The Kato theorem for abstract differential equations is applied to establish the local well-posedness of the strong solution for a nonlinear generalized Novikov equation in space C([0,T),Hs(R))∩C1([0,T),Hs-1(R)) with s>(3/2). The existence of weak solutions for the equation in lower-order Sobolev space Hs(R) with 1≤s≤(3/2) is acquired.

1. Introduction

The Novikov equation with cubic nonlinearities takes the formut-utxx+4u2ux=3uuxuxx+u2uxxx,
which was derived by Vladimir Novikov in a symmetry classification of nonlocal partial differential equations [1]. Using the perturbed symmetry approach, Novikov was able to isolate (1.1) and investigate its symmetries. A scalar Lax pair for it was discovered in [1, 2] and was shown to be related to a negative flow in the Sawada-Kotera hierarchy. Many conserved quantities were found as well as a bi-Hamiltonian structure. The scattering theory was employed by Hone et al. [3] to find nonsmooth explicit soliton solutions with multiple peaks for (1.1). This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations (see [4–10]). Ni and Zhou [11] proved that the Novikov equation associated with initial value is locally well-posedness in Sobolev space Hs with s>(3/2) by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for (1.1) were established. It is shown in [12] that the local well-posedness for the periodic Cauchy problem of the Novikov equation in Sobolev space Hs with s>(5/2). The orbit invariants are employed to get the existence of periodic global strong solution if the Sobolev index s≥3 and a sign condition holds. For analytic initial data, the existence and uniqueness of analytic solutions for (1.1) are also obtained in [12].

In this paper, motivated by the work in [7, 13], we study the following generalized Novikov equation:ut-utxx+4u2ux=3uuxuxx+u2uxxx+β∂x[(ux)N],
where N≥1 is a natural number. In fact, (1.1) has the property∫R(u2+ux2)dx=constant.

Due to the term β∂x[(ux)N] appearing in (1.2), the conservation law (1.3) for (1.2) is not valid. This brings us a difficulty to obtain the bounded estimates for the solution of (1.2). However, we will overcome this difficulty to investigate the local existence and uniqueness of the solution to (1.2) subject to initial value u0(x)∈Hs(R) with s>(3/2). Meanwhile, a sufficient condition is presented to guarantee the existence of local weak solution for (1.2).

The main tasks of this work are two-fold. Firstly, by using the Kato theorem for abstract differential equations, we establish the local existence and uniqueness of solutions for the (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 there exist local 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 [8].

2. Main Results

Firstly, some notations are presented as follows.

The space of all infinitely differentiable functions ϕ(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‖h‖Hs=(∫R(1+|ξ|2)s|ĥ(t,ξ)|2dξ)1/2<∞,
where ĥ(t,ξ)=∫Re-ixξh(t,x)dx.

For T>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.

The Cauchy problem for (1.2) is written in the formut-utxx=-43(u3)x+13∂x3u3-2∂x(uux2)+uuxuxx+β∂x[(ux)N],u(0,x)=u0(x),
which is equivalent tout+u2ux=Λ-2[-3u2ux-32∂x(uux2)-12ux3+β∂x[(ux)N]],u(0,x)=u0(x).

Now, we state our main results.

Theorem 2.1.

Let u0(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 that u0(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

Consider the abstract quasilinear evolution equationdvdt+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}.

A(y)∈L(Y,X) for y∈X with
‖(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.

QA(y)Q-1=A(y)+B(y), where B(y)∈L(X) is bounded, uniformly on bounded sets in Y. Moreover,

‖(B(y)-B(z))w‖X≤ρ2∥y-z∥Y∥w∥X,y,z∈Y,w∈X.

f:Y→Y extends to a map from X into X, is bounded on bounded sets in Y, and satisfies

‖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 [<xref ref-type="bibr" rid="B14">14</xref>])

Assume that (I), (II), and (III) hold. If v0∈Y, there is a maximal T>0 depending only on ∥v0∥Y and a unique solution v to problem (3.1) such that
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
C([0,T);Y)⋂C1([0,T);X).
For problem (2.3), we set A(u)=u2∂x, Y=Hs(R), X=Hs-1(R), Λ=(1-∂x2)1/2,
f(u)=Λ-2[-3u2ux-32∂x(uux2)-12ux3+β∂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 (Ni and Zhou [<xref ref-type="bibr" rid="B11">11</xref>]).

