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

Abstract

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.