  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].
