Nonlinear Schrödinger equations on a finite interval with point dissipation

The paper considers the initial value problem of a general type of nonlinear Schrodinger equations \begin{document}$ iu_t+u_{xx}+f(u) = 0 , \;\;\;\; u ( x, 0 ) = w_0 (x) $\end{document} posed on a finite domain \begin{document}$ x\in [0, L] $\end{document} with an \begin{document}$ L^2 $\end{document} -stabilizing feedback control law \begin{document}$ u(0, t) = \beta u(L, t), \beta u_x(0, t)-u_x(L, t) = i\alpha u(0, t), $\end{document} where \begin{document}$ L>0 $\end{document} , \begin{document}$ \alpha, \beta $\end{document} are real constants with \begin{document}$ \alpha\beta and \begin{document}$ \beta\neq \pm 1 $\end{document} , and \begin{document}$ f(u) $\end{document} is a smooth function from \begin{document}$ \mathbb{C} $\end{document} to \begin{document}$ \mathbb{C} $\end{document} satisfying some growth conditions. It is shown that for \begin{document}$ s \in \left ( \frac12, 1\right ] $\end{document} and \begin{document}$ w_0 (x) \in H^s(0, L ) $\end{document} with the boundary conditions described above, the problem is locally well-posed for \begin{document}$ u \in C([0, T]; H^s (0, L )) $\end{document} . Moreover, the solution with small initial condition exists globally and approaches to 0 as \begin{document}$ t \rightarrow + \infty $\end{document} .

1. Introduction. We consider the solutions of nonlinear Schrödinger (NLS) equation iu t + u xx + c|u| 2 u = 0 (1.1) or more generally, iu t + u xx + f (u) = 0 (1.2) on a finite interval 0 ≤ x ≤ L with t ≥ 0, where u(x, t) is a complex-valued function, c is a nonzero real constant, and f (u) satisfies certain conditions described later. The NLS equation (1.1) has many applications and was derived as a model for a considerable range of physical problems, which include propagation of light in fiber optical cables, certain types of shallow and deep surface water waves, and Langmuir waves in a hot plasma or in general forms of Bose-Einstein condensate theory. Recently, the NLS equation has been used as a popular model in attempting to explain the formation of rogue waves observed in the seas or oceans [11,29]. Here, we are only interested in the well-posedness and stabilization problems for (1.1) or (1.2) under certain boundary conditions.
Mathematical study of (1.1) or (1.2) can be traced back to several decades ago, when Zakharov and his collaborators [37,38] considered the initial value problem (IVP) of (1.1) on R using inverse scattering method. The rigorous theory on the well-posedness of the IVP of (1.1) in the classical Sobolev spaces was extensively studied afterwards and mainly focused on the pure IVP posed on the entire real line R or the periodic initial-value problem posed on one-dimensional torus T (for example, see [2,3,9,10,15,16,22,23,36] and a monograph by Cazenave [8]). Moreover, the boundary value problem of (1.1) posed in a finite interval or half line with boundary conditions has been discussed for solutions in Sobolev spaces (see [1,4,5,6,7,18,24,34], and the references therein). For the control and stabilization problems of NLS equations, Illner, et al. [20,21] applied an internal forcing to show the controllability of NLS equations posed on a finite interval with periodic boundary conditions, while Lange and Teismann [26] considered the internal controllability of NLS equations in a finite interval with Dirichlet boundary conditions. Rosier and Zhang [30] studied the exact controllability and stabilizability of (1.1) posed on a finite interval using both internal and boundary controls and showed that problems with those controls are locally exactly controllable in the Sobolev space H s for any s ≥ 0. It is also shown that the problem with an internal stabilizing forcing is locally exponentially stabilizable.
