Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term

Slim Tayachi (1) , Hatem Zaag (2)

1 Département de mathématiques [Tunis]
2 LAGA - Laboratoire Analyse, Géométrie et Applications

We consider the semilinear heat equation, to which we add a nonlinear gradient term, with a critical power. We construct a solution which blows up in finite time. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. Thanks to the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and point, we also show the stability of the constructed solution with respect to initial data. This note presents the results and the main arguments. For the details, we refer to our paper \cite{TZ15}.

Domaines: Equations aux dérivées partielles [math.AP]; Bio-Informatique, Biologie Systémique [q-bio.QM]

Slim Tayachi, Hatem Zaag. Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term. Actes du Colloque EDP Normandie, 2015. ⟨hal-02325133⟩