1. Introduction
At present, the fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in such areas as physics, chemistry, fractal, economy, and engineering. It has been found that the differential equations involving the fractional derivative are more realistic to describe many phenomena in practical cases than those of an integer order.
During the last decade, the partial differential equations with time fractional derivatives have increasingly attracted the attention of researchers both in pure and applied mathematics, because they can be taken as mathematical models of complex systems which exhibit anomalous diffusion. Evidently, fractional partial differential equations are apt to describe diffusive motions that cannot be modeled as standard Brownian ones (described by heat equation or more general parabolic equations), where the mean square displacement of a diffusive particle is a linear function of time t. The signature of an anomalous diffusion of this kind is that mean square displacement of the diffusing species scales as a nonlinear power law in time t, that is, , where is a constant. When , this is referred to as sub-diffusion; when , this is referred to as supdiffusion.
The semi-linear fractional parabolic partial differential equations have been discussed by too many authors. In 2014, Chen et al. [
1] considered the initial value problem of fractional semi-linear evolution equations with non-compact semigroups in Banach spaces
E
where
is the Caputo fractional derivative of order
,
is a closed linear operator and
generates a uniformly bounded
semigroup
in
E,
is a continuous nonlinear mapping,
. The existence of saturated mild solutions and global mild solutions is obtained under weaker conditions. Authors [
2] are concerned with non-local problems for fractional evolution equations with mixed monotone non-local terms of the form
where
E is an infinite-dimensional Banach space,
is the Caputo fractional derivative of order
,
is a closed linear operator and
generates a uniformly bounded
semigroup
in
E,
, and
g is an appropriate continuous function so that it constitutes a non-local condition. They construct a new monotone iterative method for non-local problems of fractional evolution equations with mixed monotone non-local terms and obtain the existence of coupled extremal mild
L-quasi-solutions and the mild solution between them. In 2017, Mu et al. [
3] investigated some initial-boundary value problems for time-fractional diffusion equations, and some interesting versions of maximal and spatial regularity criteria are discussed. In 2020, Chen et al. [
4] discussed the Cauchy problem to a class of nonlinear time-fractional non-autonomous integro-differential evolution equations of mixed type via a measure of non-compactness in infinite-dimensional Banach spaces; further, Reference [
5] is concerned with the existence of mild solutions, as well as approximate controllability for a class of fractional evolution equations with non-local conditions in Banach spaces. In the Reference [
6], El-Borai and Debbouche established the existence and uniqueness of local mild, then local classical solutions of a class of nonlinear fractional integro-differential equations in Banach space with analytic semigroups. Luchko [
7] first obtained the maximum principle for fractional parabolic partial differential equations. In 2020, Zhang et al. [
8] discussed a class of fractional retarded differential equations involving mixed non-local plus local initial conditions.
On the other hand, we noticed that among the previous researchers, most of the researchers focus on the case when the nonlinear terms are without gradient terms, which means that the above authors have obtained results which cannot be used when the nonlinear terms are with gradient terms. Motivated by the above mentioned aspects, in this paper, we will investigate the existence and uniqueness of the saturated classical solutions to the following problems
subject to the boundary condition
and the initial condition
where
is a bounded domain in
(
) with smooth boundary
,
is a real number, and the symbol
represents the regularized Caputo fractional derivative of order
with respect to the time variable. The unknown function
, ∇ stands for the gradient operator with respect to the spatial variable, and the
is a given function. The nonlinear term
is continuous and
is a uniformly elliptic differential operator of divergence type in
with the coefficients
and
and
on
. That is,
is a positive definite symmetric matrix for every
and there exists a constant
, such that
The rest of this paper is organized as follows. In
Section 2, we present some preliminaries on fractional calculus, the definitions and properties of the fractional powers about sectorial operator, interpolation spaces, and some lemmas, which will be used in the proof of our main result.
Section 3 states and proves the saturated classical solutions and the positive classical solutions for the initial value problem of semi-linear sub-diffusion with gradient terms (1)–(3). In
Section 4, an example is given to illustrate the feasibility of our main results. In
Section 5, we present our future work.
2. Preliminaries
Let
X be a Banach space with norm
. We denote by
the Banach space of all continuous functions from interval
into
X equipped with the norm
. Assume
. The fractional integral of order
for the function
u is defined as
The standard Caputo fractional derivative of order
for the function
u is defined by
Further,
represents the regularized Caputo fractional derivative of order
for the function
u and is defined by
For
, the regularized Caputo time-fractional derivative of the function
u can generally be written as
For more insight into the topic, see Kilbas et al. [
9].
The Hölder space with exponent
from
into
X is denoted by
. For details of the Hölder space, the usual
pth power integrable function spaces:
and the usual Sobolev spaces:
, we refer to Adams [
10], Evans [
11], and Henry [
12] which are denoted by
the Banach space of all linear bounded operators in
X endowed with the topology defined by the operator norm. The
means that the Banach space
is continuously embedded in the Banach space
.
We introduce fractional Hölder spaces:
for an arbitrary fixed time
with norms
where
For details of the fractional Hölder spaces, see [
13].
