Global solvability for abstract semilinear evolution equations |
| |
Authors: | Hirokazu Oka Naoki Tanaka |
| |
Institution: | Department of Mathematics, Faculty of Science, Shizuoka University, Shizuoka 422‐8529, Japan |
| |
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) |
| |
Keywords: | Hille‐Yosida operator evolution operator stability condition comparison function growth condition |
|
|