The Asymptotic Behavior Of A Macroscopic Quantum (fluid) 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-Hilliard equation), the diffusion process of droplet on the solid surface (thin film equation), electric charge transportation of semiconductor (QHD), and so on (see[2,3]). In this paper, we study the Cauchy problem of a parabolic equation of fourth order in one dimension derived from quantum hydrodynamics where ρ = ρ(x, t) is the density, representing the macroscopic probability density distribution of a particle in quantum mechanics . The pressure p = p(ρ) satisfies p'(ρ) > 0, ρ> 0, and ρ± > 0 are given constants, ε > 0 is the scaled Planck constant which is small in general. We study the global existence of the solutions and the large time behavior of Eq (1).By the theory of semi-classical analysis for quantum mechanics (see[19]), we can pass into limit ε �' 0 formally in (1) and we get the quasilinear parabolic equation in one-dimension (see[7])ρt = p(ρ)_xx (2)Eq.(2) has a unique self-similar solutionwith the boundary conditions W(±∞) = ρ±, ρ± > 0. Setwhere x₀ satisfiesWe prove that if |ρ+-ρ-| ≤ 1, Z₀ ∈H⁴(R), and ||z₀(x)||H⁴(R) small enough, there is unique global classical solution ρ of the Cauchy problem (1), for any T > 0 such thatand...