# The Well-Posedness of Solutions for a Generalized Shallow Water Wave Equation

Authors: Shaoyong Lai; Aiyin Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A nonlinear partial differential equation containing the famous Camassa-Holm and Degasperis-Procesi equations as special cases is investigated. The Kato theorem for abstract differential equations is applied to establish the local well-posedness of solutions for the equation in the Sobolev spaceHs(R) with s>3/2. Although the H1-norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower-order Sobolev space Hs with 1≤s≤3/2 is proved under the assumptions u0∈Hs and ∥u0x∥L∞<∞.


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## Body

## 1. Introduction

Constantin and Lannes [1] derived the shallow water equation(1.1)ut+ux+32ρuux+μ(αuxxx+βutxx)=ρμ(γuxuxx+δuuxxx),
where the constants α,β,γ,δ,ρ, and μ satisfy certain conditions. Under several restrictions on the coefficients of model (1.1), the large time well-posedness was established on a time scale O(|ρ|-1) provided that the initial value u0 belongs to Hs(R) with s>5/2, and the wave-breaking phenomena were also discussed in [1]. As stated in [1], using suitable mathematical transformations, one can turn (1.1) into the form(1.2)ut-utxx+2kux+muux=auxuxx+buuxxx,
where a,b,k, and m are constants. Obviously, (1.2) is a generalization of both the Camassa-Holm equation [2](1.3)ut-utxx+2kux+3uux=2uxuxx+uuxxx,kisaconstant
and the Degasperis-Procesi model [3](1.4)ut-uxxt+2kux+4uux=3uxuxx+uuxxx.
Equations (1.3) and (1.4) are bi-Hamiltonian and arise in the modeling of shallow water waves. These two equations pertain to waves of medium amplitude (cf. the discussions in [1, 4]) and accommodate wave-breaking phenomena. Moreover, the Camassa-Holm and Degasperis-Procesi models admit peaked (periodic as well as solitary) traveling waves capturing the main feature of the exact traveling wave solutions of the greatest height of the governing equations for water waves (cf. [5, 6]). For other dynamic properties about (1.3) and (1.4), the reader is referred to [7–20].Recently, Lai and Wu [21] investigate (1.2) in the case where k=0, m=a+b, a>0, and b>0. The well-posedness of global solutions is established in [21] in Sobolev space Hs(R) with s>3/2 under certain assumptions on the initial value. The local strong and weak solutions for (1.2) are discussed in [22] in the case where b>0, a,k, and m are arbitrary constants.Motivated by the desire to extend the work in [22], we investigate the following generalized model of  (1.2):(1.5)ut-utxx+2kux+muux=auxuxx+buuxxx+βuxn,
where b>0, a,k,m, and β are arbitrary constants, and n is a positive integer.The aim of this paper is to investigate (1.5). Since a,b,m, and β are arbitrary constants, we do not have the result that the H1 norm of the solution of (1.5) remains constant. We will apply the Kato theorem [23] to prove the existence and uniqueness of local solutions for (1.5) in the space C([0,T),Hs(R))⋂C1([0,T),Hs-1(R))(s>3/2) provided that the initial value u0(x) belongs to Hs(R)(s>3/2). Moreover, it is shown that there exists a weak solution of (1.5) in lower-order Sobolev space Hs(R) with 1≤s≤3/2.The structure of this paper is as follows. The main results are given in Section2. The existence and uniqueness of the local strong solution for the Cauchy problem (1.5) are proved in Section 3. The existence of weak solutions is established in Section 4.


