Comparison principles for the time-fractional diffusion equations with the Robin boundary conditions. Part I: Linear equations

The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type time-fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we discuss the uniqueness and existence of solutions to these initial-boundary value problems. Under some suitable conditions on the problem data, we then prove positivity of the solutions. Based on these results, several comparison principles for the solutions to the initial-boundary value problems for the linear time-fractional diffusion equations are derived.


Introduction
In this paper, we deal with a linear time-fractional diffusion equation in the form ∂ i (a ij (x)∂ j u(x, t)) b j (x, t)∂ j u(x, t) + c(x, t)u(x, t) + F (x, t), x ∈ Ω, 0 < t < T, (1.1) where ∂ α t is the Caputo fractional derivative of order α ∈ (0, 1) defined on the fractional Sobolev spaces (see Section 2 for the details) and Ω ⊂ R d , d = 1, 2, 3 is a bounded domain with a smooth boundary ∂Ω.All the functions under consideration are supposed to be real-valued.
For partial differential equations of the parabolic type that correspond to the case α = 1 in the equation (1.1), several important qualitative properties of solutions to the corresponding initial-boundary value problems are known.In particular, we mention a maximum principle and a comparison principle for the solutions to these problems ( [27], [28]).
The main purpose of this paper is the comparison principles for the linear time-fractional diffusion equation (1.1) with the Neumann or the Robin boundary conditions.
For the equations of type (1.1) with the Dirichlet boundary conditions, the maximum principles in different formulations were derived and used in [3,16,17,18,19,20,21,22,32].For a maximum principle for the time-fractional transport equations we refer to [23].In [11], a maximum principle for the more general space-and time-space-fractional partial differential equations has been derived.
Because any maximum principle involves the Dirichlet boundary values, its formulation in the case of the Neumann or Robin boundary conditions requires more cares.For this kind of the boundary conditions, both positivity of solutions and the comparison principles can be derived under some suitable restrictions on the problem data.One typical result of this sort says that the solution u to the equation (1.1) with the boundary condition (1.4) or (1.5) and an appropriately formulated initial condition is non-negative in Ω × (0, T ) if the initial value a and the non-homogeneous term F are non-negative in Ω and in Ω × (0, T ), respectively.Such positivity properties and their applications have been intensively discussed and used for the partial differential equations of parabolic type (α = 1 in the equation (1.1)), see, e.g., [4], [5], [24], or [28].
However, to the best knowledge of the authors, no results of this kind have been published for the time-fractional diffusion equations in the case of the Neumann or Robin boundary conditions.The main subject of this paper is in derivation of a positivity property and the comparison principles for the linear equation (1.1) with the boundary condition (1.4) or (1.5) and an appropriately formulated initial condition.In the subsequent paper, these result will be extended to the case of the semilinear time-fractional diffusion equations.The arguments employed in these papers rely on an operator theoretical approach to the fractional integrals and derivatives in the fractional Sobolev spaces that is an extension of the theory well-known in the case α = 1, see, e.g., [10], [25], [29].We also refer to the recent publications [2] and [15] devoted to the comparison principles for solutions to the fractional differential inequalities with the general fractional derivatives and for solutions to the ordinary fractional differential equations, respectively.The rest of this paper is organized as follows.In Section 2, some important results regarding the unique existence of solutions to the initial-boundary value problems for the linear time-fractional diffusion equations are presented.Section 3 is devoted to a proof of a key lemma that is a basis for the proofs of the comparison principles for the linear and semilinear time-fractional diffusion equations.The lemma asserts that each solution to (1.1) is non-negative in Ω × (0, T ) if a ≥ 0 and F ≥ 0, provided that u is assumed to satisfy some extra regularity.In Section 4, we prove a comparison principle that is our main result for the problem (1.1) for the linear time-fractional diffusion equation.Moreover, we establish the order-preserving properties for other problem data (the zeroth-order coefficient c of the equation and the coefficient σ of the Robin condition).Finally, a detailed proof of an important auxiliary statement is presented in an Appendix.

Well-posedness results
For x ∈ Ω, 0 < t < T , we define an operator ∂ i (a ij (x)∂ j v(x, t)+ d j=1 b j (x, t)∂ j v(x, t)+c(x, t)v(x, t) (2. 1) and assume that the conditions (1.2) for the coefficients a ij , b j , c are satisfied.
In this section, we deal with the following initial-boundary value problem for the linear time-fractional diffusion equation (1.1) with the time-fractional derivative of order α ∈ (0, 1) along with the initial condition (2.3) formulated below.
To appropriately define the Caputo fractional derivative d α t w(t), 0 < α < 1, we start with its definition on the space that reads as follows: Then we extend this operator from the domain D(d α t ) := 0 C 1 [0, T ] to L 2 (0, T ) taking into account its closability ( [31]).As have been shown in [12], there exists a unique minimum closed extension of d α t with the domain D(d α t ) = 0 C 1 [0, T ].Moreover, the domain of this extension is the closure of 0 C 1 [0, T ] in the Sobolev-Slobodeckij space H α (0, T ).Let us recall that the norm • H α (0,T ) of the Sobolev-Slobodeckij space H α (0, T ) is defined as follows ( [1]): .
By setting we obtain ( [12]) and In what follows, we also use the Riemann-Liouville fractional integral operator J β , β > 0 defined by Then, according to [9] and [12], Next we define As have been shown in [9] and [12], there exists a constant C > 0 depending only on α such that Now we can introduce a suitable form of initial condition for the problem (2.2) as follows and write down a complete formulation of an initial-boundary value problem for the linear time-fractional diffusion equation (1.1): (2.4) It is worth mentioning that the term ∂ α t (u(x, t) − a(x)) in the first line of (2.4) is well-defined due to inclusion formulated in the third line of (2.4).In particular, for 1 2 < α < 1, the Sobolev embedding leads to the inclusions ) and thus in this case the initial condition can be formulated as u(•, 0) = a in L 2 -sense.Moreover, for sufficiently smooth functions a and F , the solution to (2.4) can be proved to satisfy the initial condition in a usual sense: lim t→0 u(•, t) = a in L 2 (Ω) (see Lemma 4 in Section 3).Consequently, the third line of (2.4) can be interpreted as a generalized initial condition.
In the following theorem, a fundamental result regarding the unique existence of the solution to the initial-boundary value problem (2.4) is presented.
Moreover, there exists a constant C > 0 such that Before starting with a proof of Theorem 1, we introduce some notations and derive several helpful results needed for the proof.
For an arbitrary constant c 0 > 0, we define an elliptic operator A 0 as follows: (2.5) We recall that in the definition (2.5), σ is a smooth function, the inequality σ(x) ≥ 0, x ∈ ∂Ω holds true, and the coefficients a ij satisfy the conditions (1.2).
Henceforth, by • and (•, •) we denote the standard norm and the scalar product in L 2 (Ω), respectively.It is well-known that the operator A 0 is selfadjoint and its resolvent is a compact operator.Moreover, for a sufficiently large constant c 0 > 0, by Lemma 6 in Section 5, we can verify that A 0 is positive definite.Therefore, by choosing the constant c 0 > 0 large enough, the spectrum of A 0 consists entirely of discrete positive eigenvalues 0 < λ 1 ≤ λ 2 ≤ • • • , which are numbered according to their multiplicities and λ n → ∞ as n → ∞.Let ϕ n be an eigenvector corresponding to the eigenvalue λ n such that Aϕ n = λ n ϕ n and (ϕ n , ϕ m ) = 0 if n = m and (ϕ n , ϕ n ) = 1.Then the system {ϕ n } n∈N of the eigenvectors forms an orthonormal basis in L 2 (Ω) and for any γ ≥ 0 we can define the fractional powers A γ 0 of the operator A 0 by the following relation (see, e.g., [25]): .
