Singular solutions for space-time fractional equations in a bounded domain

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann--Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.


Introduction
This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators in bounded domains.More precisely, we want to extend the results in [6] to the fractional-in-time setting.Let Ω be a bounded smooth domain of R d and L be an elliptic operator of order 2s ∈ (0, 2) as in [6].The precise hypotheses of L are made on its Green and heat kernels in Section 2. Consider where u ⋆ is a canonically-chosen representative of a class of solutions of Lu = 0 which are singular on the boundary, which we will explain below after the statement of Theorem 2.5, and • ∂ α t (0 < α < 1) is either the Caputo derivative C ∂ α t or the Riemann-Liouville derivative R ∂ α t , which are defined as follows: These derivatives can be started at a time t 0 = 0, in which case they are denoted by C t0 ∂ α t and R t0 ∂ α t .Some authors drop R from the Riemann-Liouville derivative, but we will keep it for clarity.The initial conditions are a little trickier, as they depend on the type of time derivative.We will explain this below.For the Caputo derivative we have simply u(0, x) = u 0 (x), x ∈ Ω. (IC C ) For the Riemann-Liouville derivative, however, we set the initial condition where we define for the range α − 1 ∈ (−1, 0) This seemingly strange initial condition is motivated and justified by the Laplace transform in (3.2).
Using the notation • ∈ {C, R} we will denote the solution above by There have been a significant number of previous works for this family of problems.The problem with no boundary data H C [u 0 , f, 0] for good data u 0 , f has been studied in Gal and Warma in [11] for general operators L. The aim of this paper is to consider jointly the evolution problems with Caputo and Riemann-Liouville time derivative, and exploit the existing duality between them.Moreover, we consider the problems with singular spatial boundary data h = 0, which is only known for α = 1 (see [6]).
Our aim in this paper is to prove existence and uniqueness of suitable solutions of the problem up to finite time.Our main results are presented and explained in the next section.In Theorem 2.3 we prove well-posedness and a representation formula of spectral-type solutions for smooth data u 0 , f and h = 0.In Theorem 2.5 we show that this representation is also valid for weighted-integrable data u 0 , f and h = 0, and provide a weak notion of solution with uniqueness.Lastly, in our main result, Theorem 2.6, we show how the previously introduced functions concentrate towards the boundary to construct solutions of the general case with h = 0, and give a suitable notion for which they are also unique.
There has also been progress in other directions.Let us mention that the asymptotic behaviour for t → ∞ in the whole space was considered by [9,10].