The operator A(u)=u2∂x with u∈Hs(R), s>(3/2) belongs to G(Hs-1,1,β).

Lemma 3.2 (Ni and Zhou [<xref ref-type="bibr" rid="B11">11</xref>]).

Let A(u)=u2∂x with u∈Hs and s>(3/2). Then A(u)∈L(Hs,Hs-1) for all u∈Hs. Moreover,
‖(A(u)-A(z))w‖Hs-1≤ρ1∥u-z∥Hs-1∥w∥Hs,u,z,w∈Hs(R).

Lemma 3.3 (Ni and Zhou [<xref ref-type="bibr" rid="B11">11</xref>]).

For s>(3/2),u,z∈Hs and w∈Hs-1, it holds that B(u)=[Λ,u2∂x]Λ-1∈L(Hs-1) for u∈Hs and
‖(B(u)-B(z))w‖Hs-1≤ρ2∥u-z∥Hs∥w∥Hs-1.

Lemma 3.4.

Let r and q be real numbers such that -r<q≤r. Then
∥uv∥Hq≤c∥u∥Hr∥v∥Hq,ifr>12,∥uv∥Hr+q-1/2≤c∥u∥Hr∥v∥Hq,ifr<12,∥uv∥Hr1≤c∥u∥L∞∥v∥Hr1,ifr1≤0.

The above first two inequalities of this lemma can be found in [14, 15], and the third inequality can be found in [7].

Lemma 3.5.

Letting u,z∈Hs with s>(3/2), then f(u) is bounded on bounded sets in Hs and satisfies
‖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 space Hs0 with s0>(1/2) and s-1>(1/2), we have
‖Λ-2[∂x(uux2)-∂x(zzx2)]‖Hs≤c‖uux2-zzx2‖Hs-1≤c∥u-z∥Hs-1‖ux2‖Hs-1+∥z∥Hs-1‖ux2-zx2‖Hs-1≤c∥u-z∥Hs‖u‖Hs2+∥z∥Hs-1∥u-z∥Hs-1∥u+z∥Hs-1≤c∥u-z∥Hs(∥u∥Hs2+∥z∥Hs-1(∥u∥Hs+∥z∥Hs)),‖Λ-2∂x[(ux)N-(zx)N]‖Hs≤c‖(ux)N-(zx)N‖Hs-1≤c∥ux-zx∥Hs-1∑j=0N-1∥ux∥Hs-1N-j∥zx∥Hs-1j≤c∥u-z∥Hs∑j=0N-1∥u∥HsN-j∥z∥Hsj.
It follows from Lemma 3.4 and s-1>(1/2) that
‖Λ-2[∂x(uux2)-∂x(zzx2)]‖Hs-1≤c‖uux2-zzx2‖Hs-2≤c‖(u-z)ux2‖Hs-2+‖z(ux2-zx2)‖Hs-2≤c∥u-z∥Hs-2‖ux2‖Hs-1+∥z∥Hs-1‖ux2-zx2‖Hs-2≤c∥u-z∥Hs-1∥ux∥Hs-12+∥z∥Hs-1‖ux-zx‖Hs-2‖ux+zx‖Hs-1≤c∥u-z∥Hs-1(∥u∥Hs2+∥z∥Hs-1(∥u∥Hs+∥z∥Hs)),‖Λ-2∂x[(ux)N-(zx)N]‖Hs-1≤c‖(ux)N-(zx)N‖Hs-2≤c‖ux-zx‖Hs-2‖∑j=0N-1uxN-jzxj‖Hs-1≤c‖u-z‖Hs-1∑j=0N-1∥ux∥Hs-1N-j∥zx∥Hs-1j≤c‖u-z‖Hs-1∑j=0N-1∥u∥HsN-j∥z∥Hsj.
From (3.7), and (3.13), we know that (3.11) is valid, while inequality (3.12) follows from (3.14).

