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

Abstract

Abstract This paper is devoted to the study of the following fourth order parabolic equation ∂tu + ∂x 4u = 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 = ∂x 3u + βu = 0, on the lateral boundary of Q. The right-hand side term f of the equation lies in Lω 2 (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 Hω 1,4 (Q) = {u ∈ Lω 2 (Q) : ∂tu, ∂x ju ∈ Lω 2 (Q) , j = 1, 2, 3, 4}. The analysis is performed by using the domain decomposition method.