Differential equations in a Banach space

Крейн, С. Г.; Khazan, M. I.

doi:10.1007/bf02105398

Cited by 32 publications

(17 citation statements)

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“…For a characterization of those generators A, see Da Prato [7] and Okazawa [27]; for applications of these semigroups, see [17] and [9]). Integrating 1-times we obtain that the operators S(t) defined by S(t)x = JQ T{S)X ds define a 1-times integrated semigroup on E and i?…”

Section: If There Exists a Polynomial P(-) And A Constant W Such Thatmentioning

confidence: 99%

Integrated semigroups and their applications to the abstract Cauchy problem

Neubrander¹

1988

Pacific J. Math.

169

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This paper is concerned with characterizations of those linear, closed, but not necessarily densely defined operators A on a Banach space E with nonempty resolvent set for which the abstract Cauchy problem u'(t) = Au(t), u(0) = x has unique, exponentially bounded solutions for every initial value x e D(A n ). Investigating these operators we are led to the class of "integrated semigroups". Among others, this class contains the classes of strongly continuous semigroups and cosine families and the class of exponentially bounded distribution semigroups.The given characterizations of the generators of these integrated semigroups unify and generalize the classical characterizations of generators of strongly continuous semigroups, cosine families or exponentially bounded distribution semigroups.We indicate how integrated semigroups can be used studying second order Cauchy problems u"{t) -A\u'{t) -Aiu(t) = 0, operator valued equations U'(t) = A { U(t) + U(t)A 2 and nonautonomous equations u'{t) = A(t)u(t).

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Section: If There Exists a Polynomial P(-) And A Constant W Such Thatmentioning

confidence: 99%

Integrated semigroups and their applications to the abstract Cauchy problem

Neubrander¹

1988

Pacific J. Math.

169

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“…The basis of that theory was established in the works [9,10] by R. Phillips; see also the bibliography in [11] and Ivanov's papers [12,13] concerning semigroup perturbations in locally convex spaces. Our aim in this paper is in applying the methods of perturbation theory for operator semigroups and of the theory of degenerate semigroups to the study of problem (1.1), (1.2) in the case where ker L = {0}.…”

Section: Lu(t) = Mu(t) + Nu(t) T ∈ R +mentioning

confidence: 99%

On solvability of perturbed Sobolev type equations

Федоров

Рузакова

2009

St. Petersburg Math. J.

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Abstract. Linear Sobolev type equations Lu(t) = Mu(t) + Nu(t), t ∈ R + ,are considered, with degenerate operator L, strongly (L, p)-radial operator M , and perturbing operator N . By using methods of perturbation theory for operator semigroups and the theory of degenerate semigroups, unique solvability conditions for the Cauchy problem and Showalter problem for such equations are deduced. The abstract results obtained are applied to the study of initial boundary value problems for a class of equations, the operators in which are polynomials of elliptic selfadjoint operators, including various equations of filtration theory. Perturbed linearized systems of the phase space equations and of the Navier-Stokes equations are also considered. In all cases the perturbed operators are integral or differential. §1. IntroductionWe consider the Cauchy problemfor the Sobolev type equationThis is an abstract form of initial boundary value problems for various equations and systems of equations modeling real processes [1,2,3,4]. Here U and F are Banach spaces, L ∈ L(U; F), i.e., L is a continuous linear operator, the operators M, N belong to Cl(U; F), i.e., they are linear, closed, and densely defined in U and map U to F. The papers [1,3,5,6,7,8] are devoted to finding conditions that ensure the existence of resolving semigroups from several smoothness classes for the nonperturbed Sobolev type equation (N = 0). In particular, it was shown that in the case where ker L = {0} such semigroups are degenerate. In other words, the identity of such a semigroup has a nontrivial kernel. If the operator L −1 ∈ L(F ; U ) exists, then equation (1.2) can be reduced to the formIf the operator L −1 M generates a (C 0 )-continuous operator semigroup, then problem (1.1), (1.3) can be investigated by methods of perturbation theory for operator semigroups. The basis of that theory was established in the works [9,10] by R. Phillips; see also the bibliography in [11] and Ivanov's papers [12,13] concerning semigroup perturbations in locally convex spaces. Our aim in this paper is in applying the methods of perturbation theory for operator semigroups and of the theory of degenerate semigroups to the study of problem (1.1), (1.2) in the case where ker L = {0}. In this case the perturbed equation can be reduced to a system of two equations on mutually complementary subspaces, namely, the kernel and the image of the resolving semigroup for the 2000 Mathematics Subject Classification. Primary 34G25.

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“…The generators of these semigroups form wider sets than the class of operators with the MFPHY conditions [6,7,9,10,24,25,40,42,54], etc. This allows us to study the class of Cauchy problems which are not uniformly well-posed on D(A).…”

Section: ) With the Initial Condition Z E D(a)mentioning

confidence: 99%