9 research outputs found
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
Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions
A Probabilistic Approach to Large Time Behaviour of Viscosity Solutions of Parabolic Equations with Neumann Boundary Conditions
Large-time behavior for obstacle problems for degenerate viscous Hamilton–Jacobi equations
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 . We consider the problem posed for and with non-negative and compactly supported initial data. We take the exponent which corresponds to slow -Laplacian diffusion, and the exponent in the superlinear range . In this range the influence of the Hamilton-Jacobi term 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