Defining

$\varphi (x)=\{\begin{array}{cc}{e}^{\frac{1}{{x}^{2}-1}},\hfill & |x|<1,\hfill \\ 0,\hfill & |x|\ge 1,\hfill \end{array}$

(7)

and setting the mollifier

${\varphi}_{\epsilon}(x)={\epsilon}^{-\frac{1}{4}}\varphi ({\epsilon}^{-\frac{1}{4}}x)$ with

$0<\epsilon <\frac{1}{4}$ and

${u}_{\epsilon ,0}={\varphi}_{\epsilon}\star {u}_{0}$, we know that

${u}_{\epsilon ,0}\in {C}^{\mathrm{\infty}}$ for any

${u}_{0}\in {H}^{s}$,

$s>0$ (see Lai and Wu [

13]). In fact, choosing the mollifier properly, we have

${\parallel {u}_{\epsilon ,0}\parallel}_{{H}^{1}(R)}\le {\parallel {u}_{0}\parallel}_{{H}^{1}(R)}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{u}_{\epsilon ,0}\to {u}_{0}\phantom{\rule{1em}{0ex}}\text{in}{H}^{1}(R).$

(8)

The existence of a weak solution to Cauchy problem (3) will be established by proving the compactness of a sequence of smooth functions

${\{{u}_{\epsilon}\}}_{\epsilon >0}$ solving the following viscous problem:

$\{\begin{array}{c}\frac{\partial {u}_{\epsilon}}{\partial t}+{u}_{\epsilon}\frac{\partial {u}_{\epsilon}}{\partial x}+\frac{\partial {P}_{\epsilon}}{\partial x}=\epsilon \frac{{\partial}^{2}{u}_{\epsilon}}{\partial {x}^{2}},\hfill \\ \frac{\partial {P}_{\epsilon}}{\partial x}={\mathrm{\Lambda}}^{-2}{\partial}_{x}[f({u}_{\epsilon})-\frac{1}{2}{u}_{\epsilon}^{2}+\frac{1}{2}{(\frac{\partial {u}_{\epsilon}}{\partial x})}^{2}-\beta {u}_{\epsilon}^{2m}\frac{\partial {u}_{\epsilon}}{\partial x}]\hfill \\ \phantom{\frac{\partial {P}_{\epsilon}}{\partial x}=}+\lambda {\mathrm{\Lambda}}^{-2}{({u}_{\epsilon})}^{2N+1}+2m\beta {\mathrm{\Lambda}}^{-2}[{u}_{\epsilon}^{2m-1}{(\frac{\partial {u}_{\epsilon}}{\partial x})}^{2}],\hfill \\ {u}_{\epsilon}(0,x)={u}_{\epsilon ,0}(x).\hfill \end{array}$

(9)

Now we start our analysis by establishing the following well-posedness result for problem (9).

**Lemma 3.1** *Provided that* ${u}_{0}\in {H}^{1}(R)$,

*for any* $\sigma \ge 3$,

*there exists a unique solution* ${u}_{\epsilon}\in C([0,\mathrm{\infty});{H}^{\sigma}(R))$ *to Cauchy problem* (9).

*Moreover*,

*for any* $t>0$,

*it holds that* *Proof* For any $\sigma \ge 3$ and ${u}_{0}\in {H}^{1}(R)$, we have ${u}_{\epsilon ,0}\in C([0,\mathrm{\infty});{H}^{\sigma}(R))$. From Theorem 2.3 in [21], we conclude that problem (9) has a unique solution ${u}_{\epsilon}\in C([0,\mathrm{\infty});{H}^{\sigma}(R))$ for an arbitrary $\sigma >3$.

We know that the first equation in system (9) is equivalent to the form

$\begin{array}{rcl}\frac{\partial {u}_{\epsilon}}{\partial t}-\frac{{\partial}^{3}{u}_{\epsilon}}{\partial t{x}^{2}}+\frac{\partial f({u}_{\epsilon})}{\partial x}& =& 2\frac{\partial {u}_{\epsilon}}{\partial x}\frac{{\partial}^{2}{u}_{\epsilon}}{\partial {x}^{2}}+{u}_{\epsilon}\frac{{\partial}^{3}{u}_{\epsilon}}{\partial {x}^{3}}-\lambda {u}_{\epsilon}^{2N+1}+\beta {u}_{\epsilon}^{2m}\frac{{\partial}^{2}{u}_{\epsilon}}{\partial {x}^{2}}\\ +\epsilon (\frac{{\partial}^{2}{u}_{\epsilon}}{\partial {x}^{2}}-\frac{{\partial}^{4}{u}_{\epsilon}}{\partial {x}^{4}}),\end{array}$

(12)

from which we derive that

which completes the proof. □

From Lemma 3.1 and (8), we have

${\parallel {u}_{\epsilon}\parallel}_{{L}^{\mathrm{\infty}}(R)}\le {\parallel {u}_{\epsilon}\parallel}_{{H}^{1}(R)}\le {\parallel {u}_{\epsilon ,0}\parallel}_{{H}^{1}(R)}\le {\parallel {u}_{0}\parallel}_{{H}^{1}(R)}.$

(14)

Differentiating the first equation of problem (9) with respect to

*x* and writing

$\frac{\partial {u}_{\epsilon}}{\partial x}={q}_{\epsilon}$, we obtain

**Lemma 3.2** *Let* $0<\gamma <1$,

$T>0$ *and* $a,b\in R$,

$a<b$.

