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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 & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[5]

H. Brezis and J. L. Vázquez, Blowup solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense Madrid, 10 (1997), 443-469. 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), 973-978. 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), 259-275. 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. LIV (2001), 229-258. 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), 3741-3783. 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), 623-630. 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), 1153-1192. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  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), 63-91. 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), 407-427. 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), 155-179. 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), 40-77. 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), 3195-3223 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), 1864-1902. 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), 713?739. 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), 5457-5491. 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), 103-153. 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), 465204. 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), 308-314. 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), 2826-2833. 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), 51-68. 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), 441-476. 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), 121-139. 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), 443-469. 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), 973-978. 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), 259-275. 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. LIV (2001), 229-258. 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), 3741-3783. 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), 623-630. 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), 1153-1192. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  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), 63-91. 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), 407-427. 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), 155-179. 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), 40-77. 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), 3195-3223 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), 1864-1902. 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), 713?739. 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), 5457-5491. 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), 103-153. 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), 465204. 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), 308-314. doi: 10.1016/j.jfa.2010.03.020.  Google Scholar

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