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Applied Mathematics Colloquium
Friday, January 30, 11:30 am
Cullimore Lecture Hall II
New
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A variational
approach to front propagation in infinite cylinders
Cyrill Muratov
Department of Mathematical Sciences
New Jersey Institute of Technology
Abstract
In their classical 1937 paper, Kolmogorov, Petrovsky and Piskunov proved
that for a particular class of reaction-diffusion equations on
a line the solution of the initial
value problem with the initial data in the form of a unit step propagates at
long times with constant
velocity equal to that of a certain
special traveling wave solution. This type of a propagation result has since
been established
for a number of general classes of
reaction-diffusion-advection problems in cylinders. In this talk I will show
that actually in the
problems without advection or in
the presence of transverse advection by a potential flow these results do not
rely on the specifics of the
problem. Instead, they are a
consequence of the fact that the considered equation is a gradient flow in an
exponentially weighted
L^2 space generated by a certain functional, when the
dynamics is considered in the reference frame moving with constant velocity
along
the cylinder axis. I will show that
independently of the details of the problem only three propagation scenarios
are possible in the above
context: no propagation, a
"pulled" front, or a "pushed" front. The choice of the
scenario is completely characterized via a minimization
problem..