Our proof of Theorem 1 is similar to the one presented in [9], [12] for the case of the homogeneous Dirichlet boundary condition.In particular, we employ the operators S(t) and K(t) defined by ( [9], [12]) and (2.7) In the above formulas, E α,β (z) denotes the Mittag-Leffler function defined by a convergent series as follows: It follows directly from the definitions given above that A γ 0 K(t)a = K(t)A γ 0 a and A γ 0 S(t)a = S(t)A γ 0 a for a ∈ D(A γ 0 ).Moreover, the inequality (see, e.g., Theorem 1.6 (p.35) in [26]) ( In order to shorten the notations and to focus on the dependence on the time variable t, henceforth we sometimes omit the variable x in the functions of two variables x and t and write, say, u(t) instead of u(•, t).Due to the inequalities (2.8), the estimations provided in the formulation of Theorem 1 can be derived as in the case of the fractional powers of generators of the analytic semigroups ( [10]).To do this, we first formulate and prove the following lemma: Lemma 1 Under the conditions formulated above, the following estimates hold true for F ∈ L 2 (0, T ; L 2 (Ω)) and a ∈ L 2 (Ω): Proof We start with proving the estimate (i).By (2.7), we have Therefore, using the Parseval equality and the Young inequality for the convolution, we obtain Then we employ the representation and the complete monotonicity of the Mittag-Leffler function ( [8]) Hence, Now we proceed with proving the estimate (ii).For 0 < t < T, n ∈ N and f ∈ L 2 (0, T ), we set First we prove that (2.11) In order to prove this, we apply the Riemann-Liouville fractional integral J α to L n f and get the representation By direct calculations, using (2.9), we obtain the formula .
Therefore, we have the relation ) and thus the last formula can be rewritten in the form Using the inequality (2.10), we obtain Therefore, Thus, the estimate (2.11) is proved.Now we set f n (s) := (F (s), ϕ n ) for 0 < s < T and n ∈ N. Since we obtain By applying (2.11), we get the following chain of inequalities and equations: Thus, the proof of the estimate (ii) is completed.The estimate (iii) from Lemma 1 follows from the standard estimates of the operator S(t).It can be derived by the same arguments as those that were employed in Section 6 of Chapter 4 in [12] for the case of the homogeneous Dirichlet boundary condition and we omit here the technical details.Now we proceed to the proof of Theorem 1.
Proof In the first line of the problem (2.4), we regard the expressions d j=1 b j (x, t)∂ j u and c(x, t)u as some non-homogeneous terms.Then this problem can be rewritten in terms of the operator A 0 as follows (2.12) In its turn, the first line of (2.12) can be represented in the form ( [9], [12]) ) and the equation (2.13), then u is a solution to the problem (2.12).With the notations [6]), the estimate (2.8) implies and thus Consequently, the inclusion S(t)a ∈ L 2 (0, T ; H 2 (Ω)) holds valid.For 0 < t < T , we next estimate Rv(•, t) H 2 (Ω) for v(•, t) ∈ D(A 0 ) as follows: For derivation of this estimate, we employed the inequalities and (c(s Repeating this argumentation m-times, we obtain Applying the Young inequality to the integral at the right-hand side of the last estimate, we arrive to the inequality Employing the known asymptotic behavior of the gamma function, we obtain the relation that means that for sufficiently large m ∈ N, the mapping is a contraction.Hence, by the Banach fixed point theorem, the equation (2.13) possesses a unique fixed point.Therefore, by the first equation in (2.4), we obtain the inclusion ), we finally obtain the estimate The proof of Theorem 1 is completed.
Comparison principles for the time-fractional diffusion equations . . .

3 Key lemma
For derivation of the comparison principles for solutions to the initial-boundary value problems for the linear and semilinear time-fractional diffusion equations, we need some auxiliary results that are formulated and proved in this section.
In addition to the operator −A 0 defined by (2.5), we define an elliptic operator −A 1 with a positive zeroth-order coefficient: and min (x,t)∈Ω×[0,T ] b 0 (x, t) is sufficiently large.We also recall that for y ∈ W 1,1 (0, T ), the pointwise Caputo derivative d α t is defined by In what follows, we employ an extremum principle for the Caputo fractional derivative formulated below.
