上海理工大学学报  2022, Vol. 44 Issue (5): 477-489 PDF

Asymptotic stability of the monotone decreasing kink profile solitary-wave solution for generalized river-bed model equation
ZHANG Kun, WANG Chengwei, ZHANG Weiguo
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: The asymptotic stability of monotone decreasing kink profile solitary-wave solution for generalized river-bed model equation was mainly studied. Firstly, by using the theory of planar dynamical system, the existence of kink profile solitary-wave solution of this equation was proved. Secondly, three properties of the monotone decreasing kink profile solitary-wave solution were proved, especially the first-order and second-order derivative estimation formulas of this traveling wave solution were given. Finally, by using anti-derivative strategy, with the help of priori estimate and Young inequality, it was proved that the monotone decreasing kink profile solitary-wave solution of model equation was asymptotically stable.
Key words: generalized river-bed model equation     monotone decreasing kink profile solitary-wave solution     asymptotic stability     energy priori estimate
1 问题的提出

 ${u}_{t}+\varepsilon {u}_{xx}-\delta {u}_{xt}+{u}_{xxx}+f{\left(u\right)}_{x}=0$ (1)

 ${u}_{t}+\varepsilon {u}_{xx}-\delta {u}_{xt}+{u}_{xxx}+2\beta u{u}_{x}=0$ (2)

 $\begin{array}{c}{u}_{t}+2\beta u{u}_{x}+{u}_{xxx}=0\end{array}$

$\delta = 0$ $p = 1$ 时，方程(1)可以化为KdV-Burgers方程[7-12]

 $\begin{array}{c}{u}_{t}+2\beta u{u}_{x}+\varepsilon {u}_{xx}+{u}_{xxx}=0\end{array}$

$\delta = 0$ $p = 2$ 时，方程(1)可以化为MKdV-Burgers方程[13-16]

 $\begin{array}{c}{u}_{t}+3\beta {u}^{2}{u}_{x}+\varepsilon {u}_{xx}+{u}_{xxx}=0\end{array}$

 $u\left(x+2\text{π} \right)=u\left(x,t\right),u\left(x,0\right)={u}_{0}\left(x\right),x\in {\mathbb {R}},t\in \left[0,\tau \right]$ (3)

2 方程(1)单调递减扭状孤波解的存在性

 $\begin{split} & -c{u}{{'}}\left(\xi \right)+\varepsilon {u}{{'}{'}}\left(\xi \right)+c\delta {u}{{'}{'}}\left(\xi \right)+{u}{{'}{'}{'}}\left(\xi \right)+\\&\qquad\beta \left(p+1\right){u}^{p}\left(\xi \right){u}{{'}}\left(\xi \right)=0 \end{split}$

 $\begin{array}{c}{u}{{'}{'}}\left(\xi \right)+\left(\varepsilon +c\delta \right){u}{{'}}\left(\xi \right)-cu\left(\xi \right)+\beta {u}^{p+1}\left(\xi \right)=k\end{array}$

 ${c}{u}{{'}}\left(\xi \right),\;\;{u}{{'}{'}}\left(\xi \right)\to 0,\;\;\left|\xi \right|\to\infty$ (4)

 $\beta {x}^{p+1}-cx=0$ (5)

 ${u}{{'}{'}}\left(\xi \right)+\left(\varepsilon +c\delta \right){u}{{'}}\left(\xi \right)-cu\left(\xi \right)+\beta {u}^{p+1}\left(\xi \right)=0$ (6)

$x=u\left(\xi \right)$ ${y=u}{{'}}\left(\xi \right)$ ，则方程(6)等价于如下平面动力系统:

 $\left\{\begin{array}{l}\dfrac{\mathrm{d}x}{\mathrm{d}\xi }=y\equiv P\left(x,y\right)\\ \dfrac{\mathrm{d}y}{\mathrm{d}\xi }=-\left(\varepsilon +c\delta \right)y+cx-\beta {x}^{p+1}\equiv Q\left(x,y\right)\end{array}\right.$ (7)

a. $c > 0$ ，当 $p$ 为偶数时，方程 $f\left(x\right)$ 有3个不同的实根 ${x}_{0}$ ${x}_{1}$ ${x}_{2}$ ；当 $p$ 为奇数时，方程 $f\left(x\right)$ 有2个不同的实根 ${x}_{0}$ ${x}_{1}$

b. $c < 0$ ，当 $p$ 为偶数时，方程 $f\left(x\right)$ 仅有1个实根 ${x}_{0}$ ；当 $p$ 为奇数时，方程 $f\left(x\right)$ 有2个不同的实根 ${x}_{0}$ ${x}_{1}$

 ${\boldsymbol{J}}\left({x}_{i},0\right)=\left(\begin{array}{cc}0& 1\\ -{f}{{'}}\left({x}_{i}\right)& -\left(\varepsilon +c\delta \right)\end{array}\right),\quad i={0,\;1},\;2$

a. $c > 0$ ，当 $p$ 为偶数时，由于 $\mathrm{d}\mathrm{e}\mathrm{t}\left(\boldsymbol{J}\left({x}_{i},0\right)\right)= f{{'}}\left({x}_{i}\right) > 0$ ，当 ${\varDelta }_{1} > 0$ 时， ${P}_{i}$ ( $i=\mathrm{1,2})$ 为不稳定的结点；当 ${\varDelta }_{1} < 0$ 时， ${P}_{i}$ ( $i=\mathrm{1,2})$ 为不稳定的焦点。当 $p$ 为奇数时，由于 ${\mathrm{d}\mathrm{e}\mathrm{t}\left(\boldsymbol{J}\left({x}_{1},0\right)\right)=f}{{'}}\left({x}_{1}\right) > 0$ ，当 ${\varDelta }_{1} > 0$ 时， ${P}_{1}$ 为不稳定的结点；当 ${\varDelta }_{1} < 0$ 时， ${P}_{1}$ 为不稳定的焦点。

b. $c < 0$ ，当 $p$ 为奇数时，系统(7)有2个奇点 ${P}_{0}\left(\mathrm{0,0}\right)$ ${P}_{1}({x}_{1},0)$ 。又因为 $\mathrm{d}\mathrm{e}\mathrm{t}\left(\boldsymbol{J}\left({x}_{1},0\right)\right)={f}{{'}}\left({x}_{1}\right) < 0$ ，故 ${P}_{1}$ 为鞍点。

 图 1 ${\boldsymbol{\varepsilon +c\delta < 0}}$ 时的全局相图 Fig. 1 Global phase diagrams when ${\boldsymbol{\varepsilon +c\delta < 0 }}$

a. $\varepsilon +c\delta < -2\sqrt{pc}$ 时，方程(1)的2个有界行波解是单调的，其一表现为单调递减的扭状孤波解 $u\left(\xi \right)$ ，满足 $u\left(-\infty \right)={x}_{1}$ $u\left(+\infty \right)=0$ (对应于图1(a)中的轨线 $L({P}_{1},{P}_{0})$ )；而另一个表现为单调递增的扭状孤波解 $u\left(\xi \right)$ ，满足 $u\left(-\infty \right)={x}_{2}$ $u\left(+\infty \right)=0$ (对应于图1(a)中的轨线 $L({P}_{2},{P}_{0})$ )。

b. $-2\sqrt{pc} < \varepsilon +c\delta < 0$ 时，方程(1)的2个行波解具有振荡性。

(a) 对应图1(b)中轨线 $L({P}_{1},{P}_{0})$ 的解 $u\left(\xi \right)$ 满足 $u\left(-\infty \right)={x}_{1}$ $u\left(+\infty \right)=0$ 。该解具有性质：存在最大值点 ${\hat{\xi }}_{1}$ ，使得点 ${\hat{\xi }}_{1}$ 的右端具有单调递减性，左端具有衰减振荡性，即存在无穷多个极大值点 ${\hat{\xi }}_{i}(i= 1, 2,\cdots ,+\infty )$ 和极小值点 ${\check{\xi }}_{i}(i=\mathrm{1,2},\cdots ,+\infty )$ ，使得

