Solution of the Bitsadze-Samarskii problem for a parabolic system in a semibounded domain
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.
Petrovskii uniformly parabolic matrix operator
scientific article; zbMATH DE number 6844048
scientific article