Abstract Semilinear Differential Equations and C-regularized semigroups

C-半群與半線性方程式

張幼賢(Yu-Hsien Chang) ； 趙國欽(Guo-Chin Jau)

《師大學報：數理與科技類》 42期 (1997/10) Pp. 25-52

https://doi.org/10.6301/JNTNU.1997.42.03

摘要

本文是考慮抽象半線性微分方程式（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，f:[t0,T]× X→X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。為了證明這些結果，我們先證明（0.1）所對應的非齊次方程式（抽象非齊次微分方程式）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）有唯一的古點解、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）。

關鍵字

C-正則半群 ； 指數有界C-正則半群 ； 抽象非齊次微分方程式 ； 抽象半線性微分方程式

並列摘要

The main concern of this paper is under some suitable conditions on the forcing term and the operator A to find the unique classical solution, strong solution or mild solution for the abstract semilinear initial value problem:(abstract semilinear differential equations)where A is an infinitesimal generator of a C-semigroup {T(t):t≥0}, f:[t0,T]× X→X and X is a Banach space. We also discussed the maximum interval of the existence for the mild solutions and continuous dependence of initial data. The basic technique used in this paper is the fixed point theory for differential equations in Banach space. For this purpose, we prove first that the corresponding inhomogeneous equation(abstract in-homogeneous differential equations)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.

並列關鍵字

C-regularized semigroups ； exponentially bounded C-regularized semigroups ； abstract in-homogeneous differential equations ； abstract semilinear differential equations