1. Introduction
Dullin et al. [1] derived a new equation describing the unidirectional propagation of surface waves in a shallow water regime:(1)ut-α2uxxt+c0ux+3uux+γuxxx =α2(2uxuxx+uuxxx), x∈ℝ, t>0,
where the constants α2 and γ/c0 are squares of length scales and the constant c0>0 is the critical shallow water wave speed for undisturbed water at rest at spatial infinity. Since this equation is derived by Dullin, Gottwald, and Holm, in what follows, we call this new integrable shallow water equation (1) DGH equation.

If α=0, (1) becomes the well-known KdV equation, whose solutions are global as long as the initial data is square integrable. This is proved by Bourgain [2]. If γ=0 and α=1, (1) reduces to the Camassa-Holm equation which was derived physically by Camassa and Holm in [3] by approximating directly the Hamiltonian for Euler's equations in the shallow water regime, where u(x,t) represents the free surface above a flat bottom. The properties about the well-posedness, blow-up, global existence, and propagation speed have already been studied in recent works [4–13], and the generalized version of a family of dispersive equations related to Camassa-Holm equation was discussed in [14].

It is very interesting that (1) preserves the bi-Hamiltonian structure and has the following two conserved quantities:
(2)E(u)=12∫ℝ(u2+α2ux2)dx,F(u)=12∫ℝ(u3+α3uux2+c0u2-γux2)dx.

Recently, in [15], local well-posedness of strong solutions to (1) was established by applying Kato's theory [16], and some sufficient conditions were found to guarantee finite time blow-up phenomenon. Moreover, Zhou [17] found the best constants for two convolution problems on the unit circle via variational method and applied the best constants on (1) to give some blow-up criteria. Later, Zhou and Guo improved the results and got some new criteria for wave breaking [18].

In general, it is quite difficult to avoid energy dissipation mechanism in the real world. Ghidaglia [19] studied the long time behavior of solutions to the weakly dissipative KdV equation as a finite dimensional dynamic system. Moreover, some results on blow-up criteria and the global existence condition for the weakly dissipative Camassa-Holm equation are presented in [20], and very related work can be found in [21, 22]. In this work, we are interested in the following model, which can be viewed as the DGH equation with dissipation
(3)ut-α2uxxt+c0ux+3uux+γuxxx+λ(1-α2∂x2)u =α2(2uxuxx+uuxxx),
where x∈ℝ, t>0, λ(1-α2∂x2)u is the weakly dissipative term and λ is a positive dissipation parameter. Set Q=(1-α2∂x2)1/2, then the operator Q-2 can be expressed by
(4)Q-2f=G*f=∫ℝG(x-y)f(y)dy,
for all f∈L2(ℝ) with G(x)=(1/2α)e-|x|/α. With this in hand, we can rewrite (3) as a quasilinear equation of hyperbolic type
(5)ut+(u-γα2)ux+∂xG *(u2+α22ux2+(c0+γα2)u)+λu=0.
It is the dissipative term that causes the previous conserved quantities E(u) and F(u) to be no longer conserved for (3), and this model could also be regarded as a model of a type of a certain rate-dependent continuum material called a compressible second grade fluid [23]. Our consideration is based on this fact. Furthermore, we will show how the dissipation term affects the behavior of solutions in our forthcoming paper. As a whole, the current dissipation model is of great importance mathematically and physically, and it is worthy of being considered. In what follows, we assume that c0+γ/α2=0 and α>0 just for simplicity. Since u(x,t) is bounded by its H1-norm, a general case with c0+γ/α2≠0 does not change our results essentially, but it would lead to unnecessary technical complications. So the above equation is reduced to a simpler form as follows:
(6)ut+(u+c0)ux+∂xG*F(u)+λu=0, x∈ℝ, t>0,
where
(7)F(u)=u2+α22ux2.

The rest of this paper is organized as follows. In Section 2, we list the local well-posedness theorem for (6) with initial datum u0∈Hs, s>3/2 and collect some auxiliary results. In Section 3, we establish the condition for global existence in view of the initial potential. Persistence properties of the strong solutions are explored in Section 4. Finally, in Section 5, we give a detailed description of the corresponding solution with compactly supported initial data.

2. Preliminaries
In this section, we make some preparations for our consideration. Firstly, the local well-posedness of the Cauchy problem of (6) with initial data u0∈Hs with s>3/2 can be obtained by applying Kato's theorem [16]. More precisely, we have the following local well-posedness result.

Theorem 1.
Given that u0(x)∈Hs, s>3/2, there exist T=T(λ,∥u0∥Hs)>0 and a unique solution u to (6), such that
(8)u=u(·,u0)∈C([0,T);Hs)∩C1([0,T);Hs-1).
Moreover, the solution depends continuously on the initial data; that is, the mapping u0→u(·,u0):Hs→C([0,T);Hs)∩C1([0,T);Hs-1) is continuous, and the maximal time of existence T>0 is independent of s.

Proof.
Set A(u)=(u+c0)∂x, f(u)=-∂x(1-α2∂x2)-1F(u)-λu, Y=Hs, X=Hs-1, s>3/2, and Q=(1-α2∂x2)1/2. Applying Kato's theory for abstract quasilinear evolution equation of hyperbolic type, we can obtain the local well-posedness of (6) in Hs, s>3/2 and u∈C ([0,T);Hs)∩C1([0,T);Hs-1).

The maximal value of T in Theorem 1 is called the lifespan of the solution in general. If T<∞, that is lim supt→T ∥u(·,t)∥Hs=∞, we say that the solution blows up in finite time, otherwise, the solution exists globally in time. Next, we show that the solution blows up if and only if its first-order derivative blows up.

Lemma 2.
Given that u0∈Hs, s>3/2, the solution u=u(·,u0) of (3) blows up in finite time T<+∞ if and only if
(9)lim inft→T{infx∈ℝ[ux(x,t)]}=-∞.

Proof.
We first assume that u0∈Hs for some s∈ℕ, s≥4. Equation (6) can be written into the following form in terms of y=(1-α2∂x2)u(10)yt+(yu)x+12(u2-α2ux2)x+c0yx+λy=0,yt+(yu)x+12(u20-α2ux2)x+x∈ℝ, t>0.
Multiplying (10) by y=(1-α2∂x2)u, and integrating by parts, we have
(11)ddt∫ℝy2dx=2∫ℝyytdx=-3∫ℝuxy2dx-2λ∫ℝy2dx.
Differentiating (10) with respect to the spatial variable x, then multiplying by yx=(1-α2∂x2)ux, and integrating by parts again, we obtain
(12)ddt∫ℝyx2dx=2∫ℝyxyxtdx=-5∫ℝuxyx2dx+2α2∫ℝuxy2dx-2λ∫ℝyx2dx.
Summarizing (11) and (12), we obtain
(13)ddt(∫ℝ(y2+yx2)dx) =-(3-2α2)∫ℝuxy2dx -5∫ℝuxyx2dx-2λ(∫ℝ(y2+yx2)dx).
If ux is bounded from below on [0,T), for example, ux≥-C, C is a positive constant, then we get by (13) and Gronwall's inequality the following:
(14)∥y∥H12≤exp{(KC-2λ)t}∥y0∥H12,
where K=max {5,(3-2/α2)}. Therefore, the H3-norm of the solution to (10) does not blow up in finite time. Furthermore, similar argument shows that the Hk-norm with k≥4 does not blow up either in finite time. Consequently, this theorem can be proved by Theorem 1 and simple density argument for all s>3/2.

Lemma 3.
Let u0∈Hα1, then as long as the solution u(x,t) given by Theorem 1 exists, for any t∈[0,T), one has
(15)∥u∥Hα12=exp(-2λt)∥u0∥Hα12,
where the norm is defined as
(16)∥u∥Hα12=∫ℝ(u2+α2ux2)dx.

