## Abstract

A single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order diffusion-wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributed-order fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.

## 1. Introduction

Fractional differential equations in mathematical physics possess many interesting properties. For example, when the time derivative in the classical diffusion equation is replaced by the fractional time derivative (we call it fractionalization) of order *α*, 0<*α*<2, then a diffusion-wave equation is obtained, (see Mainardi 1996, 1997; Gorenflo *et al*. 2000; Hanyga 2002*a*,*b*; Atanackovic *et al*. 2007; Mainardi *et al*. 2008; Chechkin *et al*. 2002; Atanackovic *et al*. 2005, 2009). The fractionalization of differential equations of mathematical physics leads to the analysis of parameter *α,* which has to be determined through experimental results. Since experiments may lead to several values of the order of the derivative, it is convenient to introduce a distributed-order fractional derivative, i.e. to integrate the product of a fractional derivative of *u* and a weight function (or distribution) with respect to the order of the derivative, i.e. to evaluate. In this way, one may use several experimental results and determine *ϕ*.

Atanackovic *et al*. (2007) investigated a generalized telegraph equation(1.1)where 0<*β*<*α*<2, with the data

Hanyga (2002*a*,*b*) treated the following Cauchy problem:(1.2)where is given by (2.1); ∇^{2} denotes the Laplace space operator; *δ*(** x**),

**∈**

*x*^{3}, and

*δ*(

*t*),

*t*≥0, are space and time Dirac distributions, respectively; >0 is the diffusibility coefficient; and

*F*is a given constant. Equation (1.2) is subject to the initial conditions(1.3)where

*u*

_{0}and

*v*

_{0}are given constants. Note that (1.2) and (1.3) have to be interpreted in the sense of distributions. Because of this, part of the work by Hanyga (2002

*a*,

*b*), related to (1.2) and (1.3) via the operator theory of Prüss (1993), has to be explained in detail.

In this paper and the next (Atanackovic *et al*. 2009), we extend the results of Hanyga (2002*a*,*b*) and Atanackovic *et al*. (2007) and give an intrinsic analysis of the corresponding Volterra equation. We will summarize the results at the end of this introduction.

We use the results of Prüss (1993) and investigate equation (1.6) (see below), which generalizes (1.1) and (1.2), but involves, in the case =∇^{2}, restrictions on initial data *u*_{0} and *v*_{0} and a forcing term *F*. These restrictions were not taken into consideration in Hanyga (2002*a*,*b*).

We define the distributed-order time derivative of a vector-valued function *t*≥0, where *X* is a Banach space or distribution space ′(^{n}) as follows. If *ϕ* is a continuous function in [*μ*, *η*]⊂[0, 2] and *ϕ*(*α*)=0, *α*∈[0, 2]\[*μ*, *η*], then the distributed-order time derivative is defined as(1.4)If *ϕ*∈′() and supp *ϕ*⊂[0, 2], then the distributed-order time derivative is defined as(1.5)where the integral in (1.5) is understood in the sense of distributionsThis will be explained in §2.

Instead of Laplacian ∇^{2} in (1.2), we consider an unbounded linear operator defined on a dense subset of a suitable Banach space *X*, and instead of a forcing term *Fδ*(** x**)

*δ*(

*t*),

**∈**

*x*^{3},

*t*>0, we consider

*F*(

**,**

*x**t*)

**∈**

*x*^{n},

*t*>0, as an element of appropriate (generalized) function space. Thus, instead of (1.1) and (1.2), we consider, in a distributional setting, a time distributed-order equation, with linear differential operator , given as(1.6)subjected to(1.7)

We prove the existence and uniqueness of a solution to (1.6) and (1.7) withwhere supp *ϕ*⊂[0, 2]. Then, (1.6) becomes(1.8)

Let *X* be a Banach space and let be a linear unbounded operator defined on a dense subset of *X*. Then, *u* is a solution to (1.8) and (1.7) in the case supp *ϕ*⊂[0, 1], respectively supp *ϕ*⊂[0, 2], if, with appropriate assumptions on *F* and appropriate assumptions on *u*_{0}, respectively *u*_{0} and *v*_{0}, *u* belongs to , respectively to and that *u* satisfies (1.8) and (1.7).

The existence and uniqueness of this specific choice of a weight distribution will be proven by reducing the problem to an abstract Volterra equation. In a special case when =∇^{2}, we refer to equation (1.6) as to the time distributed-order diffusion-wave equation. Then, *X*=*L*^{2}(^{n}) and . Throughout the paper, we use notation *J*=[0, *T*], *T*>0.

Let *a*≔inf {*x*|*x*∈supp *ϕ*} and *b*≔sup {*x*|*x*∈supp *ϕ*}. The following cases can be distinguished for ⊆∇^{2}:

distributed-order diffusion-wave equation: 0≤

*a*≤1<*b*≤2;distributed-order diffusion equation:

*b*≤1; anddistributed-order wave equation:

*a*>1.

