Initial-boundary value problems for coupled systems of time-fractional diffusion equations

Abstract

This article deals with the initial-boundary value problem for a moderately coupled system of time-fractional diffusion equations. Defining the mild solution, we establish fundamental unique existence, limited smoothing property and long-time asymptotic behavior of the solution, which mostly inherit those of a single equation. Owing to the coupling effect, we also obtain the uniqueness for an inverse problem on determining all the fractional orders by the single point observation of a single component of the solution.

1 Introduction

For anomalous diffusion in heterogeneous media and viscoelastic body that conventional partial differential equations (PDEs) fail to describe, a considerable number of nonlocal models with fractional derivatives have been proposed. Especially, due to the capability of representing the time memory effect, time-fractional PDEs such as

$$\begin{aligned} (\partial _t^\alpha -\triangle )u=F \end{aligned}$$

(1.1)

with a Caputo derivative $\partial _t^\alpha$ of order $0<\alpha <2$ in time (which will be defined soon) have gathered popularity among mathematicians and multidisciplinary researchers. The past decade has witnessed a tremendous development in mathematical analysis related to (1.1) and its various generalizations: fundamental theories such as the unique existence of solutions have been established (e.g. [6, 11, 15, 20, 39]), and corresponding numerical and inverse problems have also been investigated extensively (e.g. [17, 18, 25, 28, 29]). In particular, for time-fractional diffusion equations with $\alpha <1$ in (1.1), a linear theory equivalent to their parabolic prototypes has been well constructed, and their similarity and difference have been clarified.

Among the rapidly increasing literature on fractional-related topics, however, it turns out that almost all existing researches are restricted to single and linear time-fractional PDEs. As we know, for important applications in chemistry, biology and finance, coupled systems of PDEs (represented by reaction-diffusion systems) successfully model the interaction and evolution of two or more involved components under consideration. Therefore, in the context of systems with more sophisticated mechanism due to a richer structure associated to the corresponding couplings, it is desirable to create new models based on coupled systems of time-fractional PDEs or even general nonlocal reaction-diffusion systems (e.g. [4, 5]). On the other hand, now that a linear theory for single time-fractional diffusion equations is accomplished, it seems natural to study its generalization to linear systems first and investigate them from both theoretical and practical aspects.

Keeping the above backgrounds in mind, let us set up the formulation in this article.

Let $T\in \mathbb R_+:=(0,+\infty )$ be a constant and $\Omega \subset \mathbb R^d$ ($d\in \mathbb N:=\{1,2,\dots \}$) be an open bounded domain with a smooth boundary $\partial \Omega$ (for example, of $C^2$ class). For a constant $K\in \mathbb N$, let $\alpha _1,\dots ,\alpha _K$ be constants satisfying $1>\alpha _1\ge \cdots \ge \alpha _K>0$. In this article, we consider the following initial-boundary value problem for a coupled system of time-fractional diffusion equations

$$\begin{aligned} \left\{ \begin{alignedat}{2}&\begin{aligned}&\partial _t^{\alpha _k}u_k =\textrm{div}(\varvec{A}_k(\varvec{x})\nabla u_k)+\sum _{\ell =1}^K\varvec{b}_{k\ell }(\varvec{x},t)\cdot \nabla u_\ell \\&\qquad \quad \,+\sum _{\ell =1}^K c_{k\ell }(\varvec{x},t)u_\ell +F_k(\varvec{x},t) \end{aligned}&\quad \text{ in } \Omega \times (0,T),\\&u_k=u_0^{(k)}\quad \text{ in } \Omega \times \{0\},\\&u_k=0\quad \text{ on } \partial \Omega \times (0,T), \end{alignedat}\right. \end{aligned}$$

(1.2)

$k=1,\dots ,K$, where by $\partial _t^\alpha$ ($0<\alpha <1$) we denote the Caputo derivative (e.g. Podlubny [35]) which is usually defined as

$$\begin{aligned} \partial _t^\alpha f(t):=\frac{1}{\Gamma (1-\alpha )}\int _0^t\frac{f'(\tau )}{(t-\tau )^\alpha }\,\mathrm d\tau ,\quad t>0,\ f\in C^1[0,+\infty ). \end{aligned}$$

There are several ways to define the Caputo derivative and the domain could be extended from $C^1$ to some fractional Sobolev space (e.g. [11]). Since our main concentration in this article is the coupled system, we do not discuss further on the definition of the fractional derivative. Here $\varvec{A}_k=(a_{i j}^{(k)})_{1\le i,j\le d}\in C^1(\overline{\Omega };\mathbb R_{\textrm{sym}}^{d\times d})$ ($k=1,\dots ,K$) are symmetric and strictly positive-definite matrices on $\overline{\Omega }$ . More precisely, there exists a constant $\kappa >0$ such that

$$\begin{aligned} \varvec{A}_k(\varvec{x})=(\varvec{A}_k(\varvec{x}))^\mathrm T,\quad \varvec{A}_k(\varvec{x})\varvec{\xi }\cdot \varvec{\xi }\ge \kappa |\varvec{\xi }|^2,\quad \forall \,\varvec{\xi }\in \mathbb R^d,\ \forall \,\varvec{x}\in \overline{\Omega },\ \forall \,k=1,\dots ,K, \end{aligned}$$

where $(\,\cdot \,)^\mathrm T$ denotes the transpose and $|\varvec{\xi }|^2:=\varvec{\xi }\cdot \varvec{\xi }$. Later in Section 2 we will provide further details of the assumptions on involved initial values, source terms and coefficients.

The governing equation in (1.2) is a moderately coupled system of K linear time-fractional diffusion equations, where the components are allowed to interact with each other up to the first derivative in space. The orders $\alpha _k$ of time derivatives for all components can be different from each other, and the coefficients of zeroth and first-order spatial derivatives can depend on both $\varvec{x}$ and t. Nevertheless, the coefficients of principal parts are restricted to be t-independent, since later we shall rely on the existence of eigensystems of elliptic operators $\mathcal A_k\psi =-\textrm{div}(\varvec{A}_k(\varvec{x})\nabla \psi )$. In this sense, the homogeneous Dirichlet boundary condition in (1.2) is not obligatory and can be replaced by homogeneous Neumann or Robin ones. Therefore, the formulation (1.2) covers a rather wide range of problems, which seems not yet studied in literature to the best of our knowledge.

Indeed, it reveals that existing publications, though limited, emphasize more on the nonlinear counterpart of (1.2), whereas the elliptic parts can be reasonably simple. Recently, Gal and Warma [9, Chapter 4] considered systems of fractional kinetic equations with nonlinear terms taking the form of $F_k(\varvec{x},t,u_1,\dots ,u_K)$ and discussed the existence of maximal mild and strong solutions. Very recently, Suzuki [41, 42] investigated local existence and non-existence for weakly coupled time-fractional reaction-diffusion systems with moderately and rapidly growing nonlinearities. In view of applications (e.g. [21, 32]), definitely one should consider nonlinear reaction-diffusion systems based on (1.2) and develop corresponding theories. However, as was seen in the research of traditional reaction-diffusion systems, linear coupled systems play fundamental roles especially in the discussions of super/subsolution methods and traveling waves. Hence, as the starting point, the major target of this article is to construct the basic well-posedness results for (1.2) parallel to the case of a single equation.

On the other hand, we are also interested in inverse problems associated with (1.2). Again in the framework of traditional parabolic systems, sometimes it is possible to identify multiple coefficients simultaneously from observation data of a single component due to the coupling effect. Thus, the natural curiosity is whether and to which extent such property is inherited by the fractional systems. On this direction, only Ren, Huang and Yamamoto [37] obtained the conditional stability for a coefficient inverse problem of (1.2) in the special case of $K=2$, $\alpha _1=\alpha _2=1/2$ and $d=1$ by means of Carleman estimates. Hence, in this article we also keep an eye on the minor target of studying a small inverse problem for (1.2) employing the coupling effect (see Problem 1).

For the forward problem, we prove the unique existence of a mild solution to (1.2) along with its stability with respect to initial values $u_0^{(k)}$ and source terms $F_k$ in Theorem 1. These results turn out to be parallel to those for single equations obtained in [11, 39], namely, the improvement of spatial regularity of solutions from that of $u_0^{(k)},F_k$ is at most 2. The proof generalizes the idea in [11, 22] to regard the lower order terms as a part of source terms and construct a sequence by Picard iteration, whose limit is indeed the mild solution. Further, restricting the system (1.2) to a weakly coupled one with t-independent coefficients, we investigate the long-time asymptotic behavior of the solution and obtain the sharp decay rate $t^{-\alpha _K}$ in Theorem 2, where $\alpha _K$ stands for the lowest order of the fractional derivatives. Such a result coincides with that for a multi-term time-fractional diffusion equation proved in [24]. Meanwhile, we study a parameter inverse problem on determining all the orders $\alpha _1,\dots ,\alpha _K$ by the observation of a single component $u_{k_0}$ at $\{\varvec{x}_0\}\times (0,T)$. Using the strong maximum principle for coupled elliptic systems, in Theorem 3 we show the uniqueness of this inverse problem provided that the system is not decoupled. This reflects the interaction between components as expected, which is only available for coupled systems.

The rest of this paper is organized as follows. In Section 2, we collect necessary preliminaries and give statements of the main results. Then Sections 3–5 are devoted to the proofs of Theorems 1,2 and 3, respectively. Some concluding remarks are provided in Section 6, and finally the proofs of some technical details are postponed to Appendix 1.

2 Preliminaries and main results

We start from fixing some general settings and notations. Let $L^2(\Omega )$ be the possibly complex-valued square-integrable function space and $H_0^1(\Omega )$, $H^2(\Omega )$, $H^{-1}(\Omega )$, $W^{1,\infty }(\Omega )$ etc. be the usual Sobolev spaces (e.g. [2]). The inner products of $L^2(\Omega )$ and $\mathbb C^K$ ($K\in \mathbb N$) are defined by

$$\begin{aligned}{} & {} (f,g):=\int _\Omega f(\varvec{x})\,\overline{g(\varvec{x})}\,\mathrm d\varvec{x},\quad f,g\in L^2(\Omega ),\\{} & {} \quad \varvec{\xi }\cdot \varvec{\zeta }:=\sum _{k=1}^K\xi _k\overline{\zeta _k},\quad \varvec{\xi }=(\xi _1,\dots ,\xi _K)^\mathrm T,\varvec{\zeta }=(\zeta _1,\dots ,\zeta _K)^\mathrm T\in \mathbb C^K, \end{aligned}$$

respectively, which induce the respective norms

$$\begin{aligned} \Vert f\Vert _{L^2(\Omega )}:=(f,f)^{1/2},\quad |\varvec{\xi }|:=(\varvec{\xi }\cdot \varvec{\xi })^{1/2}. \end{aligned}$$

With slight abuse of notation, we also denote the length of a multi-index $\varvec{j}=(j_1,\dots ,j_K)\in (\mathbb N\cup \{0\})^K$ by $|\varvec{j}|$, i.e., $|\varvec{j}|=\sum _{k=1}^K j_k$. Given a Banach space X, the norm of the product space $X^K$ ($K\in \mathbb N$) is defined as

$$\begin{aligned} \Vert \varvec{f}\Vert _{X^K}:=\sum _{k=1}^K\Vert f_k\Vert _X,\quad \varvec{f}=(f_1,\dots ,f_K)^\mathrm T\in X^K. \end{aligned}$$

Throughout this article, we abbreviate $\Vert \varvec{\psi }\Vert _{X^K}$ as $\Vert \varvec{\psi }\Vert _X$ for the sake of conciseness. Similarly, the inner product of $(L^2(\Omega ))^K$ is abbreviated as $(\varvec{f},\varvec{g})$:

$$\begin{aligned} (\varvec{f},\varvec{g})=\int _\Omega \varvec{f}\cdot \varvec{g}\,\mathrm d\varvec{x}=\sum _{k=1}^K(f_k,g_k), \end{aligned}$$

where

$$\begin{aligned} \varvec{f}=(f_1,\dots ,f_K)^\mathrm T,\varvec{g}=(g_1,\dots ,g_K)^\mathrm T\in (L^2(\Omega ))^K. \end{aligned}$$

For the coefficients $\varvec{b}_{k\ell }$ and $c_{k\ell }$ in (1.2), we assume

$$\begin{aligned} \begin{aligned} \varvec{b}_{k\ell }&\in (L^\infty (0,T;W^{j,\infty }(\Omega )))^d,\\ c_{k\ell }&\in L^\infty (0,T;W^{j,\infty }(\Omega )), \end{aligned} \quad k,\ell =1,\dots ,K \end{aligned}$$

(2.1)

with $j=0$ or $j=1$. Later we will see that the choice of j influences the regularity of solutions.

The governing equations in (1.2) look lengthy and we shall introduce some notations for later convenience. In the sequel, we denote $\varvec{u}:=(u_1,\dots ,u_K)^\mathrm T$ and introduce second-order elliptic operators $\mathcal A_k$ along with first-order differential operators $\mathcal P_k$ ($k=1,\dots ,K$) as

$$\begin{aligned}&\mathcal A_k:\mathcal D(\mathcal A_k):=H^2(\Omega )\cap H_0^1(\Omega )\longrightarrow L^2(\Omega ),\quad \psi \longmapsto -\textrm{div}(\varvec{A}_k(\varvec{x})\nabla \psi ),\\&\mathcal P_k:(H_0^1(\Omega ))^K\longrightarrow L^2(\Omega ),\quad \\& \varvec{\psi }=(\psi _1,\dots ,\psi _K)^\mathrm T\longmapsto \sum _{\ell =1}^K(\varvec{b}_{k\ell }(\varvec{x},t)\cdot \nabla \psi _\ell +c_{k\ell }(\varvec{x},t)\psi _\ell ). \end{aligned}$$

Then we can rewrite the governing equations in (1.2) as

$$\begin{aligned} (\partial _t^{\alpha _k}+\mathcal A_k)u_k=\mathcal P_k\varvec{u}+F_k\quad \text{ in } \Omega \times (0,T),\ k=1,\dots ,K. \end{aligned}$$

Further introducing

$$\begin{aligned}{} & {} \varvec{\alpha }:=(\alpha _1,\dots ,\alpha _K)^\mathrm T,\quad \varvec{F}(\varvec{x},t):=(F_1(\varvec{x},t),\dots ,F_K(\varvec{x},t))^\mathrm T,\\{} & {} \varvec{u}_0(\varvec{x}):=(u_0^{(1)}(\varvec{x}),\dots ,u_0^{(K)}(\varvec{x}))^\mathrm T,\quad \partial _t^{\varvec{\alpha }}\varvec{u}:=(\partial _t^{\alpha _1}u_1,\dots ,\partial _t^{\alpha _K}u_K)^\mathrm T,\\{} & {} \mathcal A\varvec{u}:=(\mathcal A_1u_1,\dots ,\mathcal A_K u_K)^\mathrm T,\quad \mathcal P\varvec{u}:=(\mathcal P_1\varvec{u},\dots ,\mathcal P_K\varvec{u})^\mathrm T, \end{aligned}$$

we can collect the above K equations to represent (1.2) in a vector form as

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t^{\varvec{\alpha }}+\mathcal A)\varvec{u}=\mathcal P\varvec{u}+\varvec{F} &{} \text{ in } \Omega \times (0,T),\\ \varvec{u}=\varvec{u}_0 &{} \text{ in } \Omega \times \{0\},\\ \varvec{u}=\varvec{0} &{} \text{ on } \partial \Omega \times (0,T). \end{array}\right. } \end{aligned}$$

(2.2)

In the sequel, we abbreviate $\varvec{u}(t)=\varvec{u}(\,\cdot ,t)$ as a vector-valued $\varvec{x}$-dependent function.

