# The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa-Holm Equation

Authors: Shaoyong Lai
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Using the Kato theorem for abstract differential equations, the local well-posedness of the solution for a nonlinear dissipative Camassa-Holm equation is established in spaceC([0,T),Hs(R))∩C1([0,T),Hs-1(R)) with s>3/2. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs(R) with 1≤s≤3/2 is developed.


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## Body

## 1. Introduction

Camassa and Holm [1] used the Hamiltonian method to derive a completely integrable wave equation(1.1)ut-uxxt+2kux+3uux=2uxuxx+uuxxx,
by retaining two terms that are usually neglected in the small amplitude, shallow water limit. Its alternative derivation as a model for water waves can be found in Constantin and Lannes [2] and Johnson [3]. Equation (1.1) also models wave current interaction [4], while Dai [5] derived it as a model in elasticity (see Constantin and Strauss [6]). Moreover, it was pointed out in Lakshmanan [7] that the Camassa-Holm equation (1.1) could be relevant to the modeling of tsunami waves (see Constantin and Johnson [8]).In fact, a huge amount of work has been carried out to investigate the dynamic properties of  (1.1). For k=0, (1.1) has traveling wave solutions of the form ce-|x-ct|, called peakons, which capture the main feature of the exact traveling wave solutions of greatest height of the governing equations (see [9–11]). For k>0, its solitary waves are stable solitons [6, 11]. It was shown in [12–14] that the inverse spectral or scattering approach was a powerful tool to handle Camassa-Holm equation. Equation (1.1) is a completely integrable infinite-dimensional Hamiltonian system (in the sense that for a large class of initial data, the flow is equivalent to a linear flow at constant speed [15]). It should be emphasized that  (1.1) gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group (see [16, 17]), and this geometric illustration leads to a proof that the Least Action Principle holds. It is worthwhile to mention that Xin and Zhang [18] proved that the global existence of the weak solution in the energy space H1(R) without any sign conditions on the initial value, and the uniqueness of this weak solution is obtained under some conditions on the solution [19]. Coclite et al. [20] extended the analysis presented in [18, 19] and obtained many useful dynamic properties to other equations (also see [21–24]). Li and Olver [25] established the local well-posedness in the Sobolev space Hs(R) with s>3/2 for (1.1) and gave conditions on the initial data that lead to finite time blowup of certain solutions. It was shown in Constantin and Escher [26] that the blowup occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. After wave breaking, the solution can be continued uniquely either as a global conservative weak solution [21] or a global dissipative solution [22]. For peakons, these possibilities are explicitly illustrated in the paper [27]. For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water models, the reader is referred to [10, 28–32] and the references therein.In this paper, motivated by the work in [25, 33], we study the following generalized Camassa-Holm equation(1.2)ut-utxx+2kux+aumux=2uxuxx+uuxxx+β∂x[(ux)N],
where m≥1 and N≥1 are natural numbers, and a, k, and β are arbitrary constants. Obviously,  (1.2) reduces to  (1.1) if we set a=3, m=1, and β=0. Actually, Wu and Yin [34] consider a nonlinearly dissipative Camassa-Holm equation which includes a nonlinearly dissipative term L(u), where L is a differential operator or a quasidifferential operator. Therefore, we can regard the term β∂x[(ux)N] as a nonlinearly dissipative term for the dissipative Camassa-Holm equation (1.2).Due to the termβ∂x[(ux)N] in  (1.2), the conservation laws in previous works [10, 25] for  (1.1) lose their powers to obtain some bounded estimates of the solution for  (1.2). A new conservation law different from those presented in [10, 25] will be established to prove the local existence and uniqueness of the solution to  (2.3) subject to initial value u0(x)∈Hs(R) with s>3/2. We should address that all the generalized versions of the Camassa-Holm equation in previous works (see [17, 25, 34]) do not involve the nonlinear term ∂x[(ux)N]. Lai and Wu [33] only studied a generalized Camassa-Holm equation in the case where β≥0 and N is an odd number. Namely,  (1.2) with β<0 and arbitrary positive integer N was not investigated in [33].The main tasks of this paper are two-fold. Firstly, by using the Kato theorem for abstract differential equations, we establish the local existence and uniqueness of solutions for (1.2) with any β and arbitrary positive integer N in space C([0,T),Hs(R))⋂C1([0,T),Hs-1(R)) with s>3/2. Secondly, it is shown that the existence of weak solutions in lower order Sobolev space Hs(R) with 1≤s≤3/2. The ideas of proving the second result come from those presented in Li and Olver [25].


