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

2025/07/19 by Jing, Tian
#34A34 #35A24 #35C06 #35K65 #35M10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.14635

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.

Citations

• The Cauchy-Dirichlet Problem for the Fast Diffusion Equation on Bounded Domains
• Higher Regularity Theory for a Mixed-Type Parabolic Equation
• Reversal in the Stationary Prandtl Equations

Related

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• Uniqueness of singular self-similar solutions of a semilinear parabolic equation
• Forward Discretely Self-Similar Solutions of the Navier-Stokes Equations
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• Linear and nonlinear parabolic forward-backward problems