# Global Dissipative Solution for an Extended Dullin–Gottwald–Holm Equation

Authors: Yan Wang; Yongsheng Yan; Rui Wu
Journal: Journal of Mathematics
 (2022)

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## Abstract

The current paper devotes to the continuation of the solutions for the Dullin–Gottwald–Holm equation with forcing. Due to forcing, this equation loses the conservation of energy and cannot be regarded as the Hamiltonian system. Thus, through employing the characteristic method and exploiting the balance law, we prove the existence of a semigroup of a global dissipative solution, which is defined for each initial valueu0∈H1ℝ and depends continuity on it.


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## Body

## 1. Introduction

In this work, we shall discuss the following Dullin–Gottwald–Holm equation with a forcing.(1)ut−a2utxx+kux+3uux−a22uxuxx+uuxxx=−γuxxx+λu,where the constants a2a>0, γ/k are the squares of length scales, k=gh denotes the linear wave speed for the undisturbed water resting at spatial infinity, here g>0 stands for the downward constant acceleration of gravity, h>0 represents the mean fluid depth, and ut,x is water’s free surface over a flat bed. In addition, λu is the forcing term, where λ is a constant. Meanwhile, the differential operators are defined as ux=∂u/∂x, ut=∂u/∂t, utxx=∂3u/∂t∂x2, uxx=∂2u/∂x2, and uxxx=∂3u/∂x3. Particularly, when k=λ=γ=0 and a=1, equation (1) is the well-known completely integrable dispersive shallow water wave equation, noted as the Camassa–Holm (C-H) equation [1–6], that was firstly reckoned as a bi-Hamiltonian partial differential equation by Fokas and Fuchssteiner [7], which possesses infinitely many conservation laws. When λ=0, equation (1) is the Dullin–Gottwald–Holm (D-G-H) equation [8, 9] derived by asymptotic expansion in the shallow water regime, from the Hamiltonian for Euler’s equation and also shown to be bi-Hamiltonian. It still preserves integrability through inverse scattering transformation and combines the nonlinear dispersion of the Camassa–Holm equation with linear dispersion of the Korteweg–deVries equation [10–13].There are numerous literature studies devoted to studying the properties of shallow water equations like equation (1). Recently, Tian [14, 15] considered the infinite propagation speed for a weakly dissipative modified two-component D-G-H system.(2)ut−α2uxxt+c0ux+3uux+βuxxx+λu−α2uxx=α22uxuxx+uuxxx+γρρ¯x,ρt+ρux+λρ=0,where λu−α2uxx and λρ are the dissipative terms, λ is the dissipative parameter, ρ=1−∂x2v, that is a modified version of the following D-G-H system:(3)ut−uxxt+c0ux+3uux+βuxxx−2uxuxxx+ρρx=0,ρt+ρux=0,which is obtained by Ivanov’s approach [16]. By taking the transformation(4)G∗fx=Λfx=∫ℝGx−ξfξdξ,f∈L2,where G=1/2αe−x/α and Λ=1−α2∂x2−1, system (2) is converted to a hyperbolic type quasilinear evolution equation system:(5)ut+λu+uux+kux=−∂xG∗2d0u+u2+α22ux2+γ2v2−γ2vx2,vt+uvx+λv=−G∗uxvxx+uxv,v0,x=v0x,u0,x=u0x,where symbol ∗ stands for the convolution, v=ρ¯−ρ¯0. Based on above system, on the one hand, the author derives the infinite propagation speed for the corresponding solution with compactly supported initial value, which does not stand in its whole lifespan. Specifically, we assume u as a nontrivial solution to system (2), that has compact support on au0,bu0 at the beginning, and T is the maximal existence time for u. Let ρ0 be also initially compactly supported on the interval aρ0,bρ0. Thus,(6)ut,x=12αE+te−x/α−d0,forx>q1t,b,12αE−tex/α−d0,forx<q1t,a,E+t=∫q1t,aq1t,beξ/αyt,s+d0dξ,E−t=∫q1t,aq1t,be−ξ/αyt,s+d0dξ,is a continuous nonvanishing function, with E+0=E−0=0, and bounded as(7)E+t≤cλ,u0xHα1,be−λt+k/αt,E−t≤cλ,u0xHα1,ae−λt−k/αt.Furthermore,E+ and E− are strictly increasing and decreasing, separately. On the other hand, for the same model, the author investigated the asymptotic behavior of the solution at infinity and illustrated some results of fast and slow decay for the strong solution in L∞ space. Regarding the fast decay, we assume u0,v0∈Hsℝ×Hs−1ℝ with s≥5/2 to be the initial value and T>0 be the maximal existence time of the strong solution u,v to system (3). If there is some θ∈0,1 conforming to(8)u0x,u0xx,v0x,v0xx∼Oe−θx,asx⟶∞,then ut,x, uxt,x∼Oe−θx, as x⟶∞, where the notation O denotes(9)fx∼Ogxasx⟶∞,iflimx⟶∞fxgx=L.For another exponential decay, we presumeu,v∈C0,T;Hsℝ×Hs−1ℝ to be the strong solution to system (2), for s≥5/2 and some T>0. If there is some θ∈1/2,1 tally with(10)u0x∼oe−x/α,u0xx,v0x,v0xx∼Oe−θ/αx,asx⟶∞,and there existst1∈0,T satisfying ut1,x∼oe−x/α, as x⟶∞, then u≡0. We should point out that oe−x/α can be replaced by Oe−x/α in the third fast decay, where the symbol o denotes(11)fx∼ogxasx⟶∞iflimx⟶∞fxgx=0.As for slow decay and other details, the readers can refer to Theorem 1.4 in [15]. When ρ=0, system (2) turns to the weakly dissipative D-G-H equation(12)ut−α2uxxt+c0ux+3uux+βuxxx+λu−α2uxx=α22uxuxx+uuxxx.Whenρ=λ=0, system (2) becomes the D-G-H equation(13)ut−α2uxxt+c0ux+3uux+βuxxx=α22uxuxx+uuxxx.Liu [17] established the local well posedness for the D-G-H equation by employing Kato’s semigroup approach [18–20] for the corresponding evolution equations that for given u0∈Hsℝ, s>3/2, there exists a maximal time T>0, and a unique solution in C0,T,Hsℝ∩C10,t,Hs−1ℝ depends continuity on initial value u0 and works out the exact rate of breaking waves for a kind of initial profiles. Furthermore, by the fiction of Constantin and Escner’s proof [21], Tian et al. in [22] investigated the global well posedness for the D-G-H equation in higher Sobolev space Hsℝs≥3, providing that the initial data u0 fulfils certain conditions. As for several special cases, Li and Olver [23] established the local well posedness in Hs for any s>3/2 of the C-H equation; more exactly, a sufficient condition for a global solution u to exist is that uxL∞ remains bounded, and they found that there is still a local solution in Hs in respect of the initial value u0, if u0x∈Hs−1 is essentially bounded when 1<s<3/2, by regularizing the equation and obtaining a solution as the limit of solutions to the regularized equations (also see [24]). It is worth mentioning that the above methods can also be employed to the C-H equation with periodic boundary conditions, i.e., solution u fulfils the condition ux,t=ux+2π,t and u0∈HsT for some s>3/2, where T is the unit circle. In addition, the researcher in [25] utilized Lagrangian coordinates to extend the previous results through lowering the common condition of the initial value for the D-G-H equation to the class of continuous differential periodic initial value, i.e., for any u0∈C1ℝ, there is some Tu0>0 and a unique solution u∈C0,T;C1ℝ∩C10,T;Cℝ. Such method is also succeeded in proving the periodic the C-H equation that lives up to the least action principle [26, 27] and