1 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 \(\Omega \) be a bounded smooth domain of \(\mathbb R^d\) and \(\textrm{L}\) be an elliptic operator of order \(2s\in (0,2)\) as in [6]. The precise hypotheses of \(\textrm{L}\) are made on its Green and heat kernels in Sect. 2. Consider

figure a

where \(u^\star \) is a canonically-chosen representative of a class of solutions of \(\textrm{L}u = 0\) which are singular on the boundary, which we will explain below after the statement of Theorem 2.5, and (\(0< \alpha < 1\)) is either the Caputo derivative or the Riemann–Liouville derivative , which are defined as follows:

and

These derivatives can be started at a time \(t_0\ne 0\), in which case they are denoted by and . 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

figure b

For the Riemann–Liouville derivative, however, we set the initial condition

figure c

where we define for the range \(\alpha - 1 \in (-1,0)\)

This seemingly strange initial condition is motivated and justified by the Laplace transform in (3.2).

Using the notation \(\bullet \in \{C,R\}\) we will denote the solution above by

$$\begin{aligned} u(t,x)=\mathcal {H}_\bullet [u_0,f,h]. \end{aligned}$$

There have been a significant number of previous works for this family of problems. The problem with no boundary data \(\mathcal H_C [u_0, f, 0]\) for good data \(u_0,f\) has been studied in Gal and Warma in [11] for general operators \(\textrm{L}\). 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 \(h \ne 0\), which is only known for \(\alpha = 1\) (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 \(h \ne 0\), 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 \(t \rightarrow \infty \) in the whole space was considered by Cortázar et al. [9, 10].

2 Main results, structure of the paper, and comparison with previous theory

Recalling the theory of elliptic problems, there is a long list of papers dealing with the continuous and bounded solutions of the elliptic problem

(2.1)

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 \(U_j \rightarrow u\), a non-trivial solution to \(\textrm{L}u = 0\) in \(\Omega \). We pick \(u^\star \) 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 \(u^\star = 1\), i.e. one recovers the solution of the homogeneous Dirichlet problem with non-homogeneous boundary data. Letting \(\delta (x) = {{\,\textrm{dist}\,}}(x, \partial \Omega )\), for the Restricted Fractional Laplacian we recover solutions of the form \(u^\star \asymp \delta ^{-s}\) whereas for the Spectral Fractional Laplacian \(u^\star \asymp \delta ^{-2(1-s)}\). In [3] (see also [1, 2]) the authors proved that the additional condition \(U/u^* = h\) can be added on the spatial boundary \(\partial \Omega \). In [6] we showed that this condition can also be added in the parabolic problem

(2.2)

In this paper we show that non-local-in-time problems also admit the additional singular (or non-singular) boundary condition

$$\begin{aligned} \lim _{x \rightarrow \zeta } \frac{u(t,x)}{u^\star (x)} = h(\zeta ). \end{aligned}$$
(BC)

We make the following assumptions on \(\textrm{L}\) throughout the paper. We assume that, for every \(f \in L^\infty (\Omega )\), (2.1) has a unique bounded solution and it is given by integral against the so-called Green kernel \(\mathbb G\) in the sense that

$$\begin{aligned} U(x) = \int \nolimits _\Omega \mathbb {G}(x,y) f(y) \textrm{d}y. \end{aligned}$$

We denote \(\mathcal {G}[f] = U\). As in [6], we will make the following assumptions on \(\mathbb {G}\):

Hypothesis 1

(Fractional structure of the Green function)

  • The Green operator \(\mathcal {G}=\textrm{L}^{-1}\) admits a symmetric kernel \(\mathbb {G}(x,y) = \mathbb {G}(y,x)\) with two-sided estimates

    $$\begin{aligned} \mathbb {G}(x,y) \asymp \frac{1}{|x-y|^{d-2s}} \left( 1 \wedge \frac{\delta (x)}{|x-y|} \right) ^\gamma \left( 1 \wedge \frac{\delta (y)}{|x-y|} \right) ^\gamma , \quad x,y\in \Omega ,\quad x\ne y, \end{aligned}$$
    (G1)

    where \(s,\gamma \in (0,1]\).

  • The Martin kernel, or the \(\gamma \)-normal derivative of the Green kernel, exists (and therefore enjoys the two-sided estimates):

    $$\begin{aligned} D_\gamma \mathbb {G}(x,\zeta ) := \lim _{\Omega \ni y\rightarrow \zeta } \frac{\mathbb {G}(x,y)}{\delta (y)^\gamma } \asymp \frac{\delta (x)^\gamma }{|x-\zeta |^{d-2s+2\gamma }}, \qquad x\in \Omega ,\quad \zeta \in \partial \Omega . \end{aligned}$$
    (G2)

  • \(\textrm{L}\) enjoys the boundary regularity that

    $$\begin{aligned} \mathcal {G}:\delta ^\gamma L^\infty (\Omega ) \rightarrow \delta ^\gamma C(\overline{\Omega }). \end{aligned}$$
    (G3)

Remark 2.1

(Spectral decomposition) Given (G1), by the Hille–Yosida theorem \(\textrm{L}\) generates a heat semigroup \(\mathcal {S}(t)\) that solves (2.2) (see [5]). Furthermore, it formally admits an \(L^2(\Omega )\) spectral decomposition with an orthogonal basis of eigenfunctions \(\varphi _j\) with eigenvalues \(\lambda _j\),

$$\begin{aligned} \textrm{L}\varphi _j = \lambda _j \varphi _j. \end{aligned}$$

The rigorous approach is that \(\varphi _j\) form the eigenbasis of \(\mathcal {G}\), namely a basis of \(L^2(\Omega )\) consisting of eigenfunctions of \({\mathcal {G}}\) (hence of \(\textrm{L}\)).

We make further assumptions on the heat semigroup \(\mathcal {S}\):

Hypothesis 2

(\(\textrm{L}\) generates a submarkovian semigroup \(\mathcal {S}(t)\))

$$\begin{aligned} 0 \le u_0\le 1 \implies 0\le \mathcal {S}(t)[u_0] \le 1. \end{aligned}$$
(S1)

In [6], under these assumptions we have proven that the heat kernel \(\mathbb {S}(t,x,y)\) exists, i.e. for every \(u_0 \in L^\infty (\Omega )\) there exists a unique bounded solution of (2.2) expressible by

$$\begin{aligned} V(t,x) = \int \nolimits _\Omega \mathbb {S}(t,x,y) u_0(y) \textrm{d}y. \end{aligned}$$

In [6] we also proved that \(\mathbb {S}\) admits a \(\gamma \)-normal derivative, certain estimates near the diagonal of \(\mathbb {S}\) and \(D_\gamma \mathbb {S}\), and a one-sided Weyl-type law for the eigenvalues of \(\textrm{L}\).

Due to Remark 2.1, we can perform the spectral decomposition

$$\begin{aligned} u(t,x) = \sum _{j=1}^\infty u_j(t) \varphi _j(x), \qquad f(t,x) = \sum _{j=1}^\infty f_j(t) \varphi _j(x), \end{aligned}$$
(2.3)

for \(u_j(t) = \langle u(t,\cdot ), \varphi _j \rangle \) and \(f_j(t) = \langle f(t,\cdot ), \varphi _j\rangle \). Thus P\(_\bullet \) can be rewritten in the eigenbasis as

We devote Sect. 3 to the study of the ordinary fractional-in-time equations

(2.4)

with the suitable initial conditions. As in [11], this spectral analysis leads to the construction of the kernels

$$\begin{aligned} \mathbb {S}_\alpha (t,x,y)&=\int \nolimits _0^\infty \Phi _\alpha (\tau ) \mathbb {S}(\tau t^\alpha ,x,y) \textrm{d}\tau , \end{aligned}$$
(2.5)
$$\begin{aligned} \mathbb {P}_\alpha (t,x,y)&=\alpha t^{\alpha -1} \int \nolimits _0^\infty \tau \Phi _\alpha (\tau ) \mathbb {S}(\tau t^\alpha ,x,y) \textrm{d}\tau , \end{aligned}$$
(2.6)

where \(\Phi _\alpha \) is the well-known Mainardi function given in (3.6). The associated integral operators are:

$$\begin{aligned} {\mathcal {S}}_\alpha (t)[u_0](x)&=\int \nolimits _{\Omega } \mathbb {S}_\alpha (t,x,y) u_0(y) \textrm{d}y, \quad {\mathcal {P}}_\alpha (t)[f](x) =\int \nolimits _{\Omega } \mathbb {P}_\alpha (t,x,y) f(y) \textrm{d}y. \end{aligned}$$

