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Global solvability for abstract semilinear evolution equations
Author(s) -
Oka Hirokazu,
Tanaka Naoki
Publication year - 2010
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200710071
Subject(s) - mathematics , lipschitz continuity , differentiable function , banach space , operator (biology) , pure mathematics , domain (mathematical analysis) , order (exchange) , nonlinear system , space (punctuation) , mathematical analysis , discrete mathematics , biochemistry , chemistry , physics , linguistics , philosophy , finance , repressor , quantum mechanics , transcription factor , economics , gene
The Cauchy problem for the abstract semilinear evolution equation u ′ ( t ) = Au ( t ) + B ( u ( t )) + C ( u ( t )) is discussed in a general Banach space X . Here A is the so‐called Hille‐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 vector‐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 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)