giving the threshold for global existence and blowup solutions [28–30]. Regarding the conservative solution of the C-H equation [1, 31, 32], Bressan and Constantin transferred the equation to the ODE system at first, in terms of a set of variables. Thus, the solution of this ODE can be derived as the fixed point of a contractive transformation. Back to the original variables, they obtained a semigroup of conservative solutions, depending continuity on the initial value. As we remark here, this conservative solution implies that the total energy of solution equals to a constant and the essential procedure during the proof is the conservation of energy i.e., Et=k (k denotes some constant). Similar approach for the dissipative C-H equation is referred to [33].However, because of the existence of the forcing termλu, the mechanism of equation (1) changes remarkably. For instance, equation (1) does not preserve integrability, and ut,x=ce−x−ct is not the peak on the solution of equation (1) anymore. The most critical problem is equation (1) cannot be considered as a solvable Hamiltonian system. Inspired by the above discussion, we are str by an interesting question: whether there exists a weak solution?, how to establish the well posedness of dissipative solutions?, and how about the existence of the global dissipative solutions?For that propose, we employ the new characteristic method [32, 34, 35] to derive the global dissipative solutions for equation (1). Through introducing a set of variables, Cauchy problem (1) is converted to a semilinear system at first. Next, we demonstrate the existence and uniqueness of global weak solutions for this semilinear system. Returning back to the original coordinates, we derive a local dissipative solution of equation (1). This local solution can be extended to all time because of the uniform bound on H1 norm. A continuous semigroup in H1 is also constructed.Since we are going to investigate the well posedness of equation (1), it is worth mentioning the following works, which are closely related to our work, to enhance the readability and comprehension of the present paper. Lately, Zhang et al. [36] investigated the Cauchy problem for a weakly dissipative D-G-H equation(14)ut−utxx+k1−∂x2ux+3uux+λ1−∂x2u=0,x∈R,t>0,ux,0=u0x,x∈R,where λ,k≥0. Firstly, the local well posedness was established by Kato’s semigroup theory. Then, the precise blowup scenario was obtained for the strong solution of equation (14). Lately, the authors discussed the blowup rate of the wave-breaking solution, which complement the result of [37]. Except D-G-H equation, Zhang [38, 39] also studied the Cauchy problem for the N−abc family of the C-H type equation with dispersion and dissipation(15)ut−utxx−cuNuxxx−buN−1uxuxx+auNux+k1−∂x2ux+λ1−∂x2u=0,x∈R,t>0,ux,0=u0x,x∈R,where a,b,c are the positive constants and satisfy a=b+c; λ,k≥0 are the dispersion and dissipation coefficients, separately; N∈Z+,N≥1. On the one hand, the authors [38] established the global well posedness of the strong solution based on certain condition of the initial datum. Then, the propagation speed with compact supported initial datum was investigated, which improves early results [40–42]. On the other hand, the authors [39] constructed blowup threshold for a positive solution in finite time under certain condition of initial datum, that complements early works [36, 37].Now, we introduce the structure of equation (1). For the integrity, we suppose γ=0 in the following discussion. Let qx=1/2ae−x/a, where x∈ℝ. After some careful analysis, we obtain several important characters of qx.(i)
ℱq=1/1+a2ξ2(ii)
ℱq−a2qxx=1 indicates q−a2qxx=δx, where δx is the Dirac function(iii)
q∗u−a2uxx=u, ℱ1−a2∂x2u=ℱu1+a2ξ2 where ℱ,∗ represents Fourier transformation and convolution in ℝ, respectively. Thus, equation (1) can be converted to(16)ut+uux+Px+kQx−λQ=0,where the operator P,Q is defined as(17)P=∫ℝqx−yu2+a22ux2ydy,Q=∫ℝqx−yuydy.The initial data is given as(18)u0,x=u0x,x∈ℝ.We assume thatu is the smooth solution of equation (16). Taking the derivative of equation (16) with respect to x, we obtain(19)uxt+uuxx+12ux2+2uxa2−u2+P+kQ+u−a2λQx=0.Multiplying both sides of equation (16) with 2ux, it can be deduced as(20)ux2t+uux2x+2uxa2−u2+P+kQ−u−a2λQx=0.Furthermore, because the third term of (20) merely includes the first-order derivative of u, it is so called the balance law, in spite of some nonlinear operators covered. In this paper, on the one hand, the first difficulty we meet is the nonlocal singular operator P and Q. On the other hand, the existence of the forcing makes equation (1) lose the conservation of energy, i.e.,(21)Et=∫ℝu2t,x+a2ux2t,xdx≠constant.We remark here the methods for the existence and uniqueness of the weak solution that Bressan and Constantin [32, 33] used, for the C-H equation, severely depending on the conservation of energy. However, the forcing term in equation (1) makes the above approach useless. Therefore, under the same manners of [43], we study the weak solution by employing the balance law (20) instead of the conservation of energy such that the methods in [32, 33] can be extended to equation (1). In this connection, we give the following lemma:Lemma 1 (see [43]).
Supposeu0∈H1ℝ is an absolutely continuous function of x. Then, the Cauchy problem (16)–(20) has a global weak solution ut,x and satisfies following identity:(22)∫Γ−uxθt−uuxθx+θa2−a2ux22−u2+P+kQ−u−a2λQxdxdt+∫ℝau0xθ0,xdx=0,for every θ∈Cc1Γ, where Γ=t,x|t∈ℝ,x∈ℝ. Additionally, the weak solution possesses the following properties:(i)
ut,x is 1/2-Hölder continuous on x and t, wheret belongs to any bounded interval(ii)
The mappingt⟶ut,⋅ is Lipchitz continuous for every fixed t∈ℝ, under L2 norm(iii)
There is a family of Radon measureμt,t∈ℝ, depending continuity on time and in respect to the weak convergence of measure in the sense of topology. For every t∈ℝ, the absolutely continuous part of μt has density ux2t,⋅ in terms of the Lebesgue measure. This provides a measure-valued solution to the balance law(23)∫Γux2θt+uux2θx+2uxa2−u2+P+kQ−u−a2λQxθdxdt−∫ℝa2u0x2θ0,xdx=0.(iv)
We consider a sequence ofu0,n such that u0,n−u0H1⟶0 when n⟶+∞. Hence, the corresponding solution unt,x converges to ut,x uniformly in terms of x and t in any bounded domain.As a result of Lemma1, we prove the existence of global dissipative solutions under a unique semigroup. What we prove here is(i)
⋅ The unique semigroup of solutions obtained by coordinate transformation belongs to entire H1R(ii)
⋅ The whole of our solutions meet the Oleinik type inequality(24)uxt,x≤C1+t−1,t>0,with a constant C depending only on u0H1For the record, a solution of the Cauchy problem (16)–(20) is described as dissipative if it follows the Oleinik type inequality (24) for some constant C, and its energy Et is nonincreasing of time.


