# Remarks on Radial Solutions of a Parabolic Gelfand-Type Equation

## Abstract

**:**

## 1. Introduction

**Theorem**

**1**

**.**Let $\mathsf{\Omega}\subset {\mathbb{R}}^{2}$ be a bounded domain with a smooth boundary $\partial \mathsf{\Omega}$. For any $\lambda >0$ and ${u}_{0}\in {H}_{0}^{1}\left(\mathsf{\Omega}\right)$ satisfying

**Theorem**

**2**

**.**Let $\mathsf{\Omega}=(0,1)$. If we replace (3) with

**Theorem**

**3.**

**Theorem**

**4**

**.**For ${u}_{0}\in X$, (6) admits a unique global solution $u=u(x,t)$, such that

## 2. Preliminary

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3**

**.**Let $A,B,\alpha >0$. Assume that $A\alpha e<1$ holds. We define

## 3. Global Solution

**Lemma**

**4.**

**Proof.**

**Proof of Theorem 3.**

**Lemma**

**5**

**.**We have

**Lemma**

**6**

**.**Let

**Proof.**

**Lemma**

**7**

**.**We have

**Proof.**

**Lemma**

**8**

**.**We have

**Proof.**

**Remark**

**1.**

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Miyasita, T.
Remarks on Radial Solutions of a Parabolic Gelfand-Type Equation. *Axioms* **2022**, *11*, 429.
https://doi.org/10.3390/axioms11090429

**AMA Style**

Miyasita T.
Remarks on Radial Solutions of a Parabolic Gelfand-Type Equation. *Axioms*. 2022; 11(9):429.
https://doi.org/10.3390/axioms11090429

**Chicago/Turabian Style**

Miyasita, Tosiya.
2022. "Remarks on Radial Solutions of a Parabolic Gelfand-Type Equation" *Axioms* 11, no. 9: 429.
https://doi.org/10.3390/axioms11090429