Asymptotic blow-up behavior for the semilinear heat equation with super-exponential nonlinearities

We consider the semilinear heat equation u t − Δ u = f ( u ) in Ω = B R ( 0 ) ⊂ R n with super-exponential nonlinearities f ( u ) = exp ⁡ ( u p ) u q ( p > 1 , q ∈ { 0 } ∪ [ 1 , ∞ ) ), nonnegative bounded radially symmetric initial data and 0-Dirichlet boundary condition. In this paper, we show the asymptotic blow-up behavior for nonnegative, radial type I blow-up solution. More precisely, we prove that if n ≤ 2 , then such blow-up solution satisfies lim t → T ⁡ T − t F ( u ( y T − t , t ) ) = 1 , where F ( u ) = ∫ u ∞ d s f ( s ) . We note that this result corresponds to the one which is proved by Liu in 1989 for the case of f ( u ) = e u , which has the scale invariance property unlike our super-exponential case. To prove the main result, we see the equation as a perturbation of the equation with f ( u ) = e u through a transformation introduced by Fujishima and Ioku in 2018 and estimate the additional term which appears after the transformation.