Abstract
In this work, we study a Dirichlet problem for the viscous Burgers equation in a domain with moving boundaries that degenerates at the initial moment. The primary method of investigation is the Galerkin method, for which we construct an orthonormal basis suitable for domains with moving boundaries. Uniform a priori estimates are obtained, and based on these, theorems on the unique solvability of the problem are proven using methods of functional analysis. The viscous Burgers equation serves asa simplified model for studying fundamental aspects of nonlinear systems. It bridges the gap between purely theoretical nonlinear equations (like the inviscid Burgers equation) and more complex systems like the Navier-Stokes equations, making it a valuable tool in mathematical and physical research.