Transition waves for lattice Fisher-KPP equations with time and space dependence

This paper is concerned with the existence results for generalized transition waves of space periodic and time heterogeneous lattice Fisher-KPP equations. By constructing appropriate subsolutions and supersolutions, we show that there is a critical wave speed such that a transition wave solution exists as soon as the least mean of wave speed is above this critical speed. Moreover, the critical speed we construct is proved to be minimal in some particular cases, such as space-time periodic or space independent.

A FUJITA-TYPE RESULT FOR A SEMILINEAR EQUATION IN HYPERBOLIC SPACE

In this paper, we study the positive solutions for a semilinear equation in hyperbolic space. Using the heat semigroup and by constructing subsolutions and supersolutions, a Fujita-type result is established.

Interaction of Traveling Curved Fronts in Bistable Reaction-Diffusion Equations in R2

We consider the interaction of traveling curved fronts in bistable reaction-diffusion equations in two-dimensional spaces. We first characterize the growth of the traveling curved fronts at infinity; then by constructing appropriate subsolutions and supersolutions, we prove that the solution of the Cauchy problem converges to a pair of diverging traveling curved fronts in R2 under appropriate initial conditions.

The Ginzburg–Landau equations in a semi-infinite superconducting film in the large κ limit

Following our preceding papers [1, 2] concerning semi-infinite superconducting films, we consider new a priori estimates on the exterior magnetic field h=A′(0), when (f, A) is a solution of the corresponding Ginzburg–Landau system. The main new results concern the limit as κ→∞, but we prove also the existence of a finite superheating field. We also discuss recent results [3] concerning the superheating field in the large κ limit, and show how to relate these formal solutions to suitable subsolutions and supersolutions giving the existence of a solution for h<1/√2 and κ large enough. We also analyse the same problem by variational techniques and get the existence of a locally stable solution for h<1/√2 and any κ>0.