On a decay property of solutions to the Haraux–Weissler equation

Abstract

We give a sufficient condition that non-radial H 1 -solutions to the Haraux-Weissler equation should belong to the weighted Sobolev space H 1 (R n ), where is the weight function exp(|x| 2 /4). Our result provides, in some sense, a connection between the solutions obtained by ODE method and those by variational approach in the space H 1 (R n ).

Three-Dimensional Stability of Burgers Vortices

“…Our main result is: Let m ∈ [0, ∞), λ ∈ C, h ∈ L 2 (∞) N , and assume that F is a continuous function satisfying|F (x, p, Q)| ≤ A(x)|p| + B(x)|Q| , for all (x, p, Q) ∈ R n × C N × C nN , (6.4)where A and B are bounded, nonnegative functions such thatlim then any solution f ∈ H 1 (m) N of (6.3) satisfies f ∈ H 1 (∞) N .Proof. The proof is a simple modification of[16, Proposition 12], which in turn is inspired by a recent work of Fukuizumi and Ozawa[6] where decay estimates are obtained for solutions of the Haraux-Weissler equation. For k ≥ 1, ǫ > 0, and θ ∈ [0, m], we define the weight functionsξ k,ǫ (x) = e (1−ǫ)k|x| 2 4k+|x| 2 , ζ θ (x) = (1 + |x| 2 ) θ , x ∈ R n .…”

On Gaussian decay estimates of solutions to some linear elliptic equations and their applications

“…Proof of Proposition 1.1. We prove Proposition 1.1 by using the idea of [8]. Let Then we can take φ = ζ l,θ ρ k,ϵ u in (1.3) and we have Here the constant C > 0 does not depend on l, k, and ϵ.…”

“…Especially, the smallness assumption on u stated in [8] is shown to be always satisfied. Remark 1.9.…”

“…Fukuizumi-Ozawa [8] discussed complex valued solutions to (1.32) and derived a condition for solutions in W 1,2 (R n ) to belong to As a consequence of Proposition 1.1 and Lemma 1.2, we can improve (1.34) as follows.…”