Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in RN

Large-time behavior of unbounded solutions of the viscous Hamilton-Jacobi equation: quadratic and subquadratic cases

“…In [8] it is shown that ergodic large-time behavior occurs assuming minimal hypotheses on f and u 0 , although in this case we are able to prove a very general comparison principle for the parabolic problem, and a similar result was already available for the ergodic problem. Regarding the data, for f it is assumed that there exists a nondecreasing function ϕ : [0, +∞) → [0, +∞) and constants α, c > 0 such that for all r ≥ 0, c −1 r α ≤ ϕ(r), and for all x ∈ R N and r = |x|,…”

“…Proof. The proof is essentially the same as that of Proposition 3 in [8] for the case m > 2. We repeat the main points of the argument for convenience.…”

“…Additionally, a constructive proof of this is provided in Lemma 2 in [8], and it is equally valid for 1 < m ≤ 2. The solution of ( 1)-( 2) can be then obtained as a locally uniform limit of the solutions u R as R → +∞ as follows.…”

“…Remark 2.2. (i) In [8], existence of solutions for all bounded-from-below u 0 ∈ C(R N ) was obtained by approximating u 0 with functions in W 1,∞ loc (R N ) and employing a result analogous to Theorem 2.1. A comparison result that is not available is a crucial tool in this procedure, since it is used to control the "error" between the corresponding solutions in terms of the error in the approximation of the initial data.…”

“…Returning to the question of large-time behavior, the authors studied this problem in the superquadratic case (m > 2) in [8], in collaboration with G. Barles.…”