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

Abstract

Let $\Omega$ be a smooth bounded domain in $\R^n$, $n\ge 5$. We consider the semilinear heat equation at the critical Sobolev exponent $$ u_t = \Delta u + u^{\frac{n+2}{n-2}} \inn \Omega\times (0,\infty), \quad u =0 \onn \pp\Omega\times (0,\infty). $$ Let $G(x,y)$ be the Dirichlet Green's function of $-\Delta$ in $\Omega$ and $H(x,y)$ its regular part. Let $q_j\in \Omega$, $j=1,\ldots,k$, 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 $$ u(x,t) \approx \sum_{j=1}^k \alpha_n \left ( \frac { \mu_j(t)} { \mu_j(t)^2 + |x-\xi_j(t)|^2 } \right )^{\frac {n-2}2} . $$ 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 $n\geq 5$.