Abstract
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)
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