## 2. Main Results

Firstly, we give some notation.The space of all infinitely differentiable functionsϕ(t,x) with compact support in [0,+∞)×R is denoted by C0∞. Lp=Lp(R)(1≤p<+∞) is the space of all measurable functions h such that ∥h∥Lpp=∫R|h(t,x)|pdx<∞. We define L∞=L∞(R) with the standard norm ∥h∥L∞=infm(e)=0supx∈R∖e|h(t,x)|. For any real number s, Hs=Hs(R) denotes the Sobolev space with the norm defined by(2.1)‖h‖Hs=(∫R(1+|ξ|2)s|ĥ(t,ξ)|2dξ)1/2<∞,
where ĥ(t,ξ)=∫Re-ixξh(t,x)dx.ForT>0 and nonnegative number s, C([0,T);Hs(R)) denotes the Frechet space of all continuous Hs-valued functions on [0,T). We set Λ=(1-∂x2)1/2.In order to study the existence of solutions for  (1.2), we consider its Cauchy problem in the form(2.2)ut-utxx=-2kux-am+1(um+1)x+2uxuxx+uuxxx+β∂x[(ux)N]=-kux-am+1(um+1)x+12∂x3u2-12∂x(ux2)+β∂x[(ux)N],u(0,x)=u0(x),
which is equivalent to(2.3)ut+uux=Λ-2[-ku-am+1(um+1)]x+Λ-2(uux)-12Λ-2∂x(ux2)+βΛ-2∂x[(ux)N],u(0,x)=u0(x).Now, we state our main results.Theorem 2.1.
Letu0(x)∈Hs(R) with s>3/2. Then problem (2.2) or problem (2.3) has a unique solution u(t,x)∈C([0,T);Hs(R))⋂C1([0,T);Hs-1(R)) where T>0 depends on ∥u0∥Hs(R).Theorem 2.2.
Suppose thatu0(x)∈Hs with 1≤s≤3/2 and ∥u0x∥L∞<∞. Then there exists a T>0 such that   (1.2) subject to initial value u0(x) has a weak solution u(t,x)∈L2([0,T],Hs) in the sense of distribution and ux∈L∞([0,T]×R).


## 3. Local Well-Posedness

We consider the abstract quasilinear evolution equation(3.1)dvdt+A(v)v=f(v),t≥0,v(0)=v0.
Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X, and let Q:Y→X be a topological isomorphism. Let L(Y,X) be the space of all bounded linear operators from Y to X. If X=Y, we denote this space by L(X). We state the following conditions in which ρ1,ρ2,ρ3, and ρ4 are constants depending on max{∥y∥Y,∥z∥Y}.(i)
A(y)∈L(Y,X) for y∈X with
(3.2)‖(A(y)-A(z))w‖X≤ρ1‖y-z‖X‖w‖Y,y,z,w∈Y,
and A(y)∈G(X,1,β) (i.e., A(y) is quasi-m-accretive), uniformly on bounded sets in Y.(ii)
QA(y)Q-1=A(y)+B(y), where B(y)∈L(X) is bounded, uniformly on bounded sets in Y. Moreover,
(3.3)‖(B(y)-B(z))w‖X≤ρ2‖y-z‖Y‖w‖X,y,z∈Y,w∈X.(iii)
f:Y→Y extends to a map from X into X is bounded on bounded sets in Y, and satisfies(3.4)‖f(y)-f(z)‖Y≤ρ3‖y-z‖Y,y,z∈Y,‖f(y)-f(z)‖X≤ρ4‖y-z‖X,y,z∈Y.Kato Theorem (see [35])
Assume that (i), (ii), and (iii) hold. Ifv0∈Y, there is a maximal T>0 depending only on ∥v0∥Y, and a unique solution v to problem (3.1) such that
(3.5)v=v(⋅,v0)∈C([0,T);Y)⋂C1([0,T);X).