Main results, structure of the paper, and comparison with previous theory
Recalling the theory of elliptic problems, there is a long list of paper dealing with the continuous and bounded solutions of the elliptic problem For a general class of integro-differential operators, it is proven [3] that there are sequences f j with support concentrating towards the boundary such that U j → u, a non-trivial solution to Lu = 0 in Ω.We pick u ⋆ a canonical representative of this class, which we will explain below after the statement of Theorem 2.5.In the case of the classical Laplacian, one such example is u ⋆ = 1, i.e. one recovers the solution of the non-homogeneous Dirichlet problem.Letting δ(x) = dist(x, ∂Ω), for the Restricted Fractional Laplacian we recover solutions of the form u ⋆ ≍ δ −s whereas for the Spectral Fractional Laplacian u ⋆ ≍ δ −2 (1−s) .In [3] (see also [1,2]) the authors proved that the additional condition U/u * = h can be added on the spatial boundary ∂Ω.In [6] we showed that in this condition can also be added in the parabolic problem In this paper we show that non-local-in-time problems also admit the additional singular (or non-singular) boundary condition We make the following assumptions on L throughout the paper.We assume that, for every f ∈ L ∞ (Ω), (2.1) has a unique bounded solution and it is given by integral against the so-called Green kernel G in the sense that We denote G[f ] = U .As in [6], we will make the following assumptions on G: Hypothesis 1 (Fractional structure of the Green function).
• The Green operator G = L −1 admits a symmetric kernel G(x, y) = G(y, x) with two-sided estimates where s, γ ∈ (0, 1]. • The Martin kernel, or the γ-normal derivative of the Green kernel, exists (and therefore enjoys the two-sided estimates): • L enjoys the boundary regularity that Remark 2.1 (Spectral decomposition).Given (G1), by the Hille-Yosida theorem L generates a heat semigroup S(t) that solves (2.2) (see [5]).Furthermore, it formally admits an L 2 (Ω) spectral decomposition with an orthogonal basis of eigenfunctions ϕ j with eigenvalues λ j Lϕ j = λ j ϕ j The rigorous approach is that ϕ j form the eigenbasis of G.
We make further assumptions on the heat semigroup S: Hypothesis 2 (L generates a submarkovian semigroup S(t)).
In [6], under these assumptions we have proven that the heat kernel S(t, x, y) exists, i.e. for every u 0 ∈ L ∞ (Ω) there exists a unique bounded solution of (2.2) expressible by V (t, x) = ˆΩ S(t, x, y)u 0 (y) dy.
In [6] we also proved that S admits a γ-normal derivative, certain estimates near the diagonal of S and D γ S, and a one-sided Weyl-type law for the eigenvalues of L.
Due to Remark 2.1, we can perform the spectral decomposition for u j (t) = u(t, •), ϕ j and f j (t) = f (t, •), ϕ j .Thus (P • ) can be rewritten in the eigenbasis as We devote Section 3 to the study of the ordinary fractional-in-time equations with the suitable initial conditions.As in [11], this spectral analysis leads to the construction of the kernels S α (t, x, y) = ˆ∞ 0 Φ α (τ )S(τ t α , x, y) dτ, (2.5) where Φ α is the well-known Mainardi function given in (3.6).The associated integral operators are: This analysis works for h = 0. To deal with the case of non-trivial singular boundary data h = 0 we need to introduce the notation for the γ-normal derivatives of P α and P α : Finally, we propose as solution to (P (2.7) Notice that the choice of Caputo or Riemann-Liouville derivative only affects the initial condition, in the sense that Hence, when u 0 = 0 we drop the sub-index C or R and denote simply u = H[0, f, h].We point out that the super-position principle (i.e.linearity) means that We make some further technical assumptions which are needed below in this paper: Hypothesis 3 (Off-diagonal bound on the heat kernel S).
Hypothesis 4 (Uniform exchange of limits between integral and D γ ).We assume that P α has the following properties: where Φ α is the Mainardi function given in (3.6).
Remark 2.2.We remark that (G1) implies (see [3]) that for some p > 1.Moreover, under (G2) and (E), In order to develop a theory of classical boundary singular solutions for time-fractional equations, we impose the following extra hypothesis: Hypothesis 5 (Uniform control of the time tail of D γ S near ∂Ω).
For any δ > 0 and In Section 4 we develop the L 2 theory using spectral analysis.For this we define the natural energy space The well-posedness result is the following.
• Riemann-Liouville derivative case: There is a unique function u such that and u satisfies of (P R ) in the spectral sense (2.4) together with the initial condition (IC R ).
In each case, this solution is given by u = H • [u 0 , f, 0] as in (2.7).
The case of Caputo derivative is already covered in [11,Theorem 2.1.7]in a slightly different functional setting.
In Section 5 we extend this theory for h = 0 outside the L 2 framework.To this end, we define a generalised notion of solution in the "optimal" domain of the Green kernel (the weighted space L 1 (Ω, δ γ )) and we prove uniform integrability estimates.We provide the following definition of weak-dual solution, which we justify by the "duality" between the Caputo and Riemann-Liouville derivatives presented in Section 3.6.
• We say that u is weak-dual solution of ( and For this notion of solution, we provide a well-posedness result: Theorem 2.5.Assume (G1), (G2), (G3), and (S1), and h = 0.Then, (P • ) admits a unique weak-dual solution and it is given by u Finally, in Section 6 we "concentrate" f towards ∂Ω to construct solutions with h = 0.In [3] the authors construct a singular solution as follows.As the authors did in [3,6] we define A j := {1/j < δ(x) < 2/j} and Under our assumptions, it then allows to show that These canonical solutions have, in some sense, "uniform" boundary conditions.Under the assumption (G1), the boundary blow-up rate is given by [3, equation (4. 2)], namely Notice that in the classical case γ = s = 1 so we recover δ 0 .In particular, when L = −∆, this yields that u ⋆ ≡ 1, the only solution of −∆U = 0 in Ω such that u = 1 on ∂Ω.Passing to the limit in the weak formulation we recover that u ⋆ satisfies the very weak formulation More general solutions are constructed by letting where P ∂Ω is the orthogonal projection onto the boundary, which is well-defined in A j for j large enough and the very weak formulation is For the parabolic case when α = 1, the same idea of picking was shown to work in [6].Now try to extend to α ∈ (0, 1) to construct H • [0, 0, h].Due to Remark 4.4, it is clear that H[0, 0, h] will be the same for both fractional time derivatives.

.11)
A comment on the hypothesis In the previous works, we have checked the hypotheses (G1), (G2), (G3), (S1) in the examples given in Appendix A. It is not difficult to check that the new hypotheses (S2), (E) also hold in those cases.
3 The Caputo and Riemann-Liouville time derivatives

The Riemann-Liouville integral
The Riemann-Liouville integral is defined for α > 0 by This operator is continuous from L 1 (0, T ) → L 1 (0, T ).It has the derivative-like properties and its Laplace transform, which we denote here by L, is given by whenever ℜ(s) > σ and w(t)e −σt ∈ L 1 (0, ∞).
Given these definitions, we point out that the Caputo derivative can be equivalently defined for α ∈ (0, 1) by This formula can be extended to α > 1.On the other hand, the Riemann-Liouville derivative is equivalently defined by where ⌈α⌉ is the ceiling function of α.
Due to this representation, the Laplace transform of these differentiation operators can be easily computed.Indeed, for the Caputo derivative we have that Similarly, the Laplace transform of the Riemann-Liouville derivative is given by

The Mittag-Leffler and Mainardi functions
For the solution of these equations we will use the Mittag-Leffler functions defined by It is also common to denote E α = E α,1 .These are entire functions.Notice, furthermore, that E 1,1 (z) = e z .The advantage of Mittag-Leffler function in the study of (2.4) can be seen in the following computation concerning the Laplace transform of its suitable moment: so long as ℜ(s), ℜ(α), ℜ(β), λ > 0. The cases β = 1 and β = α will be particularly useful.
There are many known properties of E α (−t α ) (see, e.g.[13] and the references therein).In particular ) is also non-negative and non-increasing.By manipulating the series, it is easy to see the recurrence property This implies, in particular, the global bounds We also recall the definition of the Mainardi function (or Wright-type function) It is known that Φ α ≥ 0 (see [14,Section 4]).It is in fact an entire function for complex arguments t ∈ C, and has explicit moments In particular, Φ α is a probability density function.We observe that Φ α arises as the inverse Laplace transform of the Mittag-Leffler function.In fact, we have the following relations which are easily verified via a series expansion: (3.9)

Ordinary integro-differential equations with Caputo derivative
We focus our attention on Applying (3.1), we can find the solution of (ODE C ) in the Laplace variable: where in the last equality we have (3.4) with β = 1 and β = α.Taking inverse Laplace transform we easily obtain the general solution where This has been discussed in [12].As the product of non-negative and non-increasing functions, P α (•, λ) is non-negative and non-increasing for all λ ≥ 0.

Ordinary integro-differential equations with Riemann-Liouville derivative
Consider now 2), we can solve the ODE in the Laplace space as Therefore, the general solution for (ODE R ) is given by .12)where P α (•; λ), given by (3.11), is the same as in the solution for (ODE C ).
Remark 3.1.Notice that if u is the solution of (ODE C ) and v is the solution of (ODE R ) with u 0 = v 0 = 0 and f = g, then u ≡ v.

Integration by parts with the Caputo derivative
To compute the adjoint of the Caputo derivative in L 2 (0, T ) we have Since the equation with Caputo derivative involves C ∂ α t [u] (t) and u(0), we are therefore interested in the adjoint problem This is an integro-differential equation involving the right Riemann-Liouville derivative (with final condition given by a fractional Riemann-Liouville integral of order 1 − α).Nevertheless, as for the case α = 1, we do not expect C ∂ α t * [ϕ] + Lϕ = 0 to have a solution, so we "reverse time".

