Yoshitsugu Kabeya

Global structure of solutions for equations of Brezis-Nirenberg type on the unit ball

The Brezis-Nirenberg equation and the scalar field equation on the three-dimensional unit ball are studied. Under the Robin condition, we show the existence and uniqueness of radial solutions in a unified way. In particular, it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value.

The heat kernel of a Schrödinger operator with inverse square potential

We consider the Schrodinger operator H=−Δ+V(|x|) with radial potential V which may have singularity at 0 and a quadratic decay at infinity. First, we study the structure of positive harmonic functions of H and give their precise behavior. Second, under quite general conditions we prove an upper bound for the correspond heat kernel p(x,y,t) of the type 0<p(x,y,t)⩽Ct−N/2U(min{|x|,t})U(min{|y|,t})U(t)2exp−|x−y|2Ct for all x, y∈RN and t>0, where U is a positive harmonic function of H. Third, if U2 is an A2 weight on RN, then we prove a lower bound of a similar type.

Decay Rate of L q Norms of Critical Schrödinger Heat Semigroups

Let H:=−Δ+V be a critical Schrodinger operator on L 2(R N ), where N≥3 and V is a radially symmetric function decaying quadratically at the space infinity. We study the optimal decay rate of the operator norm of the Schrodinger heat semigroup e −tH from L 2(R N ) to L q (R N ) (2≤q≤∞).

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

ABSTRACT We investigate the large time behavior of the hot spots of the solution to the Cauchy problem where and decays quadratically as . In this paper, based on the arguments in [K. Ishige and A. Mukai, to appear in Discrete Contin. Dyn. Syst.], 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 .

Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition

The blowup behaviors of solutions to a scalar-field equation with the Robin condition are discussed. For some range of the parameter, there exist at least two positive solutions to the equation. Here, the blowup rate of the large solution and the scaling properties are discussed.