## 2. An Associated Semilinear System

We now use the characteristic method to derive a semilinear system of (16) in the light of the smooth function. This approach was first utilized in [32, 33] to establish the global conservative solutions and global dissipative solutions for the C-H equation. Next, we establish the existence of the global weak solutions to this semilinear system. Eventually, we return to the dissipative problem by using the inverse transformation. Precisely speaking, for an initial data u0∈H1ℝ, we consider a new coordinate T∗,ξ as follows:(25)t,x⟶T∗,ξ,whereT∗=t,ξ=∫0x0ξ1+a2u0x2dx.The characteristic equation is(26)ddtxt,ξ=ut,xt,ξ,x0,ξ=x0ξ.For eacht,x∈ℝ×ℝ, ξ=ξt,x is the characteristic coordinate and fulfills ξt+uξx=0. We assume that Ϝ=ϜT∗,ξt,x is a smooth function; a straightforward computation shows that the following identities stand under the new coordinate T∗,ξ(27)∂Ϝ∂t=Ϝt+uϜx,∂Ϝ∂x=Ϝξξx.Taking the transformation(28)v=2arctanaux,w=1+a2ux2ξx.Thus, one has(29)a2ux21+a2ux2=sin2v2,aux1+a2ux2=sinv2,11+a2ux2=cos2v2,(30)∂x∂ξ=w1+a2ux2=wcos2v2.Next, we give the expression of the singular operatorP, Px and Q, Qx under the new coordinate. It comes from (30) that(31)xt,ξ−xt,ξ′=∫ξ′ξwt,scos2vt,s2ds.Takingx=xT∗,ξ′, it follows from (29)–(31) that(32)Pt,ξ=12a∫−∞+∞e−1/a∫ξξ′wt,scos2vt,s/2dsu2cos2v2+12sin2v2wξ′dξ′,Pxt,ξ=12a2∫ξ∞−∫ξ−∞e−1/a∫ξξ′wt,scos2vt,s/2dsu2cos2v2+12sin2v2wξ′dξ′,Qt,ξ=12a∫−∞+∞e−1/a∫ξξ′wt,scos2vt,s/2dsucos2v2wξ′dξ′,Qxt,ξ=12a2∫∞ξ−∫−∞ξe−1/a∫ξξ′wt,scos2vt,s/2dsucos2v2wξ′dξ′.Then, we investigate equation (16) under the new coordinate T∗,ξ. It is obvious that(33)∂u∂t=ut+uux=−Px−kQx+λQ.Becauseξtx+uξxx=−uxξx, (20) indicates(34)∂w∂t=−P−kQ−u+u2+a2λQx+12wasinv.Furthermore, it can be deduced from (19), (26), and (28) that(35)∂v∂t=2aux−Px−kQx+λQx1+a2ux2=1a2cos2v2−P−kQ−u+u2+a2λQx−sin2v2.In conclusion, (16) turns to(36)∂u∂t=ut+uux=−Px−kQx+λQ,∂v∂t=1a2cos2v2−P−kQ−u+u2+a2λQx−sin2v2,∂w∂t=−P−kQ−u+u2+a2λQx+12wasinv.Meanwhile, we give the initial data foru,v,w under the new coordinate T∗,ξ(37)u0,ξ=u0x0ξ,v0,ξ=2arctanau0xx0ξ,w0,ξ=1.The system (36) can be viewed as an ODE in the Banach space(38)Ω=H1ℝ×L2ℝ∩L∞ℝ×L∞ℝ,with the norm(39)u,v,wΩ=uH1+vL2+vL∞+wL∞.For the sake of deriving the global dissipative solutions, we have to modify (36). Suppose that the first wave breaking happens at t=τξ along with a fixed characteristic t↦xt,ξ. Apparently, ux⟶−∞ and v=2arctanaux as t⟶τξ. For every t≥τξ, we assume v≡−π and remove the value of u,v,w from P,Px,Q,Qx. More precisely, we redefine P,Px,Q,Qx as(40)P=12a∫vξ′>−πe−1/a∫s∈ξ,ξ′,vs>−πwt,scos2vt,s/2dsu2cos2v2+12sin2v2wξ′dξ′,(41)Px=12a2∫ξ′>ξ,vξ′>−πe−1/a∫s∈ξ,ξ′,vs>−πwt,scos2vt,s/2dsu2cos2v2+12sin2v2wξ′dξ′=−∫ξ′<ξ,vξ′>−πe−1/a∫s∈ξ,ξ′,vs>−πwt,scos2vt,s/2dsu2cos2v2+12sin2v2wξ′dξ′,(42)Qt,ξ=12a∫vξ′>−πe−1/a∫s∈ξ,ξ′,vs>−πwt,scos2vt,s/2dsucos2v2wξ′dξ′,(43)Qxt,ξ=12a2∫ξ′>ξ,vs>−πe−1/a∫s∈ξ,ξ′,vs>−πwt,scos2vt,s/2dsucos2v2wξ′dξ′=−∫ξ′<ξ,vs>−πe−1/a∫s∈ξ,ξ′,vs>−πwt,scos2vt,s/2dsucos2v2wξ′dξ′.The ODE system (36) comes to(44)∂u∂t=ut+uux=−Px−kQx+λQ,∂v∂t=2cos2v2−P−kQ−u+u2+a2λQx−sin2v21a,ifv>−π,0,ifv≤−π,∂w∂t=sinv−P−kQ−u+u2+a2λQx+12wa,ifv>−π,0,ifv≤−π.It is worth noting that equation (1) has been reduced to an ODE in the Banach space when the right side is not continuous, i.e., discontinuity happens when v=−π. However, according to the second equation of the above ODE, we are aware that vt≈−1/a as v nears −π. It is this transversality condition that ensures the well posedness of above system.


