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This article was originally published in a journal published by Elsevier, the attached copy is provided by Elsevier for the author s benefit for the benefit of the author s institution, for

1 This article was originally published in a journal published by Elsevier, the attached copy is provided by Elsevier for the author s benefit for the benefit of the author s institution, for non-commercial research educational use including without limitation use in instruction at your institution, sending it to specific colleagues that you know, providing a copy to your institution s administrator. All other uses, reproduction distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your sonal or institution s website or repository, are prohibited. For exceptions, mission may be sought for such use through Elsevier s missions site at:

Nonlinear Analysis 67 (27) 2527 2539 www.elsevier.

University of Texas Pan American Edinburg, TX 78541, USA b Department of Mathematics Statistics, Concordia University Montreal, Quebec H3G 1M8, Canada c Department of Mathematics, Champlain College

2 Nonlinear Analysis 67 (27) Asymptotic behavior of solutions to the Rosenau Burgers equation with a iodic initial boundary Abstract Liping Liu a, Ming Mei b,c,, Yau Shu Wong d a Department of Mathematics, University of Texas Pan American Edinburg, TX 78541, USA b Department of Mathematics Statistics, Concordia University Montreal, Quebec H3G 1M8, Canada c Department of Mathematics, Champlain College at St.-Lambert, St.-Lambert, Quebec J4P 3P2, Canada d Department of Mathematical Statistical Sciences, University of Alberta Edmonton, Alberta T6G 2G1, Canada Received 18 July 24; accepted 25 August 26 This study focuses on the Rosenau Burgers equation u t + u xxxxt αu xx + f (u) x with a iodic initial boundary condition. It is proved that with smooth initial value the global solution uniquely exists. Furthermore, for α >, the global solution converges time asymptotically to the average of the initial value in an exponential form, the convergence rate is optimal; while for α, the unique solution oscillates around the initial average all the time. Finally, the numerical simulations are reported to confirm the theoretical results. c 26 Elsevier Ltd. All rights reserved. MSC: 35Q53; 35B4; 76B15 Keywords: Rosenau Burgers equation; Periodic boundary condition; Convergence; Oscillation 1. Introduction main results In the study of the dynamics of dense discrete systems, the cases of wave wave wave wall interactions cannot be treated using the well-known KdV equations. To overcome this shortcoming, Rosenau [17] proposed the so-called Rosenau equation u t + u xxxxt + u x + uu x. Since then, much work has been done on the solution existence uniqueness, as well as numerical schemes by the Galerkin method; cf. [3 6,8,13 16], the references therein. On the other h, for the further consideration of the dissipation in space for the dynamic system, such as the phenomenon of bore propagation the water waves, the viscous term αu xx needs to be included: u t + u xxxxt αu xx + u x + uu x, Corresponding author at: Department of Mathematics Statistics, Concordia University, Montreal, Quebec H3G 1M8, Canada. Tel.: x3236; fax: address: mei@mathstat.concordia.ca (M. Mei) X/$ - see front matter c 26 Elsevier Ltd. All rights reserved. doi:1.116/j.na

3 2528 L. Liu et al. / Nonlinear Analysis 67 (27) with α >. This equation is usually called the Rosenau Burgers equation, because its dissipative effect is the same as that in the Burgers equation u t αu xx + u x + uu x. The asymptotic behavior of the solution for the Cauchy problem to the Rosenau Burgers equation, in particular, the stability of traveling waves diffusion waves, have been well studied in [7,1,11]. Subsequently, this study focuses on the iodic initial boundary value problem for the generalized Rosenau Burgers equation: u t + u xxxxt αu xx + f (u) x, x R, t R +, u t u (x) x R, (1.1) u (x) u (x + 2L), x R, where f (u) is a smooth function of u, L >, 2L is the iod of the initial value u (x). The main purpose is to investigate the asymptotic behavior of the solution u(x, t) to the iodic IBVP (1.1). On the basis of the 2L-iodicity, the solution u(x, t) on the whole number line, < x <, can be regarded as a 2L-iodic extension of u(t, x) on [, 2L]. Therefore, the investigation concentrates on Eq. (1.1) on the bounded interval [, 2L]. Integrating (1.1) with respect to x over the interval [, 2L], noticing the iodicity of u(x, t), we have d 2L u(t, x)dx, dt which gives u(t, x)dx u (x)dx. Let m be the average of the initial value u (x) over the interval [, 2L] m : 1 u (x)dx. 2L Then (1.2) with (1.3) implies [u(t, x) m ]dx. For the asymptotic profile of the solution u(t, x), Eq. (1.1) is linearized around m : U t + U xxxxt αu xx + f (m )U x, (1.5) where U(t, x) : u(t, x) m. It is known that (1.5) admits a solution in the form U(t, x) Ae λt+iωx, where A is a constant, λ is the wave frequency (which may be complex), ω is the wavenumber satisfying the iodic condition e iωx e iω(x+2l). Substituting (1.6) into (1.5) yields (1.2) (1.3) (1.4) (1.6) λ + (iω) 4 λ α(iω) 2 + f (m )iω, which is solved as λ αω2 i f (m )ω 1 + ω 4.

