Non linear problems in dissipative models

TL;DR

This paper analyzes a quasi-linear parabolic third order equation modeling dissipative systems, providing explicit Green functions, boundary layer estimates, and insights into nonlinear problem reduction.

Contribution

It offers a detailed qualitative analysis of nonlinear dissipative models, including explicit solutions and boundary estimates, advancing understanding of their boundary value problems.

Findings

Explicit Green functions for the linear case.

Boundary layer estimates derived.

Reduction of nonlinear problems to integral equations.

Abstract

Aim of the paper is the qualitative analysis of a quasi-linear parabolic third order equation, which describes the evolution in a large class of dissipative models. As examples of some typical boundary problems, both Dirichlet's and Neumann's type boundary conditions are examined. In the linear case, the related Green functions are explicitly determined, together with rigorous estimates of their behavior when the parameter of dissipation is vanishing. These results are basic to study the integral equations to which the non linear problems can be reduced. Moreover, boundary layer estimates can be determined too.