The main concern of this paper is the local and global well-posedness of (1.1) (or more generally (1.2)) and the asymptotic behavior of small solutions as t → +∞ using a closed-loop point dissipation process (general discussions on such problems can be found in [17,31]). This type of control problems for the KdV equation was first discussed by Russell and Zhang [32,33] using dissipative point boundary condition, which, in control theory, is called a closed-loop control process that generally refers to control synthesis via some kind of state feedback and is mainly concerned with achieving asymptotic stability of an equilibrium or invariant set. Similar problems using dissipative point boundary conditions were studied for the KdV equation in a singular case [35] and other KdV type of equations [14]. To design a dissipation mechanism for (1.1) or (1.2), we first state some conditions on f (u) in (1.2). Assumption 1.1. f (0) = 0 and f (u) ≡ 0 with f (u) ∈ C 1 (C, C) and f (u)ū ∈ R for any u ∈ C; Assumption 1.2. |f (u) − f (v)| ≤ f 0 (u, v)|u − v| where f 0 (u, v) ≤ C 0 (|u| p1−1 + |v| p1−1 ) + C 1 (|u| p2−1 + |v| p2−1 ) for any u, v ∈ C with 1 < p 1 < p 2 < ∞.
From Assumptions 1.1 and 1.2, it is easy to see that |f (u)| ≤ C 0 |u| p1 + C 1 |u| p2 . Obviously, f (u) = c|u| 2 u in (1.1) satisfies Assumptions 1.1 and 1.2. Now, we multiply both sides of (1.2) byū(x, t) and integrate it from zero to L. Then, take the complex conjugate of the equations and subtract each other to obtain d dt To make the L 2 -norm of u dissipative, we need the right side of (1. where α, β are any real constants satisfying αβ < 0. Thus, we make the following assumptions on α, β. Assumption 1.3. α, β are any real numbers satisfying αβ < 0 and β = ±1 The second condition of Assumption 1.3 on β is technical. If β = 1 or −1, the problem is singular (see also [35]). From (1.5), the boundary conditions (1.4) are considered as dissipative for (1.2) and it is reasonable to expect that the solution u(x, t) of (1.2) with (1.4) goes to zero as t → ∞.
The main result of this paper is stated as follows. Let an operator Au = iu xx with domain D(A) = u ∈ H 2 (0, L) u satisfies (1.4) and A * be its adjoint operator in L 2 = L 2 (0, L). Then, the following results are obtained.
From Theorem 1.4, the following function space can be defined. For any s ≥ 0, let Then, the solutions of (1.2) with (1.4) and initial condition u(x, 0) = w 0 (x) can be found in H s α,β .
Note that (i) is a local well-posedness result and (ii) gives the global existence and stability results for small initial data. The idea for the proof of Theorems 1.4 and 1.5 basically follows from the method introduced in the papers by Russell and Zhang [32,33], which discuss a similar problem for the KdV equation. However, it is known that the solutions of the linearized KdV equation posed in a bounded interval have smoothing property, while the solutions of the linear Schrödinger equation have no smoothing property. Therefore, the estimates of the linear Schrödinger operator together with the estimates of the corresponding semigroup are more delicate. In this paper, it is first shown that the operator A is dissipative and generates a strongly C 0 -semigroup S(t). After deriving some estimates of the resolvent operator for A, it is proved that the eigenvalues of A lie in the left half of complex plane and the semigroup S(t) is C ∞ and decays exponentially as t → ∞. Then, under Assumption 1.3 and using the properties of the eigenvalues of A, it is shown that A and A * are discrete spectral operators whose eigenfunctions form dual Riesz bases in L 2 . Furthermore, the asymptotic forms of the eigenvalues are derived, from which it can be seen that the solutions of linear problem do not have any smoothing property. The spectral properties of A are essential in obtaining the estimates of S(t) in H s α,β . The existence and uniqueness results for the nonlinear problem are deduced from those estimates using contraction mapping principle. The asymptotic behavior of the solutions of (1.2) with small initial data is then derived by use of Lyapunov techniques based upon the linear operator A and its spectral properties.
The paper is organized as follows. Section 2 discusses the properties of the operator A with its resolvent operator, the semi-group S(t) generated by the operator A, and the exponential decay property of S(t). The spectral properties of A are provided and the function spaces are defined in Section 3. In Section 4, the various estimates of solutions for the linear problem in these function spaces are derived. Section 5 proves the local well-posedness of (1.2), while the global well-posedness and decay of the solutions of (1.2) with small initial data are presented in Section 6.

