Affiliations: [a] LMPT, Fédération Denis Poisson, Université François-Rabelais Tours, France. E-mail: [email protected] | [b] IRMAR, INSA Rennes, France | [c] Dipartimento di Matematica, Università di Padova, Italy. E-mail: [email protected] | [d] Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam. E-mail: [email protected]
Abstract: We study the large time behavior of solutions of first-order convex Hamilton–Jacobi Equations of Eikonal type ut+H(x,Du)=l(x), set in the whole space RN×[0,∞). We assume that l is bounded from below but may have arbitrary growth and therefore the solutions may also have arbitrary growth. A complete study of the structure of solutions of the ergodic problem H(x,Dv)=l(x)+c is provided: contrarily to the periodic setting, the ergodic constant is not anymore unique, leading to different large time behavior for the solutions. We establish the ergodic behavior of the solutions of the Cauchy problem (i) when starting with a bounded from below initial condition and (ii) for some particular unbounded from below initial condition, two cases for which we have different ergodic constants which play a role. When the solution is not bounded from below, an example showing that the convergence may fail in general is provided.
