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The self-similarity in space and time (hereafter self-similarity), eitherdeterministic or statistical, is characterized by similarity exponents and afunction of scaled variable, called the scaling function. In the present paper,we address mainly the self-similarity in the limit of early stage, as~opposedto the latter one, and also consider the scaling functions that decay or growalgebraically, as~opposed to the rapidly decaying functions such as Gaussian orerror function. In particular, in~the case of simple diffusion, our symmetryanalysis shows a mathematical mechanism by which the rapidly decaying scalingfunctions are generated by other polynomial scaling functions. While~the formeris adapted to the self-similarity in the late-stage processes, the latter isadapted to the early stages. This paper sheds some light on the internalstructure of the family of self-similarities generated by a simple diffusionequation.  Then, we present an example of self-similarity for the late stage whosescaling function has power-law tail, and also several cases of self-similarityfor the early stages. These examples show the utility of self-similarity to awider range of phenomena other than the late stage behaviors with rapidlydecaying scaling function

Symmetry in Self-Similarity in Space and Time---Short Time Transients and Power-Law Spatial~Asymptote