I. V. Melnikova

Abstract Cauchy Problems : Three Approaches

Preface Introduction ILLUSTRATION AND MOTIVATION Heat Equation The Reversed Cauchy Problem for the Heat Equation Wave Equation SEMIGROUP METHODS C0-Semigroups Integrated Semigroups k-Convoluted Semigroups C-Regularized Semigroups Degenerate Semigroups The Cauchy Problem for Inclusions Second Order Equations ABSTRACT DISTRIBUTION METHODS The Cauchy Problem The Degenerate Cauchy Problem Ultradistributions and New Distributions REGULARIZATION METHODS The Ill-Posed Cauchy Problem Regularization and C-Regularized Semigroups Bibliographic Remark Bibliography Glossary of Notation Index

Regularization of weakly ill-posed Cauchy problems

For such ill-posed problems, called strongly ill-posed Cauchy problems, various regularizing operators were constructed both by reduction to the form (1.1) and by using the specific differential character [4, 9, 7, 11]. These are regularizing operators constructed by quasi-reversibility method, auxiliary boundary conditions method (ABC method) and others. Nevertheless, along with strongly ill-posed Cauchy problems, considerable recent attention has been given to Cauchy problems that are only slightly ill-posed. The Cauchy problem for the Schrödinger

Abstract Stochastic Equations. I. Classical and Distributional Solutions

STOCHASTIC EQUATIONS I. CLASSICAL AND DISTRIBUTION SOLUTIONS I.V. Melnikova, A.I. Filinkov, and U.A. Anufrieva

Generalized solutions to abstract stochastic problems

We prove existence and uniqueness of a solution to a generalized stochastic Cauchy problem in spaces of abstract ultra-distributions.

Regularized solutions to Cauchy problems well posed in the extended sense

Cauchy problems that are not uniformly well posed, abstract ones, and those for differential systems are under consideration. Distribution, semigroup, and Laplace and Fourier transform techniques are used for constructing regularized (in a broad sense) solutions to the problems. The relationship between existence of generalized solutions and existence of convoluted semigroups is established.

Degenerate distribution semigroups and well-posedness of the cauchy problem

The Cauchy problem for the degenerate first order equation Bu′ (t) = Au(t)t≥ 0, u(0) = x, ker B ≠ {0}, or the equivalent Cauchy problem for the inclusion u′(t) ∊ B-1 Au(t)t≥0, u(0) = x, are studied in the space of abstract distributions. Well-posedness conditions are obtained in terms distribution semigroups and integrated semigroups.

Properties of an abstract pseudoresolvent and well-posedness of the degenerate Cauchy problem

The degenerate Cauchy problem in a Banach space is studied on the basis of properties of an abstract analytical function, satisfying the Hilbert identity, and a related pair of operators A,B. 1. We consider in a Banach space X an operator-valued function of complex variable R(λ) ∈ B(X), satisfying the Hilbert identity: (1) ∀ x ∈ X R(λ)R(μ)x = R(μ)−R(λ) λ− μ x, λ, μ ∈ Ω ⊂ C. For such function kerR(λ) =: K and range R(λ) =: R do not depend on λ [1]. If K = {0}, then the function R(λ) is called by resolvent, if K 6= {0} pseudoresolvent. For the case when R(λ) is resolvent, from (1) the equality follows: (2) λ−R−1(λ) = μ−R−1(μ) =: A, D(A) = R and R(λ) = RA(λ). Let V (t) be an exponentially bounded operator-function (||V (t)|| ≤ Le) and

Laplace transform of k-semigroups and well-posedness of cauchy problems

We consider the abstract Cauchy problem has an exponential growth in some domain of the right halfplane. To study well-posedness of the problem we use the technique of ultradistributions and K-convoluted semigroups with K(t) defined by the resolvent estimate.