In §3*e*, we will separately consider solutions to the Cauchy problem (1.6) and (1.7) and to the Volterra-type equation(1.9)where and(1.10)if supp *ϕ*⊂[0, 2], or *ϕ*∈*C*([0, 2]), or(1.11)if supp *ϕ*⊂[0, 1] or *ϕ*∈*C*([0, 1]). Here, *a*, *B*_{u} and *B*_{v} are defined via Laplace transforms as follows:(1.12)Explanation for (1.9), (1.10), (1.11) and (1.12) will be given in §3*a*.

A separate analysis of the Volterra equations (1.9)–(1.12) and the Cauchy problem (1.6) and (1.7) is necessary, since a strong solution to the Volterra equation (1.9) belongs to *C*(*J*, ) and may not satisfy (1.6) and (1.7), while a solution to the Cauchy problem (1.6) and (1.7) belongs to *AC*^{1}(*J*, ) or *AC*^{2}(*J*, ).

Let us summarize the main results of this paper.

The Volterra equation (1.9) and (1.11), with assumptions supp

*ϕ*⊂[0, 1],*u*_{0}∈ and*F*∈*W*^{1,1}(*J*, ), has a unique solution*u*∈*C*(*J*, ).The Cauchy problem (1.6) and (1.7), with assumptions supp

*ϕ*⊂[0, 1],*F*∈*AC*^{2}(*J*, ),*u*_{0}≡0 has a unique solution*u*∈*AC*^{1}(*J*, ) and=∇

^{2},, where*α*>0,*u*_{0}≡0 has a unique solution*u*∈*AC*^{1}(*J*,*W*^{2,2}(^{n})).

The Volterra equation (1.9) and (1.10), with assumptions supp

*ϕ*⊂[0, 2],*u*_{0},*v*_{0}∈,*a*(*t*)∈*AC*^{1}(*J*) and , has*u*∈*C*(*J*, ) as a unique solution.The Cauchy problem (1.6) and (1.7), with assumptions supp

*ϕ*⊂[0, 2],*u*_{0}=*v*_{0}≡0 and*F*∈*AC*^{3}(*J*, ), ,∈*x*^{n}, has*u*∈*AC*^{2}(*J*, ) as a unique solution.

We note that the results of the time distributed-order diffusion-wave equation can also be found in papers by Hanyga (2007), Kochubei (2008) and Mainardi *et al*. (2008). Hanyga (2007) treated (1.6) with *μ*=*α* and *η*=1, while the weight function is assumed to be a right-continuous non-decreasing function on the interval [*α*, 1], satisfying *h*(*β*)=0, for *β*<*α*. Asymptotic analysis of the solution was carried out by Hanyga (2007) with the use of regularly varying functions. An equation similar to (1.6) was treated by Kochubei (2008), with *μ*=0, *η*=1 and *ϕ*∈*C*^{2}[0, 1], where *ϕ*(*α*)≔*α*^{ν}*ϕ*_{1}(*α*), *ϕ*_{1}(*α*)≥*ρ*>0, 0≤*α*≤1, *ν*≥0. The existence was proven by use of the asymptotic properties of Fourier and Laplace transforms and their inversions. Mainardi *et al*. (2008) solved an equation similar to (1.6), with *μ*=0, *η*=1 and a general weight function *ϕ* by direct calculation, i.e. by the application of Fourier and Laplace transforms. Special case *ϕ*(*α*)≔1 was also considered.

Let us briefly present the content of the paper. Section 2 summarizes notions and notation, as well as connections between Caputo-type fractional-order derivatives of functions, corresponding distributions and of their Laplace transforms. The main results are given in §3. We reduce the Cauchy problem (1.6) and (1.7) to an abstract Volterra equation in §3*a*. In §3*b*, we specify the form of *ϕ*. Using the theory given in Prüss (1993), we prove, in §3*c*, the existence of resolvents in the case of a general linear space operator and in the case of a Laplacian, while, in §3*d*, we prove additional properties of the resolvents in the case =∇^{2}. Finally, in §3*e*, the existence of the resolvents is used in order to obtain solutions to Cauchy problems.

## 2. Notions and notation

The aim of this section is to summarize notions and notation used in this paper. For the convenience of the reader, we include, in the references, some general books related to investigations of this paper (Vladimirov 1984; Prüss 1993). Let us first define spaces, which will be used throughout this paper.

is the space of locally integrable functions on , and denotes its subspace, consisting of functions vanishing on (−∞, 0].

*AC*^{m}(*J*), *J*=[0, *T*], *T*>0 and *m*∈, denotes the space of functions *f*, such that *f*^{(k)}, *k*∈{0,1, …, *m*−1}, are continuous and *f*^{(m−1)} is absolutely continuous, which means that *f*^{(m)}∈*L*^{1}(*J*). We write if *f*∈*AC*^{m}([0, *T*]) for every *T*>0.

