Acta Appl Math (2012) 118:25–47 DOI 10.1007/s10440-012-9676-4

Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation Chaohua Jia · Bing-Yu Zhang

Received: 12 September 2011 / Accepted: 12 November 2011 / Published online: 24 February 2012 © Springer Science+Business Media B.V. 2012

Abstract In this article, we continue our study of a system described by a class of initial boundary value problem (IBVP) of the Korteweg-de Vries (KdV) equation and the KdV Burgers (KdVB) equation posed on a finite interval with nonhomogeneous boundary conditions. While the system is known to be locally well-posed (Kramer et al. arXiv:1012.1057, 2010; Rivas et al. in Math. Control Relat. Fields 1:61–81, 2011) and its small amplitude solutions are known to exist globally, it is not clear whether its large amplitude solutions would blow up in finite time or not. This problem is addressed in this article from control theory point of view: look for some appropriate feedback control laws (with boundary value functions as control inputs) to ensure that the finite time blow-up phenomena would never occur. In this article, a simple, but nonlinear, feedback control law is proposed and the resulting closed-loop system is shown not only to be globally well-posed, but also to be locally exponentially stable for the KdV equation and globally exponentially stable for the KdVB equation. Keywords Korteweg-de Vries equation · Korteweg-de Vries-Burgers equation · Boundary stabilization · Global well-posedness

The article is dedicated to Professor Goong Chen for his 60th birthday. C. Jia School of Mathematics and Systems Science, Beihang University, Beijing, 100191, China e-mail: [email protected] C. Jia LMIB of the Ministry of Education, Beijing, 100191, China B.-Y. Zhang () Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA e-mail: [email protected] B.-Y. Zhang Yangtz Center of Mathematics, Sichuan University, Chengdu, China

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1 Introduction The well-known Korteweg-de Vries (KdV) equation, ut + ux + uux + uxxx = 0,

u ≡ u(x, t), x, t ∈ R,

(1.1)

was first derived by Korteweg and de Vries [27] in 1895 (or by Boussinesq [6] in 1876)1 as a model for propagation of some surface water waves along a channel. It has been intensively studied from various aspects of both mathematics and physics since the 1960s when solitons were discovered through solving the KdV equation, and the inverse scattering method, a socalled nonlinear Fourier transform, was invented to seek solitons [21, 32]. It turns out that the equation is not only a good model for some water waves but also a very useful approximation model in nonlinear studies whenever one wishes to include and balance a weak nonlinearity and weak dispersive effects [32]. It is now commonly accepted as a mathematical model for the unidirectional propagation of small-amplitude long waves in nonlinear dispersive systems. In particular, the KdV equation is not only used to serve as a model to study surface water waves, acoustic-gravity waves in a compressible heavy fluid, axisymmetric waves in rubber cords, and hydromagnetic waves in a cold plasma, but also has been used recently to serve as a model to study blood pressure waves in large arteries [11–14, 30, 39]. In this article we consider the Korteweg-de Vries (KdV) equation on the finite interval (0, L) ut + ux + uxxx + uux = 0,

x ∈ (0, L), t > 0,

(1.2)

and the Korteweg-de Vries-Burgers (KdVB) equation on the finite interval (0, L) ut + ux + uxxx + uux − uxx = 0,

x ∈ (0, L), t > 0

(1.3)

satisfying the initial condition u(x, 0) = φ(x),

x ∈ (0, L)

(1.4)

and the nonhomogeneous boundary conditions u(0, t) = h1 (t),

ux (L, x) = h2 (t),

uxx (L, t) = h3 (t),

x ∈ (0, L), t > 0

(1.5)

where  > 0 in (1.3) is a constant. The initial-boundary-value problem (IBVP) (1.2)–(1.4)–(1.5) of the KdV equation was first considered by Colin and Ghidaglia [10] as a model for propagation of surface water waves in the situation where a wave-maker is putting energy in a finite-length channel from the left (x = 0) while the right end (x = L) of the channel is free (corresponding to the case of h2 = h3 = 0). In particular, they studied the IBVP (1.2)–(1.4)–(1.5) for its well-posedness in the space H s (0, L) and obtained the following results. Theorem A (Colin and Ghidaglia) (i) Given hj ∈ C 1 ([0, ∞)), j = 1, 2, 3 and φ ∈ H 1 (0, L) satisfying h1 (0) = φ(0), there exists a T > 0 such that the IBVP (1.2)–(1.4)–(1.5) admits a solution (in the sense of distribution)     u ∈ L∞ 0, T ; H 1 (0, L) ∩ C [0, T ]; L2 (0, L) . 1 The interested readers are refereed to a nice article of de Jager [25] for the origin of the KdV equation.

Boundary Stabilization of the Korteweg-de Vries Equation

27

(ii) Assuming h1 = h2 = h3 ≡ 0, then for any φ ∈ L2 (0, L), there exists a T > 0 such that the IBVP (1.2)–(1.4)–(1.5) admits a unique weak solution u ∈ C([0, T ]; L2 (0, L)) ∩ L2 (0, T ; H 1 (0, L)). Their results have been improved recently by Kramer and Zhang [28], and Kramer, Rivas and Zhang [29] who obtained the following well-posedness results for the IBVP (1.2)–(1.4)– (1.5). Theorem B Let s > − 34 and T > 0 and r > 0 be given with s =

2j − 1 , 2

j = 1, 2, 3, . . . .

There exists a T ∗ > 0 such that for given s-compatible2 φ ∈ H s (0, L),

h1 ∈ H

s+1 3

s

h2 ∈ H 3 (0, T ),

(0, T ),

h3 ∈ H

s−1 3

s−1

≤ r,

(0, T )

satisfying φH s (0,L) + h1 

s+1

H 3 (0,T )

+ h2 

s

H 3 (0,T )

