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

Abstract

In this article we prove the existence and uniqueness of a (weak) solution $u$ in $L_p\left((0,T),\hat{I}{›}_{{\hat{I}}^3+m}\right)$ to the Cauchy problem \begin{align} \notag &\frac{\partial u}{\partial t}(t,x)=ψ(t,i\nabla)u(t,x)+f(t,x),\quad (t,x) \in (0,T) \times \mathbf{R}^d \label{main eqn} & u(0,x)=0, \end{align} where $d\in \mathbb{N}$, $p\in (1,\infty ]$, ${\hat{I}}^3,m\in (0,\infty )$, $\hat{I}{›}_{{\hat{I}}^3+m}$ is the Lipschitz space on ${\mathbf{R}}^d$ whose order is ${\hat{I}}^3+m$, $f\in L_p\left((0,T),\hat{I}{›}_{\hat{I}}^3\right)$, and $\ddot{I}ˆ(t,i\nabla )$ is a time measurable pseudo-differential operator whose symbol is $\ddot{I}ˆ(t,\hat{I}¾)$, i.e. $$ ψ(t,i\nabla)u(t,x)=\cF^{-1}\left[ψ(t,ξ)\cF\left[u(t,\cdot)\right](ξ)\right](x), $$ with the assumptions \begin{align*} \Re[ψ(t,ξ)] \leq -ν|ξ|^γ, \end{align*} and \begin{align*} |D_ξ^αψ(t,ξ)|\leqν^{-1}|ξ|^{γ-|α|}. \end{align*} Furthermore, we show \begin{align} \label{e 1028 1} \int_0^T \|u(t,\cdot)\|^p_{Λ_{γ+m}} dt \leq N \int_0^T \|f(t,\cdot)\|^p_{Λ_{m}} dt, \end{align} where $N$ is a positive constant depending only on $d$, $p$, ${\hat{I}}^3$, $\hat{I}½$, $m$, and $T$, The unique solvability of equation (\ref{main eqn}) in $L_p$-Hölder space is also considered. More precisely, for any $f\in L_p((0,T);C^{n+\hat{I}\pm })$, there exists a unique solution $u\in L_p((0,T);C^{{\hat{I}}^3+n+\hat{I}\pm }({\mathbf{R}}^d))$ to equation (\ref{main eqn}) and for this solution $u$, \begin{align} \label{e 1029 1} \int_0^T \|u(t,\cdot)\|^p_{C^{γ+n+α}}dt \leq N \int_0^T \|f(t,\cdot)\|^p_{C^{n+α}}dt, \end{align} where $n\in {\mathbb{Z}}_{+}$, $\hat{I}\pm \in (0,1)$, and ${\hat{I}}^3+\hat{I}\pm \notin {\mathbb{Z}}_{+}$.