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

Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation