Abstract and Applied Analysis

Cauchy-Dirichlet problem for the nonlinear degenerate parabolic equations

Ismail Kombe

Full-text: Open access


We will investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation: u/t=u+V(w)up1 in Ω×(0,T), 1#60;p#60;2, u(w,0)=u0(w)0 in Ω, u(w,t)=0 on Ω×(0,T) where is the subelliptic p-Laplacian and VLloc1(Ω).

Article information

Abstr. Appl. Anal., Volume 2005, Number 6 (2005), 607-617.

First available in Project Euclid: 3 October 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Kombe, Ismail. Cauchy-Dirichlet problem for the nonlinear degenerate parabolic equations. Abstr. Appl. Anal. 2005 (2005), no. 6, 607--617. doi:10.1155/AAA.2005.607. https://projecteuclid.org/euclid.aaa/1128345941

Export citation


  • P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), no. 1, 121--139.
  • R. Beals, B. Gaveau, and P. Greiner, On a geometric formula for the fundamental solution of subelliptic Laplacians, Math. Nachr. 181 (1996), 81--163.
  • --------, Uniforms hypoelliptic Green's functions, J. Math. Pures Appl. (9) 77 (1998), no. 3, 209--248.
  • --------, Green's functions for some highly degenerate elliptic operators, J. Funct. Anal. 165 (1999), no. 2, 407--429.
  • R. Beals, B. Gaveau, P. Greiner, and J. Vauthier, The Laguerre calculus on the Heisenberg group. II, Bull. Sci. Math. (2) 110 (1986), no. 3, 225--288.
  • X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier [Existence versus instantaneous blowup for linear heat equations with singular potentials ], C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 11, 973--978 (French).
  • L. Capogna, D. Danielli, and N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1765--1794.
  • L. D'Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132 (2004), no. 3, 725--734.
  • A. El Hamidi and M. Kirane, Nonexistence results of solutions to systems of semilinear differential inequalities on the Heisenberg group, Abstr. Appl. Anal. 2004 (2004), no. 2, 155--164.
  • A. El Hamidi and A. Obeid, Systems of semilinear higher-order evolution inequalities on the Heisenberg group, J. Math. Anal. Appl. 280 (2003), no. 1, 77--90.
  • C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, Ill, 1981) (W. Beckner, A. P. Calderón, R. Efferman, and P. W. Jones, eds.), Wadsworth Math. Ser., Wadsworth, California, 1983, pp. 590--606.
  • G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), no. 2, 373--376.
  • --------, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161--207.
  • J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), no. 2, 441--476.
  • N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Differential Equations 104 (1993), no. 1, 117--146.
  • N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 313--356.
  • --------, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41 (1992), no. 1, 71--98.
  • G. R. Goldstein, J. A. Goldstein, and I. Kombe, Nonlinear parabolic equations with singular coefficient and critical exponent, to appear in Appl. Anal.
  • J. A. Goldstein and I. Kombe, Instantaneous blow up, Advances in Differential Equations and Mathematical Physics (Birmingham, Ala, 2002), Contemp. Math., vol. 327, American Mathematical Society, Rhode Island, 2003, pp. 141--150.
  • --------, Nonlinear degenerate parabolic equations with singular lower-order term, Adv. Differential Equations 8 (2003), no. 10, 1153--1192.
  • --------, Nonlinear degenerate parabolic equations on the Heisenberg group, Int. J. Evol. Equ. 1 (2005), no. 1, 1--22.
  • J. A. Goldstein and Q. S. Zhang, On a degenerate heat equation with a singular potential, J. Funct. Anal. 186 (2001), no. 2, 342--359.
  • --------, Linear parabolic equations with strong singular potentials, Trans. Amer. Math. Soc. 355 (2003), no. 1, 197--211.
  • P. C. Greiner, A fundamental solution for a nonelliptic partial differential operator, Canad. J. Math. 31 (1979), no. 5, 1107--1120.
  • L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147--171.
  • D. S. Jerison and J. M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), no. 1, 1--13.
  • --------, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom. 29 (1989), no. 2, 303--343.
  • I. Kombe, Doubly nonlinear parabolic equations with singular lower order term, Nonlinear Anal. 56 (2004), no. 2, 185--199.
  • --------, The linear heat equation with highly oscillating potential, Proc. Amer. Math. Soc. 132 (2004), no. 9, 2683--2691.
  • --------, Nonlinear parabolic partial differential equations for Baouendi-Grushin operator, to appear in Mathematische Nachrichten.
  • --------, On the nonexistence of positive solutions to Doubly nonlinear equations for Baouendi-Grushin operator, preprint 2005.
  • --------, On the nonexistence of positive solutions to nonlinear degenerate parabolic equations with singular coefficients, to appear in Appl. Anal.
  • G. Lu, The sharp Poincaré inequality for free vector fields: an endpoint result, Rev. Mat. Iberoamericana 10 (1994), no. 2, 453--466.
  • A. Nagel, E. M. Stein, and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103--147.
  • P. Niu, H. Zhang, and Y. Wang, Hardy type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc. 129 (2001), no. 12, 3623--3630.
  • H. Zhang and P. Niu, Hardy-type inequalities and Pohozaev-type identities for a class of $p$-degenerate subelliptic operators and applications, Nonlinear Anal. 54 (2003), no. 1, 165--186.