tailieunhanh - On the solvability of the boundary problem for second order parabolic equations without an initial condition in cylinders with non smooth base

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 .