A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians

Abstract

We investigate large-time asymptotics for viscous Hamilton-Jacobi equations with possibly degenerate diffusion terms. We establish new results on the convergence, which are the first general ones concerning equations which are neither uniformly parabolic nor first order. Our method is based on the nonlinear adjoint method and the derivation of new estimates on long time averaging effects. It also extends to the case of weakly coupled systems.

Large-time behavior for obstacle problems for degenerate viscous Hamilton–Jacobi equations

“…There are cases where we do not see the obstacles at all because the solution u of (C) always stays below ψ for t large enough. The behavior of u(·, t) as t → ∞ then is basically the same as in the usual case in [9]. There are however cases where we have to take into account the obstacle ψ and in general u(·, t) touches ψ at some parts for all time t large enough.…”

“…It was established also in [9] that c H is unique, but v is not unique in general even up to some additive constants, which makes the convergence analysis very delicate. We were able to obtain (1.2) by using the stability of the viscosity solutions together with the deep fact that…”

“…It is clear that these methods are not applicable to the general degenerate viscous case (1.1) until now because of the lack of both the finite speed of propagation and the strong comparison principle. Very recently, Cagnetti, Gomes, and the authors in [9] introduced a new and unified approach to establish large time asymptotics for the general case (1.1), where A could be degenerate. Under (H1)-(H3), we proved that…”

“…There are however cases where we have to take into account the obstacle ψ and in general u(·, t) touches ψ at some parts for all time t large enough. We have to provide a different analysis for this case, which is based on the general method in [9].…”

“…Our analysis to prove Theorem 1.3 follows the method which was introduced in [9]. We rescale (C) in an appropriate way, and introduce an approximation to the rescaled problem by adding a viscosity term and a penalized term corresponding to the obstacle.…”