## 3. Local Solutions of the Semilinear System

In this part, we illustrate the existence and uniqueness of the local solutions for the ODE system (44) by means of the contracting mapping theory. For this purpose, the Cauchy problem (44) should be rewritten to a more compact form(45)∂U∂t=FUt,ξ+Gξ,Ut,⋅,ξ∈ℝ,(46)U0,ξ=Uξ,where U=u,v,w∈ℝ3, and(47)FU=0,1a2cos2v2ku+u2−sin2v2,sinvku+u2+12wa,ifv>−π,0,−1a,0,ifv≤−π,(48)Gξ,U⋅=ℸ,2aℸcos2v2,ℸwasinv,ifv>−π,ℸ,0,0,ifv≤−π.Here,ℸ=−P−kQ+λa2Qx.However, there is a slight difference between (45)–(48) and (44). For the same evolution of these two systems, we define F=0,0,0 when v≤−π. This adjustment makes sure that F is continuous across the data v=−π. Hence, as long as a solution of (45)–(48) is derived, the following mapping(49)t,ξ↦ut,ξ,maxvt,ξ,−π,wt,ξrender a solution of (44), considering (45) as an ODE in L∞ℝ,ℝ3. We should be aware of that field F:ℝ3⟶ℝ3 is Lipschitz continuous and uniformly bounded whenever u stays in a bounded space. In addition, the nonlocal term G is discontinuous because the set(50)ξ;τξ>t=ξ;vt,ξ>−πmight shrink unexpectedly at a time τ∗ so that(51)measξ;vτ∗,ξ>−π>0.The main focus of this part is to demonstrate the following theorem:Theorem 1.
Letu0∈H1ℝ with the corresponding initial data U0=u0,2aarctanu0x,1 defined at (37); then, the Cauchy problem in (45) and (46) has a unique local solution on 0,T.Proof.
We divide our proof into five steps as follows:(1)
We give some priori estimates ofF and G. For every ξ, supposing U=u,v,w∈L∞ℝ;ℝ3 has the following properties:(52)uL∞≤C,1C≤wξ≤C,(53)measξ;vt,ξ>−π,vξ≥π2≤C,whereC is some constant. Furthermore, there exists a constant κ∗=1/2a+1/a22C2+4auL∞+1/8vL∞ such that(54)PL1+PxL1≤κ∗1+uL1+vL1,PL∞+PxL∞≤κ∗,QL∞+QxL∞≤κ∗,QL1+QxL1≤κ∗uL1,GUL∞≤κ∗,FUL∞≤κ∗.Furthermore, there is a Lipchitz continuous constantκ such that if U^=u^,v^,w^ follows (52) and (53), then(55)FU−FU^L∞≤κU−U^L∞,GU−GU^L∞≤κU−U^L∞+measξ;vt,ξ>−π,v^t,ξ≤−π+measξ;v^t,ξ>−π,vt,ξ≤−π.(2)
We assumeu0∈H1ℝ. Since u0,u0x∈L2, for each ε>0, the set x∈ℝ;u0x≥ε has the finite measure. Hence, there exists a constant C>0 such that(56)u0L∞≤C2,measξ;v0≥π4≤C2.(3)
We suppose that (52) and (53) stand. Based on the first step, we can select one δ>0 such that vξ∈−π,δ−π which gives(57)∂∂tvt,ξ=1a2cos2v2−P−kQ−u+u2+a2λQx−sin2v2≤−12a.SettingΩδ=ξ∈ℝ;v0ξ∈−π,δ−π, if δ is small enough as supposed, we have(58)measΩδ≤18κ.(4)
For a givenT>0, we define ℧ as the domain of all continuous mapping t↦Ut=ut,vt,wt on 0,T↦L∞ℝ,ℝ3, and the Picard operator ν:℧↦℧ is given as(59)νU=U0+∫0tGξ,Uτ,⋅+FUτ,ξdτ.Then, we show that on a suitable domain℧∈ℏ0,T,L∞, there is a solution t↦Ut that is derived as a unique fixed point of a contractive transformation ν:℧↦℧. More precisely, for the fixed time T>0, we have(60)U0=U0,Ut−UsL∞≤2κ∗t−s,vt−vs≤s−t2,ξ∈Ωδ,0≤s<t≤T.(5)
AssumingT>0 is small as supposed, it is obvious that ν maps ℧ into ℧. Now, we give the proof that ν is a strict contraction. For this purpose, we suppose U,U^∈℧ and set(61)η=maxt∈0,TUt−U^tL∞,τ=supt∈0,T;vt,ξ>−π.
We notice that for everyξ∈Ωδ, (56) implies(62)τξ−τ^ξ≤2η,where τ^ξ is defined in the same way. For every t∈0,T, we derive(63)νUt−νU^tL∞≤∫0tGU−GU^L∞dτ+∫0tFU−FU^L∞dτ≤2κ∫0tUτ−U^τL∞dτκ∫0tmeasξ;vτ,ξ>−π,v^τ,ξ≤−πdτ+κ∫0tmeasξ;v^τ,ξ>−π,vτ,ξ≤−πdτ≤κ∫Ωδ2τ^ξ−τtdξ+2κTη≤2κ⋅measΩδη+2κTη≤η2.
This completes the proof thatν is a strict contraction. Then, there exists a unique fixed point which gives the local solution of the Cauchy problem (45) and (46).