4 L. Liu et al. / Nonlinear Analysis 67 (27) The real part of λ is Re(λ) αω2 1 + ω 4. When α >, Re(λ) <, then U(t, x) converges to zero as time t approaches infinity: (1.7) U(t, x) O(1)e Re(λ)t O(1)e αω2 1+ω 4 t. (1.8) When α, Re(λ), then U(t, x) O(1) sin(im(λ)t + ωx) + O(1) cos(im(λ)t + ωx), where Im(λ) f (m )ω/(1 + ω 4 ). Therefore, U(t, x) oscillates around zero all the time. This study, as shown later, concludes that similar results hold for the nonlinear case. Before the main results, some useful notation is introduced. Notations. Let L 2 (R) denote the space of square integrable 2L-iodic functions on R L 2 (R) {v(x) v(x) v(x + 2L) for all x R, v(x) L2 (, 2L) for x [, 2L]} with the norm ( 1/2 2L v L 2 v (x)dx) 2. The inner product of L 2 (R) is defined as φ, ψ φ(x)ψ(x)dx. H k (R) (k ) is the iodic Sobolev space of L2 -functions v(x) defined on R whose derivatives i x v (i 1,..., k) also belong to L 2 (R), with the norm ( k v H k i i x v(x) 2 dx) 1/2. For T > a Banach space B, C k (, T ; B) denotes the space of B-valued k-times-continuously differentiable functions on [, T ]. The corresponding spaces of B-valued functions on [, ) are defined similarly. Now we are ready to state the main results. Theorem 1.1 (Exponential Convergence). Let α > u (x) H 2 (R). For the iodic IBVP (1.1) there exists a unique global solution satisfying u(t, x) m C(, ; H 2 (R)) π (u m )(t) H L 2 (u m ) π H 2, (1.9) sup x [,2L] for all t [, ), where u(t, x) m O(1)e γ t, (1.1) γ αω ω 4 1 απ 2 L 2 π 4 + L 4, (1.11) ω 1 π 2 L 2 is the smallest wavenumber (eigenvalue) which satisfies e iωx e iω(x+2l) e iωx dx.

5 253 L. Liu et al. / Nonlinear Analysis 67 (27) Theorem 1.2 (Oscillatory Divergence). Let α u (x) H 2 (R). For the iodic IBVP (1.1) there exists a unique global solution satisfying u(t, x) m C(, ; H 2 (R)) (u m )(t) 2 L 2 + (u m ) xx (t) 2 L 2 (u m ) 2 L 2 + (u m ) xx 2 L 2 for all t [, ). In particular, if u (x) m, then the solution u(t, x) oscillates around m all the time. (1.12) Remark By Sobolev s embedding theorem H 2 C 1, the solution u(t, x) in C(, ; H 2 (R)) is continuous in time t differentiable in space x, i.e., u(t, x) C(, ; C 1 (R)), but it is not a classical solution to Eq. (1.1). Such a solution is called a strong solution. 2. We show the global existence of the strong solution in Theorems without the smallness assumption on the initial value, namely, we obtain the global existence of the solution for any large initial data. 3. Compared to the decay rate (1.8) of the linear case, the rate (1.1) obtained for the nonlinear case in Theorem 1.1 is optimal, which also matches the optimal convergence rate for the iodic initial boundary value problem of the 2 2 system of the BBM Burgers equations given by Bisognin et al. [2]. 4. Although the solution u(t, x) in the case α has been recognized to be oscillatory at all time around m, this does not mean that the solution is not stable. In fact, as shown in Section 4 numerically, we conjecture that u(t, x) converges to its iodic traveling wave φ(x ct). For such a wave-stability problem, this is still open at this moment. 2. Proof of Theorem 1.1 Let v(t, x) u(t, x) m. With (1.4), iodic IBVP (1.1) is reduced to v t + v xxxxt αv xx + F(v) x, x R, t R +, v t u (x) m : v (x), x R, v (x) v (x + 2L), x R, where v (x)dx, F(v) f (v + m ) f (m ). Corresponding to Theorem 1.1, the equivalent result for the new iodic IBVP (2.2) is as follows. Theorem 2.1. Let α > v (x) H 2 (R). For the iodic IBVP (2.2) there exists a unique global solution satisfying v(t, x) C(, ; H 2 (R)) L v(t) H π 2 v π H 2, sup v(t, x) O(1)e γ t x [,2L] for all t, where γ is given in (1.11). (2.1) (2.2) (2.3) (2.4) (2.5)

