# Blow-up Phenomena and Persistence Properties of Solutions to the Two-Component DGH Equation

Authors: Panpan Zhai; Zhengguang Guo; Weiming Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with blow-up phenomena and persistence properties for an integrable two-component Dullin-Gottwald-Holm shallow water system. We give sufficient conditions on the initial data which guarantee blow-up phenomena of solutions in finite time for both periodic and nonperiodic cases, respectively. Furthermore, the persistence properties of solutions to the system are investigated.


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## Body

## 1. Introduction

In 2001, Dullin et al. [1] derived a new equation by using the method of asymptotic analysis and a near-identity normal form transformation in water wave theory to describe the unidirectional propagation of surface waves in a shallow water regime, it reads as follows:
(1)mt+2ωux+umx+2mux=-γuxxx,x∈ℝ,t>0.
Since (1) was derived by Dullin et al., we call it DGH equation. Let m=u-α2uxx; then we can rewrite (1) as the following form:
(2)ut-α2uxxt+2ωux+3uux+γuxxx=α2(2uxuxx+uuxxx),
where α2 and γ/2ω are squares of length scales, the constant ω=gh/2 is the critical shallow water speed for undisturbed water at rest at spatial infinity, h is the mean fluid depth, and g is the gravitational constant. This equation is connected with two separately integrable soliton equations for shallow water waves.Whenα2=0, (2) becomes the KdV equation as follows:
(3)ut+2ωux+3uux=-γuxxx.
Bourgain proved that solutions to the KdV equation are global as long as the initial data is square integrable [2], which is also shown in [3]. Another remarkable property is that it is integrable and the solitary waves are nonlinearly stable, and when ω=0, there exists a smooth soliton solution [4].Instead, takingγ=0 in (2), it turns out to be the Camassa-Holm equation:
(4)ut-α2uxxt+2ωux+3uux=α2(2uxuxx+uuxxx),
which was derived by Camassa and Holm in [5] by approximating directly the Hamiltonian for Euler’s equations in the shallow water regime in 1993. Many researches have been carried out on the Camassa-Holm equation in recent years. Local well-posedness has been proven by several authors; see [6–9]. Blow-up phenomena have been investigated in [10–15]. Besides, in [11], global solutions were also discussed. In [16], global existence of weak solutions was proved, but uniqueness was obtained only under a priori assumption. Furthermore, the authors in [17] studied the persistence properties of solution to the Camassa-Holm equation. In [18–20], the soliton solution was studied.It is very interesting that (2) still preserves the bi-Hamiltonian structure and complete integrability. Indeed, (2) can be rewritten in two compatible Hamiltonian forms in terms of m=u-α2uxx [1] as follows,
(5)mt=-B2δEδm=-B1δFδm,
where
(6)B1=∂x-α2∂x3,B2=∂x(m+ω)+(m+ω)∂x+γ∂x3,E(u)=12∫‍(u2+α2ux2)dx,F(u)=12∫‍(u3+α3uux2+2ωu2-γux2)dx.E(u) and F(u) are two conserved quantities.In [21], Tian et al. studied the well-posedness of the Cauchy problem and the scattering problem of the DGH equation. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation was investigated, and the convergence of the solutions to DGH equation as α2→0 was studied. Besides, data of the scattering problem for the equation could be explicitly expressed. The blow-up phenomenon of solutions to the DGH equation was the subject of [21–24].In this paper, we are interested in the following Cauchy problem for an integrable two-component DGH equation. It reads as(7)ut-α2utxx+2ωux+3uux+γuxxx=α2(2uxuxx+uuxxx)-σρ-ρ-x,x∈ℝ,t>0,ρ-t+(ρ-u)x=0,x∈ℝ,t>0,u(0,x)=u0(x),x∈ℝ,ρ-(0,x)=ρ-0(x),x∈ℝ.
The variable u(x,t) describes the horizontal velocity of the fluid, and ρ¯(x,t) denotes the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. This model can be derived by Constantin and Ivanov’s approach [25] from shallow water theory, which includes the two-component Camassa-Holm system [26–32] as its special case with α=1 and γ=0 in (7). From a geometric point of view, (7) is the model for geodesic motion on the semidirect product Lie group of diffeomorphisms acting on densities, with respect to the H1-norm of velocity and the L2-norm on the density. Mathematically, (7) admits not only breaking-wave solutions but also global in time solutions. In view of the context of hydrodynamics, we have the assumptions u→0, ρ-→1 as x→∞, at any instant t. For convenience of later discussion, letting ρ=ρ--1, α>0, and σ=1, we have ρ→0 as x→∞. Then it follows that
(1)ut-α2utxx+2ωux+3uux+γuxxx-2α2uxuxx-α2uuxxx+ρρx+ρx=0,x∈ℝ,t>0,ρt+(ρu)x+ux=0,x∈ℝ,t>0,u(0,x)=u0(x),x∈ℝ,ρ(0,x)=ρ0(x),x∈ℝ.
