AI 中文总结
针对Banach空间中阶数α∈(0,1)的时间分数阶演化方程初值问题，基于X值Laplace变换建立解公式并证明适定性，可直接处理有界域上Lp空间的一致椭圆算子情形，且适用于解的正则性、反问题等方向。
AI 中文摘要
本研究的首要目标是建立Banach空间X中阶数为$\alpha\in(0,1)$的时间分数阶演化方程初值问题的框架：$$ \pppa (u(t)-a) = Au(t) + F(t), \quad 0<t<T. \eqno{(*)} $$ 其中$u:(0,T)\rrrr X$是定义在$(0,T)$上的X值函数，$a\in X$为初值。算子A满足与解析半群生成元相同的预解式衰减条件。基于X值Laplace变换，我们建立了一个解的公式，得到了方程(*)的适定性。特别地，我们可以直接处理有界域$\OOO$上$X=L^p(\OOO)$且A为一致椭圆算子的情形。我们的理论可切实应用于解的正则性、反问题和控制问题等其他方向。
英文摘要
Our first main purpose is to establish a framework for initial value problems for time-fractional evolution equation of order $α\in (0,1)$ in Banach space $X$: $$ \pppa (u(t)-a) = Au(t) + F(t), \quad 0<t<T. \eqno{(*)} $$ Here $u: (0,T) \rrrr X$ is an $X$-valued function defined in $(0,T)$, and $a \in X$ is an initial value. The operator $A$ satisfies a decay condition of resolvent which is the same as a generator of analytic semigroup. Based on $X$-valued Laplace transforms, we establish a solution formula yielding the well-posedness for (*). In particular, we can directly treat a case $X=L^p(\OOO)$ over a bounded domain $\OOO$ and a uniform elliptic operator $A$. Our theory is feasibly applicable to other topics such as regularity of solutions, inverse problems and control problems.