*W*^{p,q}(^{n}) denotes the Sobolev space of functions *f*, such that, for any , |*r*|≤*p*, , where the derivative is understood in the sense of distributions.

*BV*_{loc}(_{+}) denotes the space of functions of locally bounded variations on _{+}, i.e. the functions *f* which satisfy that, for every interval [*a*, *b*]⊂_{+}, there exists a constant *M*, such that , for every choice of *t*_{0}=*a*, …, *t*_{n}=*b*.

() is the space of smooth functions and ′() is the space of compactly supported distributions, i.e. the dual of ().

(^{n}) denotes the space of rapidly decreasing smooth functions.

′(^{n}) denotes the space of tempered distributions, i.e. the dual of (^{n}).

It is said that if *f*∈′(^{n+1}) and supp *f*⊂^{n}×[0,∞). Similarly, we define . In particular, if *n*=0, we have and, similarly, . Recall that and are convolution algebras.

If a function or a distribution *f* is restricted to some subinterval *J*⊂*I*, we will use the same notation, *f*=*f*|_{J}, and if it has an additional property, we will say that it belongs to the intersection of appropriate spaces. For example, means that and *f*|_{J}∈*C*(*J*).

Let *X* denote a Banach function space on ^{n}, with the norm ‖ ‖_{X}. We have in mind *X*=*L*^{2}(^{n}).

is the space of functions , such that, for every bounded interval *I* on , holds. If *f*(*t*)=0, for *t*≤0, then we use the notation .

*C*(*J*, *X*) denotes the space of continuous functions , with the norm defined as .

*C*_{0}(*J*, *X*) is a subspace of *C*(*J*, *X*), with the additional assumption *f*(0)=0.

Hölder type space , *α*∈[0, 1), is the space of continuous functions , such that *f*(0)=0 and .

We say that , if, for every *ψ*∈(^{n}), belongs to *AC*^{m}(*J*). We define in an appropriate way.

Notation means that *u* is a locally integrable function vanishing for *t*≤0 with the values in *X* and that it defines, for every *φ*∈(), a tempered distribution with the values in .

### (a) Caputo and Riemann–Liouville fractional derivatives

Recall (Vladimirov 1984) that(*H* is the Heaviside function and *Γ* is the Euler gamma function) is a family of tempered distributions supported by [0, ∞), which satisfy

The Caputo time-fractional derivative of order *α*∈(*m*−1,*m*], *m*∈, of a function , is defined by(2.1)Note that , *t*>0. The Riemann–Liouville time-fractional derivative of the same order *α*∈(*m*−1,*m*], *m*∈, of a function is defined by(2.2)

Let . Then, a connection between the Caputo and the Riemann–Liouville fractional derivatives is given byas can be found in a tutorial paper by Mainardi (1997).

Let and put(2.3)Notation means that we consider *u* as a distribution, i.e. regular distribution is determined by *u* (so it is not the complex conjugate).

Let . Then, we introduce the distributional fractional derivative of order *α*∈ as

Next, we derive a connection between the Caputo fractional derivative of a function *u* belonging to and the distributional fractional derivative of a distribution belonging to . Using the notation as in (2.3), for every *ϕ*∈() and *α*∈(*m*−1, *m*], *m*∈, it holds that(2.4)Since(2.4) becomesand therefore(2.5)

### (b) Laplace transform of the Caputo fractional derivative

The Laplace transform is applied in the sense of distributions as follows. Let . Then,(2.6)where . This definition does not depend on a choice of a smooth function *θ*, *θ*(*t*)=1 on *t*∈(−*T*, ∞) and *θ*(*t*)=0 on *t*∈(−∞, −2*T*), for some *T*>0. If , where *X* is a space of functions or distributions, then , defined by (2.6), is an analytic function on _{+} having values in *X*.

Applying the Laplace transform to (2.5), one can derive the Laplace transform of the Caputo derivative of an absolutely continuous function as follows (recall that denotes a regular distribution determined by *u*):(2.7)for , *m*∈. Using the notation(2.7) can be written as

### (c) Distributed-order time derivative of a distribution

Let *h* be an element of . Then, it is proved by Atanackovic *et al*. (2009), proposition 2.1, that the mappings(2.8)are smooth. We cite the next definition from the paper by Atanackovic *et al*. (2009). It is based on (2.8).

Let *ϕ*∈′(), supp *ϕ*⊂[0, 2], and . Then,is defined as an element of by(2.9)where _{D}*D*_{ϕ}*h* is called the distributed-order fractional derivative of *h*.

Let , *α*∈[0,*m*]. Then, the Caputo fractional derivative is defined on intervals *α*∈(*j*−1, *j*], *j*∈{1, …, *m*}, andRecall, , *t*>0. Thus, is continuous in intervals *α*∈(*j*−1,*j*), *j*∈{1, …, *m*}, left continuous at *j*, *j*∈{1, …, *m*}, and it has jumps that appear in the limit from the right at points *j*−1, *j*∈{1, …, *m*}. Moreover, for fixed *α*∈[0, *m*], function is locally integrable on [0, ∞).

