1 Introduction

In recent years, the backward diffusion equations (BDEs) have attracted the attention of many researchers due to its application arose in a variety of professional fields. We can refer to two most important areas of application of the BDEs which are image deblurring and hydrologic inversion. In image deblurring, the BDEs can be applied for the deblurring process in image restoration (see e.g., [2, 5, 7]). In hydrologic inversion, the BDEs are also applied to identify sources of groundwater pollution by reconstructing the contaminant plume history (see e.g., [1]).

More recently, the orders of the fractional diffusion equations which may be a function of time (t), space (x) or both of them were studied (see e.g., [11, 12]). These fractional orders help to describe the behavior of some heterogeneous diffusion processes better than the constant fractional orders. To the best of our knowledge, the fractional orders of the BDEs were only investigated as usual to be a constant. Hence, in this paper, we consider the problem with the time-dependent order as a benchmark case.

To state precisely the problem, we set up some notations. Let \(T>0\), and \(\alpha , \kappa : [0, T] \rightarrow (0, \infty )\) be two continuous positive functions. Let H be a Hilbert space, and let \({\mathbb {A}}: D({\mathbb {A}}) \subset H \rightarrow H\) be a positive, self-adjoint operator with compact inverse on H. Using the notations, we consider the problem of finding a function \(u: [0,T]\rightarrow H\) satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t + \kappa (t) {\mathbb {A}}^{\alpha (t)} u = f(t, u(t)), \quad 0< t<T, \\ u(T) = h. \end{array}\right. } \end{aligned}$$
(1)

The problem (1) is ill-posed; hence, a regularization method for this problem is in order. In the case that \(\alpha (t)\equiv \alpha \) and \(\kappa (t) \equiv \kappa \) are exactly known constants and \({\mathbb {A}}= (-\Delta )\), the regularization was studied in some recent papers. In [9], the authors introduced three methods, which are Tikhonov, Landweber iteration and iterative Lavrentiev to regularize the homogeneous problem; after that, they applied the results obtained in image deblurring. In the case of nonhomogeneous [13], the authors used the truncated method to regularize the fractional backward problem in an unbounded domain. In [8], the authors used the idea of the Tikhonov method to regularization the nonlinear backward problem in \({\mathbb {R}}^d\).

Although there are some works on this topic, the backward problem with perturbed variable diffusion coefficients and variable orders is still under investigation. In the real-world problems, the diffusion coefficients and fractional orders are established from statistical models as the Levy flight or from experiments; hence, the diffusion coefficients and orders are not exactly known. In fact, we can have a known approximations \(\alpha _\epsilon , \kappa _\epsilon \in C[0,T] \) such that

$$\begin{aligned} \Vert \alpha _\epsilon -\alpha \Vert _{C[0,T]} , \, \, \Vert \kappa _\epsilon -\kappa \Vert _{C[0,T]} \le \epsilon , \end{aligned}$$

where \(\epsilon \) is a noise level. So, the stability of solution of these problems with respect to diffusion coefficients and fractional orders is worth considering. Besides, if the variable diffusion coefficients and variable orders are inexact, then some common regularization schemes could be disabled (see in sect. 4). Even so, papers devoted to the stability are quite rare. To the best of our knowledge, we can only list some papers dealt with this question in [4, 10, 14,15,16,17]. In this paper, we would like to fill a part of this gap.

We also emphasize that solving the problem (1) is rather interesting and challenging due to the disturbance both of the variable diffusion coefficient and variable order. Motivated by the above reasons, in this paper, we study the BFDEs with perturbed both variable diffusion coefficient and variable order, and a locally Lipschitz source.

This paper is structured as follows. In the second section, we introduce symbols and definitions, and we state an essential lemma which will be used to prove the main results of the present paper. The third section is devoted to investigating the formulation and uniqueness of the solution of the problem. The fourth section proposes a truncated method to regularize solution of the backward problem. The final section presents the conclusions of the paper and some problems can be expanded in future studies.

2 Preliminaries

In this section, we introduce some symbols, definitions, and lemmas which are used throughout the rest of the present paper.

We denote the inner product in the Hilbert space H by \(\langle . ,.\rangle \) and the associated norm by \(\Vert .\Vert \). From the assumptions, the operator \({\mathbb {A}}\) has the system \((\lambda _k,\phi _k)\) of eigenvalues \(\{\lambda _k\}\) and eigenfunctions \(\{\phi _k\}\), respectively, with

$$\begin{aligned} 0< \lambda _1<\lambda _2<..<\lambda _k<.., \, \, \lim _{k \rightarrow \infty } \lambda _k = \infty . \end{aligned}$$

and the functions \(\{\phi _k\}\) being an orthonormal basis of the space H. In present paper, we follow [18, 19] to define the variable fractional order of the operator. For \(\alpha \in C[0,T]\), \(\alpha (t) > 0\) for all \(t\in [0,T]\), we put

$$\begin{aligned} {\mathbb {H}}^{\alpha } := \left\{ w\in C([0,T]; H):\ w = \sum _{k=1}^{+\infty } w_k(t)\phi _k\ \mathrm{and} \ \sup _{t\in [0,T]}\sum _{k=1}^{+\infty } \lambda _k^{\alpha (t) } | w_k(t) |^2<+\infty \right\} , \end{aligned}$$

where \(w_k(t) = \langle w(t), \phi _k \rangle \). We define the norm in \({\mathbb {H}}^{\alpha }\) by

$$\begin{aligned} \Vert w\Vert _{\alpha } = \sup _{t\in [0,T]} \left( \sum _{k=1}^{+\infty } \lambda _k^{\alpha (t)} | w_k(t) |^2 \right) ^{1/2} . \end{aligned}$$

