Initial Boundary Value Problems of Semi-Linear Sub-Diffusion with Gradient Terms

: We consider the existence and uniqueness of the saturated classical solutions and the positive classical solutions to initial boundary value problems of semi-linear sub-diffusion with gradient terms. Applying this to the fractional power of the sectorial operator theory and the imbedding theory in the interpolation spaces, where the nonlinear term satisﬁes more general conditions, we obtain the existence and uniqueness of the saturated classical solutions. The results obtained generalize the recent conclusions on this topic. Finally, an example is given to illustrate the feasibility of our main results.


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 (∆x) 2 scales as a nonlinear power law in time t, that is, (∆x) 2 ∼ K γ t γ , where K γ is a constant. When γ ∈ (0, 1), this is referred to as sub-diffusion; when γ > 1, 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 C D γ t u(t) + Au(t) = f (t, u(t)), t ∈ J = [0, a], u(0) = u 0 , where C D γ t is the Caputo fractional derivative of order γ ∈ (0, 1), A : D(A) ⊂ E → E is a closed linear operator and −A generates a uniformly bounded C 0 − semigroup T(t)(t ≥ 0) in E, f ∈ C(J × E × E, E) is a continuous nonlinear mapping, u 0 ∈ E. 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 C D γ t u(t) + Au(t) = f (t, u(t), u(t)), t ∈ J = [0, a], u(0) = g(u, u), where E is an infinite-dimensional Banach space, C D γ t is the Caputo fractional derivative of order γ ∈ (0, 1), A : D(A) ⊂ E → E is a closed linear operator and −A generates a uniformly bounded , 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 u| ∂Ω = 0, t ∈ (0, +∞), (2) and the initial condition where Ω is a bounded domain in R N (N ≥ 1) with smooth boundary ∂Ω, γ ∈ (0, 1) is a real number, and the symbol D γ t represents the regularized Caputo fractional derivative of order γ with respect to the time variable. The unknown function u = u(x, t) : Ω × (0, +∞) → R, ∇ stands for the gradient operator with respect to the spatial variable, and the ϕ(x) is a given function. The nonlinear term f : is a uniformly elliptic differential operator of divergence type in Ω with the coefficients a ij ∈ C 1+σ (Ω)(i, j = 1, 2, · · · , N) and a 0 ∈ C σ (Ω) and a 0 (x) ≥ 0 on Ω. That is, [a ij (x)] N×N is a positive definite symmetric matrix for every x ∈ Ω and there exists a constant > 0, 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.

