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We study two initial value problems of the linear diffusion equation and anonlinear diffusion equation, when Cauchy data are bounded and oscillatemildly. The latter nonlinear heat equation is the equation of the curvatureflow, when the moving curves are represented by graphs. By using an elementaryscaling technique, we show some formulas for space-time behavior of thesolution.  Keywords: scaling argument, self-similar solution, nonstabilizing solution,nontrivial dynamics, nontrivial large-time behavior, irregular behavior.

Remarks on space-time behavior in the Cauchy problems of the heat equation and the curvature flow equation with mildly oscillating initial values