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

Attouchi, Amal; Souplet, Philippe

doi:10.1090/tran/6684

Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $p$-Laplacian diffusion
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Trans. Amer. Math. Soc. 369 (2017), 935-974 Request permission

Abstract:

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.

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