We are concerned with blow-up mechanisms in a semilinear heat equation:ut = ∆u + |x| 2a u p , x ∈ R N , t > 0, where p > 1 and a > −1 are constants. As for the Fujita equation, which corresponds to a = 0, a well-known result due to M. A. Herrero and J. J. L. Velázquez, C. R. Acad. Sci. Paris Sér. I Math. (1994), states that if N ≥ 11 andwhere T is the blow-up time. We revisit the idea of their construction and obtain refined estimates for such solutions by the techniques developed in recent works and elaborate estimates of the heat semigroup in backward similarity variables. Our method is naturally extended to the case a = 0. As a consequence, we obtain an example of solutions that blow up at x = 0, the zero point of potential |x| 2a with a > 0, and behave in non-self-similar manner for N > 10 + 8a. This last result is in contrast to backward self-similar solutions previously obtained for N < 10 + 8a, which blow up at x = 0.

Let L := −∆ + V be a nonnegative Schrödinger operator on L 2 (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 ϕ, where ϕ ∈ L 2 (R N , e |x| 2 /4 dx).

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We investigate the large time behavior of the hot spots of the solution to the Cauchy problemwhere ϕ ∈ L 2 (R N , e |x| 2 /4 dx) and V = V (r) decays quadratically as r → ∞. In this paper, based on the arguments in [K. Ishige and A. Mukai, preprint (arXiv:1709.00809)], we classify the large time behavior of the hot spots of u and reveal the relationship between the behavior of the hot spots and the harmonic functions for −∆ + V .

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