# American Institute of Mathematical Sciences

July  2019, 39(7): 4073-4089. doi: 10.3934/dcds.2019164

## On the oscillation behavior of solutions to the one-dimensional heat equation

 1 Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan 2 Department of Financial Engineering, Providence University, Taichung 43301, Taiwan

Received  September 2018 Published  April 2019

We study the oscillation behavior of solutions to the one-dimensional heat equation and give some interesting examples. We also demonstrate a simple ODE method to find explicit solutions of the heat equation with certain particular initial conditions.

Citation: Dong-Ho Tsai, Chia-Hsing Nien. On the oscillation behavior of solutions to the one-dimensional heat equation. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4073-4089. doi: 10.3934/dcds.2019164
##### References:
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##### References:
 [1] P. Collet and J. -P. Eckmann, Space-time behavior in problems of hydrodynamic type: A case study, Nonlinearity, 5 (1992), 1265-1302.  doi: 10.1088/0951-7715/5/6/004. [2] S. D. Eidel'man, Parabolic System, North-Holland, Amsterdam, 1969. [3] S. Kamin, On stabilization of solutions of the Cauchy problem for parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 76/77 (1976), 43-53.  doi: 10.1017/S0308210500019478. [4] M. Nara and M. Taniguchi, The condition on the stability of stationary lines in a curvature flow in the whole plane, J. Diff. Eq., 237 (2007), 61-76.  doi: 10.1016/j.jde.2007.02.012. [5] W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, v. 82, SIAM, 2011. doi: 10.1137/1.9781611971972. [6] V. D. Repnikov and S. D. Eidel'man, A new proof of the theorem on the stabilization of the solution of the Cauchy problem for the heat equation, Math. USSR Sb., 2 (1967), 135-139.  doi: 10.1070/SM1967v002n01ABEH002328.
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