If the function y = y(t) attains its minimum over the interval [0, T ] at the point t 0 ∈ (0, T ], then d α t y(t 0 ) ≤ 0. In Lemma 2, the assumption t 0 > 0 is essential.This lemma was formulated and proved in [16] under a weaker regularity condition posed on the function y, but for our arguments we can assume that y ∈ C[0, T ] and Employing Lemma 2, we now formulate and prove our key lemma that is a basis for further derivations in this paper.
Lemma 3 (Positivity of a smooth solution) For F ∈ L 2 (0, T ; L 2 (Ω)) and a ∈ H 1 (Ω), let F (x, t) ≥ 0, (x, t) ∈ Ω × (0, T ), a(x) ≥ 0, x ∈ Ω, and min (x,t)∈Ω×[0,T ] b 0 (x, t) be a sufficiently large positive constant.Furthermore, we assume that there exists a solution u ∈ C([0, T ]; Then the solution u is non-negative: For the partial differential equations of parabolic type with the Robin boundary condition (α = 1 in (3.3)), a similar positivity property is wellknown.However, it is worth mentioning that the regularity of the solution to the problem (3.3) at the point t = 0 is a more delicate question compared to the one in the case α = 1.In particular, we cannot expect the inclusion u(x, •) ∈ C 1 [0, T ].This can be illustrated by a simple example of the equation Proof First we introduce an auxiliary function ψ ∈ C 1 ([0, T ]; C 2 (Ω)) that satisfies the conditions Proving existence of such function ψ is non-trivial.In this section, we focus on the proof of the lemma and then come back to the problem (3.4) in Appendix.Now, choosing M > 0 sufficiently large and ε > 0 sufficiently small, we set For a fixed x ∈ Ω, by the assumption on the regularity of u, we have the inclusion On the other hand, for w ∈ H α (0, T ) ∩ W 1,1 (0, T ) and w(0) = 0, the equality ∂ α t w = d α t w = d α t (w + c) holds true with any constant c (see, e.g., Theorem 2.4 of Chapter 2 in [12]).
Moreover, because of the relation ∂ νA w = ∂ νA u + ε∂ νA ψ, we obtain the following estimate: (3.9) Evaluation of the representation (3.5) at the point t = 0 immediately leads to the formula Let us assume that the inequality min w(x, t) ≥ 0 does not hold valid, that is, there exists a point (x Since M > 0 is sufficiently large and u(x, 0) is non-negative, we obtain the inequality and thus t 0 cannot be zero.Next, we show that x 0 ∈ ∂Ω.Indeed, let us assume that x 0 ∈ ∂Ω.Then the estimate (3.9) yields that ∂ νA w(x 0 , t 0 ) + σ(x 0 )w(x 0 , t 0 ) ≥ ε.By (3.10) and σ(x 0 ) ≥ 0, we obtain (3.11) Here A(x) = (a ij (x)) 1≤i,j≤d and [b] i means the i-th element of a vector b.
For sufficiently small ε 0 > 0 and x 0 ∈ ∂Ω, we now verify the inclusion Indeed, since the matrix A(x 0 ) is positive-definite, the inequality holds true.In other words, the inequality is satisfied.Because the boundary ∂Ω is smooth, the domain Ω is locally located on one side of ∂Ω.In a small neighborhood of the point x 0 ∈ ∂Ω, the boundary ∂Ω can be described in the local coordinates composed of its tangential component in R d−1 and the normal component along ν(x 0 ).Consequently, if y ∈ R d satisfies the inequality ∠(ν(x 0 ), y − x 0 ) > π 2 , then y ∈ Ω.Therefore, for a sufficiently small ε 0 > 0, the point x 0 − ε 0 A(x 0 )ν(x 0 ) is located in Ω and we have proved the inclusion (3.12).
According to (3.10), the function w attains its minimum at the point (x 0 , t 0 ).Because 0 < t 0 ≤ T , Lemma 2 yields the inequality Since x 0 ∈ Ω, the necessary condition for an extremum point leads to the equality ∇w(x 0 , t 0 ) = 0. (3.15) Moreover, because the function w attains its minimum at the point x 0 ∈ Ω, in view of the sign of the Hessian, the inequality holds true (see, e.g., the proof of Lemma 1 in Section 1 of Chapter 2 in [5]).
