25 research outputs found
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
Stable subharmonic solutions and asymptotic behavior in reaction-diffusion equations
Time-periodic reaction-diffusion equations can be discussed in the context of discrete-time strongly monotone dynamical systems. It follows from the general theory that typical trajectories approach stable periodic solutions. Among these periodic solutions, there are some that have the same period as the equation, but, possibly, there might be others with larger minimal periods (these are called subharmonic solutions). The problem of existence of stable subharmonic solutions is therefore of fundamental importance in the study of the behavior of solutions. We address this problem for two classes of reaction diffusion equations under Neumann boundary conditions. Namely, we consider spatially inhomogeneous equations, which can have stable subharmonic solutions on any domain, and spatially homogeneous equations, which can have such solutions on some (necessarily non-convex) domains
On the Nature of the Instability of Radial Power Equilibria of a Semilinear Parabolic Equation
The semilinear problem (Formula presented) has positive equilibria of the form u(x, t) = C |x|−a for many values of (N, p). Of concern is getting more information on exactly how stable or unstable these solutions are. When m = 1, the results have a qualitatively different nature for the two cases N ≤ 10 and N ≥ 11