A Remark on the Large Time Behavior of Solutions of Viscous Hamilton-jacobi Equations

are independent of x ∈ R . One of the main results of [5] is the following. Theorem A. Assume 0 < q < 2 and u0 ∈ Cb(R ). Then ω = ω. It was known that Theorem A fails for the linear heat equation and, moreover, Gilding observed that it fails for q = 2. The aim of this short note is to show that the assumption q < 2 in Theorem A is actually necessary. Theorem 1. Assume q ≥ 2. Then there exists u0 ∈ Cb(R ) such that ω < ω. Proof. It is known (see e. g. [5, Proposition H1]) that there exists v0 ∈ C(R )∩ W 1,∞(RN ) such that the solution v of the heat equation { vt −∆v = 0, t > 0, x ∈ R v(0, x) = v0(x), x ∈ R (2)