Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $p$-Laplacian diffusion

Amal Attouchi (1) , Philippe Souplet (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

We study the initial-boundary value problem for the Hamilton-Jacobi equation with nonlinear diffusion $u_t=\Delta_p u+|\nabla u|^q$ in a two-dimensional domain for $q>p>2$. It is known that the spatial derivative of solutions may become unbounded in finite time while the solutions themselves remain bounded. We show that, for suitably localized and monotone initial~data, the gradient blow-up occurs at a single point of the boundary. Such a result was known up to now only in the case of linear diffusion ($p=2$). The analysis in the case $p>2$ is considerably more delicate.

Mots clés: Single point gradient blow-up; Hamilton-Jacobi; $p$-Laplacian evolution equation; singularities; non linear diffusion operator

Domaines: Equations aux dérivées partielles [math.AP]

Amal Attouchi, Philippe Souplet. Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $p$-Laplacian diffusion. 2014. ⟨hal-00981167⟩