On a non linear third - order parabolic equation

TL;DR

This paper analyzes the qualitative behavior of solutions to a third-order nonlinear parabolic equation, providing explicit solutions in the linear case and asymptotic analysis for the nonlinear case, relevant to dissipative models.

Contribution

It offers a detailed qualitative analysis of nonlinear third-order parabolic equations, including explicit solutions for the linear case and asymptotic behavior insights for the nonlinear case.

Findings

Explicit Fourier series solutions for the linear case

Asymptotic estimates for nonlinear solutions

Relevance to dissipative physical models

Abstract

Aim of this paper is the qualitative analysis of the solution of a boundary value problem for a third-order non linear parabolic equation which describes several dissipative models. When the source term is linear, the problem is explictly solved by means of a Fourier series with properties of rapid convergence. In the non linear case,appropriate estimates of this series allow to deduce the asymptotic behaviour of the solution.