Abstract Cauchy problem for weakly continuous operators
Abstract Cauchy problem for weakly continuous operators
复制标题
弱连续算子的抽象柯西问题
DOI：
10.1016/j.jmaa.2015.10.027
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Toshitaka Matsumoto and Naoki Tanaka
中科院分区：
文献类型：
--
作者：
高橋泰嗣;加藤幹雄;三谷健一;Toshitaka Matsumoto and Naoki Tanaka
讨论了抽象柯西问题u ' (t)= A u (t)对于t> 0和u (0)= u 0∈D的可解性，其中D是连续嵌入在底层Banach空间X中的Banach空间Y的弱闭子集，A是一个从D到X的非线性算子，满足由泛函族描述的局部弱连续条件。在泛函的水平集上引入了一类新的弱紧性，给出了满足生长条件的弱连续可微解存在的充分必要条件。我们将得到的抽象结果应用于非线性Schrödinger方程的Cauchy问题和对数波动方程的混合问题。
We discuss the solvability of the abstract Cauchy problem u′(t)= A u (t) for t> 0 and u (0)= u 0∈ D, where D is a weakly closed subset of a Banach space Y continuously embedded in the underlying Banach space X and A is a nonlinear operator from D into X satisfying a local weak continuity condition described by a family of functionals. We introduce a new type of weak compactness on the level sets of functionals and give a necessary and sufficient condition for the existence of a weakly continuously differentiable solution satisfying a growth condition. We apply the abstract result obtained to the Cauchy problem for a nonlinear Schrödinger equation and a mixed problem for a logarithmic wave equation.