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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

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 and 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), 2826-2833. 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), 51-68. 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), 441-476. 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), 121-139. 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), 443-469.

[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), 973-978. 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), 259-275.

[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. LIV (2001), 229-258. 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), 3741-3783. 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), 623-630. 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), 1153-1192.

[12]

J. A. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials, Trans. Amer. Math. Soc., 355 (2003), 197-211. 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., 349 (2011), 1-57. 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, Methods and Appl., 11 (1987), 1103-1133.

[15]

N. I. Karachalios, Weyl's type estimates on the eigenvalues of critical Schrödinger operators, Lett. Math. Phys., 83 (2008), 189-199. 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), 63-91. 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), 407-427. 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), 155-179.

[19]

G. Reyes and A. Tesei, Self-similar solutions of a semilinear parabolic equation with inverse-square potential, J. Diff. Equations, 219 (2005), 40-77. 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), 3195-3223 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), 1864-1902. 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), 713?739. 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), 5457-5491. 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), 103-153. 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), 465204. 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), 308-314. 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), 2826-2833. 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), 51-68. 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), 441-476. 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), 121-139. 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), 443-469.

[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), 973-978. 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), 259-275.

[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. LIV (2001), 229-258. 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), 3741-3783. 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), 623-630. 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), 1153-1192.

[12]

J. A. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials, Trans. Amer. Math. Soc., 355 (2003), 197-211. 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., 349 (2011), 1-57. 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, Methods and Appl., 11 (1987), 1103-1133.

[15]

N. I. Karachalios, Weyl's type estimates on the eigenvalues of critical Schrödinger operators, Lett. Math. Phys., 83 (2008), 189-199. 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), 63-91. 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), 407-427. 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), 155-179.

[19]

G. Reyes and A. Tesei, Self-similar solutions of a semilinear parabolic equation with inverse-square potential, J. Diff. Equations, 219 (2005), 40-77. 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), 3195-3223 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), 1864-1902. 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), 713?739. 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), 5457-5491. 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), 103-153. 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), 465204. 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), 308-314. doi: 10.1016/j.jfa.2010.03.020.

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