Global solvability for abstract semilinear evolution equations

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)