On the backward behavior of some dissipative evolution equations

TL;DR

This paper investigates the backward-in-time blow-up phenomena in solutions of various dissipative evolution equations, contrasting behaviors across different models and providing physical interpretations and connections to energy spectra.

Contribution

It demonstrates backward blow-up in KdV-Burgers-Sivashinsky equations and explores similar phenomena in related models, offering new insights into their backward dynamics.

Findings

Solutions blow up backward in time outside the global attractor

Backward behavior varies among different dissipative equations

Physical and spectral interpretations of backward blow-up phenomena

Abstract

We prove that every solution of a KdV-Burgers-Sivashinsky type equation blows up in the energy space, backward in time, provided the solution does not belong to the global attractor. This is a phenomenon contrast to the backward behavior of the periodic 2D Navier-Stokes equations studied by Constantin-Foias-Kukavica-Majda [18], but analogous to the backward behavior of the Kuramoto-Sivashinsky equation discovered by Kukavica-Malcok [50]. Also we study the backward behavior of solutions to the damped driven nonlinear Schrodinger equation, the complex Ginzburg-Landau equation, and the hyperviscous Navier-Stokes equations. In addition, we provide some physical interpretation of various backward behaviors of several perturbations of the KdV equation by studying explicit cnoidal wave solutions. Furthermore, we discuss the connection between the backward behavior and the energy spectra of the solutions. The study of backward behavior of dissipative evolution equations is motivated by the investigation of the Bardos-Tartar conjecture stated in [5].