## 2. Main Results

Firstly, we give some notations.The space of all infinitely differentiable functionsϕ(t,x) with compact support in [0,+∞)×R is denoted by C0∞. We let Lp=Lp(R)(1≤p<+∞) be 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, we let Hs=Hs(R) denote 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. Here, we note that the norms ∥·∥Lpp,∥·∥L∞, and ∥·∥Hs depend on variable t.ForT>0 and nonnegative number s, let C([0,T);Hs(R)) denote the space of functions u:[0,T)×R→R with the properties that u(t,·)∈Hs(R) for each t∈[0,T], and the mapping u:[0,T)→Hs(R) is continuous and bounded.For simplicity, throughout this paper, we letc denote any positive constant which is independent of parameter ɛ and set Λ=(1-∂x2)1/2.In order to study the existence of solutions for (1.5), we consider its Cauchy problem in the form(2.2)ut-utxx=-∂x(2ku+m2u2)+auxuxx+buuxxx+βuxn,u(0,x)=u0(x),
where b>0,a,k,m,β, and n are arbitrary constants. Now, we give the theorem to describe the local well-posedness of solutions for problem (2.2).Theorem 2.1.
Letu0(x)∈Hs(R) with s>3/2, then the Cauchy problem (2.2) 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).For a real numbers with s>0, suppose that the function u0(x) is in Hs(R), and let uɛ0 be the convolution uɛ0=ϕɛ⋆u0 of the function ϕɛ(x)=ɛ-1/4ϕ(ɛ-1/4x) and u0 such that the Fourier transform ϕ̂ of ϕ satisfies ϕ̂∈C0∞, ϕ(ξ)̂≥0, and ϕ(ξ)̂=1 for any ξ∈(-1,1). Thus one has uɛ0(x)∈C∞. It follows from Theorem 2.1 that for each ɛ satisfying 0<ɛ<1/4, the Cauchy problem(2.3)ut-utxx=-∂x(2ku+m2u2)+auxuxx+buuxxx+βuxn,u(0,x)=uɛ0(x),x∈R
has a unique solution uɛ(t,x)∈C∞([0,Tɛ);H∞), in which Tɛ may depend on ɛ. However, one will show that under certain assumptions, there exist two constants c and T>0, both independent of ɛ, such that the solution of problem (2.3) satisfies ∥uɛx∥L∞≤c for any t∈[0,T), and there exists a weak solution u(t,x)∈L2([0,T],Hs) for problem (2.2). These results are summarized in the following two theorems.Theorem 2.2.
Ifu0(x)∈Hs(R) with s∈[1,3/2] such that ∥u0x∥L∞<∞, let uɛ0 be defined as in system (2.3), then there exist two constants c and T>0, which are independent of ɛ, such that the solution uɛ of problem (2.3) satisfies ∥uɛx∥L∞≤c for any t∈[0,T).Theorem 2.3.
Suppose thatu0(x)∈Hs with 1≤s≤3/2 and ∥u0x∥L∞<∞, then there exists a T>0 such that problem (2.2) has a weak solution u(t,x)∈L2([0,T],Hs(R)) in the sense of distribution and ux∈L∞([0,T]×R).


## 3. Proof of Theorem2.1

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 only 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 [23])
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).
In fact, problem (2.2) can be written as
(3.7)ut-utxx=-[2ku+m2u2]x+b2∂x3u2-3b-a2∂x(ux2)+βuxn,u(0,x)=u0(x),
which is equivalent to
(3.8)ut+buux=-Λ-2[(2ku+m2u2)x+buux-3b-a2∂x(ux2)+βuxn],u(0,x)=u0(x).
We setA(u)=bu∂x with constant b>0, Y=Hs(R), X=Hs-1(R), Λ=(1-∂x2)1/2, f(u)=-Λ-2(2ku+(m/2)u2)x+bΛ-2(uux)-((3b-a)/2)Λ-2∂x(ux2)+βΛ-2uxn, and Q=Λs. We know that Q is an isomorphism of Hs onto Hs-1. 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)=bu∂x with u∈Hs and s>3/2, then A(u)∈L(Hs,Hs-1) for all u∈Hs. Moreover,(3.9)‖(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
(3.10)‖(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 [24] or [25].Lemma 3.4 (see [23]).
Letr and q be real numbers such that -r<q≤r, then(3.11)‖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 and f(u)=-Λ-2(2ku+(m/2)u2)x+bΛ-2(uux)-((3b-a)/2)Λ-2∂x(ux2)+βΛ-2(uxn), then f is bounded in Hs and satisfies
(3.12)‖f(u)-f(z)‖Hs≤ρ3‖u-z‖Hs,‖f(u)-f(z)‖Hs-1≤ρ4‖u-z‖Hs-1.Proof.
Using the algebra property of the spaceHs0 with s0>1/2 and s-1>1/2, we have
(3.13)‖f(u)-f(z)‖Hs≤‖Λ-2((2ku+m2u2)x-(2kz+m2z2)x)‖Hs+‖bΛ-2(uux-zzx)‖Hs+‖3b-a2Λ-2∂x(ux2-zx2)‖Hs+‖βΛ-2(uxn-zxn)‖Hs≤c(∑j=0n-1‖u-z‖Hs-1(1+‖u‖Hs-1+‖z‖Hs-1)+‖Λ-2(u2-z2)x‖Hs+‖ux2-zx2‖Hs-1+‖ux-zx‖Hs-1∑j=0n-1‖ux‖Hs-1n-j‖zx‖Hs-1j)≤c(∑j=0n-1‖u-z‖Hs(1+‖u‖Hs+‖z‖Hs)+‖(u-z)(u+z)‖Hs-1+‖ux2-zx2‖Hs-1+‖ux-zx‖Hs-1∑j=0n-1‖u‖Hsn-j‖z‖Hsj)≤c‖u-z‖Hs(1+‖u‖Hs+‖z‖Hs+∑j=0n-1‖u‖Hsn-j‖z‖Hsj)≤cρ3‖u-z‖Hs,
from which we obtain (3.12).
Applying Lemma3.4, uux=1/2(u2)x, s>3/2, we get(3.14)‖f(u)-f(z)‖Hs-1≤c(‖2ku+m2u2-(2kz+m2z2)‖Hs-2+‖u2-z2‖Hs-2+‖(ux-zx)(ux+zx)‖Hs-2+‖uxn-zxn‖Hs-1‖2ku+m2u2-(2kz+m2z2)‖Hs-2)≤c‖u-z‖Hs-2(1+‖u‖Hs-1+‖z‖Hs-1)+c‖ux-zx‖Hs-2(‖ux‖Hs-1+‖zx‖Hs-1)+c‖ux-zx‖Hs-2∑j=0n-1‖ux‖Hs-1n-j‖zx‖Hs-1j≤c‖u-z‖Hs-1(1+‖u‖Hs+‖z‖Hs+∑j=0n-1‖u‖Hsn-j‖z‖Hsj),
which completes the proof of (3.12).Proof of Theorem2.1.
Using the Kato theorem, Lemmas3.1, 3.2, 3.3, and 3.5, we know that system (3.11) or problem (2.2) has a unique solution
(3.15)u(t,x)∈C([0,T);Hs(R))⋂C1([0,T);Hs-1).