6 L. Liu et al. / Nonlinear Analysis 67 (27) When Theorem 2.1 is proved, Theorem 1.1 is obtained automatically. Therefore, the main objective in this section is to prove Theorem 2.1. The method of continuity extension with the L 2 -energy estimates is adopted. The solution space is defined as { } X M (t 1, t 2 ) v(t, x) v(t, x) C(t 1, t 2 ; H 2 (R)), where M > t 2 t 1 are constants. sup v(t) H 2 M t [t 1,t 2 ] Proposition 2.2 (Local Existence). For α, consider the iodic IBVP at the initial time τ v t + v xxxxt αv xx + F(v) x, x R, t [τ, ), v tτ v τ (x), x R, v τ (x) v τ (x + 2L), x R, v τ (x)dx. Let v τ (x) H 2 (R) M > be such that v τ H 2 M. Then there exists a number t t (M) > such that the iodic IBVP (2.6) has a unique solution v(t, x) in X 2M (τ, τ + t ). This proposition can be proved by a stard iteration method; cf. [9,12]. Therefore the detail is omitted here. The following Poincaré inequality is needed in the a priori estimates for the solution. Lemma 2.3 (Poincaré Inequality). Consider (2.2) with α. Let T > v(t, x) C(, T ; H 2 (R)) be a solution of (2.2). Then π L k x v(t) L 2 k+1 x v(t) L 2, k, 1. (2.7) Proof. First, an orthonormal basis is constructed for the space whose functions satisfy the iodic condition { } Φ φ φ(x) φ(x + 2L), Consider the eigenvalue problem φ ω 2 φ, φ(x) φ(x + 2L), φ(x)dx. φ(x)dx, (2.6). (2.8) The eigenvalues the corresponding normalized eigenfunctions are solved in the form eigenvalues: ω k kπ L, φ 1,k 1 cos ω k x, k 1, 2,..., (2.1) eigenfunctions: L φ 2,k 1 sin ω k x, L with inner products (2.9) φ i,k, φ j,l φ i,k (x)φ j,l (x)dx { 1, for i j, k l,, otherwise. Thus, the sequence {φ i,k } (i 1, 2 k 1, 2, 3,...) forms the orthonormal basis of the space Φ defined in (2.8). (2.11)