Moreover, the map v0→v(·,v0) is a continuous map from Y to the space
(3.6)C([0,T);Y)⋂C1([0,T);X).For problem (2.3), we set A(u)=u∂x, Y=Hs(R), X=Hs-1(R), Λ=(1-∂x2)1/2, (3.7)f(u)=Λ-2[-ku-am+1(um+1)]x+Λ-2(uux)-12Λ-2∂x(ux2)+βΛ-2∂x[(ux)N],
and Q=Λ. In order to prove Theorem 2.1, we only need to check that A(u) and f(u) satisfy assumptions (i)–(iii).Lemma 3.1.
The operatorA(u)=u∂x with u∈Hs(R), s>3/2 belongs to G(Hs-1,1,β).Lemma 3.2.
LetA(u)=u∂x with u∈Hs and s>3/2. Then A(u)∈L(Hs,Hs-1) for all u∈Hs. Moreover,
(3.8)‖(A(u)-A(z))w‖Hs-1≤ρ1‖u-z‖Hs-1‖w‖Hs,u,z,w∈Hs(R).Lemma 3.3.
Fors>3/2, u,z∈Hs and w∈Hs-1, it holds that B(u)=[Λ,u∂x]Λ-1∈L(Hs-1) for u∈Hs and
(3.9)‖(B(u)-B(z))w‖Hs-1≤ρ2‖u-z‖Hs‖w‖Hs-1.Proofs of the above Lemmas3.1–3.3 can be found in [29] or [31].Lemma 3.4 (see [35]).
Letr and q be real numbers such that -r<q≤r. Then
(3.10)‖uv‖Hq≤c‖u‖Hr‖v‖Hq,ifr>12,‖uv‖Hr+q-1/2≤c‖u‖Hr‖v‖Hq,ifr<12.Lemma 3.5.
Letu,z∈Hs with s>3/2, then f(u) is bounded on bounded sets in Hs and satisfies
(3.11)‖f(u)-f(z)‖Hs≤ρ3‖u-z‖Hs,(3.12)‖f(u)-f(z)‖Hs-1≤ρ4‖u-z‖Hs-1.Proof.
Using the algebra property of the spaceHs0 with s0>1/2, we have
(3.13)‖f(u)-f(z)‖Hs≤c[‖Λ-2([-ku-am+1(um+1)]x-[-kz-am+1(zm+1)]x)‖Hs+‖Λ-2(uux-zzx)‖Hs+‖Λ-2∂x(ux2-zx2)‖Hs+‖Λ-2∂x[(ux)N]-Λ-2∂x[(zx)N]‖Hs‖Λ-2([-ku-am+1(um+1)]x-[-kz-am+1(zm+1)]x)‖Hs]≤c[‖u-z‖Hs-1+‖um+1-zm+1‖Hs-1+‖uux-zzx‖Hs-1+‖ux2-zx2‖Hs-1+‖(ux)N-(zx)N‖Hs-1]≤c‖u-z‖Hs[1+∑j=0m‖u‖Hsm-j‖z‖Hsj+‖u‖Hs+‖z‖Hs+∑j=0N-1‖ux‖Hs-1N-j‖zx‖Hs-1j]≤ρ3‖u-z‖Hs,
from which we obtain (3.11).