Caputo and Riemann-Liouville derivatives are adjoint up to reversing time
As usual, we want to reverse time ϕ(t) = φ(T − t) so that, taking τ = T − σ, we have This is precisely the (left) Riemann-Liouville fractional derivative.Notice the "initial" conditions But then we can rewrite the integration by parts formula as (3.13) Thus, if u solves (ODE C ) and v solves (ODE R ), then we have that Notice that the only remainder that we have due to "non-locality" is the second term on the left-hand side.As α → 1, we recover the classical integration by parts.
4 Time-fractional problem when h = 0.An L 2 theory We now go back to the spectral decomposition (2.3) and take advantage of the explicit solutions of (2.4) (as (3.10) and (3.12)).
Given suitable functions F : R → R and a spectral decomposition of L, it is natural to define F (L) by the linear operator such that F (L)[ϕ j ] = F (λ j )ϕ j .Therefore Recall that the solution of the local-in-time heat equation ∂ t u = −Lu 0 is given by Hence, we have the kernel representation e −λj t ϕ j (x)ϕ j (y).
In various examples we know upper and lower bounds for S.
Recall that, in this notation, the solution of the elliptic problem L −1 [f ] is given precisely by Or, equivalently written as in terms of the kernels G(x, y) = ˆ∞ 0 S(t, x, y) dt.This is well defined for µ > −λ 1 .Since the heat semigroup is non-negative, for µ ∈ (−λ 1 , ∞), for 0 ≤ f ∈ L 2 (Ω) we construct exactly one non-negative solutions in L 2 (Ω).This is the ethos behind [6].

Caputo
We would like to obtain a representation formula for H C [u 0 , f, 0].We give two equivalent expressions: one in terms of Mittag-Leffler function and through (4.1), and the other in terms of Mainardi function and the heat kernel S. Due to (2.4) and (3.10), it is clear, from the solution of the coefficients of the spectral decomposition, that we can write We denote this operator by S α (t) := E α (−t α L).Using a similar argument, it can be deduced that the solution of (P C ) is given by where This formula is already presented in [11].We emphasize that S α (t) and P α (t) do not satisfy the semigroup property in general.
L 2 theory As usual, we have that Therefore S α : L 2 (Ω) → C(0, ∞; L 2 (Ω)) and we have the estimate Remark 4.2.Notice that, due to the slow decay of E α we have that E α (−λ n t α )/E α (−λ 1 t α ) does not converge to zero as t → ∞ for n > 1.Hence, unlike in the case α = 1 we cannot simplify Similarly, we have an L 2 estimate for finite time t ∈ [0, T ], The solution for f (t, x) = f (x) is cleanly expressed as Hence its asymptotic behaviour is simply given by lim t→∞ i.e. the solution of the L-Poisson equation.
Remark 4.3.Notice that the notation L −1 for the Green operator (and more generally (L+µ) −1 for the resolvent) is consistent with the notation (4.1).

Properties of the kernel representation
In [11, Section 2.1] it is shown that, as t ց 0 we have We refer to [11, Section 2.2] for L p -L q estimates which are recovered in terms of the contractivity of S(t) = e −tL .We stress that S α (t) : L p (Ω) → L q (Ω) only when n 2s ( 1 p − 1 q ) < 1 and P α (t) :

Well-posedness. Proof of Theorem 2.3
Uniqueness of spectral solutions of either initial value problem that lie in the spaces in the statement follows directly from the theory of fractional ODEs developed.By construction, our candidate solutions (2.7) satisfy the spectral equation.The only missing detail, then, is the regularity of our candidate solutions.We have already proven that S α (t)[v], P α (t)[v] ∈ L 2 (Ω).In fact, due to the inverse linear (respectively quadratic) decay of the Mittag-Leffler functions stated in (3.5), they are also in H(Ω).
The continuity at t = 0 follows from (4.5).Due to the bounds presented before, in fact . The integral part is even easier.
The time differentiability follows from the explicit computation of du dt as in [11, Proposition 2.1.9].We point out that S ′ α = P α L. This can be done similarly for the Riemann-Liouville derivative.
Remark 4.5.In [11] the authors deal with the notion of strong solution.This is also possible in our setting, but our interest in the very weak solutions described below.

