The exact limits of all order derivatives of the global smooth solution of a general Korteweg-de Vries-Burgers equation

Linghai Zhang

doi:10.3934/dcdss.2025054

Article Contents

2025, Volume 18, Issue 12: 3846-3895. Doi: 10.3934/dcdss.2025054

This issue Previous Article On explicit solutions for coupled reaction-diffusion and Burgers-type equations with variable coefficients through a Riccati system Next Article On the large time dynamics of a rapidly rotating convection model at infinite Prandtl number

The exact limits of all order derivatives of the global smooth solution of a general Korteweg-de Vries-Burgers equation

Linghai Zhang

Department of Mathematics, Lehigh University, 17 Memorial Drive East, Bethlehem, PA 18015 USA

Received: September 2024

Revised: March 2025

Early access: April 22, 2025

Published online: April 22, 2025

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Abstract

We study the Cauchy problem for a general Korteweg-de Vries-Burgers equation. Also, we study the Cauchy problem for the corresponding linear equation. We will couple together several classical ideas (the Fourier transformation, Parseval's identity, Lebesgue's dominated convergence theorem, squeeze theorem, several well known inequalities) and existing results. We will use rigorous mathematical analysis to accomplish the exact limits of all order derivatives of the global smooth solution of the Cauchy problem for the Korteweg-de Vries-Burgers equation.

Keywords:

General Korteweg-de Vries-Burgers equation,
global smooth solution,
all order derivatives,
exact limits,
sharp rates,
ratios.

Mathematics Subject Classification: Primary: 35Q20.

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