Séminaires : IMJ-PRG PhD preprint seminar

Séminaire permettant aux doctorants et étudiants de Master de l'IMJ-PRG (de toutes les équipes) de se familiariser avec les résultats apparaissant dans les récentes prépublications de leurs collègues. L'objectif des présentations est d'expliquer de manière accessible, à l'aide d'exemples et/ou de cas particuliers plus simples, les résultats principaux d'une prépublication donnée. Le séminaire est à distinguer du Séminaire des thésards par sa volonté de se concentrer sur les résultats de recherche propres aux étudiants de l'IMJ-PRG (et non pas aux résultats généraux et définitions abstraites du domaine de recherche sous-jacent).

Aura typiquement lieu les mercredis de 17h00 à 18h30-19h00.

During the first part of this talk, we consider the semilinear heat equation with a pure power nonlinearity, which is a less complicated version of some famous PDEs, where we can observe the phenomenon of blow-up that occurs in some solutions for this equation. We will recall some famous results related to a class of these solutions, called solutions of type I, such as the existence and the notion of blow-up profile near the singularity.

In the second part, we consider a nonlinear heat equation with a nonlinear term depending on the unknown and its gradient, which is a more general case than the previous equation. In the earlier literature, we have the existence of a stable blow-up solution with an explicit profile. By pursuing a limited development, we are trapped in scales of order $\frac{1}{\log |T − t|^\nu}$ where $T$ is the blow-up time. In this work, we are able to provide a sharper description up to $|T −t|^\mu$. The price to pay is to replace the explicit profile by a non explicit profile, which is in a fact a well-prepared blow-up solution of the semilinear heat equation with a pure power nonlinearity.