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

Let Ω\Omega be a smooth bounded domain in Rn\mathbb R^n, n5n\ge 5. We consider the classical semilinear heat equation at the critical Sobolev exponent

ut=Δu+un+2n2inΩ×(0,),u=0onΩ×(0,).u_t = \Delta u + u^{\frac{n+2}{n-2}} \quad \mathrm {in} \: \Omega\times (0,\infty), \quad u =0 \quad \mathrm {on} \: \partial\Omega\times (0,\infty).

Let G(x,y)G(x,y) be the Dirichlet Green's function of Δ-\Delta in Ω\Omega and H(x,y)H(x,y) its regular part. Let qjΩq_j\in \Omega, j=1,,kj=1,\ldots,k, be points such that the matrix

[H(q1,q1)G(q1,q2)G(q1,qk)G(q1,q2)H(q2,q2)G(q2,q3)G(q3,qk)G(q1,qk)G(qk1,qk)H(qk,qk)]\left [ \begin{matrix} H(q_1, q_1) & -G(q_1,q_2) &\cdots & -G(q_1, q_k) \\ -G(q_1,q_2) & H(q_2,q_2) & -G(q_2,q_3) \cdots & -G(q_3,q_k) \\ \vdots & & \ddots& \vdots \\ -G(q_1,q_k) &\cdots& -G(q_{k-1}, q_k) & H(q_k,q_k) \end{matrix} \right ]

is positive definite. For any k1k\ge 1 such points indeed exist. We prove the existence of a positive smooth solution u(x,t)u(x,t) which blows-up by bubbling in infinite time near those points. More precisely, for large time tt, uu takes the approximate form

u(x,t)j=1kαn(μj(t)μj(t)2+xξj(t)2)n22.u(x,t) \approx \sum_{j=1}^k \alpha_n \left ( \frac { \mu_j(t)} { \mu_j(t)^2 + |x-\xi_j(t)|^2 } \right )^{\frac {n-2}2}.

Here ξj(t)qj\xi_j(t) \to q_j and 0<μj(t)00<\mu_j(t) \to 0, as tt \to \infty. We find that μj(t)t1n4\mu_j(t) \sim t^{-\frac 1{n-4}} as t+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