Very weak formulation when h = 0
Due to (3.14), for every (4.8)This allows for a very natural definition of very weak solution: This notion of solution yields uniqueness and positivity in a very standard way.It is also compatible with the L 2 theory constructed before.
Integrability properties can be directly recovered from the estimates of the kernels of S α (t), P α (t) that are directly related to those of S(t) = e −tL .Remark 4.6.Notice that we could equivalently write that for every v 0 ∈ L ∞ c (Ω) and for a.e.t > 0 we have This formulation is nicer for the L ∞ estimates in time.
5 Time-fractional problem when h = 0 beyond L 2

Weighted L 1 and L ∞ theory
When we leave the L 2 framework, we need to look beyond simple properties of E α and E α,α .It is here where the Mainardi function comes into play.
Furthermore, it is common that the first eigenfunction ϕ 1 (x) satisfies the boundary condition with a rate δ(x) γ for some γ positive.This is the case, for example with the Restricted Fractional Laplacian (γ = s), Censored Fractional Laplacian (γ = 2s − 1 which is only defined for s > 1  2 ), and the Spectral Fractional Laplacian (γ = 1).This is the expected boundary behaviour of all solution with good data, as we proved in [3] for the elliptic case and [6] for the parabolic case.In those papers, conditions are set on the Green kernel.However, it is more convenient for us now to set condition on the heat kernel.We set ourselves in a framework that covers the three main settings, where sharp estimates for the kernels are provided in Appendix A.
The canonical framework is that for good data we expect solutions in δ γ L ∞ (Ω) (a weighted space cointaining ϕ j ).The worst admissible data is in L 1 (Ω, δ γ ), a fact guaranteed by the lower estimate where c 1 > 0.
In general, under (G1), we have that Remark 5.1.In the local-in-time setting in [6] we showed the nice regularisation using the semigroup property.Since we were not interested in the operator norm, conditions on the Green kernel sufficed.Due to the memory coming from the non-locality in time, we cannot expect such regularisation.Going back to (2.5) we have that Therefore, the regularisation relies on the integrals Unfortunately, obtaining sharp estimates for such integral appears to be a non-trivial task.
We start developing the theory of very weak solutions with a compactness estimate.
Lemma 5.2 (Uniform space-time integrability in L 1 (0, T ; Here and below ω represents a modulus of continuity, i.e. a non-decreasing, non-negative function such that ω(0 + ) = 0. We denote the dependence by sub-indexes.
Proof.By splitting into positive and negative parts, we may assume that u 0 , f, u ≥ 0.
To obtain L 1 loc (Ω) compactness, we apply the above estimate to each K ⋐ Ω.
Remark 5.3.Notice that the only crucial ingredients in the proof above is that ϕ 1 ≍ δ γ and G[χ A ϕ 1 ] ≤ ω(|A|)ϕ 1 , which are minimal assumptions on G that uses only mild integrability assumptions, not the exact shape.In Lipschitz domains, where it can happen that ϕ 1 ≍ δ γ for any γ, the correct weight is ϕ 1 .
5.2 Well-posedness.Proof of Theorem 2.5 When u 0 and f are regular, we have proven that (2.7) is a spectral solution.As described in Section 4.4, this solution is a weak solution.Let u 0 , f be in the general classes of the statement.They can be approximated by u 0k , f k smooth.Because of the a priori estimates proven, H[u 0k , f k , 0] → H[u 0 , f, 0] in L 1 (0, T ; L 1 (Ω, δ γ )).Due to the regularity of H[0, φ, 0] we can pass to the limit in the definition of weak-dual solution.This guarantees existence.
Finally, we prove the uniqueness.Assume there are two weak-dual solutions.Let w be their difference.Since they share a right-hand side in Definition 2.4 we recover, for each T and φ smooth ˆT 0 ˆΩ w(t, x)φ(T − t, x) dx dt = 0.
Remark 5.4.We point out the pointwise estimate Unfortunately, in this direct computation one loses some power of t and the integrability in time.
Alternatives, such as (weighted) integral estimates, will be used to fit our purposes.
Proof.We compute Using (2.8), the latter integral is bounded uniformly for x ∈ Ω, as desired.
Proof.We estimate By Lemma 5.5, the last double integral is bounded by a uniform constant, as desired.
First we check that the solution lies in the correct weighted space.
Proof.We express Using Lemma 5.6, the last t-integral is in L 1 (Ω, δ γ ) (in variable x) and hence the result follows.
Integrating by parts, we see that the only possible solution is precisely (2.7).