 $\left\{\begin{array}{l}-\infty < \dots < {\check{\xi }}_{n} < {\hat{\xi }}_{n} < \dots < {\check{\xi }}_{1} < {\hat{\xi }}_{1} < +\infty \\ \underset{n\to\infty }{\mathrm{lim}}\,{\check{\xi }}_{n}=\underset{n\to\infty }{\mathrm{lim}}\,{\hat{\xi }}_{n}=-\infty \end{array}\right.$ (8)
 $\left\{\begin{array}{l}u\left(+\infty \right) < u\left({\check{\xi }}_{1}\right) < \dots < u\left({\check{\xi }}_{n}\right) < \dots < u\left(-\mathrm{\infty }\right) <\\\quad \dots < u\left({\hat{\xi }}_{n}\right) < \dots < u\left({\hat{\xi }}_{1}\right)\\ \underset{n\to\infty }{\mathrm{lim}}\,u({\check{\xi }}_{n})=\underset{n\to\infty }{\mathrm{lim}}\,u\left({\hat{\xi }}_{n}\right)=u\left(-\mathrm{\infty }\right)\end{array}\right.$ (9)
 $\begin{split}& \underset{n\to \infty }{\mathrm{lim}}({\check{\xi }}_{n}-{\check{\xi }}_{n+1})=\underset{n\to\infty }{\mathrm{lim}}\left({\hat{\xi }}_{n}-{\hat{\xi }}_{n+1}\right)=\\&\qquad\qquad\frac{4\text{π} }{\sqrt{4pc-{\left(\varepsilon +c\delta \right)}^{2}}} \end{split}$ (10)

(b)对应图1(b)中轨线 $L({P}_{2},{P}_{0})$ 的解 $u\left(\xi \right)$ 满足 $u\left(-\infty \right)={x}_{2}$ $u\left(+\infty \right)=0$ 。该解具有性质：存在最小值点 ${\check{\xi }}_{1}$ ，使得点 ${\check{\xi }}_{1}$ 的右端具有单调递增性，左端具有衰减振荡性，即存在无穷多个极大值点 ${\hat{\xi }}_{i}(i= \mathrm{1,2},\cdots ,+\infty )$ 和极小值点 ${\check{\xi }}_{i}(i=\mathrm{1,2},\cdots ,+\infty )$ ，使得

 $\left\{\begin{array}{l}-\infty < \dots < {\hat{\xi }}_{n} < {\check{\xi }}_{n} < \dots < {\hat{\xi }}_{1} < {\check{\xi }}_{1} < +\infty \\ \underset{n\to\infty }{\mathrm{lim}}\;{\check{\xi }}_{n}=\underset{n\to\infty }{\mathrm{lim}}\;{\hat{\xi }}_{n}=-\infty \end{array}\right.$ (11)
 $\left\{\begin{array}{l}u\left({\check{\xi }}_{1}\right) < \dots < u\left({\check{\xi }}_{n}\right) < \dots < u\left(-\mathrm{\infty }\right) < \dots <\\\qquad\quad u\left({\hat{\xi }}_{n}\right) < \dots < u\left({\hat{\xi }}_{1}\right) < u\left(+\infty \right)\\ \underset{n\to\infty }{\mathrm{lim}}u({\check{\xi }}_{n})=\underset{n\to\infty }{\mathrm{lim}}u\left({\hat{\xi }}_{n}\right)=u\left(-\mathrm{\infty }\right)\end{array}\right.$ (12)

 $\begin{split}& {V}{{'}{'}}\left(\xi \right)+\left(\varepsilon +c\delta \right){V}{{'}}\left(\xi \right)+{2}^{p}c\left(V\left(\xi \right)-\frac{1}{2}\right)\cdot\\&\qquad \left[{\left(V\left(\xi \right)-\frac{1}{2}\right)}^{p}-{\left(\frac{1}{2}\right)}^{p}\right]=0 \end{split}$ (13)

 $\begin{split}& {V}{{'}{'}}\left(\xi \right)+\left(\varepsilon +c\delta \right){V}{{'}}\left(\xi \right)+{2}^{p}cV\left(\xi \right)\cdot\\&\qquad\left(V\left(\xi \right)-\frac{1}{2}\right)\left(V\left(\xi \right)-1\right) \left[{\left(V\left(\xi \right)-\frac{1}{2}\right)}^{p-2}+\right.\\&\left.\qquad{\left(V\left(\xi \right)-\frac{1}{2}\right)}^{p-4}\frac{1}{4}+\dots +{\left(\frac{1}{2}\right)}^{p-2}\right]=0 \end{split}$ (14)

 $({\rm{A}}) \quad u\left(-\infty \right)={x}_{1} \text{，} u\left(+\infty \right)=0$
 $({\rm{B}})\quad u\left(-\infty \right)={x}_{2} \text{，} u\left(+\infty \right)=0$

 $\left({\mathrm{A}}_{1}\right)\quad {V}\left(-\infty \right)=1 \text{，} {V}\left(+\infty \right)=1/2$
 $\left({\mathrm{B}}_{1}\right)\quad {V}\left(-\infty \right)=0 \text{，} {V}\left(+\infty \right)=1/2$

 $-2\sqrt{{\rm{sup}}\;\frac{g\left(u\right)}{u}}\leqslant {r}^{*}\leqslant -2\sqrt{{g}{{'}}\left(0\right)}$ (15)

 $\left\{\begin{array}{l}{u}{{'}{'}}+r{u}{{'}}+g\left(u\right)=0\\ \left(-\infty \right)=0,u\left(+\infty \right)=1\end{array}\right.$

$w=2(1-V)$ ，即 $V=1-(1/2)w$ ，则方程(13)满足条件 $\left({\mathrm{A}}_{1}\right)$ 的解等价于定解问题

 $\left\{\begin{array}{l}{w}{{'}{'}}+\left(\varepsilon +c\delta \right){w}{{'}}+c\left(w-1\right)\left[{\left(1-w\right)}^{p}-1\right]=0\\ w\left(-\infty \right)=0,\;\;w\left(+\infty \right)=1\end{array}\right.$ (16)

 $\begin{split}& {\left(\frac{g\left(w\right)}{w}\right)}^{{'}}=\frac{c\left[{\left(1-w\right)}^{p}\left(pw+1\right)-1\right]}{{w}^{2}}=\frac{cl\left(w\right)}{{w}^{2}},\\&\quad\quad\quad\quad\quad\quad\quad\quad\forall w\in \left(\mathrm{0,1}\right) \end{split}$ (17)

 $\underset{\left(\mathrm{0,1}\right)}{\mathrm{sup}}\,\frac{g\left(w\right)}{w}=\underset{w\to 0}{\mathrm{lim}}\,\frac{g\left(w\right)}{w}=\underset{w\to 0}{\mathrm{lim}}\,{g}{{'}}\left(w\right)=pc$

 $\left\{\begin{array}{l}{\bar{w}}{{'}{'}}+\left(\varepsilon +c\delta \right){\bar{w}}{{'}}+c\left(\bar{w}-1\right)\left[{\left(1-\bar{w}\right)}^{p}-1\right]=0\\ \bar{w}\left(-\infty \right)=0,\;\bar{w}\left(+\infty \right)=1\end{array}\right.$ (18)

 图 2 ${{ {\boldsymbol{c > 0,}}{\boldsymbol{-2}}\sqrt{{\boldsymbol{pc}}} {\boldsymbol{< \varepsilon +c\delta < 0}}}}$ 时的振荡行波解示意图 Fig. 2 Schematic diagrams of oscillatory traveling wave solutions when ${{{\boldsymbol{c}} > {\boldsymbol{0}},{\boldsymbol{-2}}\sqrt{{\boldsymbol{pc}}} {\boldsymbol{< \varepsilon +c\delta < 0}}}}$

a. $\varepsilon +c\delta < -2\sqrt{pc}$ 时，方程(1)具有单调递减的扭状孤波解 $u\left(\xi \right)$ ，满足 $u\left(-\infty \right)={x}_{1}$ $u\left(+\infty \right)=0$ $u\left(\xi \right)$ 对应于图1(c)中的轨线 $L({P}_{1},{P}_{0}$ )。

b. $-2\sqrt{pc} < \varepsilon +c\delta < 0$ 时，方程(1)具有振荡行波解 $u\left(\xi \right)$ ，满足 $u\left(-\infty \right)={x}_{1}$ $u\left(+\infty \right)=0$ $u\left(\xi \right)$ 对应于图1(d)中的轨线 $L({P}_{1},{P}_{0})$

3 广义河床流体模型方程扭状孤波解的重要性质

 $U\left(\xi \right)\to {u}_{\pm } , \quad \xi \to \pm \infty$

${u}_{-}={x}_{1}$ ${u}_{+}=0$

$U\left(\xi \right)=u\left(t,x\right)(\xi =x-ct)$ 代入方程(1)，则 $U\left(\xi \right)$ 满足

 $-c{U}_{\xi }+(\varepsilon +\delta c){U}_{\xi \xi }+{U}_{\xi \xi \xi }+f{\left(U\right)}_{\xi }=0$ (19)

 $\begin{array}{c}(\varepsilon +\delta c){U}_{\xi }+{U}_{\xi \xi }=cU-f\left(U\right)+a\end{array}$
 $a=-c{u}_{\pm }+f\left({u}_{\pm }\right)$

 $\begin{array}{c}c\left({u}_+-{u}_-\right)=f\left({u}_+\right)-f\left({u}_-\right)\end{array}$

