Asymptotic behavior of blowup solutions for Henon type parabolic equations with exponential nonlinearity

Abstract

This article concerns the blow up behavior for the Henon type parabolic equation withexponential 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 behaviorsin 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.