<abstract xmlns="http://www.w3.org/1999/xhtml"> This paper is devoted to the study of the following fourth order parabolic equation ∂tu + ∂x4u = f in the non-necessarily rectangular domain Q ={(t,x) ∈ℝ2 : 0 &lt; t &lt; T, ϕ1(t) &lt; x &lt; ϕ2(t)}. The equation is subject to mixed type conditions ∂xu = ∂x3u + βu = 0, on the lateral boundary of Q. The right-hand side term f of the equation lies in Lω2 (Q) the space of square-integrable functions on Q with the measure ωdtdx. Our aim is to find sufficient conditions on the coefficient β and on the functions ϕi,i = 1, 2 and on the weight ω such that the solution of this equation belongs to the anisotropic Sobolev space Hω1,4 (Q) = {u ∈ Lω2 (Q) : ∂tu, ∂xju ∈ Lω2 (Q) , j = 1, 2, 3, 4}.The analysis is performed by using the domain decomposition method. </abstract>

This paper is devoted to the study of the following fourth order parabolic equation ∂tu + ∂x4u = f in the non-necessarily rectangular domain Q ={(t,x) ∈ℝ2 : 0 < t < T, ϕ1(t) < x < ϕ2(t)}. The equation is subject to mixed type conditions ∂xu = ∂x3u + βu = 0, on the lateral boundary of Q. The right-hand side term f of the equation lies in Lω2 (Q) the space of square-integrable functions on Q with the measure ωdtdx. Our aim is to find sufficient conditions on the coefficient β and on the functions ϕi,i = 1, 2 and on the weight ω such that the solution of this equation belongs to the anisotropic Sobolev space Hω1,4 (Q) = {u ∈ Lω2 (Q) : ∂tu, ∂xju ∈ Lω2 (Q) , j = 1, 2, 3, 4}.The analysis is performed by using the domain decomposition method.

On a Fourth Order Parabolic Equation with Mixed Type Boundary Conditions in a Nonrectangular Domain

Laboratoire des Mathématiques Appliquées, Faculté des Sciences Exactes, Université de Bejaia, 6000 Béjaia, Algeria Lab. E.D.P.N.L., Ecole Normale Supérieure, 16050-Kouba, Algiers,

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

This paper is devoted to the study of the following fourth order parabolic equation ∂tu + ∂x4u = f in the non-necessarily rectangular domain Q ={(t,x) ∈ℝ2 : 0...

On a Fourth Order Parabolic Equation with Mixed Type Boundary Conditions in a Nonrectangular Domain

Article Category: Research Article

Published Online: Feb 04, 2017

Page range: 76 - 90

Received: Apr 21, 2015

Accepted: Jul 03, 2015

DOI: https://doi.org/10.7603/s40956-015-0006-5

Keywords
Higher-order parabolic equations, Nonregular domains, Anisotropic weighted Sobolev spaces, Mixed type conditions

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On a Fourth Order Parabolic Equation with Mixed Type Boundary Conditions in a Nonrectangular Domain

Article Category: Research Article

Published Online: Feb 04, 2017

Page range: 76 - 90

Received: Apr 21, 2015

Accepted: Jul 03, 2015

DOI: https://doi.org/10.7603/s40956-015-0006-5

KeywordsHigher-order parabolic equations, Nonregular domains, Anisotropic weighted Sobolev spaces, Mixed type conditions

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Higher-order parabolic equations, Nonregular domains, Anisotropic weighted Sobolev spaces, Mixed type conditions