This analysis works for \(h = 0\). To deal with the case of non-trivial singular boundary data \(h \ne 0\) we need to introduce the notation for the \(\gamma \)-normal derivatives of \({\mathcal {P}}_\alpha \) and \(\mathbb {P}_\alpha \):

$$\begin{aligned} D_\gamma {\mathcal {P}}_\alpha [h](x)&=\int \nolimits _{\partial \Omega } D_\gamma \mathbb {P}_\alpha (t,x,\zeta ) h(\zeta ) \textrm{d}\zeta ,\\ D_\gamma \mathbb {P}_\alpha (t,x,\zeta )&=\alpha t^{\alpha -1} D_\gamma \bigg [\int \nolimits _0^\infty \tau \Phi _\alpha (\tau ) \mathbb {S}(\tau t^\alpha ,x,\cdot ) \textrm{d}\tau \bigg ](\zeta ). \end{aligned}$$

Finally, we propose as solution to P\(_\bullet \) the function \(u = \mathcal {H}_{\bullet }[u_0,f,h]\) given by

$$\begin{aligned}{} & {} \mathcal {H}_C[u_0,f,h](t)\nonumber \\{} & {} \quad =\mathcal {S}_\alpha (t)[u_0] +\int \nolimits _0^t{\mathcal {P}}_\alpha (t-\tau )[f(\tau )]\textrm{d}\tau +\int \nolimits _0^t D_\gamma {\mathcal {P}}_\alpha (t-\tau )[h(\tau )]\textrm{d}\tau ,\nonumber \\{} & {} \mathcal {H}_R[u_0,f,h](t)\nonumber \\{} & {} \quad ={\mathcal {P}}_\alpha (t)[u_0] +\int \nolimits _0^t{\mathcal {P}}_\alpha (t-\tau )[f(\tau )]\textrm{d}\tau +\int \nolimits _0^t D_\gamma {\mathcal {P}}_\alpha (t-\tau )[h(\tau )]\textrm{d}\tau . \end{aligned}$$
(2.7)

Notice that the choice of Caputo or Riemann–Liouville derivative only affects the initial condition, in the sense that

$$\begin{aligned} \mathcal {H}_C[0,f,h] = \mathcal {H}_R[0,f,h]. \end{aligned}$$

Hence, when \(u_0 = 0\) we drop the sub-index C or R and denote simply \(u = \mathcal {H}[0,f,h]\). We point out that the super-position principle (i.e. linearity) means that

$$\begin{aligned} \mathcal {H}_\bullet [u_0,f,h] = \mathcal {H}_\bullet [u_0,0,0] + \mathcal {H}[0,f,0] + \mathcal {H}[0,0,h]. \end{aligned}$$

We make some further technical assumptions which are needed below in this paper:

Hypothesis 3

(Off-diagonal bound on the heat kernel \(\mathbb {S}\))

$$\begin{aligned} \frac{\mathbb {S}(t,x,y)}{\delta (x)^\gamma } \le C(\varepsilon ) \quad \text { for } \quad |x-y| \ge \varepsilon , t \ge 0. \end{aligned}$$
(S2)

Hypothesis 4

(Uniform exchange of limits between integral and \(D_\gamma \)) We assume that \(\mathbb {P}_{\alpha }\) has the following properties:

$$\begin{aligned} \begin{aligned} D_\gamma \mathbb P_\alpha (t,x, \zeta )&= \alpha t^{\alpha -1} \int \nolimits _{0}^{\infty } \tau \Phi _\alpha (\tau ) D_\gamma \mathbb {S}(\tau t^{\alpha },x,\zeta )\textrm{d}\tau \\ D_\gamma \mathbb G(x,\zeta )&= \int \nolimits _0^\infty D_\gamma \mathbb {S}(t,x,\zeta ) \textrm{d}t, \end{aligned} \end{aligned}$$
(E)

where \(\Phi _\alpha \) is the Mainardi function given in (3.6).

Remark 2.2

We remark that (G1) implies (see [3]) that

$$\begin{aligned} \sup _{x\in \Omega } \int \nolimits _\Omega \bigg ( \frac{\mathbb {G}(x,y)}{\delta (x)^\gamma } \delta (y)^\gamma \bigg )^p \textrm{d}y \le C, \end{aligned}$$
(2.8)

for some \(p>1\). Moreover, under (G2) and (E),

$$\begin{aligned} u^{\star }(x) =\int \nolimits _{\partial \Omega } D_\gamma \mathbb G(x, \zeta ) \textrm{d}\zeta =\int \nolimits _0^\infty \int \nolimits _{\partial \Omega } D_\gamma \mathbb {S}(\tau ,x, \zeta ) \textrm{d}\zeta \textrm{d}\tau . \end{aligned}$$

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 \(D_\gamma \mathbb {S}\) near \(\partial \Omega \)) For any \(\delta > 0\) and \(\zeta _0 \in \partial \Omega \),

$$\begin{aligned} \lim _{x \rightarrow \zeta _0} \int \nolimits _0^\infty \Phi _\alpha (\tau ) \dfrac{ \int \nolimits _{\tau \delta ^\alpha }^\infty \int \nolimits _{\partial \Omega } D_\gamma \mathbb {S}(\sigma ,x,\zeta ) \textrm{d}\zeta \textrm{d}\sigma }{ \int \nolimits _0^\infty \int \nolimits _{\partial \Omega } D_\gamma \mathbb {S}(\tilde{\tau },x, \tilde{\zeta }) \textrm{d}\tilde{\zeta }\textrm{d}\tilde{\tau }} \textrm{d}\tau = 0 . \end{aligned}$$
(S3)

In Sect. 4 we develop the \(L^2\) theory using spectral analysis. For this we define the natural energy space

$$\begin{aligned} \mathrm H (\Omega ) = \left\{ u \in L^2 (\Omega ): \Vert \textrm{L}u \Vert _{L^2(\Omega )}^2 = \sum _{j=1}^{\infty } \lambda _j^2 \langle u, \varphi _j \rangle ^2 < \infty \right\} . \end{aligned}$$

The well-posedness result is the following.

Theorem 2.3

Assume (G1), (G2), (G3), and (S1). Let \(u_0 \in \mathrm H (\Omega )\), \(f \in C^1([0,T]; L^2 (\Omega ))\), and \(h = 0\). Then,

  • Caputo derivative case: There is a unique function

    $$\begin{aligned} u \in C([0,T]; L^2(\Omega )) \cap C((0,T]; \mathrm H (\Omega )) \cap C^1((0,T];L^2(\Omega )) \end{aligned}$$

    that satisfies (P\(_C\)) in the spectral sense (2.4) together with the initial condition IC\(_C\), i.e. \(u(0,\cdot ) = u_0\).

  • Riemann–Liouville derivative case: There is a unique function u such that

    $$\begin{aligned} t^{1-\alpha } u \in C([0,T]; L^2(\Omega )) \cap C((0,T]; \mathrm H (\Omega )) \cap C^1((0,T];L^2(\Omega )), \end{aligned}$$

    and u satisfies (P\(_R\)) in the spectral sense (2.4) together with the initial condition IC\(_R\).

In each case, this solution is given by \(u=\mathcal {H}_{\bullet }[u_0,f,0]\) 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 Sect. 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 \(L^1(\Omega , \delta ^\gamma )\)) 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 Sect. 3.6.

Definition 2.4

  • We say that u is weak-dual solution of (P\(_C\)) if \(u \in L^1(0,T;L^1(\Omega ,\delta ^\gamma ))\) and

    (2.9)

    for all \(\phi \in \delta ^\gamma L^\infty ((0,T)\times \Omega )\).

  • We say that u is weak-dual solution of (P\(_R\)) if \(u \in L^1(0,T;L^1(\Omega ,\delta ^\gamma ))\) and

    $$\begin{aligned}&\int \nolimits _{0}^{T}\int \nolimits _{\Omega } u(t,x)\phi (T-t,x) \textrm{d}x\textrm{d}t\\&\quad =\int \nolimits _\Omega u_0 (x) {\mathcal {H}}[0,\phi ,0](T,x) \textrm{d}x\\&\qquad +\int \nolimits _0^T \int \nolimits _\Omega f(t,x) \mathcal {H}[0,\phi ,0](T-t,x) \, \textrm{d}x \textrm{d}t\\&\qquad +\int \nolimits _0^T \int \nolimits _{\partial \Omega } h(t, \zeta ) D_\gamma \mathcal H[0,\phi ,0] (T-t, \zeta ) \textrm{d}\zeta \textrm{d}t. \end{aligned}$$

    for all \(\phi \in \delta ^\gamma L^\infty ((0,T)\times \Omega )\).

For this notion of solution, we provide a well-posedness result:

Theorem 2.5

Assume (G1), (G2), (G3), and (S1), \(u_0\in L^1(\Omega ,\delta ^\gamma )\), \(f\in L^1(0,T;L^1(\Omega ,\delta ^\gamma ))\), and \(h= 0\). Then, P\(_\bullet \) admits a unique weak-dual solution and it is given by \(u=\mathcal {H}_{\bullet }[u_0,f,0]\) as in (2.7).