Proof of Theorem <xref ref-type="statement" rid="thm2.1">2.1</xref>.

Using the Kato Theorem, Lemmas 3.1–3.3 and 3.5, we know that system (2.2) or problem (2.3) has a unique solution
u(t,x)∈C([0,T);Hs(R))⋂C1([0,T);Hs-1(R)).

4. Existence of Weak Solutions

For s≥2, using the first equation of system (1.3) derivesddt∫R(u2+ux2+2β∫0tuxN+1dτ)dx=0,
from which we have the conservation law∫R(u2+ux2+2β∫0tuxN+1dτ)dx=∫R(u02+u0x2)dx.

Lemma 4.1 (Kato and Ponce [<xref ref-type="bibr" rid="B15">15</xref>]).

If r≥0, then Hr⋂L∞ is an algebra. Moreover
∥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 [<xref ref-type="bibr" rid="B15">15</xref>]).

Letting r>0. If u∈Hr⋂W1,∞ and v∈Hr-1⋂L∞, then
‖[Λr,u]v‖L2≤c(‖∂xu‖L∞‖Λr-1v‖L2+‖Λru‖L2∥v∥L∞).

Lemma 4.3.

Let s≥(3/2) and the function u(t,x) is a solution of problem (2.2) and the initial data u0(x)∈Hs. Then the following results hold:
∥u∥L∞≤∥u∥H1≤c∥u0∥H1e|β|∫0t∥ux∥L∞N-1dτ.

For q∈(0,s-1], there is a constant c such that∫R(Λq+1u)2dx≤∫R[(Λq+1u0)2]dx+c∫0t∥u∥Hq+12(∥ux∥L∞∥u∥L∞+∥ux∥L∞2+∥ux∥L∞N-1)dτ.

For q∈[0,s-1], there is a constant c such that∥ut∥Hq≤c‖u‖Hq+1(∥u∥L∞∥u∥H1+∥u∥L∞∥ux∥L∞+∥ux∥L∞2+∥ux∥L∞N-1).

Proof.

The identity ∥u∥H12=∫R(u2+ux2)dx, (4.2), and the Gronwall inequality result in (4.5).

Using ∂x2=-Λ2+1 and the Parseval equality gives rise to∫RΛquΛq∂x3(u3)dx=-3∫R(Λq+1u)Λq+1(u2ux)dx+3∫R(Λqu)Λq(u2ux)dx.

For q∈(0,s-1], applying (Λqu)Λq to both sides of the first equation of system (2.2) and integrating with respect to x by parts, we have the identity12∫R((Λqu)2+(Λqux)2)dx=-3∫RΛquΛq(u2ux)dx-∫R(Λq+1u)Λq+1(u2ux)dx+2∫R(Λqux)Λq(uux2)dx+∫RΛquΛq(uuxuxx)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
|∫R(Λqu)Λq(u2ux)dx|=|∫R(Λqu)[Λq(u2ux)-u2Λqux]dx+∫R(Λqu)u2Λquxdx|≤c∥u∥Hq(2∥u∥L∞∥ux∥L∞∥u∥Hq+∥ux∥L∞∥u∥L∞∥u∥Hq)+∥u∥L∞∥ux∥L∞∥Λqu∥L22≤c∥u∥Hq2∥u∥L∞∥ux∥L∞.
Using the above estimate to the second term yields
|∫R(Λq+1u)Λq+1(u2ux)dx|≤c∥u∥Hq+12∥u∥L∞∥ux∥L∞.
For the third term, using the Cauchy-Schwartz inequality and Lemma 4.2, we obtain
|∫R(Λqux)Λq(uux2)dx|≤‖Λqux‖L2‖Λq(uux2)‖L2≤c∥u∥Hq+1(∥uux∥L∞∥ux∥Hq+∥ux∥L∞∥uux∥Hq)≤c∥u∥Hq+12∥ux∥L∞∥u∥L∞,
in which we have used ∥uux∥Hq≤c∥(u2)x∥Hq≤c∥u∥L∞∥u∥Hq+1.