+ h3 

H 3 (0,T )

the IBVP (1.2)–(1.4)–(1.5) admits a unique solution      u ∈ C 0, T ∗ ; H s (0, L) ∩ L2 0, T ∗ ; H s+1 (0, L) . Moreover, the solution u depends Lipschitz continuously on φ and hj , j = 1, 2, 3 in the corresponding spaces. Remark A similar proof with only minor modification shows that Theorem B also holds for the IBVP (1.3)–(1.4)–(1.5). The result is temporally local in the sense that the solution u is only guaranteed to exist on the time interval (0, T ∗ ), where T ∗ depends on the size of the initial value φ and the boundary data hj , j = 1, 2, 3 in the corresponding spaces. One may wonder naturally: does the solution exist globally? Usually, with the local well-posedness in hand, one needs to establish certain global a priori estimate of the solutions to obtain the global well-posedness. However, this task turns out to be difficult as the L2 -energy of the solution u of the IBVP (1.2)–(1.4)–(1.5) with the homogeneous boundary conditions (hj ≡ 0, j = 1, 2, 3) is not conserved as in the case of the KdV equation posed on the whole line R or on a periodic domain T. Indeed, for any smooth solution u of the IBVP (1.2)–(1.4)–(1.5) with hj ≡ 0, j = 1, 2, 3, it holds that  L 2 d u2 (x, t)dx = −u2 (L, t) − u2x (0, t) − u3 (L, t). dt 0 3 2 The reader is referred to [28] for the precise definition of s-compatibility for the IBVP (1.2). One of the sufficient conditions for φ, h1 , h2 , h3 to be s-compatible is φ ∈ H0s (0, L) and s+1 3

h1 ∈ H0

(0, T ],

s

h2 ∈ H03 (0, T ],

s−1 3

h3 ∈ H0

(0, T ].

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C. Jia, B.-Y. Zhang

The lack of an effective means to deal with the term 23 u3 (L, t) makes it hard to establish the needed global a priori estimate for the solutions of the IBVP (1.2)–(1.4)–(1.5) in the space L2 (0, L). In [10], Colin and Ghidaglia provided a partial answer to the problem by showing that the solution u of the IBVP (1.2)–(1.4)–(1.5) exists globally in H 1 (0, L) if the size of its initial value φ ∈ H 1 (0, L) and its boundary values hj ∈ Cb1 ([0, ∞)), j = 1, 2, 3 are all small. This result has also been improved recently by Rivas, Usman and Zhang [34]. The solution u given in Theorem B exists globally so long as its initial value φ and s+2−j the boundary values hj , j = 1, 2, 3 are small in the space H s (0, L) and H 3 (0, T ) j = 1, 2, 3 respectively. Moreover, if hj ≡ 0 for j = 1, 2, 3, the corresponding solution u decays exponentially in the space H s (0, L) as t → ∞. Nevertheless, while the small amplitude solutions exist globally, the question whether the large amplitude solutions exist globally remains open. Indeed, even in the case of hj ≡ 0 for j = 1, 2, 3, it is not clear whether large amplitude solutions of the IBVP (1.2)–(1.4)–(1.5) exist globally or blow up in finite time. In this article, we will address this issue from control theory point of view: take the boundary value functions hj as control inputs and look for some appropriate feedback control laws to ensure the finite time blow up would never occur and the resulting closed-loop system is globally stable. We will choose the following simple, but nonlinear feedback control laws 1 h3 (t) = − u2 (L, t) h1 = h2 = 0, 3 in (1.5). It results in the following closed-loop system, ⎧ ⎪ ⎨ut + ux + uux + uxxx − uxx = 0, x ∈ (0, L), t > 0, (1.6) u(x, 0) = φ(x), x ∈ (0, L), ⎪ ⎩ 1 2 u(0, t) = 0, ux (L, t) = 0, uxx (L, t) = − 3 u (L, t) with  ≥ 0. It is easy to verify that for this closed-loop system, its smooth solution u (if exists) satisfies d dt





L

u (x, t)dx + u 2

2

L

(L, t) + u2x (0, t) + 2

0

u2x (x, t)dx = 0 for any t ≥ 0, 0

from which one derives further that u(·, t)L2 (0,L) ≤ CφL2 (0,L) e

− 22 t L

for any t ≥ 0.

One may thus conjecture that the solutions of the system (1.6) exist globally and, moreover, in the case of  > 0, the solutions of the system (1.6) should decay exponentially as t → ∞. In other words, the system (1.6) should be globally exponentially stable. The main purpose of this paper is to confirm that this is the case. In order to describe our main results precisely, we first introduce some notations. For given s ≥ 0, t ≥ 0 and T > 0, let       s+1 s s 2 s+1 3 (t, t + T ) (0, L) ∩ L∞ Y(t,t+T ) = C [t, t + T ]; H (0, L) ∩ L t, t + T ; H x 0, L; H and

 s s YTs = u ∈ Y(t,t+T ) for any t ≥ 0 : sup uY(t,t+T ) < ∞ . t≥0

Boundary Stabilization of the Korteweg-de Vries Equation

29

s s Both Y(t,t+T ) and YT are Banach spaces equipped with the norms s uY(t,t+T := uC([t,t+T ];H s (0,L)) + uL2 (t,t+T ;H s+1 (0,L)) + u )

s+1

L∞ x (0,L;H 3 (0,T ))

and s , uYTs := sup uY(t,t+T )

t≥0

respectively. If s = 0, the superscript s will be omitted altogether, so that 0 Y(t,t+T ) = Y(t,t+T ),

YT = YT0 .

For any given s ∈ R, 0 ≤ b ≤ 1, 0 ≤ α ≤ 1 and function w ≡ w(x, t) : R2 → R, define

 s,b (w) =

 λα (w) =

∞



−∞ ∞

−∞

∞ −∞



 2b ˆ τ )|2 dξ dτ τ − ξ 3 + ξ + iξ 2 ξ 2s |w(ξ,

τ |w(ξ, ˆ τ )| dξ dτ 2α

|ξ |≤1

2

 12 , (1.7)

 12

1

where · := (1 + | · |2 ) 2 . Let Xs,b be the space of all functions w satisfying wXs,b := s,b (w) < ∞, α and let Xs,b be the space of all functions w satisfying

 2 1/2 2 α :=  < ∞. wXs,b s,b (w) + λα (w) α α The spaces Xs,b and Xs,b are Banach spaces. Note that Xs,b and Xs,b are equivalent when b ≥ α. For given T > 0 the restricted version of the Bourgain space Xs,b to the domain (0, L) × (0, T ) is defined as follows: T Xs,b = Xs,b |(0,L)×(0,T )

with the quotient norm uXT ≡ inf s,b

w∈Xs,b

  wXs,b : w(x, t) = u(x, t) on (0, L) × (0, T )

α,T for any given function u(x, t) defined on (0, L)×(0, T ). The space Xs,b is defined similarly. In addition, let   1 D3 (0, L) = ψ ∈ H 3 (0, L); ψ(0) = ψ  (L) = 0, ψ  (L) + ψ 2 (L) = 0 3

and 

D03 (0, L) = ψ ∈ H 3 (0, L);

ψ(0) = ψ  (L) = 0,

 ψ  (L) = ψ(L) = 0 .

