## [26.4.2016] Green’s function and infinite-time bubbling in the critical nonlinear heat equation

In this paper of Carmen Cortazar, Manuel del Pino, Monica Musso they studied the profile of solutions which blow-up in in-finite time of the critical nonlinear heat equation

$u_t - \Delta u = u^{\frac{n+2}{n-2}}$

in a bounded domain $\Omega\subset \mathbb R^n$ with $n\geq 5$.

The solutions are shown to have bubbling type of behaviour, that means, there exist $k$ points in $\Omega$ which are the only blow-up points of such a solution.

Moreover, such a solution can be constructed approximately by: let $q_1, q_2, \ldots, q_k$ be the bubble points, then

$u(x,t) \approx \sum\limits_{j=1}^{k}\alpha_n\left(\dfrac{\mu_j(t)}{\mu_j(t)^2 + |x- \xi_j(t)|^2}\right)^{(n-2)/2}$

where $\xi_j(t) \rightarrow q_j$ and $0<\mu_j(t) \rightarrow 0$ as $t\rightarrow +\infty$.