The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

and Applied Analysis 3 use the Kato theorem 29 to establish the local well-posedness for 1.5 with initial value u0 ∈ H with s > 3/2. Then, we present a precise blow-up scenario for 1.5 . Provided that u0 ∈ H R ⋂ L1 R and the potential y0 1−∂x u0 does not change sign, the global existence of the strong solution is shown to be true. Finally, under suitable assumptions, the existence and uniqueness of global weak solution in W1,∞ R × R ⋂ Lloc R ;H 1 R are proved. Our main ideas to prove the existence and uniqueness of the global weak solution come from those presented in Constantin and Molinet 8 and Yin 22 . 2. 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 let L L R be the space of all measurable functions h t, x such that ‖h‖PLP ∫ R |h t, x |dx < ∞. We define L∞ L∞ R with the standard norm ‖h‖L∞ infm e 0supx∈R\e|h t, x |. For any real number s, let H H R denote the Sobolev space with the norm defined by ‖h‖Hs ∫ R 1 |ξ| |ĥ t, ξ |dξ 1/2 < ∞, where ĥ t, ξ ∫ R e −ixξh t, x dx. We denote by ∗ the convolution. Let ‖ · ‖X denote the norm of Banach space X and 〈·, ·〉 the H1 R , H−1 R duality bracket. Let M R be the space of the Radon measures on R with bounded total variation and M R 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 . Note that if G x : 1/2 e−|x|, x ∈ R. Then, 1 − ∂x −1f G ∗ f for all f ∈ L2 R and G ∗ u − uxx u. Using this identity, we rewrite problem 1.5 in the form ut buux ∂xG ∗ [ a 2 u2 3b − a 2 ux 2 ] λu 0, t > 0, x ∈ R, u 0, x u0 x , x ∈ R, 2.1 which is equivalent to yt buyx ayux λy 0, t > 0, x ∈ R, y u − uxx, u 0, x u0 x . 2.2 3. Preliminaries Throughout this paper, let {ρn}n≥1 denote the mollifiers ρn x : (∫ R ρ ξ dξ )−1 nρ nx , x ∈ R, n ≥ 1, 3.1 4 Abstract and Applied Analysis where ρ ∈ C∞ c R is defined by ρ x : ⎧ ⎨ ⎩ e1/ x 2−1 for |x| < 1, 0 for |x| ≥ 1. 3.2


Introduction
The Camassa-Holm equation C-H equation as a model for wave motion on shallow water, has a bi-Hamiltonian structure and is completely integrable.After the equation was derived by Camassa and Holm 1 , a lot of works was devoted to its investigation of dynamical properties.The local well posedness of solution for initial data u 0 ∈ H s R with s > 3/2 was given by several authors 2-4 .Under certain assumptions, 1.1 has not only global strong solutions and blow-up solutions 2, 5-7 but also global weak solutions in H 1 R see 8-10 .For other methods to handle the problems related to the Camassa-Holm equation and functional spaces, the reader is referred to 11-14 and the references therein.
To study the effect of the weakly dissipative term on the C-H equation, Guo 15 and Wu and Yin 16 discussed the weakly dissipative C-H equation

