Existence and smoothing effects of the initial-boundary value problem for ∂u/∂t−Δσ(u) = 0 in time-dependent domains

• Autorzy: Nakao Mitsuhiro

Identyfikatory

DOI

10.7494/OpMath.2023.43.5.703

• Języki publikacji: EN

Abstrakty

EN

We show the existence, smoothing effects and decay properties of solutions to the initial-boundary value problem for a generalized porous medium type parabolic equations of the form ut − Δσ(u) = 0 in Q(0, T) with the initial and boundary conditions u(0) = u0 and u(t)|∂Ω(t) = 0, where Ω(t) is a bounded domain in RN for each t ≥ 0 and [formula]. Our class of σ(u) includes σ(u) = |u|mu, σ(u) = u log(1 + |u|m), 0 ≤ m ≤ 2, and [formula]. We derive precise estimates for ∥u(t)∥Ω(t),∞ and ∥∇σ(u(t))∥2Ω(t),2, t > 0, depending on ∥u0∥Ω(0),r and the movement of ∂Ω(t).

Słowa kluczowe

EN

quasilinear parabolic equation; time-dependent domain; smoothing effect

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2023
• Tom: Vol. 43, no. 5
• Strony: 703--734
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Nakao Mitsuhiro

• Faculty of Mathematics, Kyushu University, Moto-oka 819–1602, Fukuoka, Japan

Bibliografia

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• [7] G.M. Lieberman, Second Order Parabolic Differential Equations, Revised ed., World Scientific, Singapore, 2005.
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• [11] M. Nakao, Existence and smoothing effect of the initial-boundary value problem for quasilinear parabolic equations in time-dependent domains, submitted.
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• [13] J.L. Vázquez, The Porous Medium Equation, Oxford University Press, 2007.
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Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)