Next, we introduce the eigensystem $\{(\lambda _n^{(k)},\varphi _n^{(k)})\}_{n\in \mathbb N}$ of the operator $\mathcal A_k$ ($k=1,\dots ,K$). More precisely, the sequence $\{\lambda _n^{(k)}\}\subset \mathbb R_+$ satisfies

$$\begin{aligned} 0<\lambda _1^{(k)}<\lambda _2^{(k)}\le \cdots ,\quad \lim _{n\rightarrow \infty }\lambda _n^{(k)}=+\infty ,\quad k=1,\dots ,K \end{aligned}$$

and $\{\varphi _n^{(k)}\}\subset \mathcal D(\mathcal A_k)$ forms a complete orthonormal basis of $L^2(\Omega )$ and satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal A_k\varphi _n^{(k)}=\lambda _n^{(k)}\varphi _n^{(k)} &{} \text{ in } \Omega ,\\ \varphi _n^{(k)}=0 &{} \text{ on } \partial \Omega , \end{array}\right. }\quad n\in \mathbb N. \end{aligned}$$

Then for $\gamma >0$ and $k=1,\dots ,K$, one can define the fractional power $\mathcal A_k^\gamma$ as (e.g. Pazy [34])

$$\begin{aligned}&\mathcal D(\mathcal A_k^\gamma ):=\left\{ \psi \in L^2(\Omega )\left| \,\sum _{n=1}^\infty \left| (\lambda _n^{(k)})^\gamma (\psi ,\varphi _n^{(k)})\right| ^2<\infty \right. \right\} ,\\&\mathcal A_k^\gamma \psi :=\sum _{n=1}^\infty (\lambda _n^{(k)})^\gamma (\psi ,\varphi _n^{(k)})\varphi _n^{(k)}, \end{aligned}$$

which is equipped with the norm

$$\begin{aligned} \Vert \psi \Vert _{\mathcal D(\mathcal A_k^\gamma )}:=\left( \sum _{n=1}^\infty \left| (\lambda _n^{(k)})^\gamma (\psi ,\varphi _n^{(k)})\right| ^2\right) ^{1/2}. \end{aligned}$$

For $-1\le \gamma <0$ and $k=1,\dots ,K$, we define $\mathcal D(\mathcal A_k^\gamma )$ as the dual space of $\mathcal D(\mathcal A_k^{-\gamma })$ similarly with the norm

$$\begin{aligned} \Vert \psi \Vert _{\mathcal D(\mathcal A_k^\gamma )}:=\left( \sum _{n=1}^\infty \left| (\lambda _n^{(k)})^\gamma \langle \psi ,\varphi _n^{(k)}\rangle \right| ^2\right) ^{1/2}, \end{aligned}$$

where $\langle \,\cdot ,\,\cdot \,\rangle$ denotes the pairing between $\mathcal D(\mathcal A_k^\gamma )$ and $\mathcal D(\mathcal A_k^{-\gamma })$. Then $\mathcal D(\mathcal A_k^\gamma )$ is well-defined for all $\gamma \ge -1$ and we have

$$\begin{aligned} \Vert \psi \Vert _{\mathcal D(\mathcal A_k^\gamma )}=\Vert \mathcal A_k^\gamma \psi \Vert _{L^2(\Omega )},\quad \psi \in \mathcal D(\mathcal A_k^\gamma ). \end{aligned}$$

We know that $\mathcal D(\mathcal A_k^\gamma )$ is a Hilbert space and satisfies $\mathcal D(\mathcal A_k^\gamma )\subset H^{2\gamma }(\Omega )$ for $\gamma >0$. Especially, there hold $\mathcal D(\mathcal A_k^{1/2})=H_0^1(\Omega )$, $\mathcal D(\mathcal A_k^{-1/2})=H^{-1}(\Omega )$ and the norm equivalence $\Vert \cdot \Vert _{\mathcal D(\mathcal A_k^\gamma )}\sim \Vert \cdot \Vert _{H^{2\gamma }(\Omega )}$. Similarly as before, for $\varvec{\psi }=(\psi _1,\dots ,\psi _K)^\mathrm T$ and $\gamma \ge -1$, we denote

$$\begin{aligned} \mathcal A^\gamma \varvec{\psi }:=(\mathcal A_1^\gamma \psi _1,\dots ,\mathcal A_K^\gamma \psi _K)^\mathrm T\end{aligned}$$

and define $\mathcal D(\mathcal A^\gamma )$ as well as its norm $\Vert \cdot \Vert _{\mathcal D(\mathcal A^\gamma )}$ correspondingly.

Now we invoke the frequently used Mittag-Leffler function (e.g. [35])

$$\begin{aligned} E_{\alpha ,\beta }(z):=\sum _{m=0}^\infty \frac{z^m}{\Gamma (\alpha m+\beta )},\quad \alpha \in \mathbb R_+,\ \beta \in \mathbb R,\ z\in \mathbb C, \end{aligned}$$

where $\Gamma (\,\cdot \,)$ is the Gamma function. The following estimate of $E_{\alpha ,\beta }(z)$ is well-known.

Lemma 1

[see [35, Theorem 1.5]] Let $0<\alpha <2$ and $\beta >0$. Then there exists a constant $C_0>0$ depending only on $\alpha ,\beta$ such that

$$\begin{aligned} 0<|E_{\alpha ,\beta }(z)|\le C_0(1+z)^{(1-\beta )/\alpha }\exp (z^{1/\alpha }),\quad z>0. \end{aligned}$$

Employing the Mittag-Leffler functions, we further define the resolvent operator $\mathcal S_k(z):L^2(\Omega )\longrightarrow L^2(\Omega )$ as well as its termwise differentiation $\mathcal S_k'(z):L^2(\Omega )\longrightarrow L^2(\Omega )$ for $z\in \mathbb C\setminus \{0\}$ and $k=1,\dots ,K$ by

$$\begin{aligned} \begin{aligned} \mathcal S_k(z)\psi&:=\sum _{n=1}^\infty E_{\alpha _k,1}(-\lambda _n^{(k)} z^{\alpha _k})(\psi ,\varphi _n^{(k)})\varphi _n^{(k)},\\ \mathcal S_k'(z)\psi&:=- z^{\alpha _k-1}\sum _{n=1}^\infty \lambda _n^{(k)}E_{\alpha _k,\alpha _k}(-\lambda _n^{(k)} z^{\alpha _k})(\psi ,\varphi _n^{(k)})\varphi _n^{(k)}, \end{aligned}\quad \psi \in L^2(\Omega ). \end{aligned}$$

We recall the key properties of $\mathcal S_k(z)$ and $\mathcal S_k'(z)$.

Lemma 2

(see [22]) For $k=1,\dots ,K$ and $\psi \in L^2(\Omega ),$ the followings hold true.

• (i) For $0\le \gamma \le 1,$ there exists a constant $C_1>0$ depending only on $\Omega ,\varvec{\alpha },\mathcal A$ such that

  $$\begin{aligned} \Vert \mathcal A_k^\gamma \mathcal S_k(z)\psi \Vert _{L^2(\Omega )}&\le C_1\Vert \psi \Vert _{L^2(\Omega )}|z|^{-\alpha _k\gamma },\nonumber \\ \Vert \mathcal A_k^{\gamma -1}\mathcal S_k'(z)\psi \Vert _{L^2(\Omega )}&\le C_1\Vert \psi \Vert _{L^2(\Omega )}|z|^{\alpha _k(1-\gamma )-1} \end{aligned}$$

  (2.3)

  for all $z\in \{z\in \mathbb C\setminus \{0\}\mid |\arg z|<\pi /2\}$.

• (ii) There holds $\lim _{z\rightarrow 0}\Vert \mathcal S_k(z)\psi -\psi \Vert _{L^2(\Omega )}=0$.

In the same manner as before, for $\varvec{\psi }=(\psi _1,\dots ,\psi _K)^\mathrm T$ we denote

$$\begin{aligned} \mathcal S(z)\varvec{\psi }:=(\mathcal S_1(z)\psi _1,\dots ,\mathcal S_K(z)\psi _K)^\mathrm T. \end{aligned}$$

Let us close the preliminaries by considering the initial-boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t^{\varvec{\alpha }}+\mathcal A)\varvec{v}=\varvec{G} &{} \text{ in } \Omega \times (0,T),\\ \varvec{v}=\varvec{v}_0 &{} \text{ in } \Omega \times \{0\},\\ \varvec{v}=\varvec{0} &{} \text{ on } \partial \Omega \times (0,T) \end{array}\right. } \end{aligned}$$

(2.4)

for K independent equations of $\varvec{v}=(v_1,\dots ,v_K)^\mathrm T$. The following lemma is a direct consequence of the well-posedness results for single equations.

Lemma 3

Let $\varvec{v}_0\in (L^2(\Omega ))^K$ and $\varvec{G}\in (L^p(0,T;L^2(\Omega )))^K$ with $p\in [1,\infty ]$.

• (i) If $\varvec{G}\equiv \varvec{0},$ then (2.4) admits a unique solution $\varvec{v}\in L^{1/\gamma }(0,T;\mathcal D(\mathcal A^\gamma ))$ $(0\le \gamma \le 1)$ which takes the form

  $$\begin{aligned} \varvec{v}(t)=\mathcal S(t)\varvec{v}_0,\quad 0<t<T \end{aligned}$$

  (2.5)

  and satisfies

  $$\begin{aligned} \lim _{t\rightarrow 0}\Vert \varvec{v}(t)-\varvec{v}_0\Vert _{L^2(\Omega )}=0. \end{aligned}$$

  (2.6)

  Here we understand $1/\gamma =\infty$ if $\gamma =0$. Moreover,  there holds

  $$\begin{aligned} \Vert \varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\le C_1\sum _{k=1}^K\Vert v_0^{(k)}\Vert _{L^2(\Omega )}t^{-\alpha _k\gamma },\quad 0<t<T,\ 0\le \gamma \le 1, \end{aligned}$$

  (2.7)

  where $C_1$ is the constant in Lemma 2.

• (ii) If $\varvec{v}_0\equiv \varvec{0},$ then (2.4) admits a unique solution $\varvec{v}\in L^p(0,T;\mathcal D(\mathcal A^\gamma ))$ which takes the form

  $$\begin{aligned} \varvec{v}(t)=-\int _0^t\mathcal A^{-1}\mathcal S'(t-\tau )\varvec{G}(\tau )\,\mathrm d\tau ,\quad 0<t<T, \end{aligned}$$

  (2.8)

  where $\gamma =1$ if $p=2$ and $0\le \gamma <1$ if $p\ne 2$. Moreover,  there exists a constant $C_2>0$ depending only on $\Omega ,\varvec{\alpha },\mathcal A,T,\gamma$ such that

  $$\begin{aligned} \Vert \varvec{v}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}\le C_2\Vert \varvec{G}\Vert _{L^p(0,T;L^2(\Omega ))}. \end{aligned}$$

  (2.9)

The solution representations (2.5) and (2.8) as well as the convergence (2.6) are well-known in literature (see e.g. [39]), and the estimate (2.7) follows immediately from Lemma 2. As for the estimate (2.9), we refer to [11, Theorem 1.4] for the case of $p=2$ and [24, Theorem 2.2] for the case of $p\ne 2$. In particular, the constant $C_2$ in (2.9) may tend to $\infty$ as $\gamma \rightarrow 1$ if $p\ne 2$, while keeps uniform with respect to $\gamma$ if $p=2$.

Now we are well prepared to investigate the initial-boundary value problem (1.2) or equivalently (2.2). Following the same line of [11, 22] and regarding the lower order term $\mathcal P\varvec{u}$ in (2.2) as a part of the source term, we employ the representations (2.5) and (2.8) to conclude that the solution to (2.2) should formally satisfy

$$\begin{aligned} \varvec{u}=\varvec{w}+\mathcal Q\varvec{u}\quad \text{ in } \Omega \times (0,T), \end{aligned}$$

(2.10)

where

$$\begin{aligned}{} & {} \varvec{w}(t):=\mathcal S(t)\varvec{u}_0-\int _0^t\mathcal A^{-1}\mathcal S^\prime (t-\tau )\varvec{F}(\tau )\,\mathrm d\tau ,\nonumber \\{} & {} \quad \mathcal Q\varvec{u}(t):=-\int _0^t\mathcal A^{-1}\mathcal S'(t-\tau )(\mathcal P\varvec{u})(\tau )\,\mathrm d\tau . \end{aligned}$$

(2.11)

This encourages us to propose the following definition of a solution to (2.2).

Definition 1

(Mild solution) Let $\varvec{u}_0\in (L^2(\Omega ))^K,$ $\varvec{F}\in (L^p(0,T;L^2(\Omega )))^K$ with $p\in [1,\infty ]$ and assume (2.1) with $j=0$. We say that $\varvec{u}$ is a mild solution to the initial-boundary value problem (2.2) if it satisfies the integral equation (2.10).

Now we state the first main result in this article, which validates the well-posedness of the initial-boundary value problem (2.2) defined above.

Theorem 1

Under the same assumptions in Definition 1, the followings hold true.

• (i) If $\varvec{F}\equiv \varvec{0},$ then there exists a unique mild solution $\varvec{u}\in L^{1/\gamma }(0,T;\mathcal D(\mathcal A^\gamma ))$ $(0\le \gamma \le 1)$ to (2.2) satisfying

  $$\begin{aligned} \lim _{t\rightarrow 0}\Vert \varvec{u}(t)-\varvec{u}_0\Vert _{L^2(\Omega )}=0. \end{aligned}$$

  (2.12)

  Here we understand $1/\gamma =\infty$ if $\gamma =0$. Moreover,  there exist constants $C>0$ and $C_{T,\gamma }>0$ depending only on $\Omega ,\varvec{\alpha },\mathcal A,\mathcal P$ and $\Omega ,\varvec{\alpha },\mathcal A,\mathcal P,T,\gamma ,$ respectively such that

  $$\begin{aligned}{} & {} \Vert \varvec{u}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\le C\exp (C\,t)\Vert \varvec{u}_0\Vert _{L^2(\Omega )}t^{-\alpha _1\gamma },\quad 0<t<T,\ 0\le \gamma <1, \qquad \qquad \end{aligned}$$

  (2.13)
  $$\begin{aligned}{} & {} \Vert \varvec{u}\Vert _{L^{1/\gamma }(0,T;\mathcal D(\mathcal A^\gamma ))}\le C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )},\quad 0\le \gamma \le 1. \end{aligned}$$

  (2.14)

  If we further assume (2.1) with $j=1,$ then (2.13) also holds for $\gamma =1$. Moreover,  the solution $\varvec{u}:(0,T]\longrightarrow \mathcal D(\mathcal A^\gamma )$ is analytic for $\gamma \in [0,1)$.

• (ii) Let $\gamma =1$ if $p=2$ and $0\le \gamma <1$ if $p\ne 2$. If $\varvec{u}_0\equiv \varvec{0},$ then (2.2) admits a unique mild solution $\varvec{u}\in L^p(0,T;\mathcal D(\mathcal A^\gamma ))$. Moreover,  there exists a constant $C_{T,\gamma }>0$ depending only on $\Omega ,\varvec{\alpha },\mathcal A,\mathcal P,T,\gamma$ such that

  $$\begin{aligned} \Vert \varvec{u}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}\le C_{T,\gamma }\Vert \varvec{F}\Vert _{L^p(0,T;L^2(\Omega ))}. \end{aligned}$$

  (2.15)

In the above theorem, the system (2.2) is coupled up to first derivatives in space. In the sequel, we additionally assume

$$\begin{aligned} \varvec{b}_{k\ell }\equiv \varvec{0},\quad c_{k\ell }(\varvec{x},t)=c_{k\ell }(\varvec{x}),\quad k,\ell =1,\dots ,K \end{aligned}$$

(2.16)

in (2.2), that is, we restrict (2.2) to a weakly coupled system with t-independent zeroth order coefficients. For simplicity, by introducing a matrix-valued function $\varvec{C}:=(c_{k\ell })_{1\le k,\ell \le K}$, it is obvious that $\mathcal P\varvec{u}=\varvec{C}\varvec{u}$ in (2.2).

Under the above assumption, we discuss the long-time asymptotic behavior of the solution to (2.2). The same topics were considered by many authors in the case of a single equation (see e.g. [24, 39] and the references therein). On the basis of Theorem 1, we establish the following theorem.

Theorem 2

Let $T=+\infty ,$ $\varvec{F}=\varvec{0}$ and $\varvec{u}_0\in (L^2(\Omega ))^K$ in (2.2). Assume (2.16) and that $\varvec{C}=(c_{k\ell })_{1\le k,\ell \le K}\in (L^\infty (\Omega ))^{K\times K}$ is negative semi-definite in $\Omega$. Then the unique solution to (2.2) admits the asymptotic behavior

$$\begin{aligned} \Vert \varvec{u}(\,\cdot ,t)\Vert _{H^2(\Omega )}\le C\,t^{-\alpha _K}\Vert \varvec{u}_0\Vert _{L^2(\Omega )},\quad \forall \,t\ge t_0 \end{aligned}$$

for arbitrarily fixed $t_0>0$. Here the constant $C>0$ is independent of $t,\varvec{u}_0$ but depends on $t_0,K,\Omega ,\varvec{\alpha },\varvec{C}$ and $\mathcal A$. Moreover,  the decay rate $t^{-\alpha _K}$ is sharp provided that $u_0^{(K)}\not \equiv 0$ in $\Omega$.

The above theorem asserts that the decay rate of the solution $\varvec{u}$ to (2.2) is $t^{-\alpha _K}$ at best as $t\rightarrow +\infty$, where $\alpha _K$ is the lowest order of the time-fractional derivatives. This means that if $\Vert \varvec{u}(\,\cdot \,,t)\Vert _{H^2(\Omega )}=o(t^{-\alpha _K})$ as $t\rightarrow +\infty$, then $\varvec{u}$ must vanish identically in $\Omega \times \mathbb R_+$, so that we call the decay rate $t^{-\alpha _K}$ sharp. This coincides with the result in the multi-term case considered by [24] in which the decay rate of the solution was shown to be dominated only by the lowest order of the fractional derivatives.

On the same direction, next we consider the following inverse problem.

Problem 1

Let $\varvec{u}$ satisfy (2.2) and fix $T>0,$ $\varvec{x}_0\in \Omega ,$ $k_0\in \{1,\dots ,K\}$ arbitrarily. Determine the orders $\varvec{\alpha }=(\alpha _1,\dots ,\alpha _K)^\mathrm T$ of (2.2) by the single point observation of the $k_0$-th component $u_{k_0}$ of $\varvec{u}$ at $\{\varvec{x}_0\}\times (0,T)$.

The orders of the Caputo derivatives in time-fractional partial differential equations are related to important physical parameters describing e.g. the heterogeneity of media, whose determination is of great interest from both applied and mathematical aspects. Therefore, similar inverse problems have been considered intensively in the case of a single equation (see e.g. [13, 16, 23, 26, 27, 30] and the survey [25]). Here we attempt to extend the result from a single equation to a weakly coupled system. Since the components of the solution $\varvec{u}$ interact each other, in Problem 1 we ask the possibility of identifying all the orders $\varvec{\alpha }$ by only observing a single component. The following theorem gives an affirmative answer to the uniqueness for Problem 1.

Theorem 3

Let $d=1,2,3,$ $\varvec{F}=\varvec{0}$ and $\varvec{u}_0\!\in \!(L^2(\Omega ))^K\!$ satisfy $u_0^{(k)}\!\ge \!0,\not \equiv 0$ in

$\Omega$ for $k=1,\dots ,K$. Assume (2.16) and that $\varvec{C}=(c_{k\ell })_{1\le k,\ell \le K}\!\in \!(L^\infty (\Omega ))^{K\times K}$ satisfies

$$\begin{aligned}{} & {} c_{k\ell }\ge 0,\not \equiv 0\quad \text{ on } \overline{\Omega },\ k,\ell =1,\dots ,K,\ k\ne \ell , \end{aligned}$$

(2.17)

$$\begin{aligned}{} & {} \quad \sum _{\ell =1}^K c_{k\ell }\le 0\quad \text{ on } \overline{\Omega },\ k=1,\dots ,K. \end{aligned}$$

(2.18)

Further, let $\varvec{u}$ and $\varvec{v}$ be the solutions to (2.2) with the fractional orders $\varvec{\alpha }$ and $\varvec{\beta },$ respectively. Then for arbitrarily fixed $T>0,$ $\varvec{x}_0\in \Omega$ and $k_0\in \{1,\dots ,K\},$ we conclude that $u_{k_0}=v_{k_0}$ at $\{\varvec{x}_0\}\times (0,T)$ implies $\varvec{\alpha }=\varvec{\beta }$.

In the above theorem, we additionally assume that the dimension $d=1,2,3$ since we need the Sobolev embedding $H^2(\Omega )\subset C(\overline{\Omega })$ in the proof. Theorem 3 also holds true for $d>3$ provided that the coefficients in $\mathcal A,\varvec{C}$ and the initial value $\varvec{u}_0$ are sufficiently smooth, but here we omit the details. Moreover, we interpret a function $f\in L^2(\Omega )$ satisfying $f\ge 0,\not \equiv 0$ as $f\ge 0$ in $\Omega$ and the measure of $\{\varvec{x}\in \Omega \mid f(\varvec{x})>0\}$ is not zero.

Remark 1

The conditions (2.17)–(2.18) imposed on the matrix $\varvec{C}$ are sufficient but not necessary for Theorem 3. Such conditions are understood as cooperativeness in a coupled system. Actually, one can find a similar condition

$$\begin{aligned} c_{k\ell }\ge 0\quad \text{ on } \overline{\Omega },\ k,\ell =1,\dots ,K,\ k\ne \ell \end{aligned}$$

(2.19)

and (2.18) in many other papers (e.g. [8, 43]) which are known as classical sufficient conditions for the maximum principle for weakly coupled elliptic systems. However, we need a slightly stronger condition (2.17) than (2.19) in Theorem 3 because we attempt to determine all the orders by only one component measurement of the solution, which requires that the system (2.2) should not be decoupled (i.e. the decoupled case $c_{k\ell }\equiv 0$ for all $k\ne \ell$ should be excluded). On the other hand, it is readily seen from the proof that under the weaker assumption (2.19) (i.e. the decoupled case is allowed), one can obtain the uniqueness of determining the orders by observing all the components of $\varvec{u}$. Moreover, as we state in the following corollary, it is possible to choose different observation point for each component, that is, we can observe the k-th component $u_k$ at $\varvec{x}_0^{(k)}$ for each k.

Corollary 1

Under the same assumptions on $d,\varvec{F},\varvec{u}_0$ as those in Theorem 3, assume (2.16) and that $\varvec{C}=(c_{k\ell })_{1\le k,\ell \le K}\in (L^\infty (\Omega ))^{K\times K}$ satisfies the conditions (2.18) and (2.19). Further,  let $\varvec{u}$ and $\varvec{v}$ be the solutions to (2.2) with the fractional orders $\varvec{\alpha }$ and $\varvec{\beta },$ respectively. Then for arbitrarily fixed $T>0$ and $\varvec{x}_0^{(1)},\dots , \varvec{x}_0^{(K)}\in \Omega ,$ we conclude that $u_k(\varvec{x}_0^{(k)},t)=v_k(\varvec{x}_0^{(k)},t)$ for all $k=1,\dots ,K$ and $t\in (0,T)$ implies $\varvec{\alpha }=\varvec{\beta }$.

3 Proof of Theorem 1

In the integral equation (2.10), it is readily seen that $\varvec{w}$ relies only on $\varvec{u}_0$ and $\varvec{F}$, which is well analyzed in view of Lemma 3. On the other hand, the term $\mathcal Q\varvec{u}$ involves the solution $\varvec{u}$ itself, which is more essential in the discussion of the unique existence of the mild solution. To this end, we should first investigate the operator $\mathcal Q$.

Lemma 4

Let $j=0$ in the condition (2.1) and $1/2\le \gamma <1,$ $p\in [1,+\infty ]$ and

$$\begin{aligned} \mathcal Q:L^p(0,T;\mathcal D(\mathcal A^\gamma ))\longrightarrow L^p(0,T;\mathcal D(\mathcal A^\gamma )) \end{aligned}$$

be the linear operator defined by (2.11). Then for any $m\in \mathbb N$ and a.e. $t\in (0,T),$ there holds

$$\begin{aligned} \Vert \mathcal Q^m\varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\le M^m\left( \sum _{k=1}^K J^{\alpha _k(1-\gamma )}\right) ^m\Vert \varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )} \end{aligned}$$

(3.1)

for $\varvec{v}\in L^p(0,T;\mathcal D(\mathcal A^\gamma )),$ where

$$\begin{aligned}&M:=C_1L_0\max _{1\le k\le K}\Gamma (\alpha _k(1-\gamma ))=C_1L_0\,\Gamma (\alpha _K(1-\gamma )), \nonumber \\&L_0:=\max \left\{ 1,\sum _{k,\ell =1}^K\left( \Vert \varvec{b}_{k\ell }\Vert _{L^\infty (\Omega \times (0,T))}+\Vert c_{k\ell }\Vert _{L^\infty (\Omega \times (0,T))}\right) \right\} . \end{aligned}$$

(3.2)

Here $C_1$ is the constant in Lemma 2.

Proof

Let us show (3.1) by induction and first deal with the case of $m=1$. Noticing that $\mathcal P$ is a first-order differential operator with $L^\infty$ coefficients, we utilize (2.3) in Lemma 2 to estimate for a.e. $t\in (0,T)$ that

$$\begin{aligned} \Vert \mathcal Q\varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}&=\sum _{k=1}^K\left\| \int _0^t\mathcal A_k^{-1}\mathcal S_k'(t-\tau )(\mathcal P_k\varvec{v})(\tau )\,\mathrm d\tau \right\| _{\mathcal D(\mathcal A_k^\gamma )}\\&\le \sum _{k=1}^K\int _0^t\left\| \mathcal A_k^{\gamma -1}\mathcal S_k'(t-\tau )\left( \mathcal P_k\varvec{v}\right) (\tau )\right\| _{L^2(\Omega )}\mathrm d\tau \\&\le C_1\sum _{k=1}^K\int _0^t\left\| \left( \mathcal P_k\varvec{v}\right) (\tau )\right\| _{L^2(\Omega )}(t-\tau )^{\alpha _k(1-\gamma )-1}\,\mathrm d\tau \\&=C_1L_0\sum _{k=1}^K\int _0^t\Vert \varvec{v}(\tau )\Vert _{\mathcal D(\mathcal A^\gamma )}(t-\tau )^{\alpha _k(1-\gamma )-1}\,\mathrm d\tau . \end{aligned}$$

By the definition of the Riemann-Liouville fractional integral, we further derive

$$\begin{aligned} \Vert \mathcal Q\varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}&\le C_1L_0\sum _{k=1}^K\Gamma (\alpha _k(1-\gamma ))J^{\alpha _k(1-\gamma )}\left( \Vert \varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\right) \nonumber \\&\le C_1L_0\max _{1\le k\le K}\Gamma (\alpha _k(1-\gamma ))\sum _{k=1}^K J^{\alpha _k(1-\gamma )}\left( \Vert \varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\right) . \end{aligned}$$

(3.3)

This verifies (3.1) for $m=1$ by the definition (3.2) of M. For $m\ge 2$, assume that (3.1) holds for $\ell =1,\dots ,m-1$. Then for a.e. $t\in (0,T)$, we employ the inductive assumption to estimate

$$\begin{aligned} \Vert \mathcal Q^m\varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}&=\Vert \mathcal Q^{m-1}(\mathcal Q\varvec{v})(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\\&\le M^{m-1}\left( \sum _{k=1}^K J^{\alpha _k(1-\gamma )}\right) ^{m-1}\left( \Vert \mathcal Q\varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\right) \\&\le M^{m-1}\left( \sum _{k=1}^K J^{\alpha _k(1-\gamma )}\right) ^{m-1}\left( M\sum _{k=1}^K J^{\alpha _k(1-\gamma )}\Vert \varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\right) \\&=M^m\left( \sum _{k=1}^K J^{\alpha _k(1-\gamma )}\right) ^m\Vert \varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}. \end{aligned}$$

Here the last inequality is due to the estimate (3.3). This indicates that (3.1) holds for any $m\in \mathbb N$, which completes the proof. $\square$

On the basis of (2.10), we can construct a sequence $\{\varvec{u}^{(m)}\}_{m=0}^\infty$ iteratively by

$$\begin{aligned} \varvec{u}^{(m)}(t):={\left\{ \begin{array}{ll} \varvec{0}, &{} m=0,\\ \varvec{w}(t) + \mathcal Q\varvec{u}^{(m-1)}(t), &{} m=1,2,\dots . \end{array}\right. } \end{aligned}$$

(3.4)

Our strategy is to show the convergence of $\{\varvec{u}^{(m)}\}$ in some function spaces, whose limit is indeed a mild solution of (2.2). For proving the time-analyticity of the solution, it is convenient to consider the above iteration (3.4) in the complex plane. To this end, we first change variables and rewrite (3.4) as

$$\begin{aligned} \varvec{u}^{(m)}(t)=\varvec{w}(t)-t\int _0^1\mathcal A^{-1}\mathcal S'(\tau t)(\mathcal P\textbf{u}^{(m-1)}((1-\tau )t))\,\mathrm d\tau . \end{aligned}$$

Moreover, we extend the variable t from (0, T) to the sector $S:=\{z\in \mathbb C\setminus \{0\}\mid |\arg z|<\pi /2\}$ to obtain

$$\begin{aligned} \varvec{u}^{(m)}(z)&=\varvec{w}(z)-z\int _0^1\mathcal A^{-1}\mathcal S'(\tau z)(\mathcal P\textbf{u}^{(m-1)}((1-\tau )z))\,\mathrm d\tau \\&=:\varvec{w}(z)+\mathcal Q\mathcal P\textbf{u}^{(m-1)}(z),\quad m=1,2,\dots . \end{aligned}$$

To proceed, we shall give some estimates for $\varvec{u}^{(m)}$ in appropriate spaces.

