On a Second Order Linear Parabolic Equation with Variable Coefficients in a Non-Regular Domain of R³

Abstract

This paper is devoted to the study of the following variable-coefficient parabolic equation in non-divergence form ∂tu − 2 ∑ i=1 ai(t, x1, x2)∂iiu + 2 ∑ i=1 bi(t, x1, x2)∂iu + c(t, x1, x2)u = f (t, x1, x2), subject to Cauchy-Dirichlet boundary conditions. The problem is set in a non-regular domain of the form Q = { (t, x1) ∈ R 2 : 0 < t < T, φ1 (t) < x1 < φ2 (t) } × ]0, b[ , where φ k , k = 1, 2 are "smooth" functions. One of the main issues of this work is that the domain can possibly be non-regular, for instance, the singular case where φ1 coincides with φ2 for t = 0 is allowed. The analysis is performed in the framework of anisotropic Sobolev spaces by using the domain decomposition method. This work is an extension of the constant-coefficients case studied in [15].