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

Abstract

We study two initial value problems of the linear diffusion equation u_t = u_xx and the nonlinear diffusion equation u_t = (1 + u_x²)⁻¹u_xx, when Cauchy data u(x,0) = u₀(x) 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. In the case of lim_|x|→+∞|xu′₀(x)|= 0, by using an elementary scaling technique, we show
lim_t→+∞|u($\sqrt{t}$x,t) − (F(−x)u₀(−$\sqrt{t}$) + F(+ x)u₀(+ $\sqrt{t}$))| = 0
for the linear heat equation u_t = u_xx, where x ∈ R and F(z): = $\frac{1}{2\sqrt \pi}\int_{-\infty}^z e^{-\frac{y^2}{4}} dy$. Further, by combining with a theorem of Nara and Taniguchi, we have the same result for the curvature equation u_t = (1 + u_x²)⁻¹u_xx. In the case of lim_|x|→+0|xu′₀(x)| = 0 and in the case of sup_x∈R|xu′₀(x)| < +∞, respectively, we also give a similar remark for the linear heat equation u_t = u_xx.