Throughout this paper, for every function \(\gamma \in C[0, T]\) arbitrary, we denote

$$\begin{aligned} \left| \!\left| \!\left| {\gamma }\right| \!\right| \!\right| = \max _{t \in [0, T]} | \gamma (t) |, \,\, \gamma _*=\min _{0 \le t \le T} \gamma (t), \, \, \gamma ^*=\max _{0 \le t \le T} \gamma (t). \end{aligned}$$

For any positive function \(\alpha \) in C[0, T], it is clear that \({\mathbb {H}}^{\alpha ^*} \subset {\mathbb {H}}^{\alpha }\subset {\mathbb {H}}^{\alpha _*} \subset C([0,T]; H) \). We can easily verify that there exist three constants \(C_1, \ C_2, \ C_3 >0\) such that \( \sup _{0\le t\le T}\Vert w(t) \Vert \le C_1 \Vert w \Vert _{\alpha _*} \le C_2 \Vert w \Vert _{\alpha } \le C_3 \Vert w \Vert _{\alpha ^*} \) for any \(w \in {\mathbb {H}}^{\alpha ^*}\). Using the above notations, we define \({\mathbb {A}}^{\alpha (t)} w\) by

$$\begin{aligned} {\mathbb {A}}^{\alpha (t)} w = \sum _{k=1}^{+\infty } \lambda _k^{\alpha (t) } w_k(t) \phi _k. \end{aligned}$$

Suggested by ideas in [6], we also introduce the Gevrey class of functions of order \(p>0\) and index \(\ell >0\) as

$$\begin{aligned} {\mathbb {G}}^p_\ell = \left\{ v \in H: \sum _{k=1}^\infty e^{2 \ell \lambda _k^p} |v_k|^2 < \infty \right\} . \end{aligned}$$

We will justify this class in the next section.

In this paper, we deal with the source \(f=f(t,z)\) in the class of functions which are locally Lipschitz with respect to the second variable, i.e., for every \(z_1, z_2 \in H\) such that \(\Vert z_1 \Vert , \Vert z_2 \Vert \le M\), there exists an \(L(M)>0\) such that

$$\begin{aligned} \Vert f(t, z_1) - f(t, z_2) \Vert \le L(M) \Vert z_1 - z_2 \Vert \end{aligned}$$
(2)

for any \(t \in [0, T]\).

Next, we state an essential lemma which plays an important role in proving main results of this paper.

Lemma 1

Let \(T, \ a_1,\ a_2,\ b_1,\ b_2\) be positive numbers such that \( a_2 \le a_1,\ b_2 \le b_1\). Let \(\kappa _i, \alpha _i \in C[0, T]\) such that \(a_2 \le \kappa _i(t) \le a_1\) and \(b_2 \le \alpha _i(t)\le b_1\) for any \(t \in [0, T]\), \((i=1, 2)\). Fix \(c>1\), then for any \(\lambda _0\ge c\) and for every \(\lambda \in (0, \lambda _0]\), we have

$$\begin{aligned}&\left| \exp \left( \int _t^T \kappa _1(\tau ) \lambda ^{\alpha _1(\tau )} \,\mathrm{d}s\right) - \exp \left( \int _t^T \kappa _2(\tau ) \lambda ^{\alpha _2(\tau )} \,\mathrm{d}s\right) \right| \\&\quad \le A \left| \!\left| \!\left| {\alpha _1 - \alpha _2}\right| \!\right| \!\right| + B \left| \!\left| \!\left| {\kappa _1 - \kappa _2}\right| \!\right| \!\right| , \end{aligned}$$

where \(A = A_0 \exp \left( a_1 (T- t ) \lambda _0^{b_1}\right) \lambda _0^{b_1} \ln \lambda _0 \) , \( B = B_0 \exp \left( a_1 (T- t ) \lambda _0^{b_1}\right) \lambda _0^{b_1} \), and \(A_0, B_0\) independent of \(\alpha _1-\alpha _2\), \(\kappa _1-\kappa _2\), \(\lambda \), \(\lambda _0\) and t.

Proof

First, we have for \(0<a \le b\) and \(y, z \in [a, b]\), then

$$\begin{aligned}&| e^y -e^z | \le e^b| y - z|, \end{aligned}$$
(3)
$$\begin{aligned}&|\lambda ^{y} -\lambda ^z| \le \lambda ^b \ln \lambda | y - z|, \ \ \forall \lambda \ge 1, \end{aligned}$$
(4)
$$\begin{aligned}&|\lambda ^{y} -\lambda ^z| \le \lambda ^a |\ln \lambda | | y - z|, \ \ \forall \ 0<\lambda <1. \end{aligned}$$
(5)

The proofs of the three latter inequalities are elementary and omitted. Now, using these inequalities, we will prove the result of the lemma. Indeed, if \(\lambda \ge 1\), using (3)–(4), we obtain by direct computations

$$\begin{aligned}&\left| \exp \left( \int _t^T \kappa _1(\tau )\lambda ^{\alpha _1(\tau )} \,\mathrm{d}\tau \right) - \exp \left( \int _t^T \kappa _2(\tau ) \lambda ^{\alpha _2(\tau )} \,\mathrm{d}\tau \right) \right| \nonumber \\&\quad \le \exp \left( a_1 \lambda ^{b_1} (T-t) \right) \left| \int _t^T\left( \kappa _1(\tau ) \lambda ^{\alpha _1(\tau )}- \kappa _2(\tau ) \lambda ^{\alpha _2(\tau )} \right) \,\mathrm{d}\tau \right| \nonumber \\&\quad \le A_1 \left| \!\left| \!\left| {\alpha _1 - \alpha _2}\right| \!\right| \!\right| + B_1 \left| \!\left| \!\left| {\kappa _1 - \kappa _2}\right| \!\right| \!\right| , \end{aligned}$$
(6)

where \(A_1 = T a_1 \exp \left( a_1 (T- t ) \lambda _0^{b_1}\right) \lambda _0^{b_1} \ln \lambda _0 \) , \( B_1 = T \exp \left( a_1 (T- t ) \lambda _0^{b_1}\right) \lambda _0^{b_1} \).