Define Λ=(1-α2∂x2)-1; then the operator Λ can be expressed by
(9)Λf=G*f=∫ℝ‍G(x-y)f(y)dy,
where G(x) is the Green function for Λ and * denotes the convolution. It is easy to get G*(1/2)=1/2. For the periodic case, G(x)=cosh((x/α)-([x]/α)-(1/2α))/2αsinh(1/2α) is the Green function for Λ in the unit circle. While, for the nonperiodic case, the Green function for Λ in ℝ is G(x)=(1/2α)e-|x/α|. Using this identity, we can rewrite (1) as the following nonlocal form:
(10)ut+ux(u-γα2)=-∂xG*(u2+α22ux2+(2ω+γα2)u+12ρ2+ρα22ux2),(u2+α22ux2+(2ω+γα2)u+12ρ2+ρα22ux2)x∈ℝ,t>0,ρt+(ρu)x+ux=0,x∈ℝ,t>0,u(0,x)=u0(x),x∈ℝ,ρ(0,x)=ρ0(x),x∈ℝ.
We note that less deep results exist yet for this model except for the recent ones [33, 34]. Our motivation here is to explore some new blow-up conditions especially for the initial data were added on some different quantities. This paper is organized as follows. In Section 2, we recall some preliminary results on local well-posedness and blow-up scenario and present some useful conversation laws. In Section 3, we study the blow-up phenomenon of strong solutions in both periodic and nonperiodic cases. In Section 4, persistence properties of solutions to the system are studied.


## 2. Preliminaries

We can use Kato’s theory [35] to establish the following local well-posedness theorem for (10); its proof is referred to in the discussions of [33, 34], so we omit it here.Theorem 1.
Assume an initial data(u0,ρ0)∈Hs×Hs-1,s≥2. Then there exists a maximal T=T(∥(u0,ρ0)∥Hs×Hs-1)>0 and a unique solution
(11)(u,ρ)∈C([0,T);Hs×Hs-1)∩C1([0,T);Hs-1×Hs-2)
of system (10). Moreover, the solution (u,ρ) depends continuously on the initial value (u0,ρ0), and the maximal time of existence T>0 is independent of s.Next, we will give some useful conserved quantities, which are important to discuss the blow-up criteria.Proposition 2.
Suppose(u0,ρ0)∈Hs×Hs-1,s≥2; then the solution (u,ρ) of the system (10) guaranteed by Theorem 1 satisfies
(12)∫ℝ‍(u2+α2ux2+ρ2)dx=∫ℝ‍(u02+α2u0x2+ρ02)dx.Proof.
Consider(13)ddt∫ℝ‍(u2+α2ux2+ρ2)dx=∫ℝ‍(2uut+2α2uxuxt+2ρρt)dx=∫ℝ‍[2u(ut-α2uxxt)+2ρρt]dx.
From the first equation of (10), we have
(14)∫ℝ‍2u(ut-α2uxxt)dx=∫ℝ‍2u(-2ωux-3uux-γuxxx+2α2uxuxx+α2uuxxx-ρρx-ρx)dx.
Similarly, from the second equation of (10), we can get
(15)∫ℝ‍2ρρtdx=∫ℝ‍-2ρ((ρu)x+ux)dx.
So we can obtain
(16)ddt∫ℝ‍(u2+α2ux2+ρ2)dx=∫ℝ‍[2u(-2ωux-3uux-γuxxx+2α2uxuxx+α2uuxxx-ρρx-ρx)-2ρ((ρu)x+ux)]dx=∫ℝ[-2ω(u2)x-2(u3)x+γ(ux2)x‍+2α2(u2uxx)x-2(ρu)x-2(ρρu)x]dx=0.
This completes the proof.We also note that the quantities∫ℝ‍u(x,t)dx and ∫ℝ‍ρ(x,t)dx are also conserved for ρ0(x)∈L1, u0(x)∈L1. These conservation laws will play important roles in considering the behavior of solutions. In what follows, we denote ∫ℝ‍(u2+α2ux2+ρ2)dx by ∥(u,ρ)∥Hα1×L22.Moreover, using the techniques in [36], one can get the following criterion for finite time wave breaking to (10).Theorem 3 (see [34]).
Let(u0,ρ0)∈Hs×Hs-1 with s≥2, and let T>0 be the maximal time of existence of the solution (u,ρ) to the model (10) with initial data (u0,ρ0). Then the corresponding solution (u,ρ) blows up in finite time if and only if
(17)limt↑T{infx∈ℝux(t,x)}=-∞.We will then need to introduce the standard particle trajectory method for later use. Now consider the following initial value problem:(18)qt=u(t,q),t∈[0,T),q(0,x)=x,x∈ℝ,
where u∈C1([0,T),Hs-1) is the first component of the solution (u,ρ) to system (10) with initial data (u0,ρ0)∈Hs×Hs-1,(s≥2), and T>0 is the maximal time of existence. By direct calculation, we have
(19)qtx(t,x)=ux(t,q(t,x))qx(t,x).
Then,
(20)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,
(21)∥v(t,·)∥L∞=∥v(t,q(t,·))∥L∞,t∈[0,T).
Similarly,
(22)infx∈ℝv(t,x)=infx∈ℝv(t,q(t,x)),t∈[0,T),supx∈ℝv(t,x)=supx∈ℝv(t,q(t,x)),t∈[0,T).


## 3. Blow-Up Criteria

In this section, we establish sufficient conditions to guarantee the formation of singularities for the corresponding solutions to (10) in periodic and nonperiodic cases. These sufficient conditions are different from each other.Hereinafter, we investigate the blow-up phenomenon with periodic setting; that is,x∈𝕊=ℝ/ℤ. First, we introduce the following lemmas for later use.Lemma 4 (see [23]).
