Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line

Studied here is the large-time behavior of solutions of the Korteweg-de Vries equation posed on the right half-line under the effect of a localized damping. Assuming as in \cite{linares-pazoto} that the damping is active on a set $(a_0,+\infty)$ with $a_0>0$, we establish the exponential decay of the solutions in the weighted spaces $L^2((x+1)^mdx)$ for $m\in \N ^*$ and $L^2(e^{2bx}dx)$ for $b>0$ by a Lyapunov approach. The decay of the spatial derivatives of the solution is also derived.


Introduction
The Korteweg-de Vries (KdV) equation was first derived as a model for the propagation of small amplitude long water waves along a channel [9,16,17]. It has been intensively studied from various aspects for both mathematics and physics since the 1960s when solitons were discovered through solving the KdV equation, and the inverse scattering method, a so-called nonlinear Fourier transform, was invented to seek solitons [14,22]. It is now well known that the KdV equation is not only a good model for water waves but also a very useful approximation model in nonlinear studies whenever one wishes to include and balance weak nonlinear and dispersive effects.
The initial boundary value problems (IBVP) arise naturally in modeling small-amplitude long waves in a channel with a wavemaker mounted at one end [1,2,3,29]. Such mathematical formulations have received considerable attention in the past, and a satisfactory theory of global wellposedness is available for initial and boundary conditions satisfying physically relevant smoothness and consistency assumptions (see e.g. [1,4,6,7,11,12,13] and the references therein).
The analysis of the long-time behavior of IBVP on the quarter-plane for KdV has also received considerable attention over recent years, and a review of some of the results related to the issues we address here can be found in [5,7,19]. For stabilization and controllability issues on the half line, we refer the reader to [20] and [27,28], respectively.
In this work, we are concerned with the asymptotic behavior of the solutions of the IBVP for the KdV equation posed on the positive half line under the presence of a localized damping represented by the function a; that is, (1)      u t + u x + u xxx + uu x + a(x)u = 0, x, t ∈ R + , u(0, t) = 0, t > 0, u(x, 0) = u 0 (x), x > 0.
Assuming a(x) ≥ 0 a.e. and that u(., t) ∈ H 3 (R + ), it follows from a simple computation that is the total energy associated with (1). Then, we see that the term a(x)u plays the role of a feedback damping mechanism and, consequently, it is natural to wonder whether the solutions of (1) tend to zero as t → ∞ and under what rate they decay. When a(x) > a 0 > 0 almost everywhere in R + , it is very simple to prove that E(t) converges to zero as t tends to infinity. The problem of stabilization when the damping is effective only in a subset of the domain is much more subtle. The following result was obtained in [20]. Theorem 1.1 Assume that the function a = a(x) satisfies the following property (4) a ∈ L ∞ (R + ), a ≥ 0 a.e. in R + and a(x) ≥ a 0 > 0 a.e. in (x 0 , +∞) for some numbers a 0 , x 0 > 0. Then for all R > 0 there exist two numbers C > 0 and ν > 0 such that for all u 0 ∈ L 2 (R + ) with ||u 0 || L 2 (R + ) ≤ R, the solution u of (1) satisfies Actually, Theorem 1.1 was proved in [20] under the additional hypothesis that (6) a(x) ≥ a 0 a.e. in (0, δ) for some δ > 0, but (6) may be dropped by replacing the unique continuation property [20,Lemma 2.4] by [30,Theorem 1.6]. The exponential decay of E(t) is obtained following the methods in [23,25,26] which combine multiplier techniques and compactness arguments to reduce the problem to some unique continuation property for weak solutions of KdV. Along this work we assume that the real-valued function a = a(x) satisfies the condition (4) for some given positive numbers a 0 , x 0 . In this paper we investigate the stability properties of (1) in the weighted spaces introduced by Kato in [15]. More precisely, for b > 0 and m ∈ N, we prove that the solution u exponentially decays to 0 in L 2 b and L 2 (x+1) m dx (if u(0) belongs to one of these spaces), where The following weighted Sobolev spaces endowed with their usual inner products, will be used