Self-Similar Solutions to a Nonlinear Forward-Backward Parabolic Equation

TL;DR

This paper investigates a mixed-type nonlinear parabolic equation related to boundary layer separation, deriving self-similar solutions through similarity variables and analyzing a resulting nonlinear ODE system.

Contribution

It introduces a method to find self-similar solutions to a complex mixed-type PDE by reducing it to an ODE system, providing new insights into its behavior.

Findings

Existence of self-similar solutions with sign change.

Reduction of the PDE to a second-order ODE using similarity variables.

Analysis of the ODE system to establish solution existence.

Abstract

In this paper, we study the mixed-type equation u u_x = u_{yy}, which behaves as forward and backward parabolic equations depending on the sign of u. The equation arises from the study of boundary layers with separation. We seek solutions that change their type smoothly to better understand the equation. We simplify the equation into a second-order ODE using similarity variables, and prove an existence result by analyzing it as a first-order nonlinear ODE system. This provides us a self-similar solution with a sign change.