For allf∈H1(𝕊), the following inequality holds:
(23)G*(f2+α22fx2)(x)≥C0f2(x),
with
(24)C0=12+arctan(sinh(1/2α))2sinh(1/2α)+2arctan(sinh(1/2α))sinh2(1/2α).
Moreover, C0 is the optimal constant obtained by the function
(25)f0=(1+arctan(sinh(xα-[x]α-12α))×sinh(xα-[x]α-12α))×(1+arctan(sinh(12α))sinh(12α))-1.Lemma 5 (see [23]).
For allf∈H1(𝕊), the following inequality holds:
(26)maxx∈𝕊f2(x)≤C1∫𝕊‍(f2+α2fx2)dx,
where
(27)C1=cosh(1/2α)2αsinh(1/2α).
Moreover, C1 is the minimum value, so in this sense, C1 is the optimal constant which is obtained by the associated Green function
(28)G(x)=cosh((x/α)-([x]/α)-(1/2α))2αsinh(1/2α).Lemma 6 (see [37]).
Assume that a differentiable functiony(t) satisfies
(29)y′(t)≤-Cy2(t)+K
with the constants C,K>0. If the initial datum y(0)=y0<-K/C, then the solution to the previous equation goes to -∞ in finite time.Lemma 7 (see [24]).
Assumef(x)∈Hs(𝕊), s>2. If ∫𝕊‍f(x)dx=0, then
(30)∥f(x)∥L∞(𝕊)2≤112∫𝕊‍fx2(x)dx,∫𝕊‍f2(x)dx≤112∫𝕊‍fx2(x)dx,∫𝕊‍f2(x)fx2(x)dx≤112(∫𝕊‍fx2(x)dx)2.Now we give some sufficient conditions on the initial data to guarantee finite time wave breaking for the periodic case.Theorem 8.
Assume that the initial data(u0,ρ0)∈Hs×Hs-1,s≥2, satisfies
(31)∫𝕊‍u0x3dx<-∥(u0,ρ0)∥Hα1×L22α33A+6C1∥(u0,ρ0)∥Hα1×L22,
where C1 is defined as in Lemma 5, and constant A is determined later. Moreover, for convenience, we suppose that 2ω+γ/α2=0. Then the corresponding strong solution to (10) blows up in finite time.Proof.
By assumption and Theorem1, there exists a solution (u,ρ) to system (10) corresponding to the initial value (u0,ρ0)∈Hs×Hs-1,s≥2. Suppose that the statement is not true; that is, the corresponding solution does not blow up in finite time, and then due to Theorem 3, there exists M1≥0, such that
(32)inf(t,x)∈[0,T)×ℝux(t,x)≥-M1.
Let
(33)M(t)=ux(t,q(t,x)),γ(t)=ρ(t,q(t,x)),(t,q(t,x)),γ(t)=ρ(t,q(t,x))t∈[0,T),
with q(t,x) determined in (18). Then we have the following inequality for each x∈ℝ and t∈[0,T):
(34)M(t)≥-M1.
Since
(35)γ′(t)=-γ(t)M(t),
we   have, by solving it, (36)|γ(t)|=|γ(0)|e∫0t‍-M(τ)dτ.
So we can get
(37)|ρ(t,q(t,x))|=|γ(t)|=|γ(0)|e∫0t‍-M(τ)dτ≤∥ρ0∥L∞eM1t,
then we have
(38)∥ρ(t,·)∥L∞≤∥ρ0∥L∞eM1t.
Therefore, the following inequality holds:
(39)∥(ρ+1)2∥L∞≤(∥ρ∥L∞+1)2≤(∥ρ0∥L∞eM1t+1)2=˙A.
With the notation of ∂x2(G*f)=(1/α2)(G*f-f), differentiating both sides of the first equation of (10) with respect to variable x, we obtain
(40)utx+ux2+uuxx+2ωuxx=1α2(u2+α22ux2+12ρ2+ρ)-1α2G*(u2+α22ux2+12ρ2+ρ).
We note that G*(1/2)=1/2; then (40) can be rewritten as follows:
(41)utx+ux2+uuxx+2ωuxx=1α2(u2+α22ux2+12(ρ+1)2)-1α2G*(u2+α22ux2+12(ρ+1)2).
Multiplying ux2 on both sides and integrating by parts with respect to x, one obtains
(42)ddt∫𝕊‍ux3dx=-12∫𝕊‍ux4dx+3α2∫𝕊‍ux2u2dx+3α2∫𝕊‍ux2(12(ρ+1)2)dx-3α2∫𝕊‍ux2G*(u2+α22ux2+12(ρ+1)2)dx≤-12∫𝕊‍ux4dx+3α2∫𝕊‍ux2u2dx+3α2∫𝕊‍ux2(12(ρ+1)2)dx,
where we have used the following identity:
(43)∫𝕊‍ux2uuxxdx=-13∫𝕊‍ux4dx.
Since
(44)(∫𝕊‍ux3dx)2≤1α2(∫𝕊‍ux4dx)(∫𝕊‍α2ux2dx),
it follows that
(45)∫𝕊‍ux4dx≥α2∥(u0,ρ0)∥Hα1×L22(∫𝕊‍ux3dx)2.
