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Asymptotically sectional-hyperbolic attractors
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 |
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.
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. |
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. |
| [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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