-
Previous Article
Bifurcation analysis of the three-dimensional Hénon map
- DCDS-S Home
- This Issue
-
Next Article
Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay
On a hyperbolic-parabolic mixed type equation
School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China |
$\frac{\partial u}{\partial t} = Δ A(u)+\text{div}(b(u)),\ \ (x,t)∈ Ω × (0,T),$ |
References:
[1] |
M. Bendahamane and K. H. Karlsen,
Renarmonized entropy solutions for quasilinear anisotropic degenerate parabolic equations, SIAM J. Math. Anal., 36 (2004), 405-422.
doi: 10.1137/S0036141003428937. |
[2] |
H. Brezis and M. G. Crandall,
Uniqueness of solutions of the initial value problem for $u_{t}-Δ \varphi (u)=0$, J. Math.Pures et Appl., 58 (1979), 153-163.
|
[3] |
J. Carrillo,
Entropy solutions for nonlinear degenerate problems, Arch.Rational Mech. Anal., 147 (1999), 269-361.
doi: 10.1007/s002050050152. |
[4] |
G. Q. Chen and B. Perthame,
Well-Posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. I. H. Poincare-AN, 20 (2003), 645-668.
doi: 10.1016/S0294-1449(02)00014-8. |
[5] |
G. Q. Chen and E. DiBenedetto,
Stability of entropy solutions to Cauchy problem for a class of nonlinear hyperbolic-parabolic equations, SIAM J.Math. Anal., 33 (2001), 751-762.
doi: 10.1137/S0036141001363597. |
[6] |
B. Cockburn and G. Gripenberg,
Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Diff. Equ., 151 (1999), 231-251.
doi: 10.1006/jdeq.1998.3499. |
[7] |
G. Enrico, Minimal Surfaces and Functions of Bounded Variation Birkhauser, Bosten. Basel. Stuttgart Switzerland, 1984. |
[8] |
M. Escobedo, J. L. Vazquez and E. Zuazua,
Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc., 343 (1994), 829-842.
doi: 10.1090/S0002-9947-1994-1225573-2. |
[9] |
L. C. Evans, Weak convergence methods for nonlinear partial differential equations Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990.
doi: 10.1090/cbms/074. |
[10] |
G. Fichera,
Sulle equazioni differenziatli lineari ellittico-paraboliche del secondo ordine, Atti Accd, Naz. Lincei. Mem, CI. Sci. Fis. Mat. Nat. Sez.1, 5 (1956), 1-30.
|
[11] |
G. Fichera,
On a unified theory of boundary value problems for elliptic-parabolic equations of second order, in boundary problems, differential equations, Univ. of Wisconsin Press, Madison, Wis., 9 (1960), 97-120.
|
[12] |
L. Gu, Second Order Parabolic Partial Differential Equations Xiamen University Press, Xiamen, China, 2004. |
[13] |
F. R. Guarguaglini, V. Milišić and A. Terracina,
A discrete BGK approximation for strongly degenerate parabolic problems with boundary conditions, J. Diff. Equ., 202 (2004), 183-207.
doi: 10.1016/j.jde.2004.03.008. |
[14] |
K. H. Karlsen and N. H. Risebro,
On the uniqueness of entropy solutions of nonlinear degenerate parabolic equations with rough coefficient, Discrete Contain. Dye. Sys., 9 (2003), 1081-1104.
doi: 10.3934/dcds.2003.9.1081. |
[15] |
M. V. Keldyš,
On certain cases of degeneration of elliptic type on the boundary of a domain, Dokl. Akad. Aauk SSSR, 77 (1951), 181-183.
|
[16] |
K. Kobayasi and H. Ohwa,
Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle, J. Diff. Equ., 252 (2012), 137-167.
doi: 10.1016/j.jde.2011.09.008. |
[17] |
S. N. Kružkov,
First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217-243.
|
[18] |
Y. Li and Q. Wang,
Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations, J. Diff. Equ., 252 (2012), 4719-4741.
doi: 10.1016/j.jde.2012.01.027. |
[19] |
P. L. Lions, B. Perthame and E. Tadmor,
A kinetic formation of multidimensional conservation laws and related equations, J. Amer. Math. Soc., 7 (1994), 169-191.
