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 $L_p\left( (0, T); Λ_{γ+m}\right)$ to the Cauchy problem

$\begin{align}\notag&\frac{\partial u}{\partial t}(t, x) = ψ(t, i\nabla)u(t, x)+f(t, x), \;\;\;(t, x) ∈ (0, T) × {\bf{R}}^d \\& u(0, x) = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)\end{align}$

where $d ∈ \mathbb{N}$, $p ∈ (1, ∞]$, $γ, m ∈ (0, ∞)$, $Λ_{γ+m}$ is the Lipschitz space on ${\bf{R}}^d$ whose order is $γ+m$, $f ∈ L_p\left( (0, T) ; Λ_{γ} \right)$, and $ψ(t, i\nabla)$ is a time measurable pseudo-differential operator whose symbol is $ψ(t, ξ)$, i.e.

$ψ(t, i\nabla)u(t, x) = \mathcal{F}^{-1}[ψ(t, ξ){\mathcal{F}}\left[u(t, ·)\right]\left(ξ)\right](x), $

with the assumptions

$\begin{align*}\Re[ψ(t, ξ)] ≤ -ν|ξ|^{γ}, \end{align*}$

and

$\begin{align*}|D_{ξ}^{α}ψ(t, ξ)|≤ν^{-1}|ξ|^{γ-|α|}.\end{align*}$

Furthermore, we show

$\begin{align}\int_0^T \|u(t, ·)\|^p_{Λ_{γ+m}} dt ≤ N \int_0^T \|f(t, ·)\|^p_{Λ_{m}} dt, \;\;\;\;\;\;\;\;\;\;(2)\end{align}$

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 $u ∈ L_p((0, T);C^{γ+n+α}({\bf{R}}^d))$ to equation (1) and for this solution $u$,

$\begin{align}\int_0^T \|u(t, ·)\|^p_{C^{γ+n+α}}dt ≤N \int_0^T \|f(t, ·)\|^p_{C^{n+α}}dt, \;\;\;\;\;\;\;\;\;\;(3)\end{align}$

where $n ∈ \mathbb{Z}_+$, $α ∈ (0, 1)$, and $γ+α \notin \mathbb{Z}_+$.