Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A priori estimates for quasilinear degenerate parabolic equations


Authors: Maria Manfredini and Andrea Pascucci
Journal: Proc. Amer. Math. Soc. 131 (2003), 1115-1120
MSC (2000): Primary 35K55; Secondary 35K65
DOI: https://doi.org/10.1090/S0002-9939-02-06922-8
Published electronically: November 13, 2002
MathSciNet review: 1948102
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove some maximum and gradient estimates for classical solutions to a wide class of quasilinear degenerate parabolic equations, including first order ones. The proof is elementary and exploits the smallness of the domain in the time direction.


References [Enhancements On Off] (What's this?)

  • 1. F. ANTONELLI, A. PASCUCCI, On the viscosity solutions of a stochastic differential utility problem, to appear in J. Differential Equations.
  • 2. G. BARLES, A weak Bernstein method for fully non-linear elliptic equations, Differential Integral Equations 4, No.2, (1991), 241-262. MR 91k:35087
  • 3. S. BERNSTEIN, Sur la généralisation du probléme de Dirichlet, I, Math. Ann., 62, (1906), 253-271.
  • 4. G. CITTI, A. PASCUCCI, S. POLIDORO, Regularity properties of viscosity solutions of a non-Hörmander degenerate equation, J. Math. Pures Appl., 80-9, (2001), 901-918. MR 2002j:35055
  • 5. G. CITTI, M. MANFREDINI, A degenerate parabolic equation arising in image processing, to appear in Commun. Appl. Anal.
  • 6. M. ESCOBEDO, J.L. VAZQUEZ, E. ZUAZUA, Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc. 343, No.2, (1994), 829-842. MR 94h:35131
  • 7. D. GILBARG, N.S. TRUDINGER, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, (2001). MR 2001k:35004
  • 8. G. HUISKEN, Non-parametric mean curvature evolution with boundary conditions, J. Differential Equations 77, No.2, (1989), 369-378. MR 90g:35050
  • 9. A.V. IVANOV, Quasilinear degenerate and nonuniformly elliptic and parabolic equations of second order, Proceedings of the Steklov Institute of Mathematics, Issue 1 (Russian Vol. 160). Providence, Rhode Island: American Mathematical Society. XI, (1984). MR 85e:35055
  • 10. O.A. LADYZHENSKAYA, N.N. URAL'TSEVA, A boundary value problem for linear and quasilinear parabolic equations I, II, (Russian, English) Am. Math. Soc., Transl., II. Ser. 47, 217-299 (1965); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 26, 5-52, 753-780 (1962).
  • 11. O.A. LADYZHENSKAYA, N.N. URAL'TSEVA, Linear and quasi-linear equations of parabolic type, Transl. Math. Monographs 23. Providence, RI: American Mathematical Society (1968).
  • 12. G.M. LIEBERMAN, Second order parabolic differential equations, Singapore, World Scientific, (1996). MR 98k:35003
  • 13. G.M. LIEBERMAN, Gradient estimates for a new class of degenerate elliptic and parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21, no. 4, 497-522 (1994). MR 96c:35024
  • 14. A. PASCUCCI, S. POLIDORO, On the Cauchy problem for a nonlinear ultraparabolic equation, preprint
  • 15. J. SERRIN, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, ``Contributions to Nonlinear Functional Analysis", Proc. Sympos. Univ. Wisconsin, Madison 1971, 565-601. MR 53:6095

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35K55, 35K65

Retrieve articles in all journals with MSC (2000): 35K55, 35K65


Additional Information

Maria Manfredini
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
Email: manfredi@dm.unibo.it

Andrea Pascucci
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
Email: pascucci@dm.unibo.it

DOI: https://doi.org/10.1090/S0002-9939-02-06922-8
Received by editor(s): October 15, 2001
Published electronically: November 13, 2002
Additional Notes: This work was supported by the University of Bologna, funds for selected research topics
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society