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

TL;DR

This paper proves the existence of new blowup solutions for a semilinear heat equation with double power nonlinearity, highlighting the interplay between blowup and extinction phenomena in such equations.

Contribution

It introduces a novel method to construct blowup solutions by connecting solutions of related equations with different nonlinearities.

Findings

Existence of blowup solutions with finite time extinction property

Construction method linking solutions of different nonlinear equations

Identification of new blowup behavior in semilinear heat equations

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.