Linear parabolic equations with strong singular potentials
HTML articles powered by AMS MathViewer
- by Jerome A. Goldstein and Qi S. Zhang PDF
- Trans. Amer. Math. Soc. 355 (2003), 197-211 Request permission
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
- D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 607–694. MR 435594
- J. A. Aguilar Crespo and I. Peral Alonso, Global behavior of the Cauchy problem for some critical nonlinear parabolic equations, SIAM J. Math. Anal. 31 (2000), no. 6, 1270–1294. MR 1766560, DOI 10.1137/S0036141098341137
- 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.
- Pierre Baras and Jerome A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), no. 1, 121–139. MR 742415, DOI 10.1090/S0002-9947-1984-0742415-3
- Pierre Baras and Jerome A. Goldstein, Remarks on the inverse square potential in quantum mechanics, Differential equations (Birmingham, Ala., 1983) North-Holland Math. Stud., vol. 92, North-Holland, Amsterdam, 1984, pp. 31–35. MR 799330, DOI 10.1016/S0304-0208(08)73675-2
- Xavier Cabré and Yvan Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 11, 973–978 (French, with English and French summaries). MR 1733904, DOI 10.1016/S0764-4442(00)88588-2
- E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 1349825, DOI 10.1017/CBO9780511623721
- Dupaigne, L., A nonlinear elliptic PDE with the inverse square potential, J. Anal. Math. 86 (2002), 359–398.
- D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications. MR 929030
- Eugene B. Fabes, Nicola Garofalo, and Sandro Salsa, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. Math. 30 (1986), no. 4, 536–565. MR 857210
- 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 (English, with French summary). MR 1070830
- Nicola Garofalo and Duy-Minh Nhieu, 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 1404326, DOI 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
- 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. MR 1616905, DOI 10.1006/jdeq.1997.3375
- Goldstein, J. A. and Zhang, Qi S., On a degenerate heat equation with a singular potential, J. Functional Analysis, 186 (2001), 342-359.
- Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
- V. A. Liskevich and Yu. A. Semenov, Some problems on Markov semigroups, Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras, Math. Top., vol. 11, Akademie Verlag, Berlin, 1996, pp. 163–217. MR 1409835
- Milman, P. D. and Semenov, Yu. A., De-singularizing weights and heat kernel bounds, preprint, 1999
- P. Maheux and L. Saloff-Coste, Analyse sur les boules d’un opérateur sous-elliptique, Math. Ann. 303 (1995), no. 4, 713–740 (French). MR 1359957, DOI 10.1007/BF01461013
- Erich Rothe, Topological proofs of uniqueness theorems in the theory of differential and integral equations, Bull. Amer. Math. Soc. 45 (1939), 606–613. MR 93, DOI 10.1090/S0002-9904-1939-07048-1
- Juan Luis Vazquez and Enrike Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000), no. 1, 103–153. MR 1760280, DOI 10.1006/jfan.1999.3556
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
Additional Information
- Jerome A. Goldstein
- Affiliation: Department of Mathematics, University of Memphis, Memphis, Tennessee 38152
- MR Author ID: 74805
- Email: jgoldste@memphis.edu
- Qi S. Zhang
- Affiliation: Department of Mathematics, University of California Riverside, Riverside, California 92521
- MR Author ID: 359866
- Email: qizhang@math.ucr.edu
- Received by editor(s): July 22, 2001
- Received by editor(s) in revised form: January 17, 2002
- Published electronically: August 1, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 197-211
- MSC (2000): Primary 35D05, 35K05, 35R25; Secondary 35B50, 35C99, 35K15, 53C99
- DOI: https://doi.org/10.1090/S0002-9947-02-03057-X
- MathSciNet review: 1928085