Boundary value problems for second-order partial differential equations with operator coefficients (Q1599617)

The authors study the Cauchy problem for the equation \[ A(x,t)Lu = B(x,t)u + f(x,t,u,u_t), \quad (x,t) \in \Omega_T \] (\(u: \Omega_T \to H\), where \(H\) is a Hilbert space, \(A(x,t)\), \(B(x,t)\) families of linear -- possibly unbounded -- operators on \(D \subset H\), \(D\)-linear subset, dense in \(H\), not depending on \((x,t)\); \(\Omega_T = \{(x,t)\); \(0<t_0 \leq t < T\), \(\varphi_1(t) < x < \varphi_2(t)\), \(\varphi_1(t_0) = \varphi_2(t_0) \}\), where \(\varphi_1\) and \(\varphi_2\) are given functions. The boundary conditions for \(u\) and \(\partial u/\partial n\) are given on \(\Gamma_1 = \{(x,t)\); \(t_0 \leq t < T\), \(x = \varphi_i(t)\), \(i = 1, 2 \}\). Two theorems are proved from which uniqueness and stability (or stability only) under different conditions on the operators \(A\) and \(B\) follow.