On type I blowup of some nonlinear heat equations with a potential

In this paper, we are concerned with the following initial-boundary value problem$\{\begin{array}{cc}{u}_t=\Delta u+Q\left(\right|x\left|\right)|u{|}^{p-1}u,& x\in B_R,t>0\\ u(x,t)=0,& x\in \partial B_R,t>0\\ u(x,0)=u_0\left(x\right),& x\in B_R,\end{array}$ where $p\ge p_s:=\frac{N+2}{N-2}$, $u_0\in L^{\infty }\left(B_R\right)$, and $Q\left(r\right)\in C^1\left(\right[0,R\left]\right)$, $0<\underset{\_}{C}\le Q\left(r\right)\le \bar{C}<\infty ,Q^{\prime }\left(r\right)\le 0$. We extend the asymptotic behavior results, which is well-known when Q is constant according to Matano-Merle (cf. [25]), for the blow-up solutions. More precisely, we show that when $p_s\le p<p^{⁎}$, the blowup of radial solution to this problem is always of Type I. This result partially generalizes the conclusions in [25] for $Q\equiv 1$. This extension is nontrivial due to the appearance of Q. The quasi-monotonicity formula established by the third author and Cheng in [8] allows us to use an energy method to get a priori estimates on the rescaled solutions. The contraction mapping principle shows the existence of singular stationary solutions to an associated elliptic equation with a potential. In the end, the properties of zero number for solutions lead to the nonexistence of type II singularity for the problem.