In the latter discussion, we denote
and define the linear operator
with the domain
It is well-known that
A is a positive definite self-adjoint operator and
generates an analytic operator semigroup
in
X (see ([
14], Chapter 2, Sections 5 and 6)). Let
be the first eigenvalue of the operator
A under the Dirichlet boundary condition
. For any
, there is a constant
, such that
Clearly,
A is a sectorial operator. It is also well-known that each sectorial operator can define fractional powers. Next, we recall some concepts and conclusions on the fractional powers of
A. For
,
is defined by
where
is an analytic operator semigroup, which is generated by
.
is injective, and
can be defined by
with the domain
. For
, let
. We endow a norm
to
. Since
is a closed linear operator, it follows that
is a Banach space. We denote by
the Banach space
. Denote
,
, then
The space
is named the interpolation space between
and
or the fractional power space of the sectorial operator
A. For
,
is bounded and embedded into
, and the embedding
is compact when the operator
A exists for compact resolvent. For details of the properties of the fractional powers about the sectorial operator, refer to Carracedo and Alix [
15], Henry [
12], and Pazy [
14].
Lemma 1 ([
14])
. Let and , be the fractional power space of the sectorial operator A. These possess a constant , such that for every and , we haveand choosing , we have The interpolation space
and the Hölder space satisfies the following relationships:
Lemma 2 ([
12])
. Suppose Ω be an open, bounded domain in with smooth boundary . Then,- (1)
is compactly embedded in if for when
- (2)
is compactly embedded in if for when
We introduce the following Schauder theory of the linear fractional parabolic partial differential equations.
Lemma 3 ([
13])
. Let Ω be an open, bounded domain in with smooth boundary . For any fixed , the functions and and the compatibility conditionsis hold. Then the following initial boundary value problem of linear fractional parabolic type (LFP)possesses a unique classical solution: For any
, let
. The continuous embedding
implies that the map
is continuous. Now, we consider the initial value problem of the linear evolution equations in
X (LEE)
Definition 1. A function is said to be a mild solution of the LEE (5) if it satisfieswhere andare the functions of Wright type, defined in which satisfiesand For the definition of mild solution and the properties of operator families
and
, and more detailed results, the references [
2,
16,
17,
18] are available.
From the Reference [
18], we know that
and
are strongly continuous with
and
continuous in the uniform operator topology for
. From the reference [
3], it implies that
for
,
, where the mark
expresses the Riemann-Liouville fractional integral of order
with respect to
t, and
where
is a constant with respect to
and
. The proof of the Theorem 5.1 in Reference [
17] shows
and combining this with the above inequality and the mean value theorems implies
where
is a constant.
Lemma 4. Suppose and . Then, for any and , the mild solution u of the LEE (5) has the regularity:and the map is continuous from to Proof. Denote
. When
, for any
, combining with Definition 1, (9), (10), and (12) we have
Here, we see that
, that is,
is Hölder-continuous with exponent
. Next, we show that
is also Hölder-continuous. The previous proof implies
In view of the inequalities (13), (15) and the equality (14), we have
and
Therefore,
is also Hölder-continuous with exponent
. It follows that
and
Further, the map is continuous from to □
Lemma 5. Let Ω be an open, bounded domain in with smooth boundary and Then, for any and , the LEE (5) possesses a unique classical solution: And it is also a classical solution of LFP (4).
Proof. Obviously, we have the continuous embedding
It is well-known ([
6], Definition 2.1 and Theorem 4.1), when
and the
h is Hölder-continuous, the LEE (5) has a unique local classical solution:
expressed by
Taking
and
. Then, from the Lemma 4, the mild solutions of LEE (5) have the regularity:
and from (2) of Lemma 2, it follows that inclusion relation:
For the case
. We need the modified function
defined by
where for any
. From (16), we see that
Corresponding to the function
, we consider the initial boundary value problems of parabolic type
By Lemma 3, it is known that the problem (17) has a unique classical solution:
. Thus,
is also a classical solution of linear fractional evolution equations in
X
where the
is a zero element in Banach space
Then,
is a solution of problems (18). The uniqueness of the solution implies that
. Therefore,
We now deduce that
. Again, repeating the above regularization process, we obtain
The arbitrariness of
implies that
is a classical solution of LFP (4).
Setting in the process of the proof above, we directly reach the same conclusion as above. □
4. Applications
Using the result obtained in
Section 3, we can solve the initial boundary value problem to the following semi-linear sub-diffusion with gradient terms
where
is the regularized Caputo’s fractional derivative of order
,
is the Laplace operator, ∇ is the Gradient operator,
with
smooth boundary
,
and
Let
with the norm
. We define an operator
A in Banach space
E by
where
is the completion of the space
with respect to the norm
is the set of all continuous functions defined on
which have continuous partial derivatives of order less than or equal to 2,
is the completion of
with respect to the norm
, and
is the set of all functions
with compact supports on the domain
. It is well-known from [
14] that
generates a uniformly bounded analytic semigroup
in
E. From the definition of nonlinear term
f, we can easily verify that the assumptions (F1) and (F2) are satisfied with
and
. Therefore, by Theorem 2, we have the following existence result.
Theorem 3. The initial boundary value problem to the following semi-linear sub-diffusion with gradient terms (30) has a positive global solution It should be pointed out that this conclusion cannot be obtained from the known results of References [
1,
2,
3,
4,
5,
6,
8,
17].