We consider the semilinear heat equation in large dimension
on a smooth bounded domain
with Dirichlet boundary condition. In the supercritical range
, we prove the existence
of a countable family
of
solutions blowing up at time
with type II blow up:
with blow-up speed
.
The blow up is caused by the concentration of a profile
which
is a radially symmetric stationary solution:
at some point
.
The result generalizes previous works on the existence of type II blow-up solutions,
which only existed in the radial setting. The present proof uses robust nonlinear
analysis tools instead, based on energy methods and modulation techniques. This is
the first nonradial construction of a solution blowing up by concentration of a
stationary state in the supercritical regime, and it provides a general strategy to
prove similar results for dispersive equations or parabolic systems and to extend it to
multiple blow ups.