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

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

“…By Propositions 2.1, Lemma 2.1, (2.6) and (2.9), applying similar arguments as in the proof of [14,Theorem 1.4] to (2.4), we obtain Theorem 1.1.…”

“…Nonnegative Schrödinger operators and their heat semigroups appear in various fields and have been studied intensively by many authors since the pioneering work due to Simon [29] (see e.g., [3]- [6], [9]- [14], [17]- [28], [31]- [33] and references therein). The inverse square potential is a typical one appearing in the study of the Schrödinger operators and it also arises in the linearized analysis for nonlinear diffusion equations, in particular, in solid-fuel ignition phenomena which can be modeled by ∂ t u = ∆u + e u .…”

“…(See Section 2.2.) In this paper, combing the arguments in [9]- [12] and [14], we study the large time behavior of H(u(t)) in the cases (S), (S * ) and (C) and reveal the relationship between the large time behavior of H(u(t)) and the corresponding harmonic functions. We remark that L V is not necessarily subcritical.…”

“…Our arguments in this paper are based on [14], where the precise description of the large time behavior of the solution of (1.1) was discussed under condition (V). Applying the arguments in [14], we modify the arguments in [9]- [12] and study the large time behavior of the hot spots. We study the following subjects when the hots spots tend to the space infinity as t → ∞:…”