Backward Behavior and Determining Functionals for Chevron Pattern Equations

TL;DR

This paper investigates the backward behavior of solutions to chevron pattern equations, showing their asymptotic behavior is governed by a finite set of functionals and providing numerical evidence of finite-time blow-up in 1D.

Contribution

It establishes that the long-term backward behavior is determined by a finite set of functionals and demonstrates finite-time blow-up numerically in one dimension.

Findings

Asymptotic behavior is governed by finite functionals

Numerical evidence of finite-time blow-up in 1D

Complete characterization of backward dynamics

Abstract

The paper is devoted to the study of the backward behavior of solutions of the initial boundary value problem for the chevron pattern equations under homogeneous Dirichlet's boundary conditions. We prove that, as $t\rightarrow \infty$, the asymptotic behavior of solutions of the considered problem is completely determined by the dynamics of a finite set of functionals. Furthermore, we provide numerical evidence for the blow-up of certain solutions of the backward problem in finite time in 1D.