Note that D3 (0, L) is not a linear space, but D03 (0, L) is a linear subspace of H 3 (0, L). For any s ∈ [0, 3], let Hs (0, L) be the closure of D03 (0, L) in the space H s (0, L). Obviously, Hs (0, L) is a subspace of H s (0, L) and H0 (0, L) = H 0 (0, L) = L2 (0, L).

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C. Jia, B.-Y. Zhang

Moreover, because of their frequent occurrence, it is convenient to abbreviate the norms of u and h in the space H s (0, L) and H s (a, b), respectively, as us = uH s (0,L) ,

|h|s,(a,b) = hH s (a,b) ,

and |h|(a,b) = hL2 (a,b) .

u = uL2 (0,L) ,

The main results of this paper are summarized in the following two theorems. The first one states that the system (1.6) is globally well-posed in the space Hs (0, L). Theorem 1.1 (Global well-posedness) Let T > 0 and s ∈ [0, 3] with s = 2j2−1 , j = 1, 2, 3 α,T s be given. For any φ ∈ Hs (0, L), the system (1.6) admits a unique solution u ∈ Y(0,T ) ∩ X0, 1 for some α ∈ ( 12 , 1) with

2

  s + uXα,T ≤ αs,T φ φs uY(0,T ) 0, 21

where αs,T : R+ → R+ is a nondecreasing continuous function. Moreover, the solution map s is Lipschitz continuous from Hs (0, L) to the space Y(0,T ) . In the case of s = 3, the result 3 holds also for φ ∈ H (0, L) satisfying 1 φ(0) = φ  (L) = φ  (L) + φ 2 (L) = 0. 3 For any  ≥ 0, let A be an operator on L2 (0, L) defined by A f = −f  − f  + f  for any f in the domain of A   D(A ) := f ∈ H 3 (0, L) : f (0) = f  (L) = f  (L) = 0 . Let

  γ = − sup Re λ; λ ∈ σ (A )

and γ− denotes any number smaller that γ . Our second theorem states that the system is locally exponentially stable if  = 0 and is globally exponentially stable is  > 0. Theorem 1.2 (Asymptotic behavior) Let s ∈ [0, 3] with s =

2j −1 , 2

j = 1, 2, 3 be given.

(i) If  = 0 in (1.6), there exist T > 0 and δ > 0 such that if φ ∈ Hs (0, L) with φs ≤ δ, then the corresponding solution u of the system (1.6) satisfies   − s ≤ βs φ φs e−γ0 t uY(t,t+T )

for all t ≥ 0,

where βs : R+ → R+ is a nondecreasing continuous function.

Boundary Stabilization of the Korteweg-de Vries Equation

31

(ii) If  > 0 in (1.6), there exists T > 0, for any φ ∈ Hs (0, L), the corresponding solution u of the system (1.6) satisfies   − s uY(t,t+T ≤ κs φ φs e−γ t )

for all t ≥ 0,

where κs : R+ → R+ is a nondecreasing continuous function. To prove these two theorems, we use the approach developed in [1, 3–5, 28, 29, 34] dealing with various boundary value problems of the KdV equation and the KdVB equation posed on the finite interval (0, L). However, some special attention is needed to handle the nonlinear term − 13 u2 (L, t) appeared at the boundary. To overcome the difficulty caused by this nonlinear boundary term, we need to establish the following trace property for solutions of the associated linear system  ut + ux + uxxx − uxx = 0, u(x, 0) = φ(x), x ∈ (0, L), u(0, t) = 0, ux (L, t) = 0, uxx (L, t) = h(t) : −1

1

3 + for φ ∈ L2 (0, L), h1 ∈ Hloc3 (R+ ), its solution u belongs to the space L∞ x (0, L; Hloc (R )) and   u ∞ ≤ CT φ + h − 1 1

Lx (0,L;H 3 (0,T ))

H 3 (0,T )

for any T > 0. α,T Then, with the help of the restricted Bourgain spaces X0, 1 , we demonstrate this trace 2

property also holds for solutions of the nonlinear system  ut + ux + uxxx + uux − uxx = 0, u(x, 0) = φ(x), x ∈ (0, L), u(0, t) = 0, ux (L, t) = 0, uxx (L, t) = h(t). As we will see late, this trace property will play indispensable role in the proof of Theorem 1.1 and Theorem 1.2. The study of the initial-boundary-value problems of the KdV equation and the KdVB equation posed on the finite domain started as early as in 1979 by Bubnov [7] and has been intensively studied in the past twenty years for their well-posedness following the advances of the study of the pure initial value problems of the KdV equation posed either on the whole line R or on a torus T. The interested readers are referred to [3, 5, 7, 8, 10, 18, 20, 24, 28, 29] and the references therein for an overall review for the well-posedness of the IBVP of the KdV and KdVB equations posed a finite domain and [1, 4, 9, 15–17, 19, 24] for the IBVP of the KdV and KdVB equations posed on the half line R+ as well as [2, 8, 10, 22, 23, 28, 31, 34, 36–38, 40, 41] for long time asymptotic behavior of solutions of those systems. It should be pointed out that there are some problems related to what we discuss in this article which were studied earlier in the literature. In fact, as early as in 1980, Bubnov [8] studied the following system:  ut + uxxx + uux = f, u(x, 0) = φ(x), x ∈ (0, 1), u(0, t) = 0, ux (1, t) + a(t)u(1, t) = 0, uxx (1, t) + 13 u2 (1, t) + b(t)u(1, t) = h(t). (1.8) He obtained the following well-posedness and asymptotic behavior results.

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C. Jia, B.-Y. Zhang

Theorem C (Bubnov) (a) If a(t), b(t) ∈ C 1 [0, T ], 2b + a 2 ≤ 0, and h(0) = 0, φ ≡ 0, then any T > 0,   f ∈ H 1 (0, 1) × (0, T ) , and h ∈ H 1 (0, T ), the system (1.8) admits a unique solution u ∈ L∞ (0, T ; H 3 (0, 1)) with     ut ∈ L2 0, T ; H 1 (0, 1) ∩ L∞ 0, T ; L2 (0, 1) . (b) If a = b = f = h ≡ 0, φ is s smooth function, with φL2 (0,1) = c0 and there are constants , δ > 0 such that c0 9 − > 0, 2 (1 − δ − ) then corresponding to each integer k > 0 there is c(k) → ∞, k → ∞, such that the corresponding solution u of the system (1.6) satisfies sup |u(x, t)| ≤ 0≤x≤1

c(k) 1 + tk

for all t ≥ 0.