Abstract and Applied Analysis
The global existence of strong solutions and blow-up in finite time were presented in 16 provided that y 0 1 − ∂ 2 x u 0 changes sign.The sufficient conditions on the infinite propagation speed for 1.2 are offered in 15 .It is found that the local well posedness and the blow-up phenomena of 1.2 are similar to the C-H equation in a finite interval of time.However, there are differences between 1.2 and the C-H equation in their long time behaviors.For example, the global strong solutions of 1.2 decay to zero as time tends to infinite under suitable assumptions, which implies that 1.2 has no traveling wave solutions see 16 .
Degasperis and Procesi 17 derived the equation D-P equation The weakly dissipative D-P equation is investigated by several authors 25-27 to find out the effect of the weakly dissipative term on the D-P equation.The global existence, persistence properties, unique continuation and the infinite propagation speed of the strong solutions to 1.4 are studied in 26 .The blowup solution modeling wave breaking and the decay of solution were discussed in 27 .The existence and uniqueness of the global weak solution in space W 1,∞ loc R ×R L ∞ loc R ; H 1 R were proved see 25 .In this paper, we will consider the Cauchy problem for the weakly dissipative shallow water wave equation where a > 0, b > 0, and λ ≥ 0 are arbitrary constants, u is the fluid velocity in the x direction, λ u − u xx represents the weakly dissipative term.For λ 0, we notice that 1.5 is a special case of the shallow water equation derived by Constantin and Lannes 28 .Since 1.5 is a generalization of the Camassa-Holm equation and the Degasperis-Procesi equation, 1.5 loses some important conservation laws that C-H equation and D-P equation possesses.It needs to be pointed out that Lai and Wu 12 studied global existence and blow-up criteria for 1.5 with λ 0. To the best of our knowledge, the dynamical behaviors related to 1.5 with λ 0, such as the global weak solution in space not been yet investigated.The objective of this paper is to investigate several dynamical properties of solutions to 1.5 .More precisely, we firstly use the Kato theorem 29 to establish the local well-posedness for 1.5 with initial value u 0 ∈ H s with s > 3/2.Then, we present a precise blow-up scenario for 1.5 .Provided that u 0 ∈ H s R L 1 R and the potential y 0 1−∂ 2 x u 0 does not change sign, the global existence of the strong solution is shown to be true.Finally, under suitable assumptions, the existence and uniqueness of global weak solution in Our main ideas to prove the existence and uniqueness of the global weak solution come from those presented in Constantin and Molinet 8 and Yin 22 .

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 let 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 •, • the H 1 R , H −1 R duality bracket.Let M R be the space of the Radon measures on R with bounded total variation and M R 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 Using this identity, we rewrite problem 1.5 in the form which is equivalent to 2.2

Preliminaries
Throughout this paper, let {ρ n } n≥1 denote the mollifiers

3.2
Thus, we get Next, we give some useful results.
Lemma 3.2 see 8 .Let f : R → R be uniformly continuous and bounded.If g ∈ L ∞ R , then 3.7

Global Strong Solution
We firstly present the existence and uniqueness of the local solutions to the problem 2.1 .
Then, the problem 2.1 has a unique solution u, such that where T > 0 depends on u 0 H s R .
Proof.The proof of Theorem 4.1 can be finished by using Kato's semigroup theory see 29 or 4 .Here, we omit the detailed proof.
Proof.Setting y t, x u t, x − u xx t, x , we get Using system 2.2 , one has

4.5
Assume that there is a constant M > 0 such that From 4.5 , we get Using Gronwall' inequality, we deduce the u H 2 is bounded on 0, T .On the other hand, Therefore, using 4.4 leads to It shows that if u H 2 is bounded, then u x L ∞ is also bounded.This completes the proof.
We consider the differential equation where u solves 1.5 and T > 0.
Lemma 4.3.Let u 0 ∈ H s R s > 3 ; T is the maximal existence time of the corresponding solution u to 1.5 .Then, system 4.10 has a unique solution q ∈ C 1 0, T × R; R .Moreover, the map q t, • is an increasing diffeomorphism of R with

4.11
Proof.From Theorem 4.1, 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 4.10 has a unique solution q ∈ C 1 0, T × R, R .Differentiating 4.10 with respect to x leads to d dt q x bu x t, q q x , t ∈ 0, T , b > 0, 4.12 which yields q x exp t 0 bu x s, q s, x ds .

4.13
For every T < T, using the Sobolev embedding theorem gives rise to 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 y t, q t, x q 2 x t, x y 0 x exp t 0 − a − 2b u x s, q s, x − λ ds .

4.16
The proof of Lemma 4.3 is completed.
Proof.Let u 0 ∈ H s , s > 3/2, and let T > 0 be the maximal existence time of the solution u with initial date u 0 cf.Theorem 4.1 .If y 0 ≥ 0, then Theorem 4.2 ensures that y ≥ 0 for all t ∈ 0, T .Note that u 0 G * y 0 and y 0 u 0 − u 0,xx ∈ L 1 R .By Young's inequality, we get Integrating the first equation of problem 2.1 by parts, we get

4.22
Note that u G * y, y ≥ 0 on 0, T and the positivity of G. Thus, we can infer that u ≥ 0 on 0, T .From 4.22 we have From Theorem 4.2 and 4.23 , we find T ∞.This implies that problem 2.1 has a unique solution Due to y t, x ≥ 0 and u t, x ≥ 0 for all t ≥ 0, it shows that

