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 $L_p\left((0,T) , \Lambda_{\gamma+m}\right)$ to the Cauchy problem \begin{align} \notag &\frac{\partial u}{\partial t}(t,x)=\psi(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 $d \in \mathbb{N}$, $p \in (1,\infty]$, $\gamma,m \in (0,\infty)$, $\Lambda_{\gamma+m}$ is the Lipschitz space on $\mathbf{R}^d$ whose order is $\gamma+m$, $f \in L_p\left((0,T) , \Lambda_{\gamma} \right)$, and $\psi(t,i\nabla)$ is a time measurable pseudo-differential operator whose symbol is $\psi(t,\xi)$, i.e. $$ \psi(t,i\nabla)u(t,x)=\cF^{-1}\left[\psi(t,\xi)\cF\left[u(t,\cdot)\right](\xi)\right](x), $$ with the assumptions \begin{align*} \Re[\psi(t,\xi)] \leq -\nu|\xi|^{\gamma}, \end{align*} and \begin{align*} |D_{\xi}^{\alpha}\psi(t,\xi)|\leq\nu^{-1}|\xi|^{\gamma-|\alpha|}. \end{align*} Furthermore, we show \begin{align} \label{e 1028 1} \int_0^T \|u(t,\cdot)\|^p_{\Lambda_{\gamma+m}} dt \leq N \int_0^T \|f(t,\cdot)\|^p_{\Lambda_{m}} dt, \end{align} where $N$ is a positive constant depending only on $d$, $p$, $\gamma$, $\nu$, $m$, and $T$, The unique solvability of equation (\ref{main eqn}) in $L_p$-H\"older space is also considered. More precisely, for any $f \in L_p((0,T);C^{n+\alpha})$, there exists a unique solution $u \in L_p((0,T);C^{\gamma+n+\alpha}(\mathbf{R}^d))$ to equation (\ref{main eqn}) and for this solution $u$, \begin{align} \label{e 1029 1} \int_0^T \|u(t,\cdot)\|^p_{C^{\gamma+n+\alpha}}dt \leq N \int_0^T \|f(t,\cdot)\|^p_{C^{n+\alpha}}dt, \end{align} where $n \in \mathbb{Z}_+$, $\alpha \in (0,1)$, and $\gamma+\alpha \notin \mathbb{Z}_+$.