On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

TL;DR

This paper investigates the non-autonomously forced Burgers equation with both periodic and Dirichlet boundary conditions, proving the existence of unique bounded trajectories and their attracting properties, including exponential attraction in the Dirichlet case.

Contribution

It establishes the existence, uniqueness, and global attractor properties of bounded trajectories for the forced Burgers equation under different boundary conditions.

Findings

Existence of a unique bounded trajectory for all time.

Trajectories are attracted to this bounded solution in both pullback and forward senses.

Exponential attraction is proven for the Dirichlet boundary condition case.

Abstract

We study the non-autonomously forced Burgers equation $$ u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t) $$ on the space interval $(0,1)$ with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique $H^1$ bounded trajectory of this equation defined for all $t\in \mathbb{R}$. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.