Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Linear parabolic equations with strong singular potentials

Authors: Jerome A. Goldstein and Qi S. Zhang
Journal: Trans. Amer. Math. Soc. 355 (2003), 197-211
MSC (2000): Primary 35D05, 35K05, 35R25; Secondary 35B50, 35C99, 35K15, 53C99
Published electronically: August 1, 2002
MathSciNet review: 1928085
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Using an extension of a recent method of Cabré and Martel (1999), we extend the blow-up and existence result in the paper of Baras and Goldstein (1984) to parabolic equations with variable leading coefficients under almost optimal conditions on the singular potentials. This problem has been left open in Baras and Goldstein. These potentials lie at a borderline case where standard theories such as the strong maximum principle and boundedness of weak solutions fail. Even in the special case when the leading operator is the Laplacian, we extend a recent result in Cabré and Martel from bounded smooth domains to unbounded nonsmooth domains.

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

  • [A] Aronson, D. G., Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa 22 (1968), 607-694. MR 55:8553
  • [AP] Aguilar Crespo, J. A. and Peral Alonso, I., Global behavior of the Cauchy problem for some critical nonlinear parabolic equations, SIAM J. Math. Analysis, 31 (2000), 1270-1294. MR 2001d:35086
  • [BC] Brézis, Haim and Cabré, Xavier, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 2, 223-262.
  • [BG1] Baras, P. and Goldstein, J. A. The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139. MR 85f:35099
  • [BG2] Baras, P. and Goldstein, J. A., Remarks on the inverse square potential in quantum mechanics, North-Holland Math. Stud., 92 (1984), 31-35. MR 87a:35090
  • [CM] Cabré, X. and Martel, Y. Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C. R. Acad. Sci. Paris Sér I. Math., 329 (1999), 973-978. MR 2000j:35117
  • [D] Davies, E. B., Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, 42, Cambridge Univ. Press, Cambridge, United Kingdom (1995) MR 96h:47056
  • [Du] Dupaigne, L., A nonlinear elliptic PDE with the inverse square potential, J. Anal. Math. 86 (2002), 359-398. CMP 2002:11
  • [EE] Edmunds, D. E. and Evans, W. D., Spectral Theory and Differential Operators. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1987. MR 89b:47001
  • [FGS] Fabes, Eugene B., Garofalo, Nicola, and Salsa, Sandro, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations. Illinois J. Math. 30 (1986), no. 4, 536-565. MR 88d:35089
  • [GL] Garofalo, N. and Lanconelli, E., Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier(Grenoble) 40 (1990), 313-356. MR 91i:22014
  • [GN] Garofalo, Nicola and Nhieu, Duy-Minh, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math. 49 (1996), no. 10, 1081-1144. MR 97i:58032
  • [GP] Garcia Azorero, J. and Peral, I., Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476. MR 99f:35099
  • [GZ] Goldstein, J. A. and Zhang, Qi S., On a degenerate heat equation with a singular potential, J. Functional Analysis, 186 (2001), 342-359.
  • [L] Lieberman, Gary M., Second Order Parabolic Differential Equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 98k:35003
  • [LS] Liskevich, V. A. and Semenov, Yu. A., Some problems on Markov semigroups. In the book: Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, 163-217, Math. Top., 11, Akademie Verlag, Berlin, 1996. MR 97g:47036
  • [MS] Milman, P. D. and Semenov, Yu. A., De-singularizing weights and heat kernel bounds, preprint, 1999
  • [MS-C] Maheux, P. and Saloff-Coste, L., Analyse sur les boules d'un opérateur sous-elliptique. (French) [Analysis on the balls of a subelliptic operator] Math. Ann. 303 (1995), no. 4, 713-740. MR 96m:35049
  • [SC] Saloff-Coste, L., Uniformly elliptic operators on Riemannian manifolds, J. Differential Geometry 36 (1992), 417-450. MR 93:58122
  • [VZ] Vazquez, J. L. and Zuazua, E., The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential, J. Funct. Analysis, 173 (2000), 103-153. MR 2001j:35122
  • [Zi] Ziemer, William P., Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. MR 91e:46046

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35D05, 35K05, 35R25, 35B50, 35C99, 35K15, 53C99

Retrieve articles in all journals with MSC (2000): 35D05, 35K05, 35R25, 35B50, 35C99, 35K15, 53C99

Additional Information

Jerome A. Goldstein
Affiliation: Department of Mathematics, University of Memphis, Memphis, Tennessee 38152

Qi S. Zhang
Affiliation: Department of Mathematics, University of California Riverside, Riverside, California 92521

Received by editor(s): July 22, 2001
Received by editor(s) in revised form: January 17, 2002
Published electronically: August 1, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society