Nondegeneracy and Single-point-blowup for Solution of the Heat Equation with a Nonlinear Boundary Condition

This paper studies the nondegeneracy of the blowup limit and the single-point-blowup for the heat equation $u_t = Δ u$ with the nonlinear boundary condition $u_n = u^p$ on $\partial Ω × [0,T)$. Under certain blowup rate assumption (which was established recently under some assumptions on the initial data), we prove that the blowup limit is nontrivial at the blowup point. We also establish that the single-point-blowup occurs in two space dimensional radially symmetric domain with non-radially symmetric initial data with only one "hill" on the boundary