Let us finally mention that the positivity of the function b 0 from the definition of the operator −A 1 is an essential condition for validity of our proof of Lemma 3.However, in the next section, we remove this condition while deriving the comparison principles for the solutions to the initial-boundary value problem (2.4).

Comparison principles
According to the results formulated in Theorem 1, in this section, we consider the solutions to the initial-boundary value problem (2.4) that belong to the following space of functions: In what follows, by u(F, a) we denote the solution to the problem (2.4) with the initial data a and the source function F .Our first result concerning the comparison principles for the solutions to the initial-boundary value problems for the linear time-fractional diffusion equation is presented in the next theorem.
Let us emphasize that the non-negativity of the solution u to the problem (2.4) holds true for the space Y α and thus u does not necessarily satisfy the inclusions u ∈ C([0, T ]; C 2 (Ω)) and t 1−α ∂ t u ∈ C([0, T ]; C(Ω)).Therefore, Theorem 2 is widely applicable.Before presenting its proof, let us discuss one of its corollaries in form of a comparison property: Then the inequality , we immediately obtain the inequalities a(x) ≥ 0, x ∈ Ω and F (x, t) ≥ 0, (x, t) ∈ Ω × (0, T ) and Therefore, Theorem 2 implies that u(x, t) ≥ 0, (x, t) ∈ Ω × (0, T ), that is, In its turn, Corollary 1 can be applied for derivation of the lower and upper bounds for the solutions to the initial-boundary value problem (2.4) by suitably choosing the initial values and the source functions.Let us demonstrate this technique on an example.
Then the solution u(F, 0) can be estimated from below as follows: Indeed, it is easy to verify that the function Due to the inequality F (x, t) ≥ δt β , (x, t) ∈ Ω ×(0, T ), we can apply Corollary 1 to the solutions u and u and the inequality (4.2) immediately follows.
In particular, for the spatial dimensions d ≤ 3, the Sobolev embedding theorem leads to the inclusion u ∈ L 2 (0, T ; H 2 (Ω)) ⊂ L 2 (0, T ; C(Ω)) and thus the strict inequality u(F, 0)(x, t) > 0 holds true for almost all t > 0 and all x ∈ Ω.Now we proceed to the proof of Theorem 2.
Proof In the proof, we employ the operators Qv(t) and G(t) defined by (2.14).In terms of these operators, the solution u(t) := u(F, a)(t) to the initialboundary problem (2.4) satisfies the integral equation For readers' convenience, we split the proof into three parts.
I. First part of the proof: existence of a smoother solution.
In this part of the proof, we show that for a ∈ C ∞ 0 (Ω) and F ∈ C ∞ 0 (Ω × (0, T )), the solution to the problem (2.4) satisfies the regularity assumptions formulated in Lemma 3.
More precisely, we first prove the following lemma: Lemma 4 Let a ij , b j , c satisfy the conditions (1.2) and the inclusions a ∈ C ∞ 0 (Ω), F ∈ C ∞ 0 (Ω × (0, T )) hold true.Then the solution u = u(F, a) to the problem (2.4) satisfies the inclusions Proof We recall that c 0 > 0 is a positive fixed constant and ([6]).Moreover, for the operators S(t) and K(t) defined by (2.6) and (2.7), the estimates (2.8) hold true.
In what follows, we denote ∂u ∂t (•, t) by u ′ (t) = du dt (t) if there is no fear of confusion.
The solution u to the integral equation ( 4.3) can be constructed as a fixed point of the equation As already proved, this fixed point satisfies the inclusion u ∈ L 2 (0, T ; Now we derive some estimates for the norms A κ 0 u(t) , κ = 1, 2 and A 0 u ′ (t) for 0 < t < T .First we set Since F ∈ C ∞ 0 (Ω × (0, T )), we obtain the inclusion F ∈ L ∞ (0, T ; D(A 2 0 )) and the inequality D < +∞.Moreover, in view of (2.8), for κ = 1, 2, we get the estimates The regularity conditions (1.2) lead to the estimates by using the inequalities (2.8).Then

u(s) ds
Comparison principles for the time-fractional diffusion equations . . .
