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

Abstract

Let $\Omega$ be a smooth bounded domain in ${\mathbb{R}}^n$, $n\ge 5$. We consider the semilinear heat equation at the critical Sobolev exponent $$ u_t = Δu + u^{\frac{n+2}{n-2}} \inn Ω\times (0,\infty), \quad u =0 \onn \ppΩ\times (0,\infty). $$ Let $G(x,y)$ be the Dirichlet Green's function of $-\Delta$ in $\Omega$ and $H(x,y)$ its regular part. Let $q_j\in \Omega$, $j=1,\dots ,k$, be points such that the matrix $$\left[\begin{array}{ccccccccccccc}H(q_1,q_1)& -G(q_1,q_2)& \cdots & -G(q_1,q_k)-G(q_1,q_2)& H(q_2,q_2)& -G(q_2,q_3)\cdots & -G(q_3,q_k)⋮& & \ddots & ⋮-G(q_1,q_k)& \cdots & -G(q_{k-1},q_k)& H(q_k,q_k)\end{array}\right]$$ is positive definite. For any $k\ge 1$ 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 $$u(x,t)\approx \sum _{j=1}^k{\alpha }_n{\left(\frac{{\mu }_j(t)}{{\mu }_j(t{)}^2+|x-{\xi }_j(t){|}^2}\right)}^{\frac{n-2}{2}}.$$ Here ${\xi }_j(t)\to q_j$ and $0<{\mu }_j(t)\to 0$, as $t\to \infty$. We find that ${\mu }_j(t)\sim t^{-\frac{1}{n-4}}$ as $t\to +\infty$, when $n\ge 5$.