 ${f}{{'}}\left({u}_+\right) < c < {f}{{'}}\left({u}_-\right)$

 ${u}_-=u\left(-\infty \right)={x}_{1}={\left(\frac{c}{\beta }\right)}^{\frac{1}{p}},\;\;{u}_+=u\left(+\infty \right)=0$

 ${f}{{'}}\left({u}_+\right)={f}{{'}}\left(0\right)=0,\;{f}{{'}}\left({u}_-\right)=\beta (p+1){{u}_-^{p}}=c(p+1)$

 $\begin{array}{c}\left|{U}_{\xi }\left|,\left|{U}_{\xi \xi }\right|\leqslant C\right|{u}_--{u}_+\right|\end{array}$

 $(\varepsilon +\delta c){U}_{\xi }+{U}_{\xi \xi }=f\left(U\right)-f\left({u}_+\right)-c\left(U-{u}_+\right)$ (20)

 $\begin{split}& {(\varepsilon +\delta c){U}_{\xi }+{U}_{\xi \xi }=f}{{'}}\left({u}_++\theta \left(U-{u}_+\right)\right)\left(U-{u}_+\right)-\\&\qquad c\left(U-{u}_+\right)={(f}{{'}}\left(\theta \left(U-{u}_+\right)-c\right))\left(U-{u}_+\right),\\&\qquad0 < \theta < 1 \end{split}$ (21)

a. ${f}{{'}}\left(\theta \left(U-{u}_{+}\right)\right) > c$ 时，由式(21)可知， $(\varepsilon + \delta c){U}_{\xi }+ {U}_{\xi \xi } > 0$

 $\begin{array}{c}-c\left(U-{u}_+\right)\leqslant (\varepsilon +\delta c){U}_{\xi }+{U}_{\xi \xi }\leqslant f\left(U\right)-f\left({u}_+\right)\end{array}$

 $f\left(U\right)=\bar{f}\left(U\right)U < {M}_{\bar{f}} U\leqslant {M}_{f} {u}_-\leqslant C\left|{u}_--{u}_+\right|$

 ${c}-C\left|{u}_--{u}_+\right|\leqslant (\varepsilon +\delta c){U}_{\xi }+{U}_{\xi \xi }\leqslant C\left|{u}_--{u}_+\right|$ (22)

 $\begin{split}& -C{{\rm{e}}}^{(\varepsilon +\delta c)\xi }\left|{u}_--{u}_+\right|\leqslant \frac{{\rm{d}}}{{\rm{d}}\xi }\left({{\rm{e}}}^{(\varepsilon +\delta c)\xi }{U}_{\xi }\right)\leqslant \\&\qquad C{{\rm{e}}}^{(\varepsilon +\delta c)\xi }\left|{u}_--{u}_+\right| \end{split}$ (23)

 $\begin{array}{c}\left|{U}_{\xi }\left|\leqslant C\right|{u}_--{u}_+\right|\end{array}$

b. ${f}{{'}}\left(\theta \left(U-{u}_{+}\right)\right) < c$ 时，有

 $\begin{array}{c}(\varepsilon +\delta c){U}_{\xi }+{U}_{\xi \xi } < 0\end{array}$

 $-c\left(U-{u}_+\right)\leqslant (\varepsilon +\delta c){U}_{\xi }+{U}_{\xi \xi }$

 $(\varepsilon +\delta c){U}_{\xi }+{U}_{\xi \xi } < 0\leqslant U-{u}_+$

 $-c\left(U-{u}_+\right)\leqslant (\varepsilon +\delta c){U}_{\xi }+{U}_{\xi \xi }\leqslant U-{u}_+$

 $-C\left|{u}_--{u}_+\right|\leqslant (\varepsilon +\delta c){U}_{\xi }+{U}_{\xi \xi }\leqslant C\left|{u}_--{u}_+\right|$ (24)

 $\begin{split} & -C{{\rm{e}}}^{(\varepsilon +\delta c)\xi }\left|{u}_--{u}_+\right|\leqslant \frac{{\rm{d}}}{{\rm{d}}\xi }\left({{\rm{e}}}^{(\varepsilon +\delta c)\xi }{U}_{\xi }\right)\leqslant\\&\qquad C{{\rm{e}}}^{(\varepsilon +\delta c)\xi }\left|{u}_--{u}_+\right| \end{split}$ (25)

 $\left|{U}_{\xi }\left|\leqslant C\right|{u}_--{u}_+\right|$

 $\begin{split}& {U}_{\xi \xi }=\underset{\delta \to 0}{\mathrm{lim}}\;\frac{{U}_{\xi }\left(\xi +\delta \right)-{U}_{\xi }\left(\xi \right)}{\delta } \\& \left|{U}_{\xi \xi }\right|=\left|\underset{\delta \to 0}{\mathrm{lim}}\;\frac{{U}_{\xi }\left(\xi +\delta \right)-{U}_{\xi }\left(\xi \right)}{\delta }\right| \end{split}$

 $\left|{U}_{\xi \xi }\right|\leqslant \frac{1}{\left|\delta \right|}\left[\left|{U}_{\xi }\left(\xi +\delta \right)\right|+\left|{U}_{\xi }\left(\xi \right)\right|\right]\leqslant \frac{1}{\left|\delta \right|} 2C\left|{u}_--{u}_+\right|$

$\dfrac{1}{\left|\delta \right|}2C$ $C$ 中的最大者仍记为 $C$ ，则有 $|{U}_{\xi \xi }|\leqslant C|{u}_{-}-{u}_{+}|$ 。性质3得证。

4 单调递减扭状孤波解渐近稳定性定理

 $\begin{array}{c}u\left(0,x\right)={u}_{0}\left(x\right)\end{array}$

 $\begin{array}{c}{u}_{0}\left(x\right)\to {u}_{\pm },x\to \pm \infty \end{array}$

$U$ 为单调递减扭状孤波解，假定

 ${u}_{0}-U\in {H}^{1}$ (26)

 ${\mathrm{\varPhi }}_{0}\left(x\right)={\int }_{-\infty }^{x}\left({u}_{0}-U\right)\left(y\right){\rm{d}}y,\qquad{\mathrm{\varPhi }}_{0}\in {L}^{2}$ (27)

 $\begin{array}{c}\underset{x\to +\infty }{\mathrm{lim}}{\mathrm{\varPhi }}_{0}\left(x\right)={\int }_{-\infty }^{+\infty }\left({u}_{0}-U\right)\left(y\right){\rm{d}}y=0\end{array}$

 $\begin{array}{c}{N}_{0}={\|{u}_{0}-U\|}_{{H}^{1}}+{\|{\mathrm{\varPhi }}_{0}\|}_{{H}^{2}}\end{array}$

 $u-U\in {C}^{0}\left(0,\infty ;{H}^{1}\right)\cap {L}^{2}\left(0,\infty ;{H}^{2}\right)$ (28)

 $\underset{x\in {\mathbb{R}}}{\mathrm{sup}}\left|u\left(t,x\right)-U\left(x-ct\right)\right|\to 0,\quad t\to +\infty$ (29)

$u\left(t,x\right)=U\left(\xi \right)+\psi \left(t,\xi \right),\;\xi =x-ct$ 代入方程(1)，并注意到 $U\left(\xi \right)$ 是方程(1)的孤波解，可得 $\psi \left(t,\xi \right)$ 满足

 $\left\{\begin{array}{l}{\psi }_{t}-c{\psi }_{\xi }+\left(\varepsilon +\delta c\right){\psi }_{\xi \xi }-\delta {\psi }_{t\xi }+{\psi }_{\xi \xi \xi }+\\\quad{\left(f\left(U+\psi \right)-f\left(U\right)\right)}_{\xi }=0\\ \psi \left(0,\xi \right)={\psi }_{0}\left(\xi \right)=\left({u}_{0}-U\right)\left(\xi \right)\end{array}\right.$ (30)

 $\begin{array}{c}\psi ={\mathrm{\varPhi }}_{\xi }\end{array}$

 $\begin{split}& {\mathrm{\varPhi }}_{t\xi }-c{\mathrm{\varPhi }}_{\xi \xi }+\left(\varepsilon +\delta c\right){\mathrm{\varPhi }}_{\xi \xi \xi }-\delta {\mathrm{\varPhi }}_{t\xi \xi }+{\mathrm{\varPhi }}_{\xi \xi \xi \xi }+\\&\qquad{\left(f\left(U+{\mathrm{\varPhi }}_{\xi }\right)-f\left(U\right)\right)}_{\xi }=0 \end{split}$ (31)

