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Positive solutions to involving Wolff potentials
A strongly singular parabolic problem on an unbounded domain
1. | Department of Mathematics, National Technical University of Athens, 15780, Athens, Greece |
2. | Department of Mathematics & Engineering Sciences, Hellenic Army Academy, 16673, Athens |
References:
[1] |
Adimurthi, S. Filippas and A. Tertikas, On the best constant of Hardy-Sobolev inequalities,, Nonlinear An. TMA, 70 (2009), 2826.
doi: 10.1016/j.na.2008.12.019. |
[2] |
W. Arendt, G. R. Goldstein and J. A. Goldstein, Outgrowths of Hardy's inequality,, Contemporary Math. AMS, 412 (2006), 51.
doi: 10.1090/conm/412/07766. |
[3] |
J. P. García Azozero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Diff. Equations, 144 (1998), 441.
doi: 10.1006/jdeq.1997.3375. |
[4] |
P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121.
doi: 10.1090/S0002-9947-1984-0742415-3. |
[5] |
H. Brezis and J. L. Vázquez, Blowup solutions of some nonlinear elliptic problems,, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443. Google Scholar |
[6] |
X. Cabré and Y. Martel, Existence versus explosion instantané pour des equations de lachaleur linéaires avec potentiel singulier,, C.R. Acad. Sci. Paris, 329 (1999), 973.
doi: 10.1016/S0764-4442(00)88588-2. |
[7] |
L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259. Google Scholar |
[8] |
F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence) and symmetry of extremal functions,, Comm. Pure Appl. Math. \textbf{LIV} (2001), LIV (2001), 229.
doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. |
[9] |
C. Cazacu, Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results,, J. Funct. Anal., 263 (2012), 3741.
doi: 10.1016/j.jfa.2012.09.006. |
[10] |
G. R. Goldstein, J. A. Goldstein and A. Rhandi, Kolmogorov equations perturbed by an inverse-square potential,, Disc.Cont.Dyn.Syst.-Series S, 4 (2011), 623.
doi: 10.3934/dcdss.2011.4.623. |
[11] |
J. A. Goldstein and I. Kombe, Nonlinear degenerate parabolic equations with singular lower-order term,, Adv. Diff. Equat., 8 (2003), 1153. Google Scholar |
[12] |
J. A. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials,, Trans. Amer. Math. Soc., 355 (2003), 197.
doi: 10.1090/S0002-9947-02-03057-X. |
[13] |
N. Ghoussoub and A. Moradifam, Bessel potentials and optimal Hardy and Hardy-Rellich inequalities,, Math. Ann., (2011), 1.
doi: 10.1007/s00208-010-0510-x. |
[14] |
M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation,, Nonlinear Anal. Theory, 11 (1987), 1103. Google Scholar |
[15] |
N. I. Karachalios, Weyl's type estimates on the eigenvalues of critical Schrödinger operators,, Lett. Math. Phys., 83 (2008), 189.
doi: 10.1007/s11005-007-0218-3. |
[16] |
N. I. Karachalios and N. B. Zographopoulos, The semiflow of a reaction diffusion equation with a singular potential,, Manuscripta Math., 130 (2009), 63.
doi: 10.1007/s00229-009-0284-1. |
[17] |
L. Moschini and A. Tesei, Parabolic Harnack Inequality for the heat equation with inverse-square potential,, Forum Math., 19 (2007), 407.
doi: 10.1515/FORUM.2007.017. |
[18] |
L. Moschini, G. Reyes and A. Tesei, Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients,, Comm. Pure Appl. Anal., 1 (2006), 155. Google Scholar |
[19] |
G. Reyes and A. Tesei, Self-similar solutions of a semilinear parabolic equation with inverse-square potential,, J. Diff. Equations, 219 (2005), 40.
doi: 10.1016/j.jde.2005.06.031. |
[20] |
J. M. Tölle, Uniqueness of weighted sobolev spaces with weakly differentiable weights,, J. Funct. Analysis, 263 (2012), 3195.
doi: 10.1016/j.jfa.2012.08.002. |
[21] |
J. Vancostenoble and E. Zuazua, Null Controllability for the heat equation with singular inverse-square potentials,, J. Funct. Analysis, 254 (2008), 1864.
doi: 10.1016/j.jfa.2007.12.015. |
[22] |
J. L. Vázquez and N. B. Zographopoulos, Functional aspects of the Hardy inequality. Appearance of a hidden energy,, J. Evol. Equ., 12 (2012).
doi: 10.1007/s00028-012-0151-5. |
[23] |
J. L. Vázquez and N. B. Zographopoulos, Functional aspects of Hardy type inequalities,, Disc. Cont. Dyn. Syst., 33 (2013), 5457.
doi: 10.3934/dcds.2013.33.5457. |
[24] |
J. L. Vázquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, J. Funct. Anal., 173 (2000), 103.
doi: 10.1006/jfan.1999.3556. |
[25] |
N. B. Zographopoulos, Weyl's type estimates on the eigenvalues of critical Schrödinger operators using improved Hardy-Sobolev inequalities,, J. Physics A: Math. Theor., 42 (2009).