4.25
From the two identities above, we infer that u x 2 ≤ u 2 on R for all t ≥ 0. This proves i .
Due to y ≥ 0, we obtain From u ≥ 0 and the inequality above, we get On the other hand, from 4.23 , we have that u x t, x ≥ −e −λt u 0 L 1 R .This proves ii .
Multiplying the first equation of problem 2.1 by u and integrating by parts, we find 1 2

4.29
From Gronwall's inequality, one has This proves iii and completes the proof of the theorem.

Global Weak Solution
Theorem 5.1.
Proof.We split the proof of Theorem 5.1 in two parts. Let

5.1
Let us define u n 0 : Obviously, we get

5.2
Note that, for all n ≥ 1, Referring to the proof of 5.1 , we have

5.4
From the Theorem 4.4, we know that there exists a global strong solution

5.6
From Theorem 4.4 and 5.2 , we obtain

5.7
From the H ölder inequality, Theorem 4.4, and 5.2 , for all t ≥ 0 and n ≥ 1, we have

5.8
Using Young's inequality, we get where ∂ x G L 2 R is bounded, and

5.11
For fixed T > 0, from 5.7 and 5.11 , we deduce where M is a positive constant depending only on G x L 2 R , u 0 H 1 R , u 0 L 1 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 for some u ∈ H 1 0, T × R .From Theorem 4.4 and 5.2 , for fixed t ∈ 0, T , we have that the sequence

5.15
Applying Helly's theorem 31 , we infer that there exists a subsequence, denoted again by {u n k x t, • }, which converges at every point to some function ν t, • of finite variation with

5.16
From 5.14 , we get that 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 and, for a.e.t ∈ 0, T ,

5.18
By Theorem 4.4 and 5.7 , we have Abstract and Applied Analysis 13

5.19
Note that for fixed t ∈ 0, T , the sequence which converges weakly in L 2 R .From 5.14 , we infer that the weak

5.20
From 5.14 , 5.17 , and 5.20 , we have that u solves 2.1 in D 0, T × R .For fixed T > 0, note 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, and we infer that the family t → u n k ∈ H 1 R is weakly equicontinous on 0,T .An application of the Arzela-Ascoli theorem yields 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 have that u is locally and weakly continuous from 0, Since, for a.e.t ∈ R , u n k t, • u t, • weakly in H 1 R , from Theorem 4.4, we get

5.21
Inequality 5.21 shows that From Theorem 4.4, 5.1 and 5.2 , for t ∈ R , we obtain

5.23
Combining with 5.14 , we have Next, we will prove that R u t, • dx e −λt R u 0, • dx by using a regularization approach.
Since u satisfies 2.1 in distribution sense, convoluting 2.1 with ρ n , we have that, for a.e.t ∈ R , Integrating the above equation with respect to x on R, we obtain

5.26
Integration by parts gives rise to Finally, we prove that u t, from 5.18 , we get that, for a.e.t ∈ R ,

5.32
Abstract and Applied Analysis 15 The above inequality implies that, for a.e.t ∈ R , u t, • − u xx t, • ∈ M R is uniformly bounded on R. For fixed T ≥ 0, applying 5.13 and 5.14 , we have

5.34
Combining with 5.24 , it implies that u t, x ∈ W

5.35
From assumption, we know that N < ∞.Then, for all t, x ∈ R × R, 5.36

5.38
Following the same procedure as in 5.1 , we may also get that

5.45
Similarly, for the second term and the third term on the right-hand side of 5.44 , we have

5.46
For the last term on the right-hand side of 5.44 , we have

5.48
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 2.1 for u and v with ρ n,x and using Lemma 3.4, we get that, for a.e.t ∈ R and all n ≥ 1,

5.50
Using the identity ∂ 2 x G * g G * g − g for g ∈ L 2 R and Young's inequality, we estimate the forth term of the right-hand side of 5.50

5.61
Note that w 0 w x 0 0; therefore, we obtain u t, x v t, x for a.e.t, x ∈ R × R.This completes the proof of the theorem.