For the fouth term in (4.9), using u(ux2)x=(uux2)x-uxux2 results in|∫R(Λqu)Λq(uuxuxx)dx|≤12|∫RΛquxΛq(uux2)dx|+12|∫RΛquΛq[uxux2]dx|=K1+K2.
For K1, it follows from (4.12) that
K1≤c∥u∥Hq+12∥ux∥L∞∥u∥L∞.
For K2, applying Lemma 4.2 derives
K2≤c∥u∥Hq‖uxux2‖Hq≤c∥u∥Hq(∥ux∥L∞‖ux2‖Hq+∥ux∥Hq‖ux2‖L∞)≤c∥u∥Hq+12∥ux∥L∞2.

For the last term in (4.9), using Lemma 4.1 repeatedly results in|∫RΛquxΛq(ux)Ndx|≤c∥ux∥Hq‖uxN‖Hq≤c∥u∥Hq+12∥ux∥L∞N-1.

It follows from (4.10)–(4.16) that there exists a constant c such that12ddt∫R[(Λqu)2+(Λqux)2]dx≤c∥u∥Hq+12(∥ux∥L∞∥u∥L∞+∥ux∥L∞2+∥ux∥L∞N-1).
Integrating both sides of the above inequality with respect to t results in inequality (4.6).

To estimate the norm of ut, we apply the operator (1-∂x2)-1 to both sides of the first equation of system (2.2) to obtain the equationut=(1-∂x2)-1[-43(u3)x+13∂x3(u3)-2∂x(uux2)+uuxuxx+β∂x[(ux)N]].
Applying (Λqut)Λq to both sides of (4.18) for q∈[0,s-1] gives rise to
∫R(Λqut)2dx=∫R(Λqut)Λq-2[∂x(-43u3+13∂x2u3-2uux2+β(ux)N)+uuxuxx]dτ.
For the right-hand of (4.19), we have
∫R(Λqut)(1-∂x2)-1Λq∂x(-43u3-2uux2)dx≤c∥ut∥Hq(∫R(1+ξ2)q-1×[∫R[-43u2̂(ξ-η)û(η)-2uux̂(ξ-η)ux̂(η)]dη]2)1/2≤c∥ut∥Hq∥u∥H1∥u∥Hq+1∥u∥L∞.
Since
∫(Λqut)(1-∂x2)-1Λq∂x2(u2ux)dx=-∫(Λqut)Λq(u2ux)dx+∫(Λqut)(1-∂x2)-1Λq(u2ux)dx,
using Lemma 4.2, ∥u2ux∥Hq≤c∥(u3)x∥Hq≤c∥u∥L∞2∥u∥Hq+1, and ∥u∥L∞≤∥u∥H1, we have
∫(Λqut)Λq(u2ux)dx≤c∥ut∥Hq‖u2ux‖Hq≤c∥ut∥Hq∥u∥L∞∥u∥H1∥u∥Hq+1,|∫(Λqut)(1-∂x2)-1Λq(u2ux)dx|≤c∥ut∥Hq∥u∥L∞∥u∥H1∥u∥Hq+1.
Using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2 yields
|∫(Λqut)(1-∂x2)-1Λq∂x(uxN)dx|≤c∥ut∥Hq∥ux∥L∞N-1∥u∥Hq+1,|∫R(Λqut)(1-∂x2)-1Λq(uuxuxx)dx|≤c∥ut∥Hq‖uuxuxx‖Hq-2≤c∥ut∥Hq‖u(ux2)x‖Hq-2≤c∥ut∥Hq‖[u(ux2)]x-(u)xux2‖Hq-2≤c∥ut∥Hq(‖uux2‖Hq-1+‖uxux2‖Hq-2)≤c∥ut∥Hq(‖uux2‖Hq+‖uxux2‖Hq)≤c∥ut∥Hq∥u∥Hq+1(∥u∥L∞∥ux∥L∞+∥ux∥L∞2),
in which we have used inequality (4.15).