## 4. Global Solutions

In this part, we illustrate that the local solution of (44) can be extended globally. The key point is the global bound on the energy(64)Et=∫vt,ξ>−πu2t,ξcos2vt,ξ2+sin2vt,ξ2wt,ξdξ.Firstly, we show that(65)uξ=w2asinv=wasinv2cosv2.On the one hand, the definition ofP,Px,Q,Qx and the first equation of (44) illustrate(66)uξt=−Pxx−kQxxξx=wa2λQx−P−kQ−u+u2cos2v2+12sin2v2,ifvξ>−π,0,ifvξ≤−π.On the other hand, based on the last two equations of (44), when v>−π, we have(67)w2asinvt=12awtsinv+wvtcosv=12aa2λQx−P−kQ−u+u2+12sin2v−cosvsin2v2+cosv1+cosva2λQx−P−kQ−u+u2=wa2λQx−P−kQ−u+u2cos2v2+12sin2v2.Ifv=−π, we have(68)12awsinvt=0.In addition, whent=0, by (29), (30), and (37), we know(69)uξ=u0x1+a2u0x2=12asinv=w2asinv,because w0,ξ=1. Thus, the identity (65) holds at t=0. Identities (66)–(68) imply that for each ξ, we have(70)uξ−12awsinvt=0.Based on the above discussions, we know (65) stands for all t.Secondly, we estimate the extended energy(71)E˜t=∫ℝu2cos2v2+sin2v2wdξ,under the new coordinate. For convenience, we denote ℶ=ku−Q−P+a2λQx+u2+1/2. From (44) we deduce(72)dE˜tdt=∫ℝu2cos2v2+sin2v2wt+w2uutcos2v2+1−u2vtsinv2cosv2dξ=∫vξ>−πu2cos2v2+sin2v2ℶasinv+1−u22ℶcos2v2−1sinv2a+2uutcos2v2wdξ=∫ℝℶasinv−1−u22asinv+2uutcos2v2wdξ.We notice that we have integrated over the entire real line for (72). This action does not generate any difference due to the fact that cosv/2=0 when v=−π. Then, from (40)–(43), we conclude(73)Pξ=wPxcos2v2,Qξ=wQxcos2v2.Together with (65), we have(74)2u−P−kQ+a2λQxξ=−P−kQ+a2λQxwasinv+2−Px−kQx+λQuwcos2v2−2λwu2cos2v2.Moreover, we are aware that(75)u3+ku2ξ=3u2+2kuuξ.Using the above two formulas and noticing thatlimξ⟶+∞uξ=0, then (72) can be converted into(76)∫ℝℶwasinv−1−u2w2asinv+2wuutcos2v2dξ=2λ∫ℝu−P−kQ+a2Qxξdξ+∫ℝ3u2+2kuuξdξ+2∫ℝwλu2cos2v2dξ=2λ∫ℝwu2cos2v2dξ.With the assistance of theGronwall inequality, we claim that there is a constant S0:=S0E0,t>0 that gives a prior estimate for the extended energy E˜t on bounded interval of time(77)E˜t≤S02.Thirdly, we estimatewtL∞,1/wtL∞,utL∞,vtL∞. Together (65) with (77), we have(78)supξ∈ℝu2t,ξ≤∫ℝu2ξdξ=∫ℝ1ausinv2cosv2wdξ≤12a∫ℝu2cos2v2+sin2v2wdξ.This yields a uniform bound onutL∞,(79)utL∞≤S0,t∈ℝ.Recalling the definition ofP, Px, Q and Qx, based on the estimate (77) and (79), we claim(80)PtL∞≤12au2wcos2v2+12wsin2v2L1≤12aS02,PxtL∞≤12a2u2wcos2v2+12wsin2v2L1≤12a2S02,(81)QtL∞≤12ae−1/a∫ξξ′cos2v/2wdswL1uL∞≤12aS0,QxtL∞≤12a2e−1/a∫ξξ′cos2v/2wdswL1uL∞≤12a2S0.Substituting (79)–(81) into the last equation of (44), we deduce(82)wt≤12+u2−P−kQ−u+a2λQxL∞wa≤2a+12aS02+2a+12ak+λ2S0+12wa.Thus, there is a constantS1 such that(83)1S1≤wt≤S1,where S1=et/a2a+1/2aS02+2a+1/2ak+λ/2S0+1/2. From the second equation of (44), we have(84)−π<vt,ξ<π.Considering the estimate (77) and the bound on wt, for each η>0, it is clear that(85)ddtvtL∞≤2a2a+12aS02+λ2+2a+12akS0,t∈0,T.Hence, there is a constantS2=2/a2a+1/2aS02+λ/2+2a+1/2akS0 such that(86)vtL∞≤eS2t.Finally, according to the above estimates, for eachη>0, there is a constant Sη depending only on S0, S1, and S2 such that(87)measξ;vξ≥η≤Sη,t∈0,T.On the basis of (78)–(87), we need to construct a local solution with a constant C to satisfy the priori bounds (52) and (53). This C is uniformly valid over every time interval 0,T. Consequently, the local solution can be globally extended to all t>0.


