The Asymptotic Behavior Of Weak Solution To The Brinkman-Forchheimer Equation With Delay

The time delay effect and memory are originated from the boundary controllers in engineering,which means that the system is controlled by a force taking into account the history of the solutions.They also arise in physical,chemical,biological,thermal,and economic phenomena.In the population dynamics model similar to the field of mathematical biology,time delay is often considered.The delay term in nonlinear model means the system is not only dependent on the present state but also on some past occurrences,which is also a cause of instability even an arbitrarily small delay may destabilize a system.Brinkman-Forchheimer equation describes the motion of fluid in saturated porous media.But for the Brinkman-Forchheimer equation,all of them are related to the non-delayed situations.In addition,the delay term plays a very important role in fluid dynamics,so we consider the case with delay term and study it by dividing it into two parts with reference to the existing results.In the first part,we consider the three-dimensional Brinkman-Forchheimer equation with finite delay in bounded domain.After giving some suitable assumptions to the external force term with delay,we first study the existence and uniqueness of its weak solution and the existence and uniqueness of its steady-state equation.Then we study the long-time approaching behavior of Brinkman-Forchheimer equation to its steady-state equation.In the second part,we mainly study the existence,uniqueness and regularity of the solution of Brinkman-Forchheimer equation with infinite delay in bounded domain,and the existence of pull-back attractor and global attractor.In addition,we also study its asymptotic stability,that is,the asymptotic stability between Brinkman-Forchheimer equation with infinite delay and its weak solution of steady-state equation.