The Cauchy problem for a generalized Novikov equation

We establish the local well-posedness for a generalized Novikov equation in nonhomogeneous Besov spaces. Besides, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time.

1. Introduction. In this paper we consider the Cauchy problem for the following generalized Novikov equation: which can be rewritten as (1. 2) The equation (1.1) was proposed by Novikov in [41]. It was shown that (1.1) posses an infinite hierarchy of quasi-local higher symmetries in [41]. It is of the form 3) where F is a homogeneous polynomial. The most celebrated integrable member of (1.3) is the Camassa-Holm (CH) equation: The CH equation can be regarded as a shallow water wave equation [5,17]. It is completely integrable [4,7,15,18]. It also has a bi-Hamiltonian structure [27], and Compared with the above two equation, it has cubic nonlinearity. It also possesses a bi-Hamiltonian structure and admits exact peakon solutions u(t, x) = √ ce |x−ct| with c > 0 in [30]. The local well-posedness was studied in [46,47,49,50]. The global existence of strong solutions was established in [46] under some sign conditions and the blow-up phenomena of the strong solutions were shown in [50]. The global weak solutions were studied in [45].
The aim of this paper is to establish the local well-posedness for the Cauchy problem of (1.2) in Besov spaces, and to get blow-up results for strong solutions to (1.2). Our paper is organized as follows. In Section 2, we introduce some preliminaries which will be used in sequel. In Section 3, we prove the local well-posedness of (1.2) in B s p,r with s > max( 1 2 , 1 p ) or (s = 1 p , 1 ≤ p ≤ 2, r = 1) in the sense of Hadamard (i.e. (1.2) has a unique local solution in B s p,r with continuity with respect to the initial data). The main approach is based on the Littlewood-Paley theory and transport equations theory. In Section 4, we obtain a blow-up criteria and give a sufficient condition for strong solutions to blow up in finite time.
2. Preliminaries. In this section, we first recall the Littlewood-Paley decomposition and Besov spaces.
(1) B s p,r is a Banach space, and is continuously embedded in S .
Fatou property: if (u n ) n∈N is a bounded sequence in B s p,r , then an element u ∈ B s p,r and a subsequence (u n k ) k∈N exist such that lim k→∞ u n k = u in S and u B s p,r ≤ C lim inf k→∞ u n k B s p,r . (6) Let m ∈ R and f be a S m -mutiplier (i.e. f is a smooth function and satisfies that ∀α ∈ N d , We have the following continuity properties for the product of two functions: is an algebra, and a constant C = C(s, d) exists such that Here is the Osgood lemma, a generalization of the Gronwall lemma.
Lemma 2.6. [1] Let ρ be a measurable function from [t 0 , T ] to [0, a], γ a locally integrable function from [t 0 , T ] to R + , and µ a continuous and nondecreasing function from [0, a] to R + . Assume that for some c ≥ 0, the function ρ satisfies If c = 0, and µ satisfies We shall present here some results for the following transport equation.
3. Local well-posedness. In this section, we establish local well-posedness of (1.2) in Besov spaces. First we introduce the following function spaces.
Our main result is stated as follows.
Theorem 3.2. Let 1 ≤ p, r ≤ ∞, s ∈ R and let (s, p, r) satisfy the condition Then there exists a time T > 0 such that (1.2) has a unique solution u in E s p,r (T ). Moreover the solution depends continuously on the initial data.
In order to prove Theorem 3.2, we proceed as the following steps.
Step 1. Starting from m 0 0, we define by induction a sequence (m n ) n∈N of smooth functions by solving the following linear transport equations: Note that under the assumptions on (s, p, r), B s p,r and B s+1 p,r are algebras. We have , some estimates we shall use are slightly different, so we have to discuss separately.
By the fact B s p,r is an algebra and Lemma 2.5 (2), we have Obviously, (S n m 0 ) n∈N is a Cauchy sequence in B s−1 p,r . By Fatou's lemma, we obtain The Gronwall inequality entails that g(t) = 0 for all t ∈ [0, T ]. Therefore (m n ) n∈N is a Cauchy sequence in C([0, T ]; B s−1 p,r ) and converges to some limit function m ∈ C([0, T ]; B s−1 p,r ).
Hence Lemma 2. Step 5. Next we prove the uniqueness of solutions to (1.2). The proof is based on the way we have in Step 3. Suppose that (m 1 , m 2 ) are two solutions of (1.2). We obtain where for i = 1, 2, Case 1. s > max( 1 2 , 1 p ). By virtue to Lemma 2.8, we have By a similar calculation as in Step 3, we get (3.14) Plugging (3.13), (3.14) into (3.12) yields that Using Gronwall's inequality, we finally get Similarly, we deduce that (3.17) (3.18) Plugging (3.17), (3.18) into (3.16), and using the boundedness of m i with respect to t, we have dt .

