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
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.
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.