditions v0mx 0/ C K jv0m 0/jp v0m 0/ D v1m 0/ (2.85) Then for each m N, there exists a unique function um , vm / in the conditions of Theorem 2.2 So we can verify ˝ ˛ 00 hum t/, wi C humx t/, wx i C ‰r1 u0m t//, w D hf1 um , vm /, wi C hF1m t/, wi , ˆ ˆ ˆ ˝ ˛ ˆ 00 0 ˆ ˆ ˆ hvm t/, i C hvmx t/, x i C ‰r2 vm t//, C vm 0, t/ 0/ < D K‰p vm 0, t// 0/ C hf2 um , vm /, i C hF2m t/, i , ˆ ˆ ˆ ˆ for all w, / H01 V, ˆ ˆ ˆ : 0/ D v , v /, um 0/, u0m 0/ D u0m , u1m / , vm 0/, vm 0m 1m (2.86) and um , vm / L1 0, T ; H01 \ H2 V \ H2 / \ C Œ0, T ; H01 ˆ ˆ ˆ ˆ L1 0, T ; H1 V \ C Œ0, T ; L2 L2 , ˆ < u0m , vm 00 L1 0, T ; L2 L2 /, ˆ u00m , vm ˆ ˆ ˆ ˆ : ˇ ˇ r21 ˇ ˇ r22 0 0, / H1 0, T / ˇu ˇ um , ˇvm ˇ vm H1 QT /, vm m Copyright © 2013 John Wiley & Sons, Ltd V \ C Œ0, T ; L2 L2 , (2.87) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG By the same arguments used to obtain the aforementioned estimates, we obtain Sm t/ D jju0m t/jj2 Z C2 C jjvm t/jj2 t r2 Lr2 s/ vm 2 Z C jjumx t/jj C jjvmx t/jj C t Z t ds C u0m s/ r1 Lr1 ds (2.88) ˇ ˇ ˇv 0, s/ˇ2 ds Ä CT , m 8t Œ0, T , where CT is a positive constant independent of m and t On the other hand, we put Um,l D um ul , Vm,l D vm vl , from (2.86), it follows that E ˝ D ˛ ˝ ˛ 00 t/, w C U 0 ˆ U m,lx t/, wx C ‰r1 um t// ‰r1 ul t//, w ˆ m,l ˆ ˆ ˆ ˆ ˆ D hf1 um , vm / f1 ul , vl /, wi C hF1m t/ F1l t/, wi , ˆ ˆ ˆ ˝ ˛ ˆ 00 0, t/ 0/ t// ‰ v t//, ˆ ˆ hV t/, i C hVm,lx t/, x i C ‰r2 vm C Vm,l r2 l ˆ < m,l D K ‰p vm 0, t// ‰p vl 0, t// 0/ ˆ ˆ ˆ ˆ ˆ C hf2 um , vm / f2 ul , vl /, i C hF2m t/ F2l t/, i , for all w, / H01 V, ˆ ˆ ˆ ˆ 0/ D u ˆ ˆ Um,l 0/ D u0m u0l , Um,l u1l , 1m ˆ ˆ ˆ : Vm,l 0/ D v0m v0l , Vm,l 0/ D v1m v1l D u0 We take w D Um,l m u0l , D v0 D Vm,l m (2.89) vl0 , in (2.89)1,2 and integrating with respect to t, we obtain Z Sm,l t/ D Sm,l 0/ C t ˝ f1 um , vm / Z t ˝ C2 ˛ f2 ul , vl /, Vm,l s/ ds f2 um , vm / Z t ˝ F1m s/ C2 Z ˛ ˝ F1l s/, Um,l s/ C F2m s/ t ‰p vm 0, s// C 2K ˛ f1 ul , vl /, Um,l s/ ds ˛ F2l s/, Vm,l s/ ds (2.90) ‰p vl 0, s// Vm,l 0, s/ds Á Sm,l 0/ C J1 C J2 C J3 C J4 , where 0 t/jj2 C jjVm,l t/jj2 C jjUm,l x t/jj2 C jjVm,l x t/jj2 Sm,l t/ D jjUm,l Z t ˝ ˛ ‰r1 u0m s// ‰r1 u0l s//, Um,l C2 s/ ds Z C2 t ˝ ‰r2 vm s// ‰r2 vl0 s//, Vm,l s/ ds C u1l k2 C kv1m Sm,l 0/ D ku1m Z ˛ v1l k2 C ku0mx (2.91) t 0 jVm,l 0, s/j2 ds, u0lx k2 C kv0mx v0lx k2 We shall estimate respectively the integrals on the right-hand side of (2.90) as follows: Integral J1 Put à  @fi @fi j y, z/j C j y, z/j LT D max supp iD1,2jyj, jzjÄ C @y @z T (2.92) (2.93) we have Z t ˝ J1 D f1 um , vm / Z ˛ f1 ul , vl /, Um,l s/ ds t Ä 2LT kum ul k C kvm Z t vl k/ jjUm,l s/jjds (2.94) jjUm,l x s/jj C jjVm,l x s/jj jjUm,l s/jjds Ä 4LT Ä 2LT Z t Sm,l s/ds Integral J2 Similarly, Z J2 D ˝ f2 um , vm / Copyright © 2013 John Wiley & Sons, Ltd t ˛ f2 ul , vl /, Vm,l s/ ds Ä 4LT