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

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

Received  April 2013 Revised  September 2013 Published  October 2013

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.
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

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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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