Finally, in Sect. 6 we “concentrate” f towards \(\partial \Omega \) to construct solutions with \(h \ne 0\). In [3] the authors construct a singular solution as follows. As the authors did in [3, 6] we define \(A_j :=\{ 1/j< \delta (x) < 2/j\}\) and

$$\begin{aligned} f_j (x) :=\frac{|\partial \Omega |}{|A_j|} \frac{\chi _{A_j}}{\delta (x)^\gamma } . \end{aligned}$$

Under our assumptions, we will show that

$$\begin{aligned} \textrm{L}^{-1} [f_j ] \rightarrow u^\star \quad \text { in } L^1_\textrm{loc} (\Omega ). \end{aligned}$$

These canonical solutions have, in some sense, “uniform” boundary conditions. Under the assumption (G1), the boundary blow-up rate is given by Abatangelo et al. [3, Eq. (4.2)], namely

Notice that in the classical case \(\gamma = s = 1\) so we recover \(\delta ^0\). In particular, when \(\textrm{L}= -\Delta \), this yields that \(u^\star \equiv 1\), the only solution of \(-\Delta U = 0\) in \(\Omega \) such that \(u = 1\) on \(\partial \Omega \). Passing to the limit in the weak formulation

$$\begin{aligned} \int \nolimits _\Omega u_j \psi = \int \nolimits _\Omega f_j \textrm{L}^{-1}[\psi ], \quad \forall \psi \in L^\infty _\textrm{c} (\Omega ), \end{aligned}$$

we recover that \(u^\star \) satisfies the very weak formulation

$$\begin{aligned} \int \nolimits _\Omega u^\star \psi = \int \nolimits _{\partial \Omega } \lim _{x \rightarrow \zeta } \frac{\textrm{L}^{-1}[\psi ](x)}{\delta (x)^\gamma } \quad \forall \psi \in L^\infty _\textrm{c} (\Omega ). \end{aligned}$$

More general solutions are constructed by letting

$$\begin{aligned} f_j (x) :=\frac{|\partial \Omega |}{|A_j|} \frac{\chi _{A_j}}{\delta (x)^\gamma } h(P_{\partial \Omega } (x)), \end{aligned}$$

where \(P_{\partial \Omega }\) is the orthogonal projection onto the boundary, which is well-defined in \(A_j\) for j large enough and the very weak formulation is

$$\begin{aligned} \int \nolimits _\Omega u^\star \psi = \int \nolimits _{\partial \Omega } \lim _{x \rightarrow \zeta } h (\zeta ) \frac{\textrm{L}^{-1}[\psi ](x)}{\delta (x)^\gamma } \quad \forall \psi \in L^\infty _\textrm{c} (\Omega ). \end{aligned}$$

For the parabolic case when \(\alpha = 1\), the same idea of picking

$$\begin{aligned} f_j (t,x) :=\frac{|\partial \Omega |}{|A_j|} \frac{\chi _{A_j}}{\delta (x)^\gamma } h(t,P_{\partial \Omega } (x)), \end{aligned}$$
(2.10)

was shown to work in [6]. Now we extend this idea to \(\alpha \in (0,1)\) to construct \(\mathcal H_\bullet [0,0,h]\). Due to Remark 4.4, it is clear that \(\mathcal H[0,0,h]\) will be the same for both fractional time derivatives.

Theorem 2.6

Assume (G1), (G2), (G3), (S1), (S2) and (E). Let \(u_0, f = 0\), and \(h\in L^1((0,T)\times \partial \Omega )\).

  1. (i)

    Let \(f_j\) be given by (2.10), then \(\mathcal H[0,f_j,0] \rightarrow \mathcal H[0,0,h]\).

  2. (ii)

    P\(_\bullet \) admits a unique weak-dual solution and it is given by \(u=\mathcal {H}[0,0,h]\) as in (2.7).

  3. iii)

    Assume, in addition, (S3) and \(h\in C(\partial \Omega )\). Then (BC) holds in the sense that, for each \(\zeta \in \partial \Omega \),

    $$\begin{aligned} \lim _{x \rightarrow \zeta } \frac{\mathcal {H}[0,0,h] (t,x) }{ u^\star (x) } = h(t,\zeta ). \end{aligned}$$
    (2.11)

A comment on the hypotheses 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

3.1 The Riemann–Liouville integral

The Riemann–Liouville integral is defined for \(\alpha > 0\) by

$$\begin{aligned} {{\,\mathrm{\mathcal {I}}\,}}^\alpha w(t) := \frac{1}{\Gamma (\alpha )} \int \nolimits _0^t w(\xi ) (t - \xi )^{\alpha - 1} \textrm{d}\xi . \end{aligned}$$

This operator is continuous from \(L^1(0,T) \rightarrow L^1(0,T)\). It has the derivative-like properties

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} {{\,\mathrm{\mathcal {I}}\,}}^{\alpha + 1} w(t) = {{\,\mathrm{\mathcal {I}}\,}}^\alpha w(t), \qquad {{\,\mathrm{\mathcal {I}}\,}}^\alpha {{\,\mathrm{\mathcal {I}}\,}}^\beta w = {{\,\mathrm{\mathcal {I}}\,}}^{\alpha + \beta } w \end{aligned}$$

and its Laplace transform, which we denote here by \({{\,\mathrm{\mathcal {L}}\,}}\), is given by

$$\begin{aligned} {{\,\mathrm{\mathcal {L}}\,}}[{{\,\mathrm{\mathcal {I}}\,}}^\alpha w](s) = s^{-\alpha } {{\,\mathrm{\mathcal {L}}\,}}[w](s) \end{aligned}$$

whenever \(\Re (s) > \sigma \) and \(w(t) e^{-\sigma t} \in L^1(0,\infty )\).

Given these definitions, we point out that the Caputo derivative can be equivalently defined for \(\alpha \in (0,1)\) by

This formula can be extended to \(\alpha > 1\). On the other hand, the Riemann–Liouville derivative is equivalently defined by

where \(\lceil \alpha \rceil \) is the ceiling function of \(\alpha \).

Due to this representation, the Laplace transform of these differentiation operators can be easily computed. Indeed, for the Caputo derivative we have that

(3.1)

Similarly, the Laplace transform of the Riemann–Liouville derivative is given by

(3.2)

3.2 The Mittag–Leffler and Mainardi functions

For the solution of linear ODEs involving Caputo or Riemann–Liouville derivative, we will use the Mittag–Leffler functions defined by

$$\begin{aligned} E_{\alpha ,\beta } (z) =\sum _{k=0}^{\infty } \dfrac{z^k}{\Gamma (\alpha k+\beta )}. \end{aligned}$$
(3.3)

It is also common to denote \(E_\alpha = E_{\alpha ,1}\). 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 a especially relevant moment:

$$\begin{aligned} {{\,\mathrm{\mathcal {L}}\,}}[t^{\beta -1}E_{\alpha ,\beta }(-\lambda t^\alpha )](s) = \int \nolimits _0^\infty t^{\beta - 1} E_{\alpha , \beta } (-\lambda t^\alpha ) e^{-st} \textrm{d}t = \frac{s^{\alpha -\beta }}{s^\alpha +\lambda }, \end{aligned}$$
(3.4)

so long as \(\Re (s), \Re (\alpha ), \Re (\beta ),\lambda > 0\). The cases \(\beta =1\) and \(\beta =\alpha \) will be particularly useful.

There are many known properties of \(E_\alpha (-t^\alpha )\) (see, e.g. [13] and the references therein). In particular

It is left as a conjecture that \(f_\alpha (t) \le E_\alpha (-t^\alpha ) \le g_\alpha (t)\). Also, it is known that \(E_\alpha (-t^\alpha )\) is completely monotone in t, i.e. \((-1)^n \frac{\textrm{d}^n }{dt^n} E_\alpha (-t^\alpha ) \ge 0\). From (3.3), it is easily verified that \(E_{\alpha ,\alpha }(-t^\alpha )=t^{1-\alpha }\,(-1)\frac{d}{dt}E_{\alpha }(-t^{\alpha })\) is also non-negative and non-increasing. By manipulating the series, it is easy to see the recurrence property

$$\begin{aligned} zE_{\alpha ,\beta }(z)=E_{\alpha ,\beta -\alpha }(z)-\frac{1}{\beta -\alpha }. \end{aligned}$$

This implies, in particular, the global bounds

$$\begin{aligned} E_{\alpha }(-\lambda t^\alpha )\le \frac{C}{1+\Gamma (1-\alpha )\lambda t^\alpha }, \quad E_{\alpha ,\alpha }(-\lambda t^\alpha )\le \frac{C}{1+\Gamma (-\alpha )(\lambda t^\alpha )^2}, \end{aligned}$$
(3.5)

for \(t\ge 0\).

