1,721,003 research outputs found
POLYNOMIAL OPERATOR MATRICES AS SEMIGROUP GENERATORS - THE 2X2 CASE
No full textMany systems of linear evolution equations can be written as a single equation
\dot u(t)=\s A u(t),\eqno(*)
where is a function with values in a product space and
\s A =(A_{ij})_{n\times n} is a matrix whose entries are
linear operators on . In order to prove the well-posedness of
one shows that \s A generates a strongly continuous semigroup on .
In this paper we consider the case where the are polynomials
with respect to a single (unbounded) operator on and
restrict our attention to the case of matrices
POLYNOMIAL OPERATOR MATRICES AS GENERATORS - THE GENERAL-CASE
No full textOf concern are systems of linear evolution equations
\dot u(t)=\s A u(t),\qquad u(0)=u_0,\leqno({\rm ACP})
where is a function with values in a product Banach space \s E :=E^n and
\s A=(p_{ij}(A)) is a matrix whose entries are polynomials in
a fixed linear, possibly unbounded operator on . In this paper we will study
the well-posedness of , i.e., we will characterize those polynomial operator
matrices \s A generating a strongly continuous semigroup on \s E
ON SINGULAR PERTURBATIONS OF 2ND-ORDER CAUCHY-PROBLEMS
No full textWe give an explicit formula for the solution of complete second order Cauchy problems in Banach spaces. Using this formula we derive an estimate for the growth of the solution in terms of an associated scalar ODE. Finally these results are applied to singular perturbations of second order Cauchy problems
Generator property and stability for generalized difference operators
No full textWe introduce generalized difference operators, characterize their gen-
erator property and estimate the growth bound of the generated semigroup. The
results are illustrated by several examples
Systems of evolution equations
No full textIn this notes we present a rather general framework which allows to prove in a unified and systematic way that certain second order differential operators with
Wentzell-type boundary conditions generate analytic semigroups or even cosine families on
spaces of continuous functions. It is based on similarity transformations and perturbation
techniques which allow to decouple (complicated) Wentzell boundary conditions yielding
to an operator with (much simpler) abstract “Dirichlet” boundary conditions and an
abstract “Dirichlet–Neumann” operator on a “boundary space”
Spectral theory and generator property for one-sided coupled operator matrices
No full textMany initial value problems like Volterra equations, delay equations or wave equations can be reduced to an abstract Cauchy problem governed by an opercator matrix. We introduce a new class of unbounded operator matrices corresponding to these equations and study the spectral theory, compute the adjoint and analyze the generator property of its elements. The abstract results are illustrated by a series of applications
The Laplacian on with generalized Wentzell boundary conditions
No full textIn this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain Omega in R-m defined by
(1) Af := Deltaf, D(A) := {f is an element of C-n(1) ((Omega) over bar) : Deltaf is an element of C((Omega) over bar); Deltaf + betapartial derivativef/partial derivativen + gammaf = 0 on partial derivativeOmega}
generates an analytic semigroup of angle pi/2 on C((Omega) over bar) for every beta > 0 and gamma is an element of C (partial derivativeOmega) (for the 2 definition of C-n(1) ((Omega) over bar) cf. (1.3))
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