# Qualitative Properties of Solutions of Equations and Inequalities with KPZ-Type Nonlinearities

## Abstract

**:**

## 1. Introduction

#### 1.1. History and Motivation

#### 1.2. Illustrative Example: Gidas–Spruck Theorem and Its Generalizations

## 2. Parabolic Stabilization

#### 2.1. Regular Case

- for any $x\in {\mathbb{R}}^{n}$, $\underset{t\to \infty}{lim}v(x,t)$ exists if and only if $\underset{t\to \infty}{lim}\frac{n\mathsf{\Gamma}\left(\frac{n}{2}\right)}{2{\pi}^{\frac{n}{2}}{t}^{n}}\underset{\left|x\right|<t}{\int}v(x,0)dx$ exists; if those limits exist, then they are equal to each other.

**Theorem**

**1.**

**Remark**

**1.**

#### 2.2. Singular Case

**Theorem**

**2.**

**Theorem**

**3.**

**Remark**

**2.**

## 3. Elliptic Stabilization

**Theorem**

**4.**

**Theorem**

**5.**

**Theorem**

**6.**

**Remark**

**3.**

## 4. Blow-Up for Partial Differential Inequalities

#### 4.1. Elliptic Case

**Theorem**

**7.**

**Example**

**1.**

**Theorem**

**8.**

**Theorem**

**9.**

#### 4.2. Parabolic Case

**Theorem**

**10.**

**Remark**

**4.**

**Remark**

**5.**

**Example**

**2.**

## 5. Integrodifferential Blow-Up

#### 5.1. Stationary Case

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

#### 5.2. Nonstationary Case

**Proposition**

**4.**

**Proposition**

**5.**

## 6. Qualitative Properties of Solutions

#### 6.1. Parabolic Equations Admitting Degenerations at Infinity

**Theorem**

**11.**

**Theorem**

**12.**

**Theorem**

**13.**

**Proposition**

**6.**

**Proposition**

**7.**

#### 6.2. Extinction Phenomena

#### 6.2.1. Parabolic Inequalities

**Theorem**

**14.**

- (i)
- $\mathrm{supp}\phantom{\rule{0.166667em}{0ex}}u(x,t)$ is bounded for any positive $t;$
- (ii)
- $\underset{\left|x\right|\to \infty}{lim}u(x,t)=0$ uniformly with respect to t from $[0,+\infty );$
- (iii)
- there exists a positive T such that $u(x,t)\equiv 0$ in the half-space ${\mathbb{R}}^{n}\times [T,+\infty )$.

**Theorem**

**15.**

#### 6.2.2. Parabolic Equations with Potentials

**Theorem**

**16.**

**Theorem**

**17.**

- (i)
- there exists at most one classical bounded nonnegative solution of the Cauchy problem for Equation (23);
- (ii)
- if$$C(x,t)\le -\alpha min\left(1,\frac{1}{{\left|x\right|}^{2}}\right),$$
- (iii)
- if C(x,t) ≤ −α (this inequality is stronger), then there exists a positive constant a such that the inequality$$u(x,t)\le \underset{x\in {\mathbb{R}}^{n}}{sup}\phantom{\rule{-0.166667em}{0ex}}{u}_{0}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{e}^{-at}$$

**Theorem**

**18.**

#### 6.3. Singular Equations with Nonclassical Neumann Conditions

#### 6.3.1. Stationary Case

**Theorem**

**19.**

#### 6.3.2. Nonstationary Case: Blow-Up

**Theorem**

**20.**

- (i)
- $\underset{\mathsf{\Omega}}{\int}a\left(x\right)dx<0,$$\alpha >-1,$ and $p>1;$
- (ii)
- $\underset{\mathsf{\Omega}}{\int}a\left(x\right)dx>0,$$\alpha <-1,$ and $p<1.$

**Theorem**

**21.**

**Theorem**

**22.**

#### 6.3.3. Nonstationary Case: Large-Time Behavior

**Theorem**

**23.**

## 7. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Muravnik, A.B.
Qualitative Properties of Solutions of Equations and Inequalities with KPZ-Type Nonlinearities. *Mathematics* **2023**, *11*, 990.
https://doi.org/10.3390/math11040990

**AMA Style**

Muravnik AB.
Qualitative Properties of Solutions of Equations and Inequalities with KPZ-Type Nonlinearities. *Mathematics*. 2023; 11(4):990.
https://doi.org/10.3390/math11040990

**Chicago/Turabian Style**

Muravnik, Andrey B.
2023. "Qualitative Properties of Solutions of Equations and Inequalities with KPZ-Type Nonlinearities" *Mathematics* 11, no. 4: 990.
https://doi.org/10.3390/math11040990