Z t Sm,l s/ds (2.95) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Integral J3 Similarly, Z t J3 D ˝ ˛ ˝ F1l s/, Um,l s/ C F2m s/ F1m s/ Z t Ä2 F1l s/k jjUm,l s/jj C kF2m s/ kF1m s/ F1l k2L2 Q / C kF2m Ä kF1m ˛ F2l s/, Vm,l s/ ds F2l k2L2 Q / C T Z F2l s/k jjVm,l s/jj ds (2.96) t Sm,l s/ds T Integral J4 By using the inequality (2.57), it follows from (2.88) that Z t J4 D 2K ‰p vm 0, s// ‰p vl 0, s// Vm,l 0, s/ds Ä 2K.p Ä 2K.p Ä where Np D K p p Áp CT 1/ q p 1/ CT K p 2 t Z t p 1/2 CT Z Z jVm,l 0, s/jjVm,l 0, s/jds (2.97) jjVm,l x s/jjjVm,l 0, s/jds t Z jjVm,l x s/jj2 ds C t 0 jVm,l 0, s/j2 ds Ä Np Z t Sm,l s/ds C Sm,l t/, p 1/2 CT It follows from (2.90), (2.92), and (2.94)–(2.97) that Z Sm,l t/ Ä Rm,l C ÁT t Sm,l s/ds, (2.98) where ÁT D C Np C 8LT , Rm,l D 2Sm,l 0/ C kF1m F1l k2L2 Q / C kF2m T (2.99) F2l k2L2 Q / ! 0, as m, l ! T By Gronwall’s lemma, (2.98) gives Sm,l t/ Ä Rm,l exp.TÁT /, 8t Œ0, T  (2.100) Convergences of the sequences f.u0m , u1m /g and f.v0m , v1m /g imply the convergence to zero (when m, l ! 1) of terms on the right-hand side of (2.100) Therefore, we obtain u , v / ! u, v/ strongly in C Œ0, T ; H01 V/ \ C Œ0, T ; L2 L2 /, ˆ < m m ! u0 , v / strongly in Lr1 Q / Lr2 Q /, u0m , vm (2.101) T T ˆ : strongly in H1 0, T / vm 0, / ! v.0, / On the other hand, from (2.88), we deduce the existence of a subsequence of f.um , vm /g still also so denoted, such that ( um , vm / ! u, v/ in L1 0, T ; H01 V weakly*, ! u0 , v u0m , vm in L1 0, T ; L2 L2 weakly* (2.102) Similarly, by (2.101), we deduce that f1 um , vm / ! f1 u, v/ ˆ ˆ ˆ ˆ < f2 um , vm / ! f2 u, v/ ˆ ‰r1 u0m / ! ‰r1 u0 / ˆ ˆ ˆ : / ! ‰ v / ‰r2 vm r2 strongly in L2 QT /, strongly in L2 QT /, strongly in L2 QT /, strongly in L2 QT / Passing to the limit in (2.86) by (2.101)–(2.103), we have u, v/ satisfying the problem d ˝ ˛ 0 ˆ ˆ dt hu t/, wi C hux t/, wx i C ‰r1 u t//, w D hf1 u, v/, wi C hF1 t/, wi , ˆ ˆ ˛ ˝ ˆ d ˆ ˆ hv t/, i C hvx t/, x i C ‰r2 v t//, C v 0, t/ K‰p v.0, t// 0/ ˆ < dt D hf2 u, v/, i C hF2 t/, i , ˆ ˆ ˆ ˆ ˆ for all w, / H01 V, ˆ ˆ ˆ : u.0/, u0 0/ D Qu0 , uQ / , v.0/, v 0/ D Qv0 , vQ / Copyright © 2013 John Wiley & Sons, Ltd (2.103) (2.104) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Next, the uniqueness of a weak solution is obtained by using the well-known regularization procedure due to Lions Theorem 2.3 is proved completely Remark 2.4 n In case max ˛2 , ˇ2 , q.p 2.q 1/ 1/ o Ä 1, and F1 , F2 / L2 QT /, Qu0 , uQ / H01 L2 , Qv0 , vQ / V L2 , the integral inequality (2.24) leads to the following global estimation Sm t/ Ä CT , 8m N, 8t Œ0, T, 8T > (2.105) Then by applying a similar argument used in the proof of Theorem 2.3, we can obtain a global weak solution u, v/ of problems (1.1)–(1.3) satisfying Á Á (2.106) u, v/ L1 0, T; H01 V , u0 , v / L1 0, T; L2 L2 , v.0, / H1 