Applying (4.20)–(4.24) into (4.19) yields the inequality∥ut∥Hq≤c∥u∥Hq+1(∥u∥L∞∥u∥H1+∥u∥L∞∥ux∥L∞+∥ux∥L∞2+∥ux∥L∞N-1)
for a constant c>0. This completes the proof of Lemma 4.3.

Definingϕ(x)={e1/(x2-1),|x|<1,0,|x|≥1,
and setting ϕɛ(x)=ɛ-1/4ϕ(ɛ-1/4x) with 0<ɛ<(1/4) and uɛ0=ϕɛ⋆u0, we know that uɛ0∈C∞ for any u0∈Hs(R) and s>0.

It follows from Theorem 2.1 that for each ɛ the Cauchy problemut-utxx=-43(u3)x+13∂x3u3-2∂x(uux2)+uuxuxx+β∂x[(ux)N]=-43(u3)x+13∂x3u3-32∂x(uux2)-12ux3+β∂x[(ux)N],u(0,x)=uɛ0(x),x∈R,
has a unique solution uɛ(t,x)∈C∞([0,T);H∞).

Lemma 4.4.

Under the assumptions of problem (4.27), the following estimates hold for any ɛ with 0<ɛ<(1/4) and s>0:
‖uɛ0x‖L∞≤c1‖u0x‖L∞,‖uɛ0‖Hq≤c1,ifq≤s,‖uɛ0‖Hq≤c1ɛ(s-q)/4,ifq>s,‖uɛ0-u0‖Hq≤c1ɛ(s-q)/4,ifq≤s,‖uɛ0-u0‖Hs=o(1),
where c1 is a constant independent of ɛ.

The proof of this Lemma can be found in Lai and Wu [7].

Lemma 4.5.

If u0(x)∈Hs(R) with s∈[1,(3/2)] such that ∥u0x∥L∞<∞, and uɛ0 is defined as in system (4.27). Then there exist two positive constants T and c, which are independent of ɛ, such that the solution uɛ of problem (4.27) satisfies ∥uɛx∥L∞≤c for any t∈[0,T).

Proof.

Using notation u=uɛ and differentiating both sides of the first equation of problem (4.27) with respect to x give rise to
utx+12uux2+u2uxx=u3-β(ux)N-Λ-2[u3+32uux2+12(ux3)x-β(ux)N].
Integrating by parts leads to
∫Ru2uxx(ux)p+1dx=-1p+1∫Ruux2p+3dx,integerp>0,
from which we obtain
∫R(12uux2+u2uxx)(ux)p+1dx=p-12(p+1)∫Ruux2p+3dx.
Multiplying the above equation by (ux)2p+1 and then integrating the resulting equation with respect to x yield the equality
12p+2ddt∫R(ux)2p+2dx+p-12p+2∫Ru(ux)2p+3dx=∫R(ux)2p+1(u3-βuxN)dx-∫R(ux)2p+1Λ-2[u3+32uux2+12(ux3)x-βuxN]dx.
Applying the Hölder's inequality yields
12p+2ddt∫R(ux)2p+2dx≤((∫R|u3|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)+|p-12p+2|‖uux‖L∞∫R|ux|2p+2dx
or
ddt(∫R(ux)2p+2dx)1/(2p+2)≤(∫R|u3|2p+2dx)1/(2p+2)+β(∫R|uxN|2p+2dx)1/(2p+2)+(∫R|G|2p+2dx)1/(2p+2)+|p-12p+2|‖uux‖L∞(∫R|ux|2p+2dx)1/(2p+2),
where
G=Λ-2[u3+32uux2+12(ux3)x-βuxN].
Since ∥f∥Lp→∥f∥L∞ as p→∞ for any f∈L∞⋂L2, integrating both sides of the inequality (4.34) with respect to t and taking the limit as p→∞ result in the estimate
‖ux‖L∞≤‖u0x‖L∞+∫0tc(‖u‖L∞3+β‖ux‖L∞N+‖G‖L∞+∥u∥L∞‖ux‖L∞2)dτ.
Using the algebra property of Hs0(R) with s0>(1/2) yields (∥uɛ∥H(1/2)+ means that there exists a sufficiently small δ>0 such that ∥uɛ∥H(1/2)+=∥uɛ∥H(1/2)+δ):
‖G‖L∞≤c‖G‖H(1/2)+≤c‖Λ-2[u3+32uux2+12(ux3)x-βuxN]‖H(1/2)+≤c(‖u‖H13+‖Λ-2(uux2)‖H(1/2)++‖Λ-2(ux3)x‖H(1/2)++‖Λ-2(uxN)‖H(1/2)+)≤c(‖u‖H13+‖uux2‖H-1+‖ux3‖H-(1/2)++‖uxN‖H0)≤c(‖u‖H13+‖uux‖H-1‖ux‖L∞+‖ux3‖H0+‖ux‖L∞N-1‖u‖H0)≤(‖u‖H13+∥u∥H12∥ux∥L∞+∥u∥H1‖ux‖L∞2+∥u∥H1‖ux‖L∞N-1),
in which Lemma 3.4 is used. From Lemma 4.3, we get
∫0t∥G∥L∞dτ≤c∫0tec∫0τ∥ux∥L∞N-1dξ(1+∥ux∥L∞+∥ux∥L∞2+∥ux∥L∞N-1)dτ.
Using ∥u∥L∞≤∥u∥H1, from (4.36) and (4.38), it has
∥ux∥L∞≤‖u0x‖L∞+c∫0tec∫0t∥ux∥L∞N-1dτ[1+∥ux∥L∞+∥ux∥L∞2+∥ux∥L∞N-1+∥ux∥L∞N]dτ.

