UDC 517.9 REGULARITY OF SOLUTION OF THE SECOND INITIAL BOUNDARY VALUE PROBLEM FOR PARABOLIC EQUATIONS IN DOMAINS WITH CONICAL POINTS Nguyen Manh Nung, Nguyen Thanh Anh Hanoi University of Education, Vietnam The purpose of this paper is to establish the well-posedness and the regularity of solutions of the second initial boundary value problems for general higher order parabolic equations in infinite cylinders with the bases containing conical points. <...> KEY WORDS: parabolic equation, initial boundary value problem, nonsmooth domains, generalized solutions, regularity. 1. <...> INTRODUCTION We are concerned with initial boundary value problems for parabolic equations in nonsmooth domains. <...> These problems with Dirichlet boundary condition in domains containing conical points have been investigated in [6, 7]. <...> The problems with Neumann boundary condition in domains with edges have been dealt with for the classical heat equation in [10] and for general second order parabolic equations in [2]. <...> In the present paper, we consider such problems with Neumann boundary condition (the second initial boundary problems) for higher order linear parabolic equations in domains containing conical points. <...> For theequation dealt with in [10], whose coefficients are independent of the time variable, one used Fourier transform to reduce the problem to an elliptic one with a parameter. <...> In the present paper, for a general higher order linear parabolic equation with coefficients depending on both spatial and time variables in domains containing conical points we modify the approach suggested in [3, 6, 7]. <...> First, we study the unique solvability and the regularity with respect to the time variable for generalized solutions in the Sobolev space HQ m,1 () by Galerkin’s approximate method. <...> By © Nguyen Manh Nung, Nguyen Thanh Anh, 2008 170 modifying the arguments used in [6, 7], we can weaken the restrictions on the data at the initial time t = 0 imposed therein. <...> After that, we take the term containing the derivative in time of the unknown function to the right-hand side of the equation such that the problem can be considered as an elliptic one. <...> With the help of some auxiliary results we can apply the results for elliptic boundary value problems and our previous ones to deal with the regularity with respect to both of time and spatial variables of the solutions <...>