If \(0<\lambda <1\), using (3) and (5), by the same method in estimating (6), we obtain the desired result. \(\square \)

3 Formulation and Uniqueness of the Solution of the Problem

In this section, we give a uniqueness results for the problem (1). Firstly, we establish the formula solution of our problem. In fact, for each final data \(h \in H\), a function \(u \in {\mathbb {H}}^\alpha \) is called a weak solution of the problem (1) if u satisfies the weak form

$$\begin{aligned} {\left\{ \begin{array}{ll} \langle u_t, w\rangle + \kappa (t) \langle {\mathbb {A}}^{\alpha (t)/2} u, {\mathbb {A}}^{\alpha (t)/2} w \rangle =\langle f(t, u), w \rangle ,\, \, \forall \, w \in {\mathbb {H}}^\alpha ,\\ \langle u(T), w\rangle = \langle h, w\rangle , \, \, \forall \, w \in {\mathbb {H}}^\alpha . \end{array}\right. } \end{aligned}$$
(7)

It is easy to see that the system (7) can be transformed to the integral equation

$$\begin{aligned} u(t)= & {} \sum _{k=1}^\infty \exp \left( \int _t^T \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) h_k\phi _k.\nonumber \\&-\sum _{k=1}^\infty \int _t^T \exp \left( \int _t^s \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) f_k(u)(s) \,\mathrm{d}s\phi _k \end{aligned}$$
(8)

We note that if \(f\equiv 0\) and \(\alpha (t)\equiv \alpha _0>0, \kappa (t) = \kappa _0>0\) the latter formula can be rewritten as

$$\begin{aligned} u(t)=\sum _{k=1}^\infty \exp \left( \kappa _0\lambda _k^{\alpha _0}(T-t) \right) h_k\phi _k. \end{aligned}$$

Hence, in this special case, the condition \(u(0)\in H\) is equivalent to \(h\in {\mathbb {G}}^r_\ell \) with \(p=\alpha _0,\ \ell =\kappa _0 T/2\). This is the justification of the Gevrey class defined in Sect. 2.

Theorem 1

Assume that the condition (2) holds. Put

$$\begin{aligned} {\mathbb {K}}^\alpha =\{w\in C([0,T],H): \Vert w\Vert _{{\mathbb {K}}^\alpha }<\infty \} \end{aligned}$$

where

$$\begin{aligned} \Vert w\Vert ^2_{{\mathbb {K}}^\alpha }=\sup _{0\le t\le T}\sum _{k=1}^\infty \exp \left( 2 \int _t^T \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) |\langle w(t),\phi _k\rangle |^2<\infty . \end{aligned}$$

If \(u_1,u_2\in {\mathbb {K}}^\alpha \) satisfy (7) the \(u_1(t)=u_2(t)\) for all \(t\in [0,T]\).

Proof

Denote \(w=u_2-u_1\) and \(M=\max \{\Vert u_1\Vert _\alpha , \Vert u_2\Vert _\alpha \}\). We obtain in view of (8)

$$\begin{aligned} w(t)=\sum _{k=1}^\infty \left( -\int _t^T\exp \left( \int _t^s \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) (f_k(w+u_1)(s)-f_k(u_1)(s))\,\mathrm{d}s\right) \phi _k. \end{aligned}$$

Denoting \(w_k(t)=\langle w(.,t),\phi _k\rangle \), we have

$$\begin{aligned} w(t)= & {} \sum _{k=1}^N\left( -\int _t^T\exp \left( \int _t^s \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) (f_k(w+u_1)(s)-f_k(u_1)(s))\,\mathrm{d}s\right) \phi _k\\&+\sum _{k=N+1}^\infty w_k(t)\phi _k. \end{aligned}$$

Hence

$$\begin{aligned} \Vert w(t)\Vert ^2\le & {} \sum _{k=1}^N T\int _t^T \exp \left( 2\int _t^s \kappa (\tau ) \lambda _N^{\alpha (\tau )} \,\mathrm{d}\tau \right) \Vert f_k(w+u_1)(s)-f_k(u_1)(s)\Vert ^2 \,\mathrm{d}s \\&+ \sum _{k=N+1}^\infty |w_k(t)|^2. \end{aligned}$$

We note that

$$\begin{aligned} \sum _{k=1}^N\Vert f_k(w+u_1)(s)-f_k(u_1)(s)\Vert ^2\le \Vert f(w+u_1)(s)-f(u_1)(s)\Vert ^2\le L^2(M)\Vert w(s)\Vert ^2. \end{aligned}$$

Hence if we put

$$\begin{aligned} \psi (t)= & {} \exp \left( 2\int _0^t \kappa (\tau ) \lambda _N^{\alpha (\tau )} \,\mathrm{d}\tau \right) \Vert w(t)\Vert ^2,\\ R_N(t)= & {} \exp \left( 2\int _0^t \kappa (\tau ) \lambda _N^{\alpha (\tau )} \,\mathrm{d}\tau \right) \sum _{k=N+1}^\infty |w_k(t)|^2, \end{aligned}$$

then

$$\begin{aligned} \psi (t) \le TL^2(M)\int _t^T\psi (s) \,\mathrm{d}s+R_N(t). \end{aligned}$$

Using the Gronwall inequality, one gets

$$\begin{aligned} 0 \le \psi (t) \le R_N(t)+TL^2(M)\int _t^T R_N(s) e^{TL^2(M)(s-t)}ds. \end{aligned}$$

We note that \(R_N(s)\le \Vert w\Vert ^2_{{\mathbb {K}}^\alpha }\). Hence the latter inequality yields

$$\begin{aligned} \Vert w(t)\Vert \le \exp \left( -2\int _0^t \kappa (\tau ) \lambda _N^{\alpha (\tau )} \,\mathrm{d}\tau \right) \Vert w\Vert _{{\mathbb {K}}^\alpha } e^{T^2L^2(M)}. \end{aligned}$$

Letting \(N\rightarrow \infty \), we obtain \(w(t)=0\) for all \(0<t\le T\). By the continuity of w, we also have \(w(0)=0\). Therefore, we have \(u_1(t)=u_2(t)\) for all \(t\in [0,T]\). \(\square \)

4 Regularization of the Problem

In this section, we investigate a truncated method to regularization the problem (1) in the case of inexact variable diffusion coefficient and variable order.

