-
Previous Article
Random dynamics of fractional nonclassical diffusion equations driven by colored noise
- DCDS Home
- This Issue
-
Next Article
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. 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. |
| [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. 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. |
| [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. |
| [1] |
Fredi Tröltzsch, Daniel Wachsmuth. On the switching behavior of sparse optimal controls for the one-dimensional heat equation. Mathematical Control & Related Fields, 2018, 8 (1) : 135-153. doi: 10.3934/mcrf.2018006 |
| [2] |
Zhi-Xue Zhao, Mapundi K. Banda, Bao-Zhu Guo. Boundary switch on/off control approach to simultaneous identification of diffusion coefficient and initial state for one-dimensional heat equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2533-2554. doi: 10.3934/dcdsb.2020021 |
| [3] |
Alfredo Lorenzi, Eugenio Sinestrari. An identification problem for a nonlinear one-dimensional wave equation. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5253-5271. doi: 10.3934/dcds.2013.33.5253 |
| [4] |
Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61 |
| [5] |
Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319 |
| [6] |
Haibo Cui, Junpei Gao, Lei Yao. Asymptotic behavior of the one-dimensional compressible micropolar fluid model. Electronic Research Archive, 2021, 29 (2) : 2063-2075. doi: 10.3934/era.2020105 |
| [7] |
Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643 |
| [8] |
Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 |
| [9] |
Umberto Biccari, Mahamadi Warma, Enrique Zuazua. Controllability of the one-dimensional fractional heat equation under positivity constraints. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1949-1978. doi: 10.3934/cpaa.2020086 |
| [10] |
Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431 |
| [11] |
Yanan Li, Alexandre N. Carvalho, Tito L. M. Luna, Estefani M. Moreira. A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5181-5196. doi: 10.3934/cpaa.2020232 |
| [12] |
Benjamin Jourdain, Julien Reygner. Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4963-4996. doi: 10.3934/dcds.2016015 |
| [13] |
Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423 |
| [14] |
Toyohiko Aiki, Adrian Muntean. On uniqueness of a weak solution of one-dimensional concrete carbonation problem. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1345-1365. doi: 10.3934/dcds.2011.29.1345 |
| [15] |
Kai Yan, Zhaoyang Yin. On the initial value problem for higher dimensional Camassa-Holm equations. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1327-1358. doi: 10.3934/dcds.2015.35.1327 |
| [16] |
Ahmad Z. Fino, Mokhtar Kirane. The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3625-3650. doi: 10.3934/cpaa.2020160 |
| [17] |
Jie Jiang, Boling Guo. Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 167-190. doi: 10.3934/dcds.2012.32.167 |
| [18] |
Fazal Abbas, Rangarajan Sudarsan, Hermann J. Eberl. Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates. Mathematical Biosciences & Engineering, 2012, 9 (2) : 215-239. doi: 10.3934/mbe.2012.9.215 |
| [19] |
Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control & Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011 |
| [20] |
Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]