thereafter. Note that The exponential decay in L 2 (x+1) m dx is obtained by constructing a convenient Lyapunov function (which actually decreases strictly on the sequence of times {kT } k≥0 ) by induction on m. For u 0 ∈ L 2 (x+1) m dx , we also prove the following estimate is arbitrarily large; (ii) m ≥ 2 and ||u 0 || L 2 (x+1) m dx is small enough. In the situation (ii), we first establish a similar estimate for the linearized system and next apply the contraction mapping principle in a space of functions fulfilling the exponential decay. Note that (7) combines the (global) Kato smoothing effect to the exponential decay. The exponential decay in L 2 b is established for any initial data u 0 ∈ L 2 b under the additional assumption that 4b 3 + b < a 0 . Next, we can derive estimates of the form for any s ≥ 1, revealing that u(t) decays exponentially to 0 in strong norms. It would be interesting to see if such results are still true when the function a has a smaller support. It seems reasonable to conjecture that similar positive results can be derived when the support of a contains a set of the form ∪ k≥1 [ka 0 , ka 0 + b 0 ] where 0 < b 0 < a 0 , while a negative result probably holds when the support of a is a finite interval, as the L 2 norm of a soliton-like initial data may not be sufficiently dissipated over time. Such issues will be discussed elsewhere.
The plan of this paper is as follows. Section 2 is devoted to global well-posedness results in the weighted spaces L 2 b and L 2 (x+1) 2 dx . In section 3, we prove the exponential decay in L 2 (x+1) m dx and L 2 b , and establish the exponential decay of the derivatives as well. Fix any b > 0. To begin with, we apply the classical semigroup theory to the linearized system Let us consider the operator Then, the following result holds.
Lemma 2.1 The operator A defined above generates a continuous semigroup of operators (S(t)) t≥0 in L 2 b .
Proof. We first introduce the new variable v = e bx u and consider the following (IBVP) Clearly, the operator B : is densely defined and closed. So, we are done if we prove that for some real number λ the operator B − λ and its adjoint B * − λ are both dissipative in L 2 (R + ). It is readily seen that Pick any v ∈ D(B). After some integration by parts, we obtain that Analogously, we deduce that for any v ∈ D(B * ) which completes the proof.
The following linear estimates will be needed.
Lemma 2.2 Let u 0 ∈ L 2 b and u = S(·)u 0 . Then, for any T > 0 As a consequence, where C = C(T ) is a positive constant.
Proof. Pick any u 0 ∈ D(A). Multiplying the equation in (1) by u and integrating over (0, +∞) × (0, T ), we obtain (10). Then, the identity may be extended to any initial state u 0 ∈ L 2 b by a density argument. To derive (11) we first multiply the equation by (e 2bx − 1)u and integrate by parts over (0, +∞) × (0, T ) to deduce that Adding the above equality and (10) hand to hand, we obtain (11) using the same density argument. Then, Gronwall inequality, (4) and (11) imply that with C = C(T ) > 0. This estimate together with (11) gives us The global well-posedness result reads as follows: Proof. By computations similar to those performed in the proof of Lemma 2.2, we obtain that for any f ∈ C 1 ([0, T ]; L 2 b ) and any u 0 ∈ D(A), the solution u of the system   b be given. To prove the existence of a solution of (1) we introduce the map Γ defined by where N (u) = −uu x , and the space endowed with its natural norm. We shall prove that Γ has a fixed-point in some ball B R (0) of F . We need the following From Cauchy-Schwarz inequality, we get for any x ∈ R + which guarantees that Claim 1 holds. Claim 2. There exists a constant K > 0 such that for 0 < T ≤ 1 According to the previous analysis, . So, applying triangular inequality and Hölder inequality, we have Now, by Claim 1, we have (15) ||u|| . Then, combining (14) and (15), we deduce that Let T > 0, R > 0 be numbers whose values will be specified later, and let u ∈ B R (0) ⊂ F be given. Then, by Claim 2 and Lemma 2.2, Γu ∈ F and and T > 0 small enough, Γ maps B R (0) into itself. Moreover, we infer from (16) that this mapping contracts if T is small enough. Then, by the contraction mapping theorem, there exists a unique solution u ∈ B R (0) ⊂ F to the problem (1) for T small enough.
