Abstract and Applied Analysis
Volume 6 (2001), Issue 5, Pages 253-266

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

Received 9 June 2001

Copyright © 2001 Kudratillo S. Fayazov and Eberhard Schock. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let ΩT be some bounded simply connected region in 2 with ΩT=Γ¯1Γ¯2. We seek a function u(x,t)((x,t)ΩT) with values in a Hilbert space H which satisfies the equation ALu(x,t)=Bu(x,t)+f(x,t,u,ut),(x,t)ΩT, 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)=utt+a11uxx+a1ut+a2ux. The values u(x,t);u(x,t)/n are given in Γ1. 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.