Local and global analyticity for $\mu$-Camassa-Holm equations

We solve Cauchy problems for some $\mu$-Camassa-Holm integro-partial differential equations in the analytic category. The equations to be considered are $\mu$CH of Khesin-Lenells-Misio\l{}ek, $\mu$DP of Lenells-Misio\l{}ek-Ti\u{g}lay, the higher-order $\mu$CH of Wang-Li-Qiao and the non-quasilinear version of Qu-Fu-Liu. We prove the unique local solvability of the Cauchy problems and provide an estimate of the lifespan of the solutions. Moreover, we show the existence of a unique global-in-time analytic solution for $\mu$CH, $\mu$DP and the higher-order $\mu$CH. The present work is the first result of such a global nature for these equations. AMS subject classification: 35R09, 35A01, 35A10, 35G25


Introduction
We consider a functional equation µ(u t ) − u txx = −2µ(u)u x + 2u x u xx + uu xxx , x ∈ S 1 = R/Z, called the µ-Camassa-Holm equation (µCH), and its variants in the complex-analytic or real-analytic category. Here µ(u) = S 1 u dx. Multiplying by the inverse of µ−∂ 2 x , we get an evolution equation It motivates one to consider Cauchy problems, not only in Sobolev spaces but also in spaces of analytic functions. In the latter case, the solutions are analytic in both t and x. Recall that solutions to the KdV equation can be analytic in x but not in t. This is because it is 'not Kowalevskian', which means the first-order derivative u t equals a quantity involving higher derivatives. Our evolution equation mentioned above is 'Kowalevskian' in a generalized sense due to the presence of the negative order pseudodifferential operator (µ − ∂ 2 x ) −1 . Because of the nonlocal nature of (µ − ∂ 2 x ) −1 , our considerations are always global in x. So we will work with the Sobolev space H m (S 1 ) or A(δ), the space of analytic functions on S 1 ∋ x which admits analytic continuation to |y| < δ. On the other hand, we can work either locally or globally in t. Our local study will be given in Section 2 and Appendix. It is based on the Ovsyannikov theorem used in [1] and [8]. It is a kind of abstract Cauchy-Kowalevsky theorem about a scale of Banach spaces and enables us to obtain local-in-time solutions which are analytic in both t and x. Our global study will be given in Sections 3 and 4. Since global-in-time solutions are known to exist in Sobolev spaces, what remains to be done here is to prove their analyticity. We carry out this task by using the method of [9] following [3]. In the final part of the proof, we quote a result in [12], which gives a useful criterion of real analyticity. Now we explain some background and history. In the course of it, we will introduce some equations that will be studied in the present paper. All the equations mentioned below are integrable in some sense.
The original Camassa-Holm equation was introduced in [4] (shallow water wave) and in [7] (hereditary symmetries). It is known to be completely integrable and admits peaked soliton (peakon) solutions. The Cauchy problem for this equation can be formulated by introducing a pseudodifferential operator. Indeed, (0.1) can be written in the form Since (0.2) is Kowalevskian in a generalized sense, it is natural to solve this equation in the analytic setting as in the classical Cauchy-Kowalevsky theorem. In [1], the authors introduced a kind of Sobolev spaces with exponential weights consisting of holomorphic functions in a strip of the type |y| < const. Since these spaces form a scale of Banach spaces, an Ovsyannikov type argument can be applicable. It leads to the unique solvability and an estimate of the lifespan of the solution in the periodic and non-periodic cases.
There are a lot of works about solutions of the Cauchy problem for (0.1) or (0.2) in Sobolev spaces. See the references in [1] and [20]. Local well-posedness and blowup mechanism are major topics. In [1], the local unique solvability in the analytic category was proved. Moreover, there is a result about the global-in-time solvability in [3]. Indeed, according to [3], if the initial value is in H s (R), s > 5/2, and the McKean quantity m 0 = (1 − ∂ 2 x )u 0 does not change sign, then the Cauchy problem for (a generalization of) (0.2) has a unique global-in-time solution u ∈ C([0, ∞); H s (R)). See [15] for the necessity and sufficiency of the no-change-of-sign condition. Moreover, in [3], it is proved that this solution is analytic in both t and x if the initial value is in the space of analytic functions mentioned above. In the present paper, we follow [1] for local theory and the analyticity part of [3] for global theory.
