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 & 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. [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.
 [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] Alfredo Lorenzi, Eugenio Sinestrari. An identification problem for a nonlinear one-dimensional wave equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5253-5271. doi: 10.3934/dcds.2013.33.5253 [3] 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 [4] 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 [5] Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643 [6] Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431 [7] 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 - A, 2016, 36 (9) : 4963-4996. doi: 10.3934/dcds.2016015 [8] 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 [9] Toyohiko Aiki, Adrian Muntean. On uniqueness of a weak solution of one-dimensional concrete carbonation problem. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1345-1365. doi: 10.3934/dcds.2011.29.1345 [10] Kai Yan, Zhaoyang Yin. On the initial value problem for higher dimensional Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1327-1358. doi: 10.3934/dcds.2015.35.1327 [11] Jie Jiang, Boling Guo. Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 167-190. doi: 10.3934/dcds.2012.32.167 [12] 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 [13] 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 [14] Fei Guo, Bao-Feng Feng, Hongjun Gao, Yue Liu. On the initial-value problem to the Degasperis-Procesi equation with linear dispersion. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1269-1290. doi: 10.3934/dcds.2010.26.1269 [15] Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115 [16] Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 [17] 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 [18] V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731 [19] Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012 [20] Oleg V. Kaptsov, Alexey V. Schmidt. Reduction of three-dimensional model of the far turbulent wake to one-dimensional problem. Conference Publications, 2011, 2011 (Special) : 794-802. doi: 10.3934/proc.2011.2011.794

2017 Impact Factor: 1.179

Article outline