1 Introduction, Notations and Background

In the last years, the maximal regularity properties of boundary value problems (BVPs) for abstract differential equations (ADEs) have found many applications in PDE and pseudo DE with applications in physics (see [1, 2, 5,6,7, 9, 15,16,17,18, 22] and the references therein). ADEs have found many applications in fractional differential equations (FDEs), pseudo-differential equations (PsDE) and PDEs. FDEs were treated, e.g., in [4, 8, 10, 11, 13, 14, 19] and the references therein. The regularity properties of FDEs have been studied, e.g., in [10, 11, 19]. The existence and uniqueness of solution to FDEs in a Banach space studied, e.g., in [3, 10, 11]. The main objective of the present paper is to discuss the uniform \(L_{p}\left( {\mathbb {R}};H\right) \)-maximal regularity of the fractional ADE with parameter

$$\begin{aligned} aD^{\gamma }u+bD^{\nu }u+Au+\lambda u=f\left( x\right) \text {, }x\in \mathbb {R=}\left( -\infty ,\infty \right) , \end{aligned}$$
(1.1)

where a, b are complex numbers \(\lambda \) is a complex parameter, A is a linear operator in a Hilbert space H and \(D^{\nu }\) is a Riemann-Liouville-type fractional derivative of order \(\gamma \in \left( \left. 1,2\right] \text {, }\right. \nu \in \left( 1,2\right) \), i.e.,

$$\begin{aligned} D^{\gamma }u=\frac{1}{\Gamma \left( 2-\gamma \right) }\frac{\hbox {d}^{2}}{\hbox {d}x^{2}} \int \limits _{0}^{x}\frac{u\left( y\right) \hbox {d}y}{\left( x-y\right) ^{\gamma -1} }, \end{aligned}$$
(1.2)

here \(\Gamma \left( \gamma \right) \) is Gamma function for \(\gamma >0\) (see e.g., [8, 13] and \(\nu <\gamma \).

For \(\alpha _{i}\in \left[ 0,\infty \right) \) and \({\alpha }=\left( \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right) \). Here, \(D^{\alpha }=D_{1}^{\alpha _{1}}D_{2}^{\alpha _{2}},.,D_{n}^{\alpha _{n}}\). Let E be Banach space. Here, \(L_{p}\left( \Omega ;E\right) \) denotes the space of strongly measurable E-valued functions that are defined on the measurable subset \(\Omega \subset {\mathbb {R}}^{n}\) with the norm given by

$$\begin{aligned} \left\| f\right\| _{L_{p}\left( \Omega ;E\right) }=\left( \int \limits _{\Omega }\left\| f\left( x\right) \right\| _{E}^{p}\hbox {d}x\right) ^{\frac{1}{p}}\text {, }1\le p<\infty \ . \end{aligned}$$

Let \(S\left( {\mathbb {R}}^{n};E\right) \) denote the E-valued Schwartz class, i.e., the space of all E-valued rapidly decreasing smooth functions on \({\mathbb {R}}^{n}\) equipped with its usual topology generated by seminorms. Here, \(S^{\prime }\left( E\right) =S^{\prime }\left( {\mathbb {R}} ^{n};E\right) \) denotes the space of linear continuous mappings from S into E and is called E-valued Schwartz distributions. \(B\left( E_{1},E_{2}\right) \) denotes the space of bounded linear operators from \( E_{1}\) to \(E_{2}\). For \(E_{1}=E_{2}=E\), it denotes by \(B\left( E\right) \). A function \(\Psi \in C\left( {\mathbb {R}}^{n};B\left( E_{1},E_{2}\right) \right) \) is called a Fourier multiplier from \(L_{p}\left( {\mathbb {R}} ^{n};E_{1}\right) \) to \(L_{p}\left( {\mathbb {R}}^{n};E_{2}\right) \) if the map

$$\begin{aligned} u\rightarrow \Lambda u=F^{-1}\Psi \left( \xi \right) Fu,\text { }u\in S\left( {\mathbb {R}}^{n};E_{1}\right) \end{aligned}$$

is well defined and extends to a bounded linear operator

$$\begin{aligned} \Lambda :\ L_{p}\left( {\mathbb {R}}^{n};E_{1}\right) \rightarrow \ L_{p}\left( {\mathbb {R}}^{n};E_{2}\right) . \end{aligned}$$

We prove that problem (1.1) has an unique maximal regular solution \(u\in \) \(W_{p}^{\gamma }\left( {\mathbb {R}};H\left( A\right) ,H\right) \) for \(f\in \) \(L_{p}\left( {\mathbb {R}};H\right) \) and the following uniform coercive estimate holds

$$\begin{aligned} \sum \limits _{0\le s\le \gamma }\left| \lambda \right| ^{1-\frac{s }{\gamma }}\left\| D^{s}u\right\| _{L_{p}\left( {\mathbb {R}};H\right) }+\left\| Au\right\| _{L_{p}\left( {\mathbb {R}};H\right) }\le C\left\| f\right\| _{L_{p}\left( {\mathbb {R}};H\right) }. \end{aligned}$$
(1.3)

The estimate(1.3) implies that the operator O generated by (1.1) has a bounded inverse from \(L_{p}\left( {\mathbb {R}} ;H\right) \) into the space \(W_{p}^{\gamma }\left( {\mathbb {R}};H\left( A\right) ,H\right) \), which will be defined subsequently. Particularly, from the estimate (1.3) , we obtain that the operator O is uniformly positive in \(L_{p}\left( {\mathbb {R}};H\right) \). By using this property, we prove the well posedness of the Cauchy problem for the fractional parabolic ADE:

$$\begin{aligned} \partial _{t}u+aD_{x}^{\gamma }u+Au=f\left( t,x\right) \text {, }u(0,x)=0, \end{aligned}$$
(1.4)

in H-valued mixed spaces \(L_{{\mathbf {p}}}\) for \(\mathbf {p=}\left( p,p_{1}\right) \). In other words, we show that problem (1.4) has a unique solution \(u\in W_{{\mathbf {p}}}^{1,\gamma }\left( {\mathbb {R}} _{+}^{2};H\left( A\right) ,E\right) \) for \(f\in L_{{\mathbf {p}}}\left( \mathbb { R}_{+}^{2};H\right) \) satisfying the following coercive estimate