In [11], the µ-version of (0.1), namely (0. 3) µ(u t ) − u txx = −2µ(u)u x + 2u x u xx + uu xxx , x ∈ S 1 = R/Z, was introduced. The authors call this equation µHS (HS is for Hunter-Saxton), while it is called µCH in [14]. We have µ(u t ) = 0, but we keep µ(u t ) because this formulation facilitates later calculation. The interest of (0.3) lies, for example, in the fact that it describes evolution of rotators in liquid crystals with external magnetic field and self-interaction, and it is related to the diffeomorphism groups of the circle with a natural metric. Set (0.4) u t + uu x + ∂ x A −1 2µ(u)u + 1 2 u 2 x = 0.
In [11], the local well-posedness and the global existence in Sobolev spaces is demonstrated. In the global problem, the µ-McKean quantity (µ − ∂ 2 x )u 0 (x) is assumed to be free from change of sign. There are similar µ-equations. In [14, (5.3)], the following equation, called µDP (DP is for Degasperis-Procesi), was introduced: The local well-posedness in H s (S 1 ), s > 3/2, and the global existence in H s (S 1 ), s > 3, was proved in [14]. In [5], a family of higher-order Camassa-Holm equations depending on k = 2, 3, . . . was introduced. It is related to diffeomorphisms of the unit circle. In [18], the µ-version of the k = 2 case x + ∂ 4 x )ϕ, was formulated. The local well-posedness and the global existence in H s (S 1 ), s > 7/2, was proved in [18]. Notice that the no-change-of-sign condition is not imposed in this work. In [19], a different version is studied. The global existence is proved there. The modified µ-Camassa-Holm equation (modified µCH) with non-quasilinear terms 2)] (γ = 0) and [16, (2.7)], [17]. The global existence in Sobolev spaces remains open, as opposed to that of the other equations mentioned above, so it is not possible to show the global existence of analytic solutions by using the method in the present article. Because of this exceptional nature of the equation, we treat it in Appendix separately from the others. Notice that local theory for (0.8) is developed in a certain space of analytic functions as well as in Besov spaces in [17]. The outline of this article is as follows. In Section 1, we introduce some function spaces and operators and investigate their properties. In Section 2, we prove the local existence of analytic solutions of (0.4), (0.5), (0.6) and (0.7). These results are used in the proofs of the global existence theorems in Sections 3 and 4 about (0.4), (0.5) and (0.6). In Appendix, the local existence of analytic solutions of (0.8) is proved.

Function spaces and operators
In the present paper, L 2 (S 1 ) consists of real-valued square-integrable functions on S 1 = R/Z. We sometimes identify an element of it with a function on R with period 1. For a function on S 1 , we setφ(k) = S 1 ϕ(x)e −2kπix dx. We introduce a family of Hilbert spaces G δ,s (δ ≥ 0, s ≥ 0) by where k = (1+k 2 ) 1/2 (Japanese bracket). Notice that our definition is not exactly the same as that in [1,2]. In particular, the base space is T = R/(2πZ) in [1,2]. It is easy to see that we have continuous injections G δ,s → G δ ′ ,s ′ and G δ,s → G δ,s ′ if 0 ≤ δ ′ < δ, 0 ≤ s ′ < s. Their norms are 1. We have a continuous injection G δ,s → G δ ′ ,s ′ under the weaker assumption 0 ≤ δ ′ < δ. We recover the usual Sobolev spaces H s = G 0,s , H ∞ = ∩ s≥0 H s and we set ϕ s = ϕ 0,s . Notice that · 0 is the L 2 norm. The corresponding inner product is denoted by ·, · 0 . Set Λ 2 = 1 − (2π) −2 ∂ 2 x . Then we have When δ > 0, set f has an analytic continuation to S(δ) .
Remark 1. If a function is analytic on S 1 , then it belong to A(δ) for some δ > 0. See Proposition 4 below. Notice that an analogous statement does not hold true if S 1 is replaced with R.
We identify an element of A(δ) with its analytic continuation. For f ∈ A(δ), set This norm will be used in the global theory. Do not confuse · 2 (σ,s) with · δ,s .
