Initial boundary value problems for time-fractional evolution equations in Banach spaces

We consider an initial value problem for time-fractional evolution equation in Banach space $X$: $$\text{pppa}(u(t)-a)=Au(t)+F(t),0<t<T\mathrm{.}\text{eqno}(\ast )$$ Here $u:(0,T)\text{rrrr}X$ 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 common as a generator of analytic semigroup, and in particular, we can treat a case $X=L^p(\text{OOO})$ over a bounded domain $\text{OOO}$ and a uniform elliptic operator $A$ within our framework. First we construct a solution operator $(a,F)\text{rrrr}u$ by means of $X$-valued Laplace transform, and we establish the well-posedness of (*) in classes such as weak solution and strong solutions. We discuss also mild solutions local in time for semilinear time-fractional evolution equations. Finally we apply the result on the well-posedness to an inverse problem of determining an initial value and we establish the uniqueness for the inverse problem.