Complex phenomena in the engineering and scientific fields are modeled utilizing the fractional differential equations (FDEs). Nowadays, the fractional calculus is an efficient tool for describing dynamic behavior of living systems and hereditary properties of various materials: the relaxation process in polymers [1], chaotic neuron model [2], longtime memory in financial time series via fractional Langevin equations [3], and tumor growth models [4] (see also references therein). We also refer to [5, 6], where the authors propose the advanced mathematical model for oxygen delivery to tissue through a capillary in both (transverse and longitudinal) directions. In these studies, conveying oxygen from a capillary to the surrounding tissue is described by means of a subdiffusion equation having two fractional derivatives in time, that is

with 0 < μ < ν < 1. Here, C represents the concentration of oxygen, τ is the time lag in concentration of oxygen along the capillaries (in the present model, this parameter is a positive constant), k is the rate of consumption per volume of tissue, and a₀ and b_i are the diffusion coefficients of oxygen. In addition, the term DtνC−τDtμC details the net diffusion of oxygen to all tissues.

In this equation, the symbol Dtθ stands for the Caputo fractional derivative with respect to time of order θ ∈ (0, 1),

where Γ is the Euler's Gamma function. An equivalent definition of this derivative in the case of absolutely continuous functions reads

In this art, motivated by the discussion above, we focus on the analytical study of the semilinear integro-differential equation with memory terms. Let Ω ⊂ ℝⁿ, n ≥ 2, be a bounded domain with a smooth boundary ∂Ω, and for any T > 0, we set

We consider the initial-value problems to the multi-term time-fractional semilinear diffusion equation in the unknown function u = u(x, t):Ω_T → ℝ,

subject to the following initial and boundary conditions:

where the abbreviations DBC and 3BC mean the Dirichlet boundary condition and the boundary condition of the third kind, respectively.

Here, c₀ is given positive number, g, u₀, φ_i are given functions, and K1 and K are prescribed memory kernels.

Here, the symbol * stands for the usual time-convolution product on (0, t),

The operator D_t is the linear combination of Caputo fractional derivatives with respect to time, namely

where ν ∈ (0, 1) and ν_i, μ_j ∈ (0, ν) are arbitrary but fixed, and ρ = ρ(x, t), ρ_i = ρ_i(x, t) and γ_j = γ_j(x, t) are given positive functions.

Coming to the remaining operators, Li, i = 1, 2, are linear elliptic operators of the second order with time-dependent coefficients, while M is a first-order differential operator. Their precise forms will be given in Sections 3, where we detail the main assumptions in the model.

Published works concerning the multi-term fractional diffusion/wave equations, i.e., the equation with the operator

with q_i being positive, and 0 ≤ ν₁ < ν₂ < ... < ν_M, are quite limited in spite of rich literature on their single-term version. Exact solutions of linear multi-term fractional diffusion equations with q_i being positive constants on bounded domains are searched employing eigenfunction expansions in Daftardar-Gejji and Bhalekar [7] and Morales-Delgado et al. [5]. We quote Srivastava and Rai [6] and Morales-Delgado et al. [5], where certain numerical solutions are constructed to the corresponding initial-boundary value problems to evolution equations with D_t given via Equation 1.4. Finally, we mention [8], where existence and non-existence of the mild solutions to the Cauchy problem for semilinear subdiffusion equation with the operator Equation 1.4 are discussed. In particular, the authors obtain the Fujita-type and Escobedo-Herrero-type critical exponents for this equation and the system. It is worth noting that, all these works concern to evolution equations with the operator Equation 1.4 which can be rewritten in the form of a generalized fractional derivative with a non-negative locally integrable kernel N(t), that is

Coming to the initial-boundary value problems associated with Equation 1.1 with the operator D_t given by Equation 1.3, we point out two principal differences with respect to the aforementioned articles. The first deals with the presence of Caputo fractional derivatives of the product of two functions: the desired solution u and the prescribed coefficients ρ, ρ_i, γ_j. Incidentally, we recall that the well-known Leibniz rule does not work in the case of fractional derivatives. The second distinction is that the operator D_t given by Equation 1.3 (under certain assumptions on the coefficients) can be represented in the form Equation 1.4 but with a negative kernel. Indeed, setting in Equation 1.3

where C_ρ is a positive constant, we have the representation

with

being negative for t>e-Cγ [see Janno and Kinash [9], Lemma 4]; C_γ is the Euler–Mascheroni constant. In fact, the non-negativity of the kernel N is a principal assumption in the aforementioned studies.