0, T/ However, in case max C1 Œ0, T; L2 L2 n ˛ ˇ q.p 1/ , , 2.q 1/ o Ä 1, we not imply that a weak solution obtained here belongs to C Œ0, T; H01 V \ Furthermore, the uniqueness of a weak solution is also not asserted Finite time blow up In this section, we consider problems (1.1)–(1.3) corresponding to F1 D F2 D 0, q D r1 D r2 D 2, p > 2, show that the solution of this problem blows up in finite time if H.0/ D 1 1 kQu1 k2 C kQv1 k2 C kQu0x k2 C kQv0x k2 2 2 K jQv0 0/jp p Z 1 > 0, > 0, F Qu0 x/, vQ x//dx < > We shall (3.1) First, we add the following assumption H30 There exist F C R2 ; R/ and the constants ˛, ˇ > 2; d1 , d2 , dN , dN > such that (i) @F u, v/ D f1 u, v/, @F u, v/ D f2 u, v/, @u @v (ii) d1 F u, v/ Ä uf1 u, v/ C vf2 u, v/ Ä d2 F u, v/, for all u, v/ R2 , Á Á (iii) dN juj˛ C jvjˇ Ä F u, v/ Ä dN juj˛ C jvjˇ , for all u, v/ R2 Then we obtain the theorem Theorem 3.1 o n ˚ « Let H30 holds and H.0/ > Let max Kp , dN < K, d1 dN in H30 Then for any Qu0 , uQ / H01 L2 , Qv0 , vQ / V L2 , the solution u, v/ of problems (1.1)–(1.3) blows up in finite time Proof We denote by E.t/ the energy associated to the solution u, v/, defined by E.t/ D u0 t/ C v t/ C kux t/k2 C kvx t/k2 Á K jv.0, t/jp p Z F u.x, t/, v.x, t//dx, (3.2) and we put H.t/ D E.t/ (3.3) On the other hand, by multiplying (1.1) by u0 x, t/, v x, t/ and integrating over Œ0, 1, we obtain H0 t/ D u0 t/ C v t/ C ˇ ˇ ˇv 0, t/ˇ2 0, 8t Œ0, T / (3.4) Hence, we can deduce from (3.4) and H.0/ > that < H.0/ Ä H.t/, 8t Œ0, T / (3.5) By means of H30 , iii , we obtain Á Z ˇ dN ku.t/k˛L˛ C kv.t/k ˇ Ä L Copyright © 2013 John Wiley & Sons, Ltd Á ˇ F u.x, t/, v.x, t//dx Ä dN ku.t/k˛L˛ C kv.t/k ˇ L (3.6) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Hence, we deduce from (3.2), (3.5), and (3.6) that Á Á 2 u0 t/ C v t/ kux t/k2 C kvx t/k2 2 Z K C jv.0, t/jp C F u.x, t/, v.x, t//dx p Á K ˇ Ä jv.0, t/jp C dN ku.t/k˛L˛ C kv.t/k ˇ L p Á ˇ Q Ä d2 jv.0, t/jp C ku.t/k˛L˛ C kv.t/k ˇ , 8t Œ0, T /, < H.0/ Ä H.t/ D (3.7) L where dQ D max n K N p , d2 o Hence Á ˇ kux t/k2 C kvx t/k2 Ä 2dQ jv.0, t/jp C ku.t/k˛L˛ C kv.t/kLˇ , 8t Œ0, T / (3.8) Now, we define the functional Á L.t/ D H1 t/ C " t/, (3.9) where t/ D hu.t/, u0 t/i C hv.t/, v t/i C 2 ku.t/k2 C kv.t/k2 C v 0, t/, (3.10) for " small enough and < Á Ä Lemma 3.2 There exists a constant 1 ˛ ˇ , < 2˛ 2ˇ (3.11) > such that L0 t/ H.t/ C u0 t/ Cku.t/k˛L˛ C kux t/k2 C kvx t/k2 Á ˇ C kv.t/k ˇ C jv.0, t/jp C v t/ (3.12) L Proof of Lemma 3.2 By multiplying (1.1) by u.x, t/, v.x, t// and integrating over Œ0, 1, we obtain t/ D u0 t/ C v t/ kux t/k2 kvx t/k2 C K jv.0, t/jp (3.13) C hf1 u.t/, v.t//, u.t/i C hf2 u.t/, v.t//, v.t/i By taking a derivative of (3.9) and using (3.13), we obtain L0 t/ D Á/H Á t/H0 t/ C " u0 t/ C v t/ Á " kux t/k2 C kvx t/k2 Á C"K jv.0, t/jp C " hf1 