Fractional nonlinear heat equations and characterizations of some function spaces in terms of fractional Gauss-Weierstrass semi-groups

We present a new proof of the caloric smoothing related to the fractional Gauss-Weierstrass semi-group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.