$$\begin{aligned}&\left\| \partial _{t}u\right\| _{L_{_{{\mathbf {p}}}}\left( {\mathbb {R}} _{+}^{2};H\right) }+\left\| D_{x}^{\gamma }u\right\| _{L_{{\mathbf {p}} }\left( {\mathbb {R}}_{+}^{2};H\right) }+\left\| Au\right\| _{L_{\mathbf {p }};\left( {\mathbb {R}}_{+}^{2};H\right) }\le \nonumber \\&\quad M\left\| f\right\| _{L_{{\mathbf {p}}}\left( {\mathbb {R}}_{+}^{2};H\right) }, \end{aligned}$$
(1.5)

here \(L_{{\mathbf {p}}}=\) \(L_{{\mathbf {p}}}\left( {\mathbb {R}}_{+}^{2};H\right) \) denotes the space of \(H-\)valued strongly measurable functions f defined on \({\mathbb {R}}_{+}^{2}\) equipped with the mixed norm

$$\begin{aligned} \left\| f\right\| _{L_{{\mathbf {p}}}\left( {\mathbb {R}}_{+}^{2};H\right) }=\left( \int \limits _{{\mathbb {R}}}\left( \int \limits _{0}^{\infty }\left\| f\left( t,x\right) \right\| _{H}^{p_{1}}\hbox {d}t\right) ^{\frac{p}{p_{1}} }\hbox {d}x\right) ^{\frac{1}{p}}<\infty \text {, }p_{1}\text {, }p\in \left( 1,\infty \right) . \end{aligned}$$

As an application, in this paper the following are established: (a) maximal regularity properties of the anisotropic elliptic fractional ADE in mixed \( L_{{\mathbf {p}}}\), \(\mathbf {p=}\left( p_{1},p\right) \) spaces; (b) \(L_{p}-\) separability feature of the system of finite and infinite number of FDEs are established.

Let \({\mathbb {C}}\) denote the set of complex numbers and

$$\begin{aligned} S_{\varphi }=\left\{ \lambda ;\text { \ }\lambda \in {\mathbb {C}}\text {, } \left| \arg \lambda \right| \le \varphi \right\} \cup \left\{ 0\right\} \text {, }0\le \varphi <\pi .\ \end{aligned}$$

A linear operator A is said to be \(\varphi \)-positive (or positive) in a Banach space E if \(D\left( A\right) \) is dense on E and

$$\begin{aligned} \left\| \left( A+\lambda I\right) ^{-1}\right\| _{B\left( E\right) }\le M\left( 1+\left| \lambda \right| \right) ^{-1} \end{aligned}$$

for any \(\lambda \in S_{\varphi }\), where \(\varphi \in \left[ 0\right. ,\left. \pi \right) \), I is the identity operator in E, \(B\left( E\right) \). Sometimes \(A+\lambda I\) will be written \(A+\lambda \) and will be denoted by \(A_{\lambda }\). It is known [19, §1.15.1] that the powers \(A^{\theta }\), \(\theta \in \left( -\infty ,\infty \right) \) for a positive operator A exist. Let \(E\left( A^{\theta }\right) \) denote the space \(D\left( A^{\theta }\right) \) with the norm

$$\begin{aligned} \left\| u\right\| _{E\left( A^{\theta }\right) }=\left( \left\| u\right\| ^{p}+\left\| A^{\theta }u\right\| ^{p}\right) ^{\frac{1}{p }},\text { }1\le p<\infty ,\text { }0<\theta <\infty . \end{aligned}$$

For any \(\alpha =\left( \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right) \), \( \alpha _{i}\in \left[ 0,\infty \right) \), the function \(\left( i\xi \right) ^{\alpha }\) will be defined as:

$$\begin{aligned} \left( i\xi \right) ^{\alpha }=\left\{ \begin{array}{l} \left( i\xi _{1}\right) ^{\alpha _{1}},.,\left( i\xi _{n}\right) ^{\alpha _{n}}\text {, }\xi _{1}\xi _{2},.,\xi _{n}\ne 0 \\ 0\text {, }\xi _{1},\xi _{2},.,\xi _{n}=0, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} \left( i\xi _{k}\right) ^{\alpha _{k}}=\exp \left[ \alpha _{k}\left( \ln \left| \xi _{k}\right| +i\frac{\pi }{2}\text { sgn }\xi _{k}\right) \right] \text {, }k=1,2,\ldots ,n. \end{aligned}$$

The Liouville derivatives \(D^{\alpha }u\) of an E-valued function u are defined similarly to the case of scalar functions [12]

Let \(E_{0}\) and E be two Banach spaces and \(E_{0}\) be continuously and densely embedded into E. Let \(s\in {\mathbb {R}}\) and \(\xi =\left( \xi _{1},\xi _{2},\ldots ,\xi _{n}\right) \in {\mathbb {R}}^{n}\). Consider the following Liouville-Lions space

$$\begin{aligned}&W_{p}^{s}({\mathbb {R}}^{n};E_{0},E)=\text { }\\&\quad \left\{ u\right. \ u\in S^{\prime }\left( {\mathbb {R}}^{n};E_{0}\right) \text { , }F^{-1}\left( 1+\left| \xi \right| ^{2}\right) ^{\frac{s}{2}}Fu\in L_{p}\left( {\mathbb {R}}^{n};E\right) \text {,}\\&\quad \left\| u\right\| _{W_{p}^{s}\left( {\mathbb {R}}^{n};E_{0},E\right) }=\left\| u\right\| _{L_{p}\left( {\mathbb {R}}^{n};E_{0}\right) }+\text { }\left. \left\| F^{-1}\left( 1+\left| \xi \right| ^{2}\right) ^{ \frac{s}{2}}Fu\right\| _{L_{p}\left( {\mathbb {R}}^{n};E\right) }<\infty \right\} . \end{aligned}$$

Let \(\varepsilon _{k}\) are positive parameters and \(\varepsilon =\left( \varepsilon _{1},\varepsilon _{2},\ldots .,\varepsilon _{n}\right) \). Now, we define the parameterized norm in \(W_{p}^{s}({\mathbb {R}}^{n};E_{0},E)\);

$$\begin{aligned} \left\| u\right\| _{W_{p_{\varepsilon }}^{s}\left( {\mathbb {R}} ^{n};E_{0},E\right) }=\left\| u\right\| _{L_{p}\left( {\mathbb {R}} ^{n};E_{0}\right) }+\text { }\left. \varepsilon _{k}\left\| F^{-1}\left( 1+\left| \xi \right| ^{2}\right) ^{\frac{s}{2}}Fu\right\| _{L_{p}\left( {\mathbb {R}}^{n};E\right) }<\infty \right\} . \end{aligned}$$

Sometimes we use one and the same symbol C without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say \(\alpha \), we write \(C_{\alpha }\).