Remark 3. Proposition 2 is a variant of the Sobolev embedding theorem: if s > 1/2, then there is a continuous embedding H s = G 0,s → C 0 (S 1 ) as is proved by d Proposition 4. If ϕ is a real-analytic function on S 1 , then there exists δ > 0 such that ϕ ∈ G δ,s for any s. More precisely, if ϕ ∈ A(δ), then ϕ ∈ G δ ′ ,s for any δ ′ ∈]0, δ[ and any s.
Proposition 5. We have the following three estimates about products of functions: (i) Assume s > 1/2, δ ≥ 0. Then G δ,s is closed under pointwise multiplication and we have (ii) There exists a positive constant d s such that we have ϕψ 0 ≤ d s ϕ 0 ψ s for any ϕ ∈ H 0 = L 2 (S 1 ) and any ψ ∈ H s (s ≥ 1).
Proof. The proof of (i) is given in [2]. Although different formulations are used in [2] and the present article, the constant c s is the same. This is because ϕψ(k) = nφ (n)ψ(k − n) holds in either situations. The difference of the base spaces are offset by that of conventions, namely the presence or the absence of the 1/(2π) factor in the definition of the Fourier coefficients. The other three estimates follow from the boundedness of H 1 ֒→ C 0 (S 1 ) (and the Leibniz rule).
Remark 6. A better estimate than Proposition 5 (iii) can be found in [10], which implies that · 1 can be replaced with · L ∞ . It is used in [18], but (iii) is good enough in the present paper.
A derivative can be estimated in two ways: 'larger δ' or 'larger s'.
Proof. The second inequality is easy to prove. We give a proof of the first in order to clarify that the assumption δ ≤ 1 in [1,2] is superfluous. The present author thinks that the authors of [1,2] wrote δ ≤ 1 not because they really needed it for the omitted proof but for the sole reason that they were interested only in 0 < δ ≤ 1. Since It follows that A is a bounded operator from G δ,s+2 to G δ,s . It is a bijection and its inverse A −1 is a pseudodifferential operator of order −2. Therefore it is bounded from G δ,s to G δ,s+2 . On the other hand, B −1 is a pseudodifferential operator of order −4. Notice that A −1 and B −1 commute with ∂ x . The following proposition is easy to prove.
Proof. In the proof of the first part, we may assume s = 0. Let Then f (j) 0 ≤ j!e −2πjσ M and Since there exists C > 0 such that ϕ C 0 ≤ C( ϕ 0 + ϕ ′ 0 ) for any ϕ ∈ H 1 by the Sobolev embedding, we have for any x. Therefore f (x + iy) = ∞ j=0 f (j) (x)(iy) j /j! is holomorphic in |y| < e 2πσ for any σ with e 2πσ < δ. Hence f ∈ A(δ).
Next, we show the second part. Assume 0 < δ ′ < δ. It is enough to prove f 2 (σ,s) < ∞ for any σ, s with e 2πσ < δ ′ . Let S = sup S(δ ′ ) |f | < ∞. Set the periodicity of f and Goursat's formula yield For s > 0, we can prove that where e 2πσ < e 2πσ ′ < δ ′ . To see this, we may assume that s is a positive integer in view of f (σ,s1) ≤ f (σ,s2) (s 1 ≤ s 2 ). For simplicity, we explain the case of s = 1 only. (The general case follows the same line of proof, the only additional tool being the binomial expansion of powers of the Japanese bracket.) We have . On the other hand, setting ℓ = j + 1, we get The proof of the second part is over.
Corollary 11. The following four families of norms on A(δ) determine the same topology as a Fréchet space.
Proof. It is trivial that (ii) is stronger (not weaker) than (iv) and that (iii) is between (ii) and (iv). On the other hand, (1.8) implies (iv) is stronger than (ii).
The estimate (1.6) and the Taylor expansion imply that (iv) is stronger than (i). On the other hand, (1.7) implies (i) is stronger than (iv).

Local-in-time solutions
2.1. Autonomous Ovsyannikov theorem. We recall some basic facts about the autonomous Ovsyannikov theorem. Among many versions, we adopt the one in [1,2].
is a mapping satisfying the following conditions.
(a) For any u 0 ∈ X 1 and R > 0, there exist The autonomous Ovsyannikov theorem below is our main tool. For the proof, see [1].
