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

Abstract

Let Ω T be some bounded simply connected region in ℝ 2 with . We seek a function u ( x , t )(( x , t ) ∈ Ω T ) with values in a Hilbert space H which satisfies the equation A L u ( x , t ) = B u ( 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 L u ( x , t ) = u t t + a 11 u x x + a 1 u t + a 2 u 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.

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Identifiers

DOI

10.1155/s1085337501000628

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