The generalized Gronwall inequality yields the estimate which implies the inequality Next, for the space C([0, T ]; L 2 (Ω)), we can repeat the same arguments as the ones employed for the iterations R n of the operator R in the proof of Theorem 1 and apply the fixed point theorem to the equation ( 4.3) that leads to the inclusion A 0 u ∈ C([0, T ]; L 2 (Ω)).The obtained results implicate Choosing ε 0 > 0 sufficiently small, we have the equation Next, according to [6], the inclusion ) ⊂ H ). By (2.8), we obtain the inequality which leads to the estimate because of the inequality which follows from the regularity conditions (1.2) posed on the coefficients b j , c.For 0 < t < T , the generalized Gronwall inequality applied to the integral inequality (4.8) yields the estimate For the relation (4.7), we repeat the same arguments as the ones employed in the proof of Theorem 1 to estimate A Summarising the estimates derived above, we have shown that Next we estimate the norm Au ′ (t) .First, u ′ (t) is represented in the form Similarly to the arguments applied for derivation of (4.5), we obtain the inequality ) follows from the regularity conditions (1.2) and the inclusion a ∈ C ∞ 0 (Ω).Furthermore, by (2.7) and (2.8), we obtain Hence, the representation (4.10) leads to the estimate Now we consider a vector space It is easy to verify that X with the norm v X defined above is a Banach space.

2) we have the inclusions
Now we use the conditions (1.2) and (2.8) and repeat the arguments employed for derivation of (4.5) by means of (4.6) and (4.11) to obtain the estimates and the inclusion Therefore, that is, On the other hand, the estimate (4.9) implies For further arguments, we define the Schauder spaces C θ (Ω) and C 2+θ (Ω) with 0 < θ < 1 (see, e.g., [7], [13]) as follows: A function w is said to belong to the space For w ∈ C θ (Ω), we define the norm and for w ∈ C 2+θ (Ω), the norm is given by In the last formula, the notations Therefore, in view of (4.14), we obtain the inclusion h := A 0 u(•, t) ∈ C θ (Ω) for each t ∈ [0, T ].Now we apply the Schauder estimate (see, e.g., [7] or [13]) for solutions to the elliptic boundary value problem This inclusion and (4.11) yield the conclusion u ∈ C([0, T ]; C 2 (Ω)) and Finally we prove that lim t→0 u(t) − a = 0.By (2.8), we have and so for each h ∈ L ∞ (0, T ; L 2 (Ω)).Therefore by the regularity u ∈ C([0, T ]; C 2 (Ω)), we see that where R is defined in (2.14).Moreover, for justifying (4.12), we have already proved lim t→0 S(t)a − a = 0 for a ∈ L 2 (Ω).Thus the proof of Lemma 4 is complete.
II. Second part of the proof.
In this part, we weaken the regularity conditions posed on the solution u to (3.3) in Lemma 3 and prove the same results provided that u ∈ L 2 (0, T ; H 2 (Ω)) and u−a ∈ H α (0, T ; L 2 (Ω)), under the assumption that min ) and a ∈ H 1 (Ω) satisfy the inequalities F (x, t) ≥ 0, (x, t) ∈ Ω × (0, T ) and a(x) ≥ 0, x ∈ Ω.Now we apply the standard mollification procedure (see, e.g., [1]) and construct the sequences and thus Lemma 3 ensures the inequalities Since Theorem 1 holds true for the initial-boundary value problem (3.3) with F and a replaced by F − F n and a − a n , respectively, we have for almost all (x, t) ∈ Ω × (0, T ).Then the inequality (4.16) leads to the desired result, namely, to the inequality u(F, a)(x, t) ≥ 0 for almost all (x, t) ∈ Ω × (0, T ).
III. Third part of the proof.
Let the inequalities a(x) ≥ 0, x ∈ Ω and F (x, t) ≥ 0, (x, t) ∈ Ω × (0, T ) hold true for a ∈ H 1 (Ω) and F ∈ L 2 (0, T ; L 2 (Ω)) and let u = u(F, a) ∈ L 2 (0, T ; H 2 (Ω)) is a solution to the problem (2.4).In order to complete the proof of Theorem 2, we have to demonstrate the non-negativity of the solution without any assumptions on the sign of the zeroth-order coefficient.