u.t/, v.t//, u.t/i C hf2 u.t/, v.t//, v.t/i/ (3.14) As (3.4), (3.5), (3.8), (3.14), and the following inequality Z hf1 u.t/, v.t//, u.t/i C hf2 u.t/, v.t//, v.t/i d1 F u.x, t/, v.x, t//dx Á ˇ d1 dN ku.t/k˛L˛ C kv.t/k ˇ , (3.15) L the result is L0 t/ " " D" Á Á 2 ˇ 2"dQ jv.0, t/jp C ku.t/k˛L˛ C kv.t/k ˇ u0 t/ C v t/ L Ái h ˇ C" K jv.0, t/jp C d1 dN ku.t/k˛L˛ C kv.t/k ˇ L Á Á 2 ˇ p 0 Q 2"d2 jv.0, t/j C ku.t/k˛L˛ C kv.t/k ˇ u t/ C v t/ L h Ái ˇ p ˛ Q C"d1 jv.0, t/j C ku.t/kL˛ C kv.t/k ˇ Á Á L Á 2 ˇ u0 t/ C v t/ C " dQ 2dQ jv.0, t/jp C ku.t/k˛L˛ C kv.t/k ˇ , Copyright © 2013 John Wiley & Sons, Ltd (3.16) L Math Meth Appl Sci 2013 L T P NGOC AND N T LONG « ˚ where dQ D K, d1 dN Note that « ˚ 2dQ D K, d1 dN < dQ D dQ ˚ « K N , d2 < K, d1 dN p max (3.17) Using inequalities (3.7), (3.8), (3.17), and (3.16) gives L0 t/ H.t/ C u0 t/ Cku.t/k˛L˛ for C kux t/k2 C kvx t/k2 Á ˇ C kv.t/k ˇ C jv.0, t/jp , C v t/ (3.18) L is a positive constant Lemma 3.2 is proved completely Remark 3.1 From the formula of L.t/ and Lemma 3.2, we can choose " small enough such that L.0/ > 0, 8t Œ0, T / L.t/ Now we continue with proof of Theorem 3.1 Using the inequality Ãr ÂX xi Ä 6r X6 iD1 xr , iD1 i Á/ t/ Ä Const ˇ ˇ1=.1 H.t/ C ˇhu.t/, u0 t/iˇ (3.19) for all r > 1, and x1 , : : : , x6 0, (3.20) we deduce from (3.9) and (3.10) that L1=.1 Cku.t/k2=.1 Á/ Á/ ˇ ˇ1=.1 C ˇhv.t/, v t/iˇ Á/ C kv.t/k2=.1 C jv.0, t/j2=.1 Á/ Á/ Á (3.21) On the other hand, by using Young’s inequality, we have ˇ ˇ ˇhu.t/, u0 t/iˇ1=.1 Á/ Ä ku.t/k1=.1 Á/ ku0 t/k1=.1 Á/ 1=.1 Á/ Ä Const ku.t/kL˛ Ä Const ku.t/ksL˛ ku0 t/k1=.1 Á C ku0 t/k2 , Á/ (3.22) where s D 2=.1 2Á/ Ä f˛, ˇg by (3.11) Similarly, we obtain ˇ ˇ ˇhv.t/, v t/iˇ1=.1 Á/ Á Ä Const kv.t/ksLˇ C kv t/k2 (3.23) Here, we need the following lemma Lemma 3.3 Let Ä s D 2=.1 2Á/ Ä minf˛, ˇg, Ä 2=.1 Á/ Ä minf˛, ˇ, pg, (3.24) then i/ kuksL˛ C kuk2=.1 Á/ Á Ä kux k2 C kuk˛L˛ , for all u H01 , ii/ kvks ˇ C kvk2=.1 Á/ C jv.0/j2=.1 L Á/ Á ˇ Ä kvx k2 C kvk ˇ C jv.0/jp , for all v V L (3.25) Proof of (3.25, (i)) We consider two cases for kukL˛ : Case kukL˛ Ä : By Ä s Ä ˛, we have Case kukL˛ kuksL˛ Ä kuk2L˛ Ä kux k2 Ä kux k2 C kuk˛L˛ (3.26) kuksL˛ Ä kuk˛L˛ Ä kux k2 C kuk˛L˛ (3.27) : By Ä s Ä ˛, we have Copyright © 2013 John Wiley & Sons, Ltd Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Therefore, kuksL˛ Ä kux k2 C kuk˛L˛ , for any u H01 (3.28) We consider two cases for kuk : Case kuk Ä : By Ä 2=.1 Á/ Ä ˛, we have kuk2=.1 Case kuk : By Ä 2=.1 Á/ Ä kuk2 Ä kux k2 Ä kux k2 C kuk˛L˛ (3.29) Á/ Ä kuk˛ Ä kuk˛L˛ Ä kux k2 C kuk˛L˛ (3.30) Á/ Ä ˛, we have kuk2=.1 Therefore, kuk2=.1 Á/ Ä kux k2 C kuk˛L˛ , for any u H01 (3.31) Combining (3.28) and (3.31), we obtain kuksL˛ C kuk2=.1 Á/ Á Ä kux k2 C kuk˛L˛ , 8u H01 (3.32) Proof of (3.25, (ii)) It is similar to the proof of (3.25, (i)) Lemma 3.3 is