Parabolic - hyperbolic boundary layer

TL;DR

This paper investigates the asymptotic behavior of a boundary value problem involving a higher order parabolic operator with a small parameter, revealing a transition to hyperbolic behavior and the emergence of a boundary layer as parameters vary.

Contribution

The paper provides a rigorous uniform asymptotic approximation for a boundary value problem transitioning from parabolic to hyperbolic behavior as parameters change.

Findings

Asymptotic approximation valid for all t

Identification of boundary layer formation

Transition from parabolic to hyperbolic operator

Abstract

A boundary value problem related to a parabolic higher order operator with a small parameter is analized. When the small parameter tends to zero, the reduced operator is hyperbolic. When t tends to infinity a parabolic hyperbolic boundary layer appears. In this paper a rigorous asymptotic approximation uniformly valid for all t is established.