Firstly, we analyze the instability of the final value problem. For convenience, let us consider the homogeneous problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t + \kappa (t) {\mathbb {A}}^{\alpha (t)} u = 0, \quad 0< t<T, \\ u(T) = h, \end{array}\right. } \end{aligned}$$
(9)

where \({\mathbb {A}}=\Delta \) and \(u=u(x, t)\) with \(x \in {\mathbb {R}}^d\). By Weyl’s law, we known that \(\lambda _k=O\left( k^{2/d}\right) \) and \(\sum _{k=1}^\infty \frac{1}{\lambda _k^d} <+\infty . \) From (8), the problem (9) can be transformed to the following integral equation

$$\begin{aligned} u(t)=\sum _{k=1}^\infty \exp \left( \int _t^T \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) h_k. \end{aligned}$$

This gives

$$\begin{aligned} u(0)=\sum _{k=1}^\infty \exp \left( \int _0^T \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) h_k. \end{aligned}$$

We choose

$$\begin{aligned} h = \sum _{k=1}^\infty \frac{ \phi _k }{ \lambda _k^{d/2} \exp \left( \int _0^T \kappa (\tau )\lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) } . \end{aligned}$$

We have

$$\begin{aligned} \Vert u(0)\Vert ^2 = \sum _{k=1}^\infty \frac{1}{\lambda _k^d} < +\infty . \end{aligned}$$

Now we consider the truncated method as follows

$$\begin{aligned} R_N(h)(t) = \sum _{k=1}^N \exp \left( \int _t^T \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) h_k . \end{aligned}$$

Direct computations yield

$$\begin{aligned} \left\| {u(0)-R_N(h)(0)}\right\| ^2 = \sum _{k=N+1}^\infty \frac{1}{\lambda _k^d} \rightarrow 0 \ \ \text {as} \ \ N \rightarrow \infty . \end{aligned}$$

This shows that the truncated regularization is convergent. Now we perturbed the order fractional \(\alpha (t)\) by \(\alpha _N(t) =\alpha (t)+r_N\) with \(r_N = \frac{\ln \left( 1+ \frac{d \ln \lambda _N}{\kappa _* T \lambda _N^{\alpha _*}} \right) }{\ln \lambda _N}\) for some N large enough such that \(\lambda _N >1\). It is easy to see that \(r_N \rightarrow 0\) as \(N \rightarrow \infty \) and \( \exp \left( \kappa _* T (\lambda _N^{r_N} - 1) \lambda _N^{\alpha _*}\right) =\lambda _N^d \). In this case, the truncated method become

$$\begin{aligned} R_{N, inexact}(h)(t) = \sum _{k=1}^N \exp \left( \int _t^T \kappa (\tau ) \lambda _k^{\alpha _N(\tau )} \,\mathrm{d}\tau \right) h_k \phi _k , \end{aligned}$$

where \(h_k = \frac{1}{ \lambda _k^{d/2} \exp \left( \int _0^T \kappa (\tau )\lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) }\). By direct computation, we have

$$\begin{aligned}&\Vert R_{N, inexact}(h)(0)-u(0) \Vert ^2\\&\quad \ge \sum _{k=1}^N\left( \exp \left( \int _0^T \kappa (\tau ) \lambda _k^{\alpha _N(\tau )} \,\mathrm{d}\tau \right) -\exp \left( \int _0^T \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) \right) ^2h_k^2\\&\quad \ge \left( \exp \left( \int _0^T \kappa (\tau ) \lambda _N^{\alpha _N(\tau )} \,\mathrm{d}\tau \right) -\exp \left( \int _0^T \kappa (\tau ) \lambda _N^{\alpha (\tau )} \,\mathrm{d}\tau \right) \right) ^2h_N^2\\&\quad =\frac{1}{\lambda _N^d}\left( \exp \left( \left( \lambda _N^{r_N}-1\right) \int _0^T \kappa (\tau ) \lambda _N^{\alpha (\tau )} \,\mathrm{d}\tau \right) -1\right) ^2\\&\quad \ge \frac{1}{\lambda _N^d}\left( \exp \left( \kappa _* T\left( \lambda _N^{r_N}-1\right) \lambda _N^{\alpha _*}\right) -1\right) ^2\\&\quad \ge \frac{\left( \lambda _N^d-1\right) ^2}{\lambda _N^d} \rightarrow +\infty , \, \,(N \rightarrow \infty ). \end{aligned}$$

The latter inequality shows that the solution of the final value problem is instability with respect to the fractional-order and leads to the common regularization truncated strategy to failure. In other words, we need a significant improvement for the current regularization strategy when the diffusion coefficient and fractional are inexact.

