Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle

被引:43
作者
Kobayasi, Kazuo [1 ]
Ohwa, Hiroki [2 ]
机构
[1] Waseda Univ, Dept Math, Sch Educ, Shinjuku Ku, Tokyo 1698050, Japan
[2] Waseda Univ, Grad Sch Educ, Shinjuku Ku, Tokyo 1698050, Japan
基金
日本学术振兴会;
关键词
Degenerate parabolic equation; Anisotropic; Dirichlet boundary problem; Kinetic formulation; Comparison theorem; Uniqueness and existence; ENTROPY SOLUTIONS; HYPERBOLIC EQUATIONS; CONSERVATION-LAWS; WELL-POSEDNESS; COEFFICIENTS; CONVERGENCE; FORMULATION; STABILITY;
D O I
10.1016/j.jde.2011.09.008
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We study the comparison principle for anisotropic degenerate parabolic-hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and super-solution, which immediately deduces the L(1) contractivity and therefore, uniqueness of entropy solutions. The method used here is based upon the kinetic formulation and the kinetic techniques developed by Lions, Perthame and Tadmor. By adapting and modifying those methods to the case of Dirichlet boundary problems for degenerate parabolic equations we can establish a comparison property. Moreover, in the quasi-isotropic case the existence of entropy solutions is proved. (C) 2011 Elsevier Inc. All rights reserved.
引用
收藏
页码:137 / 167
页数:31
相关论文
共 29 条
[1]   Well-posedness results for triply nonlinear degenerate parabolic equations [J].
Andreianov, B. ;
Bendahmane, M. ;
Karlsen, K. H. ;
Ouaro, S. .
JOURNAL OF DIFFERENTIAL EQUATIONS, 2009, 247 (01) :277-302
[2]  
Andreianov B. P., 2009, DISCRETE DUALITY FIN
[3]  
Andreianov BP, 2007, P ROY SOC EDINB A, V137, P1119, DOI 10.1017/S0308210505001290
[4]  
[Anonymous], 1968, TRANSLATIONS MATH MO
[5]  
Bardos C., 1979, Commun. Partial Differ. Equ., V4, P1017, DOI [10.1080/03605307908820117, DOI 10.1080/03605307908820117]
[6]   Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations [J].
Bendahmane, M ;
Karlsen, KH .
SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 2004, 36 (02) :405-422
[7]  
Bendahmane M, 2005, CONTEMP MATH, V371, P1
[8]   Entropy solutions for nonlinear degenerate problems [J].
Carrillo, J .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 1999, 147 (04) :269-361
[9]   L1-framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations [J].
Chen, GQ ;
Karlsen, KH .
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2006, 358 (03) :937-963
[10]   Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients [J].
Chen, GQ ;
Karlsen, KH .
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 2005, 4 (02) :241-266