 $\begin{split}& {\mathrm{\varPhi }}_{t}-c{\mathrm{\varPhi }}_{\xi }+\left(\varepsilon +\delta c\right){\mathrm{\varPhi }}_{\xi \xi }-\delta {\mathrm{\varPhi }}_{t\xi }+{\mathrm{\varPhi }}_{\xi \xi \xi }+\\&\qquad \left(f\left(U+{\mathrm{\varPhi }}_{\xi }\right)-f\left(U\right)\right)=0 \end{split}$ (32)

 $\begin{split}& {\mathrm{\varPhi }}_{t}-c{\mathrm{\varPhi }}_{\xi }+\left(\varepsilon +\delta c\right){\mathrm{\varPhi }}_{\xi \xi }-\delta {\mathrm{\varPhi }}_{t\xi }+{\mathrm{\varPhi }}_{\xi \xi \xi }+\\&\qquad{f}{{'}}\left(U\right){\mathrm{\varPhi }}_{\xi }=F\left(U,{\mathrm{\varPhi }}_{\xi }\right) \end{split}$ (33)

 $\varPhi \left(0,\;\mathrm{\xi }\right)={\mathrm{\varPhi }}_{0}\left(\xi \right)$ (34)

 $\begin{array}{c}X\left(0,T\right)=\left\{\mathrm{\varPhi }\in {L}^{\infty }\left(0,T;{H}^{2}\right),\;\;{\mathrm{\varPhi }}_{\xi }\in {L}^{2}\left(0,T;{H}^{2}\right)\right\}\end{array}$

 ${\|\mathrm{\varPhi }\|}_{{H}^{2}}^{2}+{\|{\mathrm{\varPhi }}_{\xi }\|}_{{H}^{1}}^{2}+{\int }_{0}^{t}({\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi \xi }\|}^{2}){\rm{d}}\tau \leqslant {C}_{1}{N}_{0}^{2}$

 $N^2(t)+\int_0^t\left(\left\|\varPhi_{\xi}\right\|^2+\left\|\varPhi_{\xi \xi}\right\|^2+\left\|\varPhi_{\xi \xi \xi}\right\|^2\right) {\rm{d}} \tau \leqslant C_1 N_0^2$

 $ab\leqslant\theta {a}^{p}+C\left(\theta \right){b}^{q}$

 ${\|{D}^{j}u\|}_{r}\leqslant C{\|u\|}_{p}^{1-\lambda }{\|{D}^{m}u\|}_{q}^{\lambda }$

 $\frac{1}{r}-\frac{j}{n}=\lambda \left(\frac{1}{q}-\frac{m}{n}\right)+\left(1-\lambda \right)\frac{1}{p},\;\frac{j}{m}\leqslant \lambda \leqslant 1$

$m-\dfrac{n}{q}=j，1 < q < \infty ，\lambda \ne 1$

 $N\left(t\right)=\underset{0\leqslant \tau \leqslant t}{\mathrm{sup}}\left\{{\|\mathrm{\varPhi }\left(\tau \right)\|}_{{H}^{2}}+{\|{\mathrm{\varPhi }}_{\xi }\left(\tau \right)\|}_{{H}^{1}}\right\}\leqslant {\gamma }_{2}^{2}$ (35)

 $\underset{0\leqslant \tau \leqslant t}{\mathrm{sup}}{\|{ \partial }_{x}^{k}\mathrm{\varPhi }\left(\tau \right)\|}_{{L}^{\infty }}\leqslant C{\gamma }_{2},\quad k={0,\;1}$

 $\begin{split}& {N}^{2}\left(t\right)+{\lambda }^{*}{\int }_{0}^{t}\left({\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi \xi }\|}^{2}\right){\rm{d}}\tau \leqslant\\&\quad \quad\quad\quad\quad\quad\quad{C}_{2}{N}_{0}^{2} \end{split}$ (36)

a. 低阶先验估计。

$\mathrm{\varPhi }$ 乘以式(33)，有

 $\begin{split}& \varPhi {\mathrm{\varPhi }}_{t}-c\varPhi {\mathrm{\varPhi }}_{\xi }+\left(\varepsilon +\delta c\right)\varPhi {\mathrm{\varPhi }}_{\xi \xi }-\delta \varPhi {\mathrm{\varPhi }}_{t\xi }+\varPhi {\mathrm{\varPhi }}_{\xi \xi \xi }+\\&\qquad {f}{{'}}\left(U\right)\varPhi {\mathrm{\varPhi }}_{\xi }=\varPhi F\left(U,{\mathrm{\varPhi }}_{\xi }\right) \end{split}$ (37)

 $\mathrm{\varPhi }{\mathrm{\varPhi }}_{t}=\frac{1}{2}{\left({\mathrm{\varPhi }}^{2}\right)}_{t}$
 $-c\mathrm{\varPhi }{\mathrm{\varPhi }}_{\xi }=-\frac{c}{2}{\left({\mathrm{\varPhi }}^{2}\right)}_{\xi }$
 $-\delta \mathrm{\varPhi }{\mathrm{\varPhi }}_{t\xi }=\delta {\mathrm{\varPhi }}_{t}{\mathrm{\varPhi }}_{\xi }+{\left(-\delta \mathrm{\varPhi }{\mathrm{\varPhi }}_{t}\right)}_{\xi }$
 $\begin{split}& \left(\varepsilon +c\delta \right)\mathrm{\varPhi }{\mathrm{\varPhi }}_{\xi \xi }+\mathrm{\varPhi }{\mathrm{\varPhi }}_{\xi \xi \xi }=\mathrm{\varPhi }{\left[\left(\varepsilon +c\delta \right){\mathrm{\varPhi }}_{\xi }+{\mathrm{\varPhi }}_{\xi \xi }\right]}_{\xi } =\\& \qquad{\left[\mathrm{\varPhi }\left((\varepsilon +c\delta ){\mathrm{\varPhi }}_{\xi }+{\mathrm{\varPhi }}_{\xi \xi }\right)\right]}_{\xi }-{\mathrm{\varPhi }}_{\xi }\left[(\varepsilon +c\delta ){\mathrm{\varPhi }}_{\xi }+{\mathrm{\varPhi }}_{\xi \xi }\right] =\\& \qquad{\left[{\left(\frac{1}{2}\left(\varepsilon +c\delta \right){\mathrm{\varPhi }}^{2}\right)}_{\xi }+{\mathrm{\varPhi }\mathrm{\varPhi }}_{\xi \xi }-\frac{1}{2}{\mathrm{\varPhi }}_{\xi }^{2}\right]}_{\xi }+(\varepsilon +c\delta ){\mathrm{\varPhi }}_{\xi }^{2} \end{split}$
 ${f}{{'}}\left(U\right)\mathrm{\varPhi }{\mathrm{\varPhi }}_{\xi }=\frac{1}{2}{\left({f}{{'}}\left(U\right){\mathrm{\varPhi }}^{2}\right)}_{\xi }-\frac{1}{2}{f}{{'}{'}}\left(U\right){U}_{\xi }{\mathrm{\varPhi }}^{2}$

 $\begin{split}& \frac{1}{2}{\left({\mathrm{\varPhi }}^{2}\right)}_{t}-\left(\varepsilon +c\delta \right){\mathrm{\varPhi }}_{\xi }^{2}+\delta {\mathrm{\varPhi }}_{t}{\mathrm{\varPhi }}_{\xi }-\frac{1}{2}{f}{{'}{'}}\left(U\right){U}_{\xi }{\mathrm{\varPhi }}^{2}+\\&\qquad{\left\{{Q}_{1}\right\}}_{\xi }=\varPhi F\left(U,{\mathrm{\varPhi }}_{\xi }\right) \end{split}$ (38)

 $\begin{split}& {Q}_{1}=-\frac{c}{2}{\mathrm{\varPhi }}^{2}-\delta \mathrm{\varPhi }{\mathrm{\varPhi }}_{t}+{\left(\frac{1}{2}\left(\varepsilon +c\delta \right){\mathrm{\varPhi }}^{2}\right)}_{\xi }+\mathrm{\varPhi }{\mathrm{\varPhi }}_{\xi \xi }-\\&\qquad \frac{1}{2}{\mathrm{\varPhi }}_{\xi }^{2}+\frac{1}{2}{f}{{'}}\left(U\right){\mathrm{\varPhi }}^{2} \end{split}$

 $\begin{split}& \delta {\mathrm{\varPhi }}_{\xi }{\mathrm{\varPhi }}_{t}-\delta c{\mathrm{\varPhi }}_{\xi }^{2}+\delta \left(\varepsilon +\delta c\right){\mathrm{\varPhi }}_{\xi }{\mathrm{\varPhi }}_{\xi \xi }-{\delta }^{2}{\mathrm{\varPhi }}_{\xi }{\mathrm{\varPhi }}_{t\xi }+\delta {\mathrm{\varPhi }}_{\xi }{\mathrm{\varPhi }}_{\xi \xi \xi }+\\&\qquad \delta {f}{{'}}\left(U\right){\mathrm{\varPhi }}_{\xi }^{2}=\delta {\mathrm{\varPhi }}_{\xi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right) \end{split}$

