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 A L u ( x , t ) = B u ( 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 L u ( x , t ) = u t t + a 11 u x x + a 1 u t + a 2 u 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.