Theorem 13. Assume that the mapping F satisfies the conditions (a) and (b). For any u 0 ∈ X 1 and R > 0, set Then, for any δ ∈]0, 1[, the Cauchy problem 2.2. µCH and µDP equations. First we consider the analytic Cauchy problem for the µCH equation (0.4), namely.
Theorem 14. Let s > 1/2. If u 0 ∈ G 1,s+1 , then there exists a positive time T = T (u 0 , s) such that for every δ ∈]0, 1[, the Cauchy problem (2.5) has a unique solution which is a holomorphic function valued in G δ,s+1 in the disk D(0, T (1−δ)). Furthermore, the analytic lifespan T satisfies By Proposition 5 (i), the first inequality in Proposition 7 and (1.5), On the other hand, since we have . Next, by Proposition 5 (i) and the second inequality in Proposition 7, Then (2.6), (2.7) and (2.8) give the Lipschitz continuity of F µ : Because of Theorem 13, there exists a unique solution u = u(t) to (2.5) which is a holomorphic mapping from D(0, T (1 − δ)) to G δ,s+1 and If we set R = u 0 1,s+1 , we have In Theorem 14, we assumed the initial value u 0 was in G 1,s+1 . We can relax this assumption as in the following theorem.
Theorem 15. If u 0 is a real-analytic function on S 1 , then the Cauchy problem (2.5) has a holomorphic solution near t = 0. More precisely, we have the following: such that for every d ∈]0, 1[, the Cauchy problem (2.5) has a unique solution which is a holomorphic function valued in G ∆d,s+1 in the disk D(0, T ∆ (1 − d)). Furthermore, the analytic lifespan T ∆ satisfies Proof. The first statement is nothing but Proposition 4.
Because of Theorem 13, there exists a unique solution u = u(t) to (2.5) which is a holomorphic mapping from We can study the following Cauchy problem for the µDP equation (0.5) by using the same estimates (2.6) and (2.7). (2.12) Theorem 16. If u 0 is a real-analytic function on S 1 , then the Cauchy problem (2.12) has a holomorphic solution near t = 0. More precisely, we have the following: such that for every d ∈]0, 1[, the Cauchy problem (2.12) has a unique solution which is a holomorphic function valued in G ∆d,s+1 in the disk D(0, T ∆ (1 − d)). Furthermore, the analytic lifespan T ∆ satisfies when ∆ is fixed.
Theorem 18. Let s > 1/2. If u 0 ∈ G 1,s+3 , then there exists a positive time T = T (u 0 , s) such that for every δ ∈ (0, 1), the Cauchy problem has a unique solution which is a holomorphic function valued in G δ,s+3 in the disk D(0, T (1 − δ)). Furthermore, the analytic lifespan T satisfies Proof. The proof is almost the same as for Theorem 17. Indeed, A −2 has the same properties as those of B −1 .
Global-in-time analytic solutions of (2.5), (2.12), (2.13) and (0.7) will be studied in the following sections. The proofs will rely on known results about the global existence in Sobolev spaces. On the other hand, that kind of existence for the non-quasilinear equation (0.8) is unknown. Therefore, the argument given below is not valid for it. The local theory of (0.8) will be given in Appendix.
We will discuss global-in-time analytic solutions. These solutions are analytic in both the time and space variables. Notice that in the KdV and other cases treated in [9], solutions are analytic in the space variable only. This is due to the absence of Cauchy-Kowalevsky type theorems. Our main result about the µ-Camassa-Holm equation is the following: Theorem 21. Assume that a real-analytic function u 0 on S 1 has non-zero mean and satisfies the condition x )u 0 ≥ 0 (or ≤ 0). Then the Cauchy problem (2.5) has a unique solution u ∈ C ω (R t × S 1 x ). We have the following estimate of the radius of analyticity. Let u 0 ∈ A(r 0 ). Fix σ 0 < (log r 0 )/(2π) and set Then, for any fixed T > 0, we have u(·, t) ∈ A(e 2πσ(t) ) for t ∈ [−T, T ].