Let , *α*∈[0, 2] and be continuous in [0, 2]. Then, we define

Let , *α*_{j}∈[0, 2], *j*∈{0, …, *k*}. Then, we define

Let us derive connections between the distributed-order fractional derivative of and the corresponding distribution (in the sense of (2.3)) in the cases that are analysed above.

If *ϕ* belongs to *C*([0, 2]) and , then we start from (1.4) and use (2.3) and (2.5) to calculate , *t*>0, as follows. Let *φ*∈(), thenThus,(2.10)

Now, let , supp *ϕ*⊂[0, 2], *a*_{j}∈_{+}, .

We use the order of points because we shall separately consider two cases. The first case is when *α*_{j}≤1, *j*∈{0, …, *k*}. The second case is when some of the *α*_{j} are in (1, 2]. So, this notation is helpful from this point of view.

Since it belongs to ′(), we use definition 2.1, (2.9), to calculate , *t*>0, as(2.11)Let *l*≤*k* be chosen so that *α*_{l}>1 and *α*_{l+1}≤1. By (2.5), we obtainThe first term on the r.h.s. of the previous expression can be interpreted as *D*_{ϕ}*u*, so the expression becomes(2.12)

Both cases will be summarized by the use of (2.9) as(2.13)

If supp *ϕ*⊂[0, 1] and *u* belongs to , equation (2.13) reduces to(2.14)

### (d) Laplace transform of the distributed-order time derivative

In proposition 2.4, we apply the Laplace transform to _{D}*D*_{ϕ}*h*, . This proposition is stated by Atanackovic *et al*. (2009) for the Riemann–Liouville fractional derivative. As we have noted before, both types of derivatives coincide if they are concerned in .

*Let ϕ*∈′(), supp *ϕ*⊂[0, 2] *and* . *Then,*

*is linear and continuous mapping from**to**,*(2.15)

*and**let ϕ*∈*C*([*μ*,*η*]), [*μ*,*η*]⊂[0, 2]*and ϕ*(*α*)=0,*α*∈[0, 2]\[*μ*,*η*].*Then*,(2.16)

Furthermore, we use proposition 2.4 in order to derive the Laplace transform of a function , using the connection between its distributed-order fractional derivative and distributed-order fractional derivative of the corresponding distribution (in the sense of (2.3)). Again, we have two cases.If *ϕ* belongs to *C*([0, 2]) and , then we apply the Laplace transform to (2.10) and, using (2.16), obtain(2.17)

Now, let , supp *ϕ*⊂[0, 2], *a*_{j}∈_{+}, , *α*_{l}>1 and *α*_{l+1}≤1. Since it belongs to ′(), we apply the Laplace transform (in the sense of (2.6) to (2.11) and, using (2.15), obtainNext, we apply the Laplace transform (in the sense of (2.6)) to (2.12), where we assumed that . By using the previous expression, we obtain(2.18)Again, both (2.17) and (2.18) are summarized by(2.19)

If supp *ϕ*⊂[0, 1] and *u* belongs to , equation (2.19) reduces to(2.20)

## 3. Existence of a solution to the Cauchy problem (1.6) and (1.7)

In §2, we dealt with functions or distributions . However, our goal is to prove the existence of a solution to the Cauchy problem (1.6) and (1.7), which is a vector-valued function or distribution . Because of that, the next conclusion is essential.

All the calculations given in §2 for functions, distributions and regular distributions, defined on the real line, can be applied to , *ψ*∈*S*(^{n}), where is a distribution-valued function, or its regularization , *t*∈, or distribution-valued distribution, defined as 〈〈*u*(** x**,

*t*),

*ψ*(

**)〉,**

*x**ϕ*(

*t*)〉,

*ψ*∈

*S*(

^{n}),

*φ*∈

*S*().

Here, we implicitly use the Schwartz kernel theorem and the generalized Fubini theoremfor every .

Thus, all results of previous sections will be used in this section for vector-valued functions or distributions.

### (a) Reduction to an abstract Volterra equation

We apply the Laplace transform to (1.6) and (1.7). It is understood in the sense of distributions, as in (2.6). Let *ϕ* belong to ′(), supp *ϕ*⊂[0, 2] or supp *ϕ*⊂[0, 1]. Then, we use the Laplace transform of the distributed-order derivative given by (2.19) or (2.20). Let *X* be a Banach space of functions on ^{n} with the norm |.|, for example *X*=*L*^{2}(^{n}). We will use the notation *u*(** x**,

*t*),

**∈**

*x*^{n},

*t*>0, for the mapping , ,

**∈**

*x*^{n}. Let , with the explanation as in §2. Then, the Laplace transform of

*u*with respect to

*t*is given byThis is an analytic function in

_{+}, with values in

*X*. We will assume for

*F*the same as for

*u*in order to perform the Laplace transform. So, we assume and, later, we will give additional assumptions on

*F*.