The linear version of Equations 1.1, 1.3 subject to various type boundary conditions with the coefficients in D_t being alternating sign is discussed in Pata et al. [10] and Vasylyeva [11]. For any fixed time T, the existence and uniqueness of a solution to semilinear problem (Equations 1.1, 1.3) with the Dirichlet or the Neumann boundary conditions are analyzed in Siryk and Vasylyeva [12] and Vasylyeva [11]. Namely, if the coefficients of the operator D_t are only time-dependent and non-decreasing functions, then the well-posedness of these problems in the fractional Hölder and Sobolev spaces is established in the one-dimensional case in Siryk and Vasylyeva [12]. As for the multidimensional case, the classical solvability of the Cauchy-Dirichlet problem to semilinear (Equation 1.1) in the case of two-term fractional derivatives in D_t (i.e., either M = 1, N = 0 or M = 0, N = 1) is proved in Vasylyeva [11]. In this study, the coefficients in Equation 1.1 are time and space dependent but instead of their non-decreasing in time, they have to satisfy more complex assumption. Indeed, if M = 1, N = 0, then the function ρρ1 should be decreasing. Finally, we remark that in Vasylyeva [11], the non-linear term is the local Lipschitz.

The goal of this study is finding sufficient conditions on the coefficients of the operator D_t, the order fractional derivatives ν, μ_j, and ν_i, j, i ≥ 1, and the non-linearity f which provide one-to-one classical and strong solvability (for any fixed T) in the case of the DBC or the 3BC. Actually, we consider two types of the non-linearity f(u). The first is f satisfying the local Lipschitz condition and having the linear growth. As for the second, f is a continuous differentiable on ℝ with a super-linear growth. For example, f is a polynomial of odd degree with the positive leading coefficient (see Giorgi et al. [13]). Coming to the coefficients in the fractional operator D_t, we discuss both the non-decreasing coefficients and the coefficients satisfying the properties of Theorem 2 [11].

We notice that the key ingredient in the proof of the classical solvability is the continuation approach, based on the introduction of a family of auxiliary problems depending on a parameter λ ∈ [0, 1]. Then, one has to produce a priori estimates in the fractional Hölder spaces for the solution which are independent of λ. One of the crucial points in the arguments is concerned to the estimates of ||u||C(Ω¯T), obtained via integral iteration technique adopted to the multi-term fractional case. As for the strong solvability, it is proved via the construction of this solution as a limit of approximate smooths solutions and exploiting a priori estimates in the Sobolev spaces.

Finally, we notice that assumptions on the coefficients and the memory kernels in the one-dimensional and multidimensional cases are different. It is related with using various approaches to get a priori estimates of the solutions if n = 1 and n ≥ 2. Namely, if n ≥ 2, we relax assumptions on the coefficients of D_t, in particular, we allow coefficients depending on time and space in Equation 1.3. However, we require more regular memory kernel in Equation 1.1, K∈C1([0,T]).

Throughout this study, the symbol C will denote a generic positive constant, depending only on the structural quantities of the problem.

In the course of our study, we will exploit the fractional Hölder and Sobolev spaces. To this end, in what follows, we take two arbitrary (but fixed) parameters

For any non-negative integer l, any p ≥ 1, s ≥ 0, and any Banach space (X, ||·||_X), we consider the usual spaces

Recall that for non-integer s, the space W^{s, p} is called Sobolev-Slobodeckii space [for its definition and properties see, e.g., Adams and Fournier [14], Chapter 1].

Denoting for β ∈ (0, 1)

Then, we assert the following definition.

Definition 2.1. A function v = v(x, t) belongs to the class Cl+α,l+α2ν(Ω̄T), for l = 0, 1, 2, if the function v and its corresponding derivatives are continuous and the norms here below are finite:

In a similar way, for l = 0, 1, 2, we introduce the space Cl+α,l+α2ν(∂ΩT).

The properties of these spaces have been discussed in Krasnoschok et al. [15] (Section 2). As for the limiting case ν = 1, these classes boil down to the usual parabolic Hölder spaces.

Finally, we will say that a function v defined in Ω_T belongs to ℌps1,s2(ΩT) with p > 1 and s₁, s₂ ≥ 0, if v∈Ws1,p((0,T),Lp(Ω))∩Lp((0,T),Ws2,p(Ω)), and the norm here below is finite

The space ℌps1,s2(∂ΩT) is defined in the similar manner.

