  We present a new method to obtain weighted $L^{1}$-estimates of global
solutions to the Cauchy problem for the semilinear heat equation with a simple
power of super-critical Fujita exponent. Our approach is based on direct and
explicit computations of commutation relations between the heat semigroup and
monomial weights in $\mathbb{R}^{n}$, while it is independent of the standard
parabolic arguments which rely on the comparison principle or some compactness
arguments. We also give explicit asymptotic profiles with parabolic
self-similarity of the global solutions.