We can also get
(46)3α2∫𝕊‍ux2u2dx≤3α4C1∥(u0,ρ0)∥Hα1×L24,|3α2∫𝕊‍ux2(12(ρ+1)2)dx|=32α2∫𝕊‍ux2(ρ+1)2dx≤32α4∥(ρ+1)2∥L∞∫𝕊‍α2ux2dx≤32α4∥(ρ+1)2∥L∞∥(u0,ρ0)∥Hα1×L22.
So we obtain
(47)ddt∫𝕊‍ux3dx≤-α22∥(u0,ρ0)∥Hα1×L22(∫𝕊‍ux3dx)2+3α4C1∥(u0,ρ0)∥Hα1×L24+32α4∥(ρ+1)2∥L∞∥(u0,ρ0)∥Hα1×L22≤-α22∥(u0,ρ0)∥Hα1×L22(∫𝕊‍ux3dx)2+3α4C1∥(u0,ρ0)∥Hα1×L24+32α4A∥(u0,ρ0)∥Hα1×L22.
Due to the theory of Riccati type differential equations in Lemma 6, if (31) holds, then we have
(48)∫𝕊‍ux3dx⟶-∞,
as t tends to some T0.
On the other hand, the majorization(49)∫𝕊‍ux3dx≥infux(x,t)∫𝕊‍ux2dx>c(α)infux(x,t)∥(u0,ρ0)∥Hα1×L22
implies that
(50)limt↑T0infux(x,t)=-∞.
It contradicts the assumption
(51)inf(t,x)∈[0,T)×ℝux(t,x)≥-M1.
By Theorem 3, we know that the solution must blow up in finite time.Next, we find if the initial velocity has zero mean and initial energy is sufficiently large, wave breaking can occur.Theorem 9.
Supposez0=(u0,ρ0)∈Hs×Hs-1,s>5/2, and α satisfies 0<sinh(1/2α)/α<3. If the initial energy satisfies ∫𝕊‍(u02+α2u0x2+ρ02)dx>C0 for some positive constant C0, and then ∫𝕊‍u0(x)dx=0 holds for nontrival z0. Besides, assume that 2ω+γ/α2=0 just for convenience. Then the solution to the system (10) with initial value z0 blows up in finite time.Proof.
From the first equation of (10), we know that
(52)ddt∫𝕊‍ux3dx=-12∫𝕊‍ux4dx+3α2∫𝕊‍ux2u2dx+3α2∫𝕊‍ux2(12ρ2+ρ)dx-3α2∫𝕊‍ux2G*(u2+α22ux2+12ρ2+ρ)dx.
On the other hand, we know that ∫𝕊‍u(x,t)dx=0 in view of the hypothesis, and the following inequality holds:
(53)12αsinh(1/2α)≤G(x)≤cosh(1/2α)2αsinh(1/2α).
Using Lemma 7 and (53), we obtain
(54)ddt∫𝕊‍ux3dx=-12∫𝕊‍ux4dx+3α2∫𝕊‍ux2u2dx+3α2∫𝕊‍ux2(12(ρ+1)2)dx-3α2∫𝕊‍ux2G*(u2+α22ux2+12(ρ+1)2)dx≤-12∫𝕊‍ux4dx+14α2(∫𝕊‍ux2dx)2+3α2∫𝕊‍ux2(12(ρ+1)2)dx-32∫𝕊‍ux2(G*ux2)dx≤-12∫𝕊‍ux4dx+3α2∫𝕊‍ux2(12(ρ+1)2)dx+(14α2-34αsinh(1/2α))(∫𝕊‍ux2dx)2.
Similar to Theorem 8, assume that the solution (u,ρ) of system (10) does not blow up; that is, there is M1≥0, such that
(55)inf(t,x)∈[0,T)×ℝux(t,x)≥-M1,
then
(56)∥ρ(t,·)∥L∞≤∥ρ0∥L∞exp(M1t)=˙κ~.
So we can get
(57)ddt∫𝕊‍ux3dx≤-12∫𝕊‍ux4dx+3α2∫𝕊‍ux2(12(ρ+1)2)dx+(14α2-34αsinh(1/2α))(∫𝕊‍ux2dx)2≤-12∫𝕊‍ux4dx+3(κ~+1)22α2∫𝕊‍ux2dx+(14α2-34αsinh(1/2α))(∫𝕊‍ux2dx)2.
If 0<sinh(1/2α)/α<3, then (1/4α2)-(3/4αsinh(1/2α))<0. Since ∫𝕊‍(u02+α2u0x2+ρ02)dx>C0 for some constant C0, due to Lemma 7, there is some δ⋆>0 such that
(58)3(κ~+1)22α2∫𝕊‍ux2dx≤δ⋆(∫𝕊‍ux2dx)2,(34αsinh(1/2α)-14α2-δ⋆)>0.
Moreover, Lemma 7 also implies that
(59)∫𝕊‍ux2dx≥1212α2+1∥u(x)∥Hα1(𝕊)2.
Hence,
(60)ddt∫𝕊‍ux3dx≤-12∫𝕊‍ux4dx-144(12α2+1)2×(34αsinh(1/2α)-14α2-δ⋆)∥u(x)∥Hα1(𝕊)4.
Note that ∥u(x)∥Hα1(𝕊) is bounded. In view of Hölder’s inequality, there exists
(61)∫𝕊‍ux4dx≥(∫𝕊‍ux3dx)4/3.