The embedding theorems in vector valued spaces play a key role in the theory of DOEs. By reasoning as in [17], we obtain estimating lower-order derivatives in terms of interpolation spaces:

Theorem A\(_{1}\) Suppose H is a Hilbert space, \(0<\varepsilon _{k}\le \varepsilon _{0}<\infty \), \(1<p\le q<\infty \) and A is a positive operator in H. Then for s \(\in \left( 0,\infty \right) \) with \(\varkappa =\frac{1}{s}\left[ \left| \alpha \right| +n\left( \frac{1}{p}-\frac{1}{q}\right) \right] \le 1\), \(0\le \mu \le 1-\varkappa \) the embedding

$$\begin{aligned} D^{\alpha }W_{p}^{s}\left( {\mathbb {R}}^{n};H\left( A\right) ,H\right) \subset L_{q}\left( {\mathbb {R}}^{n};H\left( A^{1-\varkappa -\mu }\right) \right) \end{aligned}$$

is continuous and there exists a constant \(C_{\mu }\)\(>0\), depending only on \(\mu \) such that

$$\begin{aligned}&\prod \limits _{k=1}^{n}\varepsilon _{k}^{\frac{\alpha _{k}}{s}+\frac{1}{p}- \frac{1}{q}}\left\| D^{\alpha }u\right\| _{L_{q}\left( {\mathbb {R}} ^{n};H\left( A^{1-\varkappa -\mu }\right) \right) }\le \\&\quad C_{\mu }\left[ h^{\mu }\left\| u\right\| _{W_{p,\varepsilon }^{s}\left( {\mathbb {R}}^{n};H\left( A\right) ,H\right) }+h^{-\left( 1-\mu \right) }\left\| u\right\| _{L_{p}\left( {\mathbb {R}}^{n};H\right) } \right] \end{aligned}$$

for all \(u\in W_{p}^{s}\left( {\mathbb {R}}^{n};H\left( A\right) ,H\right) \) and \(0<h\le h_{0}<\infty \).

2 Elliptic FADE with Parameters

First, consider (1.1) for \(b=0\), i.e., consider the problem

$$\begin{aligned} aD^{\gamma }u+Au+\lambda u=f\left( x\right) \text {, }x\in {\mathbb {R}}, \end{aligned}$$
(2.1)

Let

$$\begin{aligned} X=L_{p}\left( {\mathbb {R}};H\right) \text {, }Y=W_{p}^{2}\left( {\mathbb {R}} ;H\left( A\right) ,H\right) . \end{aligned}$$

In this section, we prove the following:

Theorem 2.1

Assume \(a\left( i\xi \right) ^{\gamma }\in S_{\varphi _{1}}\) for all \(\xi \in {\mathbb {R}}\), \(\gamma \in \left( \left. 1,2\right] \right. \) and \(\lambda \in S_{\varphi _{2}}\). Suppose A is a positive operator in H with respect to \(\varphi \in \left( 0\right. ,\left. \pi \right] \). Then for \(f\in X\), \(0\le \varphi _{1}<\pi -\varphi _{2}\) and \( \varphi _{1}+\varphi _{2}\le \varphi \) there is a unique solution u of Eq. (2.1) belonging to Y and the following coercive uniform estimate holds

$$\begin{aligned} \sum \limits _{0\le s\le \gamma }\left| \lambda \right| ^{1-\frac{s }{\gamma }}\left\| D^{s}u\right\| _{X}+\left\| Au\right\| _{X}\le C\left\| f\right\| _{X}. \end{aligned}$$
(2.2)

Proof

By applying the Fourier transform to Eq. (2.1), we obtain

$$\begin{aligned} Q\left( \xi ,\lambda \right) {\hat{u}}\left( \xi \right) ={\hat{f}}\left( \xi \right) \text {,} \end{aligned}$$
(2.3)

where

$$\begin{aligned} Q\left( \xi ,\lambda \right) =a\left( i\xi \right) ^{\gamma }+A+\lambda . \end{aligned}$$
(2.4)

By assumption, \(\lambda +a\left( i\xi \right) ^{\gamma }\in S_{\varphi }\) for all \(\xi \in {\mathbb {R}}\) and the operator \(Q\left( \xi ,\lambda \right) \) is invertible in H for all \(\xi \in {\mathbb {R}}\) and \(\lambda \in S_{\varphi _{2}}\). So, from(2.4) we obtain that the solution of (2.3) can be represented in the form

$$\begin{aligned} u\left( x\right) =F^{-1}\left[ Q\left( \xi ,\lambda \right) \right] ^{-1} {\hat{f}}. \end{aligned}$$
(2.5)

By definition of the positive operator, the inverse of A is bounded in H . Then the operator A is a closed linear operator (as an inverse of bounded linear operator \(A^{-1}\)). By properties of the Fourier transform and by using (2.5) we have

$$\begin{aligned}&\left\| Au\right\| _{X}=\left\| F^{-1}\left[ Q\left( \xi ,\lambda \right) \right] ^{-1}{\hat{f}}\right\| _{X}\text {, }\\&\quad \left\| D^{s}u\right\| _{X}=\left\| F^{-1}\left[ \xi ^{s}Q\left( \xi ,\lambda \right) \right] ^{-1}{\hat{f}}\right\| _{X}. \end{aligned}$$

