# Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type

## Abstract

**:**

## 1. Introduction

## 2. Formulation

## 3. Existence Theorem

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

## 4. Exact Solutions. General Case

## 5. Exact Solutions. Particular Cases

## 6. Discussion

## 7. Conclusions

- The creation of a method for the numerical solution of the considered problems, based on the use of modern computing technologies. This can be, for example, the boundary element method, which the author has been developing in recent years in collaboration with colleagues. In this case, the segments of the constructed series will be used to eliminate the singularity;
- Increasing the dimension of problems; i.e., considering cases when the desired function depends on two ${x}_{1},{x}_{2}$ or three spatial variables ${x}_{1},{x}_{2},{x}_{3}$. Here, the most interesting case is when the diffusion wave front cannot be resolved with respect to one of the coordinates ${x}_{i}$ and is, for example, a cylindrical surface with a closed generatrix;
- The final stage of the research cycle should be the use of the developed model–algorithmic apparatus to solve applied problems related to the modeling of processes that occur in Lake Baikal. The lake is the largest natural reservoir of fresh water and is included in the UNESCO World Heritage List. The author lives and works close to this unique natural object and participates in scientific projects aimed at studying and saving it.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Kazakov, A.
Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type. *Symmetry* **2021**, *13*, 871.
https://doi.org/10.3390/sym13050871

**AMA Style**

Kazakov A.
Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type. *Symmetry*. 2021; 13(5):871.
https://doi.org/10.3390/sym13050871

**Chicago/Turabian Style**

Kazakov, Alexander.
2021. "Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type" *Symmetry* 13, no. 5: 871.
https://doi.org/10.3390/sym13050871