Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 42, pp. 1-19.

Title: Asymptotic behavior of blowup solutions for Henon type parabolic equations with
       exponential nonlinearity
       
Authors: Caihong Chang (Xi'an Jiaotong Univ., Xi'an, China)
         Zhengce Zhang (Xi'an Jiaotong Univ., Xi'an, China)

Abstract:     
This article concerns the blow up behavior for the Henon type parabolic equation with
exponential nonlinearity,
$$
u_t=\Delta u+|x|^{\sigma}e^u\quad \text{in } B_R\times \mathbb{R}_+,
$$
where $\sigma\geq 0$ and $B_R=\{x\in\mathbb{R}^N: |x|<R\}$.
We consider all cases in which blowup of solutions occurs, i.e. $N\geq 10+4\sigma$.
Grow up rates are established by a certain matching of different asymptotic behaviors
in the inner region (near the singularity) and the outer region (close to the boundary).
For the cases $N>10+4\sigma$ and $N=10+4\sigma$, the asymptotic expansions of stationary
solutions have different forms, so two cases are discussed separately. Moreover, different
inner region widths in two cases are also obtained.

Submitted May 3, 2021. Published June 28, 2022.
Math Subject Classifications: 35A01, 35B40, 35B44, 35K20.
Key Words: Matched expansion; weighted term; stabilization; grow up rate; degeneracy.