Exponential decay for the linear equation with boundary dissipation.
To study the solutions of (1.2) with (1.4), we may write (1.2) as u t = iu xx − if (u) and then define a linear operator A by with the domain The following properties of A hold.
Lemma 2.1. The operator A is dissipative. For any f in the range of λI − A with λ > 0, the pre-image of f , denoted by (λI − A) −1 f , is unique with its L 2 -norm bounded by λ −1 f .
Proof. By definition, A is dissipative if and only if Re (u, Au) ≤ 0 for every u ∈ D(A). It is straightforward to check that which implies that for any f in the range of λI − A and λ > 0, Theorem 2.2. The operator A generates a strongly continuous semigroup S(t) for t ≥ 0 on L 2 (0, L).
Proof. By Lumer-Phillips theorem [27] and Lemma 2.1, A generates a strongly continuous semigroup on L 2 (0, L) if the range of (λI − A) is all of L 2 (0, L). Thus, we need to prove that there exists u ∈ D(λI − A), the domain of λI − A, such that (λI − A)u = f for any f ∈ L 2 (0, L), i.e., we need to find u ∈ D(λI − A) satisfying u + iλu = if . For λ = 0, we denote two square roots of −iλ by µ 0 and µ 1 , respectively. Denote u = z and rewrite u + iλu = if to a system of first-order differential equations, We diagonalize (2.5) by a transformation using µ = (µ 0 , µ 1 ), and plug (2.6) into (2.5) to obtain where The solution of (2.7) can be written by Then, the boundary conditions (1.4) are changed to for A, corresponding to the eigenvalue λ associated with µ by −iλ = µ 2 j , j = 0, 1. For this − → v (µ, α, β, x), there is a nonzero solution u of u +iλu = 0 (i.e. (λI−A)u = 0), which contradicts to (2.4). Hence, Φ(µ, α, β) is invertible for any λ > 0 and From (2.9), we have (2.11) Therefore, for any f ∈ L 2 (0, L), we can find u = (λI − A) −1 f , which means that the range of λI − A is all the functions in L 2 (0, L) for any λ > 0. Thus, A generates a strongly continuous semigroup on L 2 (0, L). Now, we study the resolvent of A for λ on the imaginary axis. Similarly, the case for ω = 0 also impliesũ = 0. Since A has only discrete spectrum, we conclude that for any λ on the imaginary axis, R(λ, A) exists.
The following is the resolvent estimate for large λ on the imaginary axis.
Proof. First, we find the solution u of (λI − A)u = f with boundary conditions (1.4) using Green's function. If we define G(λ, x, ζ) that satisfies then the solution u is given by The Green's function G can be found as follows. From the homogeneous equation (λI − A)u = 0, it is obtained that (2. 16) By the conditions at x = ζ, The boundary conditions (2.13) and (2.14) imply that From the definitions in the proof of Theorem 2.2, we have that if c = (c 0 , c 1 ) T and c = (ĉ 0 ,ĉ 1 ) T , then and Ω(µ) defined in (2.8).
Lemma 2.4 implies the following Corollary using Corollary 4.10 in Chapter 2 of [28].
The next theorem gives the exponential decay of the semi-group for the linear Schrödinger operator (2.1) with boundary conditions given by (2.2). Theorem 2.6. There exist positive constants ξ and η such that Proof. By Lemmas 2.3 and 2.4, it is obtained that for all λ on the imaginary axis, R(λ, A) exists and is uniformly bounded for large λ. Thus, to derive the uniform exponential decay property of S(t) using the result by Huang [19], we only need to prove that R(λ, A) is bounded as λ → 0. From (2.6), (2.8) and (2.11), it is deduced differentiable with respect to λ. At λ = 0, using the Taylor expansion of e LF (0) and (F (0)) 2 = 0, we find Therefore, by the continuity of (2.21) with respect to λ, the matrix (2.21) is uniformly bounded in a small neighborhood of λ = 0 and R(λ, A) is bounded as λ → 0.