*Then there exists a positive constant* ${c}_{1}$ *depending only on* ${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$,

*γ*,

*T*,

*a*,

*b* *and the coefficients of Eq*. (1),

*but independent of ε*,

*such that the space higher integrability estimate holds* ${\int}_{0}^{T}{\int}_{a}^{b}{\left|\frac{\partial {u}_{\epsilon}(t,x)}{\partial x}\right|}^{2+\gamma}\phantom{\rule{0.2em}{0ex}}dx\le {c}_{1},$

(16)

*where* ${u}_{\epsilon}={u}_{\epsilon}(t,x)$ *is the unique solution of problem* (9).

The proof is similar to that of Proposition 3.2 presented in Xin and Zhang [10] (also see Coclite *et al.* [20]). Here we omit it.

**Lemma 3.3** *There exists a positive constant* *C* *depending only on* ${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$ *and the coefficients of Eq*. (1)

*such that* *where* ${u}_{\epsilon}={u}_{\epsilon}(t,x)$ *is the unique solution of system* (9).

Due to strong similarities with the proof of Lemma 5.1 presented in Coclite *et al.* [20], we do not prove Lemma 3.3 here.

**Lemma 3.4** *Assume that* ${u}_{\epsilon}={u}_{\epsilon}(t,x)$ *is the unique solution of* (9).

*For an arbitrary* $T>0$,

*there exists a positive constant* *C* *depending only on* ${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$ *and the coefficients of Eq*. (1)

*such that the following one*-

*sided* ${L}^{\mathrm{\infty}}$ *norm estimate on the first*-

*order spatial derivative holds*:

$\frac{\partial {u}_{\epsilon}(t,x)}{\partial x}\le \frac{4}{t}+C\phantom{\rule{1em}{0ex}}\mathit{\text{for}}\phantom{\rule{0.1em}{0ex}}(t,x)\in [0,\mathrm{\infty})\times R.$

(19)

*Proof* From (15) and Lemma 3.3, we know that there exists a positive constant

*C* depending only on

${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$ and the coefficients of Eq. (

1) such that

${\parallel {Q}_{\epsilon}(t,x)\parallel}_{{L}^{\mathrm{\infty}}(R)}\le C$. Therefore,

$\frac{\partial {q}_{\epsilon}}{\partial t}+{u}_{\epsilon}\frac{\partial {q}_{\epsilon}}{\partial x}+\frac{1}{2}{q}_{\epsilon}^{2}+\beta {({u}_{\epsilon})}^{2m}{q}_{\epsilon}={Q}_{\epsilon}(t,x)\le C.$

(20)

Let

$f=f(t)$ be the solution of

$\frac{df}{dt}+\frac{1}{2}{f}^{2}+\beta {\left({u}^{\ast}\right)}^{2m}f=C,\phantom{\rule{1em}{0ex}}t>0,\phantom{\rule{2em}{0ex}}f(0)={\parallel \frac{\partial {u}_{\epsilon ,0}}{\partial x}\parallel}_{{L}^{\mathrm{\infty}}(R)},$

(21)

where

${u}_{\epsilon}^{\ast}$ is the value of

${u}_{\epsilon}(t,x)$ when

${sup}_{x\in R}{q}_{\epsilon}(t,x)=f(t)$. From the comparison principle for parabolic equations, we get

${q}_{\epsilon}(t,x)\le f(t).$

(22)

Using (14) and

$-\beta {({u}^{\ast})}^{2m}f\le {\rho}^{2}{f}^{2}+\frac{1}{4{\rho}^{2}}{\beta}^{2}{({u}^{\ast})}^{4m}$, we derive that

$\begin{array}{rcl}\frac{df}{dt}& =& C-\frac{1}{2}{f}^{2}-\beta {\left({u}^{\ast}\right)}^{2m}f\le C-\frac{1}{2}{f}^{2}+{\rho}^{2}{f}^{2}+\frac{1}{4{\rho}^{2}}{\beta}^{2}{\left({u}^{\ast}\right)}^{4m}\\ \le & C-\frac{1}{4}{f}^{2}+{C}_{1},\end{array}$