7 2532 L. Liu et al. / Nonlinear Analysis 67 (27) Notice that the solution v(t, x) C(, T ; H 2 ) satisfies v(t, x) v(t, x + 2L) v(t, x)dx. Therefore, v(t, x) Φ, can be expressed in the form v(t, x) [a 1,k (t)φ 1,k (x) + a 2,k (t)φ 2,k (x)], (2.12) k1 where a i,k (t) (i 1, 2 k 1, 2, 3,...) are the so-called Fourier coefficients determined as a i,k (t) v(t, x), φ i,k (x) Differentiating both sides of (2.12) with respect to x yields v x (t, x) v(t, x)φ i,k (x)dx, i 1, 2, k 1, 2,.... (2.13) ω k [ a 1,k (t)φ 2,k (x) + a 2,k (t)φ 1,k (x)]. (2.14) k1 Now, taking the inner product of v(t, x) with itself using (2.11), we then have v(t) 2 L 2 v(t, x), v(t, x) [a 1,k (t)φ 1,k (x) + a 2,k (t)φ 2,k (x)], [a 1,l (t)φ 1,l (x) + a 2,l (t)φ 2,l (x)] k1 k1 l1 l1 [a 1,k (t)a 1,l (t) φ 1,k, φ 1,l + a 1,k (t)a 2,l (t) φ 1,k, φ 2,l + a 2,k (t)a 1,l (t) φ 2,k, φ 1,l + a 2,k (t)a 2,l (t) φ 2,k, φ 2,l ] [(a 1,k (t)) 2 + (a 2,k (t)) 2 ]. (2.15) k1 Similarly, from (2.14) we obtain v x (t) 2 L 2 Since ω k > ω 1 π L v x (t) 2 L 2 ωk 2 [(a 1,k(t)) 2 + (a 2,k (t)) 2 ]. (2.16) k1 for k 2, 3,..., (2.15) (2.16) imply ωk 2 [(a 1,k(t)) 2 + (a 2,k (t)) 2 ] k1 ω 2 1 [(a 1,k (t)) 2 + (a 2,k (t)) 2 ] k1 π 2 L 2 v(t) 2 L 2. Similarly, we can prove v xx (t) 2 L 2 π 2 L 2 v x(t) 2 L 2. Hence (2.7) is proved. (2.17) (2.18) Proposition 2.4 (A Priori Estimate). Let T > M > be arbitrary fixed constants, v(t, x) X M (, T ) be a solution of (2.2). Then v(t) H 2 L 2 + π 2 v π H 2, t [, T ], (2.19)

8 L. Liu et al. / Nonlinear Analysis 67 (27) sup v(t, x) O(1)e γ t, t [, T ]. (2.2) x [,2L] Proof. Let v(t, x) C(, T ; H 2 (R)) be a solution of (2.2). The a priori estimates (2.19) (2.2) will be proved by the L 2 -energy method. Since the strong solution v(t, x) lacks the regularity for v xxxxt since v (x) H 2, v(t, x) cannot be treated directly in the form of differential equation (2.2). In order to overcome this shortcoming, the iodic IBVP (2.2) is first investigated with a smooth enough initial value. The regularity of the solution v(t, x) depends on the smoothness of its initial value v (x). That is to say: the smoother the initial value, the smoother the solution. Smooth the initial value v (x) as v ε (x) J ε (x y)v (y)dy, where ε > is a constant, J ε (x) C (R) is the mollifier satisfying J ε(x) for x ε J ε(x)dx 1. It is known that v ε (x) C (R) lim ε vε + (x) v (x) for all x R; cf. [1]. The iodic conditions still hold: v ε (x + 2L) v ε (x), J ε (x + 2L y)v (y)dy [by change of variables: z y 2L] J ε (x z)v (z + 2L)dz J ε (x z)v (z)dz ( ) v ε (x)dx J ε (x y)v (y)dy dx ( ) J ε (y)v (x y)dy dx ( ) J ε (y) v (x y)dx dy ( ) 2L+y J ε (y) v (z)dz dy y [by iodicity: v (x + 2L) v (z)] [ by iodicity: +y y v (z)dz ] (2.21) (2.22) J ε (y) dy. (2.23) Let v ε (t, x) be the local solution of the iodic IBVP vt ε + vxxxxt ε αvε xx + F(vε ) x, x R, t [, T ], v ε t v ε (x), x R, v ε (x) vε (x + 2L), x R, v ε (x)dx. Since the initial value v ε (x) C (R), the solution vε (t, x) has good enough regularities in x t, (2.24) lim ε vε (t, x) v(t, x) + for all (t, x) [, T ] R. The convergence (2.25) can be proved like in [9,12], the details are omitted here. (2.25)

