29 research outputs found
Pulsating Front Speed-up and Quenching of Reaction by Fast Advection
We consider reaction-diffusion equations with combustion-type non-linearities
in two dimensions and study speed-up of their pulsating fronts by general
periodic incompressible flows with a cellular structure. We show that the
occurence of front speed-up in the sense ,
with the amplitude of the flow and the (minimal) front speed, only
depends on the geometry of the flow and not on the reaction function. In
particular, front speed-up happens for KPP reactions if and only if it does for
ignition reactions. We also show that the flows which achieve this speed-up are
precisely those which, when scaled properly, are able to quench any ignition
reaction.Comment: 16p
Quenching and Propagation of Combustion Without Ignition Temperature Cutoff
We study a reaction-diffusion equation in the cylinder , with combustion-type reaction term without
ignition temperature cutoff, and in the presence of a periodic flow. We show
that if the reaction function decays as a power of larger than three as
and the initial datum is small, then the flame is extinguished -- the
solution quenches. If, on the other hand, the power of decay is smaller than
three or initial datum is large, then quenching does not happen, and the
burning region spreads linearly in time. This extends results of
Aronson-Weinberger for the no-flow case. We also consider shear flows with
large amplitude and show that if the reaction power-law decay is larger than
three and the flow has only small plateaux (connected domains where it is
constant), then any compactly supported initial datum is quenched when the flow
amplitude is large enough (which is not true if the power is smaller than three
or in the presence of a large plateau). This extends results of
Constantin-Kiselev-Ryzhik for combustion with ignition temperature cutoff. Our
work carries over to the case , when
the critical power is , as well as to certain non-periodic flows
Propagation and blocking in periodically hostile environments
We study the persistence and propagation (or blocking) phenomena for a
species in periodically hostile environments. The problem is described by a
reaction-diffusion equation with zero Dirichlet boundary condition. We first
derive the existence of a minimal nonnegative nontrivial stationary solution
and study the large-time behavior of the solution of the initial boundary value
problem. To the main goal, we then study a sequence of approximated problems in
the whole space with reaction terms which are with very negative growth rates
outside the domain under investigation. Finally, for a given unit vector, by
using the information of the minimal speeds of approximated problems, we
provide a simple geometric condition for the blocking of propagation and we
derive the asymptotic behavior of the approximated pulsating travelling fronts.
Moreover, for the case of constant diffusion matrix, we provide two conditions
for which the limit of approximated minimal speeds is positive
On the loss of continuity for super-critical drift-diffusion equations
We show that there exist solutions of drift-diffusion equations in two
dimensions with divergence-free super-critical drifts, that become
discontinuous in finite time. We consider classical as well as fractional
diffusion. However, in the case of classical diffusion and time-independent
drifts we prove that solutions satisfy a modulus of continuity depending only
on the local norm of the drift, which is a super-critical quantity.Comment: Minor edit