A diffusive delayed predator-prey model with modified Leslie-Gower and Holling-type II schemes is considered. Local stability for each constant steady state is studied by analyzing the eigenvalues. Some simple and easily verifiable sufficient conditions for global stability are obtained by virtue of the stability of the related FDE and some monotonous iterative sequences. Numerical simulations and reasonable biological explanations are carried out to illustrate the main results and the justification...

We prove the linear and non-linear stability of oscillating Ekman boundary layers for rotating fluids in the so-called ill-prepared case under a spectral hypothesis. Here, we deal with the case where the viscosity and the Rossby number are both equal to $\epsilon$. This study generalizes the study of [23] where a smallness condition was imposed and the study of [26] where the well-prepared case was treated.

— Si presentano alcuni risultati di esistenza e molteplicità di soluzioni positive per l'equazione $\mathrm{\Delta }u+u^{2^{*}-1}=0$ in $H_0^{\left(1,2\right)}\left(\mathrm{\Omega }\right)$, dove $\mathrm{\Omega }$ è un aperto limitato di ${\mathbb{R}}^n$ con $n\ge 3$ e $2^{*}=2n/\left(n\mathrm{—}2\right)$. Si mostra che opportune perturbazioni di $\mathrm{\Omega }$ comportano l'esistenza di soluzioni positive, che convergono a zero quando la capacità delle perturbazioni tende a zero. In particolare, si ottengono risultati di esistenza e molteplicità di soluzioni positive in alcuni aperti limitati e contrattili, non necessariamente simmetrici.

Pour un système parabolique de lois de conservation, nous considérons le problème mixte, dans le domaine $x\>0$. Pour une condition de Dirichlet, le système admet en général des solutions stationnaires $U\left(x\right)$, qui tendent vers une limite en $+\infty$. Ce sont les profils des couches limites, dans l’approximation du second ordre, pour le système hyperbolique du premier ordre sous-jacent. La stabilité de cette couche limite est liée à la stabilité linéaire asymptotique de $U$. On étudie celle-ci au moyen d’une fonction d’Evans,...

Le but de l’exposé est de présenter les résultats obtenus par S. Bianchini et A. Bressan sur le problème de Cauchy pour des perturbations visqueuses ${\partial }_t{u}^{\epsilon }+{\partial }_{x}f\left(u^{\epsilon }\right)=\epsilon {\partial }_{xx}{u}^{\epsilon }$ de systèmes strictement hyperboliques ${\partial }_{t}u+{\partial }_{x}f\left(u\right)=0$ en une dimension d’espace. Ils ont en particulier montré l’existence globale ($t\ge 0$), l’unicité et la stabilité des solutions et justifié la convergence quand $\epsilon$ tend vers zéro pour des données initiales à petite variation totale. Leur analyse montre aussi que les solutions du système hyperbolique ainsi obtenues...

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${}^1$ 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 ${ℝ}^3$. The initial data is assumed to behave as $O\left(\right|x{|}^{-1-ϵ})$, and its gradient as $O\left(\right|x{|}^{-3/2-ϵ})$, when $\left|x\right|$ tends to infinity, where $ϵ$ is a fixed positive number. Then we show that the velocity $u$ decays according to the equation $\left|u(x,t)\right|=O\left(\right|x{|}^{-1})$, and its spatial gradient ${\nabla }_{x}u$ decreases with the rate ${\left|x\right|}^{-3/2}$, for $\left|x\right|$ tending to infinity, uniformly with respect to the time variable $t$. 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 ${ℝ}^2$ 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.