# Well-posedness of fractional parabolic equations

- Allaberen Ashyralyev
^{1}Email author

**2013**:31

**DOI: **10.1186/1687-2770-2013-31

© Ashyralyev; licensee Springer. 2013

**Received: **2 October 2012

**Accepted: **25 January 2013

**Published: **18 February 2013

## Abstract

In the present paper, we consider the abstract Cauchy problem for the fractional differential equation

in an arbitrary Banach space *E* with the strongly positive operators $A(t)$. The well-posedness of this problem in spaces of smooth functions is established. The coercive stability estimates for the solution of problems for 2*m* th order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in a space variable are obtained. The stable difference scheme for the approximate solution of this problem is presented. The well-posedness of the difference scheme in difference analogues of spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of difference schemes for the fractional parabolic equation with nonlocal boundary conditions in a space variable and the 2*m* th order multidimensional fractional parabolic equation are obtained.

**MSC:**65M12, 65N12.

### Keywords

fractional parabolic equation Basset problem well-posedness coercive stability## 1 Introduction

It is known that differential equations involving derivatives of noninteger order have shown to be adequate models for various physical phenomena in areas like rheology, damping laws, diffusion processes, *etc.* Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers (see, *e.g.*, [1–43] and the references given therein).

*e.g.*, [44–51]). In the present paper, the initial value problem

for the fractional differential equation in an arbitrary Banach space *E* with the linear (unbounded) operators $A(t)$ is considered. Here $u(t)$ and $f(t)$ are the unknown and the given functions, respectively, defined on $[0,T]$ with values in *E*. The derivative ${u}^{\mathrm{\prime}}(t)$ is understood as the limit in the norm of *E* of the corresponding ratio of differences. $A(t)$ is a given closed linear operator in *E* with the domain $D(A(t))=D$, independent of *t* and dense in *E*. Finally, $u(0)=0$.

Here ${D}_{t}^{\frac{1}{2}}={D}_{0+}^{\frac{1}{2}}$ is the standard Riemann-Liouville derivative of order $\frac{1}{2}$. This fractional differential equation corresponds to the Basset problem [9]. It represents a classical problem in fluid dynamics where the unsteady motion of a particle accelerates in a viscous fluid due to the gravity of force. Recently, fractional Basset equations with independent in *t* operator coefficients $A(t)=A$ have been studied extensively (see, *e.g.*, [52–56] and the references given therein).

*t*operator coefficients $A(t)$ in spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of problems for 2

*m*th order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in a space variable are obtained. The stable difference scheme for the approximate solution of initial value problem (2)

is presented. Here $\mathrm{\Gamma}(k-m+\frac{1}{2})={\int}_{0}^{\mathrm{\infty}}{t}^{k-m-\frac{1}{2}}{e}^{-t}\phantom{\rule{0.2em}{0ex}}dt$.

The paper is organized as follows. The well-posedness of problem (2) in spaces of smooth functions is established in Section 2. In Section 3 the coercive stability estimates for the solution of problems for 2*m* th order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions are obtained. The well-posedness of (3) in difference analogues of spaces of smooth functions is established and the coercive stability estimates for the solution of difference schemes for the fractional parabolic equation with nonlocal boundary conditions in a space variable and the 2*m* th order multidimensional fractional parabolic equation are obtained in Section 4.

## 2 The well-posedness of problem (2)

- (i)
$u(t)$ is continuously differentiable on the segment $[0,1]$. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.

- (ii)
The element $u(t)$ belongs to $D(A(t))$ for all $t\in [0,1]$ and the function $A(t)u(t)$ is continuous on the segment $[0,1]$.

- (iii)
$u(t)$ satisfies the equation and the initial condition (2).

*E*equipped with the norm

In this paper, positive constants, which can differ in time, are indicated with an *M*. On the other hand, $M(\alpha ,\beta ,\dots )$ is used to focus on the fact that the constant depends only on $\alpha ,\beta ,\dots $ .

is true for its solution $u(t)\in C(E)$.

*i.e.*, the following estimates

hold for some $M\in [1,+\mathrm{\infty})$, $\delta \in (0,+\mathrm{\infty})$. From this inequality it follows the operator ${A}^{-1}(t)$ exists and is bounded, and hence $A(t)$ is closed in $C(E)$.