Spectral properties of linear Schrödinger operator and function spaces.
In this section, we will discuss the spectral properties of A in (2.1)-(2.2) together with its semi-group S(t).
It is straightforward to find the adjoint operator A * of A in L 2 = L 2 (0, L) as Proposition 3.1. The operator A is a discrete spectral operator. All of its eigenvalues λ except for first few ones correspond to one-dimensional projections E(λ; T ).
Proof. The proposition is a direct consequence of Theorem 8 in Section 4 of Chapter XIX on p.2334 of Dunford and Schwartz's book [12], if we can check that the hypotheses of the theorem are satisfied. By the notations introduced there, we let Also, p = m 1 + m 2 = 0 + 1 = 1 and n = 2 with n = 2ν and ν = 1.
Thus, the coefficients are with the leading order parts in terms of µ as Hence, the hypotheses of Theorem 8 in Section 4 of Chapter XIX of Dunford and Schwartz's book [12] on p.2334 are satisfied.
In the following, for ψ ∈ L 2 , we denote ψ * as the corresponding adjoint vector of ψ in the Hilbert space L 2 , i.e., for any φ in L 2 , Proposition 3.2. The operators A, A * with their corresponding domains have compact resolvents and possess complete sets of eigenvectors, respectively. The eigenvectors form dual Riesz bases in L 2 (0, L) and satisfy (with δ kj , the Kronecker delta) ψ * j φ k = δ kj , while the eigenvalues λ of A satisfy Re λ ≤ −γ < 0 and have the asymptotic form Proof. By Lemmas 2.1 and 2.3, it is obvious that any λ with Re λ ≥ 0 is not an eigenvalue of operator A or A * . Thus, the eigenvalues must satisfy Re λ < 0. For Im λ > 0, the eigenfunction φ satisfies The general solution of (3.6) is Substituting (3.7) into the boundary conditions in (3.6), we have which give a system of equations Setting the determinant of the coefficient matrix equal to zero yields If the real part of µ 0 → ∞, (3.9) implies (β 2 + 1) = 0, which does not hold. Thus, the real part of µ 0 must be bounded. Dividing (3.9) by µ 0 gives Hence, Since cosh x ≥ 1 and |2β/(β 2 + 1)| < 1 for β = ±1, | cos(bL)| cannot approach to one for large b or sin(bL) cannot go to zero. Thus, sinh(aL) = O(1/µ 0 ), cosh(aL) = O(1), and cos(bL) = 2β where cos θ = 2β/(β 2 + 1) has been used. Thus, where τ = (−αβL)/(2(β 2 + 1)π) > 0 . Hence, Then, Rouché's theorem yields a one-to-one correspondence between the eigenvalues λ k and the indices k, k = 0, ±1, ±2, . . . . Therefore, there is a γ > 0 such that Re λ k ≤ −γ < 0. A similar argument gives that λ k , the complex conjugate of λ k , is the eigenvalue of adjoint operator A * . Hence, the eigenfunction of A for the eigenvalue λ k is ,k L c 0,k and c 1,k is uniformly bounded relative to c 0,k as |k| → ∞. The eigenfunctions ψ k (x) of adjoint operator A * take the form By the boundary conditions (2.2), it is obtained that Hence, When j = k that implies λ k = λ j , we have that L 0 φ k φ j dx = 0 and (ψ k , φ j ) = 0. When j = k, an appropriate choice of the coefficients c 0 and c 1 makes (ψ k , φ j ) = ψ k L 2 = φ k L 2 = 1. Now, we show that {ψ k } and {φ k } form Riesz bases. It was shown that both A and A * are discrete spectral operators. Then, we apply the following slightly modified Ingham-Komornik result in [25]: Let {λ k } k≥1 be a sequence of complex numbers with Reλ k uniformly bounded, {b k } k≥1 a sequence of complex numbers satisfying sup a∈R a ≤ Imλ k < a+1 |b k | 2 < ∞ .