Now, we will construct a regularization for our problem. For \(N\in {\mathbb {N}}, M>0\), we approximate the problem (1) by the problem \({\mathcal {P}}(M, N, h, \alpha , \kappa )\) as below  

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{N, t} + \kappa (t) {\mathbb {A}}^{\alpha (t)} u_N = \sum _{k=1}^N f_{M, k}(u_N)(t) \phi _k\\ u_N(T) = \sum _{k=1}^N h_k \phi _k, \end{array}\right. } \end{aligned}$$
(10)

where

$$\begin{aligned} f_M(t, u) = \left\{ \begin{array}{ll} f(t, u), &{} \text {if} ~ \Vert u \Vert \le M, \\ f\left( t, M \frac{u}{\Vert u\Vert } \right) , &{} \text {otherwise} \end{array}\right. \end{aligned}$$

and \(f_{M, k}(u_N)(t)=\langle f_M(t, u_N),\phi _k\rangle \). We can transform the problem (10) to the integral equation

$$\begin{aligned} u_N(t)= & {} \sum _{k=1}^{N} \exp \left( \int _t^T \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) h_k \phi _k. \nonumber \\&- \sum _{k=1}^{N} \left( \int _t^T \exp \left( \int _t^s \kappa (\tau ) \lambda _k^{\alpha (\tau )} \,\mathrm{d}\tau \right) f_{M, k}(u_N)(s)\,\mathrm{d}s \right) \phi _k. \end{aligned}$$
(11)

From now on, we suppose further that N is larger enough such that \(\lambda _N > 1\). Firstly, we show that, the approximate problem is well-posed.

Theorem 2

Let f be a locally Lipschitz function as in (2), and such that \(f(t, 0) \in L^2(0, T; H)\). Then

(i):

The problem \({\mathcal {P}}(M, N, h, \alpha , \kappa )\) has a unique solution in \( {\mathbb {H}}^\alpha \), and we have the following estimate

$$\begin{aligned} \int _0^T \Vert f_N(t, u_N(t)) \Vert ^2 \,\mathrm{d}t \le P_0 \exp \left( 2 \kappa T \lambda ^{\alpha ^*}_N \right) , \end{aligned}$$
(12)

where \(P_0\) is a positive number which is independent of \(\lambda _N\), t.

(ii):

If \(v_N\) is the solution of the problem \({\mathcal {P}}(M, N, \widetilde{h}, \widetilde{\alpha }, \widetilde{\kappa })\) for every \(\widetilde{h} \in H\) and \(\widetilde{\kappa }, \, \widetilde{\alpha } \in C[0, T]\), we have

$$\begin{aligned} \Vert u_N (t) - v_N(t) \Vert \le Q_1 \Vert h - \widetilde{h}\Vert + Q_2 \left| \!\left| \!\left| {\alpha - \widetilde{\alpha }}\right| \!\right| \!\right| + Q_3 \left| \!\left| \!\left| {\kappa - \widetilde{\kappa }}\right| \!\right| \!\right| , \end{aligned}$$
(13)

where \(Q_1, Q_2, Q_3\) are positive numbers independent of \(h -\widetilde{h}\), \(\alpha - \widetilde{\alpha }\) and \(\kappa -\widetilde{\kappa }\).

Proof

(i):

We can easily verify that the function \(f_M\) satisfies the condition

$$\begin{aligned} \Vert f_M(t, z_1) - f_M(t, z_2) \Vert \le 2 L(M) \Vert z_1 - z_2 \Vert \end{aligned}$$
(14)

for any \(t \in [0, T]\) and \(z_1, z_2 \in H\). On the other hand, the dimension of the approximate problem is finite; hence, we can use the Cauchy–Lipschitz–Picard theorem (see [3], chapter 7, page 184) to deduce that the problem \({\mathcal {P}}(M, N, h, \alpha , \kappa )\) has a unique solution \(u_N\) in \({\mathbb {H}}^\alpha \).

First, we verify that

$$\begin{aligned} \Vert u_N (t) \Vert ^2 \le P_1 \exp \left( 2 \kappa T \lambda ^{\alpha ^*}_N\right) , \end{aligned}$$

where \(P_1\) is a positive number which is independent of \(\lambda _N\), t. Indeed, applying the Cauchy–Schwarz inequality for Eq. (11) yields

$$\begin{aligned} \Vert u_N (t)\Vert ^2\le & {} 2 \exp \left( 2\int _t^T \kappa (\tau ) \lambda ^{\alpha (\tau )}_N \,\mathrm{d}\tau \right) \Vert h \Vert ^2\\&+ 2 T \int _t^T \exp \left( 2 \int _t^s \kappa (\tau ) \lambda _N^{\alpha (\tau )} \,\mathrm{d}\tau \right) \Vert f_{M}(s, u_N) \Vert ^2 \,\mathrm{d}s\\\le & {} 2 \exp \left( 2 \int _t^T \kappa (\tau ) \lambda ^{\alpha (\tau )}_N \,\mathrm{d}\tau \right) \left( \Vert h \Vert ^2 +2 \sigma _T \right) \\&+ 16 T L^2(M) \int _t^T \exp \left( 2 \int _t^s \kappa (\tau ) \lambda _N^{\alpha (\tau )} \,\mathrm{d}\tau \right) \Vert u_N(s) \Vert ^2 \,\mathrm{d}s, \end{aligned}$$

where \(\sigma _T = \int _0^T \Vert f(s, 0) \Vert ^2 \,\mathrm{d}s\). The latter inequality deduces

$$\begin{aligned}&\exp \left( 2\int _0^t \kappa (\tau ) \lambda ^{\alpha (\tau )}_N \,\mathrm{d}\tau \right) \Vert u_N (t)\Vert ^2\\&\quad \le 2 \exp \left( 2 \int _0^T \kappa (\tau ) \lambda ^{\alpha (\tau )}_N \,\mathrm{d}\tau \right) \left( \Vert h \Vert ^2 +2 \sigma _T \right) \\&\qquad + 16 T L^2(M) \int _t^T \exp \left( 2 \int _0^s \kappa (\tau ) \lambda _N^{\alpha (\tau )} \,\mathrm{d}\tau \right) \Vert u_N(s) \Vert ^2 \,\mathrm{d}s. \end{aligned}$$

