On the geometry of the domain of the solution of nonlinear Cauchy problem

TL;DR

This paper investigates the geometric structure of the solution domain for nonlinear second-order PDEs with parabolic degeneration, exploring inverse problems and the impact of characteristic curve singularities on solution domains.

Contribution

It introduces a geometric analysis of the solution domain for nonlinear PDEs with parabolic degeneration and addresses inverse problems with regularity conditions.

Findings

Solution domains can contain gaps due to characteristic curve singularities.

Inverse problems have solutions under certain regularity conditions.

Characteristic curves may have common envelopes or singular points.

Abstract

We consider the Cauchy problem for a second order quasi-linear partial differential equation with an admissible parabolic degeneration such that the given functions described the initial conditions are defined on a closed interval. We study also a variant of the inverse problem of the Cauchy problem and prove that the considered inverse problem has a solution under certain regularity condition. We illustrate the Cauchy and the inverse problems in some interesting examples such that the families of the characteristic curves have either common envelopes or singular points. In these cases the definition domain of the solution of the differential equation contains a gap.