Fourth-order Quasi-linear Parabolic Equations Qualitative Analysis

In this paper, we mainly consider the global existence of the solutions, large time behavior and the L¹-time decay of the fourth order parabolic equations.Let E= (?)(ρ- n)dξ, E₀ = (?)(ρ₀-n₀)dξ.Here are our main results.Theorem 1. Assume that the initial dataρ₀ > 0, n₀ > 0, (ρ₀ -ρ,n₀ -ρ)∈H³(R)×H³(R),∫_-∞^x(ρ₀-ρ)dy∈H⁴(R),∫_-∞^x(n₀ -ρ)dy∈H⁴(R),E₀∈H²(R), and‖ρ₀-ρ‖_H³(R)+‖n₀-ρ‖_H³(R)+‖E₀‖_H²(R) is sufficiently small. Then the unique global solution (ρ, n) of the IVP (0.1) exists and satisfiesMoreover, it holdsTheorem 2. Under the assumptions of Theorem 1, if the initial data also satisfiesρ₀-ρ, n₀-ρ∈L¹(R), and where I_βis the Riesz potential defined by I_βF(x) = C∫_R|x- y|^β-1F(y)dy. Then,we have the L¹-time decay rate of the solution (ρ,n) of the IVP (0.1) for large timeRemark 0.1 The method used here can be applied to deal with the high dimensional case.