August  2018, 38(8): 4041-4069. doi: 10.3934/dcds.2018176

Large time behavior of solutions of the heat equation with inverse square potential

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

* Corresponding author: Kazuhiro Ishige

Received  November 2017 Revised  March 2018 Published  May 2018

Fund Project: The first author is partially supported by the Grant-in-Aid for Scientific Research (A)(No. 15H02058) from Japan Society for the Promotion of Science.

Let $L: = -Δ+V$ be a nonnegative Schrödinger operator on $L^2({\bf R}^N)$, where $N≥ 2$ and $V$ is a radially symmetric inverse square potential. In this paper we assume either $L$ is subcritical or null-critical and we establish a method for obtaining the precise description of the large time behavior of $e^{-tL}\varphi$, where $\varphi∈ L^2({\bf R}^N, e^{|x|^2/4}\, dx)$.

Citation: Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176
References:
[1]

G. BarbatisS. Filippas and A. Tertikas, Critical heat kernel estimates for Schrödinger operators via Hardy-Sobolev inequalities, J. Funct. Anal., 208 (2004), 1-30.  doi: 10.1016/j.jfa.2003.10.002.

[2]

I. Chavel and L. Karp, Large time behavior of the heat kernel: The parabolic λ-potential alternative, Comment. Math. Helv., 66 (1991), 541-556.  doi: 10.1007/BF02566664.

[3]

F. Chiarenza and R. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova, 73 (1985), 179-190. 

[4]

D. Cruz-Uribe and C. Rios, Gaussian bounds for degenerate parabolic equations, J. Funct. Anal., 255 (2008), 283–312; Corrigendum in J. Funct. Anal., 267 (2014), 3507–3513. doi: 10.1016/j.jfa.2008.01.017.

[5]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92, Cambridge Univ. Press 1989. doi: 10.1017/CBO9780511566158.

[6]

E. B. Davies and B. Simon, Lp norms of noncritical Schrödinger semigroups, J. Funct. Anal., 102 (1991), 95-115.  doi: 10.1016/0022-1236(91)90137-T.

[7]

A. Grigor'yan, Heat Kernel and Analysis on Manifolds, AMS, Providence, RI, 2009.

[8]

A. Grigor'yan and L. Saloff-Coste, Stability results for Harnack inequalities, Ann. Inst. Fourier, 55 (2005), 825-890.  doi: 10.5802/aif.2116.

[9]

N. IokuK. Ishige and E. Yanagida, Sharp decay estimates of Lq norms of nonnegative Schrödinger heat semigroups, J. Funct. Anal., 264 (2013), 2764-2783.  doi: 10.1016/j.jfa.2013.03.009.

[10]

N. IokuK. Ishige and E. Yanagida, Sharp decay estimates in Lorentz spaces for nonnegative Schrödinger heat semigroups, J. Math. Pures Appl., 103 (2015), 900-923.  doi: 10.1016/j.matpur.2014.09.006.

[11]

K. Ishige, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J., 155 (1999), 1-26.  doi: 10.1017/S0027763000006978.

[12]

K. Ishige, Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition, J. Differential Equations, 212 (2005), 394-431.  doi: 10.1016/j.jde.2004.11.002.

[13]

K. Ishige and Y. Kabeya, Large time behaviors of hot spots for the heat equation with a potential, J. Differential Equations, 244 (2008), 2934–2962; Corrigendum in J. Differential Equations 245 (2008), 2352–2354. doi: 10.1016/j.jde.2008.07.023.

[14]

K. Ishige and Y. Kabeya, Hot spots for the heat equation with a rapidly decaying negative potential, Adv. Differential Equations, 14 (2009), 643-662. 

[15]

K. Ishige and Y. Kabeya, Hot spots for the two dimensional heat equation with a rapidly decaying negative potential, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 833-849.  doi: 10.3934/dcdss.2011.4.833.

