Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term

Abstract:

We consider a nonlinear heat equation with a double source: $|u|^{p-1}u$ and $|\nabla u|^q$.

This equation has a double interest: in ecology, it was used by Souplet (1996) as a population dynamics model; in mathematics, it was introduced by Chipot and Weissler (1989) as an intermediate equation between the semilinear heat equation and the Hamilton-Jacobi equation. Further interest in this equation comes from its lack of variational structure.

In this paper, we intend to see whether the standard blow-up dynamics known for the standard semilinear heat equation (with $|u|^{p-1}u$ as the only source) can be modified by the addition of the second source ($|\nabla u|^q$).

Here arises a nice critical phenomenon at blow-up:

- when $q<2p/(p+1)$, the second source is subcritical in size with respect to the first, and we recover the classicial blow-up profile known for the standard semilinear case;

- when $q=2p/(p+1)$, both terms have the same size, and only partial blow-up descriptions are available.

In this paper, we focus on this case, and start from scratch to:

- first, formally justify the occurrence of a new blow-up profile, which is different from the standard semilinear case;

- second, to rigorously justify the existence of a solution obeying that profile, thanks to the constructive method introduced by Bricmont and Kupiainen together with Merle and Zaag. Note that our method yields the stability of the constructed solution.

Moreover, our method is far from being a straightforward adaptation of earlier literature and should be considered as a source of novel ideas whose application goes beyond the particular equation we are considering, as we explain in the introduction.