Analysis Seminar

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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.

Synopsis:

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.