Proof.
Multiplying both sides of (10) by u and integrating by parts on ℝ, we get
(17)∫ℝuyt dx+∫ℝ(yu)xu dx+∫ℝ12(u2-α2ux2)xu dx +∫ℝc0yxu dx+∫ℝλyu dx=0.
Note that
(18)∫ℝ(yu)xu dx+∫ℝ12(u2-α2ux2)xu dx=0,∫ℝc0yxu dx=0.
Then, we have
(19)∫ℝu(ut-α2uxxt)dx+∫ℝλ(u2-α2uuxx)dx=0.
Hence,
(20)∫ℝuutdx-α2∫ℝuuxxt dx+λ∫ℝu2dx -λα2∫ℝuuxx dx=0.
Thus, we easily get
(21)∫ℝ(uut+α2uxuxt)dx+λ∫ℝ(u2+α2ux2)dx=0,
and, therefore,
(22)ddt∥u∥Hα12+2λ∥u∥Hα12=0.
By integration from 0 to t, we get
(23)∥u∥Hα12=exp(-2λt)∥u0∥Hα12, for any t∈[0,T).
Hence, the lemma is proved.

We also need to introduce the standard particle trajectory method for later use. Consider now the following initial value problem as follows:
(24)qt=u(t,q)+c0, t∈[0,T),q(0,x)=x, x∈ℝ,
where u∈C1([0,T),Hs-1) is the solution to (6) with initial data u0∈Hs, (s>3/2) and T>0 is the maximal time of existence. By direct computation, we have
(25)qtx(t,x)=ux(t,q(t,x))qx(t,x).
Then,
(26)qx(t,x)=exp(∫0tux(τ,q(τ,x))dτ)>0, t>0, x∈ℝ,
which means that q(t,·):ℝ→ℝ is a diffeomorphism of the line for every t∈[0,T). Consequently, the L∞-norm of any function v(t,·) is preserved under the family of the diffeomorphism q(t,·), that is,
(27)∥v(t,·)∥L∞=∥v(t,q(t,·))∥L∞, t∈[0,T).
Similarly,
(28)infx∈ℝ v(t,x)=infx∈ℝv(t,q(t,x)), t∈[0,T),supx∈ℝ v(t,x)=supx∈ℝ v(t,q(t,x)), t∈[0,T).
Moreover, one can verify the following important identity for the strong solution in its lifespan:
(29)ddt(y(q(x,t),t)qx2(x,t))=-λy(q(x,t),t)qx2(x,t).
We get that
(30)y(q(x,t),t)qx2(x,t)=y0(x)exp(-λt),
where y(x,t) is defined by y(x,t)=(1-α2∂x2)u(x,t), for t≥0 in its lifespan.

From the expression of u(x,t) in terms of y(x,t), for all t∈[0,T), x∈ℝ, we can rewrite u(x,t) and ux(x,t) as follows:
(31)u(x,t)=12αe-x/α∫-∞xeξ/αy(ξ,t)dξ+12αex/α∫x∞e-ξ/αy(ξ,t)dξ,
from which we get that
(32)ux(x,t)=-12α2e-x/α∫-∞xeξ/αy(ξ,t)dξ+12α2ex/α∫x∞e-ξ/αy(ξ,t)dξ.

3. Global Existence
It is shown that it is the sign of initial potential not the size of it that can guarantee the global existence of strong solutions.

Theorem 4.
Assume that u0∈Hs, s>3/2, and y0=u0-α2u0xx satisfies
(33)y0(x)≤0, x∈(-∞,x0),y0(x)≥0, x∈(x0,∞),
for some point x0∈ℝ. Then, the solution u(x,t) to (6) exists globally in time.

Proof.
From the hypothesis and (30), we obtain that y(x,t)≥0, q(x0,t)≤x<∞; y(x,t)≤0, -∞<x≤q(x0,t). According to (31) and (32), one can get that when x>x0,
(34)u(q(x,t),t)+αux(q(x,t),t) =1αeq(x,t)/α∫q(x,t)∞e-ξ/αy(ξ,t)dξ≥0,
it follows that
(35)-αux(q(x,t),t)≤u(q(x,t),t)≤∥u∥L∞≤exp(-λt)2α∥u0∥Hα1≤12α∥u0∥Hα1,
that is, ux(x,t) is bounded below. Similarly, when x<x0,
(36)u(q(x,t),t)-αux(q(x,t),t) =1αe-q(x,t)/α∫-∞q(x,t)eξ/αy(ξ,t)dξ≤0,
so -αux(q(x,t),t)≤-u(q(x,t),t). We also get the bounded below result as above. Therefore, the theorem is proved by Lemma 2.

Corollary 5.
Assume that u0∈Hs, s>3/2, and y0=u0-α2u0xx is of one sign, then the corresponding solution u(x,t) to (6) exists globally.

