Semilinear evolution equations in Banach spaces
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In this paper, the authors studied the evolution equation u′(t) = Au(t + J(u(t)), t ⩾ 0, where etA is a C 0 semi-group on a Banach space E, and J is a singular non-linear mapping defined on a subset of E. Under certain integrability conditions on the Kt, they proved existence and uniqueness of local solutions to the integral equation u(t), = etA φ + ∝0t Kt − s(u (s)) ds for all φAbout:
This article is published in Journal of Functional Analysis.The article was published on 1979-06-01 and is currently open access. It has received 163 citations till now. The article focuses on the topics: Banach space & Domain (mathematical analysis).read more
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Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system
TL;DR: In this paper, a local regular solution for the Navier-Stokes system was constructed for a class of semilinear parabolic equations with dimensionless or scaling invariant norm, where p and q are chosen so that the norm of Lq(0, T; Lp) is dimensionless.
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Existence and non-existence of global solutions for a semilinear heat equation
TL;DR: In this paper, the existence and non-existence of global solutions and the L petertodd p blow-up of non-global solutions to the initial value problem were studied under mild technical restrictions.
Journal Article
Characterizing Blow-up Using Similarity Variables
Yoshikazu Giga,Robert V. Kohn +1 more
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A nonlinear heat equation with singular initial data
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Blowup in diffusion equations: a survey
Catherine Bandle,Hermann Brunner +1 more
TL;DR: In this paper, the authors dealt with quasilinear reaction-diffusion equations for which a solution local in time exists and the problem of determining whether the solution ceases to exist for some finite time is studied.
References
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Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
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Linear Operators. Part I: General Theory.
TL;DR: Dunford and Schwartz as discussed by the authors provided a comprehensive survey of the general theory of linear operations, together with applications to the diverse fields of more classical analysis, and emphasized the significance of the relationships between the abstract theory and its applications.
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Fractional powers of operators.
TL;DR: In this paper, a definition of fractional (or complex) powers A°>, a e C, is given for closed linear operators A in a Banach space X with the resolvent set containing the negative real ray (oo, 0) and such that {λ(λ + A); 0 < λ < oo} is bounded; fundamental properties such as additivity (A*A = AP), coincidence with the iterations A* = A for integers a = n, and analytic dependence on a are discussed.