proved Combining (3.21)–(3.25), we obtain L1=.1 H.t/ C ku0 t/k2 C kv t/k2 C kux t/k2 C ku.t/k˛L˛ Á ˇ C kvx t/k2 C kv.t/k ˇ C jv.0, t/jp , 8t Œ0, T / Á/ t/ Ä Const (3.33) L Using Lemma 3.2, we deduce from (3.33), that L0 t/ where 1=.1 Á/ t/, 2L 8t Œ0, T /, (3.34) is a positive constant By integrating (3.34) over 0, t/, we deduce that LÁ=.1 Á/ t/ L 2Á Á=.1 Á/ 0/ Át , 0Ät< Á L Á Á=.1 Á/ 0/ (3.35) Therefore, (3.35) shows that L.t/ blows up in a finite time given by T D Á 2Á L Á=.1 Á/ 0/ (3.36) Theorem 3.1 is proved completely Exponential decay In this section, we also consider problems (1.1)–(1.3) corresponding to r1 D r2 D q D 2, p > We show that each solution u, v/ of R1 (1.1)–(1.3) is global and exponential decays provided that I.0/ D kQu0x k2 C kQv0x k2 K jQv0 0/jp p F Qu0 x/, vQ x//dx > and E.0/ is small enough Let u, v/ be a weak solution of problems (1.1)–(1.3) satisfying (2.10) In order to obtain the decay result, we use the functional L.t/ D E.t/ C ı t/, Copyright © 2013 John Wiley & Sons, Ltd (4.1) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG where ı is a positive constant and E.t/ D u0 t/ Z C v t/ Á C Á kux t/k2 C kvx t/k2 K jv.0, t/jp p (4.2) F u.x, t/, v.x, t//dx, t/ D hu.t/, u0 t/i C hv.t/, v t/i C 2 ku.t/k2 C kv.t/k2 C v 0, t/ (4.3) We rewrite E.t/ as follows E.t/ D u0 t/ C v t/ Á C J.t/, (4.4) where  J.t/ D p à Á kux t/k2 C kvx t/k2 C I.t/, p I.t/ D kux t/k2 C kvx t/k2 K jv.0, t/jp Z p (4.5) F u.x, t/, v.x, t//dx (4.6) We further assume the following Á Á H200 / F1 , F2 L1 RC ; L2 \ L1 RC ; L2 Then we have the following theorem Theorem 4.1 Assume that H1 /, H200 /, and H30 / hold with d2 < p in H30 Let I.0/ > 0, and the initial energy E.0/ satisfies Ã˛ Â Ãˇ Ãp   2 2p 2p 2p 5CK E E E Á D pdN C < 1, p p p where E D E.0/ C / exp.2 /, D Assume that (4.7) R 1 kF1 t/k C kF2 t/k/ dt kF1 t/k2 C kF2 t/k2 Ä Á1 exp Á2 t/ for all t where Á1 and Á2 are two positive constants Then there exist positive constants C and E.t/ Ä C exp t/, for all t 0, (4.8) such that (4.9) Proof First, we need the following lemmas Lemma 4.2 The energy functional E.t/ satisfies (i) E t/ Ä u0 t/ C v t/ Á ˇ ˇ ˇv 0, t/ˇ2 1 kF1 t/k C kF2 t/k/ C kF1 t/k C kF2 t/k/ u0 t/ 2 Á ˇ ˇ "1 Á 2 ˇv 0, t/ˇ2 u0 t/ C v t/ (ii) E t/ Ä Á C kF1 t/k2 C kF2 t/k2 , 2"1 C for all "1 > 0, and D minf C v t/ Á , (4.10) , g > Copyright © 2013 John Wiley & Sons, Ltd Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Proof of Lemma 4.2 Multiplying (1.1) by u0 x, t/, v x, t/ and integrating over Œ0, 1, we obtain E t/ D u0 t/ ˇ ˇ ˇv 0, t/ˇ2 C hF1 t/, u0 t/i C hF2 t/, v t/i v t/ (4.11) On the other hand, kF1 t/k C hF2 t/, v t/i Ä kF2 t/k C hF1 t/, u0 t/i Ä kF1 t/k u0 t/ kF2 t/k v t/ 2 , (4.12) Thus, hF1 t/, u0 t/i C hF2 t/, v t/i Ä kF1 t/k C kF2 t/k/ C kF1 t/k C kF2 t/k/ u0 t/ C v t/ Á (4.13) Combining (4.11) and (4.13), it is easy to see that (4.10)i holds Similarly, we have also kF1 t/k2 C 2"1 hF2 t/, v t/i Ä kF2 t/k2 C 2"1 hF1 t/, u0 t/i Ä "1 u t/ "1 v t/ 2 , (4.14) Combining (4.11), (4.14), it is easy to see (4.10)ii holds Lemma 4.2 is proved completely Lemma 4.3 Suppose that H1 /, H200 /, and H30 / hold with d2 < p in H30 Then if we have I.0/ > and the initial energy E.0/ satisfies (4.7), then I.t/ > 0, 8t Proof of Lemma 4.3 By the continuity of I.t/ and I.0/ > 0, there exists T1 > such that 0, 8t Œ0, T1 , I.t/ (4.15) and this implies that J.t/ D p 2p Á kux t/k2 C kvx t/k2 C I.t/ p p Á kux t/k2 C kvx t/k2 , 8t Œ0, T1  2p (4.16) It follows from (4.16) that kux t/k2 C kvx t/k2 Ä 2p p 2p J.t/ Ä p E.t/, 8t Œ0, T1  (4.17) Combining (4.10)i and (4.17), and using Gronwall’s inequality, we have kux t/k2 C kvx t/k2 Ä 2p p E.t/ Ä 2p p E Á 2p p E.0/ C / exp.2 /, 8t Œ0, T1 , (4.18) where as in (4.7) Hence, from H30 , iii//, (4.7), and (4.18), the result is K jv.0, t/jp C p Ä K kvx t/kp ÄK " 2p p 2E Ä pdN ÁÁ R1 ˇ F u.x, t/, v.x, t//dx Ä K kvx t/kp C pdN ku.t/k˛L˛ C kv.t/k ˇ L Á kvx t/k2 C pdN kux t/k˛ kux t/k2 C kvx t/kˇ kvx t/k2 Á Áp 2 2p p 2E kvx t/k C pdN 2 Á˛ 2 C 2p p 2E 2p p 2E Áˇ Á˛ 2 kux t/k2 ! 2 CK 2p p 2E Áp C # 2p p 2E Áˇ ! 2 kvx t/k kux t/k2 C kvx t/k2 (4.19) Á Á Á kux t/k2 C kvx t/k2 < kux t/k2 C kvx t/k2 , 8t Œ0, T1  Therefore, I.t/ > 0, 8t Œ0, T1  Copyright © 2013 John Wiley & Sons, Ltd Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Now, we put T D sup fT > : I.t/ > 0, 8t Œ0, Tg If T < C1; then, because of the continuity of I.t/, we have I.T / same arguments as above, we can deduce that there exists T2 > T such that I.t/ > 0, 8t Œ0, T2  This leads to I.t/ > 0, 8t Lemma 4.3 is proved completely By the Lemma 4.4 Let I.0/ > and (4.7) hold Then there exist the positive constants ˇ1 and ˇ2 such that ˇ1 E.t/ Ä L.t/ Ä ˇ2 E.t/, 8t 0, (4.20) for ı is small enough Proof of Lemma 4.4 It is easy to see that L.t/ D à Á kux t/k2 C kvx t/k2 C p ı ı C ıhu.t/, u0 t/i C ıhv.t/, v t/i C ku.t/k2 C kv.t/k2 C 2 u0 t/ C v t/ Á  C I.t/ p ı v 0, t/ (4.21) On the other hand, hu.t/, u0 t/i Ä 1 u t/ kux t/k2 C 2 , hv.t/, v t/i Ä 1 v t/ kvx t/k2 C 2 (4.22) This implies that L.t/ Á 2 ı/ u0 t/ C v t/ 2 à Á 1 ı C kux t/k2 C kvx t/k2 C I.t/ p p (4.23) ˇ1 E.t/, where < lˇ1 D 1, : ı, < ı < minf1, 1 ı p = p ; gD1 p p ( D ) ı ı, p > 0, ı is small enough, (4.24) Similarly, we can prove that L.t/ Ä Á 2 C ı/ u0 t/ C v t/ 2 à Á ı ı ı ı 1 C C C C C kux t/k2 C kvx t/k2 C I.t/ Ä ˇ2 E.t/, p 2 2 p (4.25) where ( ˇ2 D max C ı, C ı C C 2 p C / ) (4.26) Lemma 4.4 is proved completely Lemma 4.5 Let I.0/ > and (4.7) hold The functional t/ defined by (4.3) satisfies t/ Ä u t/  C 2  C v t/ à d2 K jv.0, t/jp p "2 d2 p à kux t/k2 C kvx t/k2 Á Á d2 I.t/ C kF1 t/k2 C kF2 t/k2 p 2"2 (4.27) for all "2 > Copyright © 2013 John Wiley & Sons, Ltd Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Proof of Lemma 4.5 By multiplying (1.1) by u.x, t/, v.x, t// and integrating over Œ0, 1, we obtain t/ D u0 t/ C v t/ kux t/k2 kvx t/k2 C K jv.0, t/jp (4.28) C hf1 u.t/, v.t//, u.t/i C hf2 u.t/, v.t//, v.t/i C hF1 t/, u.t/i C hF2 t/, v.t/i On the other hand, Z hf1 u.t/, v.t//, u.t/i C hf2 u.t/, v.t//, v.t/i Ä d2 and by I.t/ D kux t/k2 C kvx t/k2 K jv.0, t/jp p R1 F u.x, t/, v.x, t//dx, F u.x, t/, v.x, t//dx > 0, for all t Z hf1 u.t/, v.t//, u.t/i C hf2 u.t/, v.t//, v.t/i Ä d2 d2 h Ä kux t/k2 C kvx t/k2 p K jv.0, t/j p (4.29) 0, we have F u.x, t/, v.x, t//dx (4.30) i I.t/ By hF1 t/, u.t/i C hF2 t/, v.t/i Ä Á Á "2 kux t/k2 C kvx t/k2 C kF1 t/k2 C kF2 t/k2 , 2"2 (4.31) for all "2 > It follows from (4.28), (4.30), and (4.31) that t/ Ä u0 t/  C à  Á "2 d2 C v t/ kux t/k2 C kvx t/k2 p à Á d d2 K jv.0, t/jp I.t/ C kF1 t/k2 C kF2 t/k2 , p p 2"2 (4.32) for all "2 > Hence, Lemma 4.5 is proved by using some simple estimates Now we continue with the proof of Theorem 4.1 It follows from (4.1), (4.10)ii , and (4.27) that "1 Á Á Á ˇ ˇ ˇv 0, t/ˇ2 C kF1 t/k2 C kF2 t/k2 C v t/ 2"1 à  Á Á d " 2 2 ı C ı u0 t/ C v t/ kux t/k2 C kvx t/k2 p à  Á ı ıd d2 K jv.0, t/jp I.t/ C Cı kF1 t/k2 C kF2 t/k2 p p 2"2 Á "1 Á 2 u0 t/ C v t/ Ä ı à à   Á "2 d2 d2 ıd2 K jv.0, t/jp I.t/ ı kux t/k2 C kvx t/k2 C ı p p p  à Á 1 ı C C kF1 t/k2 C kF2 t/k2 "1 "2 L0 t/ Ä u0 t/ (4.33) Note that  ı d2 p à  K jv.0, t/jp Ä ı  Äı  Äı Copyright © 2013 John Wiley & Sons, Ltd d2 p à K kvx t/kp kvx t/k2 à  Ãp  2p d2 K E kvx t/k2 Ä ı p p à Á d2 Á kux t/k2 C kvx t/k2 p d2 p à Á kvx t/k2 (4.34) Math Meth Appl Sci 2013 L T P NGOC AND N T LONG Hence, L0 t/ Ä "1 Á  Á / ı Ä ı C  ı C "1 "2 à u0 t/ d2 p à C v t/ "2 2 Á kux t/k2 C kvx t/k2 Á ıd2 I.t/ p (4.35) Á kF1 t/k2 C kF2 t/k2 for all ı, "1 , "2 > Let d2 < p, < "2 < 2.1 Then for ı small enough, with < ı < D minf , g and if "1  Á / à (4.36) satisfies < "1 < We deduce from (4.20) and (4.35)–(4.37) that there exists a constant , < L0 t/ Ä d2 p ı/ (4.37) < Á2 , such that L.t/ C ÁN exp Á2 t/, 8t (4.38) Combining (4.20) and (4.38), (4.9) follows Theorem 4.1 is proved completely Acknowledgements The authors wish to express their sincere thanks to the referees for their valuable comments This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number B2013-18-05 References Bergounioux M, Long NT, Dinh APN Mathematical model for a shock problem involving a linear viscoelastic bar Nonlinear Analysis 2001; 43:547–561 Cavalcanti MM, Domingos VN, Prates Filho JS, Soriano JA Existence and uniform decay of solutions of a degenerate equation with nonlinear boundary damping and boundary memory source term Nonlinear Analysis 1999; 38:281–294 Cavalcanti MM, Domingos VN, Soriano JA On the existence and the uniform