Hot spots of solutions to the heat equation with inverse square potential

References

• Simon B. Large time behavior of the Lp norm of Schrödinger semigroups. J Funct Anal. 1981;40:66–83.
   Web of Science ®Google Scholar
• Chavel I, Karp L. Large time behavior of the heat kernel: the parabolic λ-potential alternative. Comment Math Helv. 1991;66:541–556.
   Web of Science ®Google Scholar
• Davies EB. Heat kernels and spectral theory. Vol. 92, Cambridge tracts in mathematics. Cambridge: Cambridge University Press; 1989.
   Google Scholar
• Davies EB, Simon B. Lp norms of noncritical Schrödinger semigroups. J Funct Anal. 1991;102:95–115.
   Web of Science ®Google Scholar
• Grigor’yan A. Heat kernel and analysis on manifolds. Providence (RI): AMS; 2009.
   Google Scholar
• Ishige K, Kabeya Y. Large time behaviors of hot spots for the heat equation with a potential. J Differ Equ. 2008;244:2934–2962; Corrigendum in J Differ Equ. 2008;245:2352--2354.
   Web of Science ®Google Scholar
• Ishige K, Kabeya Y. Hot spots for the heat equation with a rapidly decaying negative potential. Adv Differ Equ. 2009;14:643–662.
   Web of Science ®Google Scholar
• Ishige K, Kabeya Y. Hot spots for the two dimensional heat equation with a rapidly decaying negative potential. Discrete Contin Dyn Syst Ser S. 2011;4:833–849.
   Web of Science ®Google Scholar
• Ishige K, Kabeya Y. Lp norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots. J Funct Anal. 2012;262:2695–2733.
   Web of Science ®Google Scholar
• Ishige K, Kabeya Y, Ouhabaz EM. The heat kernel of a Schrödinger operator with inverse square potential. Proc Lond Math Soc. 2017;115:381–410.
   Web of Science ®Google Scholar
• Ishige K, Mukai A. Large time behavior of solutions of the heat equation with inverse square potential. Discrete Contin Dyn Syst. Forthcoming.
   Web of Science ®Google Scholar
• Liskevich V, Sobol Z. Estimates of integral kernels for semigroups associated with second-order elliptic operators with singular coefficients. Potential Anal. 2003;18:359–390.
   Web of Science ®Google Scholar
• Marchi C. The Cauchy problem for the heat equation with a singular potential. Differ Integr Equ. 2003;16:1065–1081.
   Google Scholar
• Milman PD, Semenov YA. Global heat kernel bounds via desingularizing weights. J Funct Anal. 2004;212:373–398.
   Web of Science ®Google Scholar
• Moschini L, Tesei A. Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential. Rend Mat Acc Lincei. 2005;16:171–180.
   Google Scholar
• Moschini L, Tesei A. Parabolic Harnack inequality for the heat equation with inverse-square potential. Forum Math. 2007;19:407–427.
   Web of Science ®Google Scholar
• Murata M. Structure of positive solutions to (-Δ + V)u = 0 in Rn . Duke Math J. 1986;53:869–943.
   Web of Science ®Google Scholar
• Ouhabaz EM. Analysis of heat equations on domains. Vol. 31, London mathematical society monographs series. Princeton: Princeton University Press; 2005.
   Google Scholar
• Pinchover Y. On criticality and ground states of second order elliptic equations, II. J Differ Equ. 1990;87:353–364.
   Web of Science ®Google Scholar
• Pinchover Y. Large time behavior of the heat kernel and the behavior of the Green function near criticality for nonsymmetric elliptic operators. J Funct Anal. 1992;104:54–70.
   Web of Science ®Google Scholar
• Pinchover Y. On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators. Duke Math J. 1996;85:431–445.
   Web of Science ®Google Scholar
• Pinchover Y. Large time behavior of the heat kernel. J Funct Anal. 2004;206:191–209.
   Web of Science ®Google Scholar
• Pinchover Y. Some aspects of large time behavior of the heat kernel: an overview with perspectives. In: Demuth M, Kirsch W, editors. Mathematical physics, spectral theory and stochastic analysis (Basel). Vol. 232, Operator theory: advances and applications. Berlin: Springer Verlag; 2013. p. 299–339.
   Google Scholar
• Vázquez JL, Zuazua E. The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J Funct Anal. 2000;173:103–153.
   Web of Science ®Google Scholar
• Zhang QS. Large time behavior of Schrödinger heat kernels and applications. Commun Math Phys. 2000;210:371–398.
   Web of Science ®Google Scholar
• Zhang QS. Global bounds of Schrödinger heat kernels with negative potentials. J Funct Anal. 2001;182:344–370.
   Web of Science ®Google Scholar
• Bebernes J, Eberly D. Mathematical problems from combustion theory. Vol. 83, Mathematical science. New York (NY): Springer-Verlag; 1989.
   Google Scholar
• Vazquez JL. Domain of existence and blowup for the exponential reaction-diffusion equation. Indiana Univ Math J. 1999;48:677–709.
   Web of Science ®Google Scholar
• Zel’dovich YaB, Barenblatt GI, Librovich VB, et al . The mathematical theory of combustion and explosions. New York (NY): Consultants Bureau; 1985.
   Google Scholar
• Chavel I, Karp L. Movement of hot spots in Riemannian manifolds. J Anal Math. 1990;55:271–286.
   Web of Science ®Google Scholar
• Jimbo S, Sakaguchi S. Movement of hot spots over unbounded domains in RN . J Math Anal Appl. 1994;182:810–835.
   Web of Science ®Google Scholar
• Ishige K. Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition. J Differ Equ. 2005;212:394–431.
   Web of Science ®Google Scholar
• Ishige K. Movement of hot spots on the exterior domain of a ball under the Dirichlet boundary condition. Adv Differ Equ. 2007;12:1135–1166.
   Google Scholar