An L_p-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators

In this article we prove the existence and uniqueness of a (weak) solution u in L_p ((0, T);Λ_γ+m) to the Cauchy problem (equetion presented) where d ∈ ℕ, p ∈ (1,∞], γ, m ∈ (0,∞), Λ_γ+m is the Lipschitz space on R^d whose order is γ + m, f ∈ L_p ((0, T),Λ_γ), and ψ (t, iΔ) is a time measurable pseudo-differential operator whose symbol is ψ(t, ξ), (equetion presented) with the assumptions (equetion presented) and (equetion presented): Furthermore, we show (equetion presented) where N is a positive constant depending only on d, p, γ, ν, m, and T, The unique solvability of equation (1) in L_p-Hölder space is also considered. More precisely, for any f ∈ L_p((0, T),C^n+α), there exists a unique solution u ∈ L_p((0, T),C^γ+n+α(R^d)) to equation (1) and for this solution u, (equetion presented) where n ∈ ℤ₊, α ∈ (0, 1), and γ + α ∉ ℤ₊.