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March 2014 Remarks on space-time behavior in the Cauchy problems of the heat equation and the curvature flow equation with mildly oscillating initial values
Hiroki Yagisita
Kodai Math. J. 37(1): 16-23 (March 2014). DOI: 10.2996/kmj/1396008246

Abstract

We study two initial value problems of the linear diffusion equation ut = uxx and the nonlinear diffusion equation ut = (1 + ux2)−1uxx, 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|→+∞|xu0(x)|= 0, by using an elementary scaling technique, we show

limt→+∞|u($\sqrt{t}$x,t) − (F(−x)u0(− $\sqrt{t}$) + F(+ x)u0(+ $\sqrt{t}$))| = 0

for the linear heat equation ut = uxx, where x $in$ 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 ut = (1 + ux2)−1uxx. In the case of lim|x|→+0|xu0(x)| = 0 and in the case of supx$in$R|xu0(x)| < +∞, respectively, we also give a similar remark for the linear heat equation ut = uxx.

Citation

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Hiroki Yagisita. "Remarks on space-time behavior in the Cauchy problems of the heat equation and the curvature flow equation with mildly oscillating initial values." Kodai Math. J. 37 (1) 16 - 23, March 2014. https://doi.org/10.2996/kmj/1396008246

Information

Published: March 2014
First available in Project Euclid: 28 March 2014

zbMATH: 1292.35160
MathSciNet: MR3189512
Digital Object Identifier: 10.2996/kmj/1396008246

Rights: Copyright © 2014 Tokyo Institute of Technology, Department of Mathematics

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Vol.37 • No. 1 • March 2014
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