WEIGHTED ESTIMATES AND LARGE TIME BEHAVIOR OF SMALL AMPLITUDE SOLUTIONS TO THE SEMILINEAR HEAT EQUATION

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 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.