Anisotropic Type II Blow-up for the Semi-linear Heat Equation
Speaker: Charles COLLOT, NYU AD / Courant Institute
Location: Warren Weaver Hall 1302
Date: Thursday, February 8, 2018, 11 a.m.
This talk is about solutions to the semi-linear heat equation, where nonlinear growth competes with linear diffusion. This is a model equation to study finite time singularity formation, happening here when the solution becomes unbounded approaching some finite time. Several scenarios have been revealed so far and will be recalled, sharing similarities with the singularities of various other equations. In particular, type II blow-up is related to the collapse of a stationary state by scale instability. If u(t,x) is such a solution to the d-dimensional problem, setting y a new d'-dimensional variable, then v(t,x,y)=u(t,x) is also a type II blow-up solution to the d+d'-dimensional problem. This is the natural lift to higher dimension of a lower dimensional blow-up. In d+d' dimension, a natural question is that of the existence of solutions becoming unbounded on some set of codimension d such that the above picture holds locally up to a change of variables. The talk is about a first step in that direction: the construction of type II blow-up solutions exploding by concentration of a lower-dimensional stationary state along a subspace with scaling parameters depending on the transversal variables, producing an anisotropic blow-up at a point. This is joint work with F. Merle and P. Raphael.