## 4. Proofs of Theorems2.2 and 2.3

Before establishing the proofs of Theorems2.2 and 2.3, we give several lemmas.Lemma 4.1 (Kato and Ponce [26]).
Ifr≥0, then Hr⋂L∞ is an algebra. Moreover,
(4.1)‖uv‖Hr≤c(‖u‖L∞‖v‖Hr+‖u‖Hr‖v‖L∞),
where c is a constant depending only on r.Lemma 4.2 (Kato and Ponce [26]).
Letr>0. If u∈Hr⋂W1,∞ and v∈Hr-1⋂L∞, then
(4.2)‖[Λr,u]v‖L2≤c(‖∂xu‖L∞‖Λr-1v‖L2+‖Λru‖L2‖v‖L∞),
where [Λr,u]v=Λr(uv)-uΛrv.
Using the first equation of problem (2.2) gives rise to
(4.3)ddt[∫R(u2+ux2)dx]+(a-2b)∫R(ux)3dx=2β∫Ruuxndx,
from which one has
(4.4)∫R(u2+ux2)dx+∫0t[∫R((a-2b)ux3-2βuuxn)dx]dτ=∫R(u02+u0x2)dx.Lemma 4.3.
Lets≥3/2, and the function u(t,x) is a solution of the problem (2.2) and the initial data u0(x)∈Hs, then it holds that
(4.5)‖u‖L∞≤c‖u‖H1≤c‖u0‖H1ec0∫0t(‖ux‖L∞+‖ux‖L∞n-1)dτ,
where c0=1/2max(|a-2b|,|β|).
Forq∈(0,s-1], there is a constant c depending only on q such that
(4.6)∫R(Λq+1u)2dx≤∫R(Λq+1u0)2dx+c∫0t(∥ux∥L∞+∥ux∥L∞n-1)‖u‖Hq+12dτ.
If q∈[0,s-1], there is a constant c depending only on q such that
(4.7)‖ut‖Hq≤c‖u‖Hq+1(1+‖u‖H1+‖ux‖L∞n-1).Proof.
Using∥u∥H12≤c∫R(u2+ux2)dx and (4.4) derives (4.5).
We write (1.5) in the equivalent form
(4.8)ut-utxx=-[2ku+m2u2]x+b2∂x3u2-3b-a2∂x(ux2)+βuxn.
Applying∂x2=-Λ2+1 and the Parseval’s equality gives rise to
(4.9)∫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 on both sides of (4.8), noting the above equality, and integrating the new equation with respect to x by parts, we obtain the equation
(4.10)12ddt[∫R((Λqu)2+(Λqux)2)dx]=-∫R(Λqu)Λq[2ku+m2u2]xdx-b∫R(Λq+1u)Λq+1(uux)dx+3b-a2∫R(Λqux)Λq(ux2)dx+b∫R(Λqu)Λq(uux)dx+β∫RΛquΛquxndx.
We will estimate each of the terms on the right-hand side of (4.10). For the first and the fourth terms, using integration by parts, the Cauchy-Schwartz inequality, and Lemmas 4.1-4.2, we have
(4.11)∫R(Λqu)Λq(uux)dx=∫R(Λqu)[Λq(uux)-uΛqux]dx+∫R(Λqu)uΛquxdx≤c‖u‖Hq(‖ux‖L∞‖u‖Hq+‖ux‖L∞‖u‖Hq)+12‖ux‖L∞‖Λqu‖L22≤c‖u‖Hq2‖ux‖L∞,
where c only depends on q. Using the above estimate to the second term yields
(4.12)∫R(Λq+1u)Λq+1(uux)dx≤c‖u‖Hq+12‖ux‖L∞.
For the third term, using Lemma 4.1 gives rise to
(4.13)∫R(Λqux)Λq(ux2)dx≤‖Λqux‖L2‖Λqux2‖L2≤c‖u‖Hq+1‖ux‖L∞‖ux‖Hq≤c‖u‖Hq+12‖ux‖L∞.
For the last term, using Lemma4.1 repeatedly, we get
(4.14)|∫R(Λqu)Λq(ux)ndx|≤‖u‖Hq‖uxn‖Hq≤c‖u‖Hq+12‖ux‖L∞n-1.
It follows from (4.10)–(4.14) that
(4.15)12∫R[(Λqu)2+(Λqux)2]dx-12∫R[(Λqu0)2+(Λqu0x)2]dx≤c∫0t(‖ux‖L∞+‖ux‖L∞n-1)‖u‖Hq+12dτ,
which results in (4.6). Applying the operator (1-∂x2)-1 on both sides of (4.8) yields the equation
(4.16)ut=(1-∂x2)-1[-[2ku+m2u2]x+b2∂x3u2-3b-a2∂x(ux2)+βuxn].
Multiplying both sides of (4.16) by (Λqut)Λq for q∈[0,s-1] and integrating the resultant equation by parts give rise to
(4.17)∫R(Λqut)2dx=∫R(Λqut)(1-∂x2)-1Λq[-[2ku+m2u2]x+b2∂x3u2-3b-a2∂x(ux2)+βuxn]dx.
On the right-hand side of  (4.17), we have
