An \begin{document} $L_p$\end{document} -Lipschitz theory for parabolic equations with time measurable pseudo-differential operators
An \begin{document} $L_p$\end{document} -Lipschitz theory for parabolic equations with time measurable pseudo-differential operators
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带有时间可测伪微分算子的抛物线方程的 egin{document} $L_p$end{document} -Lipschitz 理论
DOI：
10.3934/cpaa.2018130
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Ildoo Kim
中科院分区：
文献类型：
--
作者：
Ildoo Kim
In this article we prove the existence and uniqueness of a (weak) solution \begin{document}$u$\end{document} in \begin{document}$L_p\left( (0, T); Λ_{γ+m}\right)$\end{document} to the Cauchy problem \begin{document}$\begin{align}\notag&\frac{\partial u}{\partial t}(t, x) = ψ(t, i\nabla)u(t, x)+f(t, x), \;\;\;(t, x) ∈ (0, T) × {\bf{R}}^d \\& u(0, x) = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)\end{align}$ \end{document} where \begin{document}$d ∈ \mathbb{N}$\end{document} , \begin{document}$p ∈ (1, ∞]$\end{document} , \begin{document}$γ, m ∈ (0, ∞)$\end{document} , \begin{document}$Λ_{γ+m}$\end{document} is the Lipschitz space on \begin{document}${\bf{R}}^d$\end{document} whose order is \begin{document}$γ+m$\end{document} , \begin{document}$f ∈ L_p\left( (0, T) ; Λ_{γ} \right)$\end{document} , and \begin{document}$ψ(t, i\nabla)$\end{document} is a time measurable pseudo-differential operator whose symbol is \begin{document}$ψ(t, ξ)$\end{document} , i.e. \begin{document}$ψ(t, i\nabla)u(t, x) = \mathcal{F}^{-1}[ψ(t, ξ){\mathcal{F}}\left[u(t, ·)\right]\left(ξ)\right](x), $ \end{document} with the assumptions \begin{document}$\begin{align*}\Re[ψ(t, ξ)] ≤ -ν|ξ|^{γ}, \end{align*}$ \end{document} and \begin{document}$\begin{align*}|D_{ξ}^{α}ψ(t, ξ)|≤ν^{-1}|ξ|^{γ-|α|}.\end{align*}$ \end{document} Furthermore, we show \begin{document}$\begin{align}\int_0^T \|u(t, ·)\|^p_{Λ_{γ+m}} dt ≤ N \int_0^T \|f(t, ·)\|^p_{Λ_{m}} dt, \;\;\;\;\;\;\;\;\;\;(2)\end{align}$ \end{document} where \begin{document}$N$\end{document} is a positive constant depending only on \begin{document}$d$\end{document} , \begin{document}$p$\end{document} , \begin{document}$γ$\end{document} , \begin{document}$ν$\end{document} , \begin{document}$m$\end{document} , and \begin{document}$T$\end{document} , The unique solvability of equation (1) in \begin{document}$L_p$\end{document} -Holder space is also considered.More precisely, for any \begin{document}$f ∈ L_p((0, T);C^{n+α})$\end{document} , there exists a unique solution \begin{document}$u ∈ L_p((0, T);C^{γ+n+α}({\bf{R}}^d))$\end{document} to equation (1) and for this solution \begin{document}$u$\end{document} , \begin{document}$\begin{align}\int_0^T \|u(t, ·)\|^p_{C^{γ+n+α}}dt ≤N \int_0^T \|f(t, ·)\|^p_{C^{n+α}}dt, \;\;\;\;\;\;\;\;\;\;(3)\end{align}$ \end{document} where \begin{document}$n ∈ \mathbb{Z}_+$\end{document} , \begin{document}$α ∈ (0, 1)$\end{document} , and \begin{document}$γ+α \notin \mathbb{Z}_+$\end{document} .
In this article we prove the existence and uniqueness of a (weak) solution \begin{document}$u$\end{document} in \begin{document}$L_p\left( (0, T); Λ_{γ+m}\right)$\end{document} to the Cauchy problem \begin{document}$\begin{align}\notag&\frac{\partial u}{\partial t}(t, x) = ψ(t, i\nabla)u(t, x)+f(t, x), \;\;\;(t, x) ∈ (0, T) × {\bf{R}}^d \\& u(0, x) = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)\end{align}$ \end{document} where \begin{document}$d ∈ \mathbb{N}$\end{document} , \begin{document}$p ∈ (1, ∞]$\end{document} , \begin{document}$γ, m ∈ (0, ∞)$\end{document} , \begin{document}$Λ_{γ+m}$\end{document} is the Lipschitz space on \begin{document}${\bf{R}}^d$\end{document} whose order is \begin{document}$γ+m$\end{document} , \begin{document}$f ∈ L_p\left( (0, T) ; Λ_{γ} \right)$\end{document} , and \begin{document}$ψ(t, i\nabla)$\end{document} is a time measurable pseudo-differential operator whose symbol is \begin{document}$ψ(t, ξ)$\end{document} , i.e. \begin{document}$ψ(t, i\nabla)u(t, x) = \mathcal{F}^{-1}[ψ(t, ξ){\mathcal{F}}\left[u(t, ·)\right]\left(ξ)\right](x), $ \end{document} with the assumptions \begin{document}$\begin{align*}\Re[ψ(t, ξ)] ≤ -ν|ξ|^{γ}, \end{align*}$ \end{document} and \begin{document}$\begin{align*}|D_{ξ}^{α}ψ(t, ξ)|≤ν^{-1}|ξ|^{γ-|α|}.\end{align*}$ \end{document} Furthermore, we show \begin{document}$\begin{align}\int_0^T \|u(t, ·)\|^p_{Λ_{γ+m}} dt ≤ N \int_0^T \|f(t, ·)\|^p_{Λ_{m}} dt, \;\;\;\;\;\;\;\;\;\;(2)\end{align}$ \end{document} where \begin{document}$N$\end{document} is a positive constant depending only on \begin{document}$d$\end{document} , \begin{document}$p$\end{document} , \begin{document}$γ$\end{document} , \begin{document}$ν$\end{document} , \begin{document}$m$\end{document} , and \begin{document}$T$\end{document} , The unique solvability of equation (1) in \begin{document}$L_p$\end{document} -Holder space is also considered.More precisely, for any \begin{document}$f ∈ L_p((0, T);C^{n+α})$\end{document} , there exists a unique solution \begin{document}$u ∈ L_p((0, T);C^{γ+n+α}({\bf{R}}^d))$\end{document} to equation (1) and for this solution \begin{document}$u$\end{document} , \begin{document}$\begin{align}\int_0^T \|u(t, ·)\|^p_{C^{γ+n+α}}dt ≤N \int_0^T \|f(t, ·)\|^p_{C^{n+α}}dt, \;\;\;\;\;\;\;\;\;\;(3)\end{align}$ \end{document} where \begin{document}$n ∈ \mathbb{Z}_+$\end{document} , \begin{document}$α ∈ (0, 1)$\end{document} , and \begin{document}$γ+α \notin \mathbb{Z}_+$\end{document} .