From Lemma 4.4, it follows from the contraction mapping principle that there is a T>0 such that the equation‖W‖L∞=∥u0x∥L∞+c∫0tec∫0t∥W∥L∞N-1dτ[1+‖W‖L∞+‖W‖L∞2+‖W‖L∞N-1+‖W‖L∞N]dτ
has a unique solution W∈C[0,T]. Using the theorem presented at page 51 in [8] yields that there are constants T>0 and c>0 independent of ɛ such that ∥ux∥L∞≤W(t) for arbitrary t∈[0,T], which leads to the conclusion of Lemma 4.5.

Using Lemmas 4.3 and 4.5, notation uɛ=u, and Gronwall's inequality results in the inequalities∥uɛ∥Hq≤CTeCT,∥uɛt∥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 {uɛ}, denoted by {uɛn}, such that {uɛn} and their temporal derivatives {uɛnt} 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, {uɛn} is convergent to the function u strongly in the space L2([0,T],Hq(-R1,R1)) for q∈[0,s) and {uɛnt} 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 (1.2).

Proof of Theorem <xref ref-type="statement" rid="thm2.2">2.2</xref>.

From Lemma 4.5, we know that {uɛnx}(ɛn→0) is bounded in the space L∞. Thus, the sequences {uɛn} and {uɛnxN} are weakly convergent to u and uxN in L2[0,T], Hr(-R,R) for any r∈[0,s-1), respectively. Therefore, u satisfies the equation
-∫0T∫Ru(gt-gxxt)dxdt=∫0T∫R[(43u3+32uux2)gx-12ux3g(x)-13u3gxxx-β(ux)Ngx]dxdt
with u(0,x)=u0(x) and g∈C0∞. Since X=L1([0,T]×R) is a separable Banach space and {uɛnx} is a bounded sequence in the dual space X*=L∞([0,T]×R) of X, there exists a subsequence of {uɛnx}, still denoted by {uɛnx}, weakly star convergent to a function v in L∞([0,T]×R). It derives from the {uɛnx} weakly convergent to ux in L2([0,T]×R) that ux=v almost everywhere. Thus, we obtain ux∈L∞([0,T]×R).

Acknowledgment

This work is supported by the Applied and Basic Project of Sichuan Province (2012JY0020).

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