Applying the Gronwall inequality and direct computation, we obtain

$$\begin{aligned} \Vert u_N (t)\Vert ^2 \le P_1 \exp \left( 2 \kappa ^* T \lambda ^{\alpha ^*}_N\right) \end{aligned}$$
(15)

due to \(\int _0^T \kappa (\tau ) \lambda ^{\alpha (\tau )}_N \,\mathrm{d}\tau \le \kappa ^* T \lambda ^{\alpha ^*}_N\), where \( P_1 = 2 \left( \Vert h \Vert ^2 +2 \sigma _T \right) \exp \left( 16 T^2L^2(M)\right) \). Now we prove (12). It is easy to see that (14) implies

$$\begin{aligned} \Vert f_N(t, u_N(t)) \Vert ^2 \le 8 L^2(M) \Vert u_N(t) \Vert ^2 +2 \Vert f(t, 0) \Vert ^2. \end{aligned}$$

Combining the latter inequality with (15), we have

$$\begin{aligned} \int _0^T \Vert f_N(t, u_N(t)) \Vert ^2 \,\mathrm{d}t \le P_1 \exp \left( 2 \kappa ^* T \lambda ^{\alpha ^*}_N \right) + 2 \sigma _T \le P_2 \exp \left( 2 \kappa ^* T \lambda ^{\alpha ^*}_N \right) , \end{aligned}$$
(16)

where \(P_2 = P_1 + 2 \sigma _T\) and \(P_1\) defined in (15). This completes the proof of Part (i).

(ii):

Firstly, we denote by \(v_N\) and \(w_N\) the solutions of the problem \({\mathcal {P}}(M, N, h, \widetilde{\alpha }, \widetilde{\kappa })\) and \({\mathcal {P}}(M, N, \widetilde{h}, \widetilde{\alpha }, \widetilde{\kappa })\) respectively. Put \(\eta = \max \{ \alpha ^*, \, \widetilde{\alpha }^* \}\), \(\mu = \max \{\kappa ^*, \widetilde{\kappa }^*\}\) . By the same way as in Part (i), we have

$$\begin{aligned} \Vert v_N (t) - w_N(t) \Vert \le Q_1 \Vert h - \widetilde{h} \Vert , \end{aligned}$$
(17)

where \(Q_1 = \widetilde{Q_1}\exp \left( \mu T\lambda ^{\eta }_N \right) \) and \(\widetilde{Q_1} = \sqrt{2} \exp ( 4 T^2L^2(M))\).

Secondly, we will prove that

$$\begin{aligned} \Vert u_N(t) - v_N(t) \Vert \le Q_2\left| \!\left| \!\left| {\alpha - \widetilde{\alpha }}\right| \!\right| \!\right| + Q_3 \left| \!\left| \!\left| {\kappa - \widetilde{\kappa }}\right| \!\right| \!\right| , \end{aligned}$$
(18)

where \(Q_2 = Q_0 \exp \left( 2 \mu T \lambda _N^{\eta }\right) \lambda _N^{\eta } \ln \lambda _N\), \(Q_3 = Q_0 \exp \left( 2 \mu T \lambda _N^{\eta }\right) \lambda _N^{\eta }\) and \(Q_0\) independent of \(\alpha - \widetilde{\alpha }\), \(\kappa - \widetilde{\kappa }\), \(\lambda _N\) and t.

Indeed, using Lemma 1 and by direct computation, we have

$$\begin{aligned}&\Vert u_N(t) - v_N(t) \Vert ^2\nonumber \\&\quad \le (C_1 \left| \!\left| \!\left| {\alpha - \widetilde{\alpha }}\right| \!\right| \!\right| +C_2 \left| \!\left| \!\left| {\kappa - \widetilde{\kappa }}\right| \!\right| \!\right| )^2 \left( \Vert h\Vert ^2 + \int _t^T \Vert f_N(s, u_N(s))\Vert ^2 \,\mathrm{d}s \right) \nonumber \\&\qquad + 12 T L^2(M)\int _t^T\exp \left( 2 \mu (s - t) \lambda _N^{\eta }\right) \Vert u_N(s) - v_N(s)\Vert ^2\,\mathrm{d}s, \end{aligned}$$
(19)

where \(C_1 = C_{01} \exp \left( \mu (T-t) \lambda _N^{\eta }\right) \lambda _N^\eta \ln \lambda _N, C_2 = C_{02} \exp \left( \mu (T-t) \lambda _N^{\eta }\right) \lambda _N^\eta \) and \(C_{01}, C_{02}\) independent of \(\alpha - \widetilde{\alpha }\), \(\kappa - \widetilde{\kappa }\), \(\lambda _N\) and t. On the other hand, using (16), we have

$$\begin{aligned} \Vert h\Vert ^2 + \int _t^T \Vert f_N(s, u_N(s))\Vert ^2 \,\mathrm{d}s \le \Vert h\Vert ^2 + P_2 \exp \left( 2 \mu T \lambda _N^{\eta }\right) \le P_3 \exp \left( 2 \mu T \lambda _N^{\eta }\right) , \end{aligned}$$

where \(P_3\) independent of \(\lambda _N\), t. Combining the latter inequality with (19), we have

$$\begin{aligned}&\exp \left( 2\mu \lambda _N^{\eta } t\right) \Vert u_N(t) - v_N(t) \Vert ^2\\&\quad \le D_0 (D_1 \left| \!\left| \!\left| {\alpha - \widetilde{\alpha }}\right| \!\right| \!\right| +D_2 \left| \!\left| \!\left| {\kappa - \widetilde{\kappa }}\right| \!\right| \!\right| )^2 \exp \left( 2 \mu T \lambda _N^{\eta } \right) \\&\qquad + 12 T L^2(M) \int _t^T\exp \left( 2 \mu \lambda _N^{\eta } s \right) \Vert u_N(s) - v_N(s)\Vert ^2\,\mathrm{d}s. \end{aligned}$$

