6 research outputs found
Nonlinear diffusion problems with free boundaries: Convergence, transition speed and zero number arguments,
This paper continues the investigation of Du and Lou (J. European Math Soc,
to appear), where the long-time behavior of positive solutions to a nonlinear
diffusion equation of the form for over a varying
interval was examined. Here and are free
boundaries evolving according to , , and . We answer several intriguing
questions left open in the paper of Du and Lou.First we prove the conjectured
convergence result in the paper of Du and Lou for the general case that is
and . Second, for bistable and combustion types of , we
determine the asymptotic propagation speed of and in the
transition case. More presicely, we show that when the transition case happens,
for bistable type of there exists a uniquely determined such that
, and for
combustion type of , there exists a uniquely determined such that
. Our
approach is based on the zero number arguments of Matano and Angenent, and on
the construction of delicate upper and lower solutions
Localized and Expanding Entire Solutions of Reaction-Diffusion Equations
This paper is concerned with the spatio-temporal dynamics of nonnegative
bounded entire solutions of some reaction-diffusion equations in R N in any
space dimension N. The solutions are assumed to be localized in the past. Under
certain conditions on the reaction term, the solutions are then proved to be
time-independent or heteroclinic connections between different steady states.
Furthermore, either they are localized uniformly in time, or they converge to a
constant steady state and spread at large time. This result is then applied to
some specific bistable-type reactions