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

We present a new method to obtain weighted L¹-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 Rⁿ, 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. [ABSTRACT FROM AUTHOR]

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