Constructing the asymptotics of a solution of the heat equation from the known asymptotics of the initial function in three-dimensional space

An asymptotic approximation, as time increases without limit, is constructed to the solution of the Cauchy problem for the heat equation in three-dimensional space. The locally integrable initial function, which does not necessarily tend to zero at infinity, is assumed to have powerlike asymptotics. The method of introduction of an auxiliary parameter, which also involves the regularization of singularities in integrals, plays the central role in the research. The asymptotic expression for the solution is shown to have the form of a series in negative half-integer powers of the time variable, with coefficients depending on self-similar variables and the logarithm of time; the leading term is found explicitly. Using the example of the Cauchy problem for the vector Burgers equation, it is shown that to perform an asymptotic analysis of the solution by the matching method one needs to construct an asymptotic approximation to a solution of the heat equation.