Lemma 5

Under the assumptions in Theorem 1, define a sequence $\{\varvec{u}^{(m)}\}_{m=0}^\infty$ by (3.4). Let $C_1$ and $C_2$ be the constants in Lemmas 2 and 3, respectively, and M be defined by (3.2).

(i) If $\varvec{F}\equiv \varvec{0},$ then for $m=0,1,\dots$ and $1/2\le \gamma <1,$ there holds

$$\begin{aligned} \begin{aligned}&\quad \,\,\Vert (\varvec{u}^{(m+1)}-\varvec{u}^{(m)})(z)\Vert _{\mathcal D(\mathcal A^\gamma )}\\&\le C_3M^{m+1}\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\sum _{|\varvec{j}|=m}\frac{m!}{j_1!\cdots j_K!}\frac{|z|^{\varvec{\beta }\cdot \varvec{j}-\alpha _1\gamma }}{\Gamma (\varvec{\beta }\cdot \varvec{j}+1-\alpha _1\gamma )} \end{aligned} \end{aligned}$$

(3.5)

for $z\in S$ satisfying $|z|\le T,$ where $\beta _k:=\alpha _k(1-\gamma ),$ $\varvec{\beta }=(\beta _1,\dots ,\beta _K)^\mathrm T,$ $\varvec{j}=(j_1,\dots ,j_K)^\mathrm T\in (\mathbb N\cup \{0\})^K$ and

$$\begin{aligned} C_3:=C_1\max _{1\le k\le K} T^{(\alpha _1-\alpha _k)\gamma }. \end{aligned}$$

(ii) If $\varvec{u}_0\equiv \varvec{0},$ let $\gamma =1$ for $p=2$ and $0\le \gamma <1$ for $p\ne 2,$ then for $m=0,1,\dots ,$ there holds

$$\begin{aligned} \begin{aligned}&\quad \,\,\Vert \varvec{u}^{(m+1)}-\varvec{u}^{(m)}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}\\&\le C_4(M K)^m\Vert \varvec{F}\Vert _{L^p(0,T;L^2(\Omega ))}\frac{\max \{1,T^{\overline{\beta }m}\}}{\Gamma (\underline{\beta }m+1)}, \end{aligned} \end{aligned}$$

(3.6)

where $\overline{\beta }:=\max \{\beta _1,\dots ,\beta _K\},$ $\underline{\beta }:=\min \{\beta _1,\dots ,\beta _K\}$ and

$$\begin{aligned} C_4:= C_2\left( \min _{y\ge 0}\Gamma (1+y)\right) ^{-1}. \end{aligned}$$

Proof

(i) If $\varvec{F}\equiv \varvec{0}$, then $\varvec{w}(z)=\mathcal S(z)\varvec{u}_0$. Taking advantage of the estimate (2.7) in Lemma 3, we obtain

$$\begin{aligned} \Vert \varvec{w}(z)\Vert _{\mathcal D(\mathcal A^\gamma )}&\le C_1\sum _{k=1}^K\Vert u_0^{(k)}\Vert _{L^2(\Omega )}|z|^{-\alpha _k\gamma }\\&\le C_1|z|^{-\alpha _1\gamma }\sum _{k=1}^K T^{(\alpha _1-\alpha _k)\gamma }\Vert u_0^{(k)}\Vert _{L^2(\Omega )}=C_3|z|^{-\alpha _1\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )}, \end{aligned}$$

which is exactly (3.5) for $m=0$. Next, for any $m\in \mathbb N$, using a method similar to estimating $\mathcal Q\varvec{v}(t)$ in Lemma 4, we can get

$$\begin{aligned} \Vert \mathcal Q\varvec{v}(z)\Vert _{\mathcal D(\mathcal A^\gamma )}&=|z|\sum _{k=1}^K\left\| \int _0^1 \mathcal A_k^{-1}\mathcal S_k'(\tau z)(\mathcal P_k\varvec{v})((1-\tau )z)\,\mathrm d\tau \right\| _{\mathcal D(\mathcal A_k^\gamma )}\\&\le |z|\sum _{k=1}^K\int _0^1\left\| \mathcal A_k^{\gamma -1}\mathcal S_k'(\tau z)\left( \mathcal P_k\varvec{v}\right) ((1-\tau )z)\right\| _{L^2(\Omega )}\mathrm d\tau \\&\le C_1\sum _{k=1}^K|z|^{\alpha _k(1-\gamma )}\int _0^1\Vert (\mathcal P_k\varvec{v})(\tau z)\Vert _{L^2(\Omega )}(1-\tau )^{\alpha _k(1-\gamma )-1}\,\mathrm d\tau \\&\le C_1L_0\sum _{k=1}^K\int _0^{|z|}\Vert \varvec{v}(\tau z/|z|)\Vert _{\mathcal D(\mathcal A^\gamma )}(|z|-\tau )^{\alpha _k(1-\gamma )-1}\,\mathrm d\tau , \end{aligned}$$

which, combined with the inductive assumption, implies that

$$\begin{aligned}&\left\| (\varvec{u}^{(m+1)}-\varvec{u}^{(m)})(z)\right\| _{\mathcal D(\mathcal A^\gamma )}\le \left\| \mathcal Q(\varvec{u}^{(m)}-\varvec{u}^{(m-1)})(z)\right\| _{\mathcal D(\mathcal A^\gamma )}\\&\quad \le C_1L_0\sum _{k=1}^K\int _0^{|z|}\left\| (\varvec{u}^{(m)}-\varvec{u}^{(m-1)})(\tau z/|z|)\right\| _{\mathcal D(\mathcal A^\gamma )}(|z|-\tau )^{\alpha _k(1-\gamma )-1}\,\mathrm d\tau \\&\quad \le C_1L_0C_3M^m\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\\&\quad \quad \times \sum _{k=1}^K\int _0^{|z|}\!\!\!\!\!\sum _{|\varvec{j}|=m-1}\!\frac{(m-1)!}{j_1!\cdots j_K!}\frac{\tau ^{\varvec{\beta }\cdot \varvec{j}-\alpha _1\gamma }}{\Gamma (\varvec{\beta }\cdot \varvec{j}+1-\alpha _1\gamma )}(|z|-\tau )^{\alpha _k(1-\gamma )-\!1}\!\,\mathrm d\tau . \end{aligned}$$

Moreover, reminding the definition of the Riemann-Liouville integral operator, we can rewrite the above terms as

$$\begin{aligned}&\quad \,\left\| (\varvec{u}^{(m+1)}-\varvec{u}^{(m)})(z)\right\| _{\mathcal D(\mathcal A^\gamma )}\\&\le C_1L_0C_3M^m\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\max _{1\le k\le K}\Gamma (\alpha _k(1-\gamma ))\\&\quad \,\,\times \sum _{k=1}^K J^{\alpha _k(1-\gamma )}\left( \sum _{j=1}^K J^{\alpha _j(1-\gamma )}\right) ^{m-1}\!\!\!\!(|z|^{-\alpha _1\gamma })\\&=C_3M^{m+1}\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\sum _{|\varvec{j}|=m}\frac{m!}{j_1!\cdots j_K!}\frac{|z|^{\varvec{\beta }\cdot \varvec{j}-\alpha _1\gamma }}{\Gamma (\varvec{\beta }\cdot \varvec{j}+1-\alpha _1\gamma )} \end{aligned}$$

in view of the definition of the constant M in (3.2). We finish the proof of the first part of the lemma.

(ii) In the case of $\varvec{u}_0\equiv \varvec{0}$, let $\gamma =1$ for $p=2$ and $0\le \gamma <1$ for $p\ne 2$. Then the estimate (2.9) in Lemma 3 directly implies

$$\begin{aligned} \Vert \varvec{w}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}\le C_2\Vert \varvec{F}\Vert _{L^p(0,T;L^2(\Omega ))}, \end{aligned}$$

which is exactly (3.6) for $m=0$. For $m\in \mathbb N$, it is readily seen from (3.4) that $\varvec{u}^{(1)}=\varvec{w}$ and

$$\begin{aligned} \varvec{u}^{(m+1)}-\varvec{u}^{(m)}=\mathcal Q(\varvec{u}^{(m)}-\varvec{u}^{(m-1)})=\cdots =\mathcal Q^m(\varvec{u}^{(1)}-\varvec{u}^{(0)})=\mathcal Q^m\varvec{w} \end{aligned}$$

(3.7)

for $m\in \mathbb N$. Then applying Lemma 4 to (3.7) indicates

$$\begin{aligned} \Vert (\varvec{u}^{(m+1)}-\varvec{u}^{(m)})(t)\Vert _{\mathcal D(\mathcal A^\gamma )}&\le M^m\left( \sum _{k=1}^K J^{\alpha _k(1-\gamma )}\right) ^m\Vert \varvec{w}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\\&\le M^m\sum _{|\varvec{j}|=m}\frac{m!}{j_1!\cdots j_K!}J^{\varvec{\beta }\cdot \varvec{j}}\left( \Vert \varvec{w}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\right) \end{aligned}$$

for a.e. $t\in (0,T)$. Then we employ Young’s convolution inequality to obtain

$$\begin{aligned}&\quad \,\,\Vert \varvec{u}^{(m+1)}-\varvec{u}^{(m)}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}\\&\le M^m\Vert \varvec{w}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}\int _0^T\! \left| \sum _{|\varvec{j}|=m}\frac{m!}{j_1!\cdots j_K!}\frac{t^{\varvec{\beta }\cdot \varvec{j}-1}}{\Gamma (\varvec{\beta }\cdot \varvec{j})} \right| \mathrm dt\\&\le C_2M^m\Vert \varvec{F}\Vert _{L^p(0,T;L^2(\Omega ))}\sum _{|\varvec{j}|=m}\frac{m!}{j_1!\cdots j_K!}\frac{T^{\varvec{\beta }\cdot \varvec{j}}}{\Gamma (\varvec{\beta }\cdot \varvec{j}+1)}. \end{aligned}$$

Now Stirling’s formula (e.g. Abramowitz and Stegun [1, p.257])

$$\begin{aligned} \Gamma (\eta )=\sqrt{2\pi }\,\mathrm e^{-\eta }\eta ^{\eta -1/2}(1+O(\eta ^{-1}))\quad \text{ as } \eta \rightarrow +\infty \end{aligned}$$

yields

$$\begin{aligned} \Gamma (\varvec{\beta }\cdot \varvec{j}+1)\ge \widetilde{C}\,\Gamma (\underline{\beta }|\varvec{j}|+1)=\widetilde{C}\,\Gamma (\underline{\beta }m+1),\quad \underline{\beta }=\min \{\beta _1,\cdots ,\beta _K\} \end{aligned}$$

with a positive constant $\widetilde{C} = \min _{y_1\ge y_2\ge 0}(\Gamma (1+y_1)/\Gamma (1+y_2))$. Together with the multinomial theorem

$$\begin{aligned} \sum _{|\varvec{j}|=m}\frac{m!}{j_1!\cdots j_K!}=K^m, \end{aligned}$$

it follows that

$$\begin{aligned} \sum _{|\varvec{j}|=m}\frac{m!}{j_1!\cdots j_K!}\frac{T^{\varvec{\beta }\cdot \varvec{j}}}{\Gamma (\varvec{\beta }\cdot \varvec{j}+1)} \le \widetilde{C}^{-1} K^m\frac{\max \{1,T^{\overline{\beta }m}\}}{\Gamma (\underline{\beta }m+1)}, \end{aligned}$$

where $\overline{\beta }=\max \{\beta _1,\dots ,\beta _K\}$. Collecting all the above estimates, we finally arrive at

$$\begin{aligned} \Vert \varvec{u}^{(m+1)}-\varvec{u}^{(m)}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))} \le C_2\widetilde{C}^{-1}(M K)^m\frac{\max \{1,T^{\overline{\beta }m}\}}{\Gamma (\underline{\beta }m+1)} \Vert \varvec{F}\Vert _{L^p(0,T;L^2(\Omega ))}, \end{aligned}$$

which finishes the proof. $\square$

Now we are ready to complete the proof of Theorem 1. Throughout this section, by $C>0$ and $C_{T,\gamma }>0$ we refer to generic constants depending only on $\Omega ,\varvec{\alpha },\mathcal A,\mathcal P$ and $\Omega ,\varvec{\alpha },\mathcal A,\mathcal P,T,\gamma$, respectively, which may change from line to line.

Proof of Theorem 1(i)

Let $\varvec{u}_0\in (L^2(\Omega ))^K$ and $\varvec{F}\equiv \varvec{0}$.

Case 1 First we assume $\mathcal P$ is a first-order differential operator with $L^\infty$ coefficients. For clarity, we further divide the proof into three steps.

Step 1 We claim that for $1/2\le \gamma <1$, the sequence $\{\varvec{u}^{(m)}\}$ defined by (3.4) converges in $L^{1/\gamma }(0,T;\mathcal D(\mathcal A^\gamma ))$, whose limit $\varvec{u}\!\in \! L^{1/\gamma }(0,T;\mathcal D(\mathcal A^\gamma ))$ is a mild solution to (2.2) satisfying the estimates (2.13)–(2.14).

