On weak solutions to a generalized Camassa–Holm equation with solitary wave

A generalized Camassa–Holm equation proposed by Novikov is considered. The existence and uniqueness of a positive weak solution for the equation is established by using a classical method.


Introduction
Recently, Novikov [22] proposed the following integrable quasi-linear scalar evolution equation of order 2: where = 0 is a real constant.
Letting v(t, x) = u( t, x), one can transform Eq. (1) into the following form: It was shown in [22] that Eq. (2) possesses a hierarchy of local higher symmetries. Equation (2) is regarded as a generalized Camassa-Holm equation [22]. In [17], Li and Yin establish the local existence and uniqueness of strong solutions for Eq. (2) in nonhomogeneous Besov spaces by using the Littlewood-Paley theory. Under some assumptions, a blowup criterion and a global existence result for the equation are also presented in [17]. The well-posedness of (2) is studied in [11] for the periodic and the nonperiodic cases in the sense of Hadamard. In addition, nonuniform dependence is proved by using the method of approximate solutions and well-posedness estimates. To the best of our knowledge, up to now the weak solutions for Eq. (2) have not been investigated yet. The equation closest to the relevant problem (2) is the Degasperis-Procesi equation, Degasperis, Holm and Hone [12] proved the formal integrability of Eq. (2) by constructing a Lax pair. They showed that it has a bi-Hamiltonian structure and there is an infinite sequence of conserved quantities. Since the Degasperis-Procesi equation was born, much attention has been attracted by the study its dynamics. Yin proved local well-posedness of Eq. (2) on the line [24] and on the circle [25]. In addition, the precise blow-up scenario and blow-up structure for the equation were derived in [24,25]. Lenells [16] classified all weak traveling wave solutions. Matsuno [20] obtained multisolutions of Eq. (2). Escher et al. [13] investigated the blow-up phenomena and global weak solutions for Degasperis-Procesi equation. In a different direction, Coclite and Karlsen [3][4][5], and Lundmark [19] initiated a study of discontinuous solutions (shock wave) to the Degasperis-Procesi equation (2). It is shown in [2] that a new blow-up quantity along the characteristics is established for the Degasperis-Procesi equation (2). The other equations related to Eq. (2), such as the Camassa-Holm equation, the Novikov equation and the Modified Camassa-Holm equation with cubic nonlinearity, can be found in [1, 6-10, 14, 15, 18, 22, 23] and the references therein. Inspired by the ideas from [13,26], in this paper, we investigate the weak solutions for the following Cauchy problem: More precisely, we focus on the existence and uniqueness of positive weak solutions to the problem (4) using the method from [11] under the condition y 0 = u 0u 0xx ∈ M + . One of the difficult issues in our proof is how to prove that there is a subsequence of {u n } which converges pointwise a.e. to a function u ∈ H 1 loc (R + × R) that satisfies (4) in the sense of distributions, and how to show that u ∈ C w (R + ; H 1 (R)), the space of continuous functions from R + with values in H 1 (R) when the latter space is equipped with its weak topology. Luckily, using y 0 = u 0u 0xx > 0 and the estimate u(t, ·) L ∞ (R) ≤ 3 2 u 0 2 L 3 (R) t + u 0 L ∞ , we successfully overcome the problems.

Notations
The space of all infinitely differentiable functions φ(t, x) with compact support in [0, +∞)×R is denoted by C ∞ 0 . Let 1 ≤ p < +∞ and L p = L p (R) be the space of all measurable functions h(t, x) such that h P We denote by * the convolution. Let · X denote the norm of Banach space X and ·, · denote the H 1 (R), H -1 (R) duality bracket. Let M(R) be the space of Radon measures on R with bounded total variation and M + (R) be the subset of positive measures. Finally, we write BV(R) for the space of functions with bounded variation, V (f ) being the total variation of f ∈ BV(R).

Preliminaries
Throughout this paper, let {ρ n } n≥1 denote the mollifiers Thus, we get Using this identity, we rewrite problem (4) in the form which is equivalent to Next, we give some useful results.

