The Asymptotic Behavior Of A Class Of Fourth Order Parabolic Equations

Recently, the parabolic equation of fourth order has been caused research interest due to its significant application in the study of physical phenomena in modern applied science, for instance, phase transitions (Cahn-Hillianrd equation), the diffusion process of droplet on the solid surface (thin film equation), electric charge transportation of semiconductor (QHD), (see[2,6,8]), Heisenberg (quantumLangevin equation), (see[4]), and so on. In this paper, we mainly consider the global existence of the solutions and large time behavior.LetE = (?) (ρ-n- C^*)dξ. Here are our main results:Theorem(1) Supposeρ, u, E satisfyρ₀(±∞) =ρ, n₀(±∞) = n,ρ-n-C^* = 0,ρ(+∞) =ρ(-∞), n(+∞) = n(-∞), p'(ρ) > 0, q'(n) > 0, p'(ρ) = q'(n), and‖ρ₀ -ρ‖_H³(R),‖n₀-ρ‖_H³(R),‖E₀‖_L²(R) are sufficiently small. Then the global solutions(ρ, n) of theⅣP(0.1) exists and satisfies for (?)T > 0, we have; Also, we can prove the large time behavior of global solutions in L², is described byρ,Remark1: The method used here can be applied to deal with the high dimensional case.Remark2: In the paper,we assume p'(ρ) = q'(n).The next work is that studying p'(ρ)≠q'(n) ,ρ- n - c^* = 0 or p'(ρ)≠q'(n).