Boundary value problems for second‐order partial differential equations with operator coefficients

Let Ω T be some bounded simply connected region in ℝ 2 with ∂ Ω T=Γ¯1∩Γ¯2. We seek a function u(x, t)((x, t) ∈ Ω T) with values in a Hilbert space H which satisfies the equation ALu(x, t) = Bu(x, t) + f(x, t, u, u t), (x, t) ∈ Ω T, where A(x, t), B(x, t) are families of linear operators (possibly unbounded) with everywhere dense domain D (D does not depend on (x, t)) in H and Lu(x, t) = u tt + a 11u xx + a 1u t + a 2u x. The values u(x, t); ∂u(x, t)/∂n are given in Γ 1. This problem is not in general well posed in the sense of Hadamard. We give theorems of uniqueness and stability of the solution of the above problem.