Sharp non explicit blow-up profile for a nonlinear heat equation with a gradient term


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 these equations. 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

Following that, we will 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.

Please note that the seminar will take place in person in room 144 of Huxley Building.

Click here to get to the Junior Analysis Seminar webpage.

Getting here