Solution of the Bitsadze-Samarskii problem for a parabolic system in a semibounded domain
type: Item
instance of: scholarly article
title: Solution of the Bitsadze-Samarskii problem for a parabolic system in a semibounded domain
zbMATH Open document ID: 1400.35154
DOI: 10.1134/S0012266117100068
author: E. A. Baderko
author: M. F. Cherepova
published in: Differential Equations
publication date: 2018-02-27T00:00:00Z
review text: Set \[ Lu:=\partial_tu-\sum_{k=0}^2 A^{(k)}\partial^k_xu,\quad u=(u_1,\ldots, u_m)^{\text{T}},\quad m\geq 1,\quad (x,t)\in D \] for the Petrovskii uniformly parabolic matrix operator in the strip \(D=\mathbb{R}\times(0,T),\) where \(A^{(k)}=\{a^{(k)}_{ij}\}_{i,j=1}^m\) are \(m\times m\)-matrices with eigenvalues having bounded away from zero real parts and entries \(a^{(k)}_{ij}\) belonging to the anisotropic Hölder space \(H^{\alpha, \alpha/2}(\overline{D}).\) Further on, a semibounded domain of the form \(\Omega=\{(x,t)\in D:\;x>g_1(t)\}\) is considered, together with a curve \(x=g_2(t)\) in the closure \(\overline{\Omega}.\) The functions \(g_1\) and \(g_2\) are supposed to be Hölder continuous with exponent \((1+\alpha)/2\) and \(g_1(t) < g_2(t).\) The authors study classical solvavility of the system \[ Lu=0\quad \text{in}\;\Omega, \] with the initial condition \[ u(x,0)=0 \quad \text{for}\;x\geq g_1(0) \] and the nonlocal boundary condition \[ u(g_1(t),t)=u(g_2(t),t)+\psi(t)\quad t\in [0,T], \] with a suitable vector-valued function \(\psi:\;[0,T]\to\mathbb{R}^m.\) The solution is constructed by means of the boundary integral equations method, assuming that \(\psi\) has a continuous derivative of order \(1/2\) vanishing at \(t=0.\) The regularity of the solution constructed is studied as well.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6844048
zbMATH Keywords: Petrovskii uniformly parabolic matrix operator
zbMATH Keywords: initial-boundary value problem
zbMATH Keywords: nonlocal
zbMATH Keywords: classical solution
MaRDI profile type: Publication
cites work: Q5579204
cites work: Q3712561
cites work: Q3710887
cites work: ELLIPTIC PROBLEMS WITH NONLOCAL CONDITIONS NEAR THE BOUNDARY
cites work: Nonclassical boundary value problems. I. Transl. from the Russian
cites work: Nonclassical boundary-value problems. II
cites work: Elliptic functional differential equations and applications
cites work: Q4777797
cites work: The Bitsadze-Samarskij problem for parabolic equation
cites work: Uniqueness of the solution of a boundary value problem with samarskii boundary conditions for a second-order parabolic equation
cites work: Simple layer potential and the first boundary value problem for a parabolic system on the plane
cites work: The first boundary value problem for parabolic systems in plane domains with nonsmooth lateral boundaries
cites work: Bitsadze-Samarskii problem for a parabolic system on the plane
cites work: Q5548525
cites work: Q5732852
cites work: Q5560475
cites work: Q3699149
cites work: Q4765491
full work available at URL: s0012266117100068
OpenAlex ID: W2770103956
description: scientific article; zbMATH DE number 6844048