Indeed, by the completeness of $\mathcal D(\mathcal A^\gamma )$, it suffices to verify that $\{\varvec{u}^{(m)}(z)\}$ is a Cauchy sequence in $\mathcal D(\mathcal A^\gamma )$ in any compact subset of S. To this end, we pick $m,m'\in \mathbb N$ ($m<m'$) and utilize the estimate (3.5) in Lemma 5 to estimate

$$\begin{aligned}&\quad \,\,\left\| (\varvec{u}^{(m')}-\varvec{u}^{(m)})(z)\right\| _{\mathcal D(\mathcal A^\gamma )}\le \sum _{\ell =m}^{m'-1}\left\| (\varvec{u}^{(\ell +1)}-\varvec{u}^{(\ell )})(z)\right\| _{\mathcal D(\mathcal A^\gamma )}\\&\le \sum _{\ell =m}^{m'-1}C_3M^{\ell +1}\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\sum _{|\varvec{j}|=\ell }\frac{\ell !}{j_1!\cdots j_K!}\frac{|z|^{\varvec{\beta }\cdot \varvec{j}-\alpha _1\gamma }}{\Gamma (\varvec{\beta }\cdot \varvec{j}+1-\alpha _1\gamma )}\\&\le C_3M\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\,|z|^{-\alpha _1\gamma }\sum _{\ell =m}^\infty \sum _{|\varvec{j}|=\ell }\frac{\ell !}{j_1!\cdots j_K!}\frac{\prod _{k=1}^K(M\,|z|^{\beta _k})^{j_k}}{\Gamma (1-\alpha _1\gamma +\varvec{\beta }\cdot \varvec{j})}. \end{aligned}$$

Now let us invoke the definition of the multinomial Mittag-Leffler function, which was first introduced by Hadid and Luchko in [12] (see also [31])

$$\begin{aligned} E_{\varvec{\beta },\beta _0}(z_1,\dots ,z_K):=\sum _{\ell =0}^\infty \sum _{|\varvec{j}|=\ell }\frac{\ell !}{j_1!\cdots j_K!}\frac{\prod _{k=1}^K z_k^{j_k}}{\Gamma (\beta _0+\varvec{\beta }\cdot \varvec{j})}. \end{aligned}$$

Then we immediately see that the summation in the above estimate coincides with the remainder after the m-th term in the definition of the multinomial Mittag-Leffler function $E_{\varvec{\beta },1-\alpha _1\gamma }(M\,|z|^{\beta _1},\dots ,$ $M\,|z|^{\beta _K})$, which converges uniformly in any compact set of S. In other words, we conclude

$$\begin{aligned} \left\| (\varvec{u}^{(m')}-\varvec{u}^{(m)})(z)\right\| _{\mathcal D(\mathcal A^\gamma )}\longrightarrow 0\quad \text{ as } m,m'\rightarrow \infty \text{ in } \text{ any } \text{ compact } \text{ set } \text{ of } S. \end{aligned}$$

Thus, the sequence $\{\varvec{u}^{(m)}(t)\}$ converges to some limit $\varvec{u}(t)\in \mathcal D(\mathcal A^\gamma )$ for a.e. $t\in (0,T)$ and $\varvec{u}:(0,T]\longrightarrow \mathcal D(\mathcal A^\gamma )$ is analytic. Moreover, following the argument used in the proof of [22, Theorem 2.3], we further obtain

$$\begin{aligned} \Vert \varvec{u}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\le C\,t^{-\alpha _1\gamma }\exp (\widetilde{M} t)\Vert \varvec{u}_0\Vert _{L^2(\Omega )},\quad \text{ a.e. } t>0 \end{aligned}$$

(3.8)

for $0\le \gamma <1$. Therefore, it is straightforward to deduce (2.14) by

$$\begin{aligned} \Vert \varvec{u}\Vert _{L^{1/\gamma }(0,T;\mathcal D(\mathcal A^\gamma ))}&=\left( \int _0^T\Vert \varvec{u}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}^{1/\gamma }\,\mathrm dt\right) ^\gamma \\&\le C\exp (\widetilde{M} T)\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\left( \int _0^T t^{-\alpha _1}\,\mathrm dt\right) ^\gamma \\&=C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )},\quad 1/2\le \gamma <1. \end{aligned}$$

Finally, passing $m\rightarrow \infty$ in (3.4), it is readily seen that $\varvec{u}$ satisfies (2.10) and thus it is a mild solution to (2.2) according to Definition 1.

Now for $0\le \gamma <1/2$, we perform $\mathcal A^\gamma$ on both sides of (2.10) and calculate directly to find

$$\begin{aligned} \mathcal A^\gamma \varvec{u}=\mathcal A^\gamma \mathcal S(t)\varvec{u}_0+\mathcal A^\gamma \mathcal Q\varvec{u}. \end{aligned}$$

In view of the estimates in Lemma 3 and by using the argument in the proof of Lemma 4, a direct calculation yields

$$\begin{aligned}&\Vert \mathcal A^\gamma \varvec{u}(t)\Vert _{L^2(\Omega )}\\&\quad \le C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )}t^{-\alpha _1\gamma }+\left\| \int _0^t\mathcal A^{\gamma -1}\mathcal S'(t-\tau )(\mathcal P\varvec{u})(\tau )\,\mathrm d\tau \right\| _{L^2(\Omega )}\\&\quad \le C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )}t^{-\alpha _1\gamma }+\left\| \int _0^t\mathcal A^{-1/2}\mathcal S'(t-\tau )\mathcal A^{\gamma -1/2}(\mathcal P\varvec{u})(\tau )\,\mathrm d\tau \right\| _{L^2(\Omega )}\\&\quad \le C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )}t^{-\alpha _1\gamma }+C\sum _{k=1}^K\int _0^t (t-\tau )^{\alpha _k/2-1}\Vert u_k(\tau )\Vert _{\mathcal D(\mathcal A^{\gamma })}\mathrm d\tau \\&\quad \le C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )}t^{-\alpha _1\gamma }+C_{T,\gamma } \int _0^t (t-\tau )^{\alpha _K/2-1}\Vert \varvec{u}(\tau )\Vert _{\mathcal D(\mathcal A^{\gamma })}\mathrm d\tau . \end{aligned}$$

Thus, by the generalized Grönwall’s inequality (e.g. [14, Lemma 7.1.1]), we obtain

$$\begin{aligned} \Vert \varvec{u}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}&\le C_{T,\gamma } \Vert \varvec{u}_0\Vert _{L^2(\Omega )} t^{-\alpha _1\gamma } \!+\! C_{T,\gamma } \Vert \varvec{u}_0\Vert _{L^2(\Omega )} \!\int _0^t (t-\tau )^{\alpha _K/2-1}\tau ^{-\alpha _1\gamma } \mathrm d\tau \\&\le C_{T,\gamma } \Vert \varvec{u}_0\Vert _{L^2(\Omega )} t^{-\alpha _1\gamma } + C_{T,\gamma } \Vert \varvec{u}_0\Vert _{L^2(\Omega )} t^{\alpha _K/2-\alpha _1\gamma }. \end{aligned}$$

Finally, we can obtain

$$\begin{aligned} \Vert \varvec{u}\Vert _{L^{1/\gamma }(0,T;\mathcal D(\mathcal A^\gamma ))}\le C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )},\quad 0\le \gamma <1. \end{aligned}$$

Step 2 Next we demonstrate the uniqueness of the mild solution and the convergence (2.12).

For the uniqueness, assume that $\varvec{v}$ is another mild solution to (2.2). Taking difference between the integral equations (2.10) for $\varvec{u}$ and $\varvec{v}$, we obtain

$$\begin{aligned} \varvec{u}-\varvec{v}=\mathcal Q(\varvec{u}-\varvec{v}). \end{aligned}$$

Then for $\gamma \in [1/2,1)$ applying Lemma 4 with $m=1$ immediately yields

$$\begin{aligned} \Vert (\varvec{u}-\varvec{v})(t)\Vert _{\mathcal D(\mathcal A^\gamma )}&\le M\sum _{k=1}^K\int _0^t\Vert (\varvec{u}-\varvec{v})(\tau )\Vert _{\mathcal D(\mathcal A^\gamma )}(t-\tau )^{\alpha _k(1-\gamma )-1}\,\mathrm d\tau \\&\le C\int _0^t\Vert (\varvec{u}-\varvec{v})(\tau )\Vert _{\mathcal D(\mathcal A^\gamma )}(t-\tau )^{\alpha _K(1-\gamma )-1}\,\mathrm d\tau ,\ 0<t<T. \end{aligned}$$

Thus we can employ Grönwall’s inequality with a weakly singular kernel (e.g. [14, Lemma 7.1.1]) to conclude $\varvec{u}\equiv \varvec{v}$.

Now we turn to the convergence issue (2.12). We have

$$\begin{aligned} \varvec{u}(t)-\varvec{u}_0=(\varvec{w}(t)-\varvec{u}_0)+\mathcal Q\varvec{u}(t), \end{aligned}$$

where $\lim _{t\rightarrow 0}\Vert \varvec{w}(t)-\varvec{u}_0\Vert _{L^2(\Omega )}=0$ is guaranteed by (2.6) in Lemma 3. For $\mathcal Q\varvec{u}(t)$, again we apply Lemma 4 with $m=1$ and employ (3.8) with $\gamma =0$ to dominate

$$\begin{aligned} \Vert \mathcal Q\varvec{u}(t)\Vert _{L^2(\Omega )}&\le M\sum _{k=1}^K\int _0^t\Vert \varvec{u}(\tau )\Vert _{L^2(\Omega )}\frac{(t-\tau )^{\alpha _k-1}}{\Gamma (\alpha _k)}\,\mathrm d\tau \nonumber \\&\le C_{T,\gamma }K\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\int _0^t \exp (\widetilde{M}\,\tau )(t-\tau )^{\alpha _K-1}\,\mathrm d\tau \nonumber \\&\le C_{T,\gamma }K\exp (\widetilde{M}\,t)\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\int _0^t (t-\tau )^{\alpha _K-1}\,\mathrm d\tau \nonumber \\&=C_{T,\gamma }K\exp (\widetilde{M}\,t)\Vert \varvec{u}_0\Vert _{L^2(\Omega )}t^{\alpha _K}\longrightarrow 0\quad (t\rightarrow 0). \end{aligned}$$

(3.9)

Thus (2.12) is verified.

Step 3 Finally, we improve the $\varvec{x}$-regularity of $\varvec{u}$ to show $\varvec{u}\in L^1(0,T;\mathcal D(\mathcal A))$ and the estimate (2.14) for $\gamma =1$. Recall that now

$$\begin{aligned} \varvec{u}=\varvec{w}+\varvec{y} \quad \text {with } \varvec{w} = \mathcal S(\cdot ) \varvec{u}_0, \ \varvec{y}:=\mathcal Q\varvec{u}. \end{aligned}$$

According to the estimate (2.7) in Lemma 3(i), we immediately see

$$\begin{aligned} \Vert \varvec{w}\Vert _{L^{1/\gamma }(0,T;\mathcal D(\mathcal A^\gamma ))}&=\left( \int _0^T\Vert \varvec{w}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}^{1/\gamma }\,\mathrm dt\right) ^\gamma \nonumber \\&\le C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\left( \int _0^T t^{-\alpha _1}\,\mathrm dt\right) ^\gamma \nonumber \\&\le C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )},\quad 0\le \gamma \le 1. \end{aligned}$$

(3.10)

On the other hand, by the definition of $\mathcal Q$ and Lemma 3(ii), it reveals that $\varvec{y}$ satisfies the initial-boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _t^{\varvec{\alpha }}+\mathcal A)\varvec{y}=\mathcal P\varvec{u} &{} \text{ in } \Omega \times (0,T),\\ \varvec{y}=\varvec{0} &{} \text{ in } \Omega \times \{0\},\\ \varvec{y}=\varvec{0} &{} \text{ on } \partial \Omega \times (0,T). \end{array}\right. } \end{aligned}$$

(3.11)

Here the initial condition is understood in the sense of (3.9). Notice that in Step 1, we already obtained $\varvec{u}\in L^{1/\gamma }(0,T;\mathcal D(\mathcal A^\gamma ))$ along with (2.14) for $0\le \gamma <1$, and in particular

$$\begin{aligned}{} & {} \varvec{u}\in L^2(0,T;\mathcal D(\mathcal A^{1/2}))=(L^2(0,T;H_0^1(\Omega )))^K,\\{} & {} \Vert \varvec{u}\Vert _{L^2(0,T;H_0^1(\Omega ))}\le C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )}. \end{aligned}$$

Then for $\gamma =1$, we can employ Lemma 3(ii) with $p=2$ to derive

$$\begin{aligned} \Vert \varvec{y}\Vert _{L^1(0,T;\mathcal D(\mathcal A))}&\le T^{1/2}\Vert \varvec{y}\Vert _{L^2(0,T;\mathcal D(\mathcal A))}\le C_2T^{1/2}\Vert \mathcal P\varvec{u}\Vert _{L^2(0,T;L^2(\Omega ))}\\&\le C_{T,\gamma }\Vert \varvec{u}\Vert _{L^2(0,T;H_0^1(\Omega ))}\le C_{T,\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )}. \end{aligned}$$

Finally, combining the above inequality with (3.10) leads to $\varvec{u}=\varvec{w}+\varvec{y}\in L^1(0,T;\mathcal D(\mathcal A))$ and the estimate (2.14) for $\gamma =1$.

Case 2 Now we assume $\mathcal P$ is a first-order differential operator with $W^{1,\infty }$ coefficients in $\varvec{x}$. Indeed, the key estimate

$$\begin{aligned} \Vert \mathcal Q^m\varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )}\le M^m\left( \sum _{k=1}^K J^{\alpha _k/2-1}\right) ^m\Vert \varvec{v}(t)\Vert _{\mathcal D(\mathcal A^\gamma )} \end{aligned}$$

analogous to that in Lemma 4 still holds true in the case of $1/2<\gamma \le 1$. Repeating the same argument as that in Step 1, we can easily conclude that the mild solution $\varvec{u}$ belongs to $L^{1/\gamma }(0,T;\mathcal D(\mathcal A^\gamma ))$ and satisfies (2.13) also for $0\le \gamma \le 1$. This completes the proof of Theorem 1(i). $\square$

Proof of Theorem 1(ii)

Let $\varvec{u}_0\equiv \varvec{0}$ and $\varvec{F}\in (L^p(0,T;L^2(\Omega )))^K$ with $p\in [1,+\infty ]$. Throughout this proof, we restrict $0\le \gamma <1$ for $p\ne 2$ and allow $\gamma =1$ for $p=2$.

