The asymptotic stability of steady solutions of reaction-convection-diffusion equations.
In this paper we investigate the singular limiting behavior of slow invariant manifolds for a system of singularly perturbed evolution equations in Banach spaces. The aim is to prove the C stability of invariant manifolds with respect to small values of the singular parameter.
We consider the homogeneous time-dependent Oseen system in the whole space . The initial data is assumed to behave as , and its gradient as , when tends to infinity, where is a fixed positive number. Then we show that the velocity decays according to the equation , and its spatial gradient decreases with the rate , for tending to infinity, uniformly with respect to the time variable . Since these decay rates are optimal even in the stationary case, they should also be the best possible...
Hildebrand et al. (1999) proposed an adsorbate-induced phase transition model. For this model, Takei et al. (2005) found several stationary and evolutionary patterns by numerical simulations. Due to bistability of the system, there appears a phase separation phenomenon and an interface separating these phases. In this paper, we introduce the equation describing the motion of two interfaces in and discuss an application. Moreover, we prove the existence of the traveling front solution which approximates...
We study the relation between the porous medium equation ut = Δ(um), m > 1, and the eikonal equation vt = |Dv|2. Under quite general assumtions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as m↓1 to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions for the case of the Dirichlet boundary value problem.
We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species...