We also recall the definition of the Mainardi function (or Wright-type function)

(3.6)

It is known that \(\Phi _\alpha \ge 0\) (see [14, Sect. 4]). It is in fact an entire function for complex arguments \(t\in \mathbb {C}\), and has explicit moments

$$\begin{aligned} \int \nolimits _0^\infty t^p \Phi _\alpha (t) \,dt =\dfrac{ \Gamma (1+p) }{ \Gamma (1+\alpha p) }, \qquad p>-1. \end{aligned}$$

In particular, \(\Phi _\alpha \) is a probability density function. We observe that \(\Phi _\alpha \) 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:

$$\begin{aligned} \int \nolimits _{0}^{\infty } \Phi _\alpha (t) e^{-t z} \textrm{d}t&= E_\alpha (-z), \\ \int \nolimits _{0}^{\infty } t \Phi _\alpha (t) e^{-t z} \textrm{d}t&= - \frac{\textrm{d}E_\alpha }{\textrm{d}z}(-z) =\frac{1}{\alpha }E_{\alpha ,\alpha }(-z). \end{aligned}$$

3.3 Ordinary integro-differential equations with Caputo derivative

We focus our attention on

figure d

Applying (3.1), we can find the solution of ODE\(_C\) in the Laplace variable:

$$\begin{aligned} {{\,\mathrm{\mathcal {L}}\,}}[u] (s)&= \frac{s^{\alpha -1}}{s^\alpha + \lambda } u_0 + \frac{1}{s^\alpha + \lambda } {{\,\mathrm{\mathcal {L}}\,}}[f](s)\\&={{\,\mathrm{\mathcal {L}}\,}}[u_0 E_\alpha (-\lambda t^\alpha )](s) +{{\,\mathrm{\mathcal {L}}\,}}[t^{\alpha -1}E_{\alpha ,\alpha }(-\lambda t^\alpha )](s) {{\,\mathrm{\mathcal {L}}\,}}[f](s), \end{aligned}$$

where in the last equality we have used (3.4) with \(\beta =1\) and \(\beta =\alpha \). Taking inverse Laplace transform we easily obtain the general solution

$$\begin{aligned} u (t) = u_0 E_\alpha (-\lambda t^\alpha ) + \int \nolimits _0^t P_{\alpha }(t-\tau ;\lambda ) f(\tau ) \textrm{d}\tau . \end{aligned}$$
(3.7)

where

$$\begin{aligned} P_{\alpha }(t;\lambda )= & {} t^{\alpha -1} E_{\alpha ,\alpha }(-\lambda t^\alpha ) =-\frac{1}{\lambda }\frac{d}{dt} E_\alpha (-\lambda t^\alpha )\nonumber \\= & {} \frac{\alpha t^{\alpha -1}}{\lambda } \int \nolimits _{0}^{\infty } \sigma \Phi _\alpha (\sigma ) e^{-\sigma \lambda t^\alpha } \textrm{d}\sigma . \end{aligned}$$
(3.8)

This has been discussed in [12]. As the product of non-negative and non-increasing functions, \(P_\alpha (\cdot , \lambda )\) is non-negative and non-increasing for all \(\lambda \ge 0\).

3.4 Ordinary integro-differential equations with Riemann–Liouville derivative

Consider now

figure e

Applying (3.2), we can solve the ODE in the Laplace space as

$$\begin{aligned} {{\,\mathrm{\mathcal {L}}\,}}[v] (s) = \frac{1}{ s^\alpha + \lambda }v_0 + \frac{{{\,\mathrm{\mathcal {L}}\,}}[g]}{s^\alpha + \lambda }. \end{aligned}$$

Therefore, the general solution for ODE\(_R\) is given by

$$\begin{aligned} v(t) = v_0 P_{\alpha }(t;\lambda )+ \int \nolimits _0^t P_{\alpha } (t-\tau ;\lambda ) g(\tau ) \textrm{d}\tau , \end{aligned}$$
(3.9)

where \(P_{\alpha }(\cdot ;\lambda )\), given by (3.8), 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 \(u \equiv v\).

Remark 3.2

Notice that \(P_\alpha (t;\lambda ) \sim \frac{t^{\alpha -1}}{\Gamma (\alpha )}\) as \(t \rightarrow 0\). Therefore, the initial condition \(v(0^+)\) must be understood in a singular way.

3.5 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 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 \(1-\alpha \)). Nevertheless, as for the case \(\alpha =1\), we do not expect to have a solution, so we “reverse time”.

3.6 Caputo and Riemann–Liouville derivatives are adjoint up to reversing time

As usual, we want to reverse time \(\varphi (t) = \phi (T-t)\) so that, taking \(\tau = T - \sigma \), 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

Thus, if u solves ODE\(_C\) and v solves ODE\(_R\), then we have that

$$\begin{aligned}{} & {} \int \nolimits _0^T v(T-t) \Big ( - \lambda u(t) + f(t) \Big ) \textrm{d}t + u_0 \int \nolimits _0^T \frac{ t^{-\alpha } v(T-t)}{\Gamma (1-\alpha )} \textrm{d}t \nonumber \\{} & {} \quad = \int \nolimits _0^T u(t) \Big ( - \lambda v (T-t) + g(T-t) \Big ) \textrm{d}t + u(T) v_0. \end{aligned}$$
(3.10)

Notice that the only remainder that we have due to “non-locality” is the second term on the left-hand side. As \(\alpha \rightarrow 1\), 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 (3.7) and (3.9)).

Given suitable functions \(F: \mathbb R \rightarrow \mathbb R\) and a spectral decomposition of \(\textrm{L}\), it is natural to define \(F(\textrm{L})\) by the linear operator such that \(F(\textrm{L}) [\varphi _j] = F(\lambda _j) \varphi _j\). Therefore

$$\begin{aligned} F(\textrm{L}) [u] (x) = \sum _{j=1}^\infty F(\lambda _j) \langle u, \varphi _j \rangle \varphi _j (x). \end{aligned}$$
(4.1)

Recall that the solution of the local-in-time heat equation \(\partial _t u = - \textrm{L}u_0\) is given by

$$\begin{aligned} \mathcal {S}(t)v =\sum _{m=1}^{\infty } e^{-\lambda _m t} \left\langle v,\varphi _m\right\rangle \varphi _m = e^{-t \textrm{L}} v. \end{aligned}$$

Hence, we have the kernel representation

$$\begin{aligned} \mathcal {S}(t)v (x) = \int \nolimits _\Omega \mathbb {S}(t,x,y) u_0 (y) \textrm{d}y, \quad \mathbb {S}(t,x,y) = \sum _{j=0}^\infty e^{-\lambda _j t} \varphi _j (x) \varphi _j (y). \end{aligned}$$

In various examples we know upper and lower bounds for \(\mathbb {S}\).

Recall that, in this notation, the solution of the elliptic problem \(\textrm{L}^{-1} [f]\) is given precisely by

$$\begin{aligned} \textrm{L}^{-1} [f] (x)&= \sum _{j=1}^\infty \frac{ 1 }{\lambda _j } \left\langle f, \varphi _j\right\rangle \varphi _j (x) = \int \nolimits _0^\infty \sum _{j=1} ^\infty e^{-\lambda _j t} \left\langle f, \varphi _j\right\rangle \varphi _j (x) \textrm{d}t \\&= \int \nolimits _0^\infty e^{-t \textrm{L}} [f] (x) \textrm{d}t \end{aligned}$$

Or, equivalently written as in terms of the kernels

$$\begin{aligned} \mathbb {G}(x,y) = \int \nolimits _0^\infty \mathbb {S}(t,x,y) \textrm{d}t. \end{aligned}$$

Remark 4.1

Notice that the solution of \(\textrm{L}u + \mu u = f\) can be obtained similarly by

$$\begin{aligned} (\textrm{L}+ \mu )^{-1} = \int \nolimits _0^\infty e^{-\mu t} e^{-t \textrm{L}} \textrm{d}t. \end{aligned}$$

This is well defined for \(\mu > -\lambda _1\). Since the heat semigroup is non-negative, for \(\mu \in (-\lambda _1, \infty )\), for \(0 \le f \in L^2 (\Omega )\) we construct exactly one non-negative solutions in \(L^2 (\Omega )\). This is the ethos behind [6].

4.1 Caputo

We would like to obtain a representation formula for \(\mathcal {H}_{C}[u_0,f,0]\). We give two equivalent expressions: one in terms of the Mittag–Leffler functions \(E_\alpha \) and \(E_{\alpha ,\alpha }\) and through (4.1), and the other in terms of the Mainardi function \(\Phi _\alpha \) and the heat kernel \(\mathbb {S}\).