Applying Lemma3.4, uux=(1/2)(u2)x, s>3/2,∥u∥L∞≤c∥u∥Hs-1 and ∥ux∥L∞≤c∥u∥Hs, we get
(3.14)‖f(u)-f(z)‖Hs-1≤c[‖(ux-zx)∑j=0N-1uxN-1-jzxj‖‖u-z‖Hs-2+‖um+1-zm+1‖Hs-2+‖u2-z2‖Hs-2+‖(ux-zx)(ux+zx)‖Hs-2+‖(ux-zx)∑j=0N-1uxN-1-jzxj‖Hs-2]≤c‖u-z‖Hs-1[1+∑j=0m‖u‖Hs-1m-j‖z‖Hs-1j+‖u‖Hs-1+‖z‖Hs-1+‖u‖Hs+‖z‖Hs+∑j=0N-1‖ux‖Hs-1N-j‖zx‖Hs-1j]≤ρ4‖u-z‖Hs-1,
which completes the proof of (3.12).Proof of Theorem2.1.
Using the Kato Theorem, Lemmas3.1–3.3, and 3.5, we know that system (2.2) or problem (2.3) has a unique solution
(3.15)u(t,x)∈C([0,T);Hs(R))⋂C1([0,T);Hs-1(R)).


## 4. Existence of Weak Solutions

Fors≥2, using the first equation of system (2.2) derives(4.1)ddt∫R(u2+ux2+2β∫0tuxN+1dτ)dx=0,
from which we have the conservation law(4.2)∫R(u2+ux2+2β∫0tuxN+1dτ)dx=∫R(u02+u0x2)dx.Lemma 4.1 (Kato and Ponce [36]).
Ifr>0, then Hr⋂L∞ is an algebra. Moreover,
(4.3)‖uv‖r≤c(‖u‖L∞‖v‖r+‖u‖r‖v‖L∞),
where c is a constant depending only on r.Lemma 4.2 (Kato and Ponce [36]).
Letr>0. If u∈Hr⋂W1,∞ and v∈Hr-1⋂L∞, then
(4.4)‖[Λr,u]v‖L2≤c(‖∂xu‖L∞‖Λr-1v‖L2+‖Λru‖L2‖v‖L∞).Lemma 4.3.
Lets≥2 and the function u(t,x) is a solution of problem (2.2) and the initial data u0(x)∈Hs(R). Then the following inequality holds
(4.5)‖u‖L∞≤‖u‖H1≤‖u0‖H1e|β|∫0t‖ux‖L∞N-1dτ.
Forq∈(0,s-1], there is a constant c, which only depends on m, N, k, a, and β, such that
(4.6)∫R(Λq+1u)2dx≤∫R(Λq+1u0)2dx+c∫0t‖ux‖L∞‖u‖Hq+12(1+‖u‖L∞m-1)dτ+c∫0t‖u‖Hq+12‖ux‖L∞N-1dτ.
Forq∈[0,s-1], there is a constant c, which only depends on m,N,k,a, and β, such that
(4.7)‖ut‖Hq≤c‖u‖Hq+1(1+(1+‖u‖L∞m-1)‖u‖H1+‖ux‖L∞N-1).Proof.
Using∥u∥H12=∫R(u2+ux2)dx and (4.2) derives (4.5).
Using∂x2=-Λ2+1 and the Parseval equality gives rise to
(4.8)∫RΛquΛq∂x3(u2)dx=-2∫R(Λq+1u)Λq+1(uux)dx+2∫R(Λqu)Λq(uux)dx.
Forq∈(0,s-1], applying (Λqu)Λq to both sides of the first equation of system (2.3) and integrating with respect to x by parts, we have the identity
(4.9)12ddt∫R((Λqu)2+(Λqux)2)dx=-a∫R(Λqu)Λq(umux)dx-∫R(Λq+1u)Λq+1(uux)dx+12∫R(Λqux)Λq(ux2)dx+∫R(Λqu)Λq(uux)dx-β∫RΛquxΛq[(ux)N]dx.
We will estimate the terms on the right-hand side of (4.9) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2, we have
(4.10)∫R(Λqu)Λq(umux)dx=∫R(Λqu)[Λq(umux)-umΛqux]dx+∫R(Λqu)umΛquxdx≤c‖u‖Hq(m‖u‖L∞m-1‖ux‖L∞‖u‖Hq+‖ux‖L∞‖u‖L∞m-1‖u‖Hq)+12‖u‖L∞m-1‖ux‖L∞‖Λqu‖L22≤c‖u‖Hq2‖u‖L∞m-1‖ux‖L∞.
