TY - JOUR
T1 - Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball
AU - Ishige, Kazuhiro
AU - Kabeya, Yoshitsugu
N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.
PY - 2007/7
Y1 - 2007/7
N2 - We consider the initial-boundary value problem (p) {∂/∂tu = Δu - V(|x|)u in ΩL × (0,∞), μu + (1 - μ)∂/∂vu = 0 on ∂ΩL × (0, ∞), u(·, 0) = ø(·) ∈ Lp(ΩL), p ≥ 1, where ΩL = {x ∈N : |x| > L}, N ≥ 2, L > 0, 0 ≤ μ ≤ 1, v is the outer unit normal vector to ∂ΩL, and V is a nonnegative smooth function such that V(r) = O(r-2) as r → ∞. In this paper, we study the decay rates of the derivatives ▽xju of the solution u to (P) as t → ∞.
AB - We consider the initial-boundary value problem (p) {∂/∂tu = Δu - V(|x|)u in ΩL × (0,∞), μu + (1 - μ)∂/∂vu = 0 on ∂ΩL × (0, ∞), u(·, 0) = ø(·) ∈ Lp(ΩL), p ≥ 1, where ΩL = {x ∈N : |x| > L}, N ≥ 2, L > 0, 0 ≤ μ ≤ 1, v is the outer unit normal vector to ∂ΩL, and V is a nonnegative smooth function such that V(r) = O(r-2) as r → ∞. In this paper, we study the decay rates of the derivatives ▽xju of the solution u to (P) as t → ∞.
KW - Decay rates estimate
KW - Linear parabolic equation
KW - Radial solutions
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U2 - 10.2969/jmsj/05930861
DO - 10.2969/jmsj/05930861
M3 - Article
AN - SCOPUS:34548603568
VL - 59
SP - 861
EP - 898
JO - Journal of the Mathematical Society of Japan
JF - Journal of the Mathematical Society of Japan
SN - 0025-5645
IS - 3
ER -