In [31], Liu and Krsti´c studied the following system  ut + δuxxx + uux − uxx = 0, u(x, 0) = φ(x), x ∈ (0, 1), t > 0, u(0, t) = 0, ux (1, x) = 0, uxx (1, t) = k1 u3 (1, t) + k2 u(1, t)

(1.9)

where δ > 0,

 > 0,

k1 ,

k2 >

1 . 6δ

They have shown that the system (1.9) is globally well-posed in the space H j (0, 1) for j = 1, 2 and also have the following global asymptotic stability. Theorem D (Liu and Krsti´c) Set  ω/  2 2 K = 2 max 1, ω = (4δ + ), , 9 δ(4δ + )   H01 (0, 1) := f ∈ H 1 (0, 1) : φ(0) = 0

F (r) = r 2 + r 4+4ω/ ,

and   3 Hbc (0, 1) = f ∈ H 3 (0, 1) : f (0) = f  (1) = f  (1) − k1 f 3 (1) − k2 f (1) = 0 .

(a) For any φ ∈ H01 (0, 1), the corresponding solution u of the system (1.9) satisfies   u(t)2 ≤ K φ2 + φ2(1+ω/) e−2ωt , ∀ t ≥ 0 and

     u(t)1 ≤ cF φ1 exp cF φ e−ωt ,

∀ t ≥ 0.

Boundary Stabilization of the Korteweg-de Vries Equation

33

3 (b) For any φ ∈ Hbc (0, 1), the corresponding solution u of the system (1.9) satisfies

u(t)23 ≤ c

3 

     F j φ3 exp cF φ e−ωt ,

∀ t ≥ 0.

j =1

The same approach we use to prove Theorem 1.1 and Theorem 1.2 in this article can be applied with a minor modification to both systems (1.8) and (1.9) to obtain the following results. Theorem 1.3 (a) Let T > 0 be given. Assume that 1

a ∈ H 3 (0, T ),

b ∈ L2 (0, T ),

2b + a 2 ≤ 0.

For any   f ∈ L1 0, T ; L2 (0, 1) ,

1

h ∈ H − 3 (0, T )

and φ ∈ L2 (0, 1), the system (1.8) admits a unique solution u ∈ Y(0,T ) . (b) Let T > 0 be given. Assume that 4

a ∈ H 3 (0, T ),

b ∈ H 1 (0, T ),

2b + a 2 ≤ 0.

For any   f ∈ W 1,1 0, T ; L2 (0, 1) ,

2

h ∈ H 3 (0, T )

with h(0) = 0,

and φ ∈ H 3 (0, 1) ∩ H01 (0, 1)

with φ  (1) = φ  (1) = 0,

3 the system (1.8) admits a unique solution u ∈ Y(0,T ) with ut ∈ Y(0,T ) . (c) Assume a = b = f = h ≡ 0. Let s ∈ [0, 3] with s = 2j2−1 , j = 1, 2, 3 be given. There exist η > 0 and T > 0 such that φ ∈ Hs (0, 1) with

φ ≤ η, then the corresponding solution u of the system (1.8) belongs to the space YTs satisfies   − s ≤ Cs φ φs e−γ0 t uY(t,t+T )

for all t ≥ 0

where Cs : R+ → R+ is a nondecreasing continuous function. Theorem 1.4 Assume  ≥ 0 for system (1.9). Let T > 0 and s ∈ [0, 3] with s = 2j2−1 , j = s 1, 2, 3 be given. For any φ ∈ Hs (0, 1), the system (1.9) admits a unique solution u ∈ Y(0,T )∩ α,T 1 X0, 1 for some α ∈ ( 2 , 1) with 2

  s uY(0,T + uXα,T ≤ αs,T φ φs ) 0, 21

34

C. Jia, B.-Y. Zhang

where αs,T : R+ → R+ is a nondecreasing continuous function. Moreover, the solution map s is Lipschitz continuous from Hs (0, L) to the space Y(0,T ) . Furthermore, if s = 3, the above 3 result holds for φ ∈ Hbc (0, 1). Remark 1.5 It is worthwhile to point out that our approach enable us to obtain the global well-posedness of system in both cases of  = 0 (KdV system) and  > 0 (KdV-Burgers system). By contrast, the approach used by Liu and Krsti´c in [31] only works for the system (1.9) with  > 0. Theorem 1.6 Let s ∈ [0, 3] with s =

2j −1 , 2

j = 1, 2, 3 be given.

(i) If  = 0 in ( 1.9), there exist T > 0 and η > 0 such that if φ ∈ Hs (0, 1) with φ ≤ η, then the corresponding solution u of the system (1.9) satisfies   − s uY(t,t+T ≤ βs φ φs e−γ0 t )

for all t ≥ 0,

where βs : R+ → R+ is a nondecreasing continuous function. (ii) If  > 0 in (1.9), there exists a T > 0 such that for any φ ∈ Hs (0, 1) the corresponding solution u of the system (1.9) satisfies   − s ≤ κs φ φs e−γ t uY(t,t+T )

for all t ≥ 0

where κs : R+ → R+ is a nondecreasing continuous function. 3 (iii) In the case of s = 3, both (i) and (ii) hold for φ ∈ Hbc (0, 1). Remark 1.7 The decay rate γ− is strictly larger than ω in Theorem D. We believe that γ− is the largest decay rate one can have for system (1.9). The paper is organized as follows. – In Sect. 2, we consider the associated linear problem ⎧ ⎪ ⎨ut + ux + uxxx − uxx = 0, x ∈ (0, L), t > 0, u(x, 0) = φ(x), ⎪ ⎩ u(0, t) = h1 (t), ux (L, x) = h2 (t) uxx (L, t) = h3 (t).

(1.10)

Various estimates for its solutions will be recalled or established. Those estimates will play important roles in showing the nonlinear system (1.6) is globally well-posed in the space H s (0, L) and is also globally exponentially stable or locally exponentially stable depending on whether  > 0 or  = 0. – In Sect. 3, the nonlinear IBVP (1.6) will be the focus of our attention. The proofs will be provided for both Theorem 1.1 and Theorem 1.2.

