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

Carmen Cortázar; Manuel del Pino; Monica Musso

doi:10.4171/jems/922

JournalsjemsVol. 22, No. 1pp. 283–344

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

A subscription is required to access this article.

Abstract

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

u_{t} = Δ u + u^{\frac{n + 2}{n - 2}} in Ω \times (0, \infty), u = 0 on \partial Ω \times (0, \infty) .

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

H (q_{1}, q_{1}) - G (q_{1}, q_{2}) ⋮ - G (q_{1}, q_{k}) - G (q_{1}, q_{2}) H (q_{2}, q_{2}) \dots \dots - G (q_{2}, q_{3}) \dots ⋱ - G (q_{k - 1}, q_{k}) - G (q_{1}, q_{k}) - G (q_{3}, q_{k}) ⋮ H (q_{k}, q_{k})

is positive definite. For any $k \geq 1$ 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

u (x, t) \approx j = 1 \sum k α_{n} (\frac{μ _{j} ( t )}{μ _{j} ( t ) ^{2} + ∣ x - ξ _{j} ( t ) ∣ ^{2}})^{\frac{n - 2}{2}} .

Here $ξ_{j} (t) \to q_{j}$ and $0 < μ_{j} (t) \to 0$ , as $t \to \infty$ . We find that $μ_{j} (t) \sim t^{- \frac{1}{n - 4}}$ as $t \to + \infty$ .

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