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June  2011, 4(3): 623-630. doi: 10.3934/dcdss.2011.4.623

Kolmogorov equations perturbed by an inverse-square potential

Gisèle Ruiz Goldstein 1, , Jerome A. Goldstein 1, and  Abdelaziz Rhandi 2,

1. 

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States, United States
2. 

Dipartimento di Ingegneria dell'Informazione e Matematica Applicata, Università degli Studi di Salerno, Via Ponte Don Melillo, 84084 Fisciano (Sa), Italy

Received  April 2009 Revised  November 2009 Published  November 2010

In this paper we present a nonexistence result of exponentially bounded positive solutions to a parabolic equation of Kolmogorov type with a more general drift term perturbed by an inverse square potential. This result generalizes the one obtained in [8]. Next we introduce some classes of nonlinear operators, related to the filtration operators and the $p$-Laplacian, and involving Kolmogorov operators. We establish the maximal monotonicity of some of these operators. In the third part we discuss the possibility of some nonexistence results in the context of singular potential perturbations of these nonlinear operators.
Keywords: nonlinear parabolic equations, Hardy's inequality, positive solutions, $p$-Kolmogorov operator., Inverse square potential, Ornstein-Uhlenbeck operator, critical constant.
Mathematics Subject Classification: Primary: 35K15, 35K65, 35R05; Secondary: 35B25, 34G10, 47D0.
Citation: Gisèle Ruiz Goldstein, Jerome A. Goldstein, Abdelaziz Rhandi. Kolmogorov equations perturbed by an inverse-square potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 623-630. doi: 10.3934/dcdss.2011.4.623
References:
[1]

P. Baras and J. A. Goldstein, The heat equation with singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.  Google Scholar

[2]

H. Brezis, "Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espaces de Hilbert," Math. Studies, 5, North-Holland, Amsterdam, 1973.  Google Scholar

[3]

X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C.R. Acad. Sci. Paris, 329 (1999), 973-978.  Google Scholar

[4]

M. Crandall and T. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-293.  Google Scholar

[5]

J. A. Goldstein, "Semigroups of Nonlinear Operators,", book in preparation., ().   Google Scholar

[6]

J. A. Goldstein and I. Kombe, Nonlinear degenerate parabolic equations with singular lower-order term, Adv. Diff. Equat., 8 (2003), 1153-1192.  Google Scholar

[7]

J. A. Goldstein and I. Kombe, The Hardy inequality and nonlinear parabolic equation on Carnot groups, Nonlinear Anal., 69 (2008), 4643-4653.  Google Scholar

[8]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential,, submitted., ().   Google Scholar

[9]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, The Hardy inequality and nonlinear parabolic equation of Kolmogorov type,, in preparation., ().   Google Scholar

[10]

Y. Komura, Nonlinear semigroups in Hilbert spaces, J. Math. Soc. Japan, 19 (1967), 493-507.  Google Scholar

[11]

I. Miyadera, "Nonlinear Semigroups," Translations of Mathematical Monographs, Vol. 109, Amer. Math. Soc., Providence, Rhode Island, 1992.  Google Scholar

show all references

References:
[1]

P. Baras and J. A. Goldstein, The heat equation with singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.  Google Scholar

[2]

H. Brezis, "Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espaces de Hilbert," Math. Studies, 5, North-Holland, Amsterdam, 1973.  Google Scholar

[3]

X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C.R. Acad. Sci. Paris, 329 (1999), 973-978.  Google Scholar

[4]

M. Crandall and T. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-293.  Google Scholar

[5]

J. A. Goldstein, "Semigroups of Nonlinear Operators,", book in preparation., ().   Google Scholar

[6]

J. A. Goldstein and I. Kombe, Nonlinear degenerate parabolic equations with singular lower-order term, Adv. Diff. Equat., 8 (2003), 1153-1192.  Google Scholar

[7]

J. A. Goldstein and I. Kombe, The Hardy inequality and nonlinear parabolic equation on Carnot groups, Nonlinear Anal., 69 (2008), 4643-4653.  Google Scholar

[8]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential,, submitted., ().   Google Scholar

[9]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, The Hardy inequality and nonlinear parabolic equation of Kolmogorov type,, in preparation., ().   Google Scholar

[10]

Y. Komura, Nonlinear semigroups in Hilbert spaces, J. Math. Soc. Japan, 19 (1967), 493-507.  Google Scholar

[11]

I. Miyadera, "Nonlinear Semigroups," Translations of Mathematical Monographs, Vol. 109, Amer. Math. Soc., Providence, Rhode Island, 1992.  Google Scholar

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