Global weak solution
Proof The proof of (i) may be found in [17]. Now, we prove (ii).
Multiplying the first equation of problem (6) by u and integrating by parts, we find which yields where the Gronwall inequality and (i) were used. This proves (ii) and completes the proof of the lemma.
Proof We split the proof of Theorem 3.1 in two parts.
Let u 0 ∈ H 1 (R) and y 0 = u 0u 0,xx ∈ M + (R). Note that u 0 = G * y 0 . Thus, for ϕ ∈ L ∞ (R), we have Let us define u n 0 := ρ n * u 0 ∈ H ∞ (R) for n ≥ 1. Obviously, we get Note that, for all n ≥ 1, Referring to the proof of (9), we have From Lemma 3.1, we know that there exists a global strong solution, From Lemma 3.1 and (10), we obtain From the Hölder inequality, Lemma 3.1 and (10), for all t ≥ 0 and n ≥ 1, we have Using the Young inequality, we get ∂ x G * u n 2 + u n 2 x L 2 (R) where ∂ x G L 2 (R) is bounded. Applying (13)- (14) and problem (6), we have For fixed T > 0, from (12) and (15), we deduce where M is a positive constant depending only on G x L 2 (R) , u 0 H 1 (R) , u 0 L 3 (R) , y 0 M(R) and T. It follows that the sequence {u n } n≥1 is uniformly bounded in the space H 1 ((0, T) × R). Thus, we can extract a subsequence such that and u n k → u, a.e. on (0,T) × R for n k → ∞, for some u ∈ H 1 ((0, T) × R). From Lemma 3.1 and (10), for fixed t ∈ (0, T), we see that the sequence u n k Applying Helly's theorem [21], we infer that there exists a subsequence, denoted again From (18), we get, for almost all t ∈ (0, T), u n k x (t, ·) → u x (t, ·) in D (R). It follows that ν(t, ·) = u x (t, ·) for a.e. t ∈ (0, T). Therefore, we have u n k x (t, ·) → u x (t, ·) a.e. on (0,T) × R for n k → ∞, and for a.e. t ∈ (0, T), By Lemma 3.1 and (12), we have Note that, for fixed t ∈ (0, T), the sequence {(u n ) 2 + ((u n ) 2 ) x } n≥1 is uniformly bounded in L 2 (R). Therefore, it has a subsequence {(u n k ) 2 + ((u n k ) 2 ) x } n k ≥1 which converges weakly in L 2 (R). From (18), we infer that the weak L 2 (R)-limit is From (18), (19) and (21), we see that u solves Eq. (6) in D ((0, T) × R). For fixed T > 0, noticing that u n k t is uniformly bounded in L 2 (R) as t ∈ [0, T) and u n k (t) H 1 (R) is uniformly bounded for all t ∈ [0, T) and n ≥ 1, we infer that the family t → u n k ∈ H 1 (R) is weakly equicontinuous on [0,T]. An application of the Arzela-Ascoli theorem shows that {u n k } has a subsequence, denoted again {u n k }, which converges weakly in H 1 (R), uniformly in t ∈ [0, T). The limit function is u. T being arbitrary, we see that u is locally and weakly continuous from [0, ∞) into H 1 (R), i.e., u ∈ C w,loc (R + ; H 1 (R)).
Note that u(t, x) = G * (u(t, x)u xx (t, x)). Then we get Combining with (23), it implies that u(t, x) ∈ W 1,∞ (R + × R). This completes the proof of the existence of Theorem 3.1. Next, we present the uniqueness proof of Theorem 3.1. Let u, v ∈ W 1,∞ (R + × R) ∩ L ∞ loc (R + ; H 1 (R)) be two global weak solutions of problem (6) with the same initial data u 0 . Assume that (u(t, ·)-u xx (t, ·)) ∈ M + (R) and (v(t, ·)-v xx (t, ·)) ∈ M + (R) are uniformly bounded on R + and set From the assumption, we know that N < ∞. Then, for all (t, x) ∈ R + × R, and Similarly Following the same procedure as in (9), we may also get and for all (t, We define Convoluting Eq. (6) for u and v with ρ n , we get, for a.e. t ∈ R + and all n ≥ 1, and Subtracting (31) from (30) and using Lemma 2.4, integration by parts shows that for a.e. t ∈ R + and all n ≥ 1 Using (24)- (26) and the Young inequality to the first term on the right-hand of (32) yields R ρ n * (wu x ) sgn(ρ n * w) dx Similarly, we obtain For the last term on the right-hand side of (32), we have From (33)-(37), for a.e. t ∈ R + and all n ≥ 1, we find where where K is a positive constant depending on N and the H 1 (R)-norms of u(0) and v(0). In the same way, convoluting Eq. (6) for u and v with ρ n,x and using Lemma 2.4, we see that for a.e. t ∈ R + and all n ≥ 1 Using the identity ∂ 2 x (G * g) = G * gg for g ∈ L 2 (R) and the Young inequality, we estimate the fourth term of the right-hand side of (40): Using (24)- (26) and the Young inequality to the first term on the right-hand of (40) gives rise to To treat the second term of the right-hand side of (40), we note that Applying Lemma 2.1, the second expression of right-hand of (43) can be estimated by a function R n (t) belonging to (39). Making use of the Hölder inequality and (9), for a.e. t ∈ R + and all n ≥ 1, we have It follows from (43) and (44) that 2 R (ρ n * v xx w) sgn(ρ n,x * w) dx ≤ 2N R |ρ n * w| dx + 2N R |ρ n * w x | + R n (t).