An $L_q(L_p)$-theory for diffusion equations with space-time nonlocal operators

Abstract

We present an $L_q(L_p)$-theory for the equation $${\partial }_t^{\alpha }u=\varphi (\Delta )u+f,t>0,x\in {\mathbb{R}}^d;u(0,\cdot )=u_0.$$ Here $p,q>1$, $\alpha \in (0,1)$, ${\partial }_t^{\alpha }$ is the Caputo fractional derivative of order $\alpha$, and $\varphi$ is a Bernstein function satisfying the following: $\exists {\delta }_0\in (0,1]$ and $c>0$ such that \begin{equation} \label{eqn 8.17.1} c \left(\frac{R}{r}\right)^{δ_0}\leq \frac{ϕ(R)}{ϕ(r)}, \qquad 0<r<R<\infty. \end{equation} We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove \begin{align*} \| |\partial^α_t u|+|u|+|ϕ(Δ)u|\|_{L_q([0,T];L_p)}\leq N(\|f\|_{L_q([0,T];L_p)}+ \|u_0\|_{B_{p,q}^{ϕ,2-2/ αq}}), \end{align*} where $B_{p,q}^{\varphi ,2-2/\alpha q}$ is a modified Besov space on ${\mathbb{R}}^d$ related to $\varphi$. Our approach is based on BMO estimate for $p=q$ and vector-valued Calderón-Zygmund theorem for $p\ne q$. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory.

References in corpus (1)

• Analytical properties and applications of the Wright function