Hence, it suffices to show that operator-functions

$$\begin{aligned} \eta \left( \lambda ,\xi \right) =A\left[ Q\left( \xi ,\lambda \right) \right] ^{-1}\text {, }\eta _{s}\left( \lambda ,\xi \right) =\sum \limits _{0\le s\le \gamma }\left| \lambda \right| ^{1-\frac{s }{\gamma }}\xi ^{s}\left[ Q\left( \xi ,\lambda \right) \right] ^{-1} \end{aligned}$$

are collections of multipliers in \(X\ \)uniformly with respect to \(\lambda \in S_{\varphi _{2}}\). By virtue of [6, Lemma 2.3], for \(\lambda \in S_{\varphi _{2}}\) and \(\nu \in S_{\varphi _{1}}\) with \( \varphi _{1}+\varphi _{2}<\pi \), there is a positive constant C such that

$$\begin{aligned} \left| \lambda +\nu \right| \ge C\left( \left| \lambda \right| +\left| \nu \right| \right) . \end{aligned}$$
(2.6)

By using the positivity properties of operator A, we get that \(\left[ Q\left( \xi ,\lambda \right) \right] ^{-1}\) is uniformly bounded for all \( \xi \in {\mathbb {R}}\), \(\lambda \in S_{\varphi _{2}}\) and

$$\begin{aligned} \left\| \left[ Q\left( \xi ,\lambda \right) \right] ^{-1}\right\| \le C\left( 1+\left| \lambda +a\left( i\xi \right) ^{\gamma }\right| \right) ^{-1}. \end{aligned}$$

By (2.6) , we obtain that

$$\begin{aligned}&\left\| \left[ Q\left( \xi ,\lambda \right) \right] ^{-1}\right\| \le C\left( 1+\left| \lambda \right| +\left| a\right| \left| \xi \right| ^{\gamma }\right) ^{-1}\le \nonumber \\&\quad C_{2}\left[ 1+\left| \lambda \right| +\left| \xi \right| ^{2} \right] ^{-1}. \end{aligned}$$
(2.7)

Then by resolvent properties of positive operators and uniform estimate (2.7), we obtain

$$\begin{aligned}&\left\| \eta \left( \lambda ,\xi \right) \right\| \le \left\| I+\left( \lambda +a\left( i\xi \right) ^{\gamma }\right) \left[ Q\left( \xi ,\lambda \right) \right] ^{-1}\right\| \\&\quad \le 1+\left( \left| \lambda \right| +\left| a\right| \left| \xi \right| ^{\gamma }\right) \left( 1+\left| \lambda \right| +\left| a\right| \left| \xi \right| ^{\gamma }\right) ^{-1}\le C_{3}, \end{aligned}$$

where I is an identity operator in H. Moreover, by well-known inequality we have

$$\begin{aligned}&\left| \lambda \right| ^{1-\frac{s}{2}}\left| \xi \right| ^{s}=\left| \lambda \right| \left| \lambda \right| ^{-\frac{s }{\gamma }}\left| \xi \right| ^{s}\le \left| \lambda \right| \left( \left| \lambda \right| ^{-\frac{1}{\gamma } }\left| \xi \right| \right) ^{s}\le \nonumber \\&\quad \left| \lambda \right| \left[ 1+\left| \lambda \right| ^{-1}\left| \xi \right| ^{\gamma }\right] =\left| \lambda \right| +\left| \xi \right| ^{\gamma }. \end{aligned}$$
(2.8)

Hence, in view of (2.7) and (2.8) , we have

$$\begin{aligned} \left\| \eta _{s}\left( \lambda ,\xi \right) u\right\| _{H}\le \left| a\right| \sum \limits _{0\le s\le \gamma }\left| \lambda \right| ^{1-\frac{s}{\gamma }}\left| \xi \right| ^{s}\left\| \left[ Q\left( \xi ,\lambda \right) \right] ^{-1}u\right\| _{H}\le C_{2}\left\| u\right\| _{H}. \end{aligned}$$

So, we obtain that the operator functions \(D^{i}\eta \left( \lambda ,\xi \right) \) and \(D^{i}\eta _{s}\left( \lambda ,\xi \right) \) are uniformly bounded, i.e.,

$$\begin{aligned} \left\| D^{i}\eta \left( \lambda ,\xi \right) \right\| _{B\left( H\right) }\le C_{1},\left\| D^{i}\eta _{s}\left( \lambda ,\xi \right) \right\| _{B\left( E\right) }\le C_{2} \end{aligned}$$

for \(i=0,1\). Then by virtue of [21, Theorem 4.1], \( \eta \left( \lambda ,\xi \right) \) and \(\eta _{s}\left( \lambda ,\xi \right) \) are Fourier multipliers in X.

Let O denote the operator in X generated by problem (1.1) for \(\lambda =0\), i.e.,

$$\begin{aligned} D\left( O\right) \subset W_{p}^{\gamma }\left( {\mathbb {R}}^{n};H\left( A\right) ,H\right) ,\text { }Ou=aD^{\gamma }u+Au. \end{aligned}$$

Theorem 2.1 and the definition of the space \(W_{p}^{s}\left( {\mathbb {R}} ^{n};H\left( A\right) ,H\right) \) imply the following result: \(\square \)

Result 2.1

Assume all conditions of Theorem 2.1 are satisfied. Then there are positive constants \(C_{1}\) and \(C_{2}\) so that

$$\begin{aligned} C_{1}\left\| Ou\right\| _{X}\le \left\| u\right\| _{W_{p}^{2}\left( {\mathbb {R}};H\left( A\right) ,H\right) }\le C_{2}\left\| Ou\right\| _{X} \end{aligned}$$

for \(u\in Y\). Indeed, if we put \(\lambda =1\) in (2.2) , by Theorem 2.1 we get

$$\begin{aligned} \sum \limits _{0\le s\le \gamma }\left\| D^{s}u\right\| _{X}+\left\| Au\right\| _{X}\le C\left\| Ou\right\| _{X} \end{aligned}$$
(2.9)

for \(u\in Y\), i.e., the first inequality of (2.9) holds. The first inequality is equivalent to the following estimate

$$\begin{aligned}&\left\| F^{-1}A{\hat{u}}\right\| _{X}+\left\| F^{-1}a\left( i\xi \right) ^{\gamma }{\hat{u}}\right\| _{X}\le \\&\quad C\left\{ \left\| F^{-1}A{\hat{u}}\right\| _{X}+\left\| F^{-1}\left( 1+\xi ^{2}\right) {\hat{u}}\right\| _{X}\right\} . \end{aligned}$$

So, it suffices to show that the operator functions

$$\begin{aligned} A\left[ A+\left( 1+\xi ^{2}\right) \right] ^{-1},\text { }\left( \text {i}\xi \right) ^{\gamma }\left( A+1+\xi ^{2}\right) ^{-1} \end{aligned}$$

are uniform Fourier multipliers in X. This fact is proved in a similar way as in the proof of Theorem 2.1.

