Abstract and Applied Analysis
Volume 6 (2001), Issue 5, Pages 253-266
doi:10.1155/S1085337501000628

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

Kudratillo S. Fayazov1 and Eberhard Schock2

1Department of Mathematics, National University of Usbekistan, Tashkent, 700090 Tashkent, Vuzgorodok, Uzbekistan
 2Department of Mathematics, University of Kaiserslautern, P.O. Box 3049, Kaiserslautern 67653, Germany

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.