Differential Equations in Banach Spaces II. Theory of Cosine Operator Functions

Vasil'ev, V. V.; Piskarev, Sergey

doi:10.1023/b:joth.0000029697.92324.47

Cited by 58 publications

(34 citation statements)

References 166 publications

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“…Under the above assumptions, the mild solution of (1.6) is defined for all t ≥ 0 (see [48,82]). Let u n (·) = T n (·)u 0 : + → E be the only mild solution of (1.6).…”

Section: Downloaded By [University Of North Texas] At 08:02 01 Decembmentioning

confidence: 99%

A General Approximation Scheme for Attractors of Abstract Parabolic Problems

Carvalho

Piskarev

2006

Numerical Functional Analysis and Optimization

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We consider semilinear problems of the form u = Au + f (u), where A generates an exponentially decaying compact analytic C 0 -semigroup in a Banach space E , f : E → E is differentiable globally Lipschitz and bounded (E = D((−A) ) with the graph norm). Under a very general approximation scheme, we prove that attractors for such problems behave upper semicontinuously. If all equilibrium points are hyperbolic, then there is an odd number of them. If, in addition, all global solutions converge as t → ±∞, then the attractors behave lower semicontinuously. This general approximation scheme includes finite element method, projection and finite difference methods. The main assumption on the approximation is the compact convergence of resolvents, which may be applied to many other problems not related to discretization.Keywords Abstract parabolic equations; Compact convergence of resolvents; General approximation scheme; Upper and lower semicontinuity of attractors.

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“…Under the above assumptions, the mild solution of (1.6) is defined for all t ≥ 0 (see [48,82]). Let u n (·) = T n (·)u 0 : + → E be the only mild solution of (1.6).…”

Section: Downloaded By [University Of North Texas] At 08:02 01 Decembmentioning

confidence: 99%

A General Approximation Scheme for Attractors of Abstract Parabolic Problems

Carvalho

Piskarev

2006

Numerical Functional Analysis and Optimization

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“…This result is disregarded in Fattorini's monograph [20] and in the recent systematic survey on cosine function theory, due to Vasil'ev and Piskarev [46], where other results from the paper [19] are included. Note that Theorem 2.8 is a special case of our characterisation of cosine function generators on UMD spaces, Theorem 2.5.…”

Section: Recall That a Banach Space X Is Called A Umd Space If The Himentioning

confidence: 99%

Resolvent characterisation of generators of cosine functions and C 0-groups

Król

2013

J. Evol. Equ.

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Abstract. The paper provides new characterisations of generators of cosine functions and C 0 -groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C 0 -groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini's result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh's characterisation of the boundedness of the H ∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L 2 space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C 0 -semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.

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“…It is wellknown that for u 0 ∈ D(A) the unique solution u(·) of (2.1) is given by the formula u(t ) = exp(tA)u 0 for t ≥ 0. The theory of well-posed problems and numerical analysis of these problems have been extensively developed, see for instance [17,23,[29][30][31].…”

Section: Preliminariesmentioning

confidence: 99%