tailieunhanh - Dirichlet cauchy problem for second order parabolic equations in domains with edges

In this paper, we study the first initial boundary value problem for second-order parabolic equations in cylinders with piecewise smooth base. Some results on the unique existence and on the smoothness with respect to time of the solution are given. | JOURNAL OF SCIENCE OF HNUE Natural Sci. 2011 Vol. 56 No. 3 pp. 13-24 DIRICHLET-CAUCHY PROBLEM FOR SECOND-ORDER PARABOLIC EQUATIONS IN DOMAINS WITH EDGES Do Van Loi Hong Duc University E-mail 37loilinh@ Abstract. In this paper we study the first initial boundary value prob- lem for second-order parabolic equations in cylinders with piecewise smooth base. Some results on the unique existence and on the smoothness with respect to time of the solution are given. Keywords Domains with edges generalized solution regularity. 1. Introduction Initial boundary value problems for non-stationary systems and equations in cylinders with non-smooth bases have been investigated in 2-8 where some impor- tant results on the unique existence smoothness and asymptotic of the solution for the problems in Sobolev spaces were given. The initial boundary value problems for parabolic equations in a cylinder with base containing conical points 5 or in a dihe- dral angle 8 have been studied in Sobolev spaces with weights. In the present paper we consider the first initial boundary value problem for second-order parabolic equa- tions in domains with edges. We study the existence uniqueness and smoothness with respect to time of the generalized solution for this particular problem. The paper is organized as follows. In the second section we define Dirichlet- Cauchy problem for second-order parabolic equations in domains with edges. In the 3rd section we study the solvability of the problem. The last section is intended to regularity with respect to time of the Generani Zaisedzed solution. 2. Formulation of the problem Let Ω be a bounded domain in Rn n 2. Its boundary Ω is a piecewise smooth surface consisting of finitely many n 1 -dimensional smooth surfaces Γi . We assume that the surface Γi intersects only the surfaces Γi 1 Γi 1 along smooth n 2 -Dimensional manifolds li 1 li 1 . Without loss of generality we shall deal 13 Do Van Loi explicitly with the case when Ω consists two .