On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions

In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-Neumann problems by using two fairly different methods : the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the "weak KAM approach" which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry-Mather sets

Localized non-diffusive asymptotic patterns for nonlinear parabolic equations with gradient absorption

International audienceWe study the large-time behaviour of the solutions u of the evolution equation involving nonlinear diffusion and gradient absorption $\partial_t u − \Delta_p u + |\nabla u|^q = 0$. We consider the problem posed for $x\in\mathbb{R}^N$ and $t>0$ with non-negative and compactly supported initial data. We take the exponent $p > 2$ which corresponds to slow $p$-Laplacian diffusion, and the exponent $q$ in the superlinear range $1<q<p−1$. In this range the influence of the Hamilton-Jacobi term $|\nabla u|^q$ is determinant, and gives rise to the phenomenon of localization. The large time behaviour is described in terms of a suitable self-similar solution that solves a Hamilton-Jacobi equation. The shape of the corresponding spatial pattern is rather conical instead of bell-shaped or parabolic