First, the zeroth-order coefficient b 0 (x, t) in the definition (3.1) of the operator −A 1 is set to a constant b 0 > 0 that is assumed to be sufficiently large.In this case, the initial-boundary value problem (2.4) can be rewritten as follows: In what follows, we choose sufficiently large b 0 > 0 such that b 0 ≥ c C(Ω×[0,T ]) .
In the previous parts of the proof, we already interpreted the solution u as a unique fixed point for the equation (4.3).Now let us construct an appropriate approximating sequence u n , n ∈ N for the fixed point u.First we set u 0 (x, t) := 0 for (x, t) ∈ Ω × (0, T ) and u 1 (x, t) = a(x) ≥ 0, (x, t) ∈ Ω × (0, T ).Then we define a sequence u n+1 , n ∈ N of solutions to the following initial-boundary value problems with the given u n : Indeed, the inequality (4.19) holds for n = 1.Now we assume that u n (x, t) ≥ 0, (x, t) ∈ Ω×(0, T ).Then (b 0 +c(x, t))u n (x, t)+F (x, t) ≥ 0, (x, t) ∈ Ω×(0, T ), and thus by the results established in the second part of the proof of Theorem 2, we obtain the inequality u n+1 (x, t) ≥ 0, (x, t) ∈ Ω × (0, T ).By the principle of mathematical induction, the inequality (4.19) holds true for all n ∈ N. Now we rewrite the problem (4.18) as where A 0 and Q(t) are defined by (2.5) and (2.14), respectively.Next we estimate w n+1 := u n+1 − u n .By the relation (4.18), w n+1 is a solution to the problem In terms of the operator K(t) defined by (2.7), acting similarly to our analysis of the fixed point equation ( 4.3), we obtain the integral equation 0 w n (s) ds for 0 < t < T .
For their derivation, we used the norm estimates that hold true under the conditions (1.2).Thus we arrive at the integral inequality

The generalized Gronwall inequality yields now the estimate
The second term at the right-hand side of the last inequality can be represented as follows: 0 w n (s) ds.
Thus, we can choose a constant C > 0 depending on α and T , such that Recalling that and setting η n (t) := A 1 2 0 w n (t) , we can rewrite (4.20) in the form Since the Riemann-Liouville integral J 1 2 α preserves the sign and the semigroup property J β1 (J β2 η)(t) = J β1+β2 η(t) is valid for any β 1 , β 2 > 0, applying the inequality (4.21) repeatedly, we obtain the estimates .
Lemma 5 Let the elliptic operator −A be defined by (2.1) and the conditions (1.2) be satisfied.Moreover, let the inequality c(x, t) < 0 for x ∈ Ω and 0 ≤ t ≤ T hold true and there exist a constant σ 0 > 0 such that σ(x) ≥ σ 0 for all x ∈ ∂Ω.
In the formulation of this lemma, at the expense of the extra condition σ(x) > 0 on ∂Ω, we do not assume that min (−c(x, t)) is sufficiently large.This is the main difference between the conditions supposed in Lemma 5 and in Lemma 3. The proof of Lemma 5 is much simpler compared to the one of Lemma 3; it will be presented at the end of this section.Now we complete the proof of Theorem 3. Since c(x, t) < 0 for (x, t) ∈ Ω × (0, T ) and σ 1 (x) ≥ σ 0 > 0 on ∂Ω and taking into account the conditions (4.24) and (4.26), we can apply Lemma 5 to the initial-boundary value problem (4.25) and deduce the inequality w n (x, t) ≥ 0, (x, t) ∈ Ω × (0, T ), that is, u n (x, t) ≥ v n (x, t), (x, t) ∈ Ω × (0, T ) for n ∈ N. Due to the relation (4.23), we can choose a suitable subsequence of w n , n ∈ N and pass to the limit as n tends to infinity thus arriving at the inequality u(c, σ 1 )(x, t) ≥ u(c, σ 2 )(x, t) in Ω × (0, T ).The proof of Theorem 3 is completed.
