Geometric PDE and Applied Analysis Seminar (March 17, 2026)

Title: Blow-up rate for the subcritical semilinear heat equation in non-convex domains

Speaker: Dr. Erbol Zhanpeisov (Tohoku University)

Abstract: 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).