*t*in the uniform operator topology for each fixed

*s*, that is,

*t*,

*s*for $0\le s<t\le 1$, is called a fundamental solution of (2) if

- (1)
the operator $v(t,s)$ is strongly continuous in

*t*and*s*for $0\le s<t\le T$, - (2)
the following identity holds:

- (3)
the operator $v(t,s)$ maps the region

*D*into itself. The operator $u(t,s)=A(t)v(t,s){A}^{-1}(s)$ is bounded and strongly continuous in*t*and*s*for $0\le s<t\le 1$, - (4)on the region
*D*the operator $v(t,s)$ is strongly differentiable relative to*t*and*s*, while$\frac{\partial v(t,s)}{\partial t}=-A(t)v(t,s)$(8)

Now, we will give a series of interesting lemmas and estimates concerning the fundamental solution $v(t,s)$ of (2) which will be useful in the sequel.

**Lemma 2.1**

*For any*$0\le s<t\le 1$

*and*$u\in D$,

*the following identities hold*:

**Lemma 2.2**

*For any*$0\le s<t\le t+r\le 1$, $0\le \alpha \le 1$

*and*$0\le \epsilon \le 1$,

*the following estimates hold*:

**Theorem 2.1**

*Let*$A(t)$

*be a strongly positive operator in a Banach space*

*E*

*and*$f(t)\in C(E)$.

*Then for the solution*$u(t)$

*in*$C(E)$

*of initial value problem*(2),

*the following stability inequality holds*:

*Proof*Using formula (12), we get

Estimate (26) follows from estimates (29) and (31).

Estimate (22) follows from estimates (33) and (35). Theorem 2.1 is proved. □

From (6) and (7) it follows that

**Theorem 2.2** ${E}_{\alpha}(E,A(t))={E}_{\alpha}(E,A(0))$ *for all* $0<\alpha <1$ *and* $0\le t\le 1$.

Problem (2) is not well posed in $C(E)$ for arbitrary *E*. It turns out that a Banach space *E* can be restricted to a Banach space ${E}^{\mathrm{\prime}}$ in such a manner that the restricted problem (2) in ${E}^{\mathrm{\prime}}$ will be well posed in $C({E}^{\mathrm{\prime}})$. The role of ${E}^{\mathrm{\prime}}$will be played here by the fractional spaces ${E}_{\alpha}={E}_{\alpha}(A(t),E)$ ($0<\alpha <1$).

**Theorem 2.3**

*Suppose*$f(t)\in C({E}_{\alpha})$ ($0<\alpha <1$).

*Suppose that assumptions*(6)

*and*(7)

*hold and*$0<\alpha \le \epsilon <1$.

*Then for the solution*$u(t)$

*in*$C({E}_{\alpha})$

*of problem*(2),

*the coercive inequality*

*holds*.

*Proof*

Estimate (37) follows from estimates (39) and (42). Theorem 2.3 is proved. □

Let us give, without proof, the following result.

**Theorem 2.4**

*Suppose that assumption*(6)

*holds*.

*Suppose that the operator*$A(t){A}^{-1}(s)$

*is Hölder continuous in*

*t*

*in the uniform operator topology for each fixed*

*s*,

*that is*,

*where*

*M*

*and*

*ε*

*are positive constants independent of*

*t*,

*s*

*and*

*τ*

*for*$0\le t,s,\tau \le T$.

*Suppose*$f(t)\in C({E}_{\alpha})$ ($0<\alpha <1$).

*Then for the solution*$u(t)$

*in*$C({E}_{\alpha})$

*of problem*(2),

*the coercive inequality*

*holds*.

## 3 Applications

Now, we consider the applications of Theorems 2.1, 2.3 and 2.4.

*m*-order multidimensional fractional parabolic equation is considered:

where ${a}_{r}(t,x)$ and $f(t,x)$ are given as sufficiently smooth functions. Here, *σ* is a sufficiently large positive constant.

*B*and is given by

for $\xi \ne 0$. Problem (45) has a unique smooth solution. This allows us to reduce problem (45) to the abstract Cauchy problem (2) in a Banach space $E={C}^{\mu}({\mathbb{R}}^{n})$ of all continuous bounded functions defined on ${\mathbb{R}}^{n}$ satisfying the Hölder condition with the indicator $\mu \in (0,1)$ with a strongly positive operator ${A}^{t,x}={B}^{t,x}+\delta I$ defined by (52) (see [57, 58]).