doi: 10.1090/S0894-0347-1994-1201239-3. |
[20] |
C. Mascia, A. Porretta and A. Terracina,
Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Ration. Mech. Anal., 163 (2002), 87-124.
doi: 10.1007/s002050200184. |
[21] |
J. Málek, J. Necas, M. Rokyta and M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDES in Applied Mathematics and Mathematical Computation, 13. Chapman and HALL, London, 1996.
doi: 10.1007/978-1-4899-6824-1. |
[22] |
O. A. Oleinik and V. N. Samokhin, Mathematical Models in boundary, Layer Theorem Chapman and Hall/CRC, Boca Raton, London, New York, Washington, D. C., 1999. |
[23] |
O. A. Oleinik,
A problem of Fichera, Dokl. Akad. Nauk SSSR, 157 (1964), 1297-1300.
|
[24] |
O. A. Oleinik,
Linear equations of second order with nonnegative characteristic form, Math. Sb., 69 (1966), 111-140.
|
[25] |
F. Tricomi,
Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto, Rend. Reale Accad. Lincei, 14 (1923), 134-247.
|
[26] |
G. Vallet,
Dirichlet problem for a degenerated hyperbolic-parabolic equation, Advances in Mathematical Sciences and Applications, 15 (2005), 423-450.
|
[27] |
A. I. Vol'pert and S. I. Hudjaev,
On the problem for quasilinear degenerate parabolic equations of second order (Russian), Mat. Sb., 78 (1969), 374-396.
|
[28] |
A. I. Volpert,
BV space and quasilinear equations, Mat. Sb., 2 (1967), 225-302.
doi: 10.1070/SM1967v002n02ABEH002340. |
[29] |
A. I. Volpert and S. I. Hudjave, Analysis of class of discontinuous functions and the equations of mathematical physics (Russian), Izda. Nauka Moskwa, 1975. |
[30] |
Z. Wu and J. Yin,
Some properties of functions in BV$_{x}$ and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J., 5 (1989), 395-422.
|
[31] |
Z. Wu and J. Zhao,
The first boundary value problem for quasilinear degenerate parabolic equations of second order in several variables, Chin.Ann. of Math., 4 (1983), 57-76.
|
[32] |
Z. Wu and J. Zhao,
Some general results on the first boundary value problem for quasilinear degenerate parabolic equations, Chin.Ann. of Math., 4 (1983), 319-328.
|
[33] |
Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations Word Scientific Publishing, Singapore, 2001.
doi: 10.1142/9789812799791. |
[34] |
J. Yin and C. Wang,
Evolutionary weighted $p$-Laplacian with boundary degeneracy, J. of Diff. Equ., 237 (2007), 421-445.
doi: 10.1016/j.jde.2007.03.012. |
[35] |
H. Zhan, The Study of the Cauchy Problem of a Second Order Quasilinear Degenerate Parabolic Equation and the Parallelism of a Riemannian Manifold Ph. D thesis, Xiamen University, 2004. |
[36] |
H. Zhan,
The solution of a hyperbolic-parabolic mixed-type equation on half-space domain, J. Diff. Equ., 259 (2015), 1149-1181.
doi: 10.1016/j.jde.2015.03.005. |
[37] |
J. Zhao,
Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeastern Math. J., 1 (1985), 153-165.
|
[38] |
J. Zhao and H. Zhan,
Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations, Science in China Ser. A, 48 (2005), 583-593.
doi: 10.1360/03ys0269. |
show all references
References:
[1] |
M. Bendahamane and K. H. Karlsen,
Renarmonized entropy solutions for quasilinear anisotropic degenerate parabolic equations, SIAM J. Math. Anal., 36 (2004), 405-422.
doi: 10.1137/S0036141003428937. |
[2] |
H. Brezis and M. G. Crandall,
Uniqueness of solutions of the initial value problem for $u_{t}-Δ \varphi (u)=0$, J. Math.Pures et Appl., 58 (1979), 153-163.
