Parabolic equations with the second order Cauchy conditions on the boundary

TL;DR

This paper investigates ill-posed boundary value problems for parabolic equations with second order Cauchy conditions on the boundary, proposing a dense class of inputs with regularity in the frequency domain.

Contribution

It introduces a specific class of inputs that ensure regularity for these ill-posed problems, explicitly characterized in the frequency domain.

Findings

The class of inputs is dense in the space of square integrable functions.

Regularity conditions are explicitly described in the frequency domain.

The study advances understanding of boundary conditions for parabolic equations.

Abstract

The paper studies some ill-posed boundary value problems on semi-plane for parabolic equations with homogenuous Cauchy condition at initial time and with the second order Cauchy condition on the boundary of the semi-plane. A class of inputs that allows some regularity is suggested and described explicitly in frequency domain. This class is everywhere dense in the space of square integrable functions.