Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness

Abstract

We first prove the existence of a compact positively invariant set which exponentially attracts any bounded set for abstract multi-valued semidynamical systems. Then, we apply the abstract theory to handle retarded ordinary differential equations and lattice dynamical systems, as well as reactiondiffusion equations with infinite delays. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.

Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness

“…To the best of our knowledge it is the first paper dedicated to attractors of multivalued semigroups (see last section in [1]). For the related extensions later on this topic, one can see [2,3,23,24,25,26,31] and references therein. As an application, Babin and Vishik used the abstract criteria in [1] to the IBVP of the weakly damped…”

“…In such cases, the solution operators become multi-valued maps, which results in that the classical attractor theory based on the semigroup of operators becomes useless. Hence the theory on the attractor of multi-valued operators is important for it allow us to continue to study the longtime behavior of the solutions no matter we have uniqueness or not, and this topic has been extensively investigated over the last one and a half decades, one can see [3,6,9,23,24,25,26,31] and references therein.…”