Due to (2.4) and (3.7), it is clear, from the solution of the coefficients of the spectral decomposition, that we can write

$$\begin{aligned} \mathcal H_C[u_0,0,0] (t,x)&= \sum _{j=1}^\infty u_j(t) \varphi _j (x) = \sum _{j=1}^\infty E_{\alpha }(-\lambda _j t^\alpha ) \langle u_0, \varphi _j \rangle \, \varphi _j (x) \\&= E_\alpha (-t^\alpha \textrm{L}) u_0. \end{aligned}$$

We denote this operator by \(\mathcal {S}_\alpha (t) :=E_\alpha (-t^\alpha \textrm{L})\). Using a similar argument, it can be deduced that the solution of (\(P_C\)) is given by

$$\begin{aligned} \mathcal H_C [u_0,f,0] (t)&=\mathcal {S}_\alpha (t)u_0 +\int \nolimits _0^t {\mathcal {P}}_\alpha (t-\tau )f(\tau ) \textrm{d}\tau , \end{aligned}$$
(4.2)

where

$$\begin{aligned} {\mathcal {P}}_\alpha (t) :=P_\alpha (t;\textrm{L}) =t^{\alpha -1} E_{\alpha ,\alpha }(-t^\alpha \textrm{L}). \end{aligned}$$
(4.3)

This formula is already presented in [11]. We emphasize that \(\mathcal {S}_\alpha (t)\) and \({\mathcal {P}}_\alpha (t)\) do not satisfy the semigroup property in general.

\(L^2\) theory

As usual, we have that

$$\begin{aligned} \Big \Vert \mathcal H_C [u_0,0,0] (t) \Big \Vert _{L^2 (\Omega )}^2&= \sum _{j=1}^\infty E_\alpha (-\lambda _j t^\alpha )^2 \langle u_0 , \varphi _j \rangle ^2 \\&\le E_\alpha (-\lambda _1 t^\alpha )^2 \sum _{j=1}^\infty \left\langle u_0,\varphi _j\right\rangle ^2 \\&= E_\alpha (-\lambda _1 t^\alpha )^2 \Vert u_0\Vert _{L^2 (\Omega )}^2. \end{aligned}$$

Therefore \(S_\alpha : L^2 (\Omega ) \rightarrow C(0,\infty ; L^2 (\Omega ))\) and we have the estimate

$$\begin{aligned} \Big \Vert \mathcal H_C [u_0,0,0] (t) \Big \Vert _{L^2 (\Omega )} \le E_\alpha (-\lambda _1 t^\alpha ) \Vert u_0\Vert _{L^2 (\Omega )} \end{aligned}$$

Remark 4.2

Notice that, due to the slow decay of \(E_\alpha \) we have that

$$\begin{aligned} {E_\alpha (-\lambda _n t^\alpha )} / {E_\alpha (-\lambda _1 t^\alpha )} \end{aligned}$$

does not converge to zero as \(t \rightarrow \infty \) for \(n > 1\). Hence, unlike in the case \(\alpha = 1\) we cannot simplify \(\mathcal H_C [u_0,0,0] (t)\) to \(E_\alpha (-\lambda _1 t^\alpha ) \langle u_0, \varphi _1 \rangle \varphi _1\).

Similarly, we have an \(L^2\) estimate for finite time \(t\in [0,T]\),

$$\begin{aligned} \Big \Vert \mathcal H_C [0,f,0] (t) \Big \Vert _{L^2 (\Omega )}^2&\le \left( \int \nolimits _0^t \tau ^{\alpha -1} E_{\alpha ,\alpha } (-\lambda _1 \tau ^\alpha ) \textrm{d}\tau \right) \sup _{\tau \in [0,T]} \Vert f(\tau ) \Vert _{L^2(\Omega )}^2\\&=\frac{1-E(-\lambda _1 t^\alpha )}{\lambda _1} \sup _{\tau \in [0,T]} \Vert f(\tau ) \Vert _{L^2(\Omega )}^2. \end{aligned}$$

The solution for \(f(t,x) = f(x)\) is cleanly expressed as

$$\begin{aligned} \mathcal H_C [0,f,0] (t)&= \sum _{j=1}^\infty \int \nolimits _0^\infty \tau ^{\alpha -1} E_{\alpha ,\alpha } (-\lambda _j \tau ^\alpha ) \textrm{d}\tau \,\langle f, \varphi _j \rangle \varphi _j \\&= \sum _{j=1}^\infty \frac{1-E_\alpha (-\lambda _j t^\alpha )}{\lambda _j} \langle f, \varphi _j \rangle \varphi _j. \end{aligned}$$

Hence its asymptotic behaviour is simply given by

$$\begin{aligned} \lim _{t\rightarrow \infty } \mathcal H_C [0,f,0] (t)&= \sum _{j=1}^\infty \frac{1}{\lambda _j} \langle f, \varphi _j \rangle \varphi _j =\textrm{L}^{-1} f, \end{aligned}$$

i.e. the solution of the \(\textrm{L}\)-Poisson equation.

Remark 4.3

Notice that the notation \(\textrm{L}^{-1}\) for the Green operator (and more generally \((\textrm{L}+\mu )^{-1}\) for the resolvent) is consistent with the notation (4.1).

Properties of the kernel representation In [11, Sect. 2.1] it is shown that, as \(t \searrow 0\) we have

$$\begin{aligned}{} & {} \left\| \mathcal {S}_\alpha (t) [ v] - v \right\| _{L^2(\Omega )} \longrightarrow 0,\nonumber \\{} & {} \left\| \tfrac{t^{1-\alpha }}{\Gamma (\alpha )} {\mathcal {P}}_\alpha (t) [v] - v \right\| _{L^2(\Omega )} \longrightarrow 0. \end{aligned}$$
(4.4)

We refer to Gal and Warma [11, Sect. 2.2] for \(L^p\)-\(L^q\) estimates which are recovered in terms of the contractivity of \(\mathcal {S}(t)=e^{-t \textrm{L}}\). We stress that \(\mathcal {S}_\alpha (t):L^p(\Omega ) \rightarrow L^q(\Omega )\) only when \(\frac{n}{2s}(\frac{1}{p}-\frac{1}{q})<1\) and \({\mathcal {P}}_\alpha (t):L^p(\Omega ) \rightarrow L^q(\Omega )\) only when \(\frac{n}{2s}(\frac{1}{p}-\frac{1}{q})<2\).

4.2 Riemann–Liouville

Similarly, since we have already defined (4.3), going back to (3.9) we observe that

$$\begin{aligned} \mathcal H_R [u_0,f,0] (t)&={\mathcal {P}}_\alpha (t)u_0 +\int \nolimits _0^t {\mathcal {P}}_\alpha (t-\tau )f(\tau ) \textrm{d}\tau . \end{aligned}$$

Since \(P_\alpha (t; \lambda ) \sim \frac{t^{\alpha - 1}}{\Gamma (\alpha )}\) at 0 (see Remark 3.2), we cannot expect to construct a theory with \({\mathcal {P}}_\alpha : L^2 (\Omega ) \rightarrow L^\infty (0,T; L^2 (\Omega ))\). Nevertheless, we do have \({\mathcal {P}}_\alpha :L^2(\Omega ) \rightarrow t^{\alpha -1} L^\infty (0,T;L^2(\Omega ))\) with the corrected estimate

$$\begin{aligned} \Big \Vert \mathcal H_R [u_0,0,0] (t) \Big \Vert _{L^2 (\Omega )} \le t^{\alpha - 1} E_{\alpha ,\alpha } (-\lambda _1 t^\alpha ) \Vert u_0 \Vert _{L^2}. \end{aligned}$$

Remark 4.4

We point out that \( \mathcal H_R[0,f,0] (t) = \mathcal H_C[0,f,0] (t) \). Therefore, we define simply

$$\begin{aligned} \mathcal H [0,f,0] (t) :=\mathcal H_R [0,f,0] (t). \end{aligned}$$

4.3 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 \(\mathcal {S}_\alpha (t)[v], {\mathcal {P}}_\alpha (t)[v] \in L^2 (\Omega )\). In fact, due to the inverse linear (respectively quadratic) decay of the Mittag–Leffler functions stated in (3.5), they are also in \(\mathrm H(\Omega )\).

The continuity at \(t = 0\) follows from (4.4). Due to the bounds presented before, in fact \(\mathcal {S}_\alpha (\cdot )[u_0], {\mathcal {P}}_\alpha (\cdot )[u_0] \in C([\varepsilon ,T]; \mathrm H(\Omega ))\). The integral part is even easier.

The time differentiability follows from the explicit computation of \(\frac{du}{dt}\) as in [11, Proposition 2.1.9]. We point out that \(\mathcal {S}_\alpha ' = {\mathcal {P}}_\alpha \textrm{L}\). This can be done similarly for the Riemann–Liouville derivative. \(\square \)

Remark 4.5

In [11] the authors deal with the notion of strong solution. This is also possible in our setting, but our interest is in the very weak solutions described below.

