Boundary-value problems for one class of composite equations with the wave operator in the senior part
The work is devoted to the solvability of local and nonlocal boundary-value problems for composite (Sobolev-type) equations $ D^{2p+1}_t(D^2_t-\Delta u ) + Bu = f(x,t), $ where $D^k_t={\partial^k}/{\partial t^k}$, $\Delta$ is the Laplace operator acting on spatial variables, $B$ is a second-order differential operator that also acts on spatial variables, and $p$ is a nonnegative integer. For these equations, the existence and uniqueness of regular solutions (possessing all generalized derivatives in the Sobolev sense that are involved in the equation) to initial-boundary-value problems and the boundary-value problems nonlocal in the time variable. Some generalizations and refinements of the results obtained are also described.