On the Study of Global Solutions for a Nonlinear Equation

and Applied Analysis 3 The Sobolev imbedding theorem derives H(R) ∈ C(R). This means that two functions V(t, x) and V x (t, x) are bounded, Lipschitz in space and C in time. Using the existence and uniqueness theorem of ordinary differential equations, we derive that problem (12) has a unique solutionp(t, x) ∈ C ([0, T) × R). Differentiating (12) with respect to x gives rise to d dt p x = (m + 1) VmVx (t, p) px, t ∈ [0, T) , p x (0, x) = 1, (14) from which we obtain p x (t, x) = exp(∫ t 0 (m + 1) VmVx (τ, p (τ, x)) dτ) . (15) For every T < T, applying the Sobolev imbedding theorem results in sup (τ,x)∈[0,T 󸀠 )×R 󵄨󵄨󵄨Vx (τ, x) 󵄨󵄨󵄨 < ∞. (16) Therefore, we know that there exists a constant M > 0 such that p x (t, x) ≥ e for (t, x) ∈ [0, T) × R. The proof is completed. Lemma 7. Let V 0 ∈ H with s ≥ 3, and let T > 0 be the maximal existence time of the problem (2); it holds that

A lot of works have been carried out to study various dynamic properties for the Camassa-Holm and the Novikov equations.Xin and Zhang [4] proved that there exists a global weak solution for the Camassa-Holm equation in the space  1 () without the assumption of sign conditions on the initial value.Coclite et al. [5] investigated the global weak solutions for a generalized hyperelastic rod wave equation or a generalized Camassa-Holm equation.It is shown in Constantin and Escher [6] that the blowup occurs in the form of breaking waves; namely, the solution remains bounded but its slope becomes unbounded in finite time.After wave breaking, the solution can be continued uniquely either as a global conservative weak solution [7] or a global dissipative solution [8][9][10].The periodic and the nonperiodic Cauchy problems for the Novikov equation were discussed by Grayshan [11] in the Sobolev space.Using the Galerkintype approximation method, Himonas and Holliman [12] established the well-posedness for the Novikov model in the Sobolev space   () with  > 3/2 on both the line and the circle.The scattering theory was employed in Hone et al. [13] to find nonsmooth explicit soliton solutions with multiple peaks for the Novikov equation.Wu and Zhong [14] proved the existence of local strong and weak solutions for a generalized Novikov equation.
Now we give the main result of global weak solution for problem (2).
Using the first equation of system (2) derives which yields the conservation law Lemma 5 (see [1]).Let  > 3/2 and the function V(, ) is a solution of problem (2) and the initial data V 0 () ∈   .Then the following inequalities hold: For  ∈ (0,  − 1], there is a constant  such that For  ∈ [0,  − 1], there is a constant  such that Consider the differential equation where V(, ) is the solution of problem (2) and  is the maximal existence time of the solution.
Proof.From Lemma 4, we know that there exists a unique solution Abstract and Applied Analysis 3 The Sobolev imbedding theorem derives   () ∈  1 ().This means that two functions V(, ) and V  (, ) are bounded, Lipschitz in space and  1 in time.Using the existence and uniqueness theorem of ordinary differential equations, we derive that problem (12) has a unique solution (, ) ∈  1 ([0, ) × ).
Differentiating (12) with respect to  gives rise to from which we obtain For every   < , applying the Sobolev imbedding theorem results in sup Therefore, we know that there exists a constant  > 0 such that   (, ) ≥  − for (, ) ∈ [0, ) × .The proof is completed.
Proof.We have from which we have Using   (0, ) = 1 completes the proof.
Proof.We only need to prove this lemma for the case V 0 − V 0 ≥ 0 since the proof of the other case (1 −  2  )V 0 ≤ 0 is similar.It follows from Lemmas 6 and 7 that V − V  ≥ 0.
which derives On the other hand, we have The inequalities ( 22) and ( 23) derive that inequality (20) is valid.
The proof of this lemma can be found in [15,16].
From Lemma 4, it derives that the Cauchy problem has a unique solution V depending on the parameter .We write V  (, ) to represent the solution of problem (25).Using Lemma 4 derives that Lemma 10.Provided that V 0 ∈   (), 1 ≤  ≤ 3/2, and , then there exists a constant  > 0 independent of  and  such that the solution of problem (25) satisfies Proof.Using Lemmas 5 and 9, if V 0 ∈   () with 1 ≤  ≤ 3/2, we have where  is independent of  and .
From Lemma 8, we have which completes the proof.

Proof of Main Results
Proof of Theorem 1.Since ‖V‖  ∞ () ≤ ‖V‖  1 () ≤   and taking  + 1 =  in inequality (10), we have from which we obtain Applying Lemma 8 yields from which we complete the proof of Theorem 1.