分野別セミナー

We study the blow-up rate for solutions of the subcritical semilinear heat equation.
Type I blow-up means that the rate agrees with that of the associated ODE.
In the Sobolev subcritical range, type I estimates have been proved for positive solutions in convex or general domains (Giga–Kohn ’87; Quittner ’21)
and for sign-changing solutions in convex domains (Giga–Matsui–Sasayama ’04).
We extend these results to sign-changing solutions in possibly non-convex domains.
The proof uses the Giga-Kohn energy together with a geometric inequality that controls the effect of non-convexity.
As a corollary, we obtain blow-up of the scaling critical norm in the subcritical range.
Based on joint work with Hideyuki Miura and Jin Takahashi (Institute of Science Tokyo).