9 2534 L. Liu et al. / Nonlinear Analysis 67 (27) We now prove the estimates (2.19) (2.2) for v ε (t, x). Multiplying the first equation of (2.24) by v ε (x, t), we obtain 1 2 {(vε ) 2 + (v ε xx )2 } t + α(v ε x )2 + {v ε v ε xxxt vε x vε xxt αvε v ε x + G(vε )} x, (2.26) where G(v ε ) is an antiderivative of F (v ε )v ε, i.e., G (v ε ) F (v ε )v ε. Integrating (2.26) over [, 2L] [, t] with respect to x t, using the iodicity of v ε such that the last term of the left-h side of (2.26) disappears after integration with respect to x, we then have v ε (t) 2 L 2 + vxx ε (t) 2 L 2 + 2α t Applying the Poincaré inequalities (2.7), we obtain vx ε (τ) 2 L 2 dτ v ε 2 L 2 + v,xx ε 2 L 2 t [, T ]. (2.27), v ε (t) 2 L 2 + vxx ε (t) 2 L 2 v ε (t) 2 L 2 L 2 + π 2 + L 2 vε xx (t) 2 L 2 + π 2 π 2 + L 2 vε xx (t) 2 L 2 v ε (t) 2 L 2 L 2 + π 2 + L 2 π 2 L 2 vε x (t) 2 L 2 + π 2 π 2 + L 2 vε xx (t) 2 L 2 v ε (t) 2 L 2 + π 2 π 2 + L 2 vε x (t) 2 L 2 + π 2 π 2 + L 2 vε xx (t) 2 L 2 π 2 π 2 + L 2 vε (t) 2 H 2, t [, T ]. (2.28) Substituting (2.28) into (2.27) dropping the positive term 2α t vε x (τ) 2 L 2 dτ implies (2.19) for the smooth solution v ε (x, t), i.e., v ε (t) 2 H 2 π 2 + L 2 v ε 2 H 2, t [, T ]. (2.29) π 2 On the other h, again, applying the Poincaré inequality (2.7), we have vxx ε (t) 2 L 2 π 2 L 2 vε x (t) 2 L 2 (2.3) t t { 2α vx ε π 4 L 4 } (τ) 2 L 2 2α dτ π 4 + L 4 vε x (τ) 2 L 2 + π 4 + L 4 vε x (τ) 2 L 2 dτ t { π 4 L 4 2α π 4 + L 4 vε x (τ) 2 L 2 + π 4 + L 4 π 2 } L 2 vε (τ) 2 L 2 dτ 2απ 2 L 2 t { π 2 } π 4 + L 4 L 2 vε x (τ) 2 L 2 + v ε (τ) 2 L 2 dτ t { 2γ v ε (τ) 2 L 2 + π 2 } L 2 vε x (τ) 2 L 2 dτ, (2.31) where γ απ 2 L 2. Substituting (2.3) (2.31) into (2.27) yields π 4 +L 4 v ε (t) 2 L 2 + π 2 L 2 vε x (t) 2 L 2 + 2γ t Applying Gronwall s inequality to (2.32), we have v ε (t) 2 L 2 + π 2 L 2 vε x (t) 2 L 2 v ε 2 He 2γ t 2, t [, T ], { v ε (τ) 2 L 2 + π 2 } L 2 vε x (τ) 2 L 2 dτ v ε 2 H, 2 t [, T ]. (2.32)

10 L. Liu et al. / Nonlinear Analysis 67 (27) which implies v ε (t) 2 H 1 C 1 v ε 2 H 2 e 2γ t, t [, T ], (2.33) where C 1 1/ min{1, π 2 /L 2 }. Using Sobolev s embedding inequality H 1 C, we have sup v ε (x, t) C v ε H 2 e γ t, t [, T ], (2.34) x [,2L] which proves (2.2) for v ε (t, x). With (2.25), i.e., lim ε + v ε (t, x) v(t, x), (2.29) (2.34) imply (2.19) (2.2) for the strong solution v(t, x). The proof is complete. Proof of Theorem 2.1. For any given 2L-iodic initial value v (x) H 2 (R), we choose M such that Mπ v H 2 π 2 + L M, i.e., π 2 + L 2 v 2 2 H 2 M 2. (2.35) π 2 By Proposition 2.2 with τ, for the iodic IBVP (2.2) there exists a unique local solution v(t, x) in X 2M (, t ) for the time determined, t t (M) >. For such a local solution, applying Proposition 2.4, from (2.35) we have v(t) 2 H 2 π 2 + L 2 v 2 H 2 M 2, t [, t ] (2.36) π 2 sup v(t, x) O(1)e γ t, t [, t ]. x [,2L] Therefore, v(t, x) X M (, t ). Since v(t ) H 2 M (see (2.36)), we apply Proposition 2.2 with τ t to obtain the solution v in X 2M (t, 2t ), namely, we extend the existence interval of the solution v(t, x) from [, t ] to [, 2t ]. For v(t, x) X 2M (, 2t ), again we use Proposition 2.4 to obtain v(t) 2 H 2 π 2 + L 2 v 2 H 2 M 2, t [, 2t ] π 2 sup v(t, x) O(1)e γ t, t [, 2t ]. x [,2L] We prove v(t, x) X M (, 2t ). Repeating the above procedure, we prove that v(t, x) X M (, + ) v(t) 2 H 2 π 2 + L 2 v 2 H 2 M 2, π 2 sup v(t, x) O(1)e γ t, x [,2L] The proof of Theorem 2.1 is complete. 3. Proof of Theorem 1.2 t [, + ). t [, + ) With v(t, x) u(t, x) m, the iodic IBVP (1.1) with α is reduced to v t + v xxxxt + F(v) x, x R, t R +, v t u (x) m : v (x), x R, v (x) v (x + 2L), x R, v (x)dx, (3.1)