Applying Proposition 2.3 (2), it follows that
By virtue of Remark 2.7, we finally get (

3.19)
Therefore the uniqueness is a straightforward conclusion of the above inequalities (3.15) and (3.19). Moreover, an interpolation argument ensures that continuity with respect to the initial data holds for the norm C([0, T ]; B s p,r ) whenever s < s.
Step 6. Finally, we end up with a proposition about continuity until the exponent s, in the proof of which Lemma 2.11 is necessary. .
So (m n ) n∈N is bounded in L ∞ ([0, T ]; B s p,r ). We splitting m n = y n +z n with (y n , z n ) satisfying As before, we deduce that (3.21), and using the boundedness of m n , we obtain Note that y ∞ = m ∞ , z ∞ = 0. Hence we have Using Gronwall's inequality yields that Therefore we have proven that when r < ∞, z n → 0 in C([0, T ]; B s p,r ), and thus m n → m ∞ in C([0, T ]; B s p,r ). As for the case r = ∞, we have weak continuity. In fact, for fixed φ ∈ B −s p ,1 , we write By duality, we have Proof. Arguing by density, it suffices to consider the case where u ∈ C ∞ 0 (R). The equation (1.2) can be rewrite as Multiplying (4.1) by u and integrating by parts, we deduce that Next we state a blow-up criterion for (1.2).

Lemma 4.2.
Let m 0 ∈ B s p,r with (s, p, r) being as the statement of Theorem 3.2, and T * be the maximal existence time of the corresponding solution m to (1.2). If T * < ∞, then Proof. Applying Lemma 2.9, we have Note that the operator (1 − ∂ 2 x ) −1 coincides with the convolution by the function x → 1 2 e −|x| , which implies that u L ∞ , u x L ∞ and u xx L ∞ can be bounded by m L ∞ . Then As s > 0, by Lemma 2.5, we have and Plugging (4.3), (4.4) and (4.5) into (4.2), we get Hence Gronwall's inequality yields If T * is finite, and T * 0 m 2 L ∞ dt < ∞, then m ∈ L ∞ ([0, T * ); B s p,r ). By the proof of Step 1 in Theorem 3.2, we can extend the solution m beyond T * , which contradicts the assumption that T * is the maximal existence time.
Consider the ordinary differential equation: (4.7) If m ∈ B s p,r with (s, p, r) as in Theorem 3.2, then 2u 2 x − uu x − u 2 ∈ C([0, T ); C 0,1 ). By the classical results in the theory of ordinary differential equation, we infer that (4.7) has a unique solution q ∈ C 1 ([0, T ) × R; R) such that the map q(t, ·) is an increasing diffeomorphism of R with Now the following theorem shows that under particular condition for the initial data, the corresponding solution of (1.2) will blow up in finite time. Assume m 0 (x) ≥ 0 for all x ∈ R and m 0 (x 0 ) > 0, (u 0 − 4∂ x u 0 )(x 0 ) < 0 for some x 0 ∈ R, and that Then the corresponding solution m of (1.2) blows up in finite time.