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Remarks on space-time behavior in the Cauchy problems of the heat equation and the curvature flow equation with mildly oscillating initial values. (English) Zbl 1292.35160
Summary: We study two initial value problems of the linear diffusion equation \(u_t=u_{xx}\) and the nonlinear diffusion equation \(u_t=(1+u_x^2)^{-1}u_{xx}\), when Cauchy data \(u(x,0)=u_0(x)\) 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 \(\lim_{|x|\to+\infty}|xu'_0(x)|= 0\), by using an elementary scaling technique, we show \[ \lim\limits_{t\to+\infty}|u(\sqrt tx,t)-(F(-x)u_0(-\sqrt t)+F(+x)u_0(+\sqrt t))|=0 \] for the linear heat equation \(u_t=u_{xx}\), where \(x\in\mathbb R\) and \[ F(z):=\frac{1}{2\sqrt\pi}\int_{-\infty}^ze^{-y^2/4}dy. \] Further, by combining with a theorem of Nara and Taniguchi, we have the same result for the curvature equation \(u_ t=(1+u_x^2)^{-1}u_{xx}\). In the case of \(\lim_{|x|\to+0}|xu'_0(x)|=0\) and in the case of \(\sup_{x\in\mathbb R}|xu'_0(x)|<+\infty\), respectively, we also give a similar remark for the linear heat equation \(u_t=u_{xx}\).
MSC:
35K93 Quasilinear parabolic equations with mean curvature operator
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C06 Self-similar solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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