﻿ Extended Fisher-Kolmogorov系统在Direchlet边界条件下的定态分歧

# Extended Fisher-Kolmogorov系统在Direchlet边界条件下的定态分歧Steady State Bifurcation of Extended Fish-Kolmogorov System with Direchlet Boundary Condition

Abstract: In this paper, we study the bifurcation problem of extended Fisher-Kolmogorov system with Di-rechlet boundary condition. Based on normalized Lyapunov-Schmidt reduction method, we use spectral analysis and bifurcation theory to prove the existence of bifurcated solution and obtain the exact form of bifurcated solutions. Furthermore, the regularity of solutions is also discussed.

1. 引言及预备知识

${L}_{\lambda }\phi +G\left(\phi ,\lambda \right)=0$

$\lambda >{\lambda }_{0}$ 时，算子方程存在一个解 $\left({\phi }_{\lambda },\lambda \right)\ne \left(0,\lambda \right)$$\underset{\lambda \to {\lambda }_{0}}{\mathrm{lim}}\left({\phi }_{\lambda },\lambda \right)=\left(0,{\lambda }_{0}\right)$$\underset{\lambda \to {\lambda }_{0}}{\mathrm{lim}}{‖{\phi }_{\lambda }‖}_{{x}_{1}}=0$，则称算子方程

$\left(0,{\lambda }_{0}\right)$ 处发生分歧。

$\left\{\begin{array}{l}\frac{\partial u}{\partial t}=-\mu \frac{{\partial }^{4}u}{\partial {x}^{4}}+\alpha \frac{{\partial }^{2}u}{\partial {x}^{2}}+\lambda g\left(u\right),\left(x,t\right)\in \left(0,\text{π}\right)×\left(0,\infty \right)\\ u\left(0\right)=u\left(\text{π}\right)=0\\ \underset{0}{\overset{\text{π}}{\int }}u\left(x\right)\text{d}x=0,\forall t\ge 0\\ u\left(x,0\right)={u}_{0},x\in \left(0,\text{π}\right)\end{array}$ (1)

$g\left(s\right)=\underset{k=2}{\overset{p}{\sum }}{a}_{k}{s}^{k}$

$\left\{\begin{array}{l}-\mu \frac{{\partial }^{4}u}{\partial {x}^{4}}+\alpha \frac{{\partial }^{2}u}{\partial {x}^{2}}+\lambda u+g\left(u\right)=0,x\in \left(0,\text{π}\right)\\ u\left(0\right)=u\left(\text{π}\right)=0\\ \underset{\text{0}}{\overset{\text{π}}{\int }}u\left(x\right)\text{d}x=0\end{array}$ (2)

$H={L}^{2}\left(0,\text{π}\right)$

${H}_{1}=\left\{u\in {H}^{4}\left[0,\text{π}\right]|u\left(0\right)=u\left(\text{π}\right)=0,\underset{\text{0}}{\overset{\text{π}}{\int }}u\left(x\right)\text{d}x=0\right\}$

$Au=-\mu \frac{{\partial }^{4}u}{\partial {x}^{4}}+\alpha \frac{{\partial }^{2}u}{\partial {x}^{2}}$(3)

${B}_{\lambda }=\lambda u$

2. 主要结果

2.1. 定理

1) 当 ${\alpha }_{2}\ne 0$ 时，系统(2)可从 $\left(u,\lambda \right)=\left(0,\alpha +\mu \right)$ 处产生1个正则分歧解，其表达式如下：

${\stackrel{¯}{u}}_{1}=-\frac{3\text{π}\left(\lambda -\alpha -\mu \right)}{8{\alpha }_{2}}\mathrm{sin}x+O\left({|\lambda -\alpha -\mu |}^{2}\right)$(4)

2) 当 ${\alpha }_{2}=0,{\alpha }_{3}\ne 0$ 时，系统(2)从 $\left(u,\lambda \right)=\left(0,\alpha +\mu \right)$ 处产生2个正则分歧解，其表达式如下：

$\begin{array}{l}{\stackrel{¯}{u}}^{+}=\sqrt{\frac{4\left(\alpha +\mu -\lambda \right)}{3{\alpha }_{3}}}\mathrm{sin}x+O\left({|\lambda -\alpha -\mu |}^{\frac{1}{2}}\right)\\ {\stackrel{¯}{u}}^{-}=-\sqrt{\frac{4\left(\alpha +\mu -\lambda \right)}{3{\alpha }_{3}}}\mathrm{sin}x+O\left({|\lambda -\alpha -\mu |}^{\frac{1}{2}}\right)\end{array}$ (5)