From Theorem 2.1, we have:

Result 2.2

Assume that the all conditions of Theorem 2.1 are satisfied. Then, for all \(\lambda \in S_{\varphi }\) the resolvent of operator O exists and the following sharp uniform estimate holds

$$\begin{aligned} \sum \limits _{0\le s\le \gamma }\left| \lambda \right| ^{1-\frac{s }{\gamma }}\left\| D^{s}\left( O+\lambda \right) ^{-1}\right\| _{B\left( X\right) }+\left\| A\left( O+\lambda \right) ^{-1}\right\| _{B\left( X\right) }\le C. \end{aligned}$$
(2.10)

Indeed, we infer from Theorem 2.1 that the operator \(O+\lambda \) has a bounded inverse from X to Y. So, the solution u of Eq. (1.1) can be expressed as \(u\left( x\right) =\left( O+\lambda \right) ^{-1}f\) for all \(f\in X\). Then estimate (2.2) implies (2.10) .

Now consider the problem (1.1) . Here, we show the following result:

Theorem 2.2

Assume \(a\left( i\xi \right) ^{\gamma }\in S_{\varphi _{1}}\) for all \(\xi \in {\mathbb {R}}\), \(\nu \in \left( 0,1\right) \), \(\gamma \in \left( \left. 1,2\right] \text { }\right. \)with \(\nu <\gamma \) and \( \lambda \in S_{\varphi _{2}}\). Suppose A is a positive operator in H with respect to \(\varphi \in \left( 0\right. ,\left. \pi \right] \). Then for \(f\in X\), \(0\le \varphi _{1}<\pi -\varphi _{2}\) and \(\varphi _{1}+\varphi _{2}\le \varphi \) there is a unique solution u of Eq. (1.1) belonging to Y and the following coercive uniform estimate holds

$$\begin{aligned} \sum \limits _{0\le s\le \gamma }\left| \lambda \right| ^{1-\frac{s }{\gamma }}\left\| D^{s}u\right\| _{X}+\left\| Au\right\| _{X}\le C\left\| f\right\| _{X}. \end{aligned}$$
(2.11)

Proof

Let Q denote the operator in X generated by problem (1.1) for \(\lambda =0\). By using Theorem \(\hbox {A}_{{1}}\) Theorem 2.1, for all \(\delta >0\) there exists a continuous function \(C\left( \delta \right) \) such that the following estimate holds

$$\begin{aligned} \left\| bD^{\nu }u\right\| _{X}\le \delta \left\| u\right\| _{Y}+C\left( \delta \right) \left\| u\right\| _{X}\le \delta \left\| Ou\right\| _{X}+C\left( \delta \right) \left\| u\right\| _{X} \end{aligned}$$
(2.12)

for all \(u\in Y\). In view of (2.12) and the resolvent properties of the operator O, we have

$$\begin{aligned} \left\| bD^{\nu }u\right\| _{X}<\mu \left\| Ou\right\| _{X}\text { with }\mu <1. \end{aligned}$$

Then by Theorem 2.1 and by perturbation theory of linear operators, we obtain the assertion. \(\square \)

Result 2.3

Theorem 2.1 particularly implies that the operator O is positive in X. Then the operators \(O^{s}\) are generators of analytic semigroups in X for \(s\le \frac{1}{2}\) (see e.g., [20, §1.14.5].

3 The Cauchy Problem for Fractional Parabolic ADE

In this section, we shall consider the following Cauchy problem for the parabolic FDOE

$$\begin{aligned} \frac{\partial u}{\partial t}+aD_{x}^{\gamma }u+Au=f\left( t,x\right) \text { , }u(0,x)=0\text {, }t\in {\mathbb {R}}_{+}\text {, }x\in {\mathbb {R}}, \end{aligned}$$
(3.1)

where a is a complex number, \(D_{x}^{\gamma }\) is a fractional derivative in x for \(\alpha \in \left( 1,2\right) \) defined by (1.2) and A is a linear operator in a Hilbert space H. By applying Theorem 2.1, we establish the maximal regularity of the problem (3.1) in H-valued mixed \(L_{{\mathbf {p}}}\) spaces, where \(\mathbf {p=}\left( p_{1},p\right) \). Let O denote the operator generated by problem (1.1) . For this aim, we need the following result. For \(\mathbf {p=} \left( p\text {, }p_{1}\right) \), \(Z=L_{{\mathbf {p}}}\left( {\mathbb {R}} _{+}^{2};H\right) \) will denote the space of all \({\mathbf {p}}\)-summable H -valued functions on \({\mathbb {R}}_{+}^{2}\) with mixed norm, i.e., the space of all measurable H-valued functions f defined on for which

$$\begin{aligned} \left\| f\right\| _{L_{{\mathbf {p}}}\left( {\mathbb {R}}_{+}^{2};H\right) }=\left( \int \limits _{{\mathbb {R}}}\left( \int \limits _{0}^{\infty }\left\| f\left( t,x\right) \right\| _{H}^{p}\hbox {d}x\right) ^{\frac{p_{1}}{p}}\hbox {d}t\right) ^{\frac{1}{p_{1}}}<\infty . \end{aligned}$$

Let m be a positive number. \(Z^{1,\gamma }\left( A\right) =W_{{\mathbf {p}} }^{1,\gamma }\left( {\mathbb {R}}_{+}^{2};H\left( A\right) ,H\right) \) denotes the space of all functions \(u\in L_{{\mathbf {p}}}\left( {\mathbb {R}} _{+}^{n+1};H\left( A\right) \right) \) possessing the generalized derivative \( D_{t}u=\frac{\partial u}{\partial t}\in Z\) and fractional derivatives \( D_{x}^{\gamma }u\in Z\) with respect to x for \(\left| \alpha \right| \le m\) with the norm

$$\begin{aligned} \ \left\| u\right\| _{Z^{1,2}\left( A\right) }=\left\| Au\right\| _{Z}+\left\| \partial _{t}u\right\| _{Z}+\left\| D_{x}^{\gamma }u\right\| _{Z}, \end{aligned}$$

where \(\gamma \in \left( \left. 1,2\right] \right. \) and \(u=u\left( t,x\right) .\)

Now, we are ready to state the main result of this section.