First, we state additional requirements on the given data in Equations 1.1, 1.2.

where N_i is a component of the outward normal N = {N₁, ..., N_n} to Ω; the fractional operator D_t in Equation 1.1 is given by Equation 1.3.

There are positive constants 0 < δ₁ < δ₂, such that

for any (x,t,ξ)∈Ω¯T×ℝn.

Moreover, we require that

the relations hold

and there are positive constants δ, δ₃, δ₄, such that

for each (x,t)∈Ω¯T and for all i = 1, 2, ..., M, j = 1, 2, ..., N.

In addition, if N ≥ 1, then

where the function ρ0∈Cν0([0,T],C1(Ω¯)) is positive for all t ∈ [0, T] and x∈Ω¯.

Moreover, we require that the one of the following conditions holds:

(i) either ∂ρ∂t, ∂ρ0∂t, ∂ρi∂t, ∂γj∂t are non-negative for all (x,t)∈Ω¯T;

(ii) or

for all i = 1, ..., M, j = 1, ..., N, and any (x,t)∈Ω¯T.

if the DBC holds, and there is

in the 3BC case.

for any u₁, u₂ ∈ [−ϱ, ϱ]; and

there is a positive constant L, such that

Remark 3.1. It is apparent that if the positive functions ρ, ρ_i, γ_j are time-independent, then condition h3(i) boils down to h3(ii).

Example 3.1. The simplest example of the functions satisfying h3 is

where C0, Cj, C¯i are positive constants, such that

Now, we are in the position to state the one-valued classical solvability of Equations 1.1, 1.2.

Theorem 3.1. Let T > 0 be arbitrarily given, ∂Ω∈C2+α, n ≥ 2, and let assumptions h1–h6 hold. We assume that f(u) meets the requirement h7.I if N ≥ 1, while in the case of N = 0, f(u) satisfies h7. Then, initial-boundary value problem Equations 1.1, 1.2 admits a unique classical solution u = u(x, t) satisfying the regularity:

The next assertion is related to the strong solvability of Equations 1.1, 1.2.

Theorem 3.2. Let N = 0, n ≥ 2, ∂Ω∈C2+α, and let T > 0 be arbitrarily given. We assume that h1–h5 and h7 hold and

where p>max{n+2ν;1ν-νM}, r ≥ n + 1, s1∈(r-1,1), and s2∈((n+1)r-1,1). Moreover, in the DBC case, we require

Then, the initial-boundary value problem Equations 1.1, 1.2 admits a unique strong solution in the class ℌpν,2(ΩT).

Remark 3.2. Theorems 3.1 and 3.2 hold if D_tu in Equation 1.3 is changed by

where ρ, ρ_i, γ_j satisfies h3, but the requirement on the regularity of these functions can be relaxed. Namely, ρ,ρi,γj∈Cα,αν/2(Ω¯T).

The remaining part of this study is devoted to the verification of Theorems 3.1, 3.2. Here, we proceed with a detailed proof of Theorem 3.1 in the most difficult case, i.e., if N ≥ 1, M ≥ 1 in Equation 1.3. This means that the non-linear term f(u) satisfies h7.I. The verification of the remaining cases is simpler and repeats the main steps (with minor changes) in the arguments related with the cases N, M ≥ 1.

In this section, we collect some useful properties of fractional derivatives and integrals, as well as several preliminaries results that will be significant in our investigation. Throughout this art, for any θ > 0, we use the notation

and define the fractional Riemann-Liouville integral and the derivative of order θ, respectively, of a function v = v(x, t) with respect to time t as

where ⌈θ⌉ is the ceiling function of θ (i.e., the smallest integer greater than or equal to θ).

It is apparent that, for θ ∈ (0, 1), there holds

Accordingly, the Caputo fractional derivative of the order θ ∈ (0, 1) to the function v(x, t) can be represented as

provided that both derivatives exist.

At this point, we subsume [16, Proposition 4.1], [11, Proposition 1] as the following claim.

Proposition 4.1. The following hold.

(i) For any given positive numbers θ₁ and θ₂ and a summable kernel k = k(t), there are relations

Here, the positive constant C depends only on T, θ₁, and ||k||_{L1(0, T)}.

