Abstract
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 L1 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.
| Original language | English |
|---|---|
| Pages (from-to) | 137-167 |
| Number of pages | 31 |
| Journal | Journal of Differential Equations |
| Volume | 252 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2012 Jan 1 |
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Keywords
- Anisotropic
- Comparison theorem
- Degenerate parabolic equation
- Dirichlet boundary problem
- Kinetic formulation
- Uniqueness and existence
ASJC Scopus subject areas
- Analysis
Cite this
Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle. / Kobayashi, Kazuo; Ohwa, Hiroki.
In: Journal of Differential Equations, Vol. 252, No. 1, 01.01.2012, p. 137-167.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle
AU - Kobayashi, Kazuo
AU - Ohwa, Hiroki
PY - 2012/1/1
Y1 - 2012/1/1
N2 - 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 L1 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.
AB - 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 L1 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.
KW - Anisotropic
KW - Comparison theorem
KW - Degenerate parabolic equation
KW - Dirichlet boundary problem
KW - Kinetic formulation
KW - Uniqueness and existence
UR - http://www.scopus.com/inward/record.url?scp=80054100358&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=80054100358&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2011.09.008
DO - 10.1016/j.jde.2011.09.008
M3 - Article
AN - SCOPUS:80054100358
VL - 252
SP - 137
EP - 167
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 1
ER -