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Global solvability for abstract semilinear evolution equations

โœ Scribed by Hirokazu Oka; Naoki Tanaka

Publisher
John Wiley and Sons
Year
2010
Tongue
English
Weight
259 KB
Volume
283
Category
Article
ISSN
0025-584X

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โœฆ Synopsis

Abstract

The Cauchy problem for the abstract semilinear evolution equation u^โ€ฒ^(t) = Au (t) + B (u (t)) + C (u (t)) is discussed in a general Banach space X. Here A is the soโ€called Hilleโ€Yosida operator in X, B is a differentiable operator from D (A) into X, and C is a locally Lipschitz continuous operator from D (A) into itself. A vectorvalued functional defined only on X is used and appropriate conditions on the nonlinear operators B and C are imposed so that a vectorโ€valued functional defined on the domain of the operator A may be constructed in order to specify the growth of a global solution. The advantage of our formulation lies in the fact that it is possible to obtain a global solution by checking some energy inequalities concerning only low order derivatives (ยฉ 2010 WILEYโ€VCH Verlag GmbH & Co. KGaA, Weinheim)

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