Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in RN. Issue 3 (18th November 2020)

• Record Type: Journal Article
• Authors: Barles, Guy; Quaas, Alexander; Rodríguez-Paredes, Andrei
• Is Part Of: Communications in partial differential equations. Volume 46:Issue 3(2021)
• Page Start: 547
• Page End: 572
• Publication Date: 2020-11-18
• Subjects: Ergodic behavior -- Hamilton-Jacobi equations -- large-time behavior -- unbounded solutions -- viscosity solutions -- viscous Hamilton-Jacobi equation; 35B40 -- 35B51 -- 35D40 -- 35K15 -- 35K55; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2020.1846561
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space R N in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are shown in a very general framework, namely when the source term and the initial data are only bounded from below with an arbitrary growth at infinity. Our main result is that these solutions have an ergodic behavior when t → + ∞, i.e., they behave like λ * t + ϕ ( x ) where λ * is the maximal ergodic constant and ϕ is a solution of the associated ergodic problem. The main originality of this result comes from the generality of the data: in particular, the initial data may have a completely different growth at infinity from those of the solution of the ergodic problem.