For simplicity of notations, we denote by φ(t) and μ the following quantities:
(62)∫𝕊‍ux3dx,144(12α2+1)2(34αsinh(1/2α)-14α2-δ⋆),
respectively. Therefore we have
(63)dφ(t)dt≤-12φ4/3(t)-μ∥u(x)∥Hα1(𝕊)4.
First, we can easily get φ(t)≤φ(0)-μ∥u∥Hα1(𝕊)4t, and it is not difficult to find that there exists a time t0 such that φ(t0)<0. Then for all t>t0, we have
(64)dφ(t)dt≤-12φ4/3(t),withφ(t0)<0.
Solving this inequality yields
(65)φ(t)≤(φ-1/3(t0)+16(t-t0))-3,
which goes to -∞ as t tends to -6φ-1/3(t0)+t0; that is, there exists a time T≤-6φ-1/3(t0)+t0 such that
(66)limt↑T∫𝕊‍ux3dx=-∞.
Since
(67)∫𝕊‍ux3dx≥infux(x,t)∫𝕊‍ux2dx≥c(α)infux(x,t)∥u∥Hα1(𝕊)2,
it shows that
(68)limt↑Tinfx∈ℝux(x,t)=-∞.
Then it contradicts the assumption
(69)inf(t,x)∈[0,T)×ℝux(x,t)>-M1.
By Theorem 3, we know that the solution must blow up in finite time. This finishes the proof.From now on, we give some blow-up criteria for the nonperiodic case.Theorem 10.
Assume that the initial data(u0,ρ0)∈Hs×Hs-1,s≥2, satisfy
(70)u0′(x0)<-2K0,
where K0 is a constant determined later. And assume that 2ω+γ/α2=0 just for convenience. Then T is finite and the slope of u tends to negative infinity as t goes to T while u is uniformly bounded on [0,T).Proof.
Let(u,ρ) be the solution to system (10) with the initial data (u0,ρ0) and let T be the maximal existence time of solution. Defining (71)m(t)=infx∈ℝ[ux(t,x)]=ux(t,ξ(t)),γ(t)=ρ(t,ξ(t)),
one can get that m is almost everywhere differentiable on [0,T) with
(72)dmdt(t)=utx(t,ξ(t)).
Note that uxx(t,ξ(t))=0, and with (40), we have
(73)mt=-m2+1α2(u2+α22ux2+12ρ2+ρ)-1α2G*(u2+α22ux2+12ρ2+ρ),
and with the relation 1/2=G*(1/2), we can get
(74)mt=-m2+1α2(u2+α22ux2+12(ρ+1)2)-1α2G*(u2+α22ux2+12(ρ+1)2)≤-12m2+1α2(u2+12(ρ+1)2).
Since u and ρ are both bounded, then there exists a K0>0, so that (1/α2)(u2+(1/2)(ρ+1)2)<K0, and thus we have
(75)mt≤-12m2+K0.
Note that if m(0)<-2K0, then m(t)<-2K0 for all t∈[0,T). So there exists a T such that
(76)limt↑Tm(t)=-∞.
This completes the proof of the theorem.The authors in [25] claimed that the corresponding solution to the two-component Camassa-Holm equation exists globally in time provided that some initial data satisfy smallness conditions. However, we find surprisingly the smallness of initial energy can also lead to breaking wave solutions. This is stated as follows.Theorem 11.
Let(u0,ρ0)∈Hs(ℝ)×Hs-1(ℝ) with s>5/2 and let T>0 be the maximal time of existence of the solution (u,ρ) to (10) with initial data (u0,ρ0). Assume there is some x0∈ℝ such that ρ0(x0)=-1, and there holds 0<γ<1-2ωα2 for suitable α and γ. Moreover,
(77)∥(u0,ρ0)∥Hα1×L22<α5(1-2ωα2-γ)2α4+2ωα2+γ.
Then the corresponding solution to (10) blows up in finite time in the following sense, and there exists a T0 with
(78)0<T0≤2+4α7+4α7|u0,x(x0)|α5-2ωα7-γα5-(2α4+2ωα2+γ)∥(u0,ρ0)∥Hα1×L22,
such that
(79)limt↑T0inf(infx∈ℝux(x,t))=-∞.Proof.
As in Theorem8, we define
(80)M(t)=infx∈ℝ[ux(t,x)]=ux(t,ξ(t)),t∈[0,T),
so that
(81)uxx(t,ξ(t))=0,a.e.t∈[0,T).
We recall that q(t,·) defined by (18) is a diffeomorphism of the line for any t∈[0,T), so there exists an x(t)∈ℝ such that
(82)q(t,x(t))=ξ(t),t∈[0,T).
Differentiating the first equation of (10), we can get the expression of Mt as follows:
(83)Mt(t,q(t,x(t)))=-M2+1α2(u2+α22ux2+12(ρ+1)2)-1α2G*(u2+α22ux2+12(ρ+1)2)-(2ω+γα2)∂x2G*u=-12M2+1α2u2+12α2(ρ+1)2-1α2G*(u2+α22ux2)-12α2G*(ρ+1)2-(2ω+γα2)∂x2G*u.
Besides, along the trajectory of q(t,x(t)), we have
(84)d((ρ(t,q)+1)qx)dt=qx(ρt+ρxu+ρux+ux)=0.