Preliminaries
Let X be a Banach space with norm || · ||. We denote by C([0, T], X) the Banach space of all continuous functions from interval [0, T] into X equipped with the norm ||u|| C = max t∈[0,T] ||u(t)||. Assume u : [0, ∞) → X. The fractional integral of order γ > 0 for the function u is defined as The standard Caputo fractional derivative of order 0 < γ ≤ 1 for the function u is defined by Further, D γ t represents the regularized Caputo fractional derivative of order 0 < γ ≤ 1 for the function u and is defined by For u : [0, ∞) × R N → R, the regularized Caputo time-fractional derivative of the function u can generally be written as D For more insight into the topic, see Kilbas et al. [9]. The Hölder space with exponent σ from Ω into X is denoted by C m+σ (Ω, X)(m = 0, 1, 2 · · · ; 0 < σ < 1). For details of the Hölder space, the usual pth power integrable function spaces: L p (Ω)(p ≥ 1) and the usual Sobolev spaces: W m,p (Ω), we refer to Adams [10], Evans [11], and Henry [12] which are denoted by B(X) the Banach space of all linear bounded operators in X endowed with the topology defined by the operator norm. The X 1 → X 2 means that the Banach space X 1 is continuously embedded in the Banach space X 2 .
We introduce fractional Hölder spaces: for an arbitrary fixed time T > 0 with norms It is obvious fact that For details of the fractional Hölder spaces, see [13].
In the latter discussion, we denote and define the linear operator . It is well-known that A is a positive definite self-adjoint operator and −A generates an analytic operator semigroup T(t)(t ≥ 0) in X (see ([14], Chapter 2, Sections 5 and 6)). Let λ 1 be the first eigenvalue of the operator A under the Dirichlet boundary condition u| ∂Ω = 0. For any 0 < δ < λ 1 , there is a constant M ≥ 1, 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 α > 0, A −α is defined by The space X α is named the interpolation space between X 0 and X 1 or the fractional power space of the sectorial operator A. For 0 ≤ α 1 < α 2 , X α 2 is bounded and embedded into X α 1 , and the embedding X α 2 → X α 1 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 0 ≤ α 1 < α 2 ≤ 1 and X α 1 , X α 2 be the fractional power space of the sectorial operator A. These possess a constant c = c(α 1 , α 2 ), such that for every x ∈ X α 2 and ε > 0, we have The interpolation space X α and the Hölder space satisfies the following relationships: We introduce the following Schauder theory of the linear fractional parabolic partial differential equations.
) and the compatibility conditions is hold. Then the following initial boundary value problem of linear fractional parabolic type (LFP) possesses a unique classical solution: For any t ∈ [0, T], let h(t) = h(·, t). The continuous embedding C µ (Ω) → X implies that the map h : [0, T] → X is continuous. Now, we consider the initial value problem of the linear evolution where h ∈ C([0, T], X) and are the functions of Wright type, defined in (0, ∞) which satisfies For the definition of mild solution and the properties of operator families {S γ (t)} t≥0 and {P γ (t)} t≥0 , and more detailed results, the references [2,[16][17][18] are available.
From the Reference [18], we know that {S γ (t)} t≥0 and {P γ (t)} t≥0 are strongly continuous with S γ (t) and P γ (t) continuous in the uniform operator topology for t > 0. From the reference [3], it implies that for t > 0, x ∈ X, where the mark J 1−γ t expresses the Riemann-Liouville fractional integral of order 1 − γ with respect to t, and where C α,γ is a constant with respect to α and γ. 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 C γ is a constant.

Proof. Obviously, we have the continuous embedding
It is well-known ([6], Definition 2.1 and Theorem 4.1), when ϕ ∈ X α and the h is Hölder-continuous, the LEE (5) has a unique local classical solution: Taking α = 1 2 (α + N 2p ) and σ = α − N 2p . 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 σ < µ. We need the modified function where for any ε ∈ (0, T]. From (16), we see that Corresponding to the function h 1 , 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: (18) where the θ is a zero element in Banach space X. Then, φ ε u is a solution of problems (18).

The uniqueness of the solution implies that
. Again, repeating the above regularization process, we obtain Setting σ = µ in the process of the proof above, we directly reach the same conclusion as above.
Then, the problem (1)-(3) can be rewritten as the following initial value problems of the evolution equations with sectorial operator (IVPS)

(19)
It is well-known ( [6], Definition 2.1 and Theorem 4.1) that the basic theory of the evolution equations with sectorial operator implies the IVPS (19) possesses a unique local solution: Next, we need to show that u ∈ FC 2+µ,(1+ µ 2 )γ (Ω × (0, T]); however, from the Lemma 5 we know that it is obvious. Similar to the Reference [1], applied to piecewise extended method for IVPS (19), we easily obtain the FPDE (1)-(3) which possesses a unique saturated classical solution: is the interpolation space between X 0 and X 1 . This completes the proof of Theorem 1.
For case: ρ = 1, the conclusion of Theorem 2 is obvious.

Conclusions
In this work, existence and uniqueness of saturated classical solutions and the positive classical solutions to initial boundary value problems of the semi-linear fractional parabolic partial differential equations with gradient terms was obtained. Furthermore, the results obtained in this paper are a supplement to the existing literature and essentially extends some existing results in this area. Our future work will be devoted to formulating conditions that guarantee the existence and uniqueness of classical solutions and the positive classical solutions for initial boundary value problems of semi-linear sub-diffusion with gradient terms.
Author Contributions: Y.G. and Y.L. carried out the first draft of this manuscript, Y.G. prepared the final version of the manuscript. All authors have read and agreed to the published version of the manuscript.

Funding:
The research is supported by the National Natural Science Function of China (No. 11661071).