An $L_q(L_p)$-theory for time-fractional diffusion equations with nonlocal operators generated by Lévy processes with low intensity of small jumps

Abstract

We investigate an $L_q(L_p)$-regularity ($1<p,q<\infty$) theory for space-time nonlocal equations of the type ${\partial }_t^{\alpha }u=\mathcal{L}u+f$. Here, ${\partial }_t^{\alpha }$ is the Caputo fractional derivative of order $\alpha \in (0,1)$ and $\mathcal{L}$ is an integro-differential operator $$\mathcal{L}u(x)={\int }_{{\mathbb{R}}^d}\left(u(x)-u(x+y)-\nabla u(x)\cdot y{1}_{|y|\le 1}\right)j_d(|y|)dy$$ which is the infinitesimal generator of an isotropic unimodal Lévy process. We assume that the jump kernel $j_d(r)$ is comparable to $r^{-d}\ell (r^{-1})$, where $\ell$ is a continuous function satisfying $$C_1{\left(\frac{R}{r}\right)}^{{\delta }_1}\le \frac{\ell (R)}{\ell (r)}\le C_2{\left(\frac{R}{r}\right)}^{{\delta }_2}\text{for}1\le r\le R<\infty ,$$ where $0\le {\delta }_1\le {\delta }_2<2$. Hence, $\ell$ can be slowly varying at infinity. Our result covers $\mathcal{L}$ whose Fourier multiplier $\Psi (\xi )$ satisfies $\Psi (\xi )\asymp -\log (1+|\xi {|}^{\beta })$ for $\beta \in (0,2]$ and $\Psi (\xi )\asymp -(\log (1+|\xi {|}^{\beta /4}){)}^2$ for $\beta \in (0,2)$ by taking $\ell (r)\asymp 1$ and $\ell (r)\asymp \log (1+r^{\beta })$ for $r\ge 1$ respectively. In this article, we use the Calderón-Zygmund approach and function space theory for operators having slowly varying symbols.

References in corpus (3)

• Time fractional equations and probabilistic representation
• Asymptotic behaviour and estimates of slowly varying convolution semigroups
• Sobolev estimates for fractional parabolic equations with space-time non-local operators