Solvability of a Boundary-Value Problem for Degenerate Equations
- PDF / 213,075 Bytes
- 20 Pages / 594 x 792 pts Page_size
- 25 Downloads / 243 Views
SOLVABILITY OF A BOUNDARY-VALUE PROBLEM FOR DEGENERATE EQUATIONS T. Gadjiev,1 M. Kerimova,2 and G. Gasanova2
UDC 517.9
We consider a boundary-value problem for degenerate equations with discontinuous coefficients and establish the unique strong solvability (almost everywhere) of this problem in the corresponding weighted Sobolev space.
1. Introduction The aim of the present work is to prove a unique strong (almost everywhere) solvability of the first boundaryvalue problem for an equation Zu =
n X
i,j=1
aij (x, t)uij + (x, t)utt − ut = f (x, t),
(1.1) (1.2)
u|Γ(QT ) = 0, in a cylinder QT = ⌦ ⇥ (0, T ),
T 2 (0, 1),
where ⌦ is a bounded domain in Rn with boundary @⌦ ⇢ C 2 , Γ(QT ) = (@⌦ ⇥ [0, T ]) [ ⌦ ⇥ {(x, t) : t = 0} is the parabolic boundary of the domain QT , (x, t) , and the coefficients aij (x, t) tend to zero. Here, uij =
@ 2 u(x, t) , @xi @xj
utt =
@ 2 u(x, t) , @t2
ut =
@u . @t
The initial-boundary problems for this type of degenerate equations have been studied by numerous authors (see, e.g., [2–4]). In [1], Fichera considered boundary-value problems for degenerate equations in multidimensional case. He proved the existence of generalized solutions to these boundary-value problems. The boundary-value problems for the degenerate equations of this kind were studied in the stationary case in [5] and in the nonstationary case in [6]. Coercive estimates for this problem were obtained in [8]. We also mention the works [2–4] in which the property of strong solvability of the boundary-value problem (1.1), (1.2) was established for equations with smooth coefficients. Similar results for the Cordes-type discontinuous coefficients were established in [4]. Some classes of elliptic parabolic equations were considered in [9, 10]. Thus, the well-posedness of the initial-boundary-value problem for pseudoparabolic equations was studied and estimates of the generalized solution were obtained in [9]. In [10], the solvability results were obtained for the Cordes-type discontinuous coefficients. Some general problem for linear and quasilinear equations of parabolic type was considered in [11]. 1 2
Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan; e-mail: [email protected]. Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 435–451, April, 2020. Original article submitted September 3, 2016. 0041-5995/20/7204–0495
© 2020
Springer Science+Business Media, LLC
495
T. G ADJIEV, M. K ERIMOVA ,
496
AND
G. G ASANOVA
In the present paper, we consider wide classes of elliptic parabolic equations. Assume that the coefficients satisfy the conditions: |aij (x, t)| is a symmetric matrix with real measurable elements in QT and, for any (x, t) 2 QT , ⇠ 2 Rn , the following inequalities are true: 2
γ!(x)|⇠|
n X
i,j=1
aij (x, t)⇠i ⇠j γ −1 !(x)|⇠|2 ,
(1.3)
where γ 2 (0, 1], !(x) 2 Ap satisfies the Muckenhoupt condition (see [7]), and (x, t) = !(x)λ(t
Data Loading...