Boundary Stabilization of the Korteweg-de Vries Equation

35

2 Linear Estimates Consideration is first directed to the IBVP of linear equation with the nonhomogeneous boundary conditions  ut + ux + uxxx − uxx = 0, u(x, 0) = 0, x ∈ (0, L), t > 0, (2.1) u(0, t) = h1 (t), ux (L, t) = h2 (t), uxx (L, t) = h3 (t) where  ≥ 0 is a given constant. Its solution can be written as   u(x, t) = W,b (t)h (x) with h = (h1 , h2 , h3 ). Here W,b (t) is the boundary integral operator associated with the IBVP (2.1) whose explicit representation formula can be found in [29]. For any T > 0 and s ≥ 0, let     s s 2 s+1 0 (0, L) , B(0,T ) = B(0,T B(0,T ) = C [0, T ]; H (0, L) ∩ L 0, T ; H ). Lemma 2.1 Let T > 0 be given. There exist a constant C = CT depending only on T such 2−j that for any hj ∈ H 3 (0, T ), j = 1, 2, 3, the IBVP (2.1) admits a unique solution u ∈ α,T 1 B(0,T ) ∩ X0, 1 for some α ∈ ( 2 , 1) with 2

uB(0,T ) + uXα,T ≤ CT 0, 21

3 

hj 

j =1

.

2−j

H 3 (0,T )

Moreover, the solution u has the following trace property:   1−j 3 (0, T ) , ∂xj u ∈ L∞ x 0, L; H

j = 0, 1, 2

with 2   j   ∂ u x

j =0

1−j L∞ x (0,L;H 3 (0,T ))

≤ CT

3 

hj 

j =1

2−j

.

H 3 (0,T )

Proof In the case of  = 0, the proof can be found in [5, 29, 34]. The proof for the case of  > 0 is almost same with few modification and is therefore omitted.  Next we consider the IBVP of the linear equation with the homogeneous boundary conditions  ut + ux + uxxx − uxx = 0, u(x, 0) = φ(x), x ∈ (0, L), t > 0, (2.2) u(0, t) = ux (L, t) = uxx (L, t) = 0, where  ≥ 0. Its solution u can be written in the form u(x, t) = W (t)φ where W (t) is the C0 -semigroup in the space L2 (0, L) generated by the operator A g := −g  − g  + g 

36

C. Jia, B.-Y. Zhang

with domain   D(A ) = g ∈ H 3 (0, L) : g(0) = g  (L) = g  (L) = 0 . Both A and its adjoint operator A∗ are dissipative operators. We have the following spectral analysis for A . Lemma 2.2 The spectrum σ (A ) of A consists of all eigenvalues {λk }∞ 1 with k = 1, 2, 3, . . .

Re λk < 0, and

  8π 3 k 3 + O k2 λk = − √ 3 3L3

as k → ∞.

(2.3)

Lemma 2.3 The resolvent operator R(λ, A ) of A has the following asymptotic estimate on the pure imaginary axis,   R(iw, A ) = O |w|−2/3 ,

as |w| → ∞.

The proof of both lemmas in the case of  = 0 can be found [29]. The proofs presented there can be easily adapted to the case of  > 0. Thus, according to [33], the operator A generates a differential semigroup W (t) in the space L2 (0, L) and the following estimate then follows from Lemma 2.3. Lemma 2.4 Let   γ = − sup Re λ; λ ∈ σ (A ) and γ− denotes any number smaller that γ . For any φ ∈ L2 (0, L), u(t) = W (t)φ ∈ H ∞ (0, L)

for any t > 0

and −

u(t) ≤ Ce−γ t φ

for all t ≥ 0.

Moreover, we have the following estimate for solutions of the IBVP (2.2). Lemma 2.5 Let T > 0 be given. There exists a constant C = CT such that for any φ ∈ L2 (0, L), the corresponding solution u of the IBVP (2.2) belongs to the space u ∈ B(0,T ) with uB(0,T ) ≤ CT φ. α,T 1 Moreover, the solution u also belongs to the space X0, 1 for some α ∈ ( 2 , 1) with 2

uXα,T ≤ CT φ. 0, 21

Boundary Stabilization of the Korteweg-de Vries Equation

37

Proof See [5, 29, 34].



Now we turn to consider the following IBVP of the inhomogeneous equation ⎧ ⎪ ⎨ut + ux + uxxx − uxx = f, x ∈ (0, L), t > 0, u(x, 0) = 0, ⎪ ⎩ u(0, t) = ux (L, t) = uxx (L, t) = 0.

(2.4)

For any given s ∈ R, 0 ≤ b ≤ 1, 0 ≤ α ≤ 1 and function w ≡ w(x, t) : R2 → R, define

 Gs (w) =

∞ −∞

 Qs,b (w) =

 (1 + |ξ |)

2s

∞



−∞

and

∞ −∞

 Pα (w) =

∞

−∞

(1 + |ξ |)2s

∞

−∞

 |ξ |≤1

|w(ξ, ˆ τ )| dτ 1 + |τ − (ξ 3 − ξ )|

2

1/2 dξ

|w(ξ, ˆ τ )|2 dτ dξ (1 + |τ − (ξ 3 − ξ )|)2b

|w(ξ, ˆ τ )|2 dτ dξ (1 + |τ |)2(1−α)

, 1/2

1/2 .

Let Ys,b be the space of all w satisfying  1/2 < ∞, wYs,b := Gs2 (w) + Q2s,b (w) and let Yαs,b be the space of all w satisfying  1/2 wYαs,b := Pα2 (w) + Gs2 (w) + Q2s,b (w) < ∞, α and Yα,T s,b be the restriction of Ys,b on the domain (0, L) × (0, T ) for any T > 0.

Lemma 2.6 Let T > 0. There exists a constant C = CT depending on T such that for any f ∈ L1 (0, T ; L2 (0, L)) the IBVP (2.4) admits a unique solution u ∈ B(0,T ) satisfying uB(0,T ) ≤ CT f L1 (0,T ;L2 (0,L)) . α,T Moreover, there exist α ∈ ( 12 , 1) and b ∈ (0, 12 ) such that if f ∈ Y1−α,T , then u ∈ X0, 1 with 0,b 2

uXα,T ≤ CT f Y1−α,T . 0, 21

Proof See [5, 29, 34].