Then for any square-summable sequence {x k } k≥1 of complex numbers, where χ(x) is an infinitely differentiable function of compactly supported in R and C(χ) is a constant dependent upon χ.
A similar argument gives Next, we derive the relations between Sobolev norms and the norms obtained from those Riesz bases.
for any complex sequence {f k } ∈ l 2 n , where Proof. The proof of this proposition is similar to the proof of uniform l 2 -convergence property of {φ k } and {ψ k } in Proposition 3.2. By the Ingham-Komornik result stated in Proposition 3.2 (see the proof of (3.13)), it is deduced that A similar proof works for for any {g j } ∈ l 2 n . Thus, is uniform l 2 n -independent in L 2 for n ≥ 1, i.e. there exists a positiveD 2 n such that for any sequence of complex numbers {f k } ∈ l 2 n , Proof. The case n = 0 was proved in Proposition 3.2. For n = 1, from the boundary conditions (3.2), we have an identity,

JING CUI AND SHU-MING SUN
which implies that Hence, by the Sobolev embedding theorem and Proposition 3.3, it is obtained that For n = 2, since iφ = λφ, it is deduced that For n = 3, by iφ = λφ , the case for n = 1, and Sobolev interpolation inequality for any small > 0, we can use a similar argument to obtain The cases for n ≥ 4 can be handled similarly. The same proof works for {ψ (n) k } with other n ≥ 0. Now, we can define the function spaces to be used for the nonlinear problem.
if the indicated derivatives are of order ≤ n − 1. Its norm and inner product are inherited from H n [0, L] and denoted by n and (·, ·) n , respectively.

From Propositions 3.2 and 3.4, we have
The proof is completed.
By Corollary 3.6, we define a class of Banach spaces H s,p α,β . If {φ k (x)} is the Riesz basis of L 2 (0, L) given in Proposition 3.2, then for any s ≥ 0 and p ≥ 1, define By p → q for any q > p ≥ 2, it is easy to see that  = H s (0, L) are undefined and the boundary conditions are not necessary, we have that for 0 ≤ s < 1/2, H s α,β = H s . However, for 1/2 < s < 3/2, the boundary values of functions in H s are welldefined by Sobelev imbedding theorem, which implies that one boundary condition w(0) = βw(L) is needed for an H s -function w to be in H s α,β (since w x (0), w x (L) are not defined for this case, the other boundary condition involving the derivative of w is not necessary).

Properties of semi-groups generated by linear Schrödinger operators.
From the discussion of Section 2, if then we can obtain that the resolution of the identity associated with the operator which is strongly convergent in L(L 2 , L 2 ). The corresponding strongly convergent semigroup generated by A is Hence, the solution of the nonhomogeneous problem is given by First, we derive the estimates for S(t)w 0 (x).
Proposition 4.1. For any given s ≥ 0 and T > 0 and w 0 ∈ H s α,β , The following propositions will be used for the estimates of solutions corresponding to the nonhomogeneous terms. .

5.
Local well-posedness of the nonlinear problem. In this section, we consider the local well-posedness of the IVP for the nonlinear Schrödinger equation with α, β real numbers satisfying αβ < 0 and β = ±1 and f (u) satisfying the assumptions (H1) and (H2) stated in Introduction. We can rewrite the equation in . By comparing to (4.1), the inhomogeneous term f is if (u) and the solution of (5.1) is We will study the solution of (5.2) using the fixed-point theorem for the mapping 3) The following lemma gives that the Sobolev norm in H s for s > 1/2 is an algebra and the proof is well-known. Since H s α,β is a subspace of H s , Lemma 5.1 is applicable if all of f, g and f g are in H s α,β . Now, we prove that the IVP of (5.1) is well-posed in the space H s α,β for

) has a unique solution
v ∈ X T := C(0, T ; H s α,β ) and T → ∞ as w 0 s → 0; (ii) If 0 < T < T is given, there is a neighborhood U of w 0 in H s α,β and the map K : w 0 → v(x, t) from U to X T is Lipschitz continuous.

