ANNALES POLONICI MATHEMATICI 109.2 (2013) Asymptotics of solutions to the Dirichlet–Cauchy problem for parabolic equations in domains with edges by Vu Trong Luong (Sonla), Nguyen Manh Hung (Hanoi) and Do Van Loi (Thanhhoa) Abstract. This paper is concerned with the Dirichlet–Cauchy problem for second order parabolic equations in domains with edges. The asymptotic behaviour of the solution near the edge is studied. 1. Introduction. We are concerned with initial boundary value prob- lems (IBVP) for parabolic equations or systems in non-smooth domains. Such problems in domains with conical points have been studied in [3, 4, 5]; we investigated the solvability and asymptotics of solutions in a neighbour- hood of the conical point. Solonnikov [10] dealt with the Neumann problem in domains with edges for the classical heat equation. By using the Fourier transform to reduce the problem to an elliptic boundary value problem with parameter, he proved the unique solvability and obtained coercive estimates of the solution in a weighted H¨ older norm. Frolova [2] extended the solvabil- ity results of [10] to the case of boundary conditions involving derivatives with respect to both space variables and time. In the present paper, we consider the first initial boundary value problem for second order parabolic equations in domains with edges. We modify the approach suggested in [9, 3] to demonstrate the asymptotic representation of the generalized solution of the problem in a neighbourhood of the edge. Let Ω be a bounded domain in R n , n ≥ 2, with the boundary ∂Ω con- sisting of two surfaces Γ 1 , Γ 2 which intersect along a manifold l 0 . Assume that in a neighbourhood of each point of l 0 the set Ω is diffeomorphic to a dihedral angle. For any P ∈ l 0 , two half-spaces T 1 (P ) and T 2 (P ) tangent to Ω, and a two-dimensional plane π(P ) normal to l 0 , are defined. We de- note by ν (P ) the angle in the plane π(P ) (on the side of Ω) bounded by the rays R 1 = T 1 (P ) ∩ π(P ) and R 2 = T 2 (P ) ∩ π(P ), and by β (P ) the aperture 2010 Mathematics Subject Classification : 35K20, 35D30, 35B65, 35C20. Key words and phrases : asymptotics, generalized solution, regularity. DOI: 10.4064/ap109-2-2 [121] c  Instytut Matematyczny PAN, 2013