**Theorem 3.1**

*For the solution of boundary problem*(45),

*the following estimates are satisfied*:

The proof of Theorem 3.1 is based on the abstract Theorems 2.1, 2.3, 2.4 and the coercivity inequality for an elliptic operator ${A}^{t,x}$ in ${C}^{\mu}({\mathbb{R}}^{n})$ and on the following theorem on the structure of the fractional spaces ${E}_{\alpha}({C}^{\mu}({\mathbb{R}}^{n}),{A}^{t,x})$.

**Theorem 3.2** ${E}_{\alpha}({C}^{\mu}({\mathbb{R}}^{n}),{A}^{t,x})={C}^{2m\alpha +\mu}({\mathbb{R}}^{n})$ *for all* $0<2m\alpha +\mu <1$ *and* $0\le t\le 1$ [59].

where $a(t,x)$ and $f(t,x)$ are given sufficiently smooth functions and $a(t,x)\ge a>0$. Here, *σ* is a sufficiently large positive constant.

defines a positive operator ${A}^{t,x}$ acting in ${C}^{\beta}[0,1]$ with the domain ${C}^{\beta +2}[0,1]$ and satisfying the conditions $v(t,0)=v(t,1)$, ${v}_{x}(t,0)={v}_{x}(t,1)$. Therefore, we can replace the mixed problem (56) by the abstract boundary value problem (2). Using the results of Theorems 2.1, 2.3, 2.4, we can obtain the following theorem.

**Theorem 3.3**

*For the solution of mixed problem*(56),

*the following estimates are valid*:

The proof of Theorem 3.3 is based on abstract Theorems 2.1, 2.3, 2.4 and on the following theorem on the structure of the fractional spaces ${E}_{\alpha}(C[0,1],{A}^{t,x})$.

**Theorem 3.4** ${E}_{\alpha}(C[0,1],{A}^{t,x})={C}^{2\alpha}[0,1]$ *for all* $0<\alpha <\frac{1}{2}$, $0\le t\le 1$ [60].

## 4 The well-posedness of problem (3)

So, formula (66) gives the representation for the solution of problem (3).

*E*. Next on ${F}_{\tau}(E)$ we introduce the Banach space ${C}_{\tau}(E)=C({[0,1]}_{\tau},E)$ with the norm

**Theorem 4.1**

*Let*$A(t)$

*be a strongly positive operator in a Banach space*

*E*.

*Then for the solution*${u}^{\tau}={\{{u}_{k}\}}_{1}^{N}$

*in*${C}_{\tau}(E)$

*of initial value problem*(3),

*the stability inequality*

*holds*.

*Proof*Using formula (66), we get

Estimate (71) follows from estimates (75) and (80).

Estimate (68) follows from estimates (88) and (89). Theorem 4.1 is proved. □

From (73) it follows that

**Theorem 4.2** ${E}_{\alpha}^{\mathrm{\prime}}(E,A(t))={E}_{\alpha}^{\mathrm{\prime}}(E,A(0))$ *for all* $0<\alpha <1$ *and* $0\le t\le 1$.

Problem (3) is not well posed in ${C}_{\tau}(E)$ for arbitrary *E*. It turns out that a Banach space *E* can be restricted to a Banach space ${E}^{\mathrm{\prime}}$ in such a manner that the restricted problem (3) in ${E}^{\mathrm{\prime}}$ will be well posed in $C({E}^{\mathrm{\prime}})$. The role of ${E}^{\mathrm{\prime}}$ will be played here by the fractional spaces ${E}_{\alpha}={E}_{\alpha}(A(t),E)$ ($0<\alpha <1$).

**Theorem 4.3**

*Suppose that assumptions*(6)

*and*(7)

*hold and*$0<\alpha \le \epsilon <1$.

*Then for the solution*${u}^{\tau}={\{{u}_{k}\}}_{1}^{N}$

*in*${C}_{\tau}({E}_{\alpha}^{\mathrm{\prime}})$

*of initial value problem*(3),

*the coercive stability inequality*

*holds*.

*Proof*

Estimate (91) follows from estimates (93) and (96). Theorem 4.3 is proved. □

Let us give, without proof, the following result.

**Theorem 4.4**

*Suppose that assumptions*(6)

*and*(43)

*hold*.

