Lecture 1
Correct solvability of an abstract functional differential equations
with unbounded operator coefficients in Hilbert space.
Applications of these results to study of functional partial
differential equations.
At this lecture I plan to present the results about correct solvability for
an abstract Functional Differential Equations (FDE) in Hilbert space. I”ll
formulate the results obtained by R. Datko [1], G.Di Blasio, K.Kunisch, E.
Sinestrari [6]– [8], J. Wu [28] and compare it with my results in this field (see
[30]– [32] for more details). I”ll pay special attention to the equations with
variable delays and unbounded coefficients on the delay terms. Majority of
authors study the case constant delays and bounded operator coefficients on
the delay terms. Above mentioned FDE is an abstract form of partial FDE,
arising in various applications. Many interesting examples of such partial
FDE, arising in Population Ecology, in Climate Models, in Control Theory,
in Structured Population Models are presented in monograph of Jianhong
Wu [28]. I”ll speak about some of these examples. It is relevant to note that
we have several joint works with Jianhong Wu and I”ll formulate several our
joint results at this lecture (see [14], [25], [26] for more details). In addition,
I plan to demonstrate the examples of very unstable FDE with the delay in
highest derivatives terms (see [36] for details).
Lecture 2
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Description of C - semigroups generators of shift operator on the
trajectories of the solutions of functional differential equations of
neutral type.
At the second lecture I’ll present different approaches in introducing semigroups of the shift operators along the trajectories of the solutions of autonomous FDE. First of all, I mention the approach of Jack Hale and S.
Verduyn Lunel [11], M.C. Delfour [4] and our approach with my coauthor S.
Ivanov [33]. Then I plan to give the description of the generators of these
semigroups and the representation of the resolvents of these generators. This
information plays very important role in the researching the problems of completeness and basisness of the exponential solutions of autonomous FDE. In
turn completeness and basisness of exponential solutions plays significant
role in the problems of Control Theory (see the results of M.C. Delfour and
A. Manitius [2], [3], R.Rabah, G. Sklyar and Resounenko [19], [20] for more
details). In addition, I’ll speak about analogy between the classification of
functional differential equations and partial differential equations (see our
joint work with A. D. Myskis [37]).
Lecture 3
Riesz basisness of the exponential solutions system of
autonomous functional differential equations of neutral type.
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At my third lecture I’ll present the results about Riesz basisness of the
exponential solutions for autonomous neutral equations in Sobolev spaces.
Then I’ll show how on the base of Riesz basisness is possible to obtain the
sharp estimates for the strong solutions of above mentioned equations. It is
relevant to underline that these estimates are the most strict for this class of
functional differential equations now. Rather complete description of these
is given in the second part of the work [34], see also [33]. Along with the
problems of Riesz basisness of exponential solutions I plan to formulate the
results about completeness of exponential solutions (The results of S. Verduyn Lunel [21] - [23], N. Levinson and C. McCalla [13], M. C. Delfour and
A. Manitius [2], [3]) and deeply connected with the problem of existence of
nontrivial solutions for autonomous functional differential equations decaying rapidly then any exponential. Rather complete description of the results
in this field can find in well-known monograph of J. Hale and S. Verduyn
Lunel ”Introduction to Functional Differential Equations”, Springer-Verlag,
vol. 99, 1993 [11].
Lecture 4
Correct solvability of Volterra integrodifferential equations with
unbounded operator coefficients in Sobolev spaces of
vector-functions. Applications of these results to studying of
Volterra integro-partial differential equations arising in the theory
of viscoelasticity, theory of heat transfer in the materials with
memory (Gurtin-Pipkin equations).
At the fourth lecture I’ll formulate the results on correct solvability for integrodifferential equations with unbounded coefficients in Hilbert space. The
main part of these equations is an abstract hyperbolic equations disturbed
by Volterra integral operators. These integrodifferential equations are an
operator models for a wide class of integrodifferential equations with partial
derivatives on the space variables arising in various applications: viscoelasticity theory, theory of haet transfer in the materials with memory (GurtinPipkin equations see [10]). Now there exists extensive literature in this field.
We restrict our considerations by citing only two well known monographs in
this field [18], [29]. I’ll underline the specific features of our approach and
describe the difference between our results ([35], [38]–[39])and results of the
authors (L. Pandolfi [17], R. Miller [15], R. Wheeler [16], R. Decsh and R.
Miller [5]).
Lecture 5
Spectral analysis of hyperbolic Volterra integrodifferential
equations. The spectra of operator-functions which are the
symbols of these equations.
The material of this lecture is rather new and I do not know its analogous in existing literature. The main purpose is to provide the analysis the
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structure of resolvent set and spectra of the operator functions which are the
symbols for hyperbolic Volterra integrodifferential equations. The information about resolvent set and spectra of these operator-functions plays very
important role in the researching qualitative and asymptotic properties for
the solutions of equations above mentioned. It is quite understandable because of we use the inverse of Laplace transform acting on the Laplace image
of the solution. In turn, the Laplace image of the operator-functions which
are the symbols of the equations mentioned above are main part of Laplace
image for the solutions. In a sense this approach is similar to Laplace operational method using for ordinary differential equations. There is possibility
to obtain the representation for the solutions the above mentioned equations
on the base of the information about spectra, resolvent set and estimates
of operator functions which are symbols of integrodifferential equations (see
[35], [38], [39] for more details).
Lecture 6
Applications of abstract results to integro-partial differential
equations arising in the theory of viscoelasticity, theory of heat
transfer in the materials with memory etc.
In this lecture I’ll discuss in more detail the integro-partial differential
equations arising in viscoelasticity theory [29], Gurtin-Pipkin equation [10]
describing heat propagation with finite speed in the materials with memory.
I’ll describe the way following to which the abstract results about abstract
integrodifferential equations can be used to the researching specific integropartial differential equations. Moreover, I’ll interpret the results obtained
for the abstract integrodifferential equations on the examples of its concrete
realizations as integro-partial differential equations.
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