Proof. Let
where b > 0 and T > 0 will be determined later. To find the desired solution of the IVP (5.1) using a fixed point theorem, some appropriate b and T must be chosen such that the map F in (5.3) is a contraction from S T,b to S T,b . First, we note that for v ∈ C(0, T ; H s α,β ) and t fixed, it is not necessary that f (v(·, t)) ∈ H s α,β . However, for 1/2 < s ≤ 1, since v(0, t), v(L, t) are well-defined (note that v x (0, t), v x (L, t) are not defined so that the boundary condition with derivatives in the definition of H s α,β is not needed), we can let a functionf (t) = α,β and the nonlinear part in (5.3) can be rewritten as f (v(·, t)) = g(v) +f (t) with Since g(v) ∈ H s α,β , the propositions in Section 4 are applicable to g(v). Thus, the estimates of F 1 (v) in the following can be obtained using the results in Section 4.
To study F 2 (v), by the explicit form of S(t), it is deduced that

NLS EQUATIONS WITH POINT DISSIPATION 373
By the properties of φ k , it is shown that | L 0 φ k (x)dx| ≤ C|k| −1 , which gives Now, we state the Selberg's inequality, a generalization of Bessel's inequality (see Eq. 4 in [13]): Let H be a Hilbert space with its inner product (·, ·) and norm · , and z j , j = 1, 2, · · · be in H . Then, for any x ∈ H, To apply the Selberg's inequality for F 2 (v), we let H = L 2 (0, t) for fixed t > 0 with functions f (τ ) and z k = e λ k τ . Using the asymptotic form of λ k in (3.10), a straightforward calculation shows that j |(z k , z j )| ≤ C 0 where C 0 > 0 is a constant independent of k. From (5.4), Selberg's inequality, and Assumptions 1.1 and 1.2, it is obtained that Here, C may be independent of t if t 0 f (τ ) 2 dτ is uniformly bounded for any t ≥ 0. Thus, for any s ∈ ( 1 2 , 1], F 2 (v) is in C(0, T ; H 1 α,β ) and the corresponding norm is bounded by sup 0≤t≤T v(x, t) s . Therefore, in the following, for the sake of simplicity, we will always implicitly use this procedure to deal with the boundary conditions of f (v) when we apply the propositions in Section 4 directly to f (v), instead of g(v), by assuming that f (v) satisfies the boundary conditions for 1/2 < s ≤ 1. Now, applying Propositions 4.1 and 4.2 to (5.3) yields where the assumptions (H1) and (H2) on f (u) have been used. Here, we have used the facts that the norms in H s and H s α,β are equivalent if a function is in H s α,β , and f (v) is assumed to be in H s α,β . From now on, we will not emphasize this detail again.
Choose b = 2c w 0 s wherec = max{B s , c}, and T > 0 such that Then by (5.5) and the definition of Hence, F maps S T,b to S T,b . Next, we show that F is a contraction on S T,b . For To prove the second part of the theorem, it is straightforward to see that for any T ≤ T , there exists a neighborhood U of w 0 in H s α,β and the map K is well-defined from U to X T . If w 1 , w 2 ∈ U , define u 1 = Kw 1 , u 2 = Kw 2 and u = u 1 − u 2 . Then, Proposition 4.1 and the contraction property of F yield Thus K is Lipschitz continuous from U to X T .
Next, we consider the regularity of solutions of (5.1), i.e., for any given n > 1, the solution u(·, t) ∈ H n α,β if its initial state w 0 ∈ H n α,β . Note that Proposition 4.1 gives the regularity of solutions for the linear case. For the nonlinear case, the regularity is more complicated since u is defined in a special space H n α,β which is a Hilbert space inherited from Sobolev space H n with boundary conditions (1.4).
However, note that ∂ t u and ∂ 2 x u are in the same space H n and ∂ t u satisfies the boundary conditions in (5.1). Hence, we should be able to prove the regularity of the solution after proving the regularity of ∂ t u.