Using the above estimate to the second term yields
(4.11)∫R(Λq+1u)Λq+1(uux)dx≤c‖u‖Hq+12‖ux‖L∞.
For the third term, using the Cauchy-Schwartz inequality and Lemma 4.1, we obtain
(4.12)∫R(Λqux)Λq(ux2)dx≤‖Λqux‖L2‖Λq(ux2)‖L2≤c‖u‖Hq+1(‖ux‖L∞‖ux‖Hq+‖ux‖L∞‖ux‖Hq)≤c‖u‖Hq+12‖ux‖L∞.
For the last term in (4.9), using Lemma 4.1 repeatedly results in
(4.13)|∫R(Λqux)Λq(ux)Ndx|≤‖ux‖Hq‖uxN‖Hq≤c‖u‖Hq+12‖ux‖L∞N-1.
It follows from (4.9) to (4.13) that there exists a constant c depending only on m,N and the coefficients of  (1.2) such that
(4.14)12ddt∫R[(Λqu)2+(Λqux)2]dx≤c‖ux‖L∞‖u‖Hq+12(1+‖u‖L∞m-1)+c‖u‖Hq+12‖ux‖L∞N-1.
Integrating both sides of the above inequality with respect to t results in inequality (4.6).
To estimate the norm ofut, we apply the operator (1-∂x2)-1 to both sides of the first equation of system (2.3) to obtain the equation
(4.15)ut=(1-∂x2)-1[-2kux+∂x(-am+1um+1+12∂x2(u2)-12ux2)+β∂x[(ux)N]].
Applying (Λqut)Λq to both sides of  (4.15) for q∈(0,s-1] gives rise to
(4.16)∫R(Λqut)2dx=∫R(Λqut)Λq-2[∂x(-2ku-am+1um+1+12∂x2(u2)-12ux2)+β∂x[(ux)N]]dτ.
For the right-hand side of  (4.16), we have
(4.17)∫R(Λqut)Λq-2(-2kux)dx≤c‖ut‖Hq‖u‖Hq,∫R(Λqut)(1-∂x2)-1Λq∂x(-am+1um+1-12ux2)dx≤c‖ut‖Hq(∫R(1+ξ2)q-1×[∫R[-am+1um̂(ξ-η)û(η)-12ux̂(ξ-η)ux̂(η)]dη]2)1/2≤c‖ut‖Hq‖u‖H1‖u‖Hq+1(1+‖u‖L∞m-1).
Since
(4.18)∫(Λqut)(1-∂x2)-1Λq∂x2(uux)dx=-∫(Λqut)Λq(uux)dx+∫(Λqut)(1-∂x2)-1Λq(uux)dx,
using Lemma 4.1, ∥uux∥Hq≤c∥(u2)x∥Hq≤c∥u∥L∞∥u∥Hq+1 and ∥u∥L∞≤∥u∥H1, we have
(4.19)∫(Λqut)Λq(uux)dx≤c‖ut‖Hq‖uux‖Hq≤c‖ut‖Hq‖u‖H1‖u‖Hq+1,∫(Λqut)(1-∂x2)-1Λq(uux)dx≤c‖ut‖Hq‖u‖H1‖u‖Hq+1.
Using the Cauchy-Schwartz inequality and Lemma 4.1 yields
(4.20)|∫R(Λqut)(1-∂x2)-1Λq∂x(uxN)dx|≤c‖ut‖Hq‖ux‖L∞N-1‖u‖Hq+1.
Applying (4.17)–(4.20) into (4.16) yields the inequality
(4.21)‖ut‖Hq≤c‖u‖Hq+1(1+(1+‖u‖L∞m-1)‖u‖H1+‖ux‖L∞N-1).