We argue rather similarly as the previous proof and we claim that the sequence $\{\varvec{u}^{(m)}\}$ defined by (3.4) converges in $L^p(0,T;\mathcal D(\mathcal A^\gamma ))$, whose limit $\varvec{u}\in L^p(0,T;\mathcal D(\mathcal A^\gamma ))$ is the unique mild solution to (2.2) satisfying the estimate (2.15).

Indeed, we pick $m,m'\in \mathbb N$ ($m<m'$) and turn to the estimate (3.6) in Lemma 5 to estimate

$$\begin{aligned} \Vert \varvec{u}^{(m')}-\varvec{u}^{(m)}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}&\le \sum _{\ell =m}^{m'-1}\Vert \varvec{u}^{(\ell +1)}-\varvec{u}^{(\ell )}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}\\&\le \Vert \varvec{F}\Vert _{L^p(0,T;L^2(\Omega ))}\sum _{\ell =m}^{m'-1}C(M K)^\ell \frac{\max \{1,T^{\overline{\beta }\ell }\}}{\Gamma (\underline{\beta }\ell +1)}\\&\le C\Vert \varvec{F}\Vert _{L^p(0,T;L^2(\Omega ))}\sum _{\ell =m}^\infty \frac{(M K\max \{1,T^{\overline{\beta }}\})^\ell }{\Gamma (\underline{\beta }\ell +1)}. \end{aligned}$$

Again noticing that the above summation coincides with the remainder after the m-th term in the Mittag-Leffler function $E_{\underline{\beta },1}(M K\max \{1,T^{\overline{\beta }}\})$, we can easily conclude

$$\begin{aligned} \Vert \varvec{u}^{(m')}-\varvec{u}^{(m)}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}\longrightarrow 0\quad \text{ as } m,m'\rightarrow \infty \end{aligned}$$

due to the uniform convergence of $E_{\underline{\beta },1}(M K\max \{1,T^{\overline{\beta }}\})$. Therefore, $\{\varvec{u}^{(m)}\}$ is a Cauchy sequence in $L^p(0,T;\mathcal D(\mathcal A^\gamma ))$ and thus converges to some limit $\varvec{u}\in L^p(0,T;\mathcal D(\mathcal A^\gamma ))$, which is obviously a mild solution to (2.2). Moreover, similarly we deduce

$$\begin{aligned} \Vert \varvec{u}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}&\le \sum _{\ell =0}^\infty \Vert \varvec{u}^{(\ell +1)}-\varvec{u}^{(\ell )}\Vert _{L^p(0,T;\mathcal D(\mathcal A^\gamma ))}\\&\le C\Vert \varvec{F}\Vert _{L^p(0,T;L^2(\Omega ))}\sum _{\ell =0}^\infty \frac{(M K\max \{1,T^{\overline{\beta }}\})^\ell }{\Gamma (\underline{\beta }\ell +1)}\\&=C\Vert \varvec{F}\Vert _{L^p(0,T;L^2(\Omega ))}E_{\underline{\beta },1}(M K\max \{1,T^{\overline{\beta }}\})\\&\le C_{T,\gamma }\Vert \varvec{F}\Vert _{L^p(0,T;L^2(\Omega ))}, \end{aligned}$$

where we used Lemma 1 again to treat $E_{\underline{\beta },1}(M K\max \{1,T^{\overline{\beta }}\})$. This completes the proof of (2.15). The uniqueness of $\varvec{u}$ follows from identically the same argument as the proof of Theorem 1(i) and this completes the proof of Theorem 1(ii). $\square$

4 Proof of Theorem 2

Throughout this section, we assume (2.16) and consider the long-time asymptotic behavior of the solution to the following initial-boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^{\varvec{\alpha }}\varvec{u}+\mathcal A\varvec{u}=\varvec{C}\varvec{u} &{} \text{ in } \Omega \times \mathbb R_+,\\ \varvec{u}=\varvec{u}_0 &{} \text{ in } \Omega \times \{0\},\\ \varvec{u}=\varvec{0} &{} \text{ on } \partial \Omega \times \mathbb R_+, \end{array}\right. } \end{aligned}$$

(4.1)

where we recall $\varvec{C}=(c_{k\ell })_{1\le k,\ell \le K}\in (L^\infty (\Omega ))^{K\times K}$. Based on the results in the above section, we know that the solution $\varvec{u}$ to (2.2) uniquely exists in $H_0^1(\Omega )\cap H^{2\gamma }(\Omega )$ for any $t\in (0,T]$ and admits the estimate

$$\begin{aligned} \Vert \varvec{u}(t)\Vert _{\mathcal D(A^\gamma )}\le C_T\,t^{-\alpha _1\gamma }\Vert \varvec{u}_0\Vert _{L^2(\Omega )},\quad 0<t\le T \end{aligned}$$

(4.2)

with $\gamma \in [0,1)$. Thus the asymptotic behavior as $t\rightarrow 0$ is related to the largest order of the fractional derivatives. However, we know nothing about the long-time asymptotic behavior from the above estimate except that the constant in (4.2) depends on T, which may tends to infinity as $T\rightarrow +\infty$. Therefore, we shall employ the Laplace transform

$$\begin{aligned} \widehat{f}(s):=\int _{\mathbb R_+}f(t)\,\mathrm e^{-s t}\,\mathrm dt \end{aligned}$$

to investigate the long-time asymptotic behavior of the solution to (4.1).

Before giving the proof of Theorem 2, we make some necessary settings. We introduce the sector

$$\begin{aligned} S_\theta :=\{s\in \mathbb C\setminus \{0\}\mid |\arg s|<\theta \} \end{aligned}$$

with an angel $\theta \in (\frac{\pi }{2},\min \{\frac{\pi }{2\alpha _1},\pi \})$. We apply the Laplace transform to (4.1) and use the formula

$$\begin{aligned} \widehat{\partial _t^\alpha f}(s)=s^\alpha \widehat{f}(s)-s^{\alpha -1}f(0),\quad 0<\alpha <1 \end{aligned}$$

to derive the boundary value problem for a coupled elliptic system

$$\begin{aligned} {\left\{ \begin{array}{ll} (\mathcal A+s^{\varvec{\alpha }}-\varvec{C})\widehat{\varvec{u}}(\,\cdot ;s)=s^{-1}s^{\varvec{\alpha }}\varvec{u}_0 &{} \text{ in } \Omega ,\\ \widehat{\varvec{u}}(\,\cdot ;s)=\varvec{0} &{} \text{ on } \partial \Omega \end{array}\right. } \end{aligned}$$

(4.3)

with a parameter $s\in \{\textrm{Re}\,s>s_0\}$. Here we abbreviate

$$\begin{aligned} s^{\varvec{\alpha }}:=\textrm{diag}(s^{\alpha _1},\dots ,s^{\alpha _K}) \end{aligned}$$

and $s_0\ge C$ with the constant C in (2.13). As before, we simply write $\widehat{\varvec{u}}(\,\cdot \,;s)$ as $\widehat{\varvec{u}}(s)$.

We check at once that $\widehat{\varvec{u}}:\{\textrm{Re}\,s>s_0\}\longrightarrow (L^2(\Omega ))^K$ is analytic, which is clear from the property of the Laplace transform. Moreover, we claim that $\widehat{\varvec{u}}(s)$ ($\textrm{Re}\,s>s_0$) can be analytically extended to the sector $S_\theta$. Namely, the following lemma holds.

Lemma 6

Under the same assumptions in Theorem 2, for any $s\in S_\theta$ the boundary value problem (4.3) admits a unique weak solution $\widehat{\varvec{u}}(s)\in (H^2(\Omega ))^K$ and $\widehat{\varvec{u}}:S_\theta \longrightarrow (H^2(\Omega ))^K$ is analytic. Moreover,  there exists a constant $C'>0$ depending only on $d,\Omega ,\theta ,\varvec{\alpha },\varvec{C}$ and $\mathcal A$ such that

$$\begin{aligned} \Vert \widehat{\varvec{u}}(s)\Vert _{H^2(\Omega )}\le C'\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\left( \sum _{k,\ell =1}^K|s|^{\alpha _k+\alpha _\ell -1}+\sum _{k=1}^K|s|^{\alpha _k-1}\right) ,\quad \forall \,s\in S_\theta . \end{aligned}$$

(4.4)

Proof

Firstly for any fixed $s\in S_\theta$, we define a bilinear form

$$\begin{aligned} B[\,\cdot ,\,\cdot ;s]:(H_0^1(\Omega ))^K\times (H_0^1(\Omega ))^K\longrightarrow \mathbb C\end{aligned}$$

by

$$\begin{aligned} B[\varvec{\psi },\varvec{\varphi };s]:=\int _\Omega \left( \sum _{k=1}^K\varvec{A}_k\nabla \psi _k\cdot \nabla \varphi _k+(s^{\varvec{\alpha }}-\varvec{C})\varvec{\psi }\cdot \varvec{\varphi }\right) \mathrm d\varvec{x}, \end{aligned}$$

where $\varvec{\psi }=(\psi _1,\psi _2,\cdots ,\psi _K)^\mathrm T,\varvec{\varphi }=(\varphi _1,\varphi _2,\cdots ,\varphi _K)^\mathrm T\in (H_0^1(\Omega ))^K$. By using the Poincaré inequality and Hölder’s inequality, we have

$$\begin{aligned} |B[\varvec{\psi },\varvec{\varphi };s]|&\le C(s)\int _\Omega \left( \sum _{k=1}^K|\nabla \psi _k||\nabla \varphi _k|+|\varvec{\psi }||\varvec{\varphi }|\right) \mathrm d\varvec{x}\\&\le C(s)\Vert \varvec{\psi }\Vert _{H^1(\Omega )}\Vert \varvec{\varphi }\Vert _{H^1(\Omega )}. \end{aligned}$$

Here the constant $C(s)>0$ depends on s. Furthermore, taking $\varvec{\varphi }=\varvec{\psi }$ implies

$$\begin{aligned} B[\varvec{\psi },\varvec{\psi };s]=\int _\Omega \left( \sum _{k=1}^K\varvec{A}_k\nabla \psi _k\cdot \nabla \psi _k+(s^{\varvec{\alpha }}-\varvec{C})\varvec{\psi }\cdot \varvec{\psi }\right) \mathrm d\varvec{x}, \end{aligned}$$

and hence

$$\begin{aligned}&\quad \,\,\textrm{Re}(B[\varvec{\psi },\varvec{\psi };s])\\&=\int _\Omega \left\{ \sum _{k=1}^K\left( \varvec{A}_k\nabla (\textrm{Re}\,\psi _k)\!\cdot \!\nabla (\textrm{Re}\,\psi _k)\!+\!\varvec{A}_k\nabla (\textrm{Im}\,\psi _k)\!\cdot \!\nabla (\textrm{Im}\,\psi _k)\!+\!(\textrm{Re}\,s^{\alpha _k})|\psi _k|^2\right) \right. \\&\qquad \quad \,\,-\varvec{C}(\textrm{Re}\,\varvec{\psi })\cdot (\textrm{Re}\,\varvec{\psi })-\varvec{C}(\textrm{Im}\,\varvec{\psi })\cdot (\textrm{Im}\,\varvec{\psi })\bigg \}\,\mathrm d\varvec{x}. \end{aligned}$$

Since the matrix $\varvec{C}$ is assumed to be negative semi-definite, it follows that

$$\begin{aligned} -\varvec{C}(\textrm{Re}\,\varvec{\psi })\cdot (\textrm{Re}\,\varvec{\psi })\ge 0,\quad -\varvec{C}(\textrm{Im}\,\varvec{\psi })\cdot (\textrm{Im}\,\varvec{\psi })\ge 0\quad \text{ in } \Omega . \end{aligned}$$

Further noticing that $\textrm{Re}\,s^{\alpha _k}=r^{\alpha _k}\cos \alpha _k\rho >0$ ($k=1,2,\cdots ,K$) for $s=r\,\mathrm e^{\mathrm i\rho }\in S_\theta$ with $\theta \in (\frac{\pi }{2},\min \{\frac{\pi }{2\alpha _1},\pi \})$, we employ the above inequality and the ellipticity of $\mathcal A_k$ to deduce

$$\begin{aligned} \textrm{Re}(B[\varvec{\psi },\varvec{\psi };s])\ge \kappa \int _\Omega \sum _{k=1}^K\left( |\nabla (\textrm{Re}\,\psi _k)|^2+|\nabla (\textrm{Im}\,\psi _k)|^2\right) \mathrm d\varvec{x}\ge C_0\Vert \varvec{\psi }\Vert _{H^1(\Omega )}^2 \end{aligned}$$

for some positive constant $C_0$, where we used the Poincaré inequality in the last inequality. Consequently, the Lax-Milgram theorem yields that for any $s\in S_\theta$, there exists a unique $\widehat{\varvec{u}}(s)\in (H_0^1(\Omega ))^K$ such that

$$\begin{aligned} B[\widehat{\varvec{u}}(s),\varvec{\varphi };s]=s^{-1}\int _\Omega s^{\varvec{\alpha }}\varvec{u}_0\cdot \varvec{\varphi }\,\mathrm d\varvec{x},\quad \forall \,\varvec{\varphi }\in (H_0^1(\Omega ))^K, \end{aligned}$$

which implies

$$\begin{aligned} C_0\Vert \widehat{\varvec{u}}(s)\Vert _{H^1(\Omega )}^2\le |B[\widehat{\varvec{u}}(s),\widehat{\varvec{u}}(s);s]|\le \sum _{k=1}^K|s|^{\alpha _k-1}\Vert u_0^{(k)}\Vert _{L^2(\Omega )}\Vert \widehat{\varvec{u}}(s)\Vert _{H^1(\Omega )} \end{aligned}$$

in view of Hölder’s inequality, and consequently

$$\begin{aligned} \Vert \widehat{\varvec{u}}(s)\Vert _{H^1(\Omega )}\le C'\sum _{k=1}^K|s|^{\alpha _k-1}\Vert u_0^{(k)}\Vert _{L^2(\Omega )},\quad s\in S_\theta . \end{aligned}$$

(4.5)

Furthermore, since $\varvec{u}_0\in (L^2(\Omega ))^K$, then by the regularity estimate for elliptic equations (e.g. [7]), we see that $\widehat{\varvec{u}}(s)\in (H^2(\Omega ))^K$ with the estimate

$$\begin{aligned} \Vert \widehat{\varvec{u}}(s)\Vert _{H^2(\Omega )}&\le C'\left\| (s^{\varvec{\alpha }}-\varvec{C})\widehat{\varvec{u}}(s)-s^{-1}s^{\varvec{\alpha }}\varvec{u}_0\right\| _{L^2(\Omega )}\\&\le C'\sum _{k=1}^K|s|^{\alpha _k}\Vert \widehat{u_k}(s)\Vert _{L^2(\Omega )}+C'\Vert \varvec{C}\widehat{\varvec{u}}(s)\Vert _{L^2(\Omega )}\\&\quad \,\,+C'\sum _{k=1}^K|s|^{\alpha _k-1}\Vert u_0^{(k)}\Vert _{L^2(\Omega )}\\&\le C'\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\left( \sum _{k,\ell =1}^K|s|^{\alpha _k+\alpha _\ell -1}+\sum _{k=1}^K|s|^{\alpha _k-1}\right) ,\quad s\in S_\theta . \end{aligned}$$

Similar to the argument used in the proof of [36, Theorem 0.1], we can prove that the solution $\widehat{\varvec{u}}(s)$ is analytic in $s\in S_\theta$. The proof of Lemma 6 is completed. $\square$

Now we can proceed to complete the proof of Theorem 2. From Lemma 6, we see that there exists a unique analytical extension of $\widehat{\varvec{u}}(s)$ from $\{\textrm{Re}\,s>s_0\}$ to $S_\theta$. By the same notation we denote this extension.