11 2536 L. Liu et al. / Nonlinear Analysis 67 (27) where F(v) f (v + m ) f (m ) as in (2.3). The corresponding oscillatory divergence around zero of the solution v(t, x) to (3.1) is as follows. Theorem 3.1. Let α v (x) H 2 (R). For the iodic IBVP (3.1) there exists a unique global solution satisfying v(t, x) C(, ; H 2 (R)) v(t) 2 L 2 + v xx (t) 2 L 2 v 2 L 2 + v,xx 2 L 2 for all t [, ). In particular, if v (x), then the solution v(t, x) oscillates around zero at all time. Once Theorem 3.1 is proved, Theorem 1.2 can be easily obtained. As shown in the previous section, the local existence of the solution has been given in Proposition 2.2 for the case α, Theorem 3.1 can then be proved similarly by using the energy method based on the local existence the following a priori estimates. Proposition 3.2 (A Priori Estimate). Let T > M > be arbitrary fixed constants, v(t, x) X M (, T ) be a solution of (3.1), where the solution space X M (, T ) is defined as in the previous section. Then v(t) 2 L 2 + v xx (t) 2 L 2 v 2 L 2 + v,xx 2 L 2 t [, T ], (3.3), L v(t) H π 2 v π H 2, t [, T ]. (3.4) Proof. As shown in Propositions can be treated similarly. Let v(t, x) X M (, T ) be a solution of (3.1). Although it lacks regularity, a mollifier may be introduced to smooth the equation as shown in the proof of Proposition 2.4. For the sake of simplicity, the regularity of v is neglected. Multiplying Eq. (3.1) by v yields 1 2 (v2 + v xx ) t + {v xxxt v v xxt v x + G(v)} x, where G (v) F (v)v. Integrating the above equation over [, 2L] [, t] with respect to x t implies (3.3) v(t) 2 L 2 + v xx (t) 2 L 2 v 2 L 2 + v,xx 2 L 2 t [, T ]., Like for (2.28), using the Poincaré inequality (2.7) one deduces (3.4) from (3.3). Proof of Theorem 3.1. As shown in the previous section, the global existence of the solution to the iodic IBVP (3.1) can be similarly proved by the continuity argument. Its detail is omitted here. Now, let us show the oscillatory divergence of v(t, x). If v (x), by the iodicity of v(t, x) v(t, x) v(t, x + 2L), v(t, x)dx, we know that v(t, x) v(t, x) is oscillatory. To see the divergence, if the assertion is not true, i.e., v(t, x) is convergent to as t, then we have v(t) 2 L 2 v xx (t) 2 L 2 as t, (3.2) which implies from (3.2) v L 2 + v,xx L 2 as t. This is a contradiction! Therefore, v(t, x) has oscillatory divergence.

L. Liu et al. / Nonlinear Analysis 67 (27) 2527 2539 2537 4. Numerical simulations Fig. 4.1. (a) u(t, x) with α 1, u (x) sin x; (b) sup x [,2π] u(t,x) m, with u(t, x) from (a). e t/2 Fig. 4.2. u(t, x) with α 1 u (x) 1 sin x.

