On gradient estimates and other qualitative properties of solutions of nonlinear non autonomous parabolic systems



We prove several uniform \( L^1 \) -estimates on solutions of a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of degenerate type. They are uniform in the sense that they don’t depend on the coefficients, nor on the size of the spatial domain. The estimates concern the own solution or/and its spatial gradient. This paper extends some previous results by the authors to the case of nonautonomous coefficients and possibly non homogeneous boundary conditions. Moreover, an application to the asymptotic decay of the \( L^1 \) -norm of solutions, as t → +∞, is also given.


Uniform gradient estimates quasilinear parabolic onedimensional systems uniform L1-estimates independent on the spatial domain 

Mathematics Subject Classifications

35K45 35K65 

Estimaciones sobre el gradiente y otras propiedades cualitativas de las soluciones de sistemas parabólicos no lineales no aut ónomos


En este artículo se obtienen varias estimaciones uniformes en \( L^1 \) para las soluciones y su derivada espacial de ciertos sistemas parabólicos no lineales que pueden estar acoplados en los términos de difusión y que, de hecho, puede ser de tipo degenerado.

Tales estimaciones son uniformes en el sentido de que no dependen de los coeficientes del sistema, ni del tamaño del dominio espacial. Las estimaciones se refieren a la norma \( L^1 \) de la propia solución o/y de su gradiente espacial. Este trabajo extiende, al caso de coeficientes no autónomos y a posibles condiciones de contorno no homogéneas, ciertos resultados previos de los autores. Además, se ofrece una aplicación al estudio del decaimiento de la norma \( L^1 \) de la solución, cuando t → +∞.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Alt, H. W. and Luckhaus, S., (1983). Quasilinear Elliptic-Parabolic Differential Equations, Math. Z., 183, 311–341.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Amann, H., (1990). Dynamic theory of quasilinear parabolic equations: II. Reaction-diffusion systems, Diff. Int. Equ., 3, 13–75.MATHMathSciNetGoogle Scholar
  3. [3]
    Andreu, F., Caselles, V. and Mazón, J. M., (2004). Parabolic Quasilinear Equations Minimizing Linear Growth Functions, Birkhäuser, Basel.Google Scholar
  4. [4]
    Antontsev, S. N. and Díaz, J. I., (2007). Interfaces generated by discharge of a hot gas in a cold atmosphere, in Abstracts of Fourth International Conference of Applied Mathematics and Computing, August 12–18, 2007, Plovdiv, Bulgaria, 21–22.Google Scholar
  5. [5]
    Antontsev, S. N. and Díaz, J. I., (2007). Mathematical analysis of the discharge of a hot gas in a colder atmosphere, in Book of abstracts. XXII joint session of Moscow Mathematical Society and I. G. Petrovskii seminar, Moscow, May 21–26, 2007, 19–20.Google Scholar
  6. [6]
    Antontsev, S. N. and Díaz, J. I., (2007). Mathematical analysis of the discharge of a laminar hot gas in a cold atmosphere, in Abstracts of All-Russian Conference “Problems of continuum mechanics and physics of detonation”, Novosibirsk, Russia, September 17–22, 2007, 189–190.Google Scholar
  7. [7]
    Antontsev, S. N. and Díaz, J. I., (2007). Mathematical analysis of the discharge of a laminar hot gas in a colder atmosphere, RACSAM, Rev. R. Acad. Cien. Serie A. Mat, 101(1), 119–124.MATHGoogle Scholar
  8. [8]
    Antontsev, S. N. and Díaz, J. I., (2008). On thermal and stagnation interfaces generated by the discharge of a laminar hot gas in a stagnant colder atmosphere, in preparation for IFB.Google Scholar
  9. [9]
    Antontsev, S. N. and Díaz, J. I., (2008). Mathematical analysis of the discharge of a laminar hot gas in a colder atmosphere, J. Appl. Mech. Tech. Phys., (Russian version, 49, 4), 1–14.CrossRefGoogle Scholar
  10. [10]
    Antontsev, S. N. and Díaz, J. I.. Uniform \( L^1 \) -gradient estimates of solutions solutions to quasilinear parabolic systems in higher dimensions. Article in preparation.Google Scholar
  11. [11]
    Antontsev, S. N. and Díaz, J. I., New \( L^1 \) -gradient type estimates of solutions to one dimensional quasilinear parabolic systems. To appear in Contemporary Mathematics.Google Scholar