There is an analogue about the µDP equation which reads as follows: Theorem 22. Assume that a real-analytic function u 0 on S 1 has non-zero mean and satisfies the condition . Then the Cauchy problem (2.12) has a unique solution u ∈ C ω (R t × S 1 x ). We have the following estimate of the radius of analyticity. Let u 0 ∈ A(r 0 ). Fix σ 0 < (log r 0 )/(2π) and set Then, for any fixed T > 0, we have u(·, t) ∈ A(e 2πσ ′ (t) ) for t ∈ [−T, T ].
The proof of Theorem 21 will be given in the later subsections. First we will prove the analyticity in x and establish the lower bound of σ(t). Next we will establish the analyticity in (t, x). The proof of Theorem 22 is essentially contained in that of Theorem 21. We can employ Remark 26 instead of Proposition 25.

3.2.
Regularity theorem by Kato-Masuda. In [9], the authors used their theory of Liapnov families to prove a regularity result about the KdV and other equations. Later, it was applied to a generalized Camassa-Holm equation in [3]. Here we recall the abstract theorem in [9] in a weaker, more concrete form. It is good enough for our purpose. The inequality (3.1) is a special case of that in [9]. Notice (c) There exist positive constants K and L such that holds for any v ∈ O. Here ·, · (no subscript) is the pairing of X and L(X; R).
Let u ∈ C([0, T ]; O) ∩ C 1 ([0, T ]; X) be the solution to the Cauchy problem Moreover, for a fixed constant s 0 , set Roughly speaking, this theorem means that the regularity of u(t) for t ∈ [0, T ] follows from that of u 0 . Later we will use this when X = H m+2 , Z = H m+5 and Φ s is related to some variant of the Sobolev norms. We can extend Theorem 23 to t ≤ 0.

Pairing and estimates.
Recall the norm · (σ,s) in Section 1. When s = 2, It is approximated by the finite sum Later {Φ σ,m } will play the role of {Φ s } in Corollary 24. Assume w ∈ H m+2 , v ∈ H m+5 . Let DΨ j be the Fréchet derivative of Ψ j and w, DΨ j (v) be the pairing of where ·, · 2 is the H 2 inner product. Recall It is easy to see that F µ is continuous from H m+5 to H m+2 .
Remark 26. The µDP equation can be studied by using the following estimate. Set Then we have Because of (1.1), we have Therefore the Schwarz inequality and Proposition 5 (ii) imply Therefore we get, by the Schwarz inequality, Proposition 5 (ii) and Proposition 7, The estimate of I 2 is a bit complicated. We divide it into a sum of three terms as in We have Hence (3.16) |I 23 | ≤ 16π 5 d 1 v 2 w 2 2 . The three inequalities (3.14), (3.15) and (3.16) give Combining (3.11), (3.12), (3.13) and (3.17), we obtain Next we calculate m j=1 j! −2 e 4πσj Q j . Recall Combining this estimate with the Schwarz inequality, we get |Q j | ≤ 2πγ(Q j,1 + Q j,2 ), Now we set b k = k! −1 e 2πσk v (k) 2 (k = 0, 1, . . . , j). Then we have Recalling Q 0 = 0, we obtain The inequalities (3.19) and (3.20) give, together with (3.8), We consider the second term of the right-hand side of (3.7). By (1.1), we have are bounded operators whose norms are equal to 1, we have We consider the third term of the righthand side of (3.7). First, assume j = 0. Since the norm of ∂ x A −1 : This is similar to Q j in (3.10). Since the norm of ∂ 2 We can follow (3.20) for j ≥ 1 and employ (3.23) for j = 0 (Recall Q 0 = 0). Finally we obtain It completes the proof of Proposition 25.
3.4. Analyticity in the space variable. In this subsection, we prove a part of Theorem 21. We assume that u 0 ∈ A(r 0 ) is as in Theorem 21. Then we can apply Theorem 19 with arbitrarily large s. We will prove the analyticity of u(t) in x for each fixed t.

Analyticity in the space and time variables.
We continue the proof of Theorem 21. In the previous subsection, we have established the analyticity in the space variable. Here, we will prove the analyticity in the space and time variables. By convention, a real-analytic function on a closed interval is real-analytic on some open neighborhood of the closed interval.