4.4 Very weak formulation when \(h = 0\)

Due to (3.10), for every \(T > 0\), \(u_0, v_0 \in L^2 (\Omega )\) and \(f,g \in L^2((0,T) \times \Omega )\) we have that

$$\begin{aligned} \begin{aligned}&\int \nolimits _0^T \int \nolimits _\Omega \mathcal H_R [v_0,g,0](T-t,x) f(t,x) \textrm{d}x \textrm{d}t \\&\qquad +\, \int \nolimits _\Omega u_0 (x) \int \nolimits _0^T \frac{ t^{-\alpha } }{\Gamma (1-\alpha )} \mathcal H_R[v_0,g,0] (T-t,x) \textrm{d}t \textrm{d}x \\&\quad = \int \nolimits _0^T \int \nolimits _\Omega \mathcal H_C [u_0, f, 0](t , x) g(T-t,x) \textrm{d}x \textrm{d}t \\&\qquad + \,\int \nolimits _\Omega \mathcal {H}_C[u_0,f,0](T,x) v_0 (x) \textrm{d}x. \end{aligned} \end{aligned}$$

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 \(\mathcal {S}_\alpha (t)\), \({\mathcal {P}}_\alpha (t)\) that are directly related to those of \(\mathcal {S}(t)=e^{-t\textrm{L}}\).

Remark 4.6

Notice that we could equivalently write that for every \(v_0 \in L_c^\infty (\Omega )\) and for a.e. \(t > 0\) we have

$$\begin{aligned} \begin{aligned} \int \nolimits _\Omega u(t,x) v_0 (x) \textrm{d}x&= \int \nolimits _0^t \int \nolimits _\Omega f(\sigma ,x) {\mathcal {P}}_\alpha [v_0](t-\sigma ,x) \textrm{d}\sigma \textrm{d}x \\&\quad + \int \nolimits _\Omega u_0 (x) \int \nolimits _0^t \frac{ \sigma ^{-\alpha }}{\Gamma (1-\alpha )} {\mathcal {P}}_\alpha [v_0](t-\sigma ,x) \textrm{d}\sigma \textrm{d}x. \end{aligned} \end{aligned}$$

This formulation is nicer for the \(L^\infty \) estimates in time.

5 Time-fractional problem when \(h = 0\) beyond \(L^2\)

5.1 Weighted \(L^1\) and \(L^\infty \) theory

When we leave the \(L^2\) framework and consider a general functional space Y (which for us will mainly be weighted Lebesgue spaces), we need to look beyond simple properties of \(E_\alpha \) and \(E_{\alpha ,\alpha }\). It is here where the Mainardi function comes into play.

For example, [11, Proposition 2.1.3] uses the representation (2.5), (2.6), and properties of the Mainardi function to show that

$$\begin{aligned} \Vert \mathcal {S}(t) u_0 \Vert _Y \le M \Vert u_0 \Vert _Y \end{aligned}$$

for all \(u_0 \in Y\) implies

$$\begin{aligned} \Vert \mathcal {S}_\alpha (t) u_0 \Vert _Y \le C \Vert u_0 \Vert _Y \quad \text {and} \quad t^{1-\alpha } \Vert {\mathcal {P}}_\alpha (t) u_0 \Vert _Y \le C \Vert u_0 \Vert _Y \end{aligned}$$

Therefore, for suitably integrable f, similar properties hold for \(\mathcal H [0,f,0]\). In particular, the mass contractivity \(\Vert \mathcal {S}(t) u_0 \Vert _{L^1(\Omega )} \le \Vert u_0\Vert _{L^1(\Omega )}\) allows us to construct an \(L^1 (\Omega )\) theory.

Furthermore, it is common that the first eigenfunction \(\varphi _1(x)\) satisfies the boundary condition with a rate \(\delta (x)^\gamma \) for some \(\gamma \) positive. This is the case, for example with the Restricted Fractional Laplacian (\(\gamma = s\)), Censored Fractional Laplacian (\(\gamma = 2s - 1\) which is only defined for \(s > \frac{1}{2}\)), and the Spectral Fractional Laplacian (\(\gamma = 1\)). 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 conditions 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 \(\delta ^\gamma L^\infty (\Omega )\) (a weighted space containing \(\varphi _j\)). The worst admissible data is in \(L^1 (\Omega , \delta ^\gamma )\), a fact guaranteed by the lower estimate

$$\begin{aligned} \textrm{L}^{-1} [f] (x) \ge c_1 \delta (x)^\gamma \int \nolimits _\Omega f(y) \delta (y)^\gamma \textrm{d}y, \quad \forall f \ge 0, \end{aligned}$$

where \(c_1 > 0\).

In general, under (G1), we have that

$$\begin{aligned} \textrm{L}^{-1} [f] \asymp \delta ^\gamma , \qquad \forall 0 \le f \in L^\infty _c (\Omega ). \end{aligned}$$

Remark 5.1

In the local-in-time setting in [6] we showed the nice regularisation

$$\begin{aligned} \mathcal {S}(t): L^1(\Omega , \delta ^\gamma ) \rightarrow \delta ^\gamma L^\infty (\Omega ) \end{aligned}$$

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

$$\begin{aligned} \frac{ \mathcal {S}_\alpha (t)[u_0] (x) }{ \delta (x) ^{\beta _1} } = \int \nolimits _\Omega \left( \int \nolimits _{0}^{\infty } \Phi _\alpha (\tau ) \frac{\mathbb {S}({\tau t^{\alpha }} ,x,y ) }{\delta (x)^{\beta _1} \delta (y)^{\beta _2} } \textrm{d}\tau \right) u_0(y) \delta (y)^{\beta _2} \textrm{d}y. \end{aligned}$$

Therefore, the regularisation relies on the integrals

$$\begin{aligned} \int \nolimits _{0}^{\infty } \Phi _\alpha (\tau ) \frac{ \mathbb {S}({\tau t^{\alpha }} ,x,y ) }{\delta (x) ^{\beta _1} \delta (y)^{\beta _2} } \textrm{d}\tau . \end{aligned}$$

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 \(L^1(0,T;L^1(\Omega ,\delta ^\gamma ))\)) Let \(0\le t_0<t_1\le T\) and \(A\subset \Omega \). Then

$$\begin{aligned}&\int \nolimits _{t_0}^{t_1}\int \nolimits _{A} |\mathcal {H}_C[u_0,f,0]| \delta (x)^\gamma \textrm{d}x \textrm{d}t \\&\quad \le \omega _T(t_1-t_0)\omega (|A|) \left( \int \nolimits _{\Omega } |u_0(x)|\delta ^\gamma \textrm{d}x +\int \nolimits _{0}^{T}\int \nolimits _{\Omega } |f(t,x)|\delta (x)^\gamma \textrm{d}x \textrm{d}t \right) . \end{aligned}$$

Here and below \(\omega \) represents a modulus of continuity, i.e. a non-decreasing, non-negative function such that \(\omega (0^+) = 0\). We denote the dependence by sub-indexes.

Proof

By splitting into positive and negative parts, we may assume that \(u_0,f,u\ge 0\).

Step 1: Time compactness. Take \(\phi (t,x)=\chi _{[t_0,t_1]}(t)\varphi _1(x)\), so that

$$\begin{aligned}&\mathcal {H}[0,\phi ,0](t,x)\\&\quad =\int \nolimits _{0}^{t} P_\alpha (t-\tau ;\lambda _1) \phi (\tau ,x) \textrm{d}\tau =\int \nolimits _{(t-t_1)_+}^{(t-t_0)_+} -\frac{1}{\lambda _1} \frac{d}{d\tau }E_\alpha (-\lambda _1 \tau ^\alpha ) \textrm{d}\tau \,\varphi _1(x)\\&\quad =\frac{E_\alpha (-\lambda _1(t-t_1)_+^\alpha )-E_\alpha (-\lambda _1(t-t_0)_+^\alpha ))}{\lambda _1} \varphi _1(x) =\omega _T(t_1-t_0)\varphi _1(x). \end{aligned}$$

Hence,

$$\begin{aligned}&\int \nolimits _{t_0}^{t_1}\int \nolimits _{\Omega } |u(t,x)|\delta (x)^\gamma \textrm{d}x \textrm{d}t\\&\quad \le \omega _T(t_1-t_0) \left( \int \nolimits _{\Omega } |u_0(x)|\delta ^\gamma \textrm{d}x +\int \nolimits _{0}^{T}\int \nolimits _{\Omega } |f(t,x)|\delta (x)^\gamma \textrm{d}x \textrm{d}t \right) . \end{aligned}$$