Then choose ξ(0)=x0 and ρ(ξ(0))=ρ0(x0)=-1, which implies that
(85)ρ(t,ξ(t))≡-1.
Therefore we can obtain
(86)Mt=-12M2+1α2u2-1α2G*(u2+α22ux2)-12α2G*(ρ+1)2-(2ω+γα2)∂x2G*u≤-12M2+1α2u2-12α2u2-12α2G*(ρ+1)2-(2ω+γα2)∂x2G*u=-12M2+12α2u2-12α2G*(ρ+1)2-(2ω+γα2)∂x2G*u,
where we use the fact
(87)G*(u2+α22ux2)≥12u2.
We define
(88)f=12α2u2-12α2G*(ρ+1)2-(2ω+γα2)∂x2G*u.
For any x∈ℝ, then
(89)u2(x)=∫-∞x0‍uuxdx+∫∞x0‍uuxdx≤12α(∫-∞x0‍(u2+α2ux2)dx+∫∞x0‍(u2+α2ux2)dx)=12α∥u∥Hα12≤12α∥(u,ρ)∥Hα1×L22.
We also have
(90)|G*ρ|≤∥G∥L2∥ρ∥L2=12α∥ρ∥L2≤14+14α∥(u,ρ)∥Hα1×L22,|∂x2G*u|=|Gx*ux|≤∥Gx∥L2∥ux∥L2=12α3∥ux∥L2≤14+14α5∥(u,ρ)∥Hα1×L22.
Thus, we can get
(91)f=12α2u2-12α2G*ρ2-1α2G*ρ-12α2-(2ω+γα2)∂x2G*u≤12α2u2+1α2|G*ρ|-12α2+(2ω+γα2)|∂x2G*u|≤14α3∥(u,ρ)∥Hα1×L22+14α3∥(u,ρ)∥Hα1×L22+14α5(2ω+γα2)∥(u,ρ)∥Hα1×L22-14α2+14(2ω+γα2)=˙-a<0.
Therefore, when ∥(u0,ρ0)∥H1×L22<(α5-2ωα7-γα5)/(2α4+2ωα2+γ) holds, we always have f<0. In this case, we can obtain
(92)M′(t)≤-12M2-a<0,t∈[t,T),
so M(t) is decreasing strictly in such interval. Assume that the solution (u,ρ) of (10) does not blow up; that is, it exists globally in time; that is, T=∞. Then integrating (92) over [0,t1] with t1=(1+|u0,x(x0)|)/a yields
(93)M(t1)=M(0)-∫0t1‍M′(t)dt≤|u0,x(x0)|-at1≤-1.
We can also find that M′(t)≤-(1/2)M2, which leads to
(94)-ddt(1M(t))=1M2(t)ddtM(t)≤-12,t∈[t1,T).
Integrating both sides, we have
(95)-1M(t)-1≤-1M(t)+1M(t1)≤-12(t-t1),
where we use the result M(t1)≤-1 obtained previously. Solve the inequality
(96)M(t)≤2(t-t1)-2⟶-∞,ast⟶t1+2.
That implies that T≤t1+2<∞, which leads to a contradiction as t→∞. This completes the proof.


## 4. Persistence Properties

Attention now is turned to determining the persistence properties of solutions to the two-component DGH equation. We show that certain decay properties of the initial data persist in corresponding solution in later time as long as it exists. Precisely, we prove that the corresponding solution(u,ρ) and its first order spatial derivatives preserve the exponential decay as their initial values do. This is a very important investigation since we can obtain the detailed asymptotic behavior of solution from the initial values. The main idea comes from the recent works of Zhou et al. [17, 38]. Now, we state our result as follows.Theorem 12.
Assume that for someT>0 and s>5/2, (u,ρ)∈C([0,T];Hs×Hs-1) is a strong solution of the initial value problem to the system (10) and 2ω+γ/α2=0 just for convenience. If u0=u(x,0), ρ0=ρ(x,0) satisfy
(97)|u0(x)|,|∂xu0(x)|,|ρ0(x)|,|∂xρ0(x)|~O(e-θx)|∂xu0(x)|,|ρ0(x)|,|∂xρ01|~O(e-θx)asx⟶∞,
for some θ∈(0,1). Then
(98)|u(x,t)|,|ux(x,t)|,|ρ(x,t)|,|ρx(x,t)|~O(e-θx)|ux(,t)|,|ρ(x,t)|,|ρx(x,t)|~O(e-θx)asx⟶∞
uniformly in the time interval [0,T].Notation. Consider
(99)u(x,t)~O(e-θx),asx⟶∞,iflimx→∞|u(x)|e-θx=L,u(x,t)~o(e-θx),asx⟶∞,iflimx→∞|u(x)|e-θx=0.Proof.
First, we figure out the estimates on∥u(x,t)∥L∞, ∥ux(x,t)∥L∞, ∥ρ(x,t)∥L∞,  and ∥ρx(x,t)∥L∞. Then we use the weight function to obtain the desired result.
Step  1.  Estimate for ∥u(x,t)∥L∞,∥ρ(x,t)∥L∞.
Multiplying the first equation of (10) by u2n-1 with n∈ℤ and then integrating both sides with respect to x-variable, we can get
(100)∫ℝ‍u2n-1utdx+∫ℝ‍u2n-1uuxdx+∫ℝ‍u2n-12ωuxdx+∫ℝ‍u2n-1∂xp*(u2+α22ux2+12ρ2+ρ)dx=0.