By Fourier-Mellin formula (e.g. [40]) for the inverse Laplace transform, we have

$$\begin{aligned} \varvec{u}(t)=\frac{1}{2\pi \mathrm i} \int _{s_0-\mathrm i\infty }^{s_0+\mathrm i\infty } \widehat{\varvec{u}}(s)\,\mathrm e^{s t}\,\mathrm ds. \end{aligned}$$

Since $\widehat{\varvec{u}}(s)$ is analytic in the sector $S_\theta$, it follows from the residue theorem (e.g. [38]) that for $t>0$, $\varvec{u}(t)$ can be represented by an integral on the contour

$$\begin{aligned} \gamma (\varepsilon ,\theta ):=\{s\in \mathbb C\mid \arg s=\theta ,\ |s|\ge \varepsilon \}\cup \{s\in \mathbb C\mid |\arg s|\le \theta ,\ |s|=\varepsilon \} \end{aligned}$$

with $\varepsilon >0$. In other words, there holds

$$\begin{aligned} \varvec{u}(t)=\frac{1}{2\pi \mathrm i}\int _{\gamma (\varepsilon ,\theta )}\widehat{\varvec{u}}(s)\,\mathrm e^{s t}\,\mathrm ds, \end{aligned}$$

where the shift in the line of integration is justified by the estimate (4.4). Moreover, again from the estimate (4.4), we can pass $\varepsilon \rightarrow 0$ to obtain

$$\begin{aligned} \varvec{u}(t)=\frac{1}{2\pi \mathrm i}\int _{\gamma (0,\theta )}\widehat{\varvec{u}}(s)\,\mathrm e^{s t}\,\mathrm ds. \end{aligned}$$

Then we conclude from (4.4) that

$$\begin{aligned} \Vert \varvec{u}(t)\Vert _{H^2(\Omega )}&\le C\int _{\mathbb R_+}\left( \Vert \widehat{\varvec{u}}(r\,\mathrm e^{\mathrm i\theta })\Vert _{H^2(\Omega )}+\Vert \widehat{\varvec{u}}(r\,\mathrm e^{-\mathrm i\theta })\Vert _{H^2(\Omega )}\right) \mathrm e^{r t\cos \theta }\,\mathrm dr\\&\le C\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\int _{\mathbb R_+}\mathrm e^{r t\cos \theta }\left( \sum _{k,\ell =1}^K r^{\alpha _k+\alpha _\ell -1}+\sum _{k=1}^K r^{\alpha _k-1}\right) \,\mathrm dr \end{aligned}$$

and thus

$$\begin{aligned} \Vert \varvec{u}(t)\Vert _{H^2(\Omega )}\le C\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\left( \sum _{k,\ell =1}^K t^{-\alpha _k-\alpha _\ell }+\sum _{k=1}^K t^{-\alpha _k}\right) ,\quad t>0 \end{aligned}$$

in view of $\cos \theta <0$ since $\frac{\pi }{2}<\theta <\min \{\frac{\pi }{2\alpha _1},\pi \}$. Moreover, in particular we obtain

$$\begin{aligned} \Vert \varvec{u}(t)\Vert _{H^2(\Omega )}\le C\,t^{-\alpha _K}\Vert \varvec{u}_0\Vert _{L^2(\Omega )},\quad t\ge t_0 \end{aligned}$$

for any fixed $t_0>0$. Here $C>0$ is a new generic constant depending also on $t_0$.

Finally, let us turn to proving the sharpness of the decay rate $t^{-\alpha _K}$ by contradiction. If the decay rate $t^{-\alpha _K}$ is not sharp, then there exists a nonnegative and bounded function f in $[0,+\infty )$ satisfying $\lim _{t\rightarrow +\infty }f(t)=0$ and

$$\begin{aligned} \Vert \varvec{u}(t)\Vert _{H^2(\Omega )}\le C\,t^{-\alpha _K}f(t)\Vert \varvec{u}_0\Vert _{L^2(\Omega )},\quad t>0. \end{aligned}$$

Then we see that the Laplace transform $\widehat{\varvec{u}}(s)$ admits the following estimate

$$\begin{aligned} \Vert \widehat{\varvec{u}}(s)\Vert _{H^2(\Omega )}\le C\,g(s) s^{\alpha _K-1}\Vert \varvec{u}_0\Vert _{L^2(\Omega )},\quad 0<s<1, \end{aligned}$$

where

$$\begin{aligned} g(s):=s^{1-\alpha _K}\widehat{(t^{-\alpha _K}f)}(s) = s^{1-\alpha _K}\int _0^\infty t^{-\alpha _K}f(t)\,\mathrm e^{-s t}\,\mathrm dt,\quad s\ge 0. \end{aligned}$$

Moreover, we claim that

$$\begin{aligned} g \text{ is } \text{ bounded } \text{ on } [0,1]\hbox { and }\lim _{s\rightarrow 0}g(s)=0, \end{aligned}$$

(4.6)

which will be verified in Appendix 1. Then immediately we see that

$$\begin{aligned} s^{1-\alpha _K}\mathcal A\widehat{\varvec{u}}(s)\longrightarrow \varvec{0}\quad \text{ as } s\rightarrow 0. \end{aligned}$$

Now multiplying $s^{1-\alpha _K}$ on both sides of (4.3) and passing $s\rightarrow 0$ imply

$$\begin{aligned} \varvec{0}=\lim _{s\rightarrow 0}s^{-\alpha _K}s^{\varvec{\alpha }}\varvec{u}_0\quad \text{ in } \Omega \end{aligned}$$

and especially $u_0^{(K)}\equiv 0$ in $\Omega$, which is a contradiction. This completes the proof of Theorem 2.

5 Proof of Theorem 3

In this section, we prove the unique determination of the orders by the observation of $u_{k_0}$ at $\{\varvec{x}_0\}\times (0,T)$. By noting the time analyticity of the solution to (2.2), we can uniquely extend the solution $\varvec{u}$ from (0, T) to $\mathbb R_+$. Thus, by applying the Laplace transform on both sides of the equation (2.2) with $\varvec{\alpha }=(\alpha _1,\dots ,\alpha _K)^{\mathrm T}$ and $\varvec{\beta }=(\beta _1,\dots ,\beta _K)^{\mathrm T}$, respectively and recalling the notation $s^{\varvec{\alpha }}=\textrm{diag}(s^{\alpha _1},\dots ,s^{\alpha _K})$, we find

$$\begin{aligned}&{\left\{ \begin{array}{ll} (\mathcal A-\varvec{C}+s^{\varvec{\alpha }})\widehat{\varvec{u}}(s)=s^{-1}s^{\varvec{\alpha }}\varvec{u}_0 &{} \text{ in } \Omega ,\\ \widehat{\varvec{u}}(s)=\varvec{0} &{} \text{ on } \partial \Omega , \end{array}\right. }\\&{\left\{ \begin{array}{ll} (\mathcal A-\varvec{C}+s^{\varvec{\beta }})\widehat{\varvec{v}}(s)=s^{-1}s^{\varvec{\beta }}\varvec{u}_0 &{} \text{ in } \Omega ,\\ \widehat{\varvec{v}}(s)=\varvec{0} &{} \text{ on } \partial \Omega \end{array}\right. } \end{aligned}$$

for $s>0$. By taking the difference between the above problems, it turns out that $(\widehat{\varvec{u}}-\widehat{\varvec{v}})(s)$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} (\mathcal A-\varvec{C}+s^{\varvec{\alpha }})(\widehat{\varvec{u}}-\widehat{\varvec{v}})(s)=s^{-1}(s^{\varvec{\alpha }}-s^{\varvec{\beta }}) (\varvec{u}_0-s\,\widehat{\varvec{v}}(s)) &{} \text{ in } \Omega ,\\ (\widehat{\varvec{u}}-\widehat{\varvec{v}})(s)=\varvec{0} &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(5.1)

We prove by contradiction. Let us assume that $\varvec{\alpha }\ne \varvec{\beta }$. Then there exists an index $k_1\in \{1,\dots ,K\}$ such that

$$\begin{aligned} \alpha _{k_1}\ne \beta _{k_1},\quad \alpha _k=\beta _k,\quad k>k_1. \end{aligned}$$

In other words, $\alpha _{k_1}$ is the smallest order in $\varvec{\alpha }$ such that $\alpha _k\ne \beta _k$. Without loss of generality, we assume $\alpha _{k_1}<\beta _{k_1}$.

Moreover, we introduce the following auxiliary function $\varvec{w}$ defined by

$$\begin{aligned} \varvec{w}(s):=\frac{s\,(\widehat{\varvec{u}}-\widehat{\varvec{v}})(s)}{s^{\alpha _{k_1}}-s^{\beta _{k_1}}},\quad s>0, \end{aligned}$$

which shares the same sign as that of $(\widehat{\varvec{u}}-\widehat{\varvec{v}})(s)$ for each component. Multiplying both sides of the governing equation in (5.1) by $s/(s^{\alpha _{k_1}}-s^{\beta _{k_1}})$, we see that $\varvec{w}$ satisfies the boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (\mathcal A-\varvec{C}+s^{\varvec{\alpha }})\varvec{w}(s)=\varvec{\Sigma }(\varvec{u}_0-s\,\widehat{\varvec{v}}(s)) &{} \text{ in } \Omega ,\\ \varvec{w}(s)=\varvec{0} &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(5.2)

where

$$\begin{aligned} \varvec{\Sigma }:=\textrm{diag}\left( \frac{s^{\alpha _1}-s^{\beta _1}}{s^{\alpha _{k_1}}-s^{\beta _{k_1}}},\dots ,\frac{s^{\alpha _{k_1-1}}-s^{\beta _{k_1-1}}}{s^{\alpha _{k_1}}-s^{\beta _{k_1}}},1,0,\dots ,0\right) . \end{aligned}$$

From the property of the Laplace transform, it follows that $\widehat{\varvec{u}}(s)$ and $\widehat{\varvec{v}}(s)$ are analytic with respect to $s>0$, so that $\varvec{w}(s)$ is continuous for $s>0$ in view of its definition.

Next we discuss the limit of $\varvec{w}(s)$ as $s\rightarrow 0$. We claim that

$$\begin{aligned} \lim _{s\rightarrow 0}\varvec{w}(s)=\varvec{w}_0\quad \text{ in } (H^2(\Omega ))^K, \end{aligned}$$

where $\varvec{w}_0=(w_0^{(1)},\dots ,w_0^{(K)})^\mathrm T$ solves the following boundary value problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} (\mathcal A-\varvec{C})\varvec{w}_0=\varvec{D}\varvec{u}_0 &{} \text{ in } \Omega ,\\ \varvec{w}_0=\varvec{0} &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(5.3)

Here $\varvec{D}=(d_{k\ell })_{1\le k,\ell \le K}$ is a diagonal matrix with

$$\begin{aligned} d_{k k}={\left\{ \begin{array}{ll} 1, &{} (k\in I_1:=\{k<k_1\mid \alpha _k=\alpha _{k_1}\}) \text{ or } (k=k_1),\\ 0, &{} (k\in I_2:=\{k<k_1\mid \alpha _k\ne \alpha _{k_1}\}) \text{ or } (k>k_1). \end{array}\right. } \end{aligned}$$

In fact, letting $\varvec{z}(s):=\varvec{w}(s)-\varvec{w}_0$ ($s>0$), we take the difference between (5.2) and (5.3) to obtain

$$\begin{aligned} \left\{ \!\begin{alignedat}{2}&(\mathcal A-\varvec{C}+s^{\varvec{\alpha }})\varvec{z}(s) = \varvec{H}(s)&\quad&\text{ in } \Omega ,\\&\varvec{z}(s)=\varvec{0}&\quad&\text{ on } \partial \Omega \end{alignedat}\right. \end{aligned}$$

(5.4)

where

$$\begin{aligned} \varvec{H}(s):=(\varvec{\Sigma }-\varvec{D})\varvec{u}_0-s\varvec{\Sigma }\widehat{\varvec{v}}(s)-s^{\varvec{\alpha }}\varvec{w}_0. \end{aligned}$$

(5.5)

By the conditions (2.17)–(2.18) in Theorem 3, we can apply the maximum principle for coupled elliptic equations (e.g. [8]) to conclude that the operator $\mathcal {A}-\varvec{C}+s^{\varvec{\alpha }}$ is invertible for any $s>0$. Thus, the regularity estimate for elliptic equations (e.g. [7]) implies

$$\begin{aligned} \Vert \varvec{z}(s)\Vert _{H^2(\Omega )}\le C\Vert \varvec{H}(s)\Vert _{L^2(\Omega )}. \end{aligned}$$

Here the generic constant $C>0$ also depends on the $L^\infty$-norm of the coefficients in (5.4). However, henceforth we consider only small s (e.g. $0<s\le 1$) so that the constant C is independent of s. It remains to check that $\Vert \varvec{H}(s)\Vert _{L^2(\Omega )}$ vanishes as $s\rightarrow 0$. By noting the definition of the diagonal matrix $\varvec{D}$, we find that the k-th entry of the diagonal matrix $\varvec{\Sigma }-\varvec{D}$ in (5.5) reads

$$\begin{aligned} \left\{ \!\begin{alignedat}{2}&\frac{s^{\alpha _k}-s^{\beta _k}}{s^{\alpha _{k_1}}-s^{\beta _{k_1}}}-1=\frac{s^{\beta _{k_1}}-s^{\beta _k}}{s^{\alpha _{k_1}}-s^{\beta _{k_1}}}=\frac{s^{\beta _{k_1}-\alpha _{k_1}}-s^{\beta _k-\alpha _{k_1}}}{1-s^{\beta _{k_1}-\alpha _{k_1}}},&\quad&k\in I_1,\\&\frac{s^{\alpha _k}-s^{\beta _k}}{s^{\alpha _{k_1}}-s^{\beta _{k_1}}}=\frac{s^{\alpha _k-\alpha _{k_1}}-s^{\beta _k-\alpha _{k_1}}}{1-s^{\beta _{k_1}-\alpha _{k_1}}},&\quad&k\in I_2,\\&0,&\quad&k\ge k_1. \end{alignedat}\right. \end{aligned}$$

