Asymptotic Behavior For Two Types Of Parabolic Equations On Monotone Domains

With the widespread discovery of phenomena of domains change with time in various disciplines such as physics,economics,medicine,biology,and so on.the related evolutionary problems have gradually attracted widespread attention from many experts and scholars.Compared with cylindrical domain problems,the inherent non-autonomy of monotone domains makes the study of their Asymptotic behavior of the system intrinsically difficult.Two types of parabolic equations defined on monotone domains(Ωt,t≥ 0)are systematically investigated in this paper as follows:(?)and(?)Firstly,the non-linear degenerate parabolic equation with homogeneous Dirichlet boundary conditions on the monotone domains is considered,the penalty method and the idea of limit solution is used to overcome non-autonomous nature of the monotone domains and degenerate property of the equation,and the well-posedness of the variational solution which satisfies the energy equality is obtained.Furthermore,based on the well-posedness of variational solutions,a two-parameter semigroup of non-autonomous dynamical system is constructed,and the existence of asymptotically compact pullback absorbing sets for such system is proved by combining the uniform energy dissipation estimates and the contraction function method.thus,the pullback Dλ-attractor of this kind of system is established from the preceding analysis.Secondly,the non-local parabolic equation defined on monotone domains with fractional-order Laplace operators is investigated,the well-posedness of a variational solution of satisfying energy equality is obtained by using the penalty method.At the same time,based on ideas such as Stampacchia’s truncation theory and denseness approximation,the boundedness of the variational solutions and the higher-order integrability are given.Finally,a prior estimate of Nash-Moser-Alikakos-type that can establish a higherattraction pullback attractor is proposed.The existence of pullback attractors for this system was further obtained.