Step 2: Space compactness. Take \(\phi (t,x)=\chi _A(x)\varphi _1(x)\), so that

$$\begin{aligned}&\mathcal {H}[0,\phi ,0](t,x)\\&\quad =\int \nolimits _{0}^{t} P_\alpha (t-\tau ;\textrm{L})[\phi ](\tau ,x) \textrm{d}\tau =\int \nolimits _{0}^{t} P_\alpha (\tau ) \textrm{d}\tau \,[\chi _A\varphi _1](x)\\&\quad \le \int \nolimits _{0}^{\infty } P_\alpha (\tau ) \textrm{d}\tau \,[\chi _A\varphi _1](x)\\&\quad =\int \nolimits _{0}^{\infty } \left( \alpha \tau ^{\alpha -1} \int \nolimits _{0}^{\infty } \sigma \Phi _\alpha (\sigma ) \mathcal {S}(\sigma \tau ^\alpha ) \textrm{d}\sigma \right) \textrm{d}\tau \,[\chi _A\varphi _1](x)\\&\quad =\int \nolimits _{0}^{\infty } \left( \int \nolimits _{0}^{\infty } \mathcal {S}(\sigma \tau ^\alpha ) \textrm{d}(\sigma \tau ^\alpha ) \right) \Phi _\alpha (\sigma ) \textrm{d}\sigma \,[\chi _A\varphi _1](x) =\mathcal {G}[\chi _A\varphi _1](x)\\&\quad \le \omega (|A|)\delta (x)^\gamma , \end{aligned}$$

where the last estimate follows from (2.8) and the argument in [6, Lemma 7.3]. Consequently,

$$\begin{aligned}&\int \nolimits _{0}^{T}\int \nolimits _{A} |u(t,x)|\delta (x)^\gamma \textrm{d}x \textrm{d}t\\&\quad \le \omega (|A|) \left( \int \nolimits _{\Omega } |u_0(x)|\delta (x)^\gamma \textrm{d}x +\int \nolimits _{0}^{T}\int \nolimits _{\Omega } |f(t,x)|\delta (x)^\gamma \textrm{d}x \textrm{d}t \right) . \end{aligned}$$

Step 3: Space-time compactness. Using Step 1–2, we have

$$\begin{aligned}&\int \nolimits _{t_0}^{t_1}\int \nolimits _{A} |u(t,x)|\delta (x)^\gamma \textrm{d}x \textrm{d}t\\&\quad \le \left( \int \nolimits _{t_0}^{t_1}\int \nolimits _{\Omega } |u(t,x)|\delta (x)^\gamma \textrm{d}x \textrm{d}t \right) ^{\frac{1}{2}} \left( \int \nolimits _{0}^{T}\int \nolimits _{A} |u(t,x)|\delta (x)^\gamma \textrm{d}x \textrm{d}t \right) ^{\frac{1}{2}} \\&\quad \le {\omega _T}(t_1-t_0) \omega (|A|) \left( \int \nolimits _{\Omega } |u_0(x)|\delta (x)^\gamma \textrm{d}x +\int \nolimits _{0}^{T}\int \nolimits _{\Omega } |f(t,x)|\delta (x)^\gamma \textrm{d}x \textrm{d}t \right) . \end{aligned}$$

Step 4: Space-time compactness for signed data In general we split \(u_0=(u_0)_+-(u_0)_-\) and \(f=f_+-f_-\), which yields

$$\begin{aligned} u=\mathcal {H}[(u_0)_+,f_+,0]-\mathcal {H}[(u_0)_-,f_-,0]. \end{aligned}$$

Then Step 3 can be applied to each summand, completing the proof. \(\square \)

To obtain \(L^1_\textrm{loc}(\Omega )\) compactness, we apply the above estimate to each \(K\Subset \Omega \).

Remark 5.3

Notice that the only crucial ingredients in the proof above are that \(\varphi _1 \asymp \delta ^\gamma \) and \(\mathcal {G}[\chi _A \varphi _1] \le \omega (|A|) \varphi _1\), which are minimal hypotheses on \(\mathcal {G}\) that use only mild integrability assumptions, not the exact shape. In Lipschitz domains, where it can happen that \(\varphi _1 \not \asymp \delta ^\gamma \) for any \(\gamma \), the correct weight is \(\varphi _1\).

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 Sect. 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, \(\mathcal {H}[u_{0k},f_k,0] \rightarrow \mathcal {H}[u_0,f,0]\) in \( L^1(0,T; L^1(\Omega ,\delta ^\gamma ))\). Due to the regularity of \(\mathcal {H}[0,\phi ,0]\) 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 \(\phi \) smooth

$$\begin{aligned} \int \nolimits _0^T \int \nolimits _\Omega w(t,x) \phi (T-t,x) \textrm{d}x \textrm{d}t = 0. \end{aligned}$$

For any \(K \Subset \Omega \), taking

$$\begin{aligned} \phi (t,x) = \chi _K (x) {{\,\textrm{sign}\,}}w(T-t,x), \end{aligned}$$

we conclude that \(w = 0\) a.e. in \([0,T] \times \Omega \). This completes the proof. \(\square \)

5.3 Sharp boundary behaviour for good data

We derive estimates for

$$\begin{aligned} \mathbb {P}_\alpha (t,x,y) =\alpha t^{\alpha -1} \int \nolimits _0^\infty \tau \Phi _\alpha (\tau ) \mathbb {S}(\tau t^\alpha ,x,y) \textrm{d}\tau , \quad t>0,\,x,y\in \Omega , \end{aligned}$$

using the corresponding estimates of the heat kernel \(\mathbb {S}\).

Remark 5.4

We point out the pointwise estimate

$$\begin{aligned} \mathbb P_\alpha (t,x,y)&= \alpha t^{\alpha -1} \int \nolimits _0^\infty (\sigma t^{-\alpha } ) \Phi ( \sigma t^{-\alpha } ) \mathbb {S}(\sigma ,x,y) \textrm{d}\sigma t^{-\alpha } \\&= \alpha t^{-1} \int \nolimits _0^\infty (\sigma t^{-\alpha } ) \Phi ( \sigma t^{-\alpha } ) \mathbb {S}(\sigma ,x,y) \textrm{d}\sigma \\&\le \alpha t^{-1} \Vert \tau \Phi (\tau ) \Vert _{L^\infty } \mathbb {G}(x,y). \end{aligned}$$

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.

Lemma 5.5

We have that \(\mathbb {P}_\alpha (\cdot ,x,\cdot )/\delta (x)^\gamma \in L^1(0,\infty ;L^1(\Omega ,\delta ^\gamma ))\), uniformly in \(x\in \Omega \).

Proof

We compute

$$\begin{aligned}&\int \nolimits _0^\infty \int \nolimits _\Omega \frac{ \mathbb {P}_\alpha (t,x,y) }{\delta (x)^\gamma } \delta (y)^\gamma \textrm{d}y\textrm{d}t\\&\quad =\int \nolimits _0^\infty \int \nolimits _\Omega \alpha t^{\alpha -1} \int \nolimits _0^\infty \tau \Phi _\alpha (\tau ) \frac{ \mathbb {S}(\tau t^\alpha ,x,y) }{ \delta (x)^\gamma } \textrm{d}\tau \cdot \delta (y)^\gamma \textrm{d}y\textrm{d}t\\&\quad =\int \nolimits _0^\infty \int \nolimits _\Omega \int \nolimits _0^\infty \Phi _\alpha (\tau ) \frac{ \mathbb {S}(\tau t^\alpha ,x,y) }{ \delta (x)^\gamma } \alpha \tau t^{\alpha -1} \textrm{d}t \cdot \delta (y)^\gamma \textrm{d}y\textrm{d}\tau \\&\quad =\int \nolimits _0^\infty \int \nolimits _\Omega \int \nolimits _0^\infty \Phi _\alpha (\tau ) \frac{ \mathbb {S}(\sigma ,x,y) }{ \delta (x)^\gamma } \textrm{d}\sigma \cdot \delta (y)^\gamma \textrm{d}y\textrm{d}\tau \\&\quad =\int \nolimits _\Omega \frac{\mathbb {G}(x,y)}{\delta (x)^\gamma } \delta (y)^\gamma \textrm{d}y. \end{aligned}$$

Using (2.8), the latter integral is bounded uniformly for \(x\in \Omega \), as desired. \(\square \)

In [6] we proved that \(\mathcal {S}(t): \mathcal M (\Omega ,\delta ^\gamma ) \rightarrow \delta ^\gamma C(\overline{\Omega })\) is a continuous operator for any \(t > 0\). By the continuity of the linear operator \(\mathcal {S}(t)\) and the semigroup property, we deduce that

$$\begin{aligned} \frac{ \mathbb {S}(t,x,y)}{\delta (x)^\gamma } = \frac{\mathcal {S}(t)[\delta _y] (x)}{\delta (x)^\gamma }\in C((0,T] \times \overline{\Omega }\times \Omega ). \end{aligned}$$

Here \(\delta _y\) is the Dirac delta distribution centred at y, whereas \(\delta \) is the distance function. Due to this continuity, the following limit is well defined:

$$\begin{aligned} D_\gamma \mathbb {S}(t, \zeta , y) = \lim _{x \rightarrow \zeta } \frac{ \mathbb {S}(t,x,y)}{\delta (x)^\gamma }. \end{aligned}$$
(5.1)