 ${{\mathrm{\varPhi }}_{\xi }\mathrm{\varPhi }}_{\xi \xi \xi }={\left({{\mathrm{\varPhi }}_{\xi }\mathrm{\varPhi }}_{\xi \xi }\right)}_{\xi }-{\mathrm{\varPhi }}_{\xi \xi }^{2}$

 $\begin{split}& -\frac{{\delta }^{2}}{2}{\left({\mathrm{\varPhi }}_{\xi }^{2}\right)}_{t}+\delta {\mathrm{\varPhi }}_{t}{\mathrm{\varPhi }}_{\xi }+\left(\delta {f}{{'}}\left(U\right)-c\delta \right){\mathrm{\varPhi }}_{\xi }^{2}-\\&\qquad \delta {\mathrm{\varPhi }}_{\xi \xi }^{2}+{\left\{{Q}_{2}\right\}}_{\xi }=\delta {\mathrm{\varPhi }}_{\xi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right) \end{split}$ (39)

 $\begin{split}& {\left\{\frac{1}{2}{\mathrm{\varPhi }}^{2}+\frac{{\delta }^{2}}{2}{\mathrm{\varPhi }}_{\xi }^{2}\right\}}_{t}-\left(\varepsilon +\delta {f}{{'}}\left(U\right)\right){\mathrm{\varPhi }}_{\xi }^{2}+\delta {\mathrm{\varPhi }}_{\xi \xi }^{2}-\\&\qquad \frac{1}{2}{f}{{'}{'}}\left(U\right){U}_{\xi }{\mathrm{\varPhi }}^{2}+{\left\{{Q}_{1}-{Q}_{2}\right\}}_{\xi }=\\&\qquad\varPhi F\left(U,{\mathrm{\varPhi }}_{\xi }\right)-\delta {\mathrm{\varPhi }}_{\xi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right) \end{split}$ (40)

 $\begin{split}& -\left(\varepsilon +\delta {f}{{'}}\left(U\right)\right){\mathrm{\varPhi }}_{\xi }^{2}=-\left(\varepsilon +\delta {f}{{'}}\left({u}_+\right)\right){\mathrm{\varPhi }}_{\xi }^{2}+\\&\qquad\delta \left({f}{{'}}\left({u}_+\right)-{f}{{'}}\left(U\right)\right){\mathrm{\varPhi }}_{\xi }^{2} \end{split}$

 $\begin{split}& -\left(\varepsilon +\delta {f}{{'}}\left(U\right)\right){\mathrm{\varPhi }}_{\xi }^{2}=-\left(\varepsilon +\delta {f}{{'}}\left({u}_+\right)\right){\mathrm{\varPhi }}_{\xi }^{2}+\\&\qquad \delta {f}{{'}{'}}\left(U+\theta \left({u}_+-U\right)\right)\left({u}_+-U\right){\mathrm{\varPhi }}_{\xi }^{2},\;0 < \theta < 1 \end{split}$

 $\begin{split}& -\left(\varepsilon +\delta {f}{{'}}\left(U\right)\right){\mathrm{\varPhi }}_{\xi }^{2}\geqslant {\lambda }^{*}{\mathrm{\varPhi }}_{\xi }^{2}+\\& \qquad{f}{{'}{'}}\left(U+\theta \left({u}_+-U\right)\right)\delta ({u}_+-{u}_-){\mathrm{\varPhi }}_{\xi }^{2}\geqslant {\lambda }^{*}{\mathrm{\varPhi }}_{\xi }^{2}-\\& \qquad{f}{{'}{'}}\left(U+\theta \left({u}_+-U\right)\right)\delta \left|{u}_+-{u}_-\right|{\mathrm{\varPhi }}_{\xi }^{2} \end{split}$ (41)

$p\geqslant 1$ 时，由于 $f\left(U\right)$ $U$ 的多项式，在包含 $\left[{u}_{+},{u}_{-}\right]$ 的较大闭区间 $I\subset \mathbb{R}$ 上连续、可微，可设在 $I$ $\left|f\left(U\right)\right|$ $\left|{f}{{'}}\left(U\right)\right|$ $\left|{f}{{'}{'}}\left(U\right)\right|$ $\left|{f}{{'}{'}{'}}\left(U\right)\right|\leqslant {C}_{f}$ 。则不妨取 $\left|{u}_{+}-{u}_{-}\right| < \dfrac{{\lambda }^{*}}{2\delta {C}_{f}}$ ，则式(41)可以化为

 $\begin{split}& -\left(\varepsilon +\delta {f}{{'}}\left(U\right)\right){\mathrm{\varPhi }}_{\xi }^{2} > {\lambda }^{*}{\mathrm{\varPhi }}_{\xi }^{2}-{f}{{'}{'}}(U+\theta ({u}_+-\\&\qquad U))\delta \frac{{\lambda }^{*}}{2\delta {C}_{f}}{\mathrm{\varPhi }}_{\xi }^{2} > \frac{{\lambda }^{*}}{2}{\mathrm{\varPhi }}_{\xi }^{2} \end{split}$ (42)

 $\begin{split}& {\left\{\frac{1}{2}{\mathrm{\varPhi }}^{2}+\frac{{\delta }^{2}}{2}{\mathrm{\varPhi }}_{\xi }^{2}\right\}}_{t}+\frac{{\lambda }^{*}}{2}{\mathrm{\varPhi }}_{\xi }^{2}+\delta {\mathrm{\varPhi }}_{\xi \xi }^{2}-\frac{1}{2}{f}^{{'}{'}}\left(U\right){U}_{\xi }{\mathrm{\varPhi }}^{2}+\\& \qquad {\left\{{Q}_{1}-{Q}_{2}\right\}}_{\xi }\leqslant \varPhi F\left(U,{\mathrm{\varPhi }}_{\xi }\right)-\delta {\mathrm{\varPhi }}_{\xi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right)\\[-12pt] \end{split}$ (43)

 $\begin{split}& \frac{1}{2}{\|\mathrm{\varPhi }\|}^{2}+\frac{{\delta }^{2}}{2}{\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+\\&\qquad{\int }_{0}^{t}{\int }_{\mathbb{R}}\left(\frac{{\lambda }^{*}}{2}{\mathrm{\varPhi }}_{\xi }^{2}+\delta {\mathrm{\varPhi }}_{\xi \xi }^{2}-\frac{1}{2}{f}{{'}{'}}\left(U\right){U}_{\xi }{\mathrm{\varPhi }}^{2}\right){\rm{d}}\xi {\rm{d}}\tau\leqslant\\&\qquad \frac{1}{2}{\|{\mathrm{\varPhi }}_{0}\|}^{2}+\frac{{\delta }^{2}}{2}{\|{\mathrm{\varPhi }}_{0\xi }\|}^{2}+\\&\qquad{\int }_{0}^{t}{\int }_{\mathbb{R}}\left(\mathrm{\varPhi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right)-\delta {\mathrm{\varPhi }}_{\xi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right)\right){\rm{d}}\xi {\rm{d}}\tau \end{split}$

 $\begin{split}& \frac{1}{2}{\|\mathrm{\varPhi }\|}^{2}+\frac{{\delta }^{2}}{2}{\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+{\int }_{0}^{t}\left(\frac{{\lambda }^{*}}{2}{\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+\delta {\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}\right){\rm{d}}\tau\leqslant\\&\quad \frac{1}{2}{\|{\mathrm{\varPhi }}_{0}\|}^{2}+\frac{{\delta }^{2}}{2}{\|{\mathrm{\varPhi }}_{0\xi }\|}^{2}+\\&\quad{\int }_{0}^{t}{\int }_{\mathbb{R}}\left(\left|\mathrm{\varPhi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right)\right|+\left|\delta {\mathrm{\varPhi }}_{\xi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right)\right|\right){\rm{d}}\xi {\rm{d}}\tau\\[-15pt] \end{split}$ (44)

b. 高阶先验估计。

 $\begin{split}& {\mathrm{\varPhi }}_{t\xi }-c{\mathrm{\varPhi }}_{\xi \xi }+\left(\varepsilon +\delta c\right){\mathrm{\varPhi }}_{\xi \xi \xi }-\delta {\mathrm{\varPhi }}_{t\xi \xi }+\\& \qquad {\mathrm{\varPhi }}_{\xi \xi \xi \xi }+{f}{{'}}\left(U\right){\mathrm{\varPhi }}_{\xi \xi }=G \end{split}$ (45)

 $G=\left[{f}{{'}}\left(U\right)-{f}{{'}}\left(U+{\mathrm{\varPhi }}_{\xi }\right)\right]{U}_{\xi }-\left[{f}{\mathrm{{'}}}\left(U+{\mathrm{\varPhi }}_{\xi }\right)-{f}{{'}}\left(U\right)\right]{\mathrm{\varPhi }}_{\xi \xi }$

