The goal of this paper is to establish the unique existence of generalized solutions of boundary problem for second-order parabolic equations without an initial condition in cylinders with non-smooth base. | JOURNAL OF SCIENCE OF HNUE Natural Sci. 2011 Vol. 56 No. 7 pp. 18-22 ON THE SOLVABILITY OF THE BOUNDARY PROBLEM FOR SECOND-ORDER PARABOLIC EQUATIONS WITHOUT AN INITIAL CONDITION IN CYLINDERS WITH NON-SMOOTH BASE Nguyen Manh Hung and Le Thi Duyen Hanoi National University of Education E-mail Linhlinh041@ Abstract. The goal of this paper is to establish the unique existence of gen- eralized solutions of boundary problem for second-order parabolic equations without an initial condition in cylinders with non-smooth base. Keywords Generalized solutions without an initial condition. 1. Introduction The initial-boundary value problems for parabolic equations in domains with conical points were considered in 3 4 where some important results on the unique existence of solutions for these problem were given. The problem without initial condition for second-order parabolic equations in cylinders with smooth base was considered in 1 2 . In this paper we will prove the unique solvability of bound- ary problem for second-order parabolic equations without an initial condition in cylinders with non-smooth base. 2. Formulation of the problem Let G be a bounded domain in Rn n 2 with the boundary G. We suppose that S G 0 is an infinitely differentiable surface everywhere except the origin. Denote G G T S S T Gh G h T Sh S h T for each T 0 lt T . We use the following notation for each multi-index α α1 .αn N n α α1 . αn the symbol D α u α u αx11 . αxnn uxα1 1 .xαnn denotes the generalized derivative of order α with respect to x x1 . xn . We begin with recalling some functional spaces which will be used frequently in this paper. 18 On the solvability of the boundary problem. W2m G is the space consisting of all functions u x L2 G such that D α u x L2 G for almost α m with the norm m Z X 1 2 kukW2m G D αu 2 dx . α 0 G Let X Y be Banach spaces. L2 0 T X is the space consisting of all measurable functions u 0 T X with the norm ZT 21 kukL2 0 T X ku t k2X dt . 0 W21 0 T X is the .