2.2. 定理证明

$-\frac{{\text{d}}^{2}{\phi }_{k}}{\text{d}{x}^{2}}={\lambda }_{k}{\phi }_{k}$

${\phi }_{k}\left(0\right)={\phi }_{k}\left(\text{π}\right)=0$

${\int }_{0}^{\text{π}}{\phi }_{k}^{2}\text{d}x=1$ (6)

${\phi }_{k}=\sqrt{\frac{2}{\text{π}}}\mathrm{sin}kx$

$\left\{{\beta }_{k}\left(\lambda \right)=\lambda -\alpha {k}^{2}-\mu {k}^{4}|k=1,2,\cdots \right\}$

${〈{\phi }_{i},{\phi }_{j}〉}_{H}=\left\{\begin{array}{l}1,i=j,\\ 0,i\ne j.\end{array}$

${\beta }_{1}\left(\lambda \right)=\lambda -\alpha -\mu$

${\phi }_{\text{1}}=\sqrt{\frac{2}{\text{π}}}\mathrm{sin}x$

${\beta }_{j}$ 满足

${\beta }_{j}\left(\alpha +\mu \right)\ne 0,j\ge 2$

${H}_{1}={E}_{1}\oplus {E}_{\text{2}}$$H={E}_{1}\oplus {\stackrel{¯}{E}}_{\text{2}}$

${E}_{1}=span\left\{{\phi }_{1}\right\}$${E}_{\text{2}}=span\left\{{\phi }_{2},{\phi }_{3},\cdots \right\}$.

${L}_{\lambda }={L}_{\lambda }^{1}+{L}_{\lambda }^{2}$

${L}_{\lambda }^{1}:{E}_{1}\to {E}_{1}$${L}_{\lambda }^{2}:{E}_{2}\to {\stackrel{¯}{E}}_{2}$.

${u}_{1}={x}_{1}{\phi }_{1}$${u}_{2}=\underset{j=2}{\overset{\infty }{\sum }}{y}_{j}{\phi }_{j}$${y}_{j}\in R$.

${\beta }_{\text{1}}\left(\lambda \right){x}_{1}+\underset{k=2}{\overset{p}{\sum }}{\alpha }_{k}{〈{\left({x}_{1}{\phi }_{1}+\underset{j=2}{\overset{\infty }{\sum }}{y}_{j}{\phi }_{j}\right)}^{k},{\phi }_{1}〉}_{H}=0$ (7)

${\beta }_{j}\left(\lambda \right){y}_{j}+\underset{k=2}{\overset{p}{\sum }}{\alpha }_{k}{〈{\left({x}_{1}{\phi }_{1}+\underset{j=2}{\overset{\infty }{\sum }}{y}_{j}{\phi }_{j}\right)}^{k},{\phi }_{j}〉}_{H}=0,j\ge 2$ (8)

1) 当 ${\alpha }_{2}\ne 0$ 时，近似方程为

${\beta }_{\text{j}}\left(\lambda \right){y}_{i}+{\alpha }_{2}{x}_{1}^{2}{〈{\phi }_{1}^{2},{\phi }_{j}〉}_{H}+O\left({x}_{1}^{2}\right)=0,j\ge 2$

$\begin{array}{l}〈{\phi }_{1}^{2},{\phi }_{1}〉=\frac{8\sqrt{2}}{3\text{π}\sqrt{\text{π}}},\\ 〈{\phi }_{1}^{2},{\phi }_{j}〉=0,j\ge 2\end{array}$

${y}_{j}=O\left({x}_{1}^{2}\right),j\ge 2$

${\beta }_{1}\left(\lambda \right){x}_{1}+{\alpha }_{2}{x}_{1}^{2}{〈{\phi }_{1}^{2},{\phi }_{1}〉}_{H}+O\left({x}_{1}^{2}\right)=0$

${\beta }_{1}\left(\lambda \right){x}_{1}+\frac{8\sqrt{2}{\alpha }_{2}{x}_{1}^{2}}{3\text{π}\sqrt{\text{π}}}=0$(9)

${x}_{1}=-\frac{3\text{π}\sqrt{\text{π}}\left(\lambda -\alpha -\mu \right)}{8\sqrt{2}{\alpha }_{2}}$ (10)

2) 当 ${\alpha }_{2}=0$${\alpha }_{3}\ne 0$ 时，由(7)式可得

${\beta }_{1}\left(\lambda \right){x}_{1}+{\alpha }_{3}〈{x}_{1}^{3}{\phi }_{1}^{3},{\phi }_{1}〉=0$

$〈{\phi }_{1}^{3},{\phi }_{1}〉=\frac{3}{2\text{π}}$

${\beta }_{1}\left(\lambda \right){x}_{1}+\frac{3{\alpha }_{3}}{2\text{π}}{x}_{1}^{3}=0$ (11)

