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

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.