We have shown that u is analytic in t at least locally. Our next step is to show that u is analytic in t globally. Set We prove T * = ∞ by contradiction. Assume T * < ∞. By Proposition 27, u(T * ) is well-defined and there exists δ * > 0 such that By the local uniqueness, we haveû = u. Namely,û is an extension of u up to |t| ≤ T * + ε (valued in G δ ′ /2,s+1 ⊂ A(δ ′ /2)). Therefore T * + ε ∈ S. This is a contradiction.
Proposition 30. Under the situation of Theorem 21, the Cauchy problem (2.5) has a unique solution u ∈ C ω (R t × S 1 x ).
Proof. The uniqueness in H ∞ implies the uniqueness in the real-analytic category.
In the last step of the proof of Proposition 30, we have used the following theorem.
Theorem 31. ( [12]) Let Ω be a domain in R n and let P = P (∂/∂x 1 , . . . , ∂/∂x n ) be an elliptic partial differential operator of order m with constant coefficients. Then, for a function f ∈ L 2 loc (Ω) to be analytic in Ω, it is (necessary and) sufficient that 1) for every ℓ ∈ Z + , P ℓ f (in the sense of distributions) belongs to L 2 loc (Ω), and that 2) for every compact subset K ⊂ Ω, there exist positive constants M and A such that A better known result in this direction is [13], in which P = P (x, ∂/∂x 1 , . . . , ∂/∂x n ), x = (x 1, , . . . , x n ), is an elliptic operator of order m with analytic coefficients and the right-hand side of (3.29) is replaced with M ℓ+1 (mℓ)!. We can employ the result of [13] instead of Theorem 31.

Global-in-time solutions: higher-order case
In this section we consider (2.13). Global-in-time solutions in Sobolev spaces have been studied in [18]. Notice that the non-zero mean and the no-change-of-sign conditions are not imposed. Remark 33. In [18], the authors solve the Cauchy problem for t > 0 only. Since the equation is invariant under (t, u) → (−t, −u), the result for t < 0 follows immediately.
There is an analogous result about (2.24). Our main result about the higher order µCH is the following theorem.
Theorem 35. If u 0 is a real-analytic function on S 1 , then the Cauchy problem (2.13) has a unique solution u ∈ C ω (R t × S 1 x ). We have the following estimate of the radius of analyticity. Assume u 0 ∈ A(r 0 ). Fix σ 0 < (log r 0 )/(2π) and set Then, for any fixed T > 0, we have u(·, t) ∈ A(σ(t)) for t ∈ [−T, T ].
Proof. Theorem 32 implies u(t) ∈ H ∞ if u 0 ∈ H ∞ . Set Then the proof is almost the same as that of Theorem 21 and follows from Proposition 37 below. It is an analogue of Proposition 25. Notice that v 2 in (3.5) has been replaced with v 4 .
There is an analogous result about (2.24).
Theorem 36. If u 0 is a real-analytic function on S 1 , then the Cauchy problem (2.24) has a unique solution u ∈ C ω (R t × S 1 x ). The estimate of the radius of analyticity is the same as in Theorem 35.
The proof of Theorem 36 is almost the same as that of Theorem 35. One has only to replace B −1 with A −2 . The rest of this section is devoted to the proof of Theorem 35. It is enough to prove Proposition 37 below.
Proposition 37. We have where γ 1 and γ 2 are given in Theorem 35.
Proof. Recall (2.19), namely We have We compare it with (3.7). We encountered v (j) , ∂ j x (vv x ) 2 in (3.7). The follow- Therefore we can employ (3.21) and analogues of (3.22) and (3.25). Here we replace v 2 with ( v 2 ≤) v 4 . Our remaining task is to estimate The results will be given as (4.8) and (4.13) in the following subsection.
This is better than (3.10). Indeed, ∂ 4 x B −1 : H 2 → H 2 is as good as ∂ 2 x A −1 : H 2 → H 2 and the binomial coefficients have become smaller. We follow (3.20) with j ≥ 3 instead of j ≥ 0 and get Next we consider the case j = 0. We have and, by Proposition 5 (ii), These two inequalities yield We have used v 3 ≤ v 4 , because v 4 will inevitably appear later. We deal with it by modifying the definition of O, so that v 3 and v 4 are bounded there. Proposition 25 should be modified accordingly.