(a) 用 ${\mathrm{\varPhi }}_{\xi }$ 乘以式(45)，有

 $\begin{split}& {{\mathrm{\varPhi }}_{\xi }\mathrm{\varPhi }}_{t\xi }-c{{\mathrm{\varPhi }}_{\xi }\mathrm{\varPhi }}_{\xi \xi }+\left(\varepsilon +\delta c\right){{\mathrm{\varPhi }}_{\xi }\mathrm{\varPhi }}_{\xi \xi \xi }-\delta {{\mathrm{\varPhi }}_{\xi }\mathrm{\varPhi }}_{t\xi \xi }+\\&\qquad+{\mathrm{\varPhi }}_{\xi }{\mathrm{\varPhi }}_{\xi \xi \xi \xi }+{f}{{'}}\left(U\right){\mathrm{\varPhi }}_{\xi }{\mathrm{\varPhi }}_{\xi \xi }={\mathrm{\varPhi }}_{\xi }G \end{split}$ (46)

 $\begin{split}& \frac{1}{2}{\left({{\mathrm{\varPhi }}_{\xi }}^{2}\right)}_{t}-\left(\varepsilon +c\delta \right){\left({\mathrm{\varPhi }}_{\xi \xi }\right)}^{2}+\delta {\mathrm{\varPhi }}_{t\xi }{\mathrm{\varPhi }}_{\xi \xi }-\\&\qquad \frac{1}{2}{f}^{{'}{'}}\left(U\right){U}_{\xi }{{\mathrm{\varPhi }}_{\xi }}^{2}+{\left\{{Q}_{3}\right\}}_{\xi }={\mathrm{\varPhi }}_{\xi }G \end{split}$ (47)

 $\begin{split}& {Q}_{3}=-\frac{c}{2}{{\mathrm{\varPhi }}_{\xi }}^{2}-\delta {{\mathrm{\varPhi }}_{\xi }\mathrm{\varPhi }}_{t\xi }+(\varepsilon +c\delta ){{\mathrm{\varPhi }}_{\xi }\mathrm{\varPhi }}_{\xi \xi }+{{\mathrm{\varPhi }}_{\xi }\mathrm{\varPhi }}_{\xi \xi \xi }-\\& \qquad \frac{1}{2}{\left({\mathrm{\varPhi }}_{\xi \xi }\right)}^{2}+\frac{1}{2}{f}{{'}}\left(U\right){\mathrm{\varPhi }}_{\xi }^{2} \end{split}$

(b) 用 $\delta {\mathrm{\varPhi }}_{\xi \xi }$ 乘以式(45)，则可得

 $\begin{split}& {\delta {\mathrm{\varPhi }}_{\xi \xi }\mathrm{\varPhi }}_{t\xi }-c\delta {\mathrm{\varPhi }}_{\xi \xi }^{2}+\delta \left(\varepsilon +\delta c\right){\mathrm{\varPhi }}_{\xi \xi }{\mathrm{\varPhi }}_{\xi \xi \xi }-{\delta }^{2}{{\mathrm{\varPhi }}_{\xi \xi }\mathrm{\varPhi }}_{t\xi \xi }+\\&\qquad +\delta {\mathrm{\varPhi }}_{\xi \xi }{\mathrm{\varPhi }}_{\xi \xi \xi \xi }+\delta {f}^{{'}}\left(U\right){\mathrm{\varPhi }}_{\xi \xi }^{2}=\delta {\mathrm{\varPhi }}_{\xi \xi }G \end{split}$

 $\begin{split}& -\frac{{\delta }^{2}}{2}{\left({\mathrm{\varPhi }}_{\xi \xi }^{2}\right)}_{t}+{\delta {\mathrm{\varPhi }}_{\xi \xi }\mathrm{\varPhi }}_{t\xi }+\left(\delta {f}{{'}}\left(U\right)-c\delta \right){\mathrm{\varPhi }}_{\xi \xi }^{2}-\\&\qquad\delta {\mathrm{\varPhi }}_{\xi \xi \xi }^{2}+{\left\{{Q}_{5}\right\}}_{\xi }=\delta {\mathrm{\varPhi }}_{\xi \xi }G \end{split}$ (48)

 ${Q}_{4}=\frac{1}{2}\delta \left(\varepsilon +c\delta \right){\mathrm{\varPhi }}_{\xi \xi }^{2}+\delta {\mathrm{\varPhi }}_{\xi \xi }{\mathrm{\varPhi }}_{\xi \xi \xi }$

 $\begin{split}& {\left\{\frac{1}{2}{{\mathrm{\varPhi }}_{\xi }}^{2}+\frac{{\delta }^{2}}{2}{\mathrm{\varPhi }}_{\xi \xi }^{2}\right\}}_{t}-\left(\varepsilon +\delta {f}{{'}}\left(U\right)\right){\mathrm{\varPhi }}_{\xi \xi }^{2}+\delta {\mathrm{\varPhi }}_{\xi \xi \xi }^{2}-\\&\qquad\frac{1}{2}{f}{{'}{'}}\left(U\right){U}_{\xi }{{\mathrm{\varPhi }}_{\xi }}^{2}+{\left\{{Q}_{3}{-Q}_{4}\right\}}_{\xi }=\\&\qquad{\mathrm{\varPhi }}_{\xi }G-\delta {\mathrm{\varPhi }}_{\xi \xi }G \end{split}$ (49)

 $-\left(\varepsilon +\delta {f}{{'}}\left(U\right)\right) > \frac{{\lambda }^{*}}{2}$ (50)

 $\begin{split}& {\left\{\frac{1}{2}{{\mathrm{\varPhi }}_{\xi }}^{2}+\frac{{\delta }^{2}}{2}{\mathrm{\varPhi }}_{\xi \xi }^{2}\right\}}_{t}+\frac{{\lambda }^{*}}{2}{\mathrm{\varPhi }}_{\xi \xi }^{2}+\delta {\mathrm{\varPhi }}_{\xi \xi \xi }^{2}-\\&\qquad \frac{1}{2}{f}{{'}{'}}\left(U\right){U}_{\xi }{{\mathrm{\varPhi }}_{\xi }}^{2}+ +{\left\{{Q}_{3}{-Q}_{4}\right\}}_{\xi }\leqslant\\&\qquad {\mathrm{\varPhi }}_{\xi }G-\delta {\mathrm{\varPhi }}_{\xi \xi }G \end{split}$ (51)

 $\begin{split}& {\left\{\frac{1}{2}{{\mathrm{\varPhi }}_{\xi }}^{2}+\frac{{\delta }^{2}}{2}{\mathrm{\varPhi }}_{\xi \xi }^{2}\right\}}_{t}+\frac{{\lambda }^{*}}{2}{\mathrm{\varPhi }}_{\xi \xi }^{2}+\delta {\mathrm{\varPhi }}_{\xi \xi \xi }^{2}+\\&\qquad{\left\{{Q}_{3}{-Q}_{4}\right\}}_{\xi }\leqslant {\mathrm{\varPhi }}_{\xi }G-\delta {\mathrm{\varPhi }}_{\xi \xi }G \end{split}$ (52)

 $\begin{split}& \frac{1}{2}{\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+\frac{{\delta }^{2}}{2}{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}+{\int }_{0}^{t}\left(\frac{{\lambda }^{*}}{2}{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}+\delta {\|{\mathrm{\varPhi }}_{\xi \xi \xi }\|}^{2}\right){\rm{d}}\tau \leqslant\\&\qquad \frac{1}{2}{\|{\mathrm{\varPhi }}_{0\xi }\|}^{2}+\frac{{\delta }^{2}}{2}{\|{\mathrm{\varPhi }}_{0\xi \xi }\|}^{2}+\\&\qquad{\int }_{0}^{t}{\int }_{\mathbb R}\left({\mathrm{\varPhi }}_{\xi }G-\delta {\mathrm{\varPhi }}_{\xi \xi }G\right){\rm{d}}\xi {\rm{d}}\tau \\[-12pt] \end{split}$ (53)

