On a  parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary

Daniele  Castorina; Annalisa  Cesaroni; Luca Rossi

doi:10.3934/cpaa.2016.15.1251

Article Contents

2016, Volume 15, Issue 4: 1251-1263. Doi: 10.3934/cpaa.2016.15.1251

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On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary

1.
Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova
2.
Dipartimento di Scienze Statistiche, Università di Padova, Via Cesare Battisti 141, 35121 Padova
3.
Università degli Studi di Padova, Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63 - 35121 Padova

Received: August 31, 2015

Revised: December 31, 2015

Published: June 2016

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Abstract

We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in [1]. Our main assumption is an appropriate degeneracy condition on the operator at the boundary. This condition is related to the characteristic boundary points for linear operators as well as to the irrelevant points for the generalized Dirichlet problem, and implies in particular that no boundary datum has to be imposed. We prove that there exists a constant $c$ such that the solutions of the evolutive problem converge uniformly, in the reference frame moving with constant velocity $c$, to a unique steady state solving a suitable ergodic problem.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35B40, 35B45, 35K65.

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