At this point, let us mention a direction for further research in connection with the results formulated and proved in this sections.In order to remove the negativity condition posed on the coefficient c = c(x, t) in Theorem 3 (ii), one needs a unique existence result for solutions to the initial-boundary value problems of type (2.4) with non-zero Robin boundary condition similar to the one formulated in Theorem 1.There are several works that treat the case of the initial-boundary value problems with non-homogeneous Dirichlet boundary conditions (see, e.g., [30] and the references therein).However, to the best of the authors' knowledge, analogous results are not available for the initial-boundary value problems with the non-homogeneous Neumann or Robin boundary conditions.Thus, in Theorem 3 (ii), we assumed the condition c(x, t) < 0, (x, t) ∈ Ω × (0, T ), although our conjecture is that this result holds true for an arbitrary coefficient c = c(x, t).
We conclude this section with a proof of Lemma 5 that is simple because in this case we do not need the function ψ defined as in (3.4).
The inequalities c(x, t) < 0, (x, t) ∈ Ω × [0, T ] and σ(x) ≥ σ 0 > 0, x ∈ ∂Ω and the calculations similar to the ones done in the proof of Lemma 3 implicate the inequalities Based on these inequalities, the same arguments that were employed after the formula (3.10) in the proof of Lemma 3 readily complete the proof of Lemma 5.

Appendix
In the proof of Lemma 3 that is a basis for all other derivations presented in this paper, we essentially used an auxiliary function that satisfies the conditions (3.4).Thus, ensuring existence of such function is an important problem worth for detailed considerations.In this Appendix, we present a solution to this problem.
For the readers' convenience, we split our existence proof into three parts.
I. First part of the proof.
In this part, we prove the following lemma: Lemma 6 Let the conditions (1.2) be satisfied and the constant Then there exists a constant κ 1 > 0 such that In particular, Lemma 6 implies that all of the eigenvalues of the operator A 0 defined by (2.5) are positive if the constant c 0 > 0 is sufficiently large.Henceforth we employ the notation b = (b 1 , ..., b d ).
Proof By using the conditions (1.2) and the boundary condition ∂ νA v +σv = 0 on ∂Ω, integration by parts yields Here and henceforth C > 0, C ε , C δ > 0, etc. denote generic constants which are independent of the function v.
II. Second part of the proof.
Due to the estimate (5.1), we can apply Theorem 3.2 (p.137) in [13] that implicates existence of a constant θ ∈ (0, 1) such that for each t ∈ [0, T ], a solution ψ(•, t) ∈ C 2+θ (Ω) to the problem (3.4) exists uniquely, where C 2+θ (Ω) is the Schauder space defined in the proof of Lemma 4 in Section 4. Now we introduce an auxiliary function Because of the inclusion ψ(•, t) ∈ C 2+θ (Ω), the value of the function η(t) is finite for each t ∈ [0, T ].Furthermore, for an arbitrary G ∈ C θ (Ω), there exists a unique solution w = w(•, t) to the problem for each t ∈ [0, T ].Now we prove that for each t ∈ [0, T ] there exists a constant for all solutions w of the problem (5.4).In the inequality (5.5), the constant C t > 0 depends on the norms the coefficients, but not on the coefficients by themselves.Indeed, for each t ∈ [0, T ], the inequality holds true (see, e.g., the formula (3.7) on p. 137 in [13]).To obtain the desired estimate we have to eliminate the term w(•, t) C(Ω) on the right-hand side of the last inequality.This can be done by the standard compactness-uniqueness arguments.More precisely, let us assume that (5.Choosing δ := δ(t) > 0 sufficiently small, for a given t ∈ [0, T ], the estimate sup s∈I δ(t),t η(s) ≤ C 1 η(t) holds true.Varying t ∈ [0, T ], we now choose a finite number of the intervals I δ(t),t that cover the whole interval [0, T ] and thus obtain the norm estimate ψ L ∞ (0,T ;C 2+θ (Ω)) ≤ C 2 (5.9) with some constant C 2 > 0.

3 2 −
2ε0 (Ω) holds true.Now we proceed to the proof of the inclusion Q(s)u(s)