decay of a hyperbolic Southeast Asian Bulletin of Mathematics 2000; 24:183–199 Cavalcanti MM, Domingos VN, Santos ML Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary Applied Mathematics and Computation 2004; 150:439–465 Fei L, Hongjun G Global nonexistence of positive initial-energy solutions for coupled nonlinear wave equations with damping and source terms Abstract and Applied Analysis 2011:Article ID 760209, 14 pages Long NT, Dinh APN On the quasilinear wave equation: utt u C f u, ut / D associated with a mixed nonhomogeneous condition Nonlinear Analysis 1992; 19:613–623 Long NT, Dinh APN A semilinear wave equation associated with a linear differential equation with Cauchy data Nonlinear Analysis 1995; 24:1261–1279 Long NT, Diem TN On the nonlinear wave equation utt uxx D f x, t, u, ux , ut / associated with the mixed homogeneous conditions Nonlinear Analysis 1997; 29(11):1217–1230 Long NT, Dinh APN, Diem TN On a shock problem involving a nonlinear viscoelastic bar, J Boundary Value Problems Hindawi Publishing Corporation (2005); 2005(3):337–358 10 Long NT, Ngoc LTP On a nonlinear wave equation with boundary conditions of two-point type Journal of Mathematical Analysis and Applications 2012; 385(2):1070–1093 11 Messaoudi SA Blow up and global existence in a nonlinear viscoelastic wave equation Mathematische Nachrichten 2003; 260:58–66 12 Miao C, Zhu Y Global smooth solutions for a non-linear system of wave equations Nonlinear Analysis 2007; 67:3136–3151 13 Santos ML Decay rates for solutions of a system of wave equations with memory Electronic Journal of Differential Equations 2002; 38:1–17 14 Truong LX, Ngoc LTP, Dinh APN, Long N T Nonlinear Analysis, Theory, Methods & Applications, Series A: Theory and Methods 2011; 74(18):6933–6949 15 Wu J, Li S Blow-up for coupled nonlinear wave equations with damping and source Applied Mathematics Letters 2011; 24:1093–1098 16 Zhang Z, Miao X Global existence and uniform decay for wave equation with dissipative term and boundary damping Computers and Mathematics with Applications 2010; 59:1003–1018 17 Lions JL Quelques Méthodes de Ré solution des Problèmes Aux limites Nonlinéaires Dunod: Gauthier–Villars, Paris, 1969 18 Lakshmikantham V, Leela S Differential and Integral Inequalities, Vol Academic Press: New York, 1969 Copyright © 2013 John Wiley & Sons, Ltd Math Meth Appl Sci 2013 ... The aforementioned works lead to the study of the existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations associated with initial and Dirichlet boundary conditions. .. initial 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. .. type Journal of Mathematical Analysis and Applications 2012; 385(2):1070–1093 11 Messaoudi SA Blow up and global existence in a nonlinear viscoelastic wave equation Mathematische Nachrichten