This completes the proof of Lemma 4.3.Defining(4.22)ϕ(x)={e1/(x2-1),|x|<1,0,|x|≥1,
and setting ϕε(x)=ε-1/4ϕ(ε-1/4x) with 0<ε<1/4 and uε0=ϕε⋆u0, we know that uε0∈C∞ for any u0∈Hs(R) and s>0.It follows from Theorem2.1 that for each ε the Cauchy problem(4.23)ut-utxx=∂x(-2ku-am+1um+1)+12∂x3(u2)-12∂x(ux2)+β∂x[(ux)N],u(0,x)=uε0(x),x∈R,
has a unique solution uε(t,x)∈C∞([0,T);H∞).Lemma 4.4.
Under the assumptions of problem (4.23), the following estimates hold for any ε with 0<ε<1/4 and s>0(4.24)‖uε0x‖L∞≤c1‖u0x‖L∞,‖uε0‖Hq≤c1,ifq≤s,‖uε0‖Hq≤c1ε(s-q)/4,ifq>s,‖uε0-u0‖Hq≤c1ε(s-q)/4,ifq≤s,‖uε0-u0‖Hs=o(1),
where c1 is a constant independent of ε.The proof of this Lemma can be found in Lai and Wu [33].Lemma 4.5.
Ifu0(x)∈Hs(R) with s∈[1,3/2] such that ∥u0x∥L∞<∞. Let uε0 be defined as in system (4.23). Then there exist two positive constants T and c, which are independent of ε, such that the solution uε of problem (4.23) satisfies ∥uεx∥L∞≤c for any t∈[0,T).Proof.
Using notationu=uε and differentiating both sides of the first equation of problem (4.23) or  (4.15) with respect to x give rise to
(4.25)utx+12∂x2u2-12ux2=2ku+am+1um+1-12u2-βuxN-Λ-2[2ku+am+1um+1-12u2+12ux2-βuxN].
Letting p>0 be an integer and multiplying the above equation by (ux)2p+1 and then integrating the resulting equation with respect to x yield the equality
(4.26)12p+2ddt∫R(ux)2p+2dx+p2p+2∫R(ux)2p+3dx=∫R(ux)2p+1(2ku+am+1um+1-12u2-βuxN)dx-∫R(ux)2p+1Λ-2[2ku+am+1um+1-u22+12ux2-βuxN]dx.
Applying the Hölder's inequality yields
(4.27)12p+2ddt∫R(ux)2p+2dx≤{|2k|(∫R|u|2p+2dx)1/(2p+2)+am+1(∫R|um+1|2p+2dx)1/(2p+2)+12(∫R|u2|2p+2dx)1/(2p+2)+β(∫R|uxN|2p+2dx)1/(2p+2)+(∫R|G|2p+2dx)1/(2p+2)}(∫R|ux|2p+2dx)(2p+1)/(2p+2)+p2p+2‖ux‖L∞∫R|ux|2p+2dx,
or
(4.28)ddt(∫R(ux)2p+2dx)1/(2p+2)≤|2k|(∫R|u|2p+2dx)1/(2p+2)+am+1(∫R|um+1|2p+2dx)1/(2p+2)+12(∫R|u2|2p+2dx)1/(2p+2)+β(∫R|uxN|2p+2dx)1/(2p+2)+(∫R|G|2p+2dx)1/(2p+2)+p2p+2‖ux‖L∞(∫R|ux|2p+2dx)1/(2p+2),
where
(4.29)G=Λ-2[2ku+am+1um+1-u22+12ux2-βuxN].
Since ∥f∥Lp→∥f∥L∞ as p→∞ for any f∈L∞⋂L2, integrating both sides of the inequality (4.28) with respect to t and taking the limit as p→∞ result in the estimate
(4.30)‖ux‖L∞≤‖u0x‖L∞+∫0tc[(‖u‖L∞+‖u2‖L∞+‖um+1‖L∞+β‖ux‖L∞N+‖G‖L∞)+12‖ux‖L∞2]dτ.