0,b



Next we show that both solutions of the IBVP (2.2) and the IBVP (2.4) possess the following spatial trace property. Lemma 2.7 Let T > 0 be given. There exists a constant CT depending only on T such that (i) for any φ ∈ L2 (0, L), the corresponding solution u of the IBVP (2.2) has the trace property that   1−j 3 (0, T ) , j = 0, 1, 2 ∂xj u ∈ L∞ x 0, L; H

38

C. Jia, B.-Y. Zhang

and 2   j   ∂ u x

1−j

L∞ x (0,L;H 3 (0,T ))

j =0

≤ CT φ;

T and some b ∈ (0, 12 ), the corresponding solution u of the IBVP (2.4) (ii) for any f ∈ Y0,b has the trace property that

  1−j 3 (0, T ) , ∂xj u ∈ L∞ x 0, L; H

j = 0, 1, 2

and 2   j  ∂ u x

1−j

L∞ x (0,L;H 3 (0,T ))

j =0

≤ CT f Y T . 0,b

Proof Consider the initial-value problem (IVP) of the linear equation posed on the whole line R:  wt + wx + wxxx − wxx = 0, x ∈ R, t > 0, (2.5) u(x, 0) = ψ(x). Its solution can be written as w(t) = U (t)ψ where U (t) is the C0 -semigroup associated with the IVP (2.5). The solution w belongs to the space C(R+ ; L2 (R)) ∩ L2loc (R+ ; H 1 (R)) and is well-known (cf. [24, 26]) to have the spatial trace property: for any given T > 0,   1−j 3 (0, T ) , ∂xj w ∈ L∞ x R; H

j = 0, 1, 2

and 2   j  ∂ w x

1−j

L∞ x (R:H 3 (0,T ))

j =0

≤ CT ψH s (R) .

In particular, 1

q1 (t) := w(0, t) ∈ H 3 (0, T ),

q2 (t) := wx (L, t) ∈ L2 (0, T ),

1

q3 (t) := wxx (L, t) ∈ H − 3 (0, T ). For any φ ∈ L2 (0, L), let ψ ∈ L2 (R) be the zero extension of φ from (0, L) to R. Let w(t) = U (t)ψ and q = (q1 , q2 , q3 ). Then we may rewrite u(t) = W (t)φ as q. W (t)φ = U (t)ψ − W,b (t) The trace property of the solution u of the IBVP (2.2) follows from the trace property of the IVP (2.5) and the IBVP (2.1). As for the solution u of the IBVP (2.4), it can be written as  t U (t − τ )f ∗ (·, τ )dτ − W,b (t)p u(t) = 0

Boundary Stabilization of the Korteweg-de Vries Equation

39

where f ∗ ∈ Y0,b is an extension of f from (0, L) × (0, T ) to R × R with f ∗ Y0,b = f Y T 0,b and p(t)  = (p1 (t), p2 (t), p3 (t)) with p1 (t) = v(0, t),

p2 (t) = vx (L, t),

Here



t

v(x, t) :=

p3 (t) = vxx (L, t).

U (t − τ )f ∗ (x, τ )dτ.

0

According to [24, 26], qj 

2−j

H 3 (0,T )

≤ CT f ∗ Y0,b .

The trace property of the solution u of the IBVP (2.4) then follows from the trace property of the IBVP (2.1). 

3 Nonlinear Problems We first consider the well-posedness of the nonlinear system ⎧ ⎪ ⎨ut + uux + ux + uxxx − uxx = 0, x ∈ (0, L), u(x, 0) = φ(x), ⎪ ⎩ u(0, t) = 0, ux (L, t) = 0, uxx (L, t) = − 13 u2 (L, t)

(3.1)

with  ≥ 0. For T > 0, α ≥ 0 and s ∈ [0, 3], set s,α α,T s Z(0,T ) = Y(0,T ) ∩ X0, 1 . 2

We first show that (3.1) is globally well-posed in the space L2 (0, L). Theorem 3.1 Let T > 0 be given. For any φ ∈ L2 (0, L), the IBVP (3.1) admits a unique 0,α 1 solution u ∈ Z(0,T ) for some α ∈ ( 2 , 1) with   uZ0,α ≤ α0,T φ φ, (0,T )

where α0,T : R+ → R+ is a nondecreasing continuous function depending only on T . Moreover, the corresponding solution map is Lipschitz continuous. The following bilinear estimates will play important roles in the proof of Theorem 3.1. Lemma 3.2 There exist constants C > 0, δ > 0 and b ∈ (0, 12 ) such that (a) for any T > and u, v ∈ B(0,T ) , 

T

uvx (τ )dτ ≤ CT δ uB(0,T ) vB(0,T ) ; 0

40

C. Jia, B.-Y. Zhang

α,T 1 (b) for any T > 0 and u, v ∈ X0, 1 with α ∈ ( 2 , 1), 2

uvx Yα,T ≤ CT δ uXα,T vXα,T ; 0, 21

0,b

1

0, 21

1

(c) for any T > 0 and g, h ∈ H 3 (0, T ), gh ∈ H − 3 (0, T ) and gh

1

− H 3 (0,T )

≤ CT δ h

1

H 3 (0,T )

g

1

H 3 (0,T )

. 1

Proof The proof of (a) and (b) can be found in [5, 28, 29]. For (c), note that g, h ∈ H 3 (0, T ) implies that g, h ∈ L6 (0, T ) and therefore gh ∈ L3 (0, T ). We have gh

2

H

− 13

(0,T )

2

≤ cghL1 (0,T ) ≤ CT 3 ghL3 (0,T ) ≤ CT 3 g

1

H 3 (0,T )

h

1

H 3 (0,T )

.



Proof of Theorem 3.1 For any smooth solution u of the IBVP (3.1), it holds that  L  L d u2 (x, t)dx + u2 (L, t) + u2x (0, t) + 2 u2x (x, t)dx = 0, dt 0 0 from which it follows that there exists a constant C > 0 such that u(·, t) ≤ Cφe

− 22 t L

for any t ≥ 0.

(3.2)

It is thus sufficient to show that the IBVP (3.1) is locally well-posed in the space L2 (0, L): Let T > 0 be given. For any η > 0, there exists a T ∗ ∈ (0, T ] depending on η such that for any φ ∈ L2 (0, L) with φ ≤ η, 0,α the IBVP (3.1) admits a unique solution u ∈ Z(0,T ∗ ) . Moreover, the corresponding solution map is Lipschitz continuous. To this end, let φ ∈ L2 (0, L) be as given and rewrite (3.1) as  ut + ux + uxxx − uxx = −uux , u(x, 0) = φ(x), u(0, t) = 0, ux (L, t) = 0, uxx (L, t) = − 13 u2 (L, t).

