# A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Lie Symmetry of the Cauchy Problem

**Theorem**

**1.**

**Remark**

**1.**

## 3. Application of Lie Symmetry for Constructing Exact Solutions of Cauchy Problems

#### 3.1. Exact Solutions of the (1 + 1)-Dimensional Cauchy Problem

**Theorem**

**2.**

#### 3.2. Reduction and Exact Solutions of (1 + 2)-Dimensional Cauchy Problem

## 4. Lie Symmetry of the Neumann Problems

#### 4.1. Neumann Problem on the Strip

**Theorem**

**3.**

#### 4.2. Neumann Problem on Interior/Exterior of a Circle

**Theorem**

**4.**

## 5. Exact Solutions of the Neumann Problem

**Theorem**

**5.**

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Plots of the functions $\rho (0,x)$ and $\rho (t,x)$ using Formulae (19) with $\gamma =1$.

**Table 1.**maximal algebra of invariances (MAIs) of the Cauchy Problem (4).

$\mathit{\varphi}(\mathbf{x})$ | MAI | |
---|---|---|

1 | ∀ | ${G}_{1}^{\infty}$ with ${f}_{1}(0)={f}_{1}^{\prime}(0)=0,$ |

${G}_{2}^{\infty}$ with ${f}_{2}^{\prime}(0)=0,$ | ||

${X}_{S}^{\infty}$ with $g(0)=0$ | ||

2 | 0 | ${G}_{1}^{\infty}$ with ${f}_{1}^{\prime}(0)=0,$ |

${G}_{2}^{\infty}$ with ${f}_{2}^{\prime}(0)=0,$ | ||

${X}_{S}^{\infty}$ with $g(0)=0,$ | ||

${J}_{12},D$ | ||

3 | ${\lambda}_{1}x,$ ${\lambda}_{1}\ne 0$ | ${G}_{1}^{\infty}+{b}_{6}D$ with ${f}_{1}(0)=0,{f}_{1}^{\prime}(0)={b}_{6}{\lambda}_{1},$ |

${G}_{2}^{\infty}+{b}_{5}{J}_{12}$ with ${f}_{2}^{\prime}(0)={b}_{5}{\lambda}_{1},$ | ||

${b}_{1}{G}_{1}^{\infty}+{X}_{S}^{\infty}+{b}_{1}{a}_{6}D$ with $g(0)={b}_{1}{\lambda}_{1}{f}_{1}(0),$ | ||

${f}_{1}(0)\ne 0,{f}_{1}^{\prime}(0)={a}_{6}{\lambda}_{1}$ | ||

4 | ${\lambda}_{2}{x}^{2}+{\lambda}_{0},$ ${\lambda}_{2}\ne 0$ | ${G}_{1}^{\infty}$ with ${f}_{1}^{\prime}(0)=2{\lambda}_{2}{f}_{1}(0),$ |

${G}_{2}^{\infty}$ with ${f}_{2}^{\prime}(0)=0,$ | ||

${X}_{S}^{\infty}$ with $g(0)=0$ |

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Cherniha, R.; Didovych, M.
A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II. *Symmetry* **2017**, *9*, 13.
https://doi.org/10.3390/sym9010013

**AMA Style**

Cherniha R, Didovych M.
A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II. *Symmetry*. 2017; 9(1):13.
https://doi.org/10.3390/sym9010013

**Chicago/Turabian Style**

Cherniha, Roman, and Maksym Didovych.
2017. "A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II" *Symmetry* 9, no. 1: 13.
https://doi.org/10.3390/sym9010013