Asymptotic Behavior for a Nonlocal Diffusion Equation in Domains with Holes

Tuesday, December 4, 2012 - 3:15pm - 4:05pm
Keller 3-180
Noemi Wolanski (University of Buenos Aires)
We will present results on the asymptotic behavior of solutions to a
non-local diffusion equation, u_t=J*u-u:=Lu, in an exterior domain,
which excludes one or several holes, and with zero
Dirichlet data on its complement. When the space
dimension is three or more this behavior is given by a multiple of
the fundamental solution of the heat equation away from the holes.
On the other hand, if the solution is scaled according to its decay
factor, close to the holes it behaves like a function that is
L-harmonic, Lu=0, in the exterior domain and vanishes in its complement.
The height of such a function at infinity is determined through a matching
procedure with the multiple of the fundamental solution of the heat equation
representing the outer behavior. The inner and the outer behavior
can be presented in a unified way through a suitable global

The study involves a thorough understanding of the stationary solutions of the
Dirichlet problem in the exterior domain and a conservation law for the
evolution problem that gives the nontrivial final mass.

If time allows, we will comment on the differences in the case of 1 dimension where
the local decay factor differs from the global one making the study more involved.

This is joint work with C. Cortazar and M. Elgueta from PUC-Chile and F. Quiros from UAM, Spain.
MSC Code: