Well-Posedness for a Degenerate Hyperbolic Equation with Weighted Initial Data

Article

Abstract

The focus of this study is an initial-boundary value problem associated with the degenerate hyperbolic equation $t \partial_{tt}u + \frac{1}{2} \partial_{t}u - \Delta u = g$ in a bounded domain. Due to the singularity at $t=0$, standard initial conditions lead to an ill-posed problem. To achieve solvability of the problem, we introduce a "modified" Cauchy problem using weighted initial conditions for this degeneracy. The main result of the study is the proof of the well-posedness of this problem within the framework of classical Sobolev spaces, as well as the obtaining of a priori estimates of the solution. Furthermore, the general boundary conditions for the one-dimensional equation were derived by using the restriction and extension theory.