Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations

Abstract

This paper analyzes the time-dependence and backward controllability of pullback attractors for the trajectory space generated by a non-autonomous evolution equation without uniqueness. A pullback trajectory attractor is called backward controllable if the norm of its union over the past is controlled by a continuous function, and backward compact if it is backward controllable and pre-compact in the past on the underlying space. We then establish two existence theorems for such a backward compact trajectory attractor, which leads to the existence of a pullback attractor with the backward compactness and backward boundedness in two original phase spaces respectively. An essential criterion is the existence of an increasing, compact and absorbing brochette. Applying to the non-autonomous Jeffreys-Oldroyd equations with a backward controllable force, we obtain a backward compact trajectory attractor, and also a pullback attractor with backward compactness in the negative-exponent Sobolev space and backward boundedness in the Lebesgue space.