# Regularity and Decay of Global Solutions for the Generalized Benney-Lin Equation Posed on Bounded Intervals and on a Half-Line

## Abstract

**:**

## 1. Introduction

## 2. Notations and Auxiliary Facts

**Lemma 1.**

**Lemma 2.**

**Lemma 3.**

**Lemma 4.**

**Lemma**

**5.**

**Proof.**

## 3. Generalized Benney-Lin Equation Posed on Bounded Intervals

**Problem**

**1.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Estimate I.**Multiply (6) by $2u$ to obtain

**Estimate II.**

**The case $k<4$.**

**The case $k=4$.**

**Proposition 1.**

**Proof.**

**Remark**

**1.**

**Lemma**

**7.**

**Proof.**

**Benney-Lin equation posed on ${\mathbb{R}}^{+}.$**

**Theorem**

**2.**

**Proof.**

**Estimate I.**Multiplying (31) by $2u$ and acting in the same manner as by proving (13) and (14), we obtain

**Estimate II.**Differentiate (31) with respect to t, then multiply the result by $2{u}_{t}$ to get:

**The case $k<4$.**

**The case $k=4$.**

## 4. Stability Intervals. Small Solutions

**Theorem**

**3.**

**Proof.**

**Estimate I.**Multiply (43) by $2u$ to obtain

**Estimate II.**Differentiate (43) by t and multiply the result by $2{u}_{t}$ to obtain

**The case $k<8$.**

**The case $k=8$.**

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Larkin, N.A.
Regularity and Decay of Global Solutions for the Generalized Benney-Lin Equation Posed on Bounded Intervals and on a Half-Line. *Axioms* **2022**, *11*, 596.
https://doi.org/10.3390/axioms11110596

**AMA Style**

Larkin NA.
Regularity and Decay of Global Solutions for the Generalized Benney-Lin Equation Posed on Bounded Intervals and on a Half-Line. *Axioms*. 2022; 11(11):596.
https://doi.org/10.3390/axioms11110596

**Chicago/Turabian Style**

Larkin, Nikolai A.
2022. "Regularity and Decay of Global Solutions for the Generalized Benney-Lin Equation Posed on Bounded Intervals and on a Half-Line" *Axioms* 11, no. 11: 596.
https://doi.org/10.3390/axioms11110596