The first term of the previous identity is
(101)∫ℝ‍u2n-1utdx=12nddt∥u(t)∥L2n2n=∥u(t)∥L2n2n-1ddt∥u(t)∥L2n,
and the estimate of the second term is
(102)|∫ℝ‍u2n-1uuxdx|≤∥ux(t)∥L∞∥u(t)∥L2n2n.
In view of Hölder’s inequality, we can obtain the following estimate for the third and the last terms in (100):
(103)∫ℝ‍2ωu2n-1uxdx≤2ω∥u(t)∥L2n2n-1∥ux(t)∥L2n,∫ℝ‍u2n-1∂xp*(u2+α22ux2+12ρ2+ρ)dx≤∥u(t)∥L2n2n-1∥∂xp*(u2+α22ux2+12ρ2+ρ)∥L2n.
Putting all the previous inequalities into (100) yields
(104)ddt∥u(t)∥L2n≤∥ux(t)∥L∞∥u(t)∥L2n+2ω∥ux(t)∥L2n+∥∂xp*(u2+α22ux2+12ρ2+ρ)∥L2n.
Thus, according to the Sobolev embedding theorem, there exists a constant
(105)M=supt∈[0,T]∥u(x,t)∥Hs,
such that applying Gronwall’s inequality gives us
(106)∥u(t)∥L2n≤(∥∂xp*(u2+α22ux2+12ρ2+ρ)∥L2n∥u(0)∥L2n+∫0t‍2ω∥ux(τ)∥L2n+∫0t‍∥∂xp*(u2+α22ux2+12ρ2+ρ)∥L2ndτ)eMt.
For any f∈L1(ℝ)∩L∞(ℝ), we know that
(107)limq→∞∥f∥Lq=∥f∥L∞.
Thus, taking the limits in (106), we get
(108)∥u(t)∥L∞≤(∥∂xp*(u2+α22ux2+12ρ2+ρ)∥L∞∥u(0)∥L∞+∫0t‍2ω∥ux(τ)∥L∞dτ+∫0t‍∥∂xp*(u2+α22ux2+12ρ2+ρ)∥L∞dτ)eMt.
Similarly, for the second equation of (10), we have
(109)∫ℝ‍ρ2n-1ρtdx+∫ℝ‍ρ2n-1ρxudx+∫ℝ‍ρ2n-1ρuxdx+∫ℝ‍ρ2n-1uxdx=0,
and  using the same method, one can get
(110)ddt∥ρ(t)∥L2n≤∥ux∥L∞∥ρ(t)∥L2n+∥u∥L∞∥ρx∥L2n+∥ux∥L2n.
Using the Gronwall’s inequality, one gets
(111)∥ρ(t)∥L2n≤(∥ρ(0)∥L2n+∫0t‍(∥u∥L∞∥ρx∥L2n+∥ux∥L2n)dτ)eMt.
Taking the limits,  one gets
(112)∥ρ(t)∥L∞≤(∥ρ(0)∥L∞+∫0t‍(∥u∥L∞∥ρx∥L∞+∥ux∥L∞)dτ)eMt.
Step  2. Estimate for ∥ux(x,t)∥L∞, ∥ρx(x,t)∥L∞.
We will establish an estimate on∥ux(x,t)∥L∞ using the same method. Differentiating the first equation (10) with respect to x-variable produces the following equation:
(113)uxt+uuxx+ux2+2ωuxx+∂x2p*(u2+α22ux2+12ρ2+ρ)=0.
Then, multiplying (113) by ux2n-1 with n∈ℤ+, integrating the result in the x-variable, and considering the second term in the previous identity with integration by parts, one can get
(114)∫ℝ‍uuxxux2n-1dx=∫ℝ‍u(ux2n2n)xdx=-12n∫ℝ‍uxux2ndx,
so we have
(115)∫ℝ‍ux2n-1uxtdx-12n∫ℝ‍uxux2ndx+∫ℝ‍ux2n-1ux2dx+∫ℝ‍ux2n-12ωuxxdx+∫ℝ‍ux2n-1∂x2p*(u2+α22ux2+12ρ2+ρ)dx=0.
Similarly, we can get the following inequality:
(116)ddt∥ux(t)∥L2n≤2∥ux(t)∥L∞∥ux(t)∥L2n+2ω∥uxx(t)∥L2n+∥∂x2p*(u2+α22ux2+12ρ2+ρ)∥L2n.
Using Gronwall’s inequality, we can similarly get
(117)∥ux(t)∥L2n≤e2Mt(∥∂x2p*(u2+α22ux2+12ρ2+ρ)∥L2n12∥ux(0)∥L2n+∫0t‍2ω∥uxx(τ)∥L2ndτ+∫0t‍∥∂x2p*(u2+α22ux2+12ρ2+ρ)∥L2ndτ).
Taking the limits, one gets
(118)∥ux(t)∥L∞≤e2Mt(∥∂x2p*(u2+α22ux2+12ρ2+ρ)∥L∞∥ux(0)∥L∞+∫0t‍2ω∥uxx(τ)∥L∞dτ+∫0t‍∥∂x2p*(u2+α22ux2+12ρ2+ρ)∥L∞dτ).
