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

Abstract

We consider the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u-|u|^{q-1}u$ in $\mathbb{R}^n\times(0,T)$, where $n=5$, $p=\frac{n+2}{n-2}$ and $q\in(0,1)$. By the presence of $-|u|^{q-1}u$, this equation has a finite time extinction property. We show the existence of a new type of blowup solutions by using this property. In fact, we obtain such blowup solutions by connecting a specific blowup solution of $u_t=\Delta u+|u|^{p-1}u$ and a specific solution of $u_t=\Delta u-|u|^{q-1}u$, and by adding correction terms.