(23)

where

$\parallel \frac{1}{4{\rho}^{2}}{\beta}^{2}{({u}^{\ast})}^{4m}\parallel \le {C}_{1}$ and

$\rho =\frac{1}{2}$. Setting

${M}_{0}=C+{C}_{1}$, we obtain

$\frac{df}{dt}+\frac{1}{4}{f}^{2}\le {M}_{0}.$

(24)

Letting $F(t)=\frac{4}{t}+2\sqrt{{M}_{0}}$, we have $\frac{dF(t)}{dt}+\frac{1}{4}{F}^{2}(t)-{M}_{0}=\frac{4\sqrt{{M}_{0}}}{t}>0$. From the comparison principle for ordinary differential equations, we get $f(t)\le F(t)$ for all $t>0$. Therefore, by this and (22), the estimate (19) is proved. □

**Lemma 3.5** *For* ${u}_{0}\in {H}^{1}(R)$,

*there exists a sequence* ${\{{\epsilon}_{j}\}}_{j\in N}$ *tending to zero and a function* $u\in {L}^{\mathrm{\infty}}([0,\mathrm{\infty});{H}^{1}(R))\cap {H}^{1}([0,T]\times R)$ *such that*,

*for each* $T\ge 0$,

*it holds that* *where* ${u}_{\epsilon}={u}_{\epsilon}(t,x)$ *is the unique solution of* (9).

**Lemma 3.6** *There exists a sequence* ${\{{\epsilon}_{j}\}}_{j\in N}$ *tending to zero and a function* $Q\in {L}^{\mathrm{\infty}}([0,\mathrm{\infty})\times R)$ *such that for each* $1<p<\mathrm{\infty}$,

${Q}_{{\epsilon}_{j}}\to Q\phantom{\rule{1em}{0ex}}\mathit{\text{strongly in}}\phantom{\rule{0.1em}{0ex}}{L}_{\mathrm{loc}}^{p}([0,\mathrm{\infty})\times R).$

(27)

The proofs of Lemmas 3.5 and 3.6 are similar to those of Lemmas 5.2 and 5.3 in [20]. Here we omit their proofs.

Throughout this paper, we use overbars to denote weak limits (the space in which these weak limits are taken is ${L}_{\mathrm{loc}}^{r}([0,\mathrm{\infty})\times R)$ with $1<r<\frac{3}{2}$).

**Lemma 3.7** *There exists a sequence* ${\{{\epsilon}_{j}\}}_{j\in N}$ *tending to zero and two functions* $q\in {L}_{\mathrm{loc}}^{p}([0,\mathrm{\infty})\times R)$,

$\overline{{q}^{2}}\in {L}_{\mathrm{loc}}^{r}([0,\mathrm{\infty})\times R)$ *such that* *for each* $1<p<3$ *and* $1<r<\frac{3}{2}$.

*Moreover*,

${q}^{2}(t,x)\le \overline{{q}^{2}}(t,x)\phantom{\rule{1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\mathit{\text{for almost every}}\phantom{\rule{0.1em}{0ex}}(t,x)\in [0,\mathrm{\infty})\times R$

(30)

*and*
$\frac{\partial u}{\partial x}=q\phantom{\rule{1em}{0ex}}\mathit{\text{in the sense of distributions on}}\phantom{\rule{0.1em}{0ex}}[0,\mathrm{\infty})\times R.$

(31)

*Proof* (28) and (29) are a direct consequence of Lemmas 3.1 and 3.2. Inequality (30) is valid because of the weak convergence in (29). Finally, (31) is a consequence of the definition of ${q}_{\epsilon}$, Lemma 3.5 and (28). □

In the following, for notational convenience, we replace the sequence ${\{{u}_{{\epsilon}_{j}}\}}_{j\in N}$, ${\{{q}_{{\epsilon}_{j}}\}}_{j\in N}$ and ${\{{Q}_{{\epsilon}_{j}}\}}_{j\in N}$ by ${\{{u}_{\epsilon}\}}_{\epsilon >0}$, ${\{{q}_{\epsilon}\}}_{\epsilon >0}$ and ${\{{Q}_{\epsilon}\}}_{\epsilon >0}$, respectively.