Next assume j = 1. We have and, by Proposition 5 (ii) again, By (4.4), (4.5), (4.6) and (4.7), we obtain This is better than (3.10). We follow (3.20) with j ≥ 3 instead of j ≥ 0 and get Next we consider the case j = 0. We have and, by Proposition 5 (ii), These two inequalities yield (4.10) | xx ] 2 and, by Proposition 5 (ii), By using (4.9), (4.10), (4.11) and (4.12), we obtain Appendix: Local study of the non-quasilinear modified µCH equation The difficulty of (0.8) lies in the presence of the non-quasilinear terms − 1 3 u 3 x and − 1 3 µ(u 3 x ). In [2], the authors employed the power series method to deal with the non-quasilinear term au k−2 u 3 x of the k-abc-equation. In the present paper, we overcome the difficulty of non-quasilinearity by a µ-version of a classical trick used in the proof of the Cauchy-Kowalevsky theorem ([6]) following [8] and [17]. We set v = u x and differentiate (0.8) in x. It can be proved that (0.8) is equivalent to the following quasilinear modified µCH system: Of course, this trick works for the the k-abc-equation as well. We can prove unique solvability of the Cauchy problem for (4.14) and (0.8).
The Cauchy problem (0.8) for the non-quasilinear modified µCH equation can be written in the following form: We introduce the system below.
Theorem 38. The Cauchy problems (4.15) and (4.16) are equivalent to each other Proof. By differentiation with respect to x, we get the second equation in (4.16) from (4.15).
To show the converse, differentiate both sides of the first equation of (4.16) in x. By comparing it with the second equation, we get It is enough to prove that w t + ∂ x A −1 [a(t)w] = 0 and w(0, x) = 0 imply w = 0, where a(t) is a continuous function in t. Set w = k∈Z w k (t)e 2kπix . Then we have Since w k (0) = 0, we have w k (t) = 0 (k ∈ Z) for any t. It implies w = 0 and v = u x .
Remark 39. The trick of setting v = u x has been used in [8] (non-µ equation) and [17] in a different function space rather formally, i.e. without discussion corresponding to Theorem 38. In [8], this trick is applied to a quasilinear equation.
Theorem 40. If u 0 and v 0 are real-analytic functions on S 1 , then the Cauchy problem (4.16) has a holomorphic solution near t = 0. More precisely, we have the following: (i) There exists ∆ > 0 such that u 0 , v 0 ∈ G ∆,s+1 ⊂ A(∆) for any s.
We have obtained two inequalities of the Lipschitz type. Next we estimate F j (u 0 , v 0 ) (j = 1, 2). We have Since X 2 Y ≤ (X + Y ) 3 /3, X 3 ≤ (X + Y ) 3 for X, Y ≥ 0, we get Similarly we have We set R = (u 0 , v 0 ) 1,s+1 . If γ = 0, the constants corresponding to L and M in (2.1) and (2.2) are of degrees 2 and 3 respectively. Therefore T equals a constant multiple of (u 0 , v 0 ) −2 1,s+1 if γ = 0. If γ = 0, we have to consider two cases separately: large or small initial values. When the initial values are large, larger order terms are dominant and T is approximated by a constant multiple of (u 0 , v 0 ) −2 1,s+1 , while T approaches a constant as the initial values approach 0. Theorem 38 allows us to get a result about the original Cauchy problem (4.15). We assume u 0 ∈ G ∆,s+2 so that v 0 = ∂ x u 0 belongs to G ∆,s+1 .
Theorem 41. If u 0 is a real-analytic function on S 1 , then the Cauchy problem (4.15) has a holomorphic solution near t = 0. More precisely, we have the following: (i) There exists ∆ > 0 such that u 0 ∈ G ∆,s+2 for any s. (ii) If s > 1/2, there exists a positive time T ∆ = T (u 0 , s, ∆) such that for every d ∈ (0, 1), the Cauchy problem (4.15) has a unique solution which is a holomorphic function valued in G ∆d,s+2 in the disk D(0, T ∆ (1 − d)). Furthermore, if γ = 0, the analytic lifespan T ∆ satisfies , when ∆ is fixed. On the other hand, if γ = 0, we have the asymptotic behavior (large initial values), T ∆ ≈ const. (small initial values).
Remark 42. In [17], the author solved (4.15) in a space of analytic functions following [8]. What is new in the present paper is precise estimates of the lifespan.