The Initial-boundary Value Problem For A Class Of Nonlocal Fourth Order Parabolic Equations

The initial boundary value problem of a nonlocal parabolic equationis considered in this paper,where ????Rⁿ?n?5?is a bounded domain with smooth boundary,and u₀?H₀²???? L^q???,1/|x|^n-2*|u|^p=?_?|u?y?|^p/|x-y|^n-2dy=v?u?.This equation can be applied to thermal physics with nonlocal source and model population dynamics.Considering the initial-boundary value problem of the above-mentioned problem,the existence of local solutions is obtained by Banach fixed point theorem.The decay estimates and blow-up criteria of solutions under different conditions are also studied.In Chapter 1,the background,some development that related to the parabolic equations and the main work of the present dissertation are introduced.In Chapter 2,the preliminary knowledge of potential wells are considered and the relevant properties are discussed and proved.In Chapter 3,the existence of local solutions of the problem is proved by Banach fixed point theorem.The decay of global solution is studied by using the Sobolev embedding theorem and Hardy-Littlewood inequalities.In Chapter 4,when the initial energy functional satisfies different conditions,the blow-up criterias and growth estimates for the solution of the above problems are studied by using the concavity method.