Let r > 0 and 0 < θ ≤ max{1, T } be constants to be determined. Set   0,α | vZ0,α ≤ r , Sθ,r = v ∈ Z(0,θ) (0,θ)

0,α which is a bounded closed convex subset of Z(0,θ) . Define a map  on Sθ,r by

u = (v) being the unique solution of  ut + uxxx + ux − uxx = −vvx , u(x, 0) = φ(x), u(0, t) = 0, ux (L, t) = 0, uxx (L, t) = q(t) for v ∈ Sθ,r where 1 q(t) = − v 2 (L, t). 3

Boundary Stabilization of the Korteweg-de Vries Equation

41

Applying Lemma 3.2 and lemmas in Sect. 2, for any v ∈ Sθ,r , we have   (v) 0,α ≤ C0 φ + C1 θ δ v2 0,α . Z Z (0,θ)

(0,θ)

For given 0 < ν < 1, choose r > 0 and 0 < θ ≤ 1 such that r = 2C0 η

and

1 C1 θ δ r ≤ ν. 2

(3.3)

Then for any v ∈ Sθ,r ,   (v)

0,α Z(0,θ)

≤ C0 φ + C1 θ δ v2Z0,α

(0,θ)

r ≤ + C1 θ δ r 2 2 r r ≤ + = r. 2 2 Thus  : Sθ,r → Sθ,r . In addition, for any u, v ∈ Sθ,r ,  θ      (u) − (v) 0,α =   W (t − τ )[uu − vv ]dτ  x x   Z (0,θ)

0

 δ



0,α Z(0,θ)

≤ C1 θ uZ0,α + vZ0,α u − vZ0,α (0,θ)

(0,θ)

(0,θ)

≤ 2C1 θ ru − vZ0,α (0,θ)   ≤ ν (u − v)Z0,α . δ

(0,θ)

Therefore the map  is a contraction mapping on Sθ,r . Its fixed point u = (u) is the desired solution.  Next we show that the IBVP (3.1) is globally well-posed in the space H 3 (0, L). Recall that   1 D3 (0, L) = ψ ∈ H 3 (0, L); ψ(0) = ψ  (L) = 0, ψ  (L) = − ψ 2 (L) . 3 Theorem 3.3 Let T > 0 be given. For any φ ∈ D3 (0, L), the IBVP (3.1) admits a unique 3,α 1 solution u ∈ Z(0,T ) for some α ∈ ( 2 , 1) with   uZ3,α ≤ α3,T φ φ3 , (0,T )

where α3,T : R+ → R+ is a nondecreasing continuous function depending only on T . Moreover, the corresponding solution map is Lipschitz continuous. Before presenting its proof, we consider the following linear system with variable coefficients a = a(x, t) and g(t),  ut + (au)x + ux + uxxx − uxx = 0, u(x, 0) = ψ(x), x ∈ (0, L), (3.4) u(0, t) = 0, ux (L, t) = 0, uxx (L, t) = g(t)u(L, t).

42

C. Jia, B.-Y. Zhang

0,α 1 Proposition 3.4 Let T > 0 be given. Assume that a ∈ Z(0,T ) for some α ∈ ( 2 , 1) and g ∈ 1

0,α H 3 (0, T ). Then for any ψ ∈ L2 (0, L), the IBVP (3.4) admits a unique solution u ∈ Z(0,T ) satisfying   uZ0,α ≤ μ aZ0,α + g 1 ψ, (0,T )

H 3 (0,T )

(0,T )

where μ : R+ → R+ is a T -dependent continuous nondecreasing function independent of  and ψ . Proof As in the proof of Theorem 3.1, let r > 0 and 0 < θ ≤ max{1, T } be constants to be determined. Set   0,α | vZ0,α ≤ r . Sθ,r = v ∈ Z(0,θ) (0,θ)

Define a map  on Sθ,r by u = (v) being the unique solution of  ut + uxxx + ux − uxx = −(av)x , u(x, 0) = φ(x), u(0, t) = 0, ux (L, t) = 0, uxx (L, t) = q(t) for v ∈ Sθ,r where q(t) = g(t)v(L, t). There exists constants C0 and C1 such that     (v) 0,α ≤ C0 φ + C1 θ δ |g| 1 ,(0,T ) + aZ 0,α vZ 0,α Z 3

(0,θ)

(0,θ)

(0,θ)

for any v ∈ Sθ,r . For given 0 < μ < 1, choose r > 0 and 0 < θ ≤ 1 such that r = 2C0 φ

and

  1 C1 θ δ |g| 1 ,(0,T ) + aZ0,α ≤ ν. 3 (0,θ) 2

For such chosen r and θ , (v) ∈ Sr,θ and

for any v ∈ Sr,θ

  (v1 ) − (v2 )

0,α Z(0,θ)

≤ νv1 − v2 Z0,α

(0,θ)

for any v1 , v2 ∈ Sr,θ . In other words,  is a contraction mapping in Sr,θ ; its fixed point 0,α solves the IBVP (3.4). Note that the time interval (0, θ ) in which the solution u ∈ Z(0,θ) u exists depending only on |g| 1 ,(0,T ) + aZ0,α , not on φ, in particular. By a standard 3

(0,T )

continuous extension argument one can extend θ so that θ = T .



Proof of Theorem 3.3 For given φ ∈ D3 (0, L), according to Theorem 3.1, the IBVP (3.1) 0,α admits a unique solution u ∈ Z(0,T ) . Let v = ut . Then v solves 

vt + (av)x + vx + vxxx − vxx = 0, v(x, 0) = φ ∗ (x), v(0, t) = 0, vx (L, t) = 0, vxx (L, t) = g(t)v(L, t).

Boundary Stabilization of the Korteweg-de Vries Equation

43

with φ ∗ (x) = −φ  (x) − φ(x)φ  (x) + φ  (x) − φ  (x) ∈ L2 (0, L), 0,α a = u and g(t) = − 23 u(L, t). By Proposition 3.4, ut = v ∈ Z(0,T ) . Then it follows from the relation

uxxx = −ut − ux − uux + uxx 3,α that u ∈ Z(0,T ) . The proof is complete.



Finally we show that the IBVP (3.1) is globally well-posed in the space H s (0, L) for s ∈ (0, 3). Theorem 3.5 Let T > 0 and s ∈ [0, 3] with s = k2 , j = 1, 3, 5 be given. For any φ ∈ s,α 1 Hs (0, L), the IBVP (3.1) admits a unique solution u ∈ Z(0,T ) for some α ∈ ( 2 , 1) with   uZs,α ≤ αs,T φ φs (0,T )

where αs,T : R+ → R+ is a nondecreasing continuous function depending only on T . Moreover, the corresponding solution map is Lipschitz continuous. Proof According to Theorem 3.1 and Theorem 3.3, for given T > 0, the IBVP (3.1) well j,α defines a nonlinear map K from the space Hj (0, L) to the space Z(0,T ) for j = 0, 3. Moreover,     K(φ1 ) − K(φ2 ) 0,α ≤ α0,T φ1  + φ2  φ1 − φ2  Z (0,T )

for any φ1 , φ2 ∈ L (0, 1) and 2

  K(φ)

3,α Z(0,T )

  ≤ α3,T φ φ3

for any φ ∈ H3 (0, L). Then, applying Tartar’s nonlinear interpolation theory [1, 35] yields s,α that the nonlinear map K is well-defined from the space Hs (0, L) to Z(0,T ) for any s ∈ (0, 3) j with s = 2 for j = 1, 3, 5. Moreover,     K(φ) s,α ≤ αs,T φ φs Z (0,T )

for any φ ∈ Hs (0, L) with 1− s

s

3 (r). αs,T (r) = 4α0,T 3 (r)α3,T

s,α For φ ∈ Hs (0, L), u = K(φ) ∈ Z(0,T ) is the desired solution of the IBVP (3.1).



