(From the authors’ abstract:) This paper considers the semilinear parabolic equation
\(u_ t=\Delta u+f(u)\) in \(\mathbb{R}^ n\times(0,\infty)\), where \(f(u)=e^ u\) or \(f(u)=u^ p\), \(p>1\). For any initial data that is a positive, radially decreasing lower solution, and that causes the corresponding solution \(u(x,t)\) to blow up at \((0,T)\in\mathbb{R}^ n\times(0,\infty)\), the authors prove by using techniques from center manifold theory that the final time blowup profiles satisfy \[ \begin{alignedat}{2} u(x,T) &=-2\ln| x|+\ln|\ln| x||+\ln 8+o(1)\quad&&\text{for}\quad f(u)=e^ u,\\ u(x,T) &=(8\beta^ 2 p|\ln| x||/| x|^ 2)^ \beta (1+o(1))\quad&&\text{for}\quad f(u)=u^ p\text{ as } | x|\to 0.\end{alignedat} \] {}.