Using the algebra property of Hs0(R) with s0>1/2 yields (∥uε∥H(1/2)+ means that there exists a sufficiently small δ>0 such that ∥uε∥(1/2)+=∥uε∥H1/2+δ)(4.31)‖G‖L∞≤c‖G‖H(1/2)+≤c‖Λ-2[2ku+am+1um+1-u22+12ux2-βuxN]‖H(1/2)+≤c(‖u‖H1+‖u‖H12+‖u‖H1m+1+‖Λ-2(ux2)‖H(1/2)++‖Λ-2(uxN)‖H(1/2)+)≤c(‖u‖H1+‖u‖H12+‖u‖H1m+1+‖ux2‖H0+‖uxN‖H0)≤c(‖u‖H1+‖u‖H12+‖u‖H1m+1+‖ux‖L∞‖u‖H1+‖ux‖L∞N-1‖u‖H1)≤cec∫0t‖ux‖L∞N-1dτ(1+‖ux‖L∞+‖ux‖L∞N-1),
in which Lemma 4.3 is used. Therefore, we get
(4.32)∫0t‖G‖L∞dτ≤c∫0tec∫0τ‖ux‖L∞N-1dξ(1+‖ux‖L∞+‖ux‖L∞N-1)dτ.
From (4.30) and (4.32), one has
(4.33)‖ux‖L∞≤‖u0x‖L∞+c∫0t[‖ux‖L∞2+‖ux‖L∞N+ec∫0t‖ux‖L∞N-1dτ+ec∫0τ‖ux‖L∞N-1dξ(1+‖ux‖L∞+‖ux‖L∞N-1)]dτ.
From Lemma4.4, it follows from the contraction mapping principle that there is a T>0 such that the equation
(4.34)‖W‖L∞=‖u0x‖L∞+c∫0t[‖W‖L∞2+‖W‖L∞N+ec∫0t‖W‖L∞N-1dτ+ec∫0τ‖W‖L∞N-1dξ(1+‖W‖L∞+‖W‖L∞N-1)]dτ
has a unique solution W∈C[0,T]. Using the Theorem presented at page 51 in [25] or Theorem 2 in Section 1.1 presented in [37] yields that there are constants T>0 and c>0 independent of ε such that ∥ux∥L∞≤W(t) for arbitrary t∈[0,T], which leads to the conclusion of Lemma 4.5.
Using Lemmas4.3 and 4.5, notation uε=u and Gronwall's inequality results in the inequalities
(4.35)‖uε‖Hq≤CTeCT,‖uεt‖Hr≤CTeCT,
where q∈(0,s], r∈(0,s-1] and CT depends on T. It follows from Aubin's compactness theorem that there is a subsequence of {uε}, denoted by {uεn}, such that {uεn} and their temporal derivatives {uεnt} are weakly convergent to a function u(t,x) and its derivative ut in L2([0,T],Hs) and L2([0,T],Hs-1), respectively. Moreover, for any real number R1>0, {uεn} is convergent to the function u strongly in the space L2([0,T],Hq(-R1,R1)) for q∈[0,s) and {uεnt} converges to ut strongly in the space L2([0,T],Hr(-R1,R1)) for r∈[0,s-1]. Thus, we can prove the existence of a weak solution to  (2.2).Proof of Theorem2.2.
From Lemma4.5, we know that {uεnx}(εn→0) is bounded in the space L∞. Thus, the sequences {uεn} and {uεnx} are weakly convergent to u and ux in L2[0,T],Hr(-R,R) for any r∈[0,s-1), respectively. Therefore, u satisfies the equation
(4.36)-∫0T∫Ru(gt-gxxt)dxdt=∫0T∫R[(2ku+am+1um+1+12(ux2))gx-12u2gxxx-β(ux)Ngx(2ku+am+1um+1+12(ux2))]dxdt,
with u(0,x)=u0(x) and g∈C0∞. Since X=L1([0,T]×R) is a separable Banach space and {uεnx} is a bounded sequence in the dual space X*=L∞([0,T]×R) of X, there exists a subsequence of {uεnx}, still denoted by {uεnx}, weakly star convergent to a function v in L∞([0,T]×R). It derives from the {uεnx} weakly convergent to ux in L2([0,T]×R) that ux=v almost everywhere. Thus, we obtain ux∈L∞([0,T]×R).



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*Source: 285040-2011-10-01.xml*