Theorem 3.1

Assume the conditions of Theorem 2.1 hold for \( \varphi \in \left( \frac{\pi }{2},\pi \right) \). Then for \(f\in Z\) problem (3.1) has a unique solution \(u\in Z^{1,\gamma }\left( A\right) \) satisfying the following uniform coercive estimate

$$\begin{aligned} \left\| \partial _{t}u\right\| _{Z}+\left\| D_{x}^{\gamma }u\right\| _{Z}+\left\| Au\right\| _{Z}\le C\left\| f\right\| _{Z}. \end{aligned}$$

Proof

By definition of \(X=L_{p}\left( {\mathbb {R}};H\right) \) and mixed space \(L_{{\mathbf {p}}}\left( {\mathbb {R}}_{+}^{2};H\right) ,\) \(\mathbf {p=} \left( p\text {, }p_{1}\right) \), we have

$$\begin{aligned} \left\| u\right\| _{L_{p_{1}}\left( 0,\infty ;X\right) }=\left( \int \limits _{0}^{\infty }\left\| u\left( t\right) \right\| _{X}^{p_{1}}\hbox {d}t\right) ^{\frac{1}{p_{1}}}=\left( \int \limits _{0}^{\infty }\left\| u\left( t\right) \right\| _{L_{p}\left( {\mathbb {R}};H\right) }^{p_{1}}\hbox {d}t\right) ^{\frac{1}{p_{1}}}=\left\| u\right\| _{Z}. \end{aligned}$$

Therefore, the problem (3.1) can be expressed as the following Cauchy problem for the abstract parabolic equation

$$\begin{aligned} \frac{\hbox {d}u}{\hbox {d}t}+Ou\left( t\right) =f\left( t\right) ,\text { }u\left( 0\right) =0,\text { }t\in \left( 0,\infty \right) . \end{aligned}$$
(3.2)

Then, by virtue of [21, Theorem 4.2], we obtain that for \(f\in L_{p_{1}}\left( 0,\infty ;X\right) \) the problem (3.2) has a unique solution \(u\in W_{p_{1}}^{1}\left( 0,\infty ;D\left( O\right) ,X\right) \) satisfying the following estimate

$$\begin{aligned} \left\| \frac{\hbox {d}u}{\hbox {d}t}\right\| _{L_{p_{1}}\left( 0,\infty ;X\right) }+\left\| Ou\right\| _{L_{p_{1}}\left( 0,\infty ;X\right) }\le C\left\| f\right\| _{L_{p_{1}}\left( 0,\infty ;X\right) }. \end{aligned}$$

From Theorem 2.1, relation (3.2) and from the above estimate, we get the assertion.

\(\square \)

4 BVP for Anisotropic Elliptic FDE

In this section, the maximal regularity properties of the anisotropic FDE are studied.

Let \({\tilde{\Omega }}=\mathbb {R\times }\Omega \), where \(\Omega \subset \mathbb { R}^{n}\) is an open connected set with compact \(C^{2l}\)-boundary \(\partial \Omega \). Consider the BVP for the FDE

$$\begin{aligned}&aD_{x}^{\gamma }u+bD_{x}^{\nu }u+\sum \limits _{\left| \beta \right| \le 2l}\left( b_{\beta }D_{y}^{\beta }+\lambda \right) u=f\left( x,y\right) \text {, }y\in \Omega , \end{aligned}$$
(4.1)
$$\begin{aligned}&B_{j}u=\sum \limits _{\left| \beta \right| \le l_{j}}\ b_{j\beta }\left( y\right) D_{y}^{\beta }u\left( x,y\right) =0\text {, }x\in {\mathbb {R}} \text {,}\nonumber \\&\quad \text { }y\in \partial \Omega ,\text { }j=1,2,\ldots ,l, \end{aligned}$$
(4.2)

where \(u=u\left( x,y\right) \), a is a complex number, \(D_{x}^{\gamma }\) is the fractional derivative operator in x for \(\gamma \in \left( 1,2\right) \) defined by (1.2),

$$\begin{aligned} D_{j}=-i\frac{\partial }{\partial y_{j}}\text {, }y=\left( y_{1},\ldots ,y_{n}\right) \text {, }b_{\beta }=b_{\beta }\left( y\right) \text {,} \end{aligned}$$

\(\beta =\left( \beta _{1},\beta _{2},\ldots ,\beta _{n}\right) \) are nonnegative integer numbers and \(\lambda \) is a complex parameter. For \(\mathbf {p=} \left( 2,p\right) \) here, \(L_{{\mathbf {p}}}\left( {\tilde{\Omega }}\right) \) will denote the space of all \({\mathbf {p}}\)-summable scalar-valued functions with mixed norm, i.e., the space of all measurable functions f defined on \( {\tilde{\Omega }}\), for which

$$\begin{aligned} \left\| f\right\| _{L_{{\mathbf {p}}}\left( {\tilde{\Omega }}\right) }=\left( \int \limits _{{\mathbb {R}}}\left( \int \limits _{\Omega }\left| f\left( x,y\right) \right| ^{2}\hbox {d}x\right) ^{\frac{2}{p_{1}}}\hbox {d}y\right) ^{ \frac{1}{2}}<\infty . \end{aligned}$$

Analogously, \(W_{{\mathbf {p}}}^{\gamma ,2l}\left( {\tilde{\Omega }}\right) \) denotes the anisotropic fractional Sobolev space with corresponding mixed norm, i.e., \(W_{{\mathbf {p}}}^{\gamma ,2l}\left( {\tilde{\Omega }}\right) \) denotes the space of all functions \(u\in L_{{\mathbf {p}}}\left( {\tilde{\Omega }} \right) \) possessing the fractional derivatives \(D_{x}^{\gamma }u\in L_{ {\mathbf {p}}}\left( {\tilde{\Omega }}\right) \) with respect to x for \(\alpha \in \left( 1,2\right) \) and generalized derivative \(\frac{\partial ^{2l}u}{ \partial y_{k}^{2l}}\in L_{{\mathbf {p}}}\left( {\tilde{\Omega }}\right) \) with respect to y with the norm

$$\begin{aligned} \ \left\| u\right\| _{W_{{\mathbf {p}}}^{\gamma ,2l}\left( {\tilde{\Omega }} \right) }=\left\| u\right\| _{L_{{\mathbf {p}}}\left( {\tilde{\Omega }} \right) }+\left\| D_{x}^{\gamma }u\right\| _{L_{{\mathbf {p}}}\left( {\tilde{\Omega }}\right) }+\sum \limits _{k=1}^{n}\left\| \frac{\partial ^{2l}u}{\partial y_{k}^{2l}}\right\| _{L_{{\mathbf {p}}}\left( {\tilde{\Omega }} \right) }. \end{aligned}$$

