Asymptotic analysis for solutions to semilinear heat equations

[1] J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Mathematical Science, 83,

Springer-Verlag, New York, 1989.

[2] G. Barbatis, S. Filippas and A. Tertikas, Critical heat kernel estimates for Schr¨odinger operators via

Hardy-Sobolev inequalities, J. Funct. Anal. 208 (2004), 1–30.

[3] I. Chavel and L. Karp, Movement of hot spots in Riemannian manifolds, J. Analyse Math. 55 (1990),

271–286.

[4] I. Chavel and L. Karp, Large time behavior of the heat kernel: the parabolic λ-potential alternative,

Comment. Math. Helv. 66 (1991), 541–556.

[5] F. Chiarenza and R. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations,

Rend. Sem. Mat. Univ. Padova 73 (1985), 179–190.

[6] 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.

[7] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92, Cambridge Univ. Press

1989.

[8] E. B. Davies and B. Simon, Lp norms of noncritical Schr¨odinger semigroups, J. Funct. Anal. 102 (1991),

95–115.

[9] A. Grigor’yan, Heat Kernel and Analysis on Manifolds, AMS, Providence, RI, 2009.

[10] A. Grigor’yan and L. Saloff-Coste, Stability results for Harnack inequalities, Ann. Inst. Fourier 55 (2005),

825–890.

[11] N. Ioku, K. Ishige and E. Yanagida, Sharp decay estimates of Lq norms of nonnegative Schr¨odinger heat

semigroups, J. Funct. Anal. 264 (2013), 2764–2783.

[12] N. Ioku, K. Ishige and E. Yanagida, Sharp decay estimates in Lorentz spaces for nonnegative Schr¨odinger

heat semigroups, J. Math. Pures Appl. 103 (2015), 900–923.

[13] K. Ishige, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J. 155

(1999), 1–26.

[14] 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.

[15] K. Ishige, Movement of hot spots on the exterior domain of a ball under the Dirichlet boundary condition,

Adv. Differential Equations 12 (2007), 1135–1166.

[16] 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.

104

[17] K. Ishige and Y. Kabeya, Hot spots for the heat equation with a rapidly decaying negative potential,

Adv. Differential Equations 14 (2009), 643–662.

[18] 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.

[19] K. Ishige and Y. Kabeya, Lp norms of nonnegative Schr¨odinger heat semigroup and the large time

behavior of hot spots, J. Funct. Anal. 262 (2012), 2695–2733.

[20] K. Ishige and Y. Kabeya, Decay rate of Lq norms of critical Schr¨odinger heat semigroups, Geometric

properties for parabolic and elliptic PDE’s, 165–178, Springer INdAM Ser., 2, Springer, Milan, 2013.

[21] K. Ishige, Y. Kabeya and A, Mukai, Hot spots of solutions to the heat equation with inverse square

potential, Appl. Anal. 98 (2019), no.10, 1843–1861.

[22] K. Ishige, Y. Kabeya and E. M. Ouhabaz, The heat kernel of a Schr¨odinger operator with inverse square

potential, Proc. Lond. Math. Soc. 115 (2017), 381–410.

[23] K. Ishige and A. Mukai, Large time behavior of solutions of the heat equation with inverse square

potential, Discrete Contin. Dyn. Syst. 38 (2018), 4041–4069.

[24] S. Jimbo and S. Sakaguchi, Movement of hot spots over unbounded domains in RN , J. Math. Anal.

Appl. 182 (1994), 810–835.

[25] O. A. Ladyˇzenskaja, 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.

[26] 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.

[27] C. Marchi, The Cauchy problem for the heat equation with a singular potential, Differential Integral

Equations 16 (2003), 1065–1081.

[28] P. D. Milman and Y. A. Semenov, Global heat kernel bounds via desingularizing weights, J. Funct. Anal.

212 (2004), 373–398.

[29] L. Moschini and A. Tesei, Harnack inequality and heat kernel estimates for the Schr¨odinger operator

with Hardy potential, Rend. Mat. Acc. Lincei 16 (2005), 171–180.

[30] L. Moschini and A. Tesei, Parabolic Harnack inequality for the heat equation with inverse-square potential, Forum Math. 19 (2007), 407–427.

[31] N. Mizoguchi, H. Ninomiya and E. Yanagida, Critical exponent for the bipolar blowup in a semilinear

parabolic equation, J. Math. Anal. Appl. 218 (1998), 495–518.

[32] M. Murata, Positive solutions and large time behaviors of Schr¨odinger semigroups, Simon’s problem, J.

Funct. Anal. 56 (1984), 300–310.

[33] M. Murata, Structure of positive solutions to (−∆ + V )u = 0 in Rn , Duke Math. J. 53 (1986), 869–943.

[34] E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monographs, 31, Princeton

Univ. Press 2005.

[35] Y. Pinchover, On criticality and ground states of second order elliptic equations, II, J. Differential

Equations 87 (1990), 353–364.

[36] 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.

[37] Y. Pinchover, On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators, Duke Math. J. 85 (1996), 431–445.

105

[38] Y. Pinchover, Large time behavior of the heat kernel, J. Funct. Anal. 206 (2004), 191–209.

[39] 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, pp. 299–339.

[40] B. Simon, Large time behavior of the Lp norm of Schr¨odinger semigroups, J. Funct. Anal. 40 (1981),

66–83.

[41] J. L. Vazquez, Domain of existence and blowup for the exponential reaction-diffusion equation, Indiana

Univ. Math. J. 48 (1999), 677–709.

[42] J. L. V´azquez 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.

[43] Q. S. Zhang, Large time behavior of Schr¨odinger heat kernels and applications, Comm. Math. Phys. 210

(2000), 371–398.

[44] Q. S. Zhang, Global bounds of Schr¨

odinger heat kernels with negative potentials, J. Funct. Anal. 182

(2001), 344–370.

[45] Ya. B. Zel’dovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The mathematical theory

of combustion and explosions, Consultants Bureau, New York, 1985.

106

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