## 5. Continuous Dependence

The local existence theorem in Section3 was derived by denoting the solution of (45)–(47) as the fixed point of a contradiction in L∞. Thus, the continuous dependence and the uniqueness are related to the convergence of the initial value in L∞. The main focus in this part is to obtain the continuous dependence of solution for equation (16), as the initial value converges in H1 norm. Indeed, if u0,n⟶u0 in H1ℝ, assuming v=2arctanaux, we have(88)u0,n−u0L∞⟶0,v0,n−v0L2⟶0,whereas v0,n−v0L∞≠0. In this section, using the weak assumption (88), we accomplish a conclusion on the continuous dependence of solutions in (44).Theorem 2.
We consider a sequence of initial valueu0,n such that u0,n converges to u0 in H1ℝ. Let un,u be the corresponding solutions of (44) with initial value (37). For every T>0,(89)unt,ξ⟶ut,ξ,stands uniformly for t,ξ∈0,T×ℝ.Proof.
We divide our proof into seven steps as follows:(1)
Letu,v,w and u˜,v˜,w˜ be the two solutions of (44), corresponding to the initial value (37). We denote E0 as the upper bound of the energy for these two solutions. If t=0, we have(90)v0,ξ−v˜0,ξL2≤δ0,u0−u˜0L∞≤δ0.Now, we set up a priori bound(91)ut−u˜tL∞,fort∈0,T,depending only onT,E0,δ0.(2)
We introduce a set(92)Λ=ξ∈ℝ;v˜T,ξ=−π∪ξ∈ℝ;vT,ξ=−π,andα∗=measΛ is a number that depending only on E0,T.For everyξ∈Λ, let τξ be the first time when one of these solutions achieves −π; more precisely,(93)τξ=inft∈0,T;minv˜t,ξ,vt,ξ=−π.In terms of the measurable mapping,ξ↦τξ, we could constitute a measurable measure-preserving map α↦ξα in 0,α∗↦Λ, which has the following character:(94)α≤α′,if and only ifτξα≥τξα′.The inverse mapΛ↦0,α∗ (denoted as ξ↦αξ) is measure preserving too.(3)
In order to fabricate the map0,α∗↦Λ, we introduce the distance function(95)Ju˜,v˜,w˜,u,v,w=v−v˜L2+u−u˜L∞+w−w˜L2+K0∫0α∗eKαv˜ξα−vξαdα=J∗+K0Jϑ.To be brief, we define(96)Jt=Ju˜,v˜,w˜,u,v,w=J⋆+K0Jϑ.We deduce that ifK,K0,M is large as supposed (relating only to T,E0), we have(97)ddtJt≤MJt.Hence,(98)Jt≤eMtJ0.The above formulas give a priori estimate for (91).(4)
Next, we prove that (97) stands for certain constants M,K,K0. Regarding any t∈0,T, we define(99)Γt=ξ∈Λ:vt,ξ=−π,v˜t,ξ>−π∪ξ∈Λ:vt,ξ>−π,v˜t,ξ=−π,Γ+t=ξ∈Λ:v˜t,ξ=vt,ξ=−π,Γ−t=ξ∈Λ:v˜t,ξ>−π,vt,ξ>−π=ξ∈Λ:τξ>t.We should be aware that there is no intersection in these three sets such that(100)Γ+t∪Γ−t∪Γt=Λ,∀t∈0,T.In addition,ifmt=measΓ−t is set, based on (94), we have(101)Γ−t=ξα;α∈0,mt.(5)
According to the proof of Theorem1 and (44), we deduce that(102)ddtu−u˜L∞+w−w˜L2+v−v˜L2≤κu−u˜L∞+w−w˜L2+v−v˜L2+measΓt.From (101), a straightforward computation shows that(103)ddt∫0α∗eKαv˜t,ξα−vt,ξαdα=∫Γ+t∪Γ−t∪ΓteKαξ∂∂tv˜t,ξα−vt,ξαdξ=∫ΓteKαξ∂∂tv˜t,ξα−vt,ξαdξ+∫0mteKα∂∂tv˜t,ξα−vt,ξαdα,where the integral overΓ+t is zero.On the one hand, when we choose a constantδ>0 to be sufficiently small (depending only on T,E0), for ξ∈Γt,(104)v˜t,ξ−vt,ξ≤δ,means∂∂tv˜t,ξ−vt,ξ≤−12a.On the other hand, when we choose a constantκ to be sufficiently large, we derive(105)v˜t,ξ−vt,ξ≥δ,means∂∂tv˜t,ξ−vt,ξ≤−12a+κv˜t,ξ−vt,ξ.Lastly, forξ∈Γ−t, we derive the following inequality:(106)∂∂tv˜t,ξ−vt,ξ≤κu−u˜L∞+w−w˜L2+v−v˜L2+v˜t,ξ−vt,ξ+measΓt,which gives(107)∫0mteKα∂∂tv˜ξα−vξαdα≤κmeasΓt+J∗t∫0mteKαdα+κ∫Γ−teKαξv˜t,ξ−vt,ξdξ.(6)
Based on the above discussion, we rewrite (102) as(108)ddtJ∗≤κmeasΓt+J∗t,ξ∈Γt,meansαξ≥mt.According to (103)–(107), we have(109)ddtJϑt≤−12a∫ΓteKαξdξ+∫Γt−∪ΓteKαξvξα−v˜ξαdξ+κmeasΓt+J∗t∫0mteKαdα≤κJϑt−12a⋅measΓt⋅eKmt+κJ∗t∫0α∗eKαdα+κ⋅measΓt⋅eKmt∫0mteKα−mtdα≤κJϑt−14a⋅measΓt⋅eKmt+κKJ∗teKα∗t,whereK=4aκ such that(110)κ∫0mteKα−mtdα≤κK=14a.(7)
In connection with the distance functional in (95) and (96), we choose K0=4aκ. Using (108) and (109), we derive(111)ddtJ⋆+4aκJϑ≤κmeasΓt+J∗t+4a−14aeKmt⋅measΓtκKJ∗teKα∗t+κJϑt≤κJ∗t+4aκ2Jϑt+κe4aκα∗tJ∗t.
This completes the proof of (97), with M=4aκ2+κe4aκα∗J∗.