Thus, by the use of the Laplace transform, we obtain(3.1)if supp *ϕ*⊂[0, 2]. According to remark 2.5, if supp *ϕ*⊂[0, 1], then(3.2)LetBoth are analytic functions in _{+}. Furthermore, assume that and satisfyEquation (3.1) becomes(3.3)while (3.2) becomes(3.4)Next, we use (3.3) and (3.4) respectively, in order to obtain abstract Volterra equations(3.5)and(3.6)The application of the inverse Laplace transform to (3.5) and (3.6) respectively, implies new forms of (1.6)(3.7)if supp *ϕ*⊂[0, 2], and(3.8)if supp *ϕ*⊂[0, 1], where(3.9)is called a scalar kernel. It belongs to .

Equation (1.6) in the form of (3.7) or (3.8), is actually of the form of an abstract Volterra equation (see Prüss 1993, p. 30)where *w*∈*C*(*J*, *X*). However, we need to prescribe a stronger condition for *w*, since a solution to (1.6) and (1.7) should belong to *AC*^{1}(*J*, *X*), if supp *ϕ*⊂[0, 1] or to *AC*^{2}(*J*, *X*), if supp *ϕ*⊂[0, 2]. This will be explained in §3*e*. We assume that is an unbounded operator defined on a dense subset equipped with the graph norm .

Note that if supp *ϕ*⊂[0, 2], then *w* is given by(3.10)If supp *ϕ*⊂[0, 1], then *w* is given by(3.11)

Thus, we have obtained equations (1.9)–(1.12) quoted in §1.

### (b) Specifying the form of *ϕ* in (1.6)

We consider(3.12)where we assume

condition I: and

condition II: .

Note that condition I actually means that supp *ϕ*⊂[0, 1], while condition II means that supp *ϕ*⊂[0, 2].

Distribution *ϕ*, in the form (3.12), is important in viscoelasticity theory (see Mainardi 1997, p. 300). Physically, coefficients *a*_{j}, *j*∈{0, …, *k*}, may be interpreted as the relaxation times and therefore *a*_{j}∈_{+}, *j*∈{0, …, *k*}. We will show in §3*c* that this condition is in accordance with the existence of a resolvent. Functions and are given as follows:

if condition I holds, thenand

if condition II holds, then

Note that *B*_{v} is a singular distribution, but .

Now, the scalar kernel (3.9) becomes(3.13)with the same conditions as before. These functions are studied in Atanackovic *et al*. (2009), where lemma 3.1 was stated.

*Let* , *t*>0, *be defined by* (*3.13*). *Assume* , *s*∈_{+}. *Then,*

*a is locally integrable function in**with a*(*t*)=0,*t*<0*and**a is absolutely continuous in J*,*if α*_{0}−*α*_{k}>1.

The inversion of (3.13) is given by Atanackovic *et al*. (2009) in its integral representation, while in Podlubny (1999), it is given in terms of derivatives of a two-parameter Mittag-Lefler function.

In the sequel, we will investigate equation (1.6) in the form of (3.7), if is a general linear differential operator, and in the form of (3.8), if =∇^{2}. Thus, the considered equations are as follows:

a Volterra time distributed-order equation with a linear differential operator (3.14)on a Banach space

*X*anda Volterra time distributed-order diffusion-wave equation(3.15)on

*L*^{2}(^{n}).

Recall, the scalar kernel *a* is given by (3.13) and distribution *w* either by (3.10), for supp *ϕ*⊂[0, 2], or by (3.11), for supp *ϕ*⊂[0, 1].

### (c) Existence of a resolvent in a general case of operator

In the sequel, we will give assertions assuming separately that scalar kernel *a*, defined by (3.13), satisfies condition I or condition II. First, we state some properties of the scalar kernel *a*.

Recall, definition 4.5 of Prüss (1993), where the notion of a completely positive function is introduced.

*Let condition I hold for a. Then, it is a completely positive function*.

According to proposition 4.5 of Prüss (1993), in order that *a* is a completely positive function, we should have that , as well as for *s*>0, which is clear in our case, since *a* is continuous for *t*>0 and the coefficients in are positive. Moreover, one should have that , *s*>0, is a Bernstein function. By definition 4.3 of Prüss (1993), a smooth function *φ*(*s*) is a Bernstein function if it is positive for *s*>0 and *f*=*φ*′ is completely monotonic. Let us prove this. Clearly, the assumptions *α*_{j}∈[0,1] and *a*_{j}∈_{+}, *j*∈{0,…,*k*} implyfor *s*>0 and, in general, for *n*∈Thus, *f* satisfies (−1)^{n}*f*^{(n)} (*s*)≥0 for *s*>0, *α*_{j}∈[0, 1] and *a*_{j}∈_{+}, *j*∈{0, …, *k*}. ▪

A simple consequence is corollary 3.3.