Let Q denote the operator generated by problem (4.1, 4.2), i.e.,

$$\begin{aligned}&\displaystyle D\left( Q\right) =W_{{\mathbf {p}}}^{\gamma ,2l}\left( {\tilde{\Omega }} ,B_{j}\right) =\left\{ u:u\in W_{{\mathbf {p}}}^{\gamma ,2l}\left( \tilde{\Omega }\right) ,\text { }B_{j}u=0,\text { }j=1,2,\ldots l\right\} , \\&\displaystyle Qu=aD_{x}^{\gamma }u+bD_{x}^{\nu }u+\sum \limits _{\left| \alpha \right| \le 2l}b_{\alpha }D_{y}^{\alpha }u. \end{aligned}$$

Let \(\xi ^{\prime }=\left( \xi _{1},\xi _{2},\ldots ,\xi _{n-1}\right) \in {\mathbb {R}}^{\mu -1}\), \(\beta ^{\prime }=\left( \beta _{1},\beta _{2},\ldots ,\beta _{n-1}\right) \in Z^{\mu }\) and

$$\begin{aligned} A\left( y_{0},\xi ^{\prime },D_{y}\right)= & {} \sum \limits _{\left| \beta ^{\prime }\right| +j\le 2l}a_{\beta ^{\prime }}\left( y_{0}\right) \xi _{1}^{\beta _{1}}\xi _{2}^{\beta _{2}},.,\xi _{n-1}^{\beta _{n-1}}D_{n}^{j}\text { for }y_{0}\in {\bar{G}} \\ B_{j}\left( y_{0},\xi ^{\prime },D_{y}\right)= & {} \sum \limits _{\left| \beta ^{\prime }\right| +j\le l_{j}}b_{j\beta ^{\prime }}\left( y_{0}\right) \xi _{1}^{\beta _{1}}\xi _{2}^{\beta _{2}},.,\xi _{\mu -1}^{\beta _{\mu -1}}D_{\mu }^{j}\text { for }y_{0}\in \partial G. \end{aligned}$$

Condition 4.1

Let the following conditions be satisfied:

  1. (1)

    \(b_{\beta }\in C\left( {\bar{\Omega }}\right) \) for each \(\left| \beta \right| =2l\) and \(b_{\alpha }\in L_{\infty }\left( \Omega \right) +L_{r_{k}}\left( \Omega \right) \) for each \(\left| \alpha \right| =k<2l\) with \(r_{k}\ge p_{1}\), \(p_{1}\in \left( 1,\infty \right) \) and \(2l-k> \frac{l}{r_{k}};\)

  2. (2)

    \(b_{j\beta }\in C^{2l-l_{j}}\left( \partial \Omega \right) \) for each j , \(\beta \), \(l_{j}<2l\), \(p\in \left( 1,\infty \right) ,\) \(\lambda \in S_{\varphi }\), \(\varphi \in [0,\pi );\)

  3. (3)

    for \(y\in {\bar{\Omega }}\), \(\xi \in {\mathbb {R}}^{n}\), \(\sigma \in S_{\varphi _{0}}\), \(\varphi _{0}\in \left( 0,\frac{\pi }{2}\right) \), \( \left| \xi \right| +\left| \sigma \right| \ne 0\) let \( \sigma +\sum \limits _{\left| \alpha \right| =2l}b_{\alpha }\left( y\right) \xi ^{\alpha }\ne 0;\)

  4. (4)

    for each \(y_{0}\in \partial \Omega \) local BVP in local coordinates corresponding to \(y_{0},\)

    $$\begin{aligned}&\lambda +A\left( y_{0},\xi ^{\prime },D_{y}\right) \vartheta \left( y\right) =0,\\&B_{j}\left( y_{0},\xi ^{\prime },D_{y}\right) \vartheta \left( 0\right) =h_{j}\text {, }j=1,2,\ldots ,l \end{aligned}$$

has a unique solution \(\vartheta \in C_{0}\left( 0,\infty \right) \) for all \( h=\left( h_{1},h_{2},\ldots ,h_{l}\right) \in {\mathbb {C}}^{l}\) and for \(\xi ^{^{\prime }}\in {\mathbb {R}}^{n-1}\).

Suppose \(\nu =\left( \nu _{1},\nu _{2},\ldots ,\nu _{n}\right) \) are nonnegative real numbers. In this section, we present the following result:

Theorem 4.1

Assume \(a\left( i\xi \right) ^{\gamma }\in S_{\varphi _{1}}\) for all \(\xi \in {\mathbb {R}}\), \(\nu \in \left( 0,1\right) \), \(\gamma \in \left( \left. 1,2\right] \text { }\right. \)with \(\nu <\gamma \) anf \( \lambda \in S_{\varphi _{2}}\) for \(0\le \varphi _{1}<\pi -\varphi _{2}\). Moreover, suppose Conditions 4.1 are satisfied. Then for \(\ f\in L_{{\mathbf {p}} }\left( {\tilde{\Omega }}\right) \), \(\lambda \in S_{\varphi },\) \(\varphi \in \left( 0,\right. \left. \pi \right] \), \(\varphi _{1}+\varphi _{2}\le \varphi \) problem (4.1, 4.2) has a unique solution \(u\in W_{p}^{2,2l}\left( {\tilde{\Omega }}\right) \) and the following coercive uniform estimate holds

$$\begin{aligned} \sum \limits _{0\le s\le \gamma }\left| \lambda \right| ^{1-\frac{s }{\gamma }}\left\| D_{x}^{s}u\right\| _{L_{{\mathbf {p}}}\left( \tilde{ \Omega }\right) }+\sum \limits _{\left| \beta \right| \le 2l}\left\| D_{y}^{\beta }u\right\| _{L_{{\mathbf {p}}}\left( \tilde{\Omega }\right) }\le C\left\| f\right\| _{L_{{\mathbf {p}}}\left( {\tilde{\Omega }} \right) }. \end{aligned}$$

