# Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation

## 3. Existence Theorem

**Remark**

**1.**

**Remark**

**2.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

- Constructing a solution in the form of the Taylor series.
- Proving of convergence of a series by the majorant method.

**Corollary**

**1.**

**Proof**

**of**

**Corollary**

**1.**

**Remark**

**3.**

## 4. Exact Solutions

**Theorem**

**2.**

- (1)
- for $a\left(t\right)={c}_{1}{e}^{{c}_{3}t}$, if $\gamma =1$,
- (2)
- for $a\left(t\right)={c}_{3}ln({c}_{1}t+{c}_{2})$, if $\gamma =2$,
- (3)
- for $a\left(t\right)={({c}_{1}t+{c}_{2})}^{(\gamma -2)/(2\gamma -2)}$, if $\gamma \ge 3$.

**Proof**

**of**

**Theorem**

**2.**

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

## 5. Special Case

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**2.**

**Remark**

**4.**

**Example**

**1.**

**Example**

**2.**

**Remark**

**5.**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Kazakov, A.; Kuznetsov, P.; Lempert, A.
Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type. *Symmetry* **2020**, *12*, 999.
https://doi.org/10.3390/sym12060999

**AMA Style**

Kazakov A, Kuznetsov P, Lempert A.
Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type. *Symmetry*. 2020; 12(6):999.
https://doi.org/10.3390/sym12060999

**Chicago/Turabian Style**

Kazakov, Alexander, Pavel Kuznetsov, and Anna Lempert.
2020. "Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type" *Symmetry* 12, no. 6: 999.
https://doi.org/10.3390/sym12060999