Differentiating the second equation of (10), one has
(119)ρtx+ρxxu+2ρxux+ρuxx+uxx=0,
and using the same method, we can obtain
(120)∫ℝ‍ρx2n-1ρtxdx+∫ℝ‍ρx2n-1ρxxudx+2∫ℝ‍ρx2n-1ρxuxdx+∫ℝ‍ρx2n-1ρuxxdx+∫ℝ‍ρx2n-1uxxdx=0.
In a similar way, we should deal with the second term as follows:
(121)|∫ℝ‍ρx2n-1ρxxudx|=|∫ℝ‍(ρx2n2n)xudx|=|12n∫ℝ‍ρx2nuxdx|,
thus we can get the estimates of ∥ρx(x,t)∥L∞ as follows:
(122)∥ρx(t)∥L∞≤e3Mt(∥ρx(0)∥L∞+∫0t‍(∥uxx∥L∞∥ρ∥L∞+∥uxx∥L∞)dτ).
In what follows, we use the weight function to get the desired result. We shall now introduce the weight functionφN(x) with N∈ℤ+, which is independent of t as follows:
(123)φN(x)={1,x≤0,eθx,x∈(0,N),eθN,x≥N.
Note that 0≤φN′(x)≤φN(x). Multiplying the first equation of (10) and (106) by φN, we have
(124)φNut+φNuux+2ωφNux+φN∂xp*(u2+α22ux2+12ρ2+ρ)=0,φNuxt+φNuuxx+φNux2+2ωφNuxx+φN∂x2p*(u2+α22ux2+12ρ2+ρ)=0.
In order to get the estimate of uxφN, we need to eliminate the second derivatives as follows:
(125)|∫ℝ‍φNuuxx(uxφN)2n-1dx|=|∫ℝ‍u(uxφN)2n-1((uxφN)x-uxφN′)dx|=|∫ℝ‍u((uxφN)2n2n)xdx-∫ℝ‍uuxφN′(uxφN)2n-1dx|≤2(∥u(t)∥L∞+∥ux(t)∥L∞)∥uxφN∥L2n2n,
where we use the fact 0≤φN′(x)≤φN(x). Hence, as in the weightless case, we get the following inequality in view of the estimates of ∥u(x,t)∥L∞ and ∥ux(x,t)∥L∞:
(126)∥u(t)φN∥L∞+∥ux(t)φN∥L∞≤e2Mt(∥u(0)φN∥L∞+∥ux(0)φN∥L∞)+e2Mt∫0t‍∥φN∂xp*(u2+α22ux2+12ρ2+ρ)∥L∞dτ+e2Mt∫0t‍∥φN∂x2p*(u2+α22ux2+12ρ2+ρ)∥L∞dτ.
Similarly, we get estimates for ρφN and ρxφN as follows:
(127)∥ρφN∥L∞+∥ρxφN∥L∞≤e2Mt(∥ρ(0)φN∥L∞+∥ρx(0)φN∥L∞).
On the other hand, a simple calculation shows that there exists C>0, depending only on θ∈(0,1) such that for any N∈ℤ+,
(128)φN(x)∫ℝ‍e-|(x-y)/α|1φN(y)dy≤C.
Therefore for any appropriate function g, one can get
(129)|φN∂xG(x)*g2(x)|=|12αφN(x)∫ℝ‍e-|(x-y)/α|g2(y)dy|=12αφN(x)∫ℝ‍e-|(x-y)/α|1φN(y)φN(y)g(y)g(y)dy≤12α(φN(x)∫ℝ‍e-|(x-y)/α|1φN(y)dy)∥gφN∥L∞∥g∥L∞≤C′∥gφN∥L∞∥g∥L∞,
and using the same method,
(130)|φN∂x2G(x)*g2(x)|≤C′∥gφN∥L∞∥g∥L∞.
Besides,
(131)|φN∂x2p*ρ|=|φNpx*ρx|≤|φN|∥px∥L2∥ρx∥L2≤C′∥φNρx∥L∞.
In a similar way, we have
(132)|φN∂xp*ρ|≤C′∥φNρ∥L∞.
Then we can get the following estimate:
(133)∥u(t)φN∥L∞+∥ux(t)φN∥L∞+∥ρ(t)φN∥L∞+∥ρx(t)φN∥L∞≤κ0(∥u(0)φN∥L∞+∥ux(0)φN∥L∞+∥ρ(0)φN∥L∞+∥ρx(0)φN∥L∞)+κ0(∫0t‍(∥u(t)φN∥L∞+∥ux(t)φN∥L∞+∥ρ(t)φN∥L∞+∥ρx(t)φN∥L∞)dt∫0t‍).
Using Gronwall’s inequality of integral form, one can get
(134)∥u(t)φN∥L∞+∥ux(t)φN∥L∞+∥ρ(t)φN∥L∞+∥ρx(t)φN∥L∞≤κ0(∥u(0)φN∥L∞+∥ux(0)φN∥L∞+∥ρ(0)φN∥L∞+∥ρx(0)φN∥L∞).
Taking the limit as N goes to infinity, we can obtain
(135)|u(x,t)eθx|+|ux(x,t)eθx|+|ρ(x,t)eθx|+|ρx(x,t)eθx|≤κ0(∥u(0)eθx∥+∥ux(0)eθx∥+∥ρ(0)eθx∥+∥ρx(0)eθx∥).
This completes the proof.



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*Source: 750315-2013-06-15.xml*