*Let condition II hold for a*. *Suppose that*(3.16)*where β*_{p}∈[0, 1), *b*_{p}∈_{+}, *p*∈{0, …, *l*} (2*β*_{0}=*α*_{0}∈(1, 2)). *Then, a can be written as a*≔*c***c*, *where**is a completely positive function*.

Theorems that are to follow state the existence of a resolvent and it is the basic result for this section.

*Let condition I hold for a*. *Let* *be a linear differential operator on a Banach space X and let it generate an exponentially bounded C*_{0}-*semigroup T in X, such that* |*T*(*t*)|≤*M*e^{ωt}, *t*>0, *for M*>0, *ω*≥0. *Then, the Volterra time distributed-order equation with linear differential operator* , (*3.14*), *admits an exponentially bounded resolvent S*.

*Moreover, the growth bound ω _{a} is defined as*(3.17)

*If ω*=0,

*then ω*

_{a}=0 (

*because*

*does not hold*).

*If ω*>0 and ,

*then we have two cases determined by*(

*3.17*).

*The*

*resolvent S is of type*(

*M*

_{a},

*ω*

_{a}).

*If ω*>0

*and*,

*then the resolvent is of type*(

*M*

_{ϵ},

*ϵ*)

*for every ϵ*>0.

The proof follows from theorem 4.2 of Prüss (1993), since, by proposition 3.2, *a* is a completely positive function. ▪

*Let condition II and* (*3.16*) *hold for a*. *Let* *be a linear differential operator on a Banach space X and let it generate an exponentially bounded cosine family T in X*, *such that* |*T*(*t*)|≤*M*e^{ωt}, *t*>0, *for some M*>0, *ω*≥0. *Then, the Volterra time distributed-order equation with linear differential operator* , (*3.14*), *admits an exponentially bounded resolvent S*.

*Moreover, the growth bound ω*_{a} *is defined as*(3.18)

The proof follows from theorem 4.3 (iii) of Prüss (1993), since *a*=*c***c*, where *c* is a completely positive function. ▪

*Let X*=*L*^{2}(^{n}) *and* =∇^{2}. The *Volterra time distributed-order diffusion-wave equation* (*3.15*) *admits an exponentially bounded resolvent S as follows:*

*If condition I holds, then the resolvent S is of type**if α*_{k}>0*or if α*_{k}=0*and a*_{k}<*ω*,*or of type*(*M*_{a}, 0)*if α*_{k}=0*and a*_{k}<*ω*,*or of type*(*M*_{ϵ},*ϵ*)*for every ϵ*>0*if α*_{k}=0*and a*_{k}<*ω*.*If condition II holds, then the resolvent S has the growth bound**if α*_{k}>0*or if α*_{k}=0*and a*_{k}<ω^{2},*or ω*_{a}=0*if α*_{k}=0*and a*_{k}<*ω*^{2}.

Let us first prove (i). It is known, example 3.7.6 of Arendt *et al*. (2001), that the Laplacian generates a bounded holomorphic *C*_{0}-semigroup of angle *π*/2 on *L*^{2}(^{n}). Therefore by proposition 3.7.2 of Arendt *et al*. (2001), it generates a *C*_{0}-semigroup of type (*M*,*ω*), *M*>0, *ω*>0, on *L*^{2}(^{n}). From (3.13), we seeLet *α*_{k}=0. Since *ω*>0, by theorem 3.4, (3.17), if *a*_{k}<*ω*, then , by (3.13) and if *a*_{k}<*ω*, then *ω*_{a}=0. Also, if *a*_{k}<*ω*, then every *ϵ*>0 is the growth bound. Let *α*_{k}>0, then *ω*>0 implies that is satisfied in (3.17) and therefore .

We now prove (ii). By theorem 8.3.12 of Arendt *et al*. (2001), the Laplacian generates a cosine family on *L*^{2}(^{n}). Let *α*_{k}=0. Since *ω*>0, by theorem 3.5, (3.18), if *a*_{k}<*ω*^{2}, then , by (3.13) and if *a*_{k}>*ω*^{2}, then *ω*_{a}=0. Let *α*_{k}>0, then *ω*>0 implies that is satisfied in (3.18) and therefore ▪

### (d) Additional properties if =∇^{2} and condition I holds

In this case, we have the additional properties of (3.15).

*Let condition I hold for a*. *Then, the Volterra time distributed-order diffusion-wave equation* (*3.15*) *is parabolic*.

