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Initial Boundary-Value Problems for Parabolic Systems in Dihedral Domains

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Ukrainian Mathematical Journal Aims and scope

We present some facts about the smoothness of solutions of the initial-boundary-value problems for the parabolic system of partial differential equations

$$ {\displaystyle \begin{array}{c}{u}_t-{\left(-1\right)}^mP\left(x,t,{D}_x\right)u=f\left(x,t\right)\kern1em \mathrm{in}\kern1em \Omega \times \left(0,T\right),\\ {}\frac{\partial^ju}{\partial {v}^j}=0\kern1em \mathrm{on}\kern1em \left(\mathrm{\partial \Omega}\backslash M\right)\times \left(0,T\right),\\ {}u\left(x,0\right)=0,\end{array}} $$

in a domain of dihedral type ΩT , 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.

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Authors and Affiliations

  1. Hanoi University of Education, Hanoi, Vietnam

    P. T. Duong

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  1. P. T. Duong
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Correspondence to P. T. Duong.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 7, pp. 903–917, July, 2020. Ukrainian DOI: 10.37863/umzh.v72i7.1094.

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Duong, P.T. Initial Boundary-Value Problems for Parabolic Systems in Dihedral Domains. Ukr Math J 72, 1051–1067 (2020). https://doi.org/10.1007/s11253-020-01840-8

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  • DOI: https://doi.org/10.1007/s11253-020-01840-8

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