Existence of weak solutions for a degenerate Goursat type linear problem

TL;DR

This paper proves the existence and uniqueness of weak solutions for a generalized Goursat problem involving a degenerate Gellerstedt operator on a Tricomi domain, extending classical methods to mixed-type boundary conditions.

Contribution

It introduces new existence and uniqueness results for a generalized Goursat problem with mixed boundary conditions on Tricomi domains, using energy integral methods.

Findings

Proved existence and uniqueness of weak solutions.

Extended classical energy methods to degenerate operators.

Applied results to a generalized Goursat problem.

Abstract

For a generalization of the Gellerstedt operator with mixed-type Dirichlet boundary conditions to a suitable Tricomi domain, we prove the existence and uniqueness of weak solutions of the linear problem and for a generalization of this problem. The classical method introduced by Didenko, which study the energy integral argument, will be used to prove estimates for a specific Tricomi domain.