 $\begin{split}& \frac{1}{2}{\|\mathrm{\varPhi }\|}^{2}+\left(\frac{{\delta }^{2}}{2}+\frac{1}{2}\right){\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+\frac{{\delta }^{2}}{2}{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2} +\\&\qquad {\int }_{0}^{t}\left(\frac{{\lambda }^{*}}{2}{\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+\left(\delta +\frac{{\lambda }^{*}}{2}\right){\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}+\delta {\|{\mathrm{\varPhi }}_{\xi \xi \xi }\|}^{2}\right){\rm{d}}\tau \leqslant\\&\qquad \frac{{C}_{2}}{2}\left\{{\|{\mathrm{\varPhi }}_{0}\|}^{2}+({\delta }^{2}+1){\|{\mathrm{\varPhi }}_{0\xi }\|}^{2}+{\delta }^{2}{\|{\mathrm{\varPhi }}_{0\xi \xi }\|}^{2}\right. +\\&\qquad {\int }_{0}^{t}{\int }_{\mathbb R}\Big(\left|\mathrm{\varPhi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right)\right| + \left|\delta {\mathrm{\varPhi }}_{\xi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right)\right| +\Big.\\ &\qquad\left.\left. \left|{\mathrm{\varPhi }}_{\xi }G\right| + \left|\delta {\mathrm{\varPhi }}_{\xi \xi }G\right|\right){\rm{d}}\xi {\rm{d}}\tau \right\}\\[-12pt] \end{split}$ (54)

$p\geqslant 1$ 时，由于 $f\left(U\right)$ $U$ 的多项式，在包含 $\left[{u}_{+},{u}_{-}\right]$ 的较大闭区间 $I\subset \mathbb R$ 上连续、可微，可设在 $I$ $\left|f\left(U\right)\right|$ $\left|{f}{{'}}\left(U\right)\right|$ $\left|{f}{{'}{'}}\left(U\right)\right|$ $\left|{f}{{'}{'}{'}}\left(U\right)\right|\leqslant {C}_{h}$ 。于是运用性质3和Young不等式对式(54)右端积分中的4项有下列估计：

 $\begin{split}& F\left(U,{\mathrm{\varPhi }}_{\xi }\right)=-\left(f\left(U+{\mathrm{\varPhi }}_{\xi }\right)-f\left(U\right)\right)+{f}{{'}}\left(U\right){\mathrm{\varPhi }}_{\xi } =\\& \qquad-{f}{{'}}\left(U+\theta {\mathrm{\varPhi }}_{\xi }\right){\mathrm{\varPhi }}_{\xi }+{f}{{'}}\left(U\right){\mathrm{\varPhi }}_{\xi } =\\& \qquad-{f}^{{'}{'}}\left(U+\eta \theta {\mathrm{\varPhi }}_{\xi }\right)\theta {\mathrm{\varPhi }}_{\xi }^{2},\quad 0 < \eta ，\theta < 1 \end{split}$

 ${\int }_{\mathbb R}\mathrm{\varPhi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right){\rm{d}}\xi ={\int }_{\mathbb R}{-f}{{'}{'}}\left(U+\eta \theta {\mathrm{\varPhi }}_{\xi }\right){\theta \mathrm{\varPhi }\mathrm{\varPhi }}_{\xi }^{2}{\rm{d}}\xi$

 $\begin{split}& {\int }_{\mathbb R}\left|\mathrm{\varPhi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right)\right|{\rm{d}}\xi \leqslant {\int }_{\mathbb R}\left|\mathrm{\varPhi }\right|\left|{f}{{'}{'}}\left(U+\eta \theta {\mathrm{\varPhi }}_{\xi }\right)\right|{\mathrm{\varPhi }}_{\xi }^{2}{\rm{d}}\xi \leqslant\\& \qquad {C}_{3}N\left(t\right){\int }_{\mathbb R}{\mathrm{\varPhi }}_{\xi }^{2}{\rm{d}}\xi \leqslant {C}_{3}N\left(t\right){\|{\mathrm{\varPhi }}_{\xi }\|}^{2}\\[-12pt] \end{split}$ (55)

 $\begin{split}& {\int }_{\mathbb R}\left|{\mathrm{\varPhi }}_{\xi }F\left(U,{\mathrm{\varPhi }}_{\xi }\right)\right|{\rm{d}}\xi \leqslant {\int }_{R}\left|{f}{{'}{'}}\left(U+\eta \theta {\mathrm{\varPhi }}_{\xi }\right){\mathrm{\varPhi }}_{\xi }\right|{\mathrm{\varPhi }}_{\xi }^{2}{\rm{d}}\xi \leqslant\\&\qquad {C}_{4}N\left(t\right){\int }_{\mathbb R}{\mathrm{\varPhi }}_{\xi }^{2}{\rm{d}}\xi \leqslant {C}_{4}N\left(t\right){\|{\mathrm{\varPhi }}_{\xi }\|}^{2}\\[-12pt] \end{split}$ (56)

 $\begin{split}& G={F}_{\xi }=-\{{f}{{'}}\left(U+{\mathrm{\varPhi }}_{\xi }\right)\left({U}_{\xi }+{\mathrm{\varPhi }}_{\xi \xi }\right)-{f}{{'}}\left(U\right){U}_{\xi }-\\&\qquad{f}{{'}}\left(U\right){\mathrm{\varPhi }}_{\xi \xi }-{f}{{'}{'}}\left(U\right){U}_{\xi }{\mathrm{\varPhi }}_{\xi }\} =\\&\qquad-\{{f}{{'}{'}}\left(U+\theta {\mathrm{\varPhi }}_{\xi }\right){{U}_{\xi }\mathrm{\varPhi }}_{\xi }+{f}{{'}{'}}\left(U+\theta {\mathrm{\varPhi }}_{\xi }\right){\mathrm{\varPhi }}_{\xi }{\mathrm{\varPhi }}_{\xi \xi }-\\&\qquad{f}{{'}{'}}\left(U\right){{U}_{\xi }\mathrm{\varPhi }}_{\xi }\},\quad 0 < \theta < 1 \end{split}$

 $\begin{array}{c}{\int }_{\mathbb R}\left|{f}{{'}{'}}\left(U+\theta {\mathrm{\varPhi }}_{\xi }\right){{U}_{\xi }\mathrm{\varPhi }}_{\xi }^{2}\right|{\rm{d}}\xi \leqslant {C}_{5}N\left(t\right)\left|{u}_--{u}_+\right|{\|{\mathrm{\varPhi }}_{\xi }\|}^{2}\end{array}$
 $\begin{split}& {\int }_{\mathbb R}\left|{f}{{'}{'}}\left(U+\theta {\mathrm{\varPhi }}_{\xi }\right){{\mathrm{\varPhi }}_{\xi }\mathrm{\varPhi }}_{\xi }{\mathrm{\varPhi }}_{\xi \xi }\right|{\rm{d}}\xi \leqslant {C}_{5}N\left(t\right){\int }_{\mathbb R}{\mathrm{\varPhi }}_{\xi }{\mathrm{\varPhi }}_{\xi \xi }{\rm{d}}\xi\leqslant\\&\qquad {C}_{6}N\left(t\right)\left({\eta }_{2}{\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+C\left({\eta }_{2}\right){\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}\right) \end{split}$
 ${\int }_{\mathbb R}\left|-{f}^{{'}{'}}\left(U\right){{U}_{\xi }\mathrm{\varPhi }}_{\xi }^{2}\right|{\rm{d}}\xi \leqslant {C}_{h}\left|{u}_--{u}_+\right|{\|{\mathrm{\varPhi }}_{\xi }\|}^{2}$

 $\begin{split}& {\int }_{\mathbb R}\left|{\mathrm{\varPhi }}_{\xi }G\right|{\rm{d}}\xi \leqslant {C}_{7}N\left(t\right)\left({\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}\right)+\\&\qquad {C}_{8}\left|{u}_--{u}_+\right|{\|{\mathrm{\varPhi }}_{\xi }\|}^{2} \end{split}$ (57)

 $\begin{split}& {\int }_{\mathbb R}\left|{\mathrm{\varPhi }}_{\xi \xi }{G}_{\xi }\right|{\rm{d}}\xi \leqslant {C}_{9}N\left(t\right)\left({\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi \xi }\|}^{2}\right) +\\&\qquad {C}_{10}\left|{u}_--{u}_+\right|\left({\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}\right)\\[-12pt] \end{split}$ (58)

 $\begin{split} &\int_0^t \int_{\mathbb R}\left(\left|\varPhi F\left(U, \varPhi_{\xi}\right)\right|+\left|\delta \varPhi_{\xi} F\left(U, \varPhi_{\xi}\right)\right|+\left|\varPhi_{\xi} G\right|+\left|\delta \varPhi_{\xi \xi} G\right|\right)\cdot\\ &\qquad {\rm{d}} \xi {\rm{d}} \tau\} \leqslant C_{11} N(t) \int_0^t\left(\left\|\varPhi_{\xi}\right\|^2+\left\|\varPhi_{\xi \xi}\right\|^2+\left\|\varPhi_{\xi \xi \xi}\right\|^2\right) {\rm{d}} \tau+ \\ &\qquad C_{12}\left|u_{-}-u_{+}\right| \int_0^t\left(\left\|\varPhi_{\xi}\right\|^2+\left\|\varPhi_{\xi \xi}\right\|^2\right){\rm{ d}} \tau\\[-12pt] \end{split}$ (59)