In fact, if x0 is regarded as ±∞, we prove this corollary immediately from Theorem 4.

4. Persistence Properties
In this section, we will investigate the following property for the strong solutions to (6) in L∞-space which behave algebraically at infinity as their initial profiles do. The main idea comes from the recent work of Himonas and his collaborators [7].

Theorem 6.
Assume that for some T>0 and s>3/2, u∈C([0,T];Hs) is a strong solution of the initial value problem associated to (6), and that u0(x)=u(x,0) satisfies
(37)|u0(x)|, |u0x(x)|~O(x-θ/α) x↑∞,
for some θ∈(0,1) and α≥1. Then,
(38)|u(x,t)|, |ux(x,t)|~O(x-θ/α) x↑∞,
uniformly in the time interval [0,T].

Proof.
The proof is organized as follows. Firstly, we will estimate ∥u(x,t)∥L∞ and ∥ux(x,t)∥L∞. Then, we apply the weight function to obtain the desired result. In the following proof, we denote some constants by c; they may be different from instance to instance, changing even within the same line.

Multiplying (6) by u2n-1 with n∈ℤ+, then integrating both sides with respect to x variable, we can get
(39)∫ℝu2n-1utdx+∫ℝu2n-1(u+c0)uxdx +∫ℝu2n-1∂xG*F(u)dx=-λ∫ℝu2ndx.
The first term of the above identity is
(40)∫ℝu2n-1utdx=12nddt∥u(t)∥L2n2n=∥u(t)∥L2n2n-1ddt∥u(t)∥L2n,
and the estimates of the second term is
(41)∫ℝu2n-1uuxdx≤∥ux(t)∥L∞∥u(t)∥L2n2n,c0∫ℝu2n-1uxdx=c0∫ℝ(u2n2n)xdx=0.
In view of Hölder's inequality, we can obtain the following estimate for the third term in (39)
(42)|∫ℝu2n-1∂xG*F(u)dx|≤∥u(t)∥L2n2n-1∥∂xG*F(u)∥L2n.
For the last term
(43)|∫ℝu2n-1λu dx|≤λ∥u(t)∥L2n2n,
putting all the inequalities above into (39) yields
(44)ddt∥u(t)∥L2n≤(∥ux(t)∥L2n+λ)∥u(t)∥L2n +∥∂xG*F(u)∥L2n.
Using the Sobolev embedding theorem, there exists a constant
(45)M=supt∈[0,T]∥u(x,t)∥Hs,
such that we have by applying Gronwall's inequality
(46)∥u(t)∥L2n≤ceMt(∥u(0)∥L2n+∫0t∥∂xG*F(u)∥L2ndτ).
For any f∈L1(ℝ)∩L∞(ℝ), we know that
(47)limq↑∞∥f∥Lq=∥f∥L∞.
Taking the limits in (46) (notice that G∈L1 and F(u)∈L1∩L∞) from (47), we get
(48)∥u(t)∥L∞≤ceMt(∥u(0)∥L∞+∫0t∥∂xG*F(u)∥L∞dτ).
Then, differentiating (6) with respect to variable x produces the following equation:
(49)uxt+uuxx+c0uxx+ux2+∂x2G*F(u)+λux=0.
Again, multiplying (49) by ux2n-1 with n∈ℤ+, integrating the result in x variable, and considering the second term and the third term in the above identity with integration by parts, one gets
(50)∫ℝuuxxux2n-1dx=∫ℝu(ux2n-12n)xdx=-12n∫ℝuxux2ndx,c0∫ℝuxxux2n-1dx=c0∫ℝ(ux2n-12n)xdx=0,
so, we have
(51)∫ℝuxtux2n-1dx-12n∫ℝuxux2ndx+∫ℝux2n+1dx =-∫ℝux2n-1∂x2G*F(u)dx-λ∫ℝux2n-1uxdx.
Similarly, the following inequality holds
(52)ddt∥ux(t)∥L2n≤(2∥ux(t)∥L∞+λ)∥ux(t)∥L2n +∥∂x2G*F(u)(t)∥L2n,
and therefore as before, we obtain
(53)∥ux(t)∥L2n≤ce2Mt(∥ux(0)∥L2n+∫0t∥∂x2G*F(u)∥L2ndτ).
Taking the limits in(53), we obtain(54)∥ux(t)∥L∞≤ce2Mt(∥ux(0)∥L∞+∫0t∥∂x2G*F(u)∥L∞dτ).

