Reference request: fractional heat equation $\partial_t u=(-\Delta)^s u$ with $s>1$

Question

I was wondering if there are any good reference on the fractional heat equation $\partial_t u=(-\Delta)^s u$ with the fractional exponent $s>1$. I have found many references in the case $0<s<1$ since a maximum principle is available, but almost none in the former case. I am particularly interested in the heat kernel estimates and Schauder-type estimates.

Answer 1

Basics are theory of pseudo diffential operators https://en.wikipedia.org/wiki/Pseudo-differential_operator. A well known example is the complex Schrödinger equation, that generates two real equation for Re and Im with $(-\Delta)^2 $