Initial-Boundary Value Problems with Generalized Samarskii–Ionkin Condition for Parabolic Equations with Arbitrary Evolution Direction

We study the solvability of boundary value problems nonlocal with respect to the spatial variable with the generalized Samarskii–Ionkin condition for parabolic equations (Formula Presented.) where x ∈ (0, 1), t ∈ (0, T) and h(t), a(x), c(x, t), f(x, t) are given functions. If a(x) is positive, then the function h(t) can have different signs at different points of [0, T] or even vanish on a set of positive measure in [0, T]. We prove the existence and uniqueness of regular solutions, i.e., solutions possessing all weak derivatives (in the sense of Sobolev) occurring in the corresponding equation. The obtained results are new even for the classical Samarskii–Ionkin problem for the heat equation. © 2023, Springer Nature Switzerland AG.