 $\begin{split}& \frac{1}{2}\left({\|\mathrm{\varPhi }\|}^{2}+({\delta }^{2}+1){\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+{\delta }^{2}{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}\right) +\\& \qquad {\int }_{0}^{t}\left(\frac{{\lambda }^{*}}{2}{\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+\left(\delta +\frac{{\lambda }^{*}}{2}\right){\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}+\delta {\|{\mathrm{\varPhi }}_{\xi \xi \xi }\|}^{2}\right){\rm{d}}\tau \leqslant\\& \qquad \frac{{C}_{2}}{2}\left\{{\|{\mathrm{\varPhi }}_{0}\|}^{2}+({\delta }^{2}+1){\|{\mathrm{\varPhi }}_{0\xi }\|}^{2}+{\delta }^{2}{\|{\mathrm{\varPhi }}_{0\xi \xi }\|}^{2}\right\} +\\& \qquad\frac{1}{2}{C}_{2}{C}_{11}N\left(t\right){\int }_{0}^{t}\left({\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi \xi }\|}^{2}\right){\rm{d}}\tau +\\& \qquad\frac{1}{2}{C}_{2}{C}_{12}\left|{u}_--{u}_+\right|{\int }_{0}^{t}\left({\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}\right){\rm{d}}\tau \end{split}$

 $\mathrm{\varPhi }\in X\left(0,\infty \right)=\left\{\mathrm{\varPhi }\in {C}^{0}\left(0,\infty ;{H}^{2}\right),{\mathrm{\varPhi }}_{\xi }\in {L}^{2}\left(0,\infty ;{H}^{2}\right)\right\}$

b.由式(36)可知， ${‖{\mathrm{\varPhi }}_{\xi }‖}_{{H}^{1}}$ 关于 $t$ 连续， ${ \displaystyle \int }_{0}^{+\infty }{\|{\mathrm{\varPhi }}_{\xi }\|}_{{H}^{2}}^{2}{\rm{d}}\tau$ 存在，可以推出

 ${\int }_{0}^{t}\left({\|{\mathrm{\varPhi }}_{\xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi }\|}^{2}+{\|{\mathrm{\varPhi }}_{\xi \xi \xi }\|}^{2}\right){\rm{d}}\tau \leqslant {C}_{2}{N}_{0}^{2}$

 ${\int }_{0}^{t}{\|{\mathrm{\varPhi }}_{\xi }\|}_{{H}^{2}}^{2}{\rm{d}}\tau \leqslant {C}_{2}{N}_{0}^{2}$

 $\underset{t\to \infty }{\mathrm{lim}}{\;\|{\mathrm{\varPhi }}_{\xi }\left(t\right)\|}_{{H}^{2}}^{2}=0$ (60)

 $\left\|\varPhi_{\xi}\right\|_{L^{\infty}(\mathbb R)} \leqslant C\left\|\varPhi_{\xi}\right\|_{L^2(\mathbb R)}^{\frac{1}{2}} \left\|\varPhi_{\xi \xi}\right\|_{L^2(\mathbb R)}^{\frac{1}{2}}$

 $\underset{x\in \mathbb R}{\mathrm{sup}}\;\left|u\left(t,x\right)-U\left(x-ct\right)\right|=\underset{x\in \mathbb R}{\mathrm{s}\mathrm{u}\mathrm{p}}\;\left|\psi \right|\to 0 ,\quad t\to \infty$

 [1] GANSER G H, DREW D A. Nonlinear periodic waves in a two-phase flow model[J]. SIAM Journal on Applied Mathematics, 1987, 47(4): 726-736. DOI:10.1137/0147050 [2] GANSER G H, DREW D A. Nonlinear stability analysis of a uniformly fluidized bed[J]. International Journal of Multiphase Flow, 1990, 16(3): 447-460. DOI:10.1016/0301-9322(90)90075-T [3] ABIA L, CHRISTIE I, SANZ-SERNA J M. Stability of schemes for the numerical treatment of an equation modelling fluidized beds[J]. Mathematical Modelling and Numerical Analysis, 1989, 23(2): 191-204. DOI:10.1051/m2an/1989230201911 [4] LÓPEZ-MARCOS J C, SANZ-SERNA J M. A definition of stability for nonlinear problems[M]//STREHMEL K. Numerical Treatment of Differential Equations. Leipzig: B. G. Teubner, 1988: 216–226. [5] 冯民富, 明平兵. 河床流体模型方程的稳定化有限差分法及非线性稳定性分析[J]. 高等学校计算数学学报, 1997, 19(4): 298-311. [6] KORTEWEG D J, DE VRIES G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[J]. Philosophical Magazine, 2011, 91(6): 1007-1028. DOI:10.1080/14786435.2010.547337 [7] BONA J L, SCHONBEK M E. Travelling-wave solutions to the Korteweg-de Vries-Burgers equation[J]. Proceedings of the Royal Society of Edinburgh Section A:Mathematics, 1985, 101(3/4): 207-226. [8] CANOSA J, GAZDAG J. The Korteweg-de Vries-Burgers equation[J]. Journal of Computational Physics, 1977, 23(4): 393-403. DOI:10.1016/0021-9991(77)90070-5 [9] DEMIRAY H. A note on the exact travelling wave solution to the KdV-Burgers equation[J]. Wave Motion, 2003, 38(4): 367-369. DOI:10.1016/S0165-2125(03)00067-2 [10] FENG Z S. The first-integral method to study the Burgers-Korteweg-de Vries equation[J]. Journal of Physics A: Mathematical and General, 2002, 35(2): 343-349. DOI:10.1088/0305-4470/35/2/312 [11] GUAN K Y, GAO G. Qualitative research of traveling wave solution of mixed Burgers-KdV equation[J]. Acta Scientia Sinica, 1987, 17(1): 64-73. [12] XIONG S L. An analytic solution of Burgers-KdV equation[J]. Chinese Science Bulletin, 1989, 34(14): 1158-1162. [13] BIKBAEV R F. Shock waves in the modified Korteweg-de Vries-Burgers equation[J]. Journal of Nonlinear Science, 1995, 5(1): 1-10. DOI:10.1007/BF01869099 [14] JACOBS D, MCKINNEY B, SHEARER M. Traveling wave solutions of the modified Korteweg-deVries-Burgers equation[J]. Journal of Differential Equations, 1995, 116(2): 448-467. DOI:10.1006/jdeq.1995.1043 [15] FENG Z S. On traveling wave solutions to modified Burgers-Korteweg-de Vries equation[J]. Physics Letters A, 2003, 318(6): 522-525. DOI:10.1016/j.physleta.2003.09.057 [16] BEKIR A. On traveling wave solutions to combined KdV-mKdV equation and modified Burgers-KdV equation[J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(4): 1038-1042. DOI:10.1016/j.cnsns.2008.03.014 [17] ZHANG W G, BIAN L Y, ZHAO Y. Qualitative analysis and solutions of bounded travelling waves for the fluidized-bed modelling equation[J]. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2010, 140(2): 241-257. DOI:10.1017/S030821050900064X [18] PEGO R L. Remarks on the stability of shock profiles for conservation laws with dissipation[J]. Transactions of the American Mathematical Society, 1985, 291(1): 353-361. DOI:10.1090/S0002-9947-1985-0797065-0 [19] NEMYTSKII V V, STEPANOV V V. Qualitative theory of differential equations[M]. New York: Dover Publications, 1989. [20] ZHANG Z F, DING T R, HUANG W Z, et al. Qualitative theory of differential equations[M]. Providence: American Mathematical Society, 1992. [21] 马知恩, 周义仓. 常微分方程定性与稳定性方法[M]. 北京: 科学出版社, 2001. [22] ARONSON D G, WEINBERGER H F. Multidimensional nonlinear diffusion arising in population genetics[J]. Advances in Mathematics, 1978, 30(1): 33-76. DOI:10.1016/0001-8708(78)90130-5 [23] FIFE P C. Mathematical aspects of reacting and diffusing systems[M]. Berlin, Heidelberg: Springer, 1979. [24] YE Q X, LI Z Y. Introduction to reaction diffusion equations[M]. Beijing: Science Press, 1990. [25] BELYTSCHKO T, LU Y Y, GU L. Element-free Galerkin methods[J]. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229-256. DOI:10.1002/nme.1620370205 [26] KAWASHIMA S. Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics[D]. Kyoto: Kyoto University, 1984. [27] DIAZ J B, METCALF F T. An analytic proof of Young's inequality[J]. The American Mathematical Monthly, 1970, 77(6): 603-609. DOI:10.1080/00029890.1970.11992547 [28] EVANS L C. Partial differential equations[M]. 2nd ed. Berkeley: American Mathematical Society, 2010.