Abstract and Applied Analysis

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

Abstract

Let $\Omega_T$ be some bounded simply connected region in $\mathbb{R}^2$ with $\partial\Omega_{T} = \bar{\Gamma}_{1}\cap\bar{\Gamma}_{2}$. We seek a function $u(x,t),((x,t)\in\Omega_{T})$ with values in a Hilbert space $H$ which satisfies the equation $ALu(x,t) = Bu(x,t) + f(x,t,u,u_{t}),(x,t)\in\Omega_{T}$, 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 $Lu(x,t)= u_{tt}+ a_{11}u_{xx}+ a_{1}u_{t}+ a_{2}u_{x}$. The values $u(x,t);\partial u(x,t)/\partial n$ are given in $\Gamma_1$. 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.

Article information

Source
Abstr. Appl. Anal., Volume 6, Number 5 (2001), 253-266.

Dates
First available in Project Euclid: 13 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1050266864

Digital Object Identifier
doi:10.1155/S1085337501000628

Mathematical Reviews number (MathSciNet)
MR1879825

Zentralblatt MATH identifier
1016.35001

Subjects
Primary: 35A07

Citation

Fayazov, Kudratillo S.; Schock, Eberhard. Boundary value problems for second-order partial differential equations with operator coefficients. Abstr. Appl. Anal. 6 (2001), no. 5, 253--266. doi:10.1155/S1085337501000628. https://projecteuclid.org/euclid.aaa/1050266864

References

• A. L. Buchgeim, Ill-posed problems of the numerical theory and tomography, Siberian Math. J. 33 (1992), 27–41.
• K. S. Fayazov and M. M. Lavrent'ev, Cauchy problem for partial differential equations with operator coefficients in space, J. Inverse Ill-Posed Probl. 2 (1994), no. 4, 283–296.
• S. G. Kreĭn, Lineikhye Differentsialnye Uravneniya V Banakhovom Prostranstve. [Linear Differential Equations in a Banach Space ], Izdat. “Nauka”, Moscow, 1967 (Russian).
• M. M. Lavrent'ev, On the problem of Cauchy for linear elliptic equations of the second order, Dokl. Akad. Nauk SSSR (N. S.) 112 (1957), 195–197 (Russian).
• H. A. Levine, Logarithmic convexity and the Cauchy problem for some abstract second order differential inequalities, J. Differential Equations 8 (1970), 34–55.