Proof. Consider the initial boundary value problem which has a solution From (5.8) and (5.9), we have Then by Propositions 4.1 and 4.2, The proof of Lemma 5.3 is completed.
Now we discuss the regularity of solutions for (5.1) with the initial value If we can show that for any w 0 ∈ X there is a unique solution v ∈ Y T , then (ii) If 0 < T < T is given, then there is a neighborhood U of w 0 in X and the map K : w 0 → v from U to Y T is Lipschitz continuous.
Proof. Define where T > 0 and b > 0 will be determined later. First, we want to prove the map It is deduced that for any v ∈ S T,b ,

JING CUI AND SHU-MING SUN
To find ∂ ∂t S(t)w 0 , we know that u = S(t)w 0 is the solution of Let z = ∂ t u, which implies that z = iu xx and z(0, x) = iu xx (0, x). Thus, z is the solution of Thus, where B = max(c, c ). If we let B ( w 0 s + w 0 p1 s + w 0 p2 s + w 0 s+2 ) = b 2 and choose T > 0 small such that 2BT Similar to the proof of Theorem 5.2, it is obtained that Hence, Thus, F is a contraction on S T,b , which implies that the map F has a fixed point u ∈ Y T and (5.1) has a unique solution. Since u t ∈ C(0, T ; H s α,β ) and u t = iu xx + if (u), we have u ∈ C(0, T ; H s+2 ). In addition, u satisfies the boundary conditions, which implies that u ∈ C(0, T ; H 2 α,β ). Therefore, u ∈ C(0, T ; H s+2 ∩ H 2 α,β ). To prove K is Lipschitz continuous, let u 1 = Kw 1 , u 2 = Kw 2 and u = u 1 − u 2 . Then, 6. Global existence and exponential decay of small amplitude solutions. Section 5 provides the proof of existence and uniqueness of the solution of (5.1) in a finite time interval [0, T ) with T depending upon the size of initial value w 0 . In this section, we prove that for small initial data, T can be infinite, called global well-posedness, and give the behavior of the solution as t → ∞.
Theorem 6.1. Let w 0 ∈ H s α,β with 1 2 < s ≤ 1 be given. If the solution does not exist for all t > 0, then there is a finite T * > 0 such that (5.1) has a unique solution w ∈ C [0, T * ); H s α,β with lim t→T * w(x, t) s = ∞. Here, T * is called the lifespan of the solution.
Proof. For any given w 0 ∈ H s α,β , Theorem 5.2 implies that there exists a T = T ( w 0 s ) > 0 such that (5.1) has a unique solution u on (0, T ). For bounded u s , a standard extension argument can be used to extend the time interval of existence of the solution. Thus, if the solution does not exist for all time t > 0 with u s finite at each t, then u s must blow up at some T * > 0.   Choosing b > 0 and δ > 0 such that c(b p1 + b p2 ) ≤ 1 2 b and cδ ≤ 1 2 b, we obtain that sup 0≤t<∞ F v s ≤ b if w 0 s ≤ δ. Thus, F is a mapping defined on S b,∞ .
Thus, the contraction property of the map F is obtained, which gives a desired solution. The proof is completed. Theorem 6.3. If s ∈ 1 2 , 1 , then there is a η > 0 such that for any given w 0 ∈ H s α,β with w 0 s < η, the unique solution of (5.1) satisfies u(x, t) L 2 ≤ ce −ρt w 0 L 2 , t ≥ 0 (6.1) where c > 0 and ρ > 0 are constants and independent of w 0 .
Proof. First, we define an operator Y : where the series is strongly convergent and ψ k is the normalized eigenvector of A * given in Section 3. Let w ∈ L 2 and w = j c j φ j . Then, where Proposition 3.3 has been used. Hence, Y is bounded and positive on L 2 . Define X : L 2 → L 2 by where −γ ≥ Re λ k ≥ −c 0 with c 0 , γ > 0. Thus, by ζ k = −(2 Re λ k ) −1 and the definition of A in (2.1) and (2.2), it is obtained that