Ｃ－半群與半線性方程式

Abstract Semilinear Differential Equations and C-regularized Semigroups

本文是考慮抽象半線性微分方程式其中A是一個在Banach空間X上之C-半群的生成元，為一個函數。我們給予函數f某些適當的條件，使得以上之抽象半線性方程式（0.1）有唯一的古典解、強解或弱解。我們也找出弱解存在的最大時間範圍，並探討此解在趨近邊界時的行為；此外，我們也證明了解對初值條件的連續性。為了證明這結果，我們先證明（0.1）所對應的非齊次方程式在給予非齊次項函數C某些適當的條件，使得以上之抽象非齊次微分方程式（0.2）有唯一的古典解、強解或弱解。本文最大之特色是無需假設這個C-半群是指數有界（exponential bounded）。

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 equationhas 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.