[16]

K. Ishige and Y. Kabeya, Lq norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots, J. Funct. Anal., 262 (2012), 2695-2733.  doi: 10.1016/j.jfa.2011.12.024.

[17]

K. Ishige and Y. Kabeya, Decay rate of Lq norms of critical Schrödinger heat semigroups, Geometric Properties for Parabolic and Elliptic PDE's, 165–178, Springer INdAM Ser., 2, Springer, Milan, 2013. doi: 10.1007/978-88-470-2841-8_11.

[18]

K. Ishige, Y. Kabeya and A, Mukai, Hot spots of solutions to the heat equation with inverse square potential to appear in Applicable Anal., (2018), https://doi.org/10.1080/00036811.2018.1466284 doi: 10.1080/00036811.2018.1466284.

[19]

K. IshigeY. Kabeya and E. M. Ouhabaz, The heat kernel of a Schrödinger operator with inverse square potential, Proc. Lond. Math. Soc., 115 (2017), 381-410.  doi: 10.1112/plms.12041.

[20]

O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society Translations, vol. 23, American Mathematical Society, Providence, RI, 1968.

[21]

V. Liskevich and Z. Sobol, Estimates of integral kernels for semigroups associated with second-order elliptic operators with singular coefficients, Potential Anal., 18 (2003), 359-390.  doi: 10.1023/A:1021877025938.

[22]

P. D. Milman and Y. A. Semenov, Global heat kernel bounds via desingularizing weights, J. Funct. Anal., 212 (2004), 373-398.  doi: 10.1016/j.jfa.2003.12.008.

[23]

N. MizoguchiH. Ninomiya and E. Yanagida, Critical exponent for the bipolar blowup in a semilinear parabolic equation, J. Math. Anal. Appl., 218 (1998), 495-518.  doi: 10.1006/jmaa.1997.5815.

[24]

L. Moschini and A. Tesei, Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential, Rend. Mat. Acc. Lincei, 16 (2005), 171-180. 

[25]

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.

[26]

M. Murata, Positive solutions and large time behaviors of Schrödinger semigroups, Simon's problem, J. Funct. Anal., 56 (1984), 300-310.  doi: 10.1016/0022-1236(84)90079-X.

[27]

M. Murata, Structure of positive solutions to (-Δ+V)u = 0 in Rn, Proceedings of the Conference on Spectral and Scattering Theory for Differential Operators (Fujisakura-so, 1986), (1986), 64-108.  doi: 10.1215/S0012-7094-86-05347-0.

[28]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monographs, 31, Princeton Univ. Press 2005.

[29]

Y. Pinchover, On criticality and ground states of second order elliptic equations, Ⅱ, J. Differential Equations, 87 (1990), 353-364.  doi: 10.1016/0022-0396(90)90007-C.

[30]

Y. Pinchover, Large time behavior of the heat kernel and the behavior of the Green function near criticality for nonsymmetric elliptic operators, J. Funct. Anal., 104 (1992), 54-70.  doi: 10.1016/0022-1236(92)90090-6.

[31]

Y. Pinchover, On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators, Duke Math. J., 85 (1996), 431-445.  doi: 10.1215/S0012-7094-96-08518-X.

[32]

Y. Pinchover, Large time behavior of the heat kernel, J. Funct. Anal., 206 (2004), 191-209.  doi: 10.1016/S0022-1236(03)00110-1.

[33]

Y. Pinchover, Some aspects of large time behavior of the heat kernel: An overview with perspectives, Mathematical Physics, Spectral Theory and Stochastic Analysis (Basel) (M. Demuth and W. Kirsch, eds.), Operator Theory: Advances and Applications, vol. 232, Springer Verlag, 2013,299–339. doi: 10.1007/978-3-0348-0591-9_6.

[34]

B. Simon, Large time behavior of the Lp norm of Schrödinger semigroups, J. Funct. Anal., 40 (1981), 66-83.  doi: 10.1016/0022-1236(81)90073-2.