where \(D_1 = \exp \left( \mu T \lambda _N^{\eta }\right) \lambda _N^\eta \ln \lambda _N, D_2 = \exp \left( \mu T \lambda _N^{\eta }\right) \lambda _N^\eta \) and \(D_0\) independent of \(\alpha - \widetilde{\alpha }\), \(\kappa - \widetilde{\kappa }\), \(\lambda _N\) and t. Applying the Gronwall inequality yields

$$\begin{aligned} \Vert u_N(t) - v_N(t) \Vert \le Q_2 \left| \!\left| \!\left| {\alpha - \widetilde{\alpha }}\right| \!\right| \!\right| + Q_3 \left| \!\left| \!\left| {\kappa - \widetilde{\kappa }}\right| \!\right| \!\right| , \end{aligned}$$
(20)

where \(Q_2 = Q_0\exp \left( 2 \mu T \lambda _N^{\eta }\right) \lambda _N^\eta \ln \lambda _N, Q_3 = Q_0\exp \left( 2 \mu T \lambda _N^{\eta }\right) \lambda _N^\eta \) and

\(Q_0= \sqrt{D_0} \exp (6T^2 L^2(M))\). The proof of the inequality (18) is completed.

Lastly, by using the triangle inequality, combining (17) and (18), we obtain the result of Part (ii). This completes the proof of the Theorem 2. \(\square \)

In the next theorem, we give a regularization result for the problem (1) under some regularity assumptions of its exact solution.

Theorem 3

Suppose that the assumptions in Theorem 2 hold. Suppose further that the problem (1) has a solution \(u \in C \left( [0, T]; {\mathbb {G}}^{\alpha ^*}_\theta \right) \) for \(\theta > \kappa ^* T\), and \( \sup _{t \in [0, T]}\Vert u(t) \Vert \le M\) for some positive number M. Let \(\epsilon \in (0, 1)\), \(h_\epsilon \in H\) and \(\kappa _\epsilon , \alpha _\epsilon \in C[0, T]\) be the measurement data such that

$$\begin{aligned} \Vert h - h_\epsilon \Vert + \left| \!\left| \!\left| {\alpha - \alpha _\epsilon }\right| \!\right| \!\right| +\left| \!\left| \!\left| {\kappa - \kappa _\epsilon }\right| \!\right| \!\right| \le \epsilon , \end{aligned}$$
(21)

where \(h=u(T)\). Then there exist \(M_0\) independent of \(\epsilon \) such that

$$\begin{aligned} \Vert u(t) - u_\epsilon (t) \Vert \le M_0 \epsilon ^{\rho _0}, \end{aligned}$$

where \(u_\epsilon \) is the solution of problem \({\mathcal {P}}(M, N, h_\epsilon , \alpha _\epsilon , \kappa _\epsilon )\), and \(\rho _0=(\theta -\kappa ^*T) r_0/(1+(\theta -\kappa ^*T) r_0+2\kappa ^*T)\) with \(r_0=e^{-1/e}\).  

Proof

For any \(N \in {\mathbb {N}}\), assign the solution of the problem \({\mathcal {P}}(M, N, h, \alpha , \kappa )\) by \(u_N\). Due to \(\sup _{t \in [0, T}\Vert u(t) \Vert \le M\), this implies that \(f_M(t, u (t)) = f(t, u(t))\), then by straightforward, we have

$$\begin{aligned} \Vert u(t) - u_N(t) \Vert ^2\le & {} T L^2(M) \int _t^T \exp \left( 2 \kappa ^* \lambda _N^{\alpha ^*} (s - t)\right) \Vert u(s) - u_N(s) \Vert ^2 \,\mathrm{d}s\nonumber \\&+ \sum _{k=N+1}^{\infty } |u_k (t) |^2. \end{aligned}$$
(22)

Now, we estimate the second term in the right-hand side of the inequality (22). Since \( u \in C \left( [0, T]; {\mathbb {G}}^{\alpha ^*}_\theta \right) \), we have

$$\begin{aligned} \sum _{k=N+1}^{\infty } |u_k (t) |^2 \le \exp \big ( - 2 \theta \lambda _N^{\alpha ^*}\big ) \sum _{k=N+1}^{\infty } \exp \big ( 2 \theta \lambda _k^{\alpha ^*}\big ) |u_k (t) |^2 = \Lambda \exp \left( - 2 \theta \lambda _N^{\alpha ^*}\right) . \end{aligned}$$

Substituting the latter inequality into (22) and by direct computation gives

$$\begin{aligned}&\exp \left( 2 \kappa ^* \lambda _N^{\alpha ^*} t\right) \Vert u(t) - u_N(t) \Vert ^2\\&\quad \le 4 T L^2(M) \int _t^T \exp \left( 2 \kappa ^* \lambda _N^{\alpha ^*} s\right) \Vert u(s) - u_N(s) \Vert ^2 \,\mathrm{d}s + \Lambda \exp \left( - 2 r \lambda _N^{\alpha ^*}\right) , \end{aligned}$$

where \(r = \theta - \kappa ^* T\). Applying the Gronwall inequality yields

$$\begin{aligned} \Vert u(t) - u_N(t) \Vert \le \Lambda _0 \exp \left( - r \lambda _N^{\alpha ^*}\right) , \end{aligned}$$
(23)

where \(\Lambda _0 = \sqrt{\Lambda } \exp (2 T^2 L^2(M))\).