Next, we will introduce the weight function to get our desired result. This function φN(x) with N∈ℤ+ is independent of t as the following:
(55)φN(x)={1,x≤1,xθ/α,x∈(1,N),Nθ/α,x≥N.
From (6) and (49), we get the following two equations:
(56)φNut+φNuux+φNc0ux+φN∂xG*F(u)+λφNu=0,φNuxt+φNuuxx+φNc0uxx+φNux2 +φN∂x2G*F(u)+λφNux=0.
We need some tricks to deal with the following term as in [18]:
(57)∫ℝ(φN)2n-1u2n-1φNuxdx =∫ℝ(φNu)2n-1[(uφN)x-u(φN)x]dx =∫ℝ(φNu)2n-1d(φNu)-∫ℝ(φNu)2n-1u(φN)xdx ≤∫ℝ(φNu)2ndx,
where we have used the fact 0≤φN′(x)≤φN(x), a.e. x∈ℝ. Similar technique is used for the term ∫ℝ(φN)2n-1ux2n-1φNuxxdx. Hence, as in the weightless case, we get the following inequality in view of (48) and (54) as follows:
(58)∥u(t)φN∥L∞+∥ux(t)φN∥L∞ ≤ce2Mt(∥u(0)φN∥L∞+∥ux(0)φN∥L∞)+ce2Mt ×(∫0t(∥φN∂xG*F(u)∥L∞+∥φN∂x2G*F(u)∥L∞)dτ).
On the other hand, a simple calculation shows that there exists C>0, depending only on α and θ such that for any N∈ℤ+,
(59)φN(x)∫ℝe-|x-y|/α1φN(y)dy≤C.
Therefore, for any appropriate function g, one obtains that
(60)|φN∂xG*g2(x)| =|12αφN(x)∫ℝe-|x-y|/αg2(y)dy| ≤12αφN(x)∫ℝe-|x-y|/α1φN(y)φN(y)g(y)g(y)dy ≤12α(φN(x)∫ℝe-|x-y|/α1φN(y)dy)∥gφN∥L∞∥g∥L∞ ≤Cα∥gφN∥L∞∥g∥L∞,
and similarly, |φN∂x2G*g2(x)|≤(C/α)∥gφN∥L∞∥g∥L∞. Using the same method, we can estimate the following two terms
(61)|φNG*g(x)|≤Cα∥gφN∥L∞,|φN∂xG*g(x)|≤Cα∥gφN∥L∞.
Therefore, it follows that there exists a constant C1(M,T,α,λ)>0 such that
(62)∥u(t)φN∥L∞+∥ux(t)φN∥L∞ ≤C1(∥u(0)φN∥L∞+∥ux(0)φN∥L∞) +C1∫0t((∥u(τ)∥L∞+∥ux(τ)∥L∞) ·(∥φNu(τ)∥L∞+∥φNux(τ)∥L∞))dτ ≤C1(∫0t∥u(0)φN∥L∞+∥ux(0)φN∥L∞ +∫0t(∥φNu(τ)∥L∞+∥φNux(τ)∥L∞)dτ).
Hence, the following inequality is obtained for any N∈ℤ+ and any t∈[0,T]:
(63)∥u(t)φN∥L∞+∥ux(t)φN∥L∞ ≤C1(∥u(0)φN∥L∞+∥ux(0)φN∥L∞) ≤C1(∥u(0)max(1,xθ/α)∥L∞ +∥ux(0)max(1,xθ/α)∥L∞).
Finally, taking the limit as N goes to infinity in the above inequality, we can find that for any t∈[0,T],
(64)(|u(x,t)xθ/α|+|ux(x,t)xθ/α|) ≤C1(∥u(0)max(1,xθ/α)∥L∞ +∥ux(0)max(1,xθ/α)∥L∞),
which completes the proof of Theorem 6.

5. Asymptotic Description
The following result is to give a detailed description on the corresponding strong solution u(x,t) in its lifespan with u0(x) being compactly supported.

