Science in China Series A: Mathematics

, Volume 48, Issue 5, pp 583–593 | Cite as

Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations



The uniqueness and existence of BV-solutions for Cauchy problem of the form
$$\frac{{\partial u}}{{\partial t}} = \Delta A\left( u \right) + \sum\limits_{i = 1}^N {\frac{{\partial b_i \left( u \right)}}{{\partial x_i }},A'\left( u \right) \geqslant 0,u\left( {x,0} \right) = u_0 } $$
are proved.


Cauchy problem degenerate parabolic equation uniqueness of solution 


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  1. 1.
    Oleinik, O. A., Samokhin,V. N., Mathematical Models in Boundary Layer Theorem, Boca Raton; Chapman and Hall/CRC, 1999.Google Scholar
  2. 2.
    Volpert, A.I., Hudjaev, S. I., On the problem for quasilinear degenerate parabolic equations of second order (Russian), Mat.Sb., 1967, 3: 374–396.MathSciNetGoogle Scholar
  3. 3.
    Zhao, J., Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeastern Math. J., 1985, 1(2): 153–165.MATHGoogle Scholar
  4. 4.
    Wu, Z., Yin, J., Some properties of functions in BVx and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J., 1989,5(4): 395–422.MathSciNetGoogle Scholar
  5. 5.
    Brezis, H., Crandall, M. G., Uniqueness of solutions of the initial value problem for {ie593-01}, J. Math.Pures et Appl., 1979, 58: 153–163.MATHMathSciNetGoogle Scholar
  6. 6.
    Kružkov, S. N., First order quasilinear equations in several independent varaiables, Math. USSR- Sb., 1970, 10: 217–243.CrossRefGoogle Scholar
  7. 7.
    Cockburn, B., Gripenberg, G., Continious dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Diff. Equatiaons, 1999, 151: 231–251.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Volpert, A.I., BV space and quasilinear equations, Mat. Sb., 1967, 73: 255–302.MathSciNetGoogle Scholar
  9. 9.
    Volpert, A.I., Hudjave, S. I., Analysis of class of discontinuous functions and the equations of mathematical physics (Russian), Izda. Nauka Moskwa, 1975.Google Scholar
  10. 10.
    Evans, L.C., Weak convergence methods for nonlinear partial differential equations, Conference Board of the Mathematical Sciences, Regional Conferences Series in Mathematics Number 74, 1998.Google Scholar
  11. 11.
    Wu, Z., Zhao, J., Yin, J., et al., Nonlinear Diffusion Equations, Singapore: Word Scientific, 2001.MATHGoogle Scholar

Copyright information

© Science in China Press 2005

Authors and Affiliations

  1. 1.Department of MathematicsXiamen UniversityXiamenChina
  2. 2.School of ScienceJimei UniversityXiamenChina

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