## 6. Global Dissipative Solutions

Focusing on the original variablet,x, we indicate that the global weak solutions of (44) yield a global dissipative solution to equation (16). Regarding any initial data u0∈H1ℝ, let u,v,w be the corresponding solution of (44) with initial value (37). We set(112)Yt,ξ=Y0ξ+∫0tuτ,ξdτ.For everyξ, t↦Yt,ξ provides a solution to the Cauchy problem(113)∂∂tYt,ξ=ut,ξ,Y0,ξ=Y0ξ.We show that one solution of (16) can be derived by assuming(114)ut,x=ut,ξ,ifYt,ξ=x.Theorem 3.
Letu,v,w be a global solution to the Cauchy problem (37) and (44). Then, the function u=ut,x defined by (113) and (114) provides a solution to the Cauchy problem (16)–(18) for the D-G-H equation with a forcing.Proof.
We divide our proof into five steps as follows:(1)
(79) and (112) imply that(115)ut,ξ≤S0,Y0ξ−S0t≤Yt,ξ≤Y0ξ+S0t.For eacht, we have(116)limξ⟶±∞Y0ξ=±∞.Hence, the image of the mapt,ξ↦t,Yt,ξ covers 0,∞×ℝ.(2)
We begin with proving thatut,x is well defined. For simplicity, we collect the relations between variables t,ξ and t,x(117)∂∂ξYt,ξ=wt,ξcos2v2,∂u∂ξt,ξ=12awsinv,(118)uxt,x=1atanv2ifYt,ξ=x,cos2v2≠−1.Moreover, presumingξ1<ξ2 and Yt,ξ2=Yt,ξ1, for ξ∈ξ1,ξ2, we derive(119)∫ξ1ξ2wt,scos2vt,s2ds=∫ξ1ξ2Yξt,sds=0.Thus,cosvt,ξ/2≡0 through the interval of integration, and (117) yields(120)ut,ξ1−ut,ξ2=12a∫ξ1ξ2wt,ssinvt,s2cosvt,s2ds=0.Thus,u=ut,x in (114) is well defined for t,x∈0,+∞×ℝ.(3)
Now, we illustrate the regularity of the energy equation andut,x. For each t, the image of the singular set, where v=−π, has measure zero. Actually,(121)measYt,ξ;vt,ξ=−π=∫vt,ξ=−πYξt,ξdξ=∫vt,ξ=−πwt,ξcos2vt,ξ2dξ=0.Next, using (118), we compute(122)∫ℝu2t,x+a2ux2t,xdx=∫vt,ξ>−πu2cos2v2+sin2v2wdξ≤S02.Firstly, by theSobolev inequality uC0,1/2≤CuH1 and (77), we have uC0,1/2≤CuH1≤CS0. This implies that u is uniformly Hölder continuous with exponent 1/2 of ξ. Secondly, based on (44) and (79)–(81), it follows that utL∞≤CPxL∞+kQxL∞+λQL∞≤CS0. Indeed, the map t↦ut,Yt is uniformly Lipschitz continuous along each characteristic curve t↦Yt. Consequently, ut,x is Hölder continuous in every bounded domain of t.(4)
Next, we proveutL2 is Lipchitz continuous in any bounded domain of t. Regarding every interval τ,τ+h, for a fixed point x, we choose ξ∈ℝ such that the characteristic curve t↦Yt passes through τ,x. Because u satisfies uL∞≤CuH1≤CS0, we have(123)uτ,x−uτ+h,x≤uτ+h,Yτ+h,ξ−uτ,x+uτ+h,x−uτ+h,Yτ+h,ξ≤supY−x≤CS0huτ+h,x−ut+h,Y+∫ττ+hPx+kQx+λQdt.Integrating over the real line, we derive(124)∫ℝuτ,x−uτ+h,x2dx≤C∫ℝ∫x−CS0hx+CS0huxτ+h,YdY2dx+C∫ℝ∫ττ+hPx+kQx+λQdt2wdξ≤C∫ℝ2S0h∫x−CS0hx+CS0huxτ+h,Y2dYdx+C∫ℝh∫ττ+hPx2+kQx2+λQ2dtwτL∞dξ≤4CS02h2uxτL22+Ch2wτL∞PxL22+kQxL22+λQL22≤CS0h2.This indicates thatt↦ut is Lipchitz continuous.(5)
In order to estimateP, Px, Q, Qx, we need the following lemma:Lemma 2.
Letf∈Lqℝ and q≥1. If wcos2vs/2L1≤S02, then(125)∫−∞+∞e−1/a∫ξξ′wscos2vs/2dsfξ′dξ′Lq≤jL1fLq,where jξ=min1,e1/2aS12S1S02−ξ, jL1=4S1S02+4aS1.Proof.
Indeed, whenπ/4≤vY/2≤3π/4, it follows 2sin2vξ/2≥1 such that(126)measξ∈ℝ|vξ2≥π4≤2∫ξ∈ℝ|w/wsin2vξ/2≥1/2sin2vξ2dξ≤2S1S02.
For anyξ1<ξ2, we have(127)∫ξ1ξ2cos2vξ2⋅wξdξ≥12S1∫ξ1ξ2dξ−∫ξ∈ξ1,ξ2|vξ/2≥π/4dξ≥12S1ξ2−ξ1−2S1S02.
Settingjξ=min1,e1/2aS12S1S02−ξ, for each q≥1, we have(128)∫−∞+∞e−1/a∫ξξ′wscos2vs/2dsfξ′dξ′≤j∗fξ.
Here,∗ denotes the convolution. Based on the above discussions and Young's inequality, we know that (125) stands. The computation of jL1 is given as(129)jL1=∫−2S1S022S1S02dξ+∫2S1S02+∞e2S1S02−ξ/2aS1dξ+∫−∞−2S1S02e2S1S02+ξ/2aS1dξ=4S1S02+4aS1.
Then, using (125), we have the following estimates:(130)QL2≤S12ajL1uL2,QξL2≤S12a2jL1uL2,QxL2≤S12a2jL1uL2,QxξL2≤S1a3+S12a2jL1uL2,PL2≤S12ajL1uL∞uL2+vL28,PξL2≤S12a2jL1uL∞uL2+vL28,PxL2≤S1a2jL1uL∞uL2+vL28,PxξL2≤1a3S1+S12jL1uL∞uL2+vL28.
According to above estimates andSoblev's inequality uL∞≤CuH1, we know that P,Px,Q,Qx∈H1ℝ.(6)
Finally, we prove that the functionu generates a weak solution of (16). We set Γ=t,x|t∈ℝ,x∈ℝ. For every smooth function θ∈Cc1Γ, the first formula of (44) has the following weak form:(131)0=∫Γutξ+Pxξ+kQxξ+λQξθdξdt=∫Γutξθ+P−u2−a22ux2+kQ−u−λQxwθa2+a4ux2dξdt=∫Γ−uξθt+−u2cos2v2−12sin2v2+P+kQ−u−λQxcos2v2wθa2dξdt=∫Γ−uxθtwcos2v2+−u2cos2v2−12sin2v2+P+kQ−u−λQxcos2v2wθa2dξdt=∫Γ−uxθt+uθx+−u2−12ux2+P+kQ−u−λQxθa2dxdt.
This indicates that the weak form (22) holds.
We define the set of times(132)τ=t≥0|measξ∈ℝ;vt,ξ=−π>0.
Then, we show that measτ=0. For one thing, in terms of the second equation of (44), we have vt=−1/a cross cosv=−1. Hence, there exists δ>0 such that vt≤N<0 wherever 1+cosv<δ. For another thing, it is due to the fact that vtL2 is bounded in any bounded domain of t such that meas τ=0. If not, it is clear that ∫∫vt,ξ=−πvtdtdξ<0 which contradicts to the fact vt=0 a.e. on vt,ξ=−π, based on the absolute continuity of the map t⟶vt,ξ at each fixed ξ∈ℝ.
For completeness, we introduce a set of Radon measureμt,t∈ℝ, for the Lebesgue measurable set x∈A. We assume that the preimage of μt,t∈ℝ takes the form ξ∈GA, which gives(133)μtA=∫GAwsin2v2t,ξdξ.
Regarding eacht∈ℝ\N, the absolutely continuous part of μt with respect to the Lebesgue measure has the density ux2t,⋅ in view of (65). In terms of (36), for each test function θt,x∈Cc1Γ, we obtain(134)−∫ℝ+∫ℝθt+uθxdμtdt=−∫vt,ξ>−πθtwsin2v2dξdt=∫vt,ξ>−πθwsin2v2tdξdt=∫vt,ξ>−π2wθau2−P−kQ−u+a2λQxcosv2sinv2dξdt=∫vt,ξ>−π2θuxu2−P−kQ−u+a2λQxdxdt.
It is apparent that (23) holds because measτ=0. This completes the proof of Theorem 3.s