Knowing that the solutions of the IBVP (3.1) exist globally, we turn to study their long time asymptotic behavior. First we consider the linear system,  ut + (au)x + ux + uxxx − uxx = 0, u(x, 0) = φ(x), (3.5) u(0, t) = 0, ux (L, t) = 0, uxx (L, t) = g(t)u(L, t). Recall that  ≥ 0.

44

C. Jia, B.-Y. Zhang

Proposition 3.6 There exist T > 0 and δ > 0 such that if a ∈ ZT0,α for some α ∈ ( 12 , 1) with aZ0,α + |g| 1 ,T ≤ δ, 3

T

then for any φ ∈ L2 (0, L), the corresponding solution u of (3.5) satisfies −

uZ0,α

(t,t+T )

≤ Ce−γ t φ

for any t ≥ 0 where C is a positive constant independent of φ. Here |g| 1 ,T := sup g 3

t≥0

1

H 3 (t,t+T )

.

Proof The proof is similar to that of Theorem 2.8 in [34] and we therefore provide just a sketch. Rewrite (3.4) in its integral form 

t

u(t) = W (t)φ + W,b (t)g(t)u(L, t) −

W (t − τ )(au)x (τ )dτ. 0

For any T > 0,    u(·, T ) ≤ Ce−γ− T φ + CT a 0,α + g Z (0,T )

≤ Ce

−βT

 1 H 3 (0,T )

uZ0,α

(0,T )

φ + CT δμ(δ)φ.

Note that in the above estimate, the constant C is independent of T . Let yn = u(·, nT )

for n = 0, 1, 2, . . .

and let v be the solution of the IBVP  vt + vx + (av)x + vxxx = 0, v(x, 0) = yn (x), v(0, t) = 0, vx (L, t) = 0, vxx (L, t) = g(t + nT )v(L, t).

(3.6)

Thus yn+1 (x) = v(x, T ) by the semigroup property of the system (3.5). Consequently, we have the following estimate for yn+1 : −

yn+1  ≤ Ce−γ T yn  + CT δμ(δ)yn  for n = 0, 1, 2, . . . . Choose T and δ such that −T

Ce−γ

= η < 1,

η + CT δμ(δ) := r < 1.

Then, yn+1  ≤ ryn  for all n ≥ 0 if aYT + |g| 1 ,T ≤ δ, from which the conclusion of Proposition 3.6 follows.  3

Now we show the small amplitude solution of the IBVP (3.1) decays exponentially.

Boundary Stabilization of the Korteweg-de Vries Equation

45

Theorem 3.7 Let s ∈ [0, 3] with s = j2 , j = 1, 3, 5 be given. There exist T > 0 and δ > 0 such that for any φ ∈ Hs (0, L) satisfying φ ≤ δ, the corresponding solution u of the system (3.1) satisfies   − s ≤ κs φ φs e−γ t uY(t,t+T )

for any t ≥ 0,

(3.7)

where κ : R= → R+ is continuous nondecreasing function. Proof First we consider the case of s = 0. For φ ∈ L2 (0, L), according to Theorem 3.1, the corresponding solution u of (3.1) exists globally and   u(·, t) ≤ φ for any t ≥ 0 and

  uZ0,α ≤ α0,T φ φ (0,T )

for any T > 0. By the semigroup property of the system (3.1),   uZ0,α ≤ α0,T φ φ. T

We may view the nonlinear system (3.1) as the linear system (3.5) with a = 12 u and g(t) = − 13 u(L, t). Thus if δ is chosen small enough, the condition of Proposition 3.6 is satisfied and estimate (3.7) with s = 0 follows consequently. In the case of s = 3, let v = ut . Then v solves the linear system (3.5) with a = u and g(t) = − 23 u(L, t). Again if δ is chosen small enough, the condition of Proposition 3.6 is satisfied and we have ut Y 3

(t,t+T )

≤ Ce−rt φ3

from which estimate (3.7) with s = 3 follows. Finally, the general case of s ∈ (0, 3) follows by invoking Tartar’s nonlinear interpolation theory [1, 35].  The above theorem shows that the system (3.1) (with  ≥ 0) is locally exponentially stable. Our next theorem shows when  > 0 the system is globally exponentially stable. Theorem 3.8 Assume  > 0 and let s ∈ [0, 3] with s = j2 , j = 1, 3, 5 be given. There exists a T > 0 such that for any φ ∈ Hs (0, L), the corresponding solution u of the system (3.1) satisfies   − s ≤ νs φ φs e−γ t for any t ≥ 0 (3.8) uY(t,t+T ) where νs : R+ → R+ is continuous nondecreasing function. Proof When  > 0, for any φ ∈ L2 (0, L), the corresponding solution u of the system (3.1) satisfies   u(·, t) ≤ Cφe−t for any t ≥ 0

46

C. Jia, B.-Y. Zhang

where C is a constant independent of t and φ. Thus for any δ > 0, there exists a t0 > 0 such that   u(·, t0 ) ≤ δ. Then considering system (3.1) with initial time started at t0 with u(x, t0 ) as an initial value, Theorem 3.8 follows by applying Theorem 3.7.  Acknowledgements Chaohua Jia was partially supported by State Scholarship Fund of China (No. 2010602510) from China Scholarship Council (CSC), the National Natural Science Foundation of China (No.10626002) and the Fundamental Research Funds YWF-10-01-A15 for the Central Universities. Bing-Yu Zhang was partially supported by a grant from the Simons Foundation (#201615 to Bingyu Zhang), the Taft Memorial Fund at the University of Cincinnati and the Chunhui program (State Education Ministry of China) under grant Z007-1-61006. This work was conducted while Chaohua Jia was visiting Department of Mathematical Science, University of Cincinnati from February 2011 to February 2012 under the State Scholarship Fund of China. He would like to thank the University of Cincinnati for its hospitality.

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