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

TL;DR

This paper investigates the space-time behavior of solutions to linear and nonlinear heat equations with mildly oscillating initial data, revealing formulas and dynamics that describe their long-term evolution.

Contribution

It introduces an elementary scaling technique to derive formulas for the space-time behavior of solutions to these diffusion equations with oscillating initial conditions.

Findings

Formulas for long-term solution behavior

Identification of nonstabilizing solutions

Description of irregular and large-time dynamics

Abstract

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