In order to prove that this solution is global, we need some a priori estimates. So, we proceed as in the proof of Lemma 2.2 to obtain for the solution u of (1) First, observe that Combined to Claim 1, this yields On the other hand, it follows from (17) that for all T > 0, which gives the global well-posedness. (1), and such that for some b > 0 and some sequence {u n,0 } ⊂ L 2 b we have

Global well-posedness in
u n denoting the solution of (1) emanating from u n,0 at t = 0.
Theorem 2.5 For any u 0 ∈ L 2 (x+1) 2 dx and any T > 0, there exists a unique mild solution u ∈ Proof. We prove the existence and the uniqueness in two steps.
Step 1. Existence Since the embedding L 2 b ⊂ L 2 (x+1) 2 dx is dense, for any given u 0 ∈ L 2 (x+1) 2 dx we may construct a sequence {u n,0 } ⊂ L 2 b such that u n,0 → u 0 in L 2 (x+1) 2 dx as n → ∞. For each n, let u n denote the solution of (1) emanating from u n,0 at t = 0, which is given by Theorem 2.3. Then Multiplying (19) by (x + 1) 2 u n and integrating by parts, we obtain Scaling in (19) by u n gives which, combined to (22), gives An application of Gronwall's lemma yields ).
Therefore, there exists a subsequence of {u n }, still denoted by {u n }, such that , hence by Aubin's lemma, we have (after extracting a subsequence if needed) This gives that u n u n,x → uu x in the sense of distributions, hence the limit u ∈ L ∞ (0, and Young inequality we get where ε > 0 is arbitrarily chosen and c ε denotes some positive constant. . On the other hand, u(0, t) = 0 for t ∈ (0, T ) and u x (0, .) ∈ L 2 (0, T ). Scaling in (1) by (x + 1) 2 u yields for all t 1 , t 2 ∈ (0, T ) Step 2. Uniqueness Here, C will denote a universal constant which may vary from line to line.
and also v n,0 → u 0 strongly in L 2 (x+1) 2 dx , We shall prove that w = u − v vanishes on R + × [0, T ] by providing some estimate for w n = u n − v n . Note first that w n solves the system Scaling in (34) by (x + 1)w n yields Since ||w n (t)|| L 2 (R + ) ≤ ||w n (t)|| L 2 (x+1)dx , this yields for T < 1/10 It remains to estimate We have that By Sobolev embedding, we have that On the other hand, we have that Gathering together (37), (38) and (39), we conclude that for T < 1/10 and C denotes a universal constant. The following claim is needed.
Clearly, it is sufficient to prove the claim for the sequence {u n } only. From (27) applied with u = u n on [0, T ], we obtain Combined to (23)- (24), this gives It follows from Gronwall lemma that Using (43) in (42) and taking the limit sup as n → ∞ gives for a.e. T ||u(T )|| 2 As u is continuous from R + to L 2 (x+1) 2 dx , we infer that The claim is proved. Therefore, we have that for T > 0 small enough and n large enough, K n (T ) < where d m−1 > 0 is chosen sufficiently large (see below). Suppose first that m = 1 and put V = V 1 . Multiplying the first equation in (1) by u and integrating by parts over R + × (0, T ), we obtain Now, multiplying the equation by xu, we deduce that Combining (46) and (47) it follows that The next step is devoted to estimate the nonlinear term in the left hand side of (48). To do that, we first assume that ||u 0 || L 2 ≤ 1.