(ii) Let k(t)∈C1([0,T]), θ ∈ (0, 1), θ₁ ≥ 1, v=v(t)∈Cθ([0,T]), Dtθv(t)∈C([0,T]), w=w(t)∈Cθ1([0,T]). Then, the equality holds

The next result is key inequalities in the fractional calculus and includes [12, Proposition 5.1, Corollaries 5.2-5.3].

Proposition 4.2. The following holds.

(i) Let θ, θ₁ ∈ (0, 1) and θ₁ > θ/2, v∈Cθ1([0,T]). For any even integer p ≥ 2, the inequalities are true

If v is non-negative, then these bounds hold for any integer odd p.

(ii) Let 0 < θ₁ < θ ≤ 1, θ₂ ∈ (θ₁, 1), and v∈Cθ2([0,T]). Then, there is positive value T₁ = T₁(θ), such that the following inequalities hold:

where p meets requirements of (i).

At this point, for given functions w₁ and w₂, we define

and assert the results obtained in ([12], Proposition 5.5) and related to the fractional differentiation of the product.

Proposition 4.3. Let θ ∈ (0, 1), w1∈C1([0,T]), w2∈C([0,T]).

(i) If Dtθw2 belongs either to C([0,T]) or to L_p(0, T), p ≥ 2, then, there are equalities:

and Dtθ(w1w2), ∂tθ(w1w2) have the regularity:

(ii) For any θ₁ ≥ θ > 0 and each t ∈ [0, T], there hold

First, recasting step-by-step the proof of ([11], Theorem 1) and additionally exploiting [17, Theorem 3.4] and arguments leading to ([18], Theorem 4.1) in the 3BC case, we claim the following result.

Lemma 5.1. Let f(u) ≡ 0, n ≥ 2, ν, μ_j, ν_i satisfy h1, and

We require that

Moreover, in the DBC case, we additionally assume

Under assumptions h2–h5, the classical solution u∈C2+α,2+α2ν(Ω¯T) of Equations 1.1, 1.2 satisfies the estimate

where

Here, the generic constant C is independent of the right-hand sides in Equations 1.1, 1.2.

Our next result connects with a priori estimates in the fractional Hölder space to the function u satisfying the family of equations for each λ ∈ [0, 1]:

and homogeneous conditions Equation 1.2.

Lemma 5.2. Let assumptions of Theorem 3.1 hold, and

We assume also u∈C2+α,2+α2ν(Ω¯T) be solution to Equations 5.1, 5.2. Then, for any λ ∈ [0, 1], there are the following estimates:

The positive constant C is independent of λ and the right-hand sides of Equations 5.1, 5.2 and depends only on T and the structural parameters in the model.

First of all, we notice that estimate Equation 5.3 in this claim is verified with the standard Schauder technique and by means of ([10], Theorem 4.1) and bound Equation 5.2 in this art.

We focus on the proof of Equation 5.2 if DBC holds, the case of 3BC is analyzed by collecting the similar arguments with techniques leading to ([15], Lemma 5.3). We preliminary observe that verification of Equation 5.2 in the case of the absence of Dtμj(γju), j = 1, 2, ..., N, (i.e., N = 0) is simpler and recasts the main steps (with minor changes) in arguments related with N ≥ 1. Thus, here, we assume the presence of at least one fractional derivative Dtμj(γju) in the operator D_tu. Then, we will exploit the following strategy. Keeping in mind assumption h3, the homogeneous initial condition and relation Equation 4.1, we rewrite D_tu in the more suitable form:

where

Appealing to (ii) in Proposition 4.2, we introduce

such that the function Nj is strictly positive for all t∈[0,Tj*].

After that, for each fixed T₀:

we obtain the estimates

with the positive constant being independent of λ and T₀.

Then, we discuss the extension of these bounds to the interval (T₀, T] and reach the estimate Equation 5.2. It is worth noting that this step is absent in the case of N = 0, due to the proof of Equation 5.2 and consequently Equation 5.6 are carried out immediately on the entire time interval [0, T].

Step 1: Verification of Equation 5.6. Here, we focus on the obtaining of Equation 5.6 if h3(i) holds, the case of h3(ii) is analyzed with the similar arguments and is left to the interested readers.

Let K̄ be the conjugate kernel to K, its properties are described in ([15], Proposition 4.4), in particular,

Setting

and exploiting [15, Proposition 4.4] and Proposition 4.1, we rewrite Equation 5.1 in more suitable form

where