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

2017/07/15 by Kim, Ildoo
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1707.04694

Abstract

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 amp;(∂ u)/(∂ t)(t,x)=ψ(t,i∇)u(t,x)+f(t,x), (t,x) ∈ (0,T) × R^d amp; u(0,x)=0, 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,i∇) is a time measurable pseudo-differential operator whose symbol is ψ(t,ξ), i.e. ψ(t,i∇)u(t,x)=\cF^-1[ψ(t,ξ)\cF[u(t,⋅)](ξ)](x), with the assumptions \Re[ψ(t,ξ)] ≤ -ν|ξ|^γ, and |D_ξ^αψ(t,ξ)|≤ν^-1|ξ|^γ-|α|. Furthermore, we show ∫₀^T ‖u(t,⋅)‖^p_Λ_γ+m dt ≤ N ∫₀^T ‖f(t,⋅)‖^p_Λ_m dt, where N is a positive constant depending only on d, p, γ, ν, m, and T, The unique solvability of equation (\refmain eqn) 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 (\refmain eqn) and for this solution u, ∫₀^T ‖u(t,⋅)‖^p_C^γ+n+αdt ≤ N ∫₀^T ‖f(t,⋅)‖^p_C^n+αdt, where n ∈ ℤ₊, α∈ (0,1), and γ+α∉ ℤ₊.

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