  12. [12]
    Antontsev, S. N., Díaz, J. I., and S. Shmarev, (2002). Energy Methods for Free Boundary Problems: Applications to Non-linear PDEs and Fluid Mechanics, Bikhäuser, Boston.Google Scholar
  13. [13]
    Benilan, Ph., (1972). Equation d’evolution dans un espace de Banach quelconque et applications, Thése, Université de Paris-Sud.Google Scholar
  14. [14]
    Benilan, Ph., Crandall, M. G. and Pazy, A., Evolution Equations Governed by Accretive Operators. Book in preparation.Google Scholar
  15. [15]
    Brezis, H. (1983). Analyse fonctionnelle. Théorie et applications, Masson, Paris.MATHGoogle Scholar
  16. [16]
    Brezis, H. and Strauss, W. A., (1973). Semilinear second-order elliptic equations in L1, J. Math.Soc. Japan, 25, 565–590.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Chipot, M., (2002). l goes to plus infinity. Birkhäuser, Bassel.Google Scholar
  18. [18]
    Díaz, J. I., (1985). Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London.MATHGoogle Scholar
  19. [19]
    Díaz, J. I. and Padial, J. F., (1996). Uniqueness and existence of solutions in the BV (Q) space to a doublyPadial, J. F., (1996). Uniqueness and existence of solutions in the BV (Q) space to a doubly nonlinear parabolic problem, Publications Matematiques, 40, 527–560.MATHGoogle Scholar
  20. [20]
    Egorov, Yu. V., Kondratév, V. A. and Oleinik, O. A., (1998). Asymptotic behavior of solutions to nonlinear elliptic and parabolic systems in cylindrical domains, Mat. Sb., 3, 45–68.Google Scholar
  21. [21]
    Evans, L. C. and Gariepy, R. F., (1992). Measure Theory and Fine Properties of Functions, CRC Press.Google Scholar
  22. [22]
    Gilding, B. H., (1989). Improved Theory for a Nonlinear Degenerate Parabolic Equation, Annali Scu. Norm. Sup. Pisa, Serie IV, 14, 165–224.MathSciNetGoogle Scholar
  23. [23]
    Hopf, E., (1950). The partial differential equation ut + uux = μuxx, Comm. Pure Appl. Math, 3, 201–230.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Kalashnikov, A. S., (1987). Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk, 42, 135–176.MathSciNetGoogle Scholar
  25. [25]
    Kruzhkov, S. N., (1970). First-order quasilinear equations in several independent variables, Mat. Sbornik., 81, 228–255.Google Scholar
  26. [26]
    Ladyženskaja, O. A., Solonnikov, V. A. And Ural’Tseva, N. N., (1967). Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, R. I. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23.Google Scholar
  27. [27]
    Lions, J.-L., (1969). Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod.Google Scholar
  28. [28]
    Málek, J., Neĉas, J., Rokyta, M. and R⫲žiĉka, M., (1996). Weak and measure-valued solutions to evolutionary PDEs, Chapman Hall, London.MATHGoogle Scholar
  29. [29]
    Pai, S., (1949). Two-dimensional jet mixing of a compressible fluid, J. Aeronaut. Sci., 16, 463–469.MathSciNetGoogle Scholar
  30. [30]
    Pai, S., (1952). Axially symmetrical jet mixing of a compressible fluid, Quart. Appl. Math., 10, 141–148.MATHMathSciNetGoogle Scholar
  31. [31]
    Pai, S., (1954). Fluid dynamics of jets, D. Van Nostrand Company, Inc., Toronto-New York-London, Publishers, New York-London.MATHGoogle Scholar
  32. [32]
    Quittner, P. and Souplet, Ph., (2007). Superlinear Parabolic Problems, Birkhäuser, Basel.MATHGoogle Scholar
  33. [33]
    Sánchez-Sanz, M., Sánchez, A. and Liñán, A., (2006). Front solutions in high-temperature laminar gas jets, J. Fluid. Mech., 547, 257–266.MATHCrossRefGoogle Scholar
  34. [34]
    Volpert, A. I. and Khudayaev, S. I., (1985). Analysis in classes of discontinuous functions and equations of mathematical physics, Nijhoff, Dordrecht.Google Scholar
  35. [35]
    Wu, Z., Zhao, J. Yin, J. and Li, H., (2001). Nonlinear Diffusion Equations, World Scientific, New Jersey.MATHCrossRefGoogle Scholar

Copyright information

© Springer 2009

Authors and Affiliations

  1. 1.CMAFUniversidade de LisboaPortugal
  2. 2.Departamento de Matemática AplicadaUniversidad Complutense de MadridSpain

Personalised recommendations