Let us denote the solution of the problem \({\mathcal {P}}(M, N, h_\epsilon , \alpha _\epsilon , \kappa _\epsilon )\) by \(u_\epsilon \) and \(\nu =\max \{\alpha ^*, \, \alpha _\epsilon ^* \}\) , \(\upsilon =\max \{\kappa ^*, \, \kappa _\epsilon ^* \}\). Thank to (13), we have

$$\begin{aligned} \Vert u_N(t) - u_\epsilon (t) \Vert \le Q_1 \Vert h - h_\epsilon \Vert + Q_2 \left| \!\left| \!\left| {\alpha - \alpha _\epsilon }\right| \!\right| \!\right| + Q_3 \left| \!\left| \!\left| {\kappa -\kappa _\epsilon }\right| \!\right| \!\right| , \end{aligned}$$
(24)

where \(Q_1 = \widetilde{Q_1}\exp \left( \upsilon T\lambda ^{\eta }_N \right) \), \( Q_2 = Q_0 \exp \left( 2 \upsilon T \lambda _N^{\eta }\right) \lambda _N^{\eta } \ln \lambda _N \), and

\(Q_3 = Q_0 \exp \big (2 \upsilon T \lambda _N^{\eta }\big ) \lambda _N^{\eta }\). Herein \(\widetilde{Q_1}\) defined in (17) and \(Q_0\) defined in (20).

Since \(\exp \left( \upsilon T\lambda ^{\nu }_N \right) \le \exp \left( 2 \upsilon T \lambda _N^{\nu }\right) \lambda _N^{\nu } \le \exp \left( 2 \upsilon T \lambda _N^{\nu }\right) \lambda _N^{\nu } \ln \lambda _N\) for N larger enough; hence, we can combine (21) with (24) to obtain

$$\begin{aligned} \Vert u_N(t) - u_\epsilon (t) \Vert \le Q_4 \epsilon \exp \left( 2 \upsilon T \lambda _N^{\nu }\right) \lambda _N^{\nu } \ln \lambda _N , \end{aligned}$$
(25)

where \(Q_4 = \max \left\{ \widetilde{Q_1}, Q_0 \right\} \). Combining (23) with (25) and applying the triangle inequality, we obtain

$$\begin{aligned} \Vert u(t) - u_\epsilon (t) \Vert \le Q_4 \left( \epsilon \exp \left( 2 \upsilon T \lambda _N^{\nu }\right) \lambda _N^{\nu } \ln \lambda _N + \exp \left( - r \lambda _N^{\alpha ^*}\right) \right) , \end{aligned}$$
(26)

where \(Q_4 = \max \{ \Lambda _0, Q_3 \}\). Since \(\ln \lambda _N \le \lambda _N\) for any \(N \ge 1\) and \(\lambda ^k e^{-\lambda } \le k^k e^{-k} \le k^k\) for any \(k>0\), we deduce

$$\begin{aligned} \lambda _N^{\nu } \ln \lambda _N \le \lambda _N^{\nu + 1} =e^{\lambda _N^{\nu }} e^{- \lambda _N^{\nu }} \lambda _N^{\nu ( 1+ 1/\nu )} \le e^{\lambda _N^{\nu }} ( 1+ 1/\alpha ^*)^{( 1+ 1/\alpha ^*)} . \end{aligned}$$

From the latter inequality and (26), we obtain

$$\begin{aligned} \Vert u(t) - u_\epsilon (t) \Vert \le Q_5 \left( \epsilon \exp \left( (2 \upsilon T + 1 ) \lambda _N^{\nu }\right) +\exp \left( - r \lambda _N^{\alpha ^*}\right) \right) , \end{aligned}$$
(27)

where \(Q_5=Q_4( 1+ 1/\alpha ^*)^{( 1+ 1/\alpha ^*)}\). Now we can choose the parameter \(\lambda _N\) (or N) such that the right-hand side of (27) convergence to zero as \(\epsilon \) to 0. For example, we put \(\rho =(1+r r_0+2\kappa ^*T)^{-1}\), and choose \( \lambda _N^\nu = \rho \ln (1/\epsilon ) \). We have

$$\begin{aligned} \epsilon \exp \left( (2 \upsilon T + 1 )\lambda _N^{\nu }\right)&= \epsilon \epsilon ^{-(2 \upsilon T + 1)\rho } \nonumber \\&\le \epsilon ^{1 - \rho (1+2\kappa ^* T)} \epsilon ^{-2 T ( \upsilon -\kappa ^*) \rho } \le Q_6 \epsilon ^{1 - \rho (1+2\kappa ^* T)}, \end{aligned}$$
(28)

where \(Q_6 = \epsilon ^{-2 T \rho \epsilon } < + \infty \). On the other hand, for any \(\epsilon \le e^{- 1/\rho }\), we have \(\lambda _N^{-\epsilon } = (\rho \ln (1/\epsilon ))^{-\epsilon } \ge (\ln (1/\epsilon ))^{-\epsilon } \ge \epsilon ^{\epsilon } \ge e^{-1/e} :=r_0\). This lead to

$$\begin{aligned} \exp \left( - r \lambda _N^{\alpha ^*}\right) \le \exp \left( - r \lambda _N^\nu \lambda _N^{-\epsilon }\right) \le \epsilon ^{r r_0 \rho } . \end{aligned}$$
(29)

Since \(r r_0 \rho = 1 - \rho (1+2\kappa ^* T)\), we can substitute (28) and (29) into (27) to obtain the desired result. \(\square \)

5 Conclusions

In this work, we presented a truncated method to regularized solution of the nonlinear backward fractional diffusion problem with inexact variable diffusion coefficient and variable order. Under appropriate regularity assumptions of the exact solution, we obtained the order of convergence is \(O\left( \epsilon ^{\rho _0}\right) \). It would be interesting to extend this work for problems with the diffusion coefficient and fractional order dependent on both x and t.