Boundary value problems for second-order partial differential equations with operator coefficients. (25th November 2001) - Fayazov, Kudratillo S.; Schock, Eberhard
Abstract : LetΩ   T be some bounded simply connected region inℝ   2 with∂   Ω   T = Γ ¯ 1 ∩ Γ ¯ 2 . We seek a functionu ( x, t ) ( ( x, t ) ∈ Ω   T ) with values in a Hilbert spaceH which satisfies the equationA L u ( x, t ) = B u ( x, t ) + f ( x, t, u, u   t ), ( x, t ) ∈ Ω   T, whereA ( x, t ), B ( x, t ) are families of linear operators (possibly unbounded) with everywhere dense domainD (D does not depend on( x, t ) ) inH andL u ( x, t ) = u   t t + a   11 u   x x + a   1 u   t + a   2 u   x . The valuesu ( x, t ) ; ∂ u ( x, t ) / ∂ n are given inΓ   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.