doi: 10.1088/1751-8113/42/46/465204. |
[26] |
N. B. Zographopoulos, Existence of extremal functions for a Hardy-Sobolev inequality,, J. Funct. Anal., 259 (2010), 308.
doi: 10.1016/j.jfa.2010.03.020. |
show all references
References:
[1] |
Adimurthi, S. Filippas and A. Tertikas, On the best constant of Hardy-Sobolev inequalities,, Nonlinear An. TMA, 70 (2009), 2826.
doi: 10.1016/j.na.2008.12.019. |
[2] |
W. Arendt, G. R. Goldstein and J. A. Goldstein, Outgrowths of Hardy's inequality,, Contemporary Math. AMS, 412 (2006), 51.
doi: 10.1090/conm/412/07766. |
[3] |
J. P. García Azozero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Diff. Equations, 144 (1998), 441.
doi: 10.1006/jdeq.1997.3375. |
[4] |
P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121.
doi: 10.1090/S0002-9947-1984-0742415-3. |
[5] |
H. Brezis and J. L. Vázquez, Blowup solutions of some nonlinear elliptic problems,, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443. Google Scholar |
[6] |
X. Cabré and Y. Martel, Existence versus explosion instantané pour des equations de lachaleur linéaires avec potentiel singulier,, C.R. Acad. Sci. Paris, 329 (1999), 973.
doi: 10.1016/S0764-4442(00)88588-2. |
[7] |
L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259. Google Scholar |
[8] |
F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence) and symmetry of extremal functions,, Comm. Pure Appl. Math. \textbf{LIV} (2001), LIV (2001), 229.
doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. |
[9] |
C. Cazacu, Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results,, J. Funct. Anal., 263 (2012), 3741.
doi: 10.1016/j.jfa.2012.09.006. |
[10] |
G. R. Goldstein, J. A. Goldstein and A. Rhandi, Kolmogorov equations perturbed by an inverse-square potential,, Disc.Cont.Dyn.Syst.-Series S, 4 (2011), 623.
doi: 10.3934/dcdss.2011.4.623. |
[11] |
J. A. Goldstein and I. Kombe, Nonlinear degenerate parabolic equations with singular lower-order term,, Adv. Diff. Equat., 8 (2003), 1153. Google Scholar |
[12] |
J. A. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials,, Trans. Amer. Math. Soc., 355 (2003), 197.
doi: 10.1090/S0002-9947-02-03057-X. |
[13] |
N. Ghoussoub and A. Moradifam, Bessel potentials and optimal Hardy and Hardy-Rellich inequalities,, Math. Ann., (2011), 1.
doi: 10.1007/s00208-010-0510-x. |
[14] |
M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation,, Nonlinear Anal. Theory, 11 (1987), 1103. Google Scholar |
[15] |
N. I. Karachalios, Weyl's type estimates on the eigenvalues of critical Schrödinger operators,, Lett. Math. Phys., 83 (2008), 189.
doi: 10.1007/s11005-007-0218-3. |
[16] |
N. I. Karachalios and N. B. Zographopoulos, The semiflow of a reaction diffusion equation with a singular potential,, Manuscripta Math., 130 (2009), 63.
doi: 10.1007/s00229-009-0284-1. |
[17] |
L. Moschini and A. Tesei, Parabolic Harnack Inequality for the heat equation with inverse-square potential,, Forum Math., 19 (2007), 407.
doi: 10.1515/FORUM.2007.017. |
[18] |
L. Moschini, G. Reyes and A. Tesei, Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients,, Comm. Pure Appl. Anal., 1 (2006), 155. Google Scholar |
[19] |
G. Reyes and A. Tesei, Self-similar solutions of a semilinear parabolic equation with inverse-square potential,, J. Diff. Equations, 219 (2005), 40.
doi: 10.1016/j.jde.2005.06.031. |
[20] |
J. M. Tölle, Uniqueness of weighted sobolev spaces with weakly differentiable weights,, J. Funct. Analysis, 263 (2012), 3195.
doi: 10.1016/j.jfa.2012.08.002. |
[21] |
J. Vancostenoble and E. Zuazua, Null Controllability for the heat equation with singular inverse-square potentials,, J. Funct. Analysis, 254 (2008), 1864.
doi: 10.1016/j.jfa.2007.12.015. |
[22] |
J. L. Vázquez and N. B. Zographopoulos, Functional aspects of the Hardy inequality. Appearance of a hidden energy,, J. Evol. Equ., 12 (2012).
doi: 10.1007/s00028-012-0151-5. |
[23] |
J. L. Vázquez and N. B. Zographopoulos, Functional aspects of Hardy type inequalities,, Disc. Cont. Dyn. Syst., 33 (2013), 5457.
doi: 10.3934/dcds.2013.33.5457. |
[24] |
J. L. Vázquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, J. Funct. Anal., 173 (2000), 103.
doi: 10.1006/jfan.1999.3556. |
[25] |
N. B. Zographopoulos, Weyl's type estimates on the eigenvalues of critical Schrödinger operators using improved Hardy-Sobolev inequalities,, J. Physics A: Math. Theor., 42 (2009).
doi: 10.1088/1751-8113/42/46/465204. |
[26] |
N. B. Zographopoulos, Existence of extremal functions for a Hardy-Sobolev inequality,, J. Funct. Anal., 259 (2010), 308.
doi: 10.1016/j.jfa.2010.03.020. |
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