# Green's function and infinite-time bubbling in the critical nonlinear heat equation

### Carmen Cortázar

Pontificia Universidad Católica de Chile, Santiago, Chile### Manuel del Pino

University of Bath, UK and Universidad de Chile, Santiago, Chile### Monica Musso

University of Bath, UK and Pontificia Universidad Católica de Chile, Santiago, Chile

## Abstract

Let $Ω$ be a smooth bounded domain in $R_{n}$, $n≥5$. We consider the classical semilinear heat equation at the critical Sobolev exponent

Let $G(x,y)$ be the Dirichlet Green's function of $−Δ$ in $Ω$ and $H(x,y)$ its regular part. Let $q_{j}∈Ω$, $j=1,…,k$, be points such that the matrix

is positive definite. For any $k≥1$ such points indeed exist. We prove the existence of a positive smooth solution $u(x,t)$ which blows-up by bubbling in infinite time near those points. More precisely, for large time $t$, $u$ takes the approximate form

Here $ξ_{j}(t)→q_{j}$ and $0<μ_{j}(t)→0$, as $t→∞$. We find that $μ_{j}(t)∼t_{−n−41}$ as $t→+∞$.

## Cite this article

Carmen Cortázar, Manuel del Pino, Monica Musso, Green's function and infinite-time bubbling in the critical nonlinear heat equation. J. Eur. Math. Soc. 22 (2020), no. 1, pp. 283–344

DOI 10.4171/JEMS/922