Abstract Semilinear Differential Equations and C- regularized semigroups

Author: Yu-Hsien Chang (Department of Mathematics, National Taiwan Normal University), Guo-Chin Jau (Department of Mathematics, National Taiwan Normal University)

Vol.&No.：Vol. 42 (New Version)
Date：October 1997
Pages：25-52
DOI：10.6301/JNTNU.1997.42.03

Abstract：

The main concern of this paper is under some suitable conditions on the forcing term and the op-erator A to find the unique classical solution, strong solution or mild solution for the abstract semilinear initial value problem:
　　
where A is an infinitesimal generator of a C-semigroup is a Banach space. We also discussed the maximum interval of the existence for the mild solutions and contin-uous dependence of initial data. The basic technique used in this paper is the fixed point theory for dif-ferential equations in Banach space. For this purpose, we prove first that the corresponding inhomoge-neous equation
　　
has a unique classical solution, strong solution or mild solution. However, the most enjoy here is that we do not need to assume that the C-semigroup is exponentially bounded.

Keywords：C-regularized semigroups, exponentially bounded C-regularized semigroups, abstract in-homogeneous differential equations, abstract semilinear differential equations

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