Initial Boundary-Value Problems for Parabolic Systems in Dihedral Domains
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INITIAL BOUNDARY-VALUE PROBLEMS FOR PARABOLIC SYSTEMS IN DIHEDRAL DOMAINS P. T. Duong
UDC 517.9
We present some facts about the smoothness of solutions of the initial-boundary-value problems for the parabolic system of partial differential equations ut − (−1)m P (x, t, Dx )u = f (x, t) @j u =0 @⌫ j
on
in
⌦T := ⌦ ⇥ (0, T ),
(@⌦\M ) ⇥ (0, T ),
u(x, 0) = 0, in a domain of dihedral type ⌦T , where P is an elliptic operator with variable coefficients. It is shown that the regularity of solutions depends on the distribution of eigenvalues of the corresponding spectral problems. The obtained results can be useful for understanding the asymptotics of weak solutions near the singular edges of dihedral domains.
1. Introduction Boundary-value problems in nonsmooth domains have been systematically studied starting from Kondrat’ev’s work [2] devoted to the investigation of elliptic equations in conic and angular domains and the analysis of the properties of regularity and asymptotic representations of their solutions near the conic point. Some well-known results for the elliptic boundary-value problems in domains with more general singularities (edges or spikes) were obtained by Maz’ya and Plamenevskii [4] and Maz’ya and Rossmann [5]. In the cited works, the authors revealed the relationship between the spectral properties of the operator pencil connected with boundary-value problems in different domains with singularities and the well-posedness of the boundary-value problem. Nonstationary problems in conic domains were considered by Nguyen Manh Hung [6] with an aim to study their solvability and analyze the asymptotic behavior of the generalized solutions of hyperbolic systems near conic points. In the present paper, we investigate the initial-boundary-value problem for a strongly parabolic system in a cylinder whose base has the form of a dihedron. By using the results on solvability, uniqueness, and differentiability with respect to the time variable obtained in [7, 8], we prove the regularity of the generalized solution both with respect to the space variables x and with respect to time. We also present an a priori estimate in weight Sobolev spaces for a special case of the system of second-order partial differential equations. The paper is organized as follows: In Sec. 2, we introduce the required notation. Our main problem is formulated in Sec. 3. Section 4 contains some basic facts about elliptic equations in dihedral domains. In Sec. 5, we study the regularity of generalized solutions of parabolic systems. Hanoi University of Education, Hanoi, Vietnam; e-mail: [email protected]. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 7, pp. 903–917, July, 2020. Ukrainian DOI: 10.37863/umzh.v72i7.1094. Original article submitted October 12, 2019. 0041-5995/20/7207–1051
© 2020
Springer Science+Business Media, LLC
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P. T. D UONG
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2. Necessary Notation We use the notation introduced in [4]. A dihedron is defined as the product D = K ⇥ Rn−2 , where K is an angle specified in the polar coor
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