Portugaliae Mathematica   EMIS ELibM Electronic Journals PORTUGALIAE
Vol. 59, No. 3, pp. 335-349 (2002)

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Kuramoto--Sivashinsky Equation in Domains with Moving Boundaries

Alfredo Tadeu Cousin and Nickolai Andreevitch Larkine

Department of Mathematics, State University of Maringá,
Av. Colombo, 5790 -- CEP: 87020-900, Maringá, PR -- BRAZIL
E-mail: atcousin@uem.br
The Institute of Theoretical and Applied Mechanics,
Novosibirsk -- 90, 630090 -- RUSSIA
E-mail: nalarkin@dma.uem.br

Abstract: In the non-cylindrical domain ${Q}=\{(x,t); \alpha_1(t)<x<\alpha_2(t), t\in(0,T)\}$ we consider the initial-boundary value problem for the one-dimensional Kuramoto--Sivashinsky equation $$ u_t+u\,u_x+\beta\,u_{xx}+\delta\,u_{xxxx}=0. $$ We prove the existence and uniqueness of global weak, strong and smooth solutions. The exponential decay of the solutions is also proved.

Keywords: Kuramoto--Sivashinsky equation; noncylindrical domains; Galerkin method.

Classification (MSC2000): 35Q35, 35Q53.

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Electronic version published on: 9 Feb 2006. This page was last modified: 27 Nov 2007.

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