[35]

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.

[36]

Q. S. Zhang, Large time behavior of Schrödinger heat kernels and applications, Comm. Math. Phys., 210 (2000), 371-398.  doi: 10.1007/s002200050784.

[37]

Q. S. Zhang, Global bounds of Schrödinger heat kernels with negative potentials, J. Funct. Anal., 182 (2001), 344-370.  doi: 10.1006/jfan.2000.3737.

show all references

References:
[1]

G. BarbatisS. Filippas and A. Tertikas, Critical heat kernel estimates for Schrödinger operators via Hardy-Sobolev inequalities, J. Funct. Anal., 208 (2004), 1-30.  doi: 10.1016/j.jfa.2003.10.002.

[2]

I. Chavel and L. Karp, Large time behavior of the heat kernel: The parabolic λ-potential alternative, Comment. Math. Helv., 66 (1991), 541-556.  doi: 10.1007/BF02566664.

[3]

F. Chiarenza and R. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova, 73 (1985), 179-190. 

[4]

D. Cruz-Uribe and C. Rios, Gaussian bounds for degenerate parabolic equations, J. Funct. Anal., 255 (2008), 283–312; Corrigendum in J. Funct. Anal., 267 (2014), 3507–3513. doi: 10.1016/j.jfa.2008.01.017.

[5]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92, Cambridge Univ. Press 1989. doi: 10.1017/CBO9780511566158.

[6]

E. B. Davies and B. Simon, Lp norms of noncritical Schrödinger semigroups, J. Funct. Anal., 102 (1991), 95-115.  doi: 10.1016/0022-1236(91)90137-T.

[7]

A. Grigor'yan, Heat Kernel and Analysis on Manifolds, AMS, Providence, RI, 2009.

[8]

A. Grigor'yan and L. Saloff-Coste, Stability results for Harnack inequalities, Ann. Inst. Fourier, 55 (2005), 825-890.  doi: 10.5802/aif.2116.

[9]

N. IokuK. Ishige and E. Yanagida, Sharp decay estimates of Lq norms of nonnegative Schrödinger heat semigroups, J. Funct. Anal., 264 (2013), 2764-2783.  doi: 10.1016/j.jfa.2013.03.009.

[10]

N. IokuK. Ishige and E. Yanagida, Sharp decay estimates in Lorentz spaces for nonnegative Schrödinger heat semigroups, J. Math. Pures Appl., 103 (2015), 900-923.  doi: 10.1016/j.matpur.2014.09.006.

[11]

K. Ishige, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J., 155 (1999), 1-26.  doi: 10.1017/S0027763000006978.

[12]

K. Ishige, Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition, J. Differential Equations, 212 (2005), 394-431.  doi: 10.1016/j.jde.2004.11.002.

[13]

K. Ishige and Y. Kabeya, Large time behaviors of hot spots for the heat equation with a potential, J. Differential Equations, 244 (2008), 2934–2962; Corrigendum in J. Differential Equations 245 (2008), 2352–2354. doi: 10.1016/j.jde.2008.07.023.

[14]

K. Ishige and Y. Kabeya, Hot spots for the heat equation with a rapidly decaying negative potential, Adv. Differential Equations, 14 (2009), 643-662. 

[15]

K. Ishige and Y. Kabeya, Hot spots for the two dimensional heat equation with a rapidly decaying negative potential, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 833-849.  doi: 10.3934/dcdss.2011.4.833.

[16]

K. Ishige and Y. Kabeya, Lq norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots, J. Funct. Anal., 262 (2012), 2695-2733.  doi: 10.1016/j.jfa.2011.12.024.

[17]

K. Ishige and Y. Kabeya, Decay rate of Lq norms of critical Schrödinger heat semigroups, Geometric Properties for Parabolic and Elliptic PDE's, 165–178, Springer INdAM Ser., 2, Springer, Milan, 2013. doi: 10.1007/978-88-470-2841-8_11.