12 L. Liu et al. / Nonlinear Analysis 67 (27) Numerical simulations Fig (a) u(t, x) with α 1, u (x) sin x; (b) sup x [,2π] u(t,x) m, with u(t, x) from (a). e t/2 Fig u(t, x) with α 1 u (x) 1 sin x. In this section, the numerical simulations are reported to confirm the theoretical results: Theorems Here, the nonlinear function is f (u) u 3 /3 the iod 2L 2π. Case A. α >. Without loss of generality, let α 1. The small initial value is chosen to be u (x) sin x, its average over [, 2π] is m 2π sin xdx. The numerical results in Fig. 4.1(a) (b) show convergence of the solution u(t, x) to its initial average m. In particular, Fig. 4.1(b) shows sup u(t, x) m x [,2π] e t/2 1, which indicates that u(t, x) converges to m at the optimal rate e γ t e t/2 (see (1.11)), namely, γ απ 2 L 2 1 π 2 π 2 1 π 4 +π 4 2. This confirms Theorem 1.1. π 4 +L 4 The large initial value is chosen to be u (x) 1 sin x. The initial average is still m, but the turbation of u (x) around the initial average m is large, sup u (x) m 1. For this case, convergence is also obtained; see Fig. 4.2, which confirms the convergence of Theorem 1.1 in the case of the large initial value. Unlike the case with small initial turbation around its average m shown in Fig. 4.1 in which the solution u(t, x) decays smoothly to m, Fig. 4.2 exhibits a lot of irregular sharp oscillations within the initial short time from to 2.5, then converging smoothly to the initial average m.

2538 L. Liu et al. / Nonlinear Analysis 67 (27) 2527 2539 Fig. 4.3. (a) u(t, x) with α u (x) sin x; (b) u(t, ) with α u (x) sin x. Fig. 4.4. u(t, x) with α u (x) 1 sin x. Case B. α. Like for Case A, the small initial value is set as u (x) sin x, whose average over [, 2π] is m.

A similar numerical result for the large initial value is presented in Fig. 4.4, in which the initial value is u (x) 1 sin x. Acknowledgment L. Liu Y.S.

13 2538 L. Liu et al. / Nonlinear Analysis 67 (27) Fig (a) u(t, x) with α u (x) sin x; (b) u(t, ) with α u (x) sin x. Fig u(t, x) with α u (x) 1 sin x. Case B. α. Like for Case A, the small initial value is set as u (x) sin x, whose average over [, 2π] is m. The numerical results in Fig. 4.3 (a) (b) show the oscillatory divergence of the solution u(t, x) around its initial average m at all time. The numerical eximent presented here validates Theorem 1.2. A similar numerical result for the large initial value is presented in Fig. 4.4, in which the initial value is u (x) 1 sin x. Acknowledgment L. Liu Y.S. Wong would like to acknowledge the support received from the National Sciences Engineering Research Council of Canada. References [1] R.A. Adams, Sobolev Spaces, in: Pure Applied Mathematics, vol. 65, Academic Press, New York, London, [2] E. Bisognin, V. Bisognin, G. Perla Menzala, Uniform stabilization space-iodic solutions of a nonlinear dissive system, Dyn. Contin. Discrete Impuls. Syst. 7 (2) [3] P. Chatzipantelidis, Explicit multistep methods for nonstiff partial differential equations, Appl. Numer. Math. 27 (1998) [4] S.K. Chung, Finite difference approximate solutions for the Rosenau equation, Appl. Anal. 69 (1998) [5] S.K. Chung, S.N. Ha, Finite element Galerkin solutions for the Rosenau equation, Appl. Anal. 54 (1994) [6] H.Y. Lee, M.R. Ohm, J.Y. Shin, The convergence of fully discrete Galerkin approximations of the Rosenau equation, Korean J. Comput. Appl. Math. 6 (1999) [7] L. Liu, M. Mei, A better asymptotic profile of Rosenau Burgers equation, Appl. Math. Comput. 131 (22) [8] S.A.V. Manickam, A.K. Pani, S.K. Chung, A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation, Numer. Methods Partial Differential Equations 14 (1998)

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Author(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1)

Author(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1) Title On the stability of contact Navier-Stokes equations with discont free b Authors Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 4 Issue 4-3 Date Text Version publisher URL