Global Strong Solution to a Nonlinear Dispersive Wave Equation

under the Creative Commons Attribution License, which permits unrestricted use, distribution , and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, compared with the previous results, a new global existence for strong solutions to the equation is acquired provided that the potential (1−∂ 2 x)u 0 changes sign on R, which improves considerably the previous result.

For a = 3, b = 1 and λ > 0 in Eq.( 1), Eq.( 1) is read as weakly dissipative Degasperis-Procesi equation [7] In [8], Lai and Wu study global existence and blow-up to Eq.( 1) with λ = 0 assumed that the potential (1 − ∂ 2 x )u 0 does not change sign on R. To our best knowledge, the global strong solution of Eq.( 1) under the condition y 0 ≤ 0 for x ≤ x 0 and y 0 ≥ 0 for x ≥ x 0 seems not have been investigated.Present paper is mainly concerned with global strong solution to Eq.(1) provided that the potential (1 − ∂ 2 x )u 0 changes sign on R. Since Eq.( 1) is a generalization of Camassa-Holm equation and Degasperis-Procesi equation, Eq.(1) loses some important conservation laws that they possess.In the paper, we mainly depend on some useful prior estimates from the equation and the good method presented in Liu and Yin [6] to obtain the blow-up solution for the equation.

Preliminary Notes
We denote by * the convolution.Note that if G(x Using this identity, the Cauchy problem of Eq.( 1) becomes which is equivalent to We firstly establish the local well-posedness of solution and blow-up scenario for problem (6).
Lemma 2.1.[9] Given u 0 ∈ H s (s > 3 2 ), there exist a maximal T = T (u 0 ) and a unique solution u to problem (6), such that Lemma 2.2.[9] Let u 0 ∈ H s , s ≥ 2, and u be the corresponding solution to problem ( 6) with time T .Then T < ∞ if and only if

Global strong solution
In this section, we discuss the global existence of solution to problem (6).
Lemma 3.1.Let u 0 ∈ H s (R)(s > 3), and T be the maximal existence time of the corresponding solution u to Eq.(1).Then system (18) has a unique solution q ∈ C 1 ([0, T ) × R; R).Moreover, the map q(t, •) is an increasing diffeomorphism of R with Proof.From Theorem 2.1, we have u ∈ C 1 ([0, T ); H s−1 (R)) and H s−1 ∈ C 1 (R).We conclude that both functions u(t, x) and u x (t, x) are bounded, Lipschitz in space and C 1 in time.Applying the existence and uniqueness theorem of ordinary differential equations implies that system (18) has a unique solution q ∈ C 1 ([0, T ) × R, R).
Differentiating (18) with respect to x leads to which yields For every T < T , using the Sobolev embedding theorem gives rise to sup It is inferred that there exists a constant K 0 > 0 such that q x ≥ e −K 0 t for (t, x) ∈ [0, T ) × R.
By computing directly, we derive which results in The proof of Lemma 3.1 is completed.
Proof.Since q(t, x) is an increasing diffeomorphism of R with q x (t, x) > 0 with respect to time t.We deduce from the assumption that for t ∈ [0, T ), and y(t, q(t, x 0 )) = 0. Integrating the first equation of problem (6) with respect to x in interval (−∞, q(t, x 0 )] yields On the other hand, integrating the first equation of problem (6) with respect to x in interval [q(t, x 0 ), +∞) leads to Subtracting ( 12) from ( 13), one has Hence, we get

This proves (i).
Using u = G * y, one has from which we obtain from ( 14), ( 15) and (11), it follows that and From ( 16) and (17), we get −u x ≤| u | on R. Multiplying Eq.( 1) by u and integrating by parts, we have Applying the Gronwall's inequality, one has This completes the proof of Lemma 3.2.
for some point x 0 ∈ R. Then problem (6) has a global strong solution satisfying Remark.If x 0 goes to ±∞, when λ = 0 in Eq.( 1), we recover the Theorem 3 obtained by Lai and Wu [8], where the assumption y 0 ≥ 0 on R or y 0 ≤ 0 on R is needed.Therefore, Theorem 3 in [8] is a special case of Theorem 3.3.
Before proving the theorem, we firstly consider the differential equation where u solves Eq.( 1) and T > 0.
Proof.Here, we assume s = 3 to prove the theorem.Let T be the maximal time of existence of the solution u to problem (6) with initial data u 0 ∈ H 3 .