June  2017, 10(3): 605-624. doi: 10.3934/dcdss.2017030

On a hyperbolic-parabolic mixed type equation

School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China

* Corresponding author

Received  December 2015 Revised  October 2016 Published  February 2017

Fund Project: This work was partially supported by NSF of China (No. 11371297), NSF of Fujian Province in China (No. 2015J01592).

In this paper, the hyperbolic-parabolic mixed type equation
$\frac{\partial u}{\partial t} = Δ A(u)+\text{div}(b(u)),\ \ (x,t)∈ Ω × (0,T),$
with the homogeneous boundary condition is considered. We find that only a part of the boundary condition is able to ensure the posedness of the solutions. By introducing a new kind of entropy solution matching the part boundary condition in a special way, we obtain the existence of the solution by the $BV$ estimate method, and establish the stability of the solutions by the Kruzkov bi-variables method.
Citation: Huashui Zhan. On a hyperbolic-parabolic mixed type equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 605-624. doi: 10.3934/dcdss.2017030
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. EscobedoJ. 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. GuarguagliniV. 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. LionsB. 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. MasciaA. 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. EscobedoJ. 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. GuarguagliniV. 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. LionsB. 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. MasciaA. 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 & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883

[2]

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 & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[3]

Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 363-396. doi: 10.3934/dcds.2018018

[4]

Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227

[5]

Alfredo Lorenzi, Eugenio Sinestrari. Identifying a BV-kernel in a hyperbolic integrodifferential equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1199-1219. doi: 10.3934/dcds.2008.21.1199

[6]

Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601

[7]

Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191

[8]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[9]

Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic & Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031

[10]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473

[11]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497

[12]

Xingwen Hao, Yachun Li, Qin Wang. A kinetic approach to error estimate for nonautonomous anisotropic degenerate parabolic-hyperbolic equations. Kinetic & Related Models, 2014, 7 (3) : 477-492. doi: 10.3934/krm.2014.7.477

[13]

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 & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759

[14]

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 & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279

[15]

Mahamadi Warma. Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1881-1905. doi: 10.3934/cpaa.2013.12.1881

[16]

J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176

[17]

Hyun-Jung Kim. Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution. Communications on Pure & Applied Analysis, 2019, 18 (2) : 795-807. doi: 10.3934/cpaa.2019038

[18]

G. Métivier, K. Zumbrun. Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 205-220. doi: 10.3934/dcds.2004.11.205

[19]

Raluca Clendenen, Gisèle Ruiz Goldstein, Jerome A. Goldstein. Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 651-660. doi: 10.3934/dcdss.2016019

[20]

Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1163-1182. doi: 10.3934/cpaa.2013.12.1163

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (10)
  • HTML views (2)
  • Cited by (2)

Other articles
by authors

[Back to Top]