On periodic solutions of the Benjamin-Bona-Mahony-Burgers equation

TL;DR

This work analyzes temporal-periodic solutions for the BBM-Burgers equation on a finite interval with Dirichlet boundaries, establishing existence, uniqueness, and stability in high-regularity spaces $H^ll$ and extending to a pseudo-parabolic regularization. The authors develop a linear semigroup theory for the dissipative generator $A$, derive a priori estimates, and treat the nonlinear term as a perturbation via a fixed-point argument, achieving local well-posedness for $ll=1,2$ and a bootstrap extension to $ll=3$. Under time-periodic forcing, they prove the existence of a time-periodic solution with period $ heta$ and demonstrate local and global stability of this attractor; periodicity is shown to persist in the higher-regularity pseudo-parabolic extension as well. The results provide a rigorous account of how dissipation and nonlinear convection interact to yield exponential convergence to periodic states, with implications for long-time wave propagation models in confined geometries.

Abstract

In this paper, we would establish the existence and stability of periodic solutions to the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation in $H^1_0([0, 1])$, whose medium interior is applied with time-periodic force $f(x, t)$ with period $θ$. High regularity analysis has been conducted in Hilbert spaces $H^\ell, \ell>1$. We also consider periodic solution to same IBVP scenario of a pseudo-parabolic-regularized equation as an extension of the BBM-Burgers in $\mathcal{H}^{\ell}, \ell=\{1, 2\}$.

Theorems & Definitions (69)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Lemma 3.1
• proof