Boundary value problems for second-order partial differential equations with operator coefficients. (25th November 2001)

• Record Type: Journal Article
• Authors: Fayazov, Kudratillo S.; Schock, Eberhard
• Is Part Of: Abstract and applied analysis. Volume 6:Number 5(2001)
• Page Start: 253
• Page End: 266
• Publication Date: 2001-11-25
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/S1085337501000628
• Languages: English
• ISSNs: 1085-3375

Abstract

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.