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 classical semilinear heat equation at the critical Sobolev exponent 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) 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 .

Locations

• arXiv (Cornell University) - View - PDF
• Pure (University of Bath) - View - PDF
• Journal of the European Mathematical Society - View