Initial boundary-value problem for parabolic systems in dihedral domains


UDC 517.9

We present some results on the smoothness of the solution of the initial boundary-value problem for the parabolic system of partial differential equations $$u_t -(-1)^m P(x,t,D_x )u = f(x,t)\quad \text{in } \Omega_T := \Omega\times(0,T),$$ $$\frac{\partial^j u}{\partial \nu^j } = 0 \quad \text{on } (\partial\Omega \backslash M) \times (0, T)$$ $$u(x,0)=0, $$ in the domain $\Omega_T$ of dihedral type, where $P$ is an elliptic operator with variable coefficients. The dependence of the regularity of solutions on the distribution of eigenvalues for the corresponding spectral problems is shown. The obtained results are useful for understanding the asymptotics of the weak solution near the singular edge of dihedral domains.


M. S. Agranovich, M. I. Vishik, Эллиптические задачи с параметром и параболические задачи общего вида (Russian) [[Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obshhego vida]], Uspekhi mat. nauk, 19, vyp. 3(117), 53 – 161 (1964)

V. A. Kondrat'ev, Краевые задачи для эллиптических уравнений в областях с коническими или условными точками (Russian) [[Kraevye zadachi dlya ellipticheskikh uravnenij v oblastyakh s konicheskimi ili uslovnymi tochkami]], Tr. Mosk. mat. o-va, 16, 209 – 292 (1967)

V. A. Kozlov, V. G. Maz'ya, J. Rossmann, Elliptic boundary value problems in domains with point singularities, Math. Surveys and monographs 52, American Mathematical Society, Providence, Rhode Island x+414 pp. ISBN: 0-8218-0754-4 (1997).

V. G. Maz'ya, B. A. Plamenevskii, $L_p$ estimates of solutions of elliptic boundary value problems in domains with edges, Trans. Moscow Math. Soc., № 1, 49 – 97 (1980).

V. G. Maz'ya, J. Rossmann, Weighted $L_p$ estimate of solutions to boundary value problems for second order elliptic systems in polyhedral domains, Z. angew. Math. und Mech., 83, № 7, 435 – 467 (2003) DOI:

N. M. Hung, On the smoothness of solution of Dirichlet problem for hyperbolic systems in domains with conical or angular points, Dokl., Acad. Nauk, 362, № 2, 161 – 164 (1998).

N. M. Hung, P. T. Duong, On the smoothness with respect to time variable of generalized solution of the first initial boundary value problem for strongly parabolic systems in the cylinder with nonsmooth base ; translated from Ukraïn. Mat. Zh. 56 (2004), no. 1, 78--87 Ukrainian Math. J. 56 (2004), no. 1, 96--108 DOI:

N. M. Hung, P. T. Duong, On the smoothness of generalized solution for parabolic system in domains with conic points on boundary ; translated from Ukraïn. Mat. Zh. 56 (2004), no. 6, 857--864 Ukrainian Math. J. 56 (2004), no. 6, 1023--1032 DOI:

How to Cite
Duong, P. T. “Initial Boundary-Value Problem for Parabolic Systems in Dihedral Domains”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 7, July 2020, pp. 903-17, doi:10.37863/umzh.v72i7.1094.
Research articles