The well-posedness problem of a hyperbolic–parabolic mixed type equation on an unbounded domain

  • Huashui ZhanEmail author


To study the well-posedness problem of a hyperbolic–parabolic mixed type equation, the usual boundary value condition is overdetermined. Since the equation is with strong nonlinearity, the optimal partially boundary value condition can not be expressed by Fichera function. By introducing the weak characteristic function method, a different but reasonable partial boundary value condition is found first time, basing on it, the stability of the entropy solutions is established.


Hyperbolic–parabolic mixed type equation Unbounded domain Optimal partially boundary value condition The weak characteristic function method 

Mathematics Subject Classification

35L65 35K85 35R35 


Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.


  1. 1.
    Vol’pert, A.I., Hudjaev, S.I.: On the problem for quasilinear degenerate parabolic equations of second order. Mat. Sb. 3, 374–396 (1967). (Russian) Google Scholar
  2. 2.
    Wu, Z., Zhao, J., Yin, J., Li, H.: Nonlinear Diffusion Equations. Word Scientific Publishing, Singapore (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Zhao, J.: Uniqueness of solutions of quasilinear degenerate parabolic equations. Northeast. Math. J. 1(2), 153–165 (1985)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Volpert, A.I., Hudjave, S.I.: Analysis of Class of Discontinuous Functions and the Equations of Mathematical Physics. Nauka Moskwa, Martinus Nijhoff Publishers, Moscow, Izda (1975). (in Russian)Google Scholar
  5. 5.
    Wu, J., Yin, J.: Some properties of functions in BV\(_{x}\) and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations. Northeast. Math. J. 5(4), 395–422 (1989)MathSciNetGoogle Scholar
  6. 6.
    Carrillo, J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147, 269–361 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zhao, J., Zhan, H.: Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations. Sci. China Math. 48, 583–593 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Zhan, H.: The study of the Cauchy problem of a second order quasilinear degenerate parabolic equation and the parallelism of a Riemannian manifold. Doctoral Thesis, Xiamen University, China (2004)Google Scholar
  9. 9.
    Cockburn, B., Gripenberg, G.: Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differ. Equ. 151, 231–251 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, G.Q., DiBendetto, E.: Stability of entropy solutions to the Cauchy problem for a class of nonlinear hyperbolic–parabolic equations. SIAM J. Math. Anal. 33(4), 751–762 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, G.Q., Perthame, B.: Well–Posedness for non-isotropic degenerate parabolic–hyperbolic equations. Ann. Inst. Henri Poincare AN 20(4), 645–668 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Karlsen, K.H., Risebro, N.H.: On the uniqueness of entropy solutions of nonlinear degenerate parabolic equations with rough coefficient. Discrete Contain. Dye. Syst. 9(5), 1081–1104 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bendahamane, M., Karlsen, K.H.: Reharmonized entropy solutions for quasilinear anisotropic degenerate parabolic equations. SIAM J. Math. Anal. 36(2), 405–422 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Li, Y., Wang, Q.: Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations. J. Differ. Equ. 252, 4719–4741 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lions, P.L., Perthame, B., Tadmor, E.: A kinetic formation of multidimensional conservation laws and related equations. J. Am. Math. Soc. 7, 169–191 (1994)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kobayashi, K., Ohwa, H.: Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle. J. Differ. Equ. 252, 137–167 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wu, Z., Zhao, J.: The first boundary value problem for quasilinear degenerate parabolic equations of second order in several variables. Chin. Annal. Math. 4B(1), 57–76 (1983)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Zhan, H.: On a hyperbolic–parabolic mixed type equation. Discrete Contin. Dyn. Syst. Ser. S 10(3), 605C624 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhan, H.: The boundary degeneracy theory of a strongly degenerate parabolic equation. Bound. Value Probl. 2016, 15 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhan, H.: The solutions of a hyperbolic-parabolic mixed type equation on half-space domain. J. Differ. Equ. 259, 1449–1481 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. Conference Board of the Mathematical Sciences, Regional Conferences Series in Mathematics, vol. 74. American Mathematical Society, Providence (1998)Google Scholar
  22. 22.
    Gu, L.: Second Order Parabolic Partial Differential Equations. Xiamen University Press, Xiamen (2004)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Applied MathematicsXiamen University of TechnologyXiamenChina

Personalised recommendations