To cite this version:Mohammed Abderrahman Ebde, Hatem Zaag. Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term. 2010. hal-00570159Construction and stability of a blow up solution for a nonlinear heat equation with a gradient termMohamed Abderrahman Ebde LAGA Université Paris 13. Hatem Zaag CNRS LAGA Université Paris 13
RésuméWe consider a nonlinear heat equation with a gradient term. We construct a blow-up solution for this equation with a prescribed blow-up profile. For that, we translate the question in selfsimilar variables and reduce the problem to a finite dimensional one. We then solve the finite dimensional problem using index theory. The interpretation of the finite dimensional parameters allows us to derive the stability of the constructed solution with respect to initial data.
Abstract. This article is devoted to the construction of a mathematical model describing the early formation of atherosclerotic lesions. Following the work of El Khatib, Genieys and Volpert [2], we model atherosclerosis as an inflammatory disease. We consider that the inflammatory process starts with the penetration of Low Density Lipoproteins cholesterol in the intima. This phenomenon is related to the local blood flow dynamics. Using a system of reaction-diffusion equations, we first provide a one-dimensional model of lesion growth. Then we perform numerical simulations on a two-dimensional geometry mimicking the carotid artery. We couple the previous mathematical model with blood flow and we provide a model in which the lesion appears in the area of lower shear stress.Résumé. Cet article est consacréà la construction d'un modèle mathématique décrivant la formation de lésions précoces dans l'athérosclérose. Suite au travail de El Khatib, Genieys and Volpert [2], nous modélisons l'athérosclérose comme une maladie inflammatoire. Nous considérons que le procesus inflammatoire débute avec la pénétration de "Low Density Lipoproteins cholesterol" dans l'intima. Ce phénomène est reliéà la dynamique locale de l'écoulement du sang. En utilisant un système d'équations de réaction-diffusion, nous donnons tout d'abord un modèle monodimensionel de croissance de lésion. Puis, dans le cas d'une bifurcation bidimensionnelle, nous couplons le modèle mathématique précédent avec l'écoulement du sang et nous fournissons un modèle dans lequel la lésion apparaît dans la zone de plus faible cisaillement.
This paper is devoted to the analysis of the classical Keller-Segel system over R d , d ≥ 3. We describe as much as possible the dynamics of the system characterized by various criteria, both in the parabolic-elliptic case and in the fully parabolic case. The main results when dealing with the parabolic-elliptic case are: local existence without smallness assumption on the initial density, global existence under an improved smallness condition and comparison of blow-up criteria. A new concentration phenomenon criteria for the fully parabolic case is also given. The analysis is completed by a visualization tool based on the reduction of the parabolic-elliptic system to a finite-dimensional dynamical system of gradient flow type, sharing features similar to the infinitedimensional system.
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