[18]

K. Ishige, Y. Kabeya and A, Mukai, Hot spots of solutions to the heat equation with inverse square potential to appear in Applicable Anal., (2018), https://doi.org/10.1080/00036811.2018.1466284 doi: 10.1080/00036811.2018.1466284.

[19]

K. IshigeY. Kabeya and E. M. Ouhabaz, The heat kernel of a Schrödinger operator with inverse square potential, Proc. Lond. Math. Soc., 115 (2017), 381-410.  doi: 10.1112/plms.12041.

[20]

O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society Translations, vol. 23, American Mathematical Society, Providence, RI, 1968.

[21]

V. Liskevich and Z. Sobol, Estimates of integral kernels for semigroups associated with second-order elliptic operators with singular coefficients, Potential Anal., 18 (2003), 359-390.  doi: 10.1023/A:1021877025938.

[22]

P. D. Milman and Y. A. Semenov, Global heat kernel bounds via desingularizing weights, J. Funct. Anal., 212 (2004), 373-398.  doi: 10.1016/j.jfa.2003.12.008.

[23]

N. MizoguchiH. Ninomiya and E. Yanagida, Critical exponent for the bipolar blowup in a semilinear parabolic equation, J. Math. Anal. Appl., 218 (1998), 495-518.  doi: 10.1006/jmaa.1997.5815.

[24]

L. Moschini and A. Tesei, Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential, Rend. Mat. Acc. Lincei, 16 (2005), 171-180. 

[25]

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.

[26]

M. Murata, Positive solutions and large time behaviors of Schrödinger semigroups, Simon's problem, J. Funct. Anal., 56 (1984), 300-310.  doi: 10.1016/0022-1236(84)90079-X.

[27]

M. Murata, Structure of positive solutions to (-Δ+V)u = 0 in Rn, Proceedings of the Conference on Spectral and Scattering Theory for Differential Operators (Fujisakura-so, 1986), (1986), 64-108.  doi: 10.1215/S0012-7094-86-05347-0.

[28]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monographs, 31, Princeton Univ. Press 2005.

[29]

Y. Pinchover, On criticality and ground states of second order elliptic equations, Ⅱ, J. Differential Equations, 87 (1990), 353-364.  doi: 10.1016/0022-0396(90)90007-C.

[30]

Y. Pinchover, Large time behavior of the heat kernel and the behavior of the Green function near criticality for nonsymmetric elliptic operators, J. Funct. Anal., 104 (1992), 54-70.  doi: 10.1016/0022-1236(92)90090-6.

[31]

Y. Pinchover, On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators, Duke Math. J., 85 (1996), 431-445.  doi: 10.1215/S0012-7094-96-08518-X.

[32]

Y. Pinchover, Large time behavior of the heat kernel, J. Funct. Anal., 206 (2004), 191-209.  doi: 10.1016/S0022-1236(03)00110-1.

[33]

Y. Pinchover, Some aspects of large time behavior of the heat kernel: An overview with perspectives, Mathematical Physics, Spectral Theory and Stochastic Analysis (Basel) (M. Demuth and W. Kirsch, eds.), Operator Theory: Advances and Applications, vol. 232, Springer Verlag, 2013,299–339. doi: 10.1007/978-3-0348-0591-9_6.

[34]

B. Simon, Large time behavior of the Lp norm of Schrödinger semigroups, J. Funct. Anal., 40 (1981), 66-83.  doi: 10.1016/0022-1236(81)90073-2.

[35]

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.

[36]

Q. S. Zhang, Large time behavior of Schrödinger heat kernels and applications, Comm. Math. Phys., 210 (2000), 371-398.  doi: 10.1007/s002200050784.

[37]

Q. S. Zhang, Global bounds of Schrödinger heat kernels with negative potentials, J. Funct. Anal., 182 (2001), 344-370.  doi: 10.1006/jfan.2000.3737.

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