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

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 L_ω² (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 ϕ_i,i = 1, 2 and on the weight ω such that the solution of this equation belongs to the anisotropic Sobolev space

H_ω^1,4 (Q) = {u ∈ L_ω² (Q) : ∂_tu, ∂_x^ju ∈ L_ω² (Q) , j = 1, 2, 3, 4}.

The analysis is performed by using the domain decomposition method.