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

TL;DR

This paper constructs solutions to the critical nonlinear heat equation that blow up infinitely often near specified points, using Green's functions and bubbling analysis in high dimensions.

Contribution

It establishes the existence of infinite-time bubbling solutions near prescribed points in a bounded domain for the critical heat equation.

Findings

Solutions blow up infinitely often near chosen points.

Blow-up rate of parameters is polynomial in time.

Method applies to dimensions n ≥ 5.

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 $n\geq 5$.