(4.18)|∫R(Λqut)(1-∂x2)-1Λq(-2kux)dx|≤|2k|‖ut‖Hq‖u‖Hq,(4.19)|∫R(Λqut)(1-∂x2)-1Λq∂x(-m2u2-3b-a2ux2)dx|≤‖ut‖Hq×(∫R(1+ξ2)q-1dξ(∫R[m2û(ξ-η)û(η)+3b-a2ux̂(ξ-η)ux̂(η)]dη)2)1/2≤c‖ut‖Hq(∫Rc(‖u‖Hq‖u‖L2+‖ux‖L2‖ux‖Hq)1+ξ2dξ)1/2≤c‖ut‖Hq‖u‖H1‖u‖Hq+1,
in which we have used Lemma 4.1. As
(4.20)∫R(Λqut)(1-∂x2)-1Λq∂x2(uux)dx=-∫R(Λqut)Λq(uux)dx+∫R(Λqut)(1-∂x2)-1Λq(uux)dx,
by using ∥uux∥Hq≤c∥u2∥Hq+1≤c∥u∥L∞∥u∥Hq+1≤c∥u∥H1∥u∥Hq+1, we have
(4.21)|∫R(Λqut)Λq(uux)dx|≤c‖ut‖Hq‖uux‖Hq≤c‖ut‖Hq‖u‖H1‖u‖Hq+1,(4.22)|∫R(Λqut)(1-∂x2)-1Λq(uux)dx|≤c‖ut‖Hq‖u‖H1‖u‖Hq+1.
Using the Cauchy-Schwartz inequality and Lemma4.1 yields
(4.23)|∫R(Λqut)(1-∂x2)-1Λq(uxn)dx|≤c‖ut‖Hq‖ux‖L∞n-1‖u‖Hq+1.
Applying (4.18)–(4.23) to (4.17) yields the inequality
(4.24)‖ut‖Hq≤c‖u‖Hq+1(1+‖u‖H1+‖ux‖n-1),
for a constant c>0.Lemma 4.4.
Fors>0,u0∈Hs(R), and uɛ0=ϕɛ⋆u0, the following estimates hold for any ɛ with ɛ∈(0,1/4):(4.25)‖uɛ0x‖L∞≤c‖u0x‖L∞,‖uɛ0‖Hq≤cifq≤s,∥uɛ0∥Hq≤cɛ(s-q)/4ifq>s,‖uɛ0-u0‖Hq≤cɛ(s-q)/4ifq≤s,‖uɛ0-u0‖Hs=o(1),
where c is a constant independent of ɛ.The proof of this lemma can be found in [21].Applying Lemmas4.3 and 4.4, we can now state the following lemma, which plays an important role in proving existence of weak solutions.Lemma 4.5.
Fors≥1 and u0∈Hs(R), there exists a constant c independent of ɛ, such that the solution uɛ of problem (2.3) satisfies
(4.26)‖uɛ‖H1≤cec0∫0t(‖uɛx‖L∞+‖uɛx‖L∞n-1)dτfort∈[0,Tɛ),
where c0=1/2max(|a-2b|,|β|).Proof.
The proof can be directly obtained from Lemma4.4 and inequality (4.5).Proof of Theorem2.2.
Using notationu=uɛ and differentiating (4.16) with respect to x give rise to
(4.27)uxt+buuxx+a-b2ux2=2ku+m-b2u2-Λ-2[2ku+m-b2u2+3b-a2ux2-β(uxn)x].
Using
(4.28)∫Ruuxx(ux)2p+1dx=∫Ru(ux)2p+1dux=-∫Rux[(ux)2p+2+(2p+1)u(ux)2puxx]dx,
we get
(4.29)∫Ruuxx(ux)2p+1dx=-12p+2∫R(ux)2p+3dx.
Lettingp>0 be an integer and multiplying (4.27) by (ux)2p+1 and then integrating the resulting equation with respect to x yield the equality
(4.30)12p+2ddt∫R(ux)2p+2dx+(a-b)p+a-2b2p+2∫R(ux)2p+3dx=∫R(ux)2p+1(2ku+m-b2u2)dx-∫R(ux)2p+1Λ-2[2ku+m-b2u2+3b-a2ux2-β(uxn)x]dx.
Applying the Hölder’s inequality, we get
(4.31)12p+2ddt∫R(ux)2p+2dx≤{(∫R|2ku+m-b2u2|2p+2dx)1/(2p+2)+(∫R|G|2p+2dx)1/(2p+2)}(∫R|ux|2p+2dx)(2p+1)/(2p+2)+|(a-b)p+a-2b2p+2|‖ux‖L∞∫R|ux|2p+2dx,
or
(4.32)ddt(∫R(ux)2p+2dx)1/(2p+2)≤{(∫R|2ku+m-b2u2|2p+2dx)1/(2p+2)+(∫R|G|2p+2dx)1/(2p+2)}+|(a-b)p+a-2b2p+2|‖ux‖L∞(∫R|ux|2p+2dx)1/(2p+2),
where
(4.33)G=Λ-2[2ku+m-b2u2+3b-a2ux2-β(uxn)x].
Since ∥f∥Lp→∥f∥L∞ as p→∞ for any f∈L∞⋂L2, integrating (4.32) with respect to t and taking the limit as p→∞ result in the estimate
(4.34)‖ux‖L∞≤‖u0x‖L∞+∫0t[c(‖2ku+m-b2u2‖L∞+‖G‖L∞)+a-b2‖ux‖L∞2]dτ.
Using the algebraic property of Hs0(R) with s0>1/2 and Lemma 4.5 leads to
(4.35)‖2ku+m-b2u2‖L∞≤c‖2ku+m-b2u2‖H1/2+≤c(‖u‖H1+‖u‖H12)≤ce2c0∫0t(‖ux‖L∞+‖ux‖L∞n-1)dτ,(4.36)‖G‖L∞=‖Λ-2[2ku+m-b2u2+3b-a2ux2-β(uxn)x]‖L∞≤c(‖Λ-2(2ku+m-b2u2)‖H1/2++‖Λ-2ux2‖H1/2++‖Λ-2(uxn)x‖H1/2+)≤c(‖2ku+m-b2u2‖H1/2++‖u‖H1‖ux‖L∞+‖uxn‖H0)≤c(‖u‖H1+‖u‖H12+‖u‖H1‖ux‖L∞+‖u‖H1‖ux‖L∞n-1)≤c(1+‖ux‖L∞+‖ux‖L∞n-1)e2c0∫0t(‖ux‖L∞+‖ux‖L∞n-1)dτ,