According to definition 3.1 of Prüss (1993), the equation is parabolic if the scalar kernel *a* satisfies

and , and

there exists a constant

*M*≥1, such that ,*s*∈_{+}, satisfies

We first prove (P1). By writing *s*=*r*e^{iϕ}, the Laplace transform of *a* isIt is different from zero for *s*∈_{+}, since cos*α*_{j}*φ*>0 for *φ*∈(−*π*/2,*π*/2) and *α*_{j}∈[0, 1]. Next, we prove that . The resolvent set of the Laplacian is and therefore we should haveIt holds as , *s*∈_{+}, for *α*_{j}∈[0,1], *a*_{j}∈_{+}, *j*∈{0,…,*k*}.

To prove (P2), we use the Fourier transform for *Hv*, where *v* is any function and the Parseval formula. Thus,(3.19)We havei.e.Therefore, (3.19) leads toand, by the Parseval formula, we haveThus, (P2) is satisfied with *M*=1. ▪

Recall, definition 3.3 of Prüss (1993), where the notion of an *r*-regular, *r*∈, function is introduced.

*Let condition I hold for a*. *Then, it is r-regular*, *r*∈.

According to proposition 3.2 of Prüss (1993), in order to prove that *a* is *an r*-regular kernel, *r*∈, one has to prove that , *s*∈_{+}, admits an analytic extension to sector , *ϕ*>(*π*/2) and that there exists *θ*∈(0,∞), such that , *s*∈*Σ*(0,*ϕ*). By taking *ϕ*=*π*, we have that admits an analytic extension to *Σ*(0,*π*), since , *s*∈_{+}. Next, by taking *θ*=*π*, we have , since *α*_{j}∈[0, 1] and *a*_{j}∈_{+}, *j*∈{0, …, *k*}. Therefore, holds as well. Similar arguments hold in the case ; therefore, it can be concluded that ▪

*Let condition I hold for a*. *Then, the Volterra time distributed-order diffusion-wave equation* (*3.15*) *admits the resolvent* .

*Moreover, for every r*∈, *there is M*_{r}>0 *such that*

Note that the Laplacian is a closed linear operator in *L*^{2}(^{n}) with the dense domain . Moreover, by proposition 3.7, equation (3.15) is parabolic and *a*, given by (3.13), is an *r*-regular, *r*∈, scalar kernel, as is shown in proposition 3.8. Thus, theorem 3.9 follows from theorem 3.1 of Prüss (1993). ▪

Moreover, it is shown that the Volterra time distributed-order diffusion-wave equation (3.15) admits an analytic resolvent, as stated in theorem 3.10. Recall, that a sector in the complex plane is defined as .

*Let condition I hold for a and let α*_{0}<1. *Then, the Volterra time distributed-order diffusion-wave equation* (*3.15*) *admits an analytic resolvent S of analyticity type* (0, *θ*_{0}), *θ*_{0}<(*π*/2) ((1/*α*_{0})−1).

*Moreover, for every ω*>0 *and θ*<*θ*_{0}, *there exists M*=*M*(*ω*, *θ*), *such that*

Let us show that *a* satisfies the conditions (A1)–(A3) (cf. Prüss 1993, theorem 2.1).

has meromorphic extension to . It is clear that can only have poles in .

and , for all . First, we determine

*θ*_{0}so that ,*j*∈{0, …,*k*}. Since*α*_{0}is the greatest, the condition reduces to*α*_{0}((*π*/2)+*θ*_{0})<*π*/2, i.e.*θ*_{0}<(*π*/2)((1/*α*_{0})−1). Then holds since , where*θ*_{0}is determined so that ,*j*∈{0,…,*k*}.In order to prove that there exists

*C*=*C*(*ξ*,*θ*), such that, for all*θ*<*θ*_{0}and*ξ*>0, operator satisfies(3.20)we start with (3.19),(3.21)Let us show that . We haveFor*α*_{j}≤*α*_{0}<1 and , it holds that cos*α*_{j}*φ*>0. Therefore, . Thus, we have proved that and (3.21) holds with . This givesNow, we will prove that (3.20) holds for every*θ*<*θ*_{0}and*ξ*>0. Fix*θ*<*θ*_{0}and*ξ*>0 and let us define circles*K*_{1}and*K*_{2}as and . Assume that*s*∈ is out of both circles, i.e. |*s*|≥|*ξ*| and |*s*−*ξ*|≥|*ξ*|. We havewhich impliesNext, assume*s*∈ is inside both circles. Since*s*also belongs to ,*r*_{0}≤|*s*|<|*ξ*| and |*s*−*ξ*|<|*ξ*| holds. We haveWe conclude thatand therefore, for ,*θ*<*θ*_{0},*ξ*>0,Thus, by the Parseval formula for every , (3.20) holds.

Since the Laplacian is an unbounded, closed linear operator in *L*^{2}(^{n}), with the dense domain , the assertion follows from theorem 2.1 and corollary 2.1 of Prüss (1993). ▪

### (e) Solution to (1.6) and (1.7) via a resolvent

Theorems 3.4 and 3.5 state that there exists a resolvent to the Volterra time distributed-order equation (3.14), with linear differential operator , in the cases when *a*, given by (3.13), satisfies condition I or condition II. Moreover, for the Volterra time distributed-order diffusion-wave equation (3.15) with the same scalar kernel *a*, theorems 3.9, 3.10 and corollary 3.6 state the existence of the resolvent. Therefore, it is possible to construct a solution to (1.6) and (1.7) by the use of the resolvent.

