A strongly singular parabolic problem on an unbounded domain

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March 2014, 13(2): 789-809. doi: 10.3934/cpaa.2014.13.789

A strongly singular parabolic problem on an unbounded domain

G. P. Trachanas ^1, and Nikolaos B. Zographopoulos ^2,

1.	Department of Mathematics, National Technical University of Athens, 15780, Athens, Greece
2.	Department of Mathematics & Engineering Sciences, Hellenic Army Academy, 16673, Athens

Received April 2013 Revised September 2013 Published October 2013

Abstract
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We study the well-posedness and describe the asymptotic behavior of solutions of a strongly singular equation for the Cauchy problem on $R^N$. The strong singularity is exactly the critical case of the Caffarelli-Kohn-Nirenberg inequality. Moreover, we show the stabilization towards a radially symmetric solution in self-similar variables with a polynomial decay rate. This equation is closely related to a heat equation with inverse-square potential, posed on $R^N$. In this case we have the appearance of the Hardy singularity energy.

Keywords: Critical inequalities, asymptotic behavior., unbounded domain.

Mathematics Subject Classification: Primary: 35K67, 35P15; Secondary: 35A2.

Citation: G. P. Trachanas, Nikolaos B. Zographopoulos. A strongly singular parabolic problem on an unbounded domain. Communications on Pure & Applied Analysis, 2014, 13 (2) : 789-809. doi: 10.3934/cpaa.2014.13.789

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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