*Then for the solution*${u}^{\tau}={\{{u}_{k}\}}_{1}^{N}$

*in*${C}_{\tau}({E}_{\alpha}^{\mathrm{\prime}})$

*of initial value problem*(3),

*the coercive stability inequality*

*holds*.

Note that by passing to the limit for $\tau \to 0$, one can recover Theorems 2.1-2.3 and 2.4.

## 5 Applications

Now, we consider the applications of Theorems 4.1, 4.3 and 4.4.

where ${e}_{k}$ is the unit vector of the axis ${x}_{k}$.

*λ*th order ($\lambda >0$) approximation of the differential operator ${A}^{t,x}$ if the inequality

holds for any smooth function $\phi (x)$. The coefficients ${b}_{s}^{t,x}$ are chosen in such a way that the operator ${A}_{h}^{t,x}$ approximates in a specified way the operator ${A}^{t,x}$. It will be assumed that the operator ${A}_{h}^{t,x}$ approximates the differential operator ${A}^{t,x}$ with any prescribed order [57, 58].

The function ${A}^{t,x}(\xi h,h)$ is obtained by replacing the operator ${\mathrm{\Delta}}_{k\pm}$ in the right-hand side of equality (99) with the expression $\pm (exp\{\pm i{\xi}_{k}h\}-1)$, respectively, and is called the symbol of the difference operator ${B}_{h}^{t,x}$.

*x*, the symbol ${A}^{t,x}(\xi h,h)$ of the operator ${B}_{h}^{t,x}={A}_{h}^{t,x}-\sigma {I}_{h}$ satisfies the inequalities

for an infinite system of ordinary differential equations.

**Theorem 5.1**

*Suppose that assumptions*(102)

*and*(103)

*for the operator*${A}_{h}^{k,x}$

*hold*.

*Then*,

*the solutions of difference scheme*(105)

*satisfy the following stability estimates*:

The proof of Theorem 5.1 is based on the abstract Theorems 4.1, 4.3, 4.4 and the strong positivity of the operator ${A}_{h}^{x}$ defined by (114) in ${C}_{h}^{\mu}$ and on the following two theorems on the coercivity inequality for the solution of the elliptic difference equation in ${C}_{h}^{\beta}$ and on the structure of the fractional space ${E}_{\alpha}^{\mathrm{\prime}}({C}_{h},{A}_{h}^{x})$.

**Theorem 5.2**

*Suppose that assumptions*(102)

*and*(103)

*for the operator*${A}_{h}^{k,x}$

*hold*.

*Then for the solutions of the elliptic difference equation*

*the estimates*[54]

*are valid*.

**Theorem 5.3** *Suppose that assumptions* (102) *and* (103) *for the operator* ${A}_{h}^{k,x}$ *hold*. *Then for any* $0<\alpha <\frac{1}{2m}$, *the norms in the spaces* ${E}_{\alpha}^{\mathrm{\prime}}({C}_{h},{A}_{h}^{x})$ *and* ${C}_{h}^{2m\alpha}$ *are equivalent uniformly in* *h* [51].

*A*generated by problem (56), we assign the difference operator ${A}_{h}^{x}$ by the formula

**Theorem 5.4**

*Let*

*τ*

*and*

*h*

*be sufficiently small numbers*.

*Then*,

*the solutions of difference scheme*(116)

*satisfy the following stability estimates*:

The proof of Theorem 5.4 is based on the abstract Theorems 4.1, 4.3, 4.4 and the strong positivity of the operator ${A}_{h}^{t,x}$ defined by (114) in ${C}_{h}^{\mu}$ and on the following theorem on the structure of the fractional space ${E}_{\alpha}^{\mathrm{\prime}}({C}_{h},{A}_{h}^{t,x})$.

**Theorem 5.5** *For any* $0<\alpha <\frac{1}{2}$, *the norms in the spaces* ${E}_{\alpha}^{\mathrm{\prime}}({C}_{h},{A}_{h}^{t,x})$ *and* ${C}_{h}^{2\alpha}$ *are equivalent uniformly in* *h* *and* $t\in [0,1]$ [60].

## Declarations

### Acknowledgements

The author would like to thank Prof. P. E. Sobolevskii for his helpful suggestions to the improvement of this paper.

## Authors’ Affiliations

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