## 7. A Semigroup of Dissipative Solutions

For each initial valueu0∈H1ℝ, we denote the corresponding global solutions constituted in Theorem 3 as ut=Stu0 for equation (1). The purpose of this part is to prove the following theorem:Theorem 4.
The mapS:H1×0,∞↦H1 is a semigroup. Additionally, the following properties stand as follows:(i)
The total energy is nonincreasing of time(135)utH1ℝ≤ut′H1ℝ,whenever0≤t′≤t.(ii)
There exists a constant C, depending only onu0H1 such that(136)uxt,x≤C1+t−1,t>0,x∈ℝ.(iii)
For a sequence of initial datau0,n, u0,n−u0H1ℝ⟶0. Therefore, the corresponding solution unt,x converges to ut,x uniformly in every bounded domain of t,x.Proof.
We divide our proof into six steps as follows:(1)
To prove (135), for every ξ∈ℝ, where τξ is defined in (132), a straightforward computation shows that(137)utH1ℝ=∫vt,ξ>−πu2t,ξcos2vt,ξ2+sin2vt,ξ2wt,ξdξ≤S02−∫τξ≤tu2t,ξcos2vt,ξ2+sin2vt,ξ2wt,ξdξ≤S02−∫t′≤τξ≤t∪τξ≤t′wτξ,ξsin2vτξ,ξ2dξ≤S02−∫τξ≤t′wτξ,ξsin2vτξ,ξ2dξ=ut′H1ℝ.(2)
Now, we prove theOleinik estimate (136). According to the proof of Theorem 1, we pick out a constant δ>0 such that(138)v∈π−δ,πmeans∂∂tvt,ξ≤−12a.Thus,(139)vt,ξ<minπ−t2a,π−δ.Using (118) and choosing a suitable constant C, we know (136) is true.(3)
Letu0,n be a sequence of initial datum which converges to u0 in H1ℝ. With the assistance of Theorem 2, we know that unt,ξ⟶ut,ξ is uniform for ξ,t in the compact set. Let us restrain that our focus to the original coordinates t and x has(140)Ynt,ξ⟶Yt,ξ,unt,x⟶ut,x,uniformly on the bounded domain.(4)
In order to accomplish the proof, we establish some properties for the semigroup. We fix thatu0∈H1ℝ and τ>0, for t>0, we need to indicate that(141)StSτu0=Sτ+tu0.We assume thatt,ξ↦u,v,wt,ξ is the corresponding solution of (44) with the initial value (37). We set u^=Stu0. In order to constitute the trajectory t↦Stu∗, we introduce a new variable σ and the map ξ↦σξ as a solution to the ODE(142)ddξσξ=wτ,ξ,ifvτ,ξ>−π,0,ifvτ,ξ=−π,with the initial value(143)σξ0=0,whereξ0 is selected such that Yτ,ξ0=0. We set(144)u^=uτ+t,ξσ,v^=vτ+t,ξσ,w^=wτ+t,ξσwτ,ξσ,whereξσ renders an a.e. inverse to the map in (142) and (143), namely,(145)ξσ∗=sups;σs≤σ∗.(5)
For eachσ, former definitions indicate(146)∫0Yτ,ξ1+a2ux2τ,xdx=σξ.In connection with (143), on the one hand, we are aware that the above formula stands as ξ=ξ0, σ=0. On another hand, using (117) and (29) one has(147)∂∂ξYτ,ξ⋅1+a2ux2τ,Yτ,σξ=wτ,ξ=ddξσξ.Taking advantage of (142) and by an integration, this establishes (146).(6)
Now, it is suffices to illustrate that (144) provides a solution to the ODE (44). Firstly, we give the following identity:(148)wτ+t,ξdξ=dσξdξ⋅w^t,σξwτ,σξ=w^t,σξdσ.
This comes to the corresponding integral terms of (44) to fulfill(149)P^xt,σ=Pxτ+t,ξσ,P^t,σ=Pτ+t,ξσ.
The conclusion comes from (144) and (149) because the third equation in (44) is linear with respect to w.
It should be pointed out that the semigroupS rendered by the Cauchy problem (1) is not continuous as a map with values in H1. Indeed, a single trajectory t↦ut,⋅ might be failed to be continuous from 0,∞↦H1.



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