Proof

Let \(H=L_{2}\left( \Omega \right) \). It is known [4] that \(L_{2}\left( \Omega \right) \) is an UMD space. Consider the operator A defined by

$$\begin{aligned} D\left( A\right) =W_{p_{1}}^{2l}\left( \Omega ;B_{j}u=0\right) ,\text { } Au=\sum \limits _{\left| \beta \right| \le 2l}b_{\beta }\left( x\right) D^{\beta }u\left( y\right) . \end{aligned}$$

Therefore, the problem (4.1, 4.2) can be rewritten in the form of (2.2) , where \(u\left( x\right) =u\left( x,.\right) ,\) \(f\left( x\right) =f\left( x,.\right) \) are functions with values in \(H=L_{2}\left( \Omega \right) \). From [5, Theorem 8.2], we get that the following problem

$$\begin{aligned}&\eta u\left( y\right) +\sum \limits _{\left| \beta \right| \le 2l}b_{\beta }\left( y\right) D^{\beta }u\left( y\right) =f\left( y\right) \text {, }\nonumber \\&\quad B_{j}u=\sum \limits _{\left| \beta \right| \le l_{j}}\ b_{j\beta }\left( y\right) D^{\beta }u\left( y\right) =0\text {, }j=1,2,\ldots ,l \end{aligned}$$
(4.3)

has a unique solution for \(f\in L_{2}\left( \Omega \right) \) and arg \(\eta \in S\left( \varphi _{1}\right) ,\) \(\left| \eta \right| \rightarrow \infty \). Moreover, the operator A generated by (4.3) is positive in \(L_{2}\). Then from Theorem 2.1, we obtain the assertion. \(\square \)

5 The System of Parabolic FDE of Infinite Many Order

Consider the following system of FDEs

$$\begin{aligned}&\displaystyle \partial _{t}u_{i}+a\partial _{x}^{\gamma }u_{i}+\sum \limits _{j=1}^{\infty }\left( a_{ij}+\lambda \right) u_{j}\left( x\right) =f_{i}\left( t,x\right) \text {, }t\in {\mathbb {R}}_{+}\text {, }x\in {\mathbb {R}},\text { }\nonumber \\&\displaystyle u_{i}\left( 0,x\right) =0\text {, }i=1,2,\ldots \infty , \end{aligned}$$
(5.1)

where a is a complex number, \(a_{ij}\) are real numbers and \(\partial _{x}^{\gamma }\) is the fractional differential operator in x for \(\gamma \), \(\nu \in \left( 1,2\right) \) defined by (1.2) . Let

$$\begin{aligned}&l_{2}\left( A\right) =\left\{ u\in l_{2},\left\| u\right\| _{l_{q}\left( A\right) }=\left\| Au\right\| _{l_{2}}=\right. \\&\quad \left( \sum \limits _{i=1}^{\infty }\left| \left( Au\right) _{i}\right| ^{2}\right) ^{\frac{1}{2}}=\left. \left( \sum \limits _{i=1}^{\infty }\left| \sum \limits _{j=1}^{\infty }a_{ij}u_{j}\right| ^{2}\right) ^{\frac{1}{2}}<\infty \right\} ,\\&u=\left\{ u_{j}\right\} ,\text { }Au=\left\{ \sum \limits _{j=1}^{\infty }a_{ij}u_{j}\right\} ,\text { }i,\text { } j=1,2,\ldots \infty . \end{aligned}$$

Condition 5.1

Let

$$\begin{aligned} a_{ij}=a_{ji}\text {, }\sum \limits _{i,j=1}^{\infty }a_{ij}\xi _{i}\xi _{j}\ge C_{0}\left| \xi \right| ^{2}\text { for }\xi \ne 0. \end{aligned}$$

Let

$$\begin{aligned} f\left( t,x\right) =\left\{ f_{i}\left( t,x\right) \right\} _{1}^{\infty } \text {, }u=\left\{ u_{i}\left( t,x\right) \right\} _{1}^{\infty },\text { } Y_{2}=L_{{\mathbf {p}}}\left( {\mathbb {R}}_{+}^{2};l_{2}\right) . \end{aligned}$$

Theorem 5.1

Assume \(a\left( i\xi \right) ^{\gamma }\in S_{\varphi } \) for all \(\xi \in {\mathbb {R}}\) and \(\nu <\gamma .\) Suppose the Condition 5.1 is satisfied. Then, for \(f\left( t,x\right) \in L_{{\mathbf {p}} }\left( {\mathbb {R}}_{+}^{2};l_{2}\right) \) problem (5.1) has a unique solution \(u\in \) \(W_{{\mathbf {p}}}^{\gamma ,2}\left( {\mathbb {R}} _{+}^{2},l_{2}\left( A\right) ,l_{2}\right) \) and the following uniform coercive estimate holds

$$\begin{aligned} \left\| \partial _{t}u\right\| _{Y_{2}}+\left\| D_{x}^{\gamma }u\right\| _{Y_{2}}+\left\| u\right\| _{Y_{2}}\le C\left\| f\right\| _{Y_{2}} \end{aligned}$$

Proof

Let \(H=l_{2}\) and A be a matrix such that \(A=\left[ a_{ij}\right] \), i, \(j=1,2,\ldots \infty \). It is easy to see that

$$\begin{aligned} B\left( \lambda \right) =\lambda \left( A+\lambda \right) ^{-1}=\frac{ \lambda }{D\left( \lambda \right) }\left[ A_{ji}\left( \lambda \right) \right] \text {, }i\text {, }j=1,2,\ldots \infty , \end{aligned}$$

where \(D\left( \lambda \right) =\det \left( A-\lambda I\right) \), \( A_{ji}\left( \lambda \right) \) are entries of the corresponding adjoint matrix of \(A-\lambda I.\) Since the matrix A is symmetric and positive definite, it generates a positive operator in \(l_{2}.\)i.e., the operator A is positive in \(l_{q}\). So, from Theorem 3.1, we obtain the assertion.