Using (28), we conclude that for any convex function

$\eta \in {C}^{1}(R)$ with

${\eta}^{\prime}$ being bounded and Lipschitz continuous on

*R* and for any

$1<p<3$, we get

Multiplying Eq. (

15) by

${\eta}^{\prime}({q}_{\epsilon})$ yields

**Lemma 3.8** *For any convex* $\eta \in {C}^{1}(R)$ *with* ${\eta}^{\prime}$ *being bounded and Lipschitz continuous on R*,

*it holds that* $\overline{\frac{\partial \eta (q)}{\partial t}}+\frac{\partial}{\partial x}\left(u\overline{\eta (q)}\right)\le \overline{q\eta (q)}-\frac{1}{2}\overline{{\eta}^{\prime}(q){q}^{2}}-\beta {u}^{2m}\overline{q{\eta}^{\prime}(q)}+Q(t,x)\overline{{\eta}^{\prime}(q)}$

(35)

*in the sense of distributions on* $[0,\mathrm{\infty})\times R$. *Here* $\overline{q\eta (q)}$ *and* $\overline{{\eta}^{\prime}(q){q}^{2}}$ *denote the weak limits of* ${q}_{\epsilon}\eta ({q}_{\epsilon})$ *and* ${q}_{\epsilon}^{2}{\eta}^{\prime}({q}_{\epsilon})$ *in* ${L}_{\mathrm{loc}}^{r}([0,\mathrm{\infty})\times R)$, $1<r<\frac{3}{2}$, *respectively*.

*Proof* In (34), by the convexity of *η*, (14), Lemmas 3.5, 3.6 and 3.7, taking limit for $\epsilon \to 0$ gives rise to the desired result. □

**Remark 3.9** From (28) and (29), we know that

$q={q}_{+}+{q}_{-}=\overline{{q}_{+}}+\overline{{q}_{-}},\phantom{\rule{2em}{0ex}}{q}^{2}={({q}_{+})}^{2}+{({q}_{-})}^{2},\phantom{\rule{2em}{0ex}}\overline{{q}^{2}}=\overline{{({q}_{+})}^{2}}+\overline{{({q}_{-})}^{2}}$

(36)

almost everywhere in

$[0,\mathrm{\infty})\times R$, where

${\xi}_{+}:={\xi}_{\chi [0,+\mathrm{\infty})}(\xi )$,

${\xi}_{-}:={\xi}_{\chi (-\mathrm{\infty},0]}(\xi )$ for

$\xi \in R$. From Lemma 3.4 and (28), we have

${q}_{\epsilon}(t,x),\phantom{\rule{2em}{0ex}}q(t,x)\le \frac{4}{t}+C\phantom{\rule{1em}{0ex}}\text{for}t0,x\in R,$

(37)

where *C* is a constant depending only on ${\parallel {u}_{0}\parallel}_{{H}^{1}(R)}$ and the coefficients of Eq. (1).

**Lemma 3.10** *In the sense of distributions on* $[0,\mathrm{\infty})\times R$,

*it holds that* $\frac{\partial q}{\partial t}+\frac{\partial}{\partial x}(uq)=\frac{1}{2}\overline{{q}^{2}}-\beta {u}^{2m}q+Q(t,x).$

(38)

*Proof* Using (15), Lemmas 3.5 and 3.6, (28), (29) and (31), the conclusion (38) holds by taking limit for $\epsilon \to 0$ in (15). □

The next lemma contains a generalized formulation of (38).

**Lemma 3.11** *For any* $\eta \in {C}^{1}(R)$ *with* ${\eta}^{\prime}\in {L}^{\mathrm{\infty}}(R)$,

*it holds that* $\frac{\partial \eta (q)}{\partial t}+\frac{\partial}{\partial x}(u\eta (q))=q\eta (q)+(\frac{1}{2}\overline{{q}^{2}}-{q}^{2}){\eta}^{\prime}(q)-\beta {u}^{2m}q{\eta}^{\prime}(q)+Q(t,x){\eta}^{\prime}(q)$

(39)

*in the sense of distributions on* $[0,\mathrm{\infty})\times R$.

*Proof* Let

${\{{\omega}_{\delta}\}}_{\delta}$ be a family of mollifiers defined on

*R*. Denote

${q}_{\delta}(t,x):=(q(t,\cdot )\star {\omega}_{\delta})(x)$, where the ⋆ is the convolution with respect to

*x* variable. Multiplying (38) by

${\eta}^{\prime}({q}_{\delta})$ yields

$\begin{array}{rcl}\frac{\partial \eta ({q}_{\delta})}{\partial t}& =& {\eta}^{\prime}({q}_{\delta})\frac{\partial {q}_{\delta}}{\partial t}\\ =& {\eta}^{\prime}({q}_{\delta})[\frac{1}{2}\overline{{q}^{2}}\star {\omega}_{\delta}-\beta {u}^{2m}{q}_{\delta}+Q(t,x)\star {\omega}_{\delta}-{q}^{2}\star {\omega}_{\delta}-u\frac{\partial q}{\partial x}\star {\omega}_{\delta}]\end{array}$

(40)

and

$\frac{\partial}{\partial x}(u\eta ({q}_{\delta}))=q\eta ({q}_{\delta})+u{\eta}^{\prime}({q}_{\delta})\left(\frac{\partial {q}_{\delta}}{\partial x}\right).$

(41)

Using the boundedness of *η*, ${\eta}^{\prime}$ and letting $\delta \to 0$ in the above two equations, we obtain (39). □