Formally, applying the Laplace transform to the Volterra time distributed-order equation (3.14), with linear differential operator , and by solving an algebraic equation with respect to , one obtains(3.22)where the resolvent *S* is given by its Laplace transformApplying the inverse Laplace transform to (3.22), the solution is expressed via the resolvent operator by(3.23)where *w* belongs to an appropriate space. The analysis of the formal expression (3.23) will be the subject of the next two subsections. We shall consider strong solutions to the Volterra-type equation and solutions to the Cauchy problem (1.6) and (1.7). In the first case, solutions belong to *C*(*J*, ) and, in the second case, solutions belong to an appropriate space so that the distributed-order derivative can be applied to them. Recall that *w* is given by (3.11) if supp *ϕ*⊂[0, 1], i.e. if condition I holds and by (3.10) if supp *ϕ*⊂[0, 2], i.e. if condition II holds. Now, we separately consider these two cases.

#### (i) Condition I holds

In this section, we assume that generates an exponentially bounded *C*_{0}-semigroup. We have proved that also generates an exponentially bounded resolvent (corollary 3.6). Moreover, if =∇^{2}, then it generates a smooth (*C*^{∞}) resolvent (theorem 3.9), if supp *ϕ*⊂[0, 1], and an analytic resolvent, with appropriate bounds on its derivatives (theorem 3.10), if supp *ϕ*⊂[0, 1).

Now, by proposition 2.1 of Prüss (1993), we have theorem 3.11.

*Assume that* *generates an exponentially bounded C*_{0}-*semigroup. Let u*_{0}∈ *and F*∈*W*^{1,1}(*J*, ). *Then**,**is a unique strong solution to* (*3.14*), *where S is the resolvent of* (*3.14*). *It belongs to C*(*J*, ).

However, in order to have *u*∈*AC*^{1}(*J*, ), we need additional assumptions.

*Assume that* *generates an exponentially bounded C*_{0}-*semigroup*. *Let u*_{0}≡0 *and F*∈*AC*^{2}(*J*, ), *F*(** x**, 0)=0,

**∈**

*x*^{n}.

*Then,*

*is a unique strong solution to*(

*1.6*)

*belonging to AC*

^{1}(

*J*, ).

*Assume* =∇^{2}. *Let u*_{0}≡0 and *for some α*>0. *Then**,**is a unique strong solution to* (*1.6*). *Moreover*, .

The assumption implies that . Now, by theorem 2.4 (iii) of Prüss (1993), the condition . ▪

#### (ii) Condition II holds

Note thatsee (2.18), is locally integrable on [0, ∞).

In order to have an exponentially bounded resolvent *S*, must generate an exponentially bounded cosine family, as we have proved in theorem 3.5 and corollary 3.6. So with this, we have theorem 3.14.

*Assume that a*=*c***c satisfies* (*3.16*), *with α*_{0}−*α*_{k}>1 (*see* *lemma 3.1*) *and that* *generates an exponentially bounded cosine family* (*see* *theorem 3.5*). *Let u*_{0}, *v*_{0}∈, *a*∈*AC*^{1}(*J*) (*see* *lemma 3.1*) *and* . *Then*,*is a unique strong solution to* (*3.14*), *where S is the resolvent of* (*3.14*) *with the properties of* *theorem 3.5*.

Bearing in mind theorem 3.5, this theorem is a consequence of the proposition 1.2 (iv) of Prüss (1993). ▪

In order to have *u*∈*AC*^{2}(*J*, ), we have to assume additional conditions. Since, in theorem 3.14, we cannot extend the assumption on *a* (again see lemma 3.1), we have to make additional assumptions on the resolvent and on *F*.

*Assume that a*=*c***c satisfies* (*3.16*) *and that* *generates an exponentially bounded cosine family* (*see* *theorem 3.5*). *Let u*_{0}=*v*_{0}≡0 and *F*∈*AC*^{3}(*J*, ), , ** x**∈

^{n}.

*Then*

*,*

*is a unique strong solution to*(

*1.6*)

*and it belongs to AC*

^{2}(

*J*, ).

*Moreover*, *if* =∇^{2} *and* =*W*^{2,2}(^{n}), *then u is a strong unique solution to* (*1.6*) *and it belongs to AC*^{2}(*J*, *W*^{2,2}(^{n})).

## Acknowledgments

This research was supported by grant 144019A (T.M.A. and D.Z.) and grant 144016 (S.P.) of the Serbian Ministry of Sciences.

## Footnotes

- Received October 31, 2008.
- Accepted February 20, 2009.

- © 2009 The Royal Society