Theorem 7.
Assume that the initial datum 0≢u0(x)∈Hs with s>5/2 is compactly supported in [a,c], then the corresponding solution u(x,t)∈C([0,T);Hs) to (6) has the following property: for any t∈(0,T),
(65)u(x,t)=L(t)e-x/α as x>q(c,t),u(x,t)=l(t)ex/α as x<q(a,t),
where q(x,t) is defined by (24) and T is its lifespan. Furthermore, L(t) and l(t) denote continuous nonvanishing functions, with L(t)>0 and l(t)<0 for t∈(0,T). Moreover, L(t) is a strictly increasing function, while l(t) is strictly decreasing.

Remark 8.
This is an interesting phenomenon for our model; it implies that the strong solution does not have compact x-support for any t>0 in its lifespan anymore, although the corresponding u0(x) is compactly supported. No matter that the initial profile u0(x) is (no matter it is positive or negative), for any t>0 in its lifespan, the nontrivial solution u(x;t) is always positive at infinity and negative at negative infinity. Moreover, we found that the dissipative coefficient does not affect this behavior.

Proof.
First, since u0(x) has a compact support, so does y0(x)=(1-α2∂x2)u0(x). Equation (30) tells us that y=(1-α2∂x2)u(x,t)=((1-α2∂x2)u0(q-1(x,t))exp(-λt))/(∂xq-1((x,t),t))2 is compactly supported in [q(a,t),q(c,t)] in its lifespan. Hence, the following functions are well defined
(66)E(t)=∫ℝeξ/αy(ξ,t)dξ, F(t)=∫ℝe-ξ/αy(ξ,t)dξ,
with
(67)E(0)=∫ℝeξ/αy0(ξ)dξ=∫ℝeξ/αu0(ξ)dξ-α2∫ℝeξ/αu0xx(ξ)dξ=0.
And F(0)=0 by integration by parts.

Then, for x>q(c,t), we have
(68)u(x,t)=12αe-|x|/α*y(x,t)=12αe-x/α∫q(a,t)q(c,t)eξ/αy(ξ,t)dξ=12αe-x/αE(t),
where (66) is used.

Similarly, when x<q(a,t), we get
(69)u(x,t)=12αe-|x|/α*y(x,t)=12αex/α∫q(a,t)q(c,t)e-ξ/αy(ξ,t)dξ=12αex/αF(t).
Because y(x,t) has a compact support in x in the interval [q(a,t),q(c,t)] for any t∈[0,T], we get y(x,t)=u(x,t)-α2uxx(x,t)=0, for x>q(c,t) or x<q(a,t). Hence, as consequences of (68) and (69), we have
(70)u(x,t)=-αux(x,t)=α2uxx(x,t)=12αe-x/αE(t), as x>q(c,t),u(x,t)=αux(x,t)=α2uxx(x,t)=12αex/αF(t), as x<q(a,t).
On the other hand,
(71)dE(t)dt=∫ℝeξ/αyt(ξ,t)dx.

Substituting the identity (10) into dE(t)/dt, we obtain
(72)dE(t)dt=-∫ℝeξ/α[(yu)x+12(u2-α2ux2)x+c0yx+λy]dξ=1α∫ℝeξ/αyu dξ+12α∫ℝeξ/α(u2-α2ux2)dξ +c0α∫ℝeξ/αy dξ+α2∫ℝeξ/αλuxx dξ -∫ℝeξ/αλu dξ=32α∫ℝeξ/αu2dξ+α2∫ℝeξ/αux2 dξ +∫ℝeξ/αuux dξ=∫ℝeξ/α(1αu2+α2ux2)dξ>0,
where we used (70). Therefore, in the lifespan of the solution, we have that E(t) is an increasing function with E(0)=0; thus, it follows that E(t)>0 for t∈(0,T]; that is,
(73)E(t)=∫0t∫ℝeξ/α(1αu2+α2ux2)(ξ,τ)dξ dτ>0.
By similar argument, one can verify that the following identity for F(t) is true:
(74)F(t)=-∫0t∫ℝe-ξ/α(1αu2+α2ux2)(ξ,τ)dξ dτ<0.
In order to finish the proof, it is sufficient to let L(t)=(1/2α)E(t), and to let l(t)=(1/2α)F(t), respectively.