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

2012/03/20 by Hiroki Yagisita, Yagisita, Hiroki
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1203.4315

submitted to Kodai Mathematical Journal

arxiv created 2012/03/20 · openalex publication_date 2012/03/20 · arxiv updated 2012/03/21 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

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.

Citations

• COMPLEXITY OF LARGE TIME BEHAVIOUR OF EVOLUTION EQUATIONS WITH BOUNDED DATA

Related

• Asymptotic behavior of scalar convection-diffusion equations
• Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion
• Stability of Self-similar Solutions to Geometric Flows
• Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion
• Mean curvature flow of an entire graph evolving away from the heat flow