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

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

Dong-Ho Tsai 1, and  Chia-Hsing Nien 2,

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.

Keywords: Oscillation behavior, convergence behavior, one-dimensional heat equation, Cauchy problem, initial value problem.
Mathematics Subject Classification: 35K05, 35K15.
Citation: Dong-Ho Tsai, Chia-Hsing Nien. On the oscillation behavior of solutions to the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4073-4089. doi: 10.3934/dcds.2019164
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.  Google Scholar

[2]

S. D. Eidel'man, Parabolic System, North-Holland, Amsterdam, 1969. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

S. D. Eidel'man, Parabolic System, North-Holland, Amsterdam, 1969. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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