Research On Dynamical Behavior Of Gradient System With Small Time Delay

This doctoral thesis investigates the dynamical behavior of the following generalnonlinear gradient-like evolutionary equations with small time delayWe prove that each bounded solution of the delayed equation will converge to someequilibrium as tâ†'∞, provided the delay is sufciently small. The approach here ismainly based on the Morse structure of invariant sets of gradient system and somegeometric analysis of evolutionary equations.Specifically, there are four main parts in all. First, the local existence and unique-ness are obtained by the fixed point principle. After that, we make use of extendedtheorem and Gronwall inequality, with the aid of the condition on the nonlinear term,to obtain the global existence and uniqueness. Later, we also get compactness theoremof solutions and smoothing action of the diferential equation; Secondly, by means offundamental theorem on sectorial operators, we obtain the preparation lemma used inAubin-Lions lemma. Then with the hypothesis of gradient system, finite and isolatedequilibria, we prove that there exists a sufciently small delay such that any boundedsolution of the delayed equation will ultimately enter and stay in the neighborhood ofone equilibrium. Our method essentially makes use of the Morse structure of invariantsets of gradient system. By the Aubin-Lions lemma, we complete the transition of thelimit; Thirdly, with the hypothesis of hyperbolic equilibrium, we utilize exponential di-chotomies and a series estimates to prove that there exists ε>0and Ï„>0sufcientlysmall such that any solution of the delayed equation lying in the ε neighborhood of oneequilibrium will converge to this equilibrium as tâ†'∞; Finally, we give two examplesto illustrate the abstract results by considering the heat conduct equation and waveequation with small time delay.