|
[3] |
J. Carrillo,
Entropy solutions for nonlinear degenerate problems, Arch.Rational Mech. Anal., 147 (1999), 269-361.
doi: 10.1007/s002050050152. |
[4] |
G. Q. Chen and B. Perthame,
Well-Posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. I. H. Poincare-AN, 20 (2003), 645-668.
doi: 10.1016/S0294-1449(02)00014-8. |
[5] |
G. Q. Chen and E. DiBenedetto,
Stability of entropy solutions to Cauchy problem for a class of nonlinear hyperbolic-parabolic equations, SIAM J.Math. Anal., 33 (2001), 751-762.
doi: 10.1137/S0036141001363597. |
[6] |
B. Cockburn and G. Gripenberg,
Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Diff. Equ., 151 (1999), 231-251.
doi: 10.1006/jdeq.1998.3499. |
[7] |
G. Enrico, Minimal Surfaces and Functions of Bounded Variation Birkhauser, Bosten. Basel. Stuttgart Switzerland, 1984. |
[8] |
M. Escobedo, J. L. Vazquez and E. Zuazua,
Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc., 343 (1994), 829-842.
doi: 10.1090/S0002-9947-1994-1225573-2. |
[9] |
L. C. Evans, Weak convergence methods for nonlinear partial differential equations Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990.
doi: 10.1090/cbms/074. |
[10] |
G. Fichera,
Sulle equazioni differenziatli lineari ellittico-paraboliche del secondo ordine, Atti Accd, Naz. Lincei. Mem, CI. Sci. Fis. Mat. Nat. Sez.1, 5 (1956), 1-30.
|
[11] |
G. Fichera,
On a unified theory of boundary value problems for elliptic-parabolic equations of second order, in boundary problems, differential equations, Univ. of Wisconsin Press, Madison, Wis., 9 (1960), 97-120.
|
[12] |
L. Gu, Second Order Parabolic Partial Differential Equations Xiamen University Press, Xiamen, China, 2004. |
[13] |
F. R. Guarguaglini, V. Milišić and A. Terracina,
A discrete BGK approximation for strongly degenerate parabolic problems with boundary conditions, J. Diff. Equ., 202 (2004), 183-207.
doi: 10.1016/j.jde.2004.03.008. |
[14] |
K. H. Karlsen and N. H. Risebro,
On the uniqueness of entropy solutions of nonlinear degenerate parabolic equations with rough coefficient, Discrete Contain. Dye. Sys., 9 (2003), 1081-1104.
doi: 10.3934/dcds.2003.9.1081. |
[15] |
M. V. Keldyš,
On certain cases of degeneration of elliptic type on the boundary of a domain, Dokl. Akad. Aauk SSSR, 77 (1951), 181-183.
|
[16] |
K. Kobayasi and H. Ohwa,
Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle, J. Diff. Equ., 252 (2012), 137-167.
doi: 10.1016/j.jde.2011.09.008. |
[17] |
S. N. Kružkov,
First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217-243.
|
[18] |
Y. Li and Q. Wang,
Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations, J. Diff. Equ., 252 (2012), 4719-4741.
doi: 10.1016/j.jde.2012.01.027. |
[19] |
P. L. Lions, B. Perthame and E. Tadmor,
A kinetic formation of multidimensional conservation laws and related equations, J. Amer. Math. Soc., 7 (1994), 169-191.
doi: 10.1090/S0894-0347-1994-1201239-3. |
[20] |
C. Mascia, A. Porretta and A. Terracina,
Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Ration. Mech. Anal., 163 (2002), 87-124.
doi: 10.1007/s002050200184. |
[21] |
J. Málek, J. Necas, M. Rokyta and M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDES in Applied Mathematics and Mathematical Computation, 13. Chapman and HALL, London, 1996.
doi: 10.1007/978-1-4899-6824-1. |
[22] |
O. A. Oleinik and V. N. Samokhin, Mathematical Models in boundary, Layer Theorem Chapman and Hall/CRC, Boca Raton, London, New York, Washington, D. C., 1999. |
[23] |
O. A. Oleinik,
A problem of Fichera, Dokl. Akad. Nauk SSSR, 157 (1964), 1297-1300.