Then the norm of the first term in $\varvec{H}(s)$ admits

$$\begin{aligned}&\quad \,\,\Vert (\varvec{\Sigma }-\varvec{D})\varvec{u}_0\Vert _{L^2(\Omega )}\\&\le \Vert \varvec{u}_0\Vert _{L^2(\Omega )}\left( \sum _{k\in I_1}\frac{s^{\beta _{k_1}-\alpha _{k_1}}-s^{\beta _k-\alpha _{k_1}}}{1-s^{\beta _{k_1}-\alpha _{k_1}}}+\sum _{k\in I_2}\frac{s^{\alpha _k-\alpha _{k_1}}-s^{\beta _k-\alpha _{k_1}}}{1-s^{\beta _{k_1}-\alpha _{k_1}}}\right) . \end{aligned}$$

Moreover, in view of the definition of $\varvec{w}$, we estimate the norm of the second term in $\varvec{H}(s)$ as follows

$$\begin{aligned} \Vert -s\varvec{\Sigma }\widehat{\varvec{v}}(s)\Vert _{L^2(\Omega )}&\le \Vert s\widehat{\varvec{v}}(s)\Vert _{L^2(\Omega )}\sum _{k=1}^{k_1}\frac{s^{\alpha _k-\alpha _{k_1}}-s^{\beta _k-\alpha _{k_1}}}{1-s^{\beta _{k_1}-\alpha _{k_1}}}\\&\le C\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\sum _{\ell =1}^K s^{\alpha _\ell }\sum _{k=1}^{k_1}\frac{s^{\alpha _k-\alpha _{k_1}}-s^{\beta _k-\alpha _{k_1}}}{1-s^{\beta _{k_1}-\alpha _{k_1}}}. \end{aligned}$$

Here in the last inequality we used the estimate (4.5) for $\widehat{\varvec{v}}$. Noting that $\varvec{w}_0$ is the solution to (5.3), again by the regularity estimate for elliptic equations, we readily find that

$$\begin{aligned} \left\| -s^{\varvec{\alpha }}\varvec{w}_0\right\| _{L^2(\Omega )}\le C\Vert \varvec{u}_0\Vert _{L^2(\Omega )}\sum _{k=1}^K s^{\alpha _k}. \end{aligned}$$

Combining the above estimates and noticing the fact that

$$\begin{aligned} \alpha _{k_1}<\beta _{k_1}\le \beta _k,\ k\in I_1\cup I_2, \quad \alpha _{k_1}<\alpha _k,\ k\in I_2, \end{aligned}$$

we obtain $\lim _{s\rightarrow 0}\Vert \varvec{H}(s)\Vert _{L^2(\Omega )}= 0$ and hence $\lim _{s\rightarrow 0}\Vert \varvec{z}(s)\Vert _{H^2(\Omega )}= 0$.

From Lemma 7 in Appendix 1, we conclude that $\varvec{w}_0>\varvec{0}$ in $\Omega$, that is, $w_0^{(k)}>0$ in $\Omega$ for all $k=1,\dots ,K$. Then at the observation point $\varvec{x}_0\in \Omega$, from the relation $\varvec{w}_0=\lim _{s\rightarrow 0}\varvec{w}(s)$ in $(H^2(\Omega ))^K\subset (C(\overline{\Omega }))^K$, we have

$$\begin{aligned} \lim _{s\rightarrow 0}w_k(\varvec{x}_0;s)=w_0^{(k)}(\varvec{x}_0)>0,\quad k=1,\dots ,K. \end{aligned}$$

This indicates that we can choose a small $\varepsilon >0$ such that $w_k(\varvec{x}_0;s)>0$ for any $s\in (0,\varepsilon )$ and $k=1,\dots ,K$, which implies that

$$\begin{aligned} \widehat{u}_k(\varvec{x}_0;s)-\widehat{v}_k(\varvec{x}_0;s)>0,\quad 0<s<\varepsilon ,\ k=1,\dots ,K. \end{aligned}$$

Therefore, we have $\widehat{\varvec{u}}(\varvec{x}_0;s)>\widehat{\varvec{v}}(\varvec{x}_0;s)$ for all $s\in (0,\varepsilon )$, that is,

$$\begin{aligned} \widehat{u}_k(\varvec{x}_0;s)>\widehat{v}_k(\varvec{x}_0;s),\quad 0<s<\varepsilon ,\ k=1,\dots ,K. \end{aligned}$$

This yields a contradiction since $u_{k_0}=v_{k_0}$ at $\{\varvec{x}_0\}\times (0,T)$ implies $u_{k_0}=v_{k_0}$ at $\{\varvec{x}_0\}\times \mathbb R_+$ by the time analyticity, and hence $\widehat{u_{k_0}}(\varvec{x}_0;s)=\widehat{v_{k_0}}(\varvec{x}_0;s)$ for any $s>0$. The proof of Theorem 3 is completed.

6 Conclusions

In this paper, we investigate the fundamental properties of the initial-boundary value problem (1.2) for a moderately coupled system of linear time-fractional diffusion equations. In view of the unique existence, limited smoothing property and long-time asymptotic behavior, it reveals that the coupled system inherits almost the same properties as those of a single equation, especially those of a multi-term time-fractional diffusion equation. Employing the coupling effect, we also established the uniqueness for an inverse problem on the simultaneous determination of multiple parameters by the observation of a single component of the solution. These indicate the similarity and difference between a single equation and a coupled system of time-fractional PDEs.

Parallel to the case of single equations, several topics can be enumerated as future topics. For instance, in order to construct the super/subsolution methods for time-fractional reaction-diffusion systems, it is essential to develop the comparison principle or, if possible, even the strong maximum principle similarly to that of traditional ones. Regarding the unique continuation property, it worths considering the possibility of assuming the vanishing of only a part of components.

It is an interesting question to ask what happens when we allow one or several orders in (1.2) to be exactly 1. Actually one can readily check from Sect. 5 that we can choose some of the orders as 1 in the part of inverse problem. Although one has to reconsider some estimates in the part of forward problem, we conjecture that the results proved in this article still hold true even though the largest order equals to 1.

We close this article with mentioning another direction of generalizing the orders of time derivatives to (1, 2), i.e., coupled time-fractional wave systems modeling viscoelastic wave propagation (e.g. [3, 19]). In these models, the orders $\alpha _k$ can be the same, whereas the systems should be strongly coupled up to the second spatial derivatives for elasticity. Regardless of their difficulty, it is important to understand the mechanism between viscosity and elasticity.

Appendix: technical details

1.1 Strong maximum principle

We consider a strong maximum principle for the following weakly coupled elliptic system

$$\begin{aligned} {\left\{ \begin{array}{ll} (\mathcal A-\varvec{C})\varvec{w}=\varvec{F} &{} \text{ in } \Omega ,\\ \varvec{w}=\varvec{0} &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(7.1)

where $\mathcal A$ is the elliptic operator defined in Sect. 1, $\varvec{C}=(c_{k\ell })_{1\le k,\ell \le K}\in (L^\infty (\Omega ))^{K\times K}$ is a matrix-valued function and $\varvec{F}=(f_1,\dots ,f_K)^{\mathrm T}$ is a vector-valued function. Then we have the following strong maximum principle.

Lemma 7

(Strong maximum principle) Assume that $\varvec{C}$ satisfies the conditions (2.17)–(2.18) in Theorem 3 and $\varvec{F}\ge \varvec{0},\not \equiv \varvec{0}$ in $\Omega ,$ that is,  $f_k\ge 0$ in $\Omega$ for $k=1,\dots ,K$ and there exists $k_0\in \{1,\dots ,K\}$ such that $f_{k_0}\ge 0,\not \equiv 0$ in $\Omega$. Then the solution to (7.1) satisfies $\varvec{w}>\varvec{0}$ in $\Omega ,$ that is,  $w_k>0$ in $\Omega$ for all $k=1,\dots ,K$.

We refer to [33, 43] and the references therein for some results on the (strong) maximum principle for the weakly coupled elliptic systems where the settings on the coupling matrix $\varvec{C}$ are different. For the completeness, we apply the maximum principle in [8] and a strong maximum principle for a single elliptic equation to prove the above lemma.

Proof

Such a strong maximum principle was also mentioned in [8, Remark 1.7], but the statement is not clear because the assumption on the coupling coefficients was not correctly written. Thus, here we aim at giving a concise proof by using the maximum principle [8, Theorem 1.1] and a strong maximum principle (e.g. [7, Theorem 4, Chapter 6], [10, Theorem 8.19]).

Since we assume the coupling matrix $\varvec{C}$ satisfies the conditions (2.17)–(2.18) in Theorem 3 and $\varvec{F}\ge \varvec{0}$ in $\Omega$, we can apply Theorem 1.1 in [8] to obtain $\varvec{w}\ge \varvec{0}$ in $\Omega$, that is, $w_k\ge 0$ in $\Omega$ for $k=1,\dots ,K$. Moreover, the assumption $\varvec{F}\ge \varvec{0},\not \equiv \varvec{0}$ implies that there exists an index $k_0\in \{1,\dots ,K\}$ such that $f_{k_0}\ge 0,\not \equiv 0$ in $\Omega$. Then we pick up the $k_0$-th equation in (7.1), which reads

$$\begin{aligned} (\mathcal A_{k_0}-c_{k_0k_0})w_{k_0}=f_{k_0}+\sum _{\begin{array}{c} \ell =1,\dots ,K\\ \ell \ne k_0 \end{array}}c_{k_0\ell }w_\ell =:g. \end{aligned}$$

According to the conditions (2.17)–(2.18), we have $c_{k_0k_0}\le 0$ and $c_{k_0\ell }\ge 0$ for $\ell \ne k_0$, and hence $g\ge 0,\not \equiv 0$. Therefore, we can employ a strong maximum principle for a single elliptic equation to conclude that $w_{k_0}>0$ in $\Omega$. It remains to prove the other components of $\varvec{w}$ are also positive. To this end, we rewrite the equations (7.1) in each component as follows

$$\begin{aligned} (\mathcal A_k-c_{k k})w_k=f_k+\sum _{\begin{array}{c} \ell =1,\dots ,K\\ \ell \ne k \end{array}}c_{k\ell }w_\ell =:g_k,\quad k=1,\dots ,K. \end{aligned}$$

(7.2)

For each $k\ne k_0$, by the condition (2.17) and $\varvec{w}\ge \varvec{0}$, we have

$$\begin{aligned} f_k+\sum _{\begin{array}{c} \ell =1,\dots ,K\\ \ell \ne k,k_0 \end{array}}c_{k\ell }w_\ell \ge 0. \end{aligned}$$

On the other hand, the condition (2.17) reads $c_{k k_0}\ge 0,\not \equiv 0$, which, together with $w_{k_0}>0$ in $\Omega$, implies $c_{k k_0}w_{k_0}\ge 0,\not \equiv 0$. Therefore, we obtain $g_k\ge 0,\not \equiv 0$ for any $k\in \{1,\dots ,K\}{\setminus }\{k_0\}$. This completes the proof by applying the strong maximum principle (e.g. [7, Theorem 4, Chapter 6]) for (7.2). $\square$

Remark 2

According to the proof, we readily find that one can prove another strong maximum principle by releasing the condition (2.17) (to include the decoupled case) and assume stronger assumption on $\varvec{F}$. That is, we have

Lemma 8

(Another strong maximum principle) Assume that $\varvec{C}$ satisfies the conditions (2.19) in Remark 1, (2.18) in Theorem 3 and $f_k\ge 0,\not \equiv 0$ in $\Omega$ for all $k\in \{1,\dots ,K\}$. Then the solution to (7.1) satisfies $\varvec{w}>\varvec{0}$ in $\Omega ,$ that is,  $w_k>0$ in $\Omega$ for all $k=1,\dots ,K$.

1.2 Proof of (4.6)

We will prove g(s) is bounded on [0, 1] and satisfies $\lim _{s\rightarrow 0}g(s)=0$.

Proof

Recall that

$$\begin{aligned} g(s)=s^{1-\alpha _K}\int _0^\infty t^{-\alpha _K}f(t)\,\mathrm e^{-s t}\,\mathrm dt. \end{aligned}$$

By direct calculation, it is not difficult to prove the boundedness of the above defined function g(s) on [0, 1]. It remains to show $\lim _{s\rightarrow 0}g(s)=0$. For this, in view of the fact that $\lim _{t\rightarrow +\infty }f(t)=0$, it follows that for any $\varepsilon >0$, there exists $N=N(\varepsilon )>0$ such that $f(t)\le \varepsilon$ for any $t>N$. Therefore, we break the integral in the definition of g(s) into two parts: [0, N] and $(N,\infty )$, we arrive at

$$\begin{aligned} g(s)=s^{1-\alpha _K}\int _0^N t^{-\alpha _K}f(t)\,\mathrm e^{-s t}\,\mathrm dt+s^{1-\alpha _K}\int _N^\infty t^{-\alpha _K}f(t)\,\mathrm e^{-s t}\,\mathrm dt=:I_1+I_2. \end{aligned}$$

We give estimates for $I_1$ and $I_2$ separately. For $I_2$, we see that

$$\begin{aligned} I_2\le \varepsilon \,s^{1-\alpha _K}\!\!\int _N^\infty t^{-\alpha _K}\mathrm e^{-s t}\,\mathrm dt\le \varepsilon \,s^{1-\alpha _K}\!\!\int _0^\infty t^{-\alpha _K}\mathrm e^{-s t}\,\mathrm dt\le \varepsilon \,s^{1-\alpha _K}s^{\alpha _K-1}=\varepsilon . \end{aligned}$$

For $I_1$, we choose sufficiently small $\delta >0$ so that $\delta <\frac{1}{N} \varepsilon ^{\frac{1}{1-\alpha _K}}$, and we conclude from the boundedness of the function f(t) that

$$\begin{aligned} I_1\le C\,s^{1-\alpha _K}\int _0^N t^{-\alpha _K}\,\mathrm e^{-s t}\,\mathrm dt\le C\,s^{1-\alpha _K}\frac{N^{1-\alpha _K}}{1-\alpha _K} \le \frac{C\,\varepsilon }{1-\alpha _K},\quad \forall \,s<\delta . \end{aligned}$$

Collecting all the above estimates, we finally obtain for any $\varepsilon >0$, there exists $\delta <\frac{1}{N}\varepsilon ^{\frac{1}{1-\alpha _K}}$ such that

$$\begin{aligned} g(s)\le C\,\varepsilon ,\quad \forall \,s<\delta , \end{aligned}$$

that is $\lim _{s\rightarrow 0}g(s)=0$. We finish the proof of (4.6). $\square$