where c is independent of ɛ, c0=1/2max(|a-2b|,|β|) and H1/2+ means that there exists a sufficiently small δ>0 such that ∥uɛ∥H1/2+=∥uɛ∥H1/2+δ. From Lemma 4.5, we have
(4.37)∫0t‖G‖L∞dτ≤c∫0t(1+‖ux‖L∞+‖ux‖L∞n-1)×e2c0∫0τ(‖ux‖L∞+‖ux‖L∞n-1)dξdτ.
Applying (4.25), (4.34), (4.35), and (4.37) and writing out the subscript ɛ of u, we obtain
(4.38)‖uɛx‖L∞≤c‖u0x‖L∞+c∫0t[(1+‖uɛx‖L∞+‖uɛx‖L∞n-1)×e2c0∫0τ(‖uɛx‖L∞+‖uɛx‖L∞n-1)dξ+∥uɛx∥L∞2]dτ.
It follows from the contraction mapping principle that there is aT>0 such that the equation
(4.39)‖W‖L∞=c‖u0x‖L∞+c∫0t[(1+‖W‖L∞+‖W‖L∞n-1)×e2c0∫0τ(‖W‖L∞+‖W‖L∞n-1)dξ+‖W‖L∞2]dτ
has a unique solution W∈C[0,T]. From (4.39), we know that the variable T only depends on c and ∥u0x∥L∞. Using the theorem presented on page  51 in [16] or Theorem  2 in Section 1.1 in [27] derives that there are constants T>0 and c>0 independent of ɛ such that ∥uɛx∥L∞≤W(t) for arbitrary t∈[0,T], which leads to the conclusion of Theorem  2.Remark 4.6.
Under the assumptions of Theorem2.2, there exist two constants T and c, both independent of ɛ, such that the solution uɛ of problem (2.3) satisfies ∥uɛx∥L∞≤c for any t∈[0,T]. This states that in Lemma 4.5, there exists a T independent of ɛ such that (4.26) holds.
Using Theorem2.2, Lemma 4.5, (4.6), (4.7), notation uɛ=u, and Gronwall’s inequality results in the inequalities
(4.40)‖uɛ‖Hq≤cexp⁡[c∫0t(‖uɛx‖L∞+‖uɛx‖L∞n-1)dτ]≤c,‖uɛt‖Hr≤‖uɛ‖Hr+1(1+‖uɛ‖H1+‖uɛx‖L∞n-1)≤c,
where q∈(0,s],r∈(0,s-1], and t∈[0,T). It follows from Aubin’s compactness theorem that there is a subsequence of {uɛ}, denoted by {uɛn1}, such that {uɛn1} and their temporal derivatives {uɛn1t} 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, {uɛn1} is convergent to the function u strongly in the space L2([0,T],Hq(-R1,R1)) for q∈(0,s], and {uɛn1t} converges to ut strongly in the space L2([0,T],Hr(-R1,R1)) for r∈[0,s-1].Proof of Theorem2.3.
From Theorem2.2, we know that {uɛn1x}(ɛn1→0) is bounded in the space L∞. Thus, the sequences{uɛn1}, {uɛn1x}, {uɛn1x2},and {uɛn1xn} are weakly convergent to u, ux, ux2, and uxn in L2([0,T],Hr(-R1,R1)) for any r∈[0,s-1), separately. Hence, u satisfies the equation
(4.41)-∫0T∫Ru(gt-gxxt)dxdt=∫0T∫R[(2ku+m2u2+3b-a2ux2)gx-b2u2gxxx-βuxng]dxdt,
with u(0,x)=u0(x) and g∈C0∞. Since X=L1([0,T]×R) is a separable Banach space and {uɛn1x} is a bounded sequence in the dual space X*=L∞([0,T]×R) of X, there exists a subsequence of {uɛn1x}, still denoted by {uɛn1x}, weakly star convergent to a function v in L∞([0,T]×R). As {uɛn1x} weakly converges to ux in L2([0,T]×R), it results that ux=v almost everywhere. Thus, we obtain ux∈L∞([0,T]×R).



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*Source: 872187-2012-05-29.xml*