The Cauchy problem for time-fractional linear nonlocal diffusion equations

This manuscript is dedicated to study the Cauchy problem for time-fractional linear nonlocal diffusion problems in the whole RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{N}$$\end{document}, including the existence and uniqueness of solutions, their asymptotic behaviour as t goes to infinity, and the analysis of the corresponding rescaled problems by rescaling the convolution kernel J in some appropriate ways. Two time-fractional models will be considered in our work, one is related to the simplest linear nonlocal diffusion operator of the form J∗u-u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J*u-u$$\end{document}, and the other is proposed as a nonlocal analogy of higher-order evolution equations.