Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball

Kazuhiro Ishige; Yoshitsugu Kabeya

doi:10.2969/jmsj/05930861

Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball

Kazuhiro Ishige, Yoshitsugu Kabeya

Research output: Contribution to journal › Article › peer-review

10 Citations (Scopus)

Abstract

We consider the initial-boundary value problem (p) {∂/∂tu = Δu - V(|x|)u in Ω_L × (0,∞), μu + (1 - μ)∂/∂vu = 0 on ∂Ω_L × (0, ∞), u(·, 0) = ø(·) ∈ L^p(Ω_L), p ≥ 1, where Ω_L = {x ∈^N : |x| > L}, N ≥ 2, L > 0, 0 ≤ μ ≤ 1, v is the outer unit normal vector to ∂Ω_L, 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 ▽_x^ju of the solution u to (P) as t → ∞.

Original language	English
Pages (from-to)	861-898
Number of pages	38
Journal	Journal of the Mathematical Society of Japan
Volume	59
Issue number	3
DOIs	https://doi.org/10.2969/jmsj/05930861
Publication status	Published - 2007 Jul
Externally published	Yes

Keywords

Decay rates estimate
Linear parabolic equation
Radial solutions

ASJC Scopus subject areas

Mathematics(all)

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10.2969/jmsj/05930861

Cite this

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title = "Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball",

abstract = "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) = {\o}(·) ∈ 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 → ∞.",

keywords = "Decay rates estimate, Linear parabolic equation, Radial solutions",

author = "Kazuhiro Ishige and Yoshitsugu Kabeya",

year = "2007",

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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

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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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Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball

Abstract

Keywords

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