$\begin{array}{l}{x}_{1}^{+}=\sqrt{\frac{2\text{π}\left(\alpha +\mu -\lambda \right)}{3{\alpha }_{3}}},\\ {x}_{1}^{-}=-\sqrt{\frac{2\text{π}\left(\alpha +\mu -\lambda \right)}{3{\alpha }_{3}}}\end{array}$ (12)

${\beta }_{1}\left(\lambda \right)+\frac{16\sqrt{2}{\alpha }_{2}}{3\text{π}\sqrt{\text{π}}}{x}_{1}=-\left(\lambda -\alpha -\mu \right)$ (13)

${\beta }_{1}\left(\lambda \right)+\frac{3{\alpha }_{3}}{2\text{π}}{x}_{1}^{2}=-2\left(\lambda -\alpha -\mu \right)$ (14)

${\stackrel{¯}{u}}_{1}=-\frac{3\text{π}\left(\lambda -\alpha -\mu \right)}{8{\alpha }_{2}}\mathrm{sin}x+O\left({|\lambda -\alpha -\mu |}^{2}\right)$

$\begin{array}{l}{\stackrel{¯}{u}}^{+}=\sqrt{\frac{4\left(\alpha +\mu -\lambda \right)}{3{\alpha }_{3}}}\mathrm{sin}x+O\left({|\lambda -\alpha -\mu |}^{\frac{1}{2}}\right)\\ {\stackrel{¯}{u}}^{-}=-\sqrt{\frac{4\left(\alpha +\mu -\lambda \right)}{3{\alpha }_{3}}}\mathrm{sin}x+O\left({|\lambda -\alpha -\mu |}^{\frac{1}{2}}\right)\end{array}$

NOTES

*通讯作者。

[1] Coullet, P., Elphick, C. and Repaux, D. (1897) The Nature of Spatial Chaos. Physical Review Letters, 58, 431-434.
https://doi.org/10.1103/PhysRevLett.58.431

[2] Dee, G. and Saarloose, W. (1998) Bistable Systems with Prop-agating Fronts Leading to Pattern Formation. Physical Review Letters, 60, 2641-2644.
https://doi.org/10.1103/PhysRevLett.60.2641

[3] Tersians, C.J. (2001) Periodic and Homoclinic Solutions of Extended Fisher-Kolmogorov Equations. Journal of Mathematical Analysis and Applications, 2, 490-506.
https://doi.org/10.1006/jmaa.2001.7470

[4] 李军燕, 柴沙沙. 一类浮游生物捕食系统的定态分歧[J]. 平顶山学院学报: 自然科学版, 2014, 29(2): 19-23.

[5] 戴婉仪. 一类具有交叉互惠系统的定态分歧与稳定性[J]. 华南师范大学学报: 自然科学版, 2005(3): 112-117.

[6] Kwapisz, J. (2000) Uniqueness of the Stationary Wave for the Extended Fisher-Kolmogorov Equation. Journal of Differential Equations, 1, 235-253.
https://doi.org/10.1006/jdeq.1999.3750

[7] 罗宏, 蒲志林. Extended Fisher-Kolmogorov系统的整体吸引子及其分形维数估计[J]. 四川师范大学学报: 自然科学版, 2004, 27(2): 135-138.

[8] 马天, 汪守宏. 非线性演化方程的稳定性与分歧[M]. 北京: 科学出版社, 2007.

[9] 帅鲲, 蒲志林, 潘志刚. 一类带平均值约束的二元方程组的定态分歧[J]. 四川师范大学学报: 自然科学版, 2013, 36(6): 820-823.

[10] Ma, T. and Wang, S. (2005) Bifurcation Theory and Applications. World Scientific, Singapore.
https://doi.org/10.1142/9789812701152

[11] 周钰谦, 刘倩. 一类非线性磁流变阻尼系统的局部分岔[J]. 四川大学学报: 自然科学版, 2008, 45(2): 241-244.

[12] 钟承奎, 范先令, 陈文塬. 非线性泛函分析引论[M]. 兰州: 兰州大学出版社, 1998.

[13] 张强, 张正丽. 一类反应扩散方程的定态分歧[J]. 四川大学学报: 自然科学版, 2010, 47(3): 461-463.

[14] 张强, 雷开洪, 向丽. Fisher-Kolmogorov-Petrovskii-Piskunov方程的定态分歧[J]. 四川大学学报: 自然科学版, 2013, 50(1): 6-10.

[15] 张强, 曾艳, 李桂花, 张黔川. 带Neumann边界条件的Extended Fisher-Kolmogorov系统的定态分歧[J]. 四川师范大学学报: 自然科学板, 2014, 37(2): 188-191.

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