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

Kolmogorov equations perturbed by an inverse-square potential

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.
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. Google Scholar

[2]

H. Brezis, "Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espaces de Hilbert,", Math. Studies, 5 (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. 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. 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. Google Scholar

[7]

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

[11]

I. Miyadera, "Nonlinear Semigroups,", Translations of Mathematical Monographs, 109 (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. Google Scholar

[2]

H. Brezis, "Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espaces de Hilbert,", Math. Studies, 5 (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. 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. 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. Google Scholar

[7]

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

[11]

I. Miyadera, "Nonlinear Semigroups,", Translations of Mathematical Monographs, 109 (1992). Google Scholar

[1]

Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451

[2]

Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049

[3]

Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871

[4]

Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049

[5]

Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649

[6]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377

[7]

Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143

[8]

Giuseppe Da Prato. Schauder estimates for some perturbation of an infinite dimensional Ornstein--Uhlenbeck operator. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 637-647. doi: 10.3934/dcdss.2013.6.637

[9]

Yucheng Bu, Yujun Dong. Existence of solutions for nonlinear operator equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4429-4441. doi: 10.3934/dcds.2019180

[10]

Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142

[11]

Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651

[12]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013

[13]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527

[14]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033

[15]

Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583

[16]

Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128

[17]

Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571

[18]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427

[19]

Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3831-3866. doi: 10.3934/dcds.2017162

[20]

Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (7)

[Back to Top]