POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION*

doi:10.1007/s10473-023-0520-x

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 2279-2290.doi: 10.1007/s10473-023-0520-x

POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION^*

Tianjie YANG¹, Guangwei YUAN^2†

1. Graduate School of China Academy of Engineering Physics, Beijing 100088, China;
2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

收稿日期:2021-10-22 修回日期:2023-04-23 出版日期:2023-10-26 发布日期:2023-10-25

POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION^*

Tianjie YANG¹, Guangwei YUAN^2†

1. Graduate School of China Academy of Engineering Physics, Beijing 100088, China;
2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Received:2021-10-22 Revised:2023-04-23 Online:2023-10-26 Published:2023-10-25
Contact: †Guangwei YUAN, E-mail: yuan_guangwei@iapcm.ac.cn
About author:Tianjie YANG, E-mail: 690820370@qq.com
Supported by:
National Natural Science Foundation of China (11971069 and 12126307).

摘要/Abstract

摘要： In this paper, for a bounded $C^2$ domain, we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive $C^2$ boundary data. We have a non-existence result, which is the justification for taking into account the restricted boundary data. There is a smooth positive boundary datum that precludes the existence of the positive classical solution.

关键词: Dirichlet problem, steady relativistic heat equation, classical solution

Abstract: In this paper, for a bounded $C^2$ domain, we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive $C^2$ boundary data. We have a non-existence result, which is the justification for taking into account the restricted boundary data. There is a smooth positive boundary datum that precludes the existence of the positive classical solution.

Key words: Dirichlet problem, steady relativistic heat equation, classical solution

中图分类号:

35K65

Tianjie YANG, Guangwei YUAN. POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION^*[J]. 数学物理学报, 2023, 43(5): 2279-2290.

Tianjie YANG, Guangwei YUAN. POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION^*[J]. Acta mathematica scientia,Series A, 2023, 43(5): 2279-2290.

参考文献

[1] Levermore C.Chapman-enskog approach to flux-limited diffusion theory[R]. California Univ, 1979
[2] Levermore C, Pomraning G. A flux-limited diffusion theory. The Astrophysical Journal, 1981, 248: 321-334
[3] Levermore C. Relating eddington factors to flux limiters. Journal of Quantitative Spectroscopy and Radiative Transfer, 1984, 31(2): 149-160
[4] Rosenau P. Tempered diffusion: A transport process with propagating fronts and inertial delay. Physical Review A, 1992, 46(12): R7371
[5] Brenier Y. Optimal transportation and applications. Extended Monge-Kantorovich Theory, 2003, 1813: 91-121
[6] Andreu F, Caselles V, Mazón J M, Moll S. Finite propagation speed for limited flux diffusion equations. Archive for Rational Mechanics and Analysis, 2006, 182(2): 269-297
[7] Andreu F, Caselles V, Mazón J M. A strongly degenerate quasilinear elliptic equation. Nonlinear Analysis: Theory, Methods & Applications, 2005, 61(4): 637-669
[8] Andreu F, Caselles V, Mazón J M. The cauchy problem for a strongly egenerate quasilinear equation. Journal of the European Mathematical Society, 2005, 7(3): 361-393
[9] Andreu F, Caselles V, Mazón J M, Moll S. The dirichlet problem associated to the relativistic heat equation. Mathematische Annalen, 2010, 347(1): 135-199
[10] Caselles V. On the entropy conditions for some flux limited diffusion equations. Journal of Differential Equations, 2011, 250(8): 3311-3348
[11] Calvo J, Campos J, Caselles V, Sánchez O, Soler J. Pattern formation in a flux limited reaction-diffusion equation of porous media type. Inventiones Mathematicae, 2016, 206(1): 57-108
[12] Andreu F, Caselles V, Mazón J. Some regularity results on the 'relativistic' heat equation. Journal of Differential Equations, 2008, 245(12): 3639-3663
[13] Andreu F, Caselles V, Mazón J M, Soler J, Verbeni M. Radially symmetric solutions of a tempered diffusion equation. a porous media, flux-limited case. SIAM Journal on Mathematical Analysis, 2012, 44(2): 1019-1049
[14] Carrillo J A, Caselles V, Moll S. On the relativistic heat equation in one space dimension. Proceedings of the London Mathematical Society, 2013, 107(6): 1395-1423
[15] Calvo J, Caselles V. Local-in-time regularity results for some flux-limited diffusion equations of porous media type. Nonlinear Analysis: Theory, Methods & Applications, 2013, 93: 236-272
[16] Calvo J, Campos J, Caselles V, Sánchez ó, Soler J. Qualitative behaviour for flux-saturated mechanisms: travelling waves, waiting time and smoothing effects. Journal of the European Mathematical Society, 2017, 19(2): 441-472
[17] Jenkins H, Serrin J. The Dirichlet problem for the minimal surface equation in higher dimensions. Journal für die Reine und Angewandte Mathematik, 1968, 229: 170-187
[18] Gilbarg D, Trudinger N S.Elliptic Partial Differential Equations of Second Order. Berlin: Springer, 2015
[19] Poisson S. Note sur la surface dont l'aire est un minimum entre des limites données. Journal für die Reine und Angewandte Mathematik, 1832, 9: 361-362
[20] Korn A.Über Minimalflächen, deren Randkurven wenig von ebenen Kurven abweichen, Vol 1909. Königl Akademie der Wissenschaften, 1909
[21] Williams G H. The dirichlet problem for the minimal surface equation with lipschitz continuous boundary data. J Reine Ange Math, 1984, 354: 123-140
[22] Schulz F, Williams G. Barriers and existence results for a class of equations of mean curvature type. Analysis, 1987, 7(3/4): 359-374
[23] Lau C P. Quasilinear elliptic equations with small boundary data. Manuscripta Mathematica, 1985, 53(1): 77-99
[24] Hayasida K, Nakatani M. On the dirichlet problem of prescribed mean curvature equations without H-convexity condition. Nagoya Mathematical Journal, 2000, 157: 177-209
[25] Tsukamoto Y. The dirichlet problem for a prescribed mean curvature equation. Hiroshima Mathematical Journal, 2020, 50(3): 325-337
[26] Liang Z, Yang Y. Radial convex solutions of a singular dirichlet problem with the mean curvature operator in minkowski space. Acta Mathematica Scientia, 2019, 39B(2): 395-402

POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION^*

POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION^*

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