|
[24] |
O. A. Oleinik,
Linear equations of second order with nonnegative characteristic form, Math. Sb., 69 (1966), 111-140.
|
[25] |
F. Tricomi,
Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto, Rend. Reale Accad. Lincei, 14 (1923), 134-247.
|
[26] |
G. Vallet,
Dirichlet problem for a degenerated hyperbolic-parabolic equation, Advances in Mathematical Sciences and Applications, 15 (2005), 423-450.
|
[27] |
A. I. Vol'pert and S. I. Hudjaev,
On the problem for quasilinear degenerate parabolic equations of second order (Russian), Mat. Sb., 78 (1969), 374-396.
|
[28] |
A. I. Volpert,
BV space and quasilinear equations, Mat. Sb., 2 (1967), 225-302.
doi: 10.1070/SM1967v002n02ABEH002340. |
[29] |
A. I. Volpert and S. I. Hudjave, Analysis of class of discontinuous functions and the equations of mathematical physics (Russian), Izda. Nauka Moskwa, 1975. |
[30] |
Z. Wu and J. Yin,
Some properties of functions in BV$_{x}$ and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J., 5 (1989), 395-422.
|
[31] |
Z. Wu and J. Zhao,
The first boundary value problem for quasilinear degenerate parabolic equations of second order in several variables, Chin.Ann. of Math., 4 (1983), 57-76.
|
[32] |
Z. Wu and J. Zhao,
Some general results on the first boundary value problem for quasilinear degenerate parabolic equations, Chin.Ann. of Math., 4 (1983), 319-328.
|
[33] |
Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations Word Scientific Publishing, Singapore, 2001.
doi: 10.1142/9789812799791. |
[34] |
J. Yin and C. Wang,
Evolutionary weighted $p$-Laplacian with boundary degeneracy, J. of Diff. Equ., 237 (2007), 421-445.
doi: 10.1016/j.jde.2007.03.012. |
[35] |
H. Zhan, The Study of the Cauchy Problem of a Second Order Quasilinear Degenerate Parabolic Equation and the Parallelism of a Riemannian Manifold Ph. D thesis, Xiamen University, 2004. |
[36] |
H. Zhan,
The solution of a hyperbolic-parabolic mixed-type equation on half-space domain, J. Diff. Equ., 259 (2015), 1149-1181.
doi: 10.1016/j.jde.2015.03.005. |
[37] |
J. Zhao,
Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeastern Math. J., 1 (1985), 153-165.
|
[38] |
J. Zhao and H. Zhan,
Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations, Science in China Ser. A, 48 (2005), 583-593.
doi: 10.1360/03ys0269. |
[1] |
Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic and Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883 |
[2] |
Francesca R. Guarguaglini. Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1057-1087. doi: 10.3934/cpaa.2020049 |
[3] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[4] |
Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 363-396. doi: 10.3934/dcds.2018018 |
[5] |
Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227 |
[6] |
Bopeng Rao, Xu Zhang. Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2789-2809. doi: 10.3934/cpaa.2021119 |
[7] |
Alfredo Lorenzi, Eugenio Sinestrari. Identifying a BV-kernel in a hyperbolic integrodifferential equation. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1199-1219. doi: 10.3934/dcds.2008.21.1199 |
[8] |
Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic and Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601 |
[9] |
Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure and Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 |
[10] |
Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 |
[11] |
Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic and Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031 |
[12] |
Majid Bani-Yaghoub, Chunhua Ou, Guangming Yao. Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2509-2535. doi: 10.3934/dcdss.2020195 |
[13] |
M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473 |
[14] |
Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012 |
[15] |
R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 |
[16] |
Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340 |
[17] |
Xingwen Hao, Yachun Li, Qin Wang. A kinetic approach to error estimate for nonautonomous anisotropic degenerate parabolic-hyperbolic equations. Kinetic and Related Models, 2014, 7 (3) : 477-492. doi: 10.3934/krm.2014.7.477 |
[18] |
Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure and Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759 |
[19] |
Mahamadi Warma. Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1881-1905. doi: 10.3934/cpaa.2013.12.1881 |
[20] |
G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]