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

Let $Ω$ be a smooth bounded domain in $\R^n$, $n\ge 5$. 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 $-Δ$ in $Ω$ and $H(x,y)$ its regular part. Let $q_j\in Ω$, $j=1,\ldots,k$, be points such that the matrix $\left [ \begin{matrix} H(q_1, q_1) & -G(q_1,q_2) &\cdots & -G(q_1, q_k) -G(q_1,q_2) & H(q_2,q_2) & -G(q_2,q_3) \cdots & -G(q_3,q_k) \vdots & & \ddots& \vdots -G(q_1,q_k) &\cdots& -G(q_{k-1}, q_k) & H(q_k,q_k) \end{matrix} \right ]$ is positive definite. For any $k\ge 1$ 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 $u(x,t) \approx \sum_{j=1}^k α_n \left ( \frac { μ_j(t)} { μ_j(t)^2 + |x-ξ_j(t)|^2 } \right )^{\frac {n-2}2} .$ 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 $t\to +\infty$, when $n\geq 5$.

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