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Speed-up of reaction-diffusion fronts by a line of fast diffusion

Henri Berestycki, Anne-Charline Coulon, Jean-Michel Roquejoffre, Luca Rossi (2013/2014)

Séminaire Laurent Schwartz — EDP et applications

In these notes, we discuss a new model, proposed by H. Berestycki, J.-M. Roquejoffre and L. Rossi, to describe biological invasions in the plane when a strong diffusion takes place on a line. This model seems relevant to account for the effects of roads on the spreading of invasive species. In what follows, the diffusion on the line will either be modelled by the Laplacian operator, or the fractional Laplacian of order less than 1. Of interest to us is the asymptotic speed of spreading in the direction...

Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials

Jaeyoung Byeon, Zhi-Qiang Wang (2006)

Journal of the European Mathematical Society

For singularly perturbed Schrödinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeroes of the potentials. The results generalize some recent work of Ambrosetti–Malchiodi–Ni [3] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in [3].

Spread Pattern Formation of H5N1-Avian Influenza and its Implications for Control Strategies

R. Liu, V. R. S. K. Duvvuri, J. Wu (2008)

Mathematical Modelling of Natural Phenomena

Mechanisms contributing to the spread of avian influenza seem to be well identified, but how their interplay led to the current worldwide spread pattern of H5N1 influenza is still unknown due to the lack of effective global surveillance and relevant data. Here we develop some deterministic models based on the transmission cycle and modes of H5N1 and focusing on the interaction among poultry, wild birds and environment. Some of the model parameters are obtained from existing literatures, and others...

Spreading and vanishing in nonlinear diffusion problems with free boundaries

Yihong Du, Bendong Lou (2015)

Journal of the European Mathematical Society

We study nonlinear diffusion problems of the form u t = u x x + f ( u ) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special f ( u ) of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any f ( u ) which is C 1 and satisfies f ( 0 ) = 0 , we show that the omega limit set ω ( u ) of every bounded positive solution is determined by a stationary solution....

Square functions associated to Schrödinger operators

I. Abu-Falahah, P. R. Stinga, J. L. Torrea (2011)

Studia Mathematica

We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, L p and BMO of classical ℒ-square functions.

Square functions of Calderón type and applications.

Steve Hofmann, John L. Lewis (2001)

Revista Matemática Iberoamericana

We establish L2 and Lp bounds for a class of square functions which arises in the study of singular integrals and boundary value problems in non-smooth domains. As an application, we present a simplified treatment of a class of parabolic smoothing operators which includes the caloric single layer potential on the boundary of certain minimally smooth, non-cylindrical domains.

Square roots of perturbed subelliptic operators on Lie groups

Lashi Bandara, A. F. M. ter Elst, Alan McIntosh (2013)

Studia Mathematica

We solve the Kato square root problem for bounded measurable perturbations of subelliptic operators on connected Lie groups. The subelliptic operators are divergence form operators with complex bounded coefficients, which may have lower order terms. In this general setting we deduce inhomogeneous estimates. In case the group is nilpotent and the subelliptic operator is pure second order, we prove stronger homogeneous estimates. Furthermore, we prove Lipschitz stability of the estimates under small...

Stabilisation d’une poutre. Étude du taux optimal de décroissance de l’énergie élastique

Francis Conrad, Fatima-Zahra Saouri (2002)

ESAIM: Control, Optimisation and Calculus of Variations

On se propose d’étudier la stabilité d’une poutre flexible homogène, encastrée à une extrémité. À l’autre extrémité est attachée une masse ponctuelle où on applique un moment proportionnel à la vitesse de déplacement angulaire. On montre par une analyse spectrale que le taux optimal de décroissance de l’énergie est déterminé par l’abscisse spectrale du générateur infinitésimal du semi-groupe associé au problème.

Stabilisation d'une poutre. Étude du taux optimal de décroissance de l'énergie élastique

Francis Conrad, Fatima-Zahra Saouri (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the stability of a flexible beam clamped at one end. A mass is attached at the other end, where a control moment is applied. The boundary control is proportional to the angular velocity at the end. By spectral analysis, we prove that the optimal decay rate of the energy is given by the spectrum of the generator of the semigroup associated to the system.

Stabilisation exponentielle d’une équation des poutres d’Euler-Bernoulli à coefficients variables

My Driss Aouragh, Naji Yebari (2009)

Annales mathématiques Blaise Pascal

Dans ce travail, nous étudions la propriété de base de Riesz et la stabilisation exponentielle pour une équation des poutres d’Euler-Bernoulli à coefficients variables sous un contrôle frontière linéaire dépendant de la position (resp. l’angle de rotation), de la vitesse et de la vitesse de rotation dans le contrôle force (resp. moment). Nous montrons qu’il existe une suite de fonctions propres généralisées qui forme une base de Riesz de l’espace d’énergie considéré, et qu’il y a stabilité exponentielle...

Stabilisation polynomiale et analytique de l’équation des ondes sur un rectangle

Ammar Moulahi, Salsabil Nouira (2010)

Annales mathématiques Blaise Pascal

On considère l’équation des ondes sur un rectangle avec un feedback de type Dirichlet. On se place dans le cas où la condition de contrôle géométrique n’est pas satisfaite (BLR Condition), ce qui implique qu’on n’a pas stabilité exponentielle dans l’espace d’énérgie. On prouve qu’on peut trouver un sous espace de l’espace d’énergie tel qu’on a stabilité exponentielle. De plus, on montre un résultat de décroissance polynomiale pour toute donnée initiale régulière.

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