Induction Hypothesis: There exist c > 0 and The next steps are devoted to estimate the terms in the above identity. First, combining (4) and (50) we infer the existence of a positive constant c > 0 such that where we used ( * ) m−1 . In the same way where c > 0 is a positive constant. Moreover, assuming V m−1 (u 0 ) ≤ ρ with ρ > 0 small enough (so that by exponential decay of V m−1 (u(t)) we have ∞ 0 (x + 1) m−1 |u(x, t)| 2 dx ≤ 1 for all t ≥ 0) and proceeding as in the case m = 1, we obtain the existence of ε > 0 and c ε > 0 satisfying Then, if we return to (58) and take ε < 9/2 and d m−1 > 0 large enough, from (59)-(61) if follows that This yields ( * ) m , by ( * ) m−1 . Let us now check ( * * ) m . It remains to estimate the terms in the right hand side of (63). We multiply the first equation in (1) by (T − t)(x + 1) m u to obtain Then, proceeding as above, we deduce that Combined to ( * * ) m−1 , this yields ( * * ) m . This completes the construction of the sequence {V m } m≥1 by induction.
Let us now check the exponential decay of V m for m ≥ 2. It follows from ( * ) m − ( * * ) m that where c > 0, which completes the proof when V m−1 (u 0 ) ≤ ρ. The global result (V m−1 (u 0 ) ≤ R) is obtained as above for m = 1.
Proof. We first prove estimates for the linearized problem and next apply a perturbation argument to extend them to the nonlinear problem (1). Let us denote by W (t)u 0 = u(t) the solution of (72)-(74). By computations similar to those performed in the proof of Theorem 3.1, we have that We need the Claim 6. Let k ∈ {0, ..., 3}. Then there exists a constant C k > 0 such that for any Using (72), this gives This proves (75) for k = 3. The fact that (75) is valid for k = 1, 2 follows from a standard interpolation argument, for Lemma 3.4 Pick any number µ ∈ (0, ν). Then there exists some constant C = C(µ) > 0 such that for any u 0 ∈ L 2 Proof. Let u 0 ∈ L 2 (x+1) m dx and set u(t) = W (t)u 0 for all t ≥ 0. By scaling in (72) by (x + 1) m u, we see that for some constant C K = C K (T ) This implies that u(t) ∈ H 1 (x+1) m dx for a.e. t ∈ (0, 1) which, combined to (75), gives that u(t) ∈ H 1 (x+1) m dx for all t > 0. Pick any T ∈ (0, 1]. Note that, by (75), Integrating with respect to t in (77) yields and hence ∀t ∈ (0, 1).
Let us return to the proof of Corollary 3.3. Fix a number µ ∈ (0, ν), where ν is as in (75), and let us introduce the space < ∞} endowed with its natural norm. Note that (1) may be recast in the following integral form where N (u) = −uu x . We first show that (79) has a solution in F provided that ≤ r 0 and ||u|| F ≤ R, r 0 and R being chosen later. We introduce the map Γ defined by We shall prove that Γ has a fixed point in the closed ball B R (0) ⊂ F provided that r 0 > 0 is small enough.
Corollary 3.5 Assume that a(x) satisfies (4) and that ∂ k x a ∈ L ∞ (R + ) for all k ≥ 0. Pick any u 0 ∈ L 2 (x+1) m dx . Then for all ε > 0, all T > ε, and all k ∈ {1, ..., m}, there exists a constant C = C(ε, T, k) > 0 such that Proof. The proof is very similar to the one in [18, Lemma 5.1] and so we only point out the small changes. First, it should be noticed that the presence in the KdV equation of the extra terms u x and a(x)u does not cause any serious trouble. On the other hand, choosing a cut-off function in x of the form η(x) = ψ 0 (x/ε) (instead of η(x) = ψ 0 (x − x 0 + 2) as in [18]) where ψ 0 ∈ C ∞ (R, [0, 1]) satisfies ψ 0 (x) = 0 for x ≤ 1/2 and ψ 0 (x) = 1 for x ≥ 1, allows to overcome the fact that u is a solution of (1) on the half-line only.

Decay in
This section is devoted to the exponential decay in L 2 b . Our result reads as follows: Theorem 3.6 Assume that the function a = a(x) satisfies (4) with 4b 3 + b < a 0 . Then, for all R > 0, there exist C > 0 and ν > 0, such that for any solution u given by Theorem 2.3.