On the Long Time Existence of a Fractional KdV-BBM Type Equation

TL;DR

This paper investigates the long-term existence of solutions to a fractional KdV-BBM type equation, extending the lifespan of solutions with small initial data through analytical and numerical methods.

Contribution

It introduces a combined analytical and numerical approach to extend the existence time of solutions for a fractional KdV-BBM equation.

Findings

Extended solution lifespan from 1/ε to 1/ε² for small initial data.

Numerical evidence of solutions existing beyond hyperbolic time scale.

Analytical and numerical analysis of smooth solution existence.

Abstract

We consider a fractional Korteweg de Vries-Benjamin Bona Mahony (KdV-BBM) type equation including both fractional dispersive terms of fractional KdV and fractional BBM equations. We aim to enhance the existence time of solutions with small initial data $|| u_0||_{H^{N+\alpha/2}}= \epsilon$ from $\frac{1}{\epsilon}$ to $\frac{1}{\epsilon^2}$. The proof relies on the combination of a modified energy method with Fourier techniques. In addition, the long time existence issues are investigated numerically. Numerical observations of the lifespan give an evidence of existence of solutions beyond the hyperbolic time scale. This study provides a detailed analysis from both analytical and numerical aspects for the existence of smooth solutions.