Existence of blowup solutions to the semilinear heat equation with double power nonlinearity

We are concerned with the semilinear heat equation \(u_t=\Delta u+|u|^{p-1}u-|u|^{q-1}u\) $u_t=\mathrm{\Delta }u+{|u|}^{p-1}u-{|u|}^{q-1}u$ in \({\mathbb {R}}^n\times (0,T)\) ${\mathbb{R}}^n\times (0,T)$ , where \(n=5\) $n=5$ , \(p=\frac{n+2}{n-2}\) $p=\frac{n+2}{n-2}$ , and \(q\in (0,1)\) $q\in (0,1)$ . A goal of this paper is to show the existence of a new type of blowup solutions for this equation. This blowup solution is obtained by connecting a specific blowup solution of \(u_t=\Delta u+|u|^{p-1}u\) $u_t=\mathrm{\Delta }u+{|u|}^{p-1}u$ and a specific extinct solution of \(u_t=\Delta u-|u|^{q-1}u\) $u_t=\mathrm{\Delta }u-{|u|}^{q-1}u$ .