# A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations

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## Abstract

**:**

## 1. Introduction

**non-stationary**solutions of a given degenerate diffusion equation which also violate the SMP. PDEs with degenerate diffusion, such as

**nonlocal**diffusion, as we will show in this paper. We will exhibit large classes of degenerate diffusion equations for which the only solutions violating the strong maximum principle are stationary. We characterize such solutions as having zero flux, which generalizes the classical conclusion of solutions being constant.

**Notation**

**1.****Norms in a single variable.**- (i)
- $C\left(\overline{\Omega}\right)$: those continuous functions of x on $\Omega $, which are continuous up to the boundary $\phantom{\rule{0.166667em}{0ex}}\partial \Omega $, with norm ${sup}_{\Omega}\left|f\left(x\right)\right|$.
- (ii)
- ${L}^{p}(\Omega )$: those Lebesgue measurable functions of x on $\Omega $ whose p-th power has a finite Lebesgue integral, with norm ${\left({\int}_{\Omega}{\left|f\left(x\right)\right|}^{p}dx\right)}^{1/p}$. Here, $1\le p<\infty $.
- (iii)
- ${L}^{\infty}(\Omega )$: those Lebesgue measurable functions of x on $\Omega $ which are bounded up to a Lebesgue null set, with norm $\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}{sup}_{\Omega}\left|f\left(x\right)\right|$. Here, the essential supremum is the infimum of those constants $C\ge 0$, such that $\left|f\right(x\left)\right|\le C$ almost everywhere.

**2.****Spacetime norms.**We let ${C}^{1}([0,T];{L}^{\infty}(\Omega ))$ be bounded functions of space which are continuously differentiable in time, with norm

**3.****Two-point norms.**We let $C(\overline{\Omega};C\left(\overline{\Omega}\right))$ be continuous functions of y that continuously vary in x, both up to the boundary, with norm

## 2. Strong Maximum Principles for Degenerate Diffusion Operators

#### 2.1. Space-Continuous Solutions

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

**Theorem**

**2.**

**Proof.**

#### 2.2. Merely Bounded Solutions

**Theorem**

**3.**

**Proof.**

#### 2.3. Examples of Degenerate Fluxes with Maximum Principles

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 3. Counterexamples to the Strong Maximum Principle

**Proposition**

**2.**

**Proof.**

## 4. The Inconclusive Cases

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

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Hartland, T.; Shankar, R.
A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations. *Axioms* **2023**, *12*, 1059.
https://doi.org/10.3390/axioms12111059

**AMA Style**

Hartland T, Shankar R.
A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations. *Axioms*. 2023; 12(11):1059.
https://doi.org/10.3390/axioms12111059

**Chicago/Turabian Style**

Hartland, Tucker, and Ravi Shankar.
2023. "A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations" *Axioms* 12, no. 11: 1059.
https://doi.org/10.3390/axioms12111059