Lemma 3 .1 see 8 .
Let f : R → R be uniformly continuous and bounded.If μ ∈ M R , then

R ρ n
* w t sgn ρ n * w dx −b R ρ n * wu x sgn ρ n * w dx − b R ρ n * vw x sgn ρ n * w dx − a 2 R ρ n * ∂ x G * w u v sgn ρ n * w dx − 3b − a 2 R ρ n * ∂ x G * w x u x v x sgn ρ n * w dx − λ R ρ n * w sgn ρ n * w dx.5.44Using 5.36 -5.38 and Young's inequality to the first term on the right-hand side of 5.44 yields, R ρ n * wu x sgn ρ n * w dx ≤ R ρ n * wu x dx ≤ R ρ n * w ρ n * u x dx R ρ n * wu x − ρ n * w ρ n * u x dx 36 -5.38 and Young's inequality to the first term on the right-hand side of 5.50 gives rise to− b R ρ n * w x u x v x sgn ρ n,x * w dx ≤ b R ρ n * w x u x v x dx ≤ b R ρ n * w x ρ n * u x v x dx b R ρ n * w x u x v x − ρ n * w x ρ n * u x v x dx ≤ bN R ρ n * w x dx R n .5.52To treat the second term of the right-hand side of 5.50 , we note that b R ρ n * v xx w sgn ρ n,x * w dx ≤ b R ρ n * w ρ n * v xx dx b R ρ n * v xx w − ρ n * w ρ n * v xx dx.

1 ,
the second expression of the right-hand side of 5.53 can be estimated by a function R n t belonging to 5.49 .Making use of the H ölder inequality and 5.1 , for a.e.t ∈ R and all n ≥ 1, we haveR ρ n * w ρ n * v xx dx ≤ ρ n * w L ∞ R ρ n * v xx L 1 R ≤ ρ n * w W 1,1 R v xx M R .5.54It follows from 5.53 and 5.54that b R ρ n * v xx w sgn ρ n,x * w dx ≤ bN R ρ n * w dx bN R ρ n * w x R n t .5.55Now, we deal with the third term on the right-hand side of 5.50− b R ρ n * uw xx sgn ρ n,x * w dx −b R ρ n * u ρ n * w xx sgn ρ n,x * w dx − b R ρ n * uw xx − ρ n * u ρ n * w xx sgn ρ n,x * w dx uw xx − ρ n * u ρ n * w xx dx b R ρ n * u x ρ n * w x dx R n .5.56Therefore, 5.56 implies that, for a.e.t ∈ R and all n ≥ 1,−b R ρ n * uw xx sgn ρ n,x * w dx ≤ bN R ρ n * w x dx R n .5.57From 5.51 , 5.52 , 5.55 , and 5.57 , for a.e.t ∈ R and all n ≥ 1, we deduce thatd dt R ρ n * w x dx ≤ a b N R ρ n * w dx 6b a N λ R ρ n * w x dx R n .w x dx R n ≤ 2a 8b N λ R ρ n * w ρ n * w x dx R n .5.59It follows from Gronwall' inequality that, for a.e.t ∈ R and all n ≥ 1, w ρ n * w x 0, x dx e 2a 8b N λ t .5.60Fix t > 0, and let n → ∞ in 5.60 .Since w u−v ∈ W 1,1 R and relation 5.49 holds, making use of Lebesgue's dominated convergence theorem yields R |w| |w x | dx ≤ R |w| |w x | 0, x dx e 2a 8b N λ t .
R, 1.3 as a model for shallow water dynamics.Although the D-P equation 1.3 has a similar form to the C-H equation 1.1 , it should be addressed that they are truly different see 18 .In fact, many researchers have paid their attention to the study of solutions to 1.3 .Constantin et al. 19 developed an inverse scattering approach for smooth localized solutions to 1.3 .Liu and Yin 20 and Yin 21, 22 investigated the global existence of strong solutions and global weak solutions to 1.3 .Henry 23 showed that the smooth solutions to 1.3 have infinite speed of propagation.Coclite and Karlsen 24 obtained global existence results for entropy solutions in with the standard norm h L ∞ inf m e 0 sup x∈R\e |h t, x |.For any real number s, let H s H s R denote the Sobolev R e −ixξ h t, x dx.
g are a.e.equal to a function continuous from 0, T into L 2 R and Rρ n * u t sgn ρ n * u dx, 1,∞ R × R .This completes the proof of the existence of Theorem 5.1.Next, we present the uniqueness proof of the Theorem 5.1.Let u, v ∈ W 1,∞ R ×R ∩L ∞ loc R ; H 1 R be two global weak solutions of problem 2.1 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