In this article we prove the existence and uniqueness of a (weak) solution u in Lp ((0, T);Λγ+m) to the Cauchy problem (equetion presented) where d ∈ ℕ, p ∈ (1,∞], γ, m ∈ (0,∞), Λγ+m is the Lipschitz space on Rd whose order is γ + m, f ∈ Lp ((0, T),Λγ), 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 Lp-Hölder space is also considered. More precisely, for any f ∈ Lp((0, T),Cn+α), there exists a unique solution u ∈ Lp((0, T),Cγ+n+α(Rd)) to equation (1) and for this solution u, (equetion presented) where n ∈ ℤ+, α ∈ (0, 1), and γ + α ∉ ℤ+.
- Cauchy problem
- L-Lipschitz estimate
- Time measurable pseudo-differential operator
ASJC Scopus subject areas
- Applied Mathematics