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

TL;DR

The paper develops a general, operator-theoretic framework for time-fractional evolution equations $d_t^{\alpha}(u-a)=Au(t)+F(t)$ in Banach spaces with $0<\alpha<1$, using an $X$-valued Laplace transform to construct a solution operator $G(t)$ and kernel $K(t)$. It establishes existence and uniqueness of strong, weak, and mild solutions, provides a concrete representation $u(t)=G(t)a+\int_{0}^{t}K(t-s)F(s)\,ds$, and proves decay and smoothing estimates via $(-A)^{\beta}$-based bounds and holomorphy properties of $G(z)$. The work extends to semilinear problems with local-in-time well-posedness and to inverse problems, including uniqueness results for determining the initial value from interior observations, even in non-Hilbert Banach spaces. The approach yields maximal-regularity-type results in Banach spaces and applies to $X=L^{p}$-spaces with elliptic operators, broadening the applicability of time-fractional diffusion theory beyond classical Hilbert-space settings.

Abstract

We consider an initial value problem for time-fractional evolution equation 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 common as a generator of analytic semigroup, and in particular, we can treat a case $X=L^p(\OOO)$ over a bounded domain $\OOO$ and a uniform elliptic operator $A$ within our framework. First we construct a solution operator $(a, F) \rrrr u$ 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.