Lemma 5.6

For any \(\zeta \in \partial \Omega \),

$$\begin{aligned} D_\gamma \mathbb {P}_\alpha (\cdot ,\zeta ,\cdot ):=\lim _{\Omega \ni x\rightarrow \zeta } \frac{\mathbb {P}_\alpha (\cdot ,x,\cdot )}{\delta (x)^\gamma } \end{aligned}$$

exists in \(L^1(0,\infty ;L^1(\Omega ,\delta ^\gamma ))\) and is equal to

$$\begin{aligned} D_\gamma \mathbb {P}_\alpha (t,\zeta ,y) =\alpha t^{\alpha -1} \int \nolimits _0^\infty \tau \Phi _\alpha (\tau ) D_\gamma \mathbb {S}(\tau t^\alpha ,\zeta ,y) \textrm{d}\tau , \quad t>0,\,\zeta \in \partial \Omega ,\,y\in \Omega . \end{aligned}$$

Proof

We estimate

$$\begin{aligned}&\int \nolimits _0^\infty \int \nolimits _\Omega \bigg | \frac{ \mathbb {P}_\alpha (t,x,y) }{\delta (x)^\gamma } -\alpha t^{\alpha -1} \int \nolimits _0^\infty \tau \Phi _\alpha (\tau ) D_\gamma \mathbb {S}(\tau t^\alpha ,\zeta ,y) \textrm{d}\tau \bigg | \delta (y)^\gamma \textrm{d}y\textrm{d}t\\&\quad \le \int \nolimits _0^\infty \int \nolimits _\Omega \alpha t^{\alpha -1} \int \nolimits _0^\infty \tau \Phi _\alpha (\tau ) \bigg | \frac{ \mathbb {S}(\tau t^\alpha ,x,y) }{ \delta (x)^\gamma } -D_\gamma \mathbb {S}(\tau t^\alpha ,\zeta ,y) \bigg | \textrm{d}\tau \cdot \delta (y)^\gamma \textrm{d}y\textrm{d}t\\&\quad =\int \nolimits _0^\infty \int \nolimits _\Omega \int \nolimits _0^\infty \Phi _\alpha (\tau ) \bigg | \frac{ \mathbb {S}(\tau t^\alpha ,x,y) }{ \delta (x)^\gamma } -D_\gamma \mathbb {S}(\tau t^\alpha ,\zeta ,y) \bigg | \alpha \tau t^{\alpha -1} \textrm{d}t \cdot \delta (y)^\gamma \textrm{d}y\textrm{d}\tau \\&\quad =\int \nolimits _0^\infty \int \nolimits _\Omega \int \nolimits _0^\infty \Phi _\alpha (\tau ) \bigg | \frac{ \mathbb {S}(\sigma ,x,y) }{ \delta (x)^\gamma } -D_\gamma \mathbb {S}(\sigma ,\zeta ,y) \bigg | \textrm{d}\sigma \cdot \delta (y)^\gamma \textrm{d}y\textrm{d}\tau \\&\quad =\int \nolimits _\Omega \int \nolimits _0^\infty \bigg | \frac{ \mathbb {S}(\sigma ,x,y) }{ \delta (x)^\gamma } -D_\gamma \mathbb {S}(\sigma ,\zeta ,y) \bigg | \textrm{d}\sigma \cdot \delta (y)^\gamma \textrm{d}y. \end{aligned}$$

By Chan et al. [6, Theorem 6.1], the quotient \(\mathbb {S}(\sigma ,x,y)/\delta (x)^\gamma \) is continuous in x up to the boundary for each \((\sigma ,y)\in (0,\infty )\times \Omega \). In particular, the last integrand tends to 0 a.e. Moreover, this integrand is dominated by \( \frac{ \mathbb {S}(\sigma ,x,y) }{ \delta (x)^\gamma } +D_\gamma \mathbb {S}(\sigma ,\zeta ,y) \) which is integrable in \(L^1(0,\infty ;L^1(\Omega ,\delta ^\gamma ))\) as shown in Lemma 5.5. By Dominated Convergence Theorem, the last integral tends to zero as \(x \rightarrow \zeta \).

\(\square \)

Now we prove that if \(\phi \in \delta ^\gamma L^\infty ( (0,T)\times \Omega ) \), then so is \(\mathcal {H}[0,\phi ,0]\), and in addition \(D_\gamma \mathcal {H}[0,\phi ,0]\) exists.

Lemma 5.7

For any \(\phi \in \delta ^\gamma L^\infty ((0,T)\times \Omega )\),

$$\begin{aligned} \mathcal {H}[0,\phi ,0](t,x) =\int \nolimits _0^t\int \nolimits _\Omega \mathbb {P}_\alpha (t-\tau ,x,y) \phi (\tau ,y) \textrm{d}y \textrm{d}\tau \end{aligned}$$

lies also in \(\delta ^\gamma L^\infty ((0,T)\times \Omega )\).

Proof

We estimate

$$\begin{aligned} \frac{\mathcal {H}[0,\phi ,0](t,x)}{\delta (x)^\gamma }&=\int \nolimits _0^t\int \nolimits _\Omega \frac{\mathbb {P}_\alpha (t-\tau ,x,y)}{\delta (x)^\gamma } \phi (\tau ,y) \textrm{d}y \textrm{d}\tau \\&\le \left\| \frac{\phi }{\delta ^\gamma } \right\| _{L^\infty ((0,T)\times \Omega )} \int \nolimits _0^t\int \nolimits _\Omega \frac{\mathbb {P}_\alpha (t-\tau ,x,y)}{\delta (x)^\gamma } \delta (y)^\gamma \textrm{d}y \textrm{d}\tau \\&\le \left\| \frac{\phi }{\delta ^\gamma } \right\| _{L^\infty ((0,T)\times \Omega )} \int \nolimits _0^\infty \int \nolimits _\Omega \frac{\mathbb {P}_\alpha (\tau ,x,y)}{\delta (x)^\gamma } \delta (y)^\gamma \textrm{d}y \textrm{d}\tau . \end{aligned}$$

By Lemma 5.5, the last double integral is bounded by a uniform constant, as desired. \(\square \)

Lemma 5.8

For any \(\phi \in \delta ^\gamma L^\infty ((0,T)\times \Omega )\),

$$\begin{aligned} D_\gamma \mathcal {H}[0,\phi ,0](t,\zeta ):=\lim _{\Omega \ni x\rightarrow \zeta } \frac{\mathcal {H}[0,\phi ,0](t,x)}{\delta (x)^\gamma } \end{aligned}$$

exists in \(L^\infty ((0,T)\times \partial \Omega )\) and is equal to

$$\begin{aligned} D_\gamma \mathcal {H}[0,\phi ,0](t,\zeta ) =\int \nolimits _0^t\int \nolimits _\Omega D_\gamma \mathbb {P}_\alpha (t-\tau ,\zeta ,y) \phi (\tau ,y) \textrm{d}y \textrm{d}\tau . \end{aligned}$$

Proof

We estimate

$$\begin{aligned}&\bigg | \frac{\mathcal {H}[0,\phi ,0](t,x)}{\delta (x)^\gamma } -\int \nolimits _0^t\int \nolimits _\Omega D_\gamma \mathbb {P}_\alpha (t-\tau ,\zeta ,y) \phi (\tau ,y) \textrm{d}y \textrm{d}\tau \bigg |\\&\quad \le \left\| \frac{\phi }{\delta ^\gamma } \right\| _{L^\infty ((0,T)\times \Omega )} \int \nolimits _0^t\int \nolimits _\Omega \bigg | \frac{\mathbb {P}_\alpha (t-\tau ,x,y)}{\delta (x)^\gamma } -D_\gamma \mathbb {P}_\alpha (t-\tau ,\zeta ,y) \bigg | \,\delta (y)^\gamma \textrm{d}y \textrm{d}\tau \\&\quad \le \left\| \frac{\phi }{\delta ^\gamma } \right\| _{L^\infty ((0,T)\times \Omega )} \int \nolimits _0^\infty \int \nolimits _\Omega \bigg | \frac{\mathbb {P}_\alpha (\tau ,x,y)}{\delta (x)^\gamma } -D_\gamma \mathbb {P}_\alpha (\tau ,\zeta ,y) \bigg | \,\delta (y)^\gamma \textrm{d}y \textrm{d}\tau . \end{aligned}$$

By Lemma 5.6, the last integral converges to 0 as \(x\rightarrow \zeta \). \(\square \)

6 Singular boundary condition when \(h \ne 0\)

6.1 Concentration of f towards singular boundary data

Definition 6.1

We define a very weak solution for \(u_0 = 0, f = 0, \) and \(h \ne 0\) as a function \(u \in L^1 (0,T; L^1(\Omega ,\delta ^\gamma ))\) which satisfies

$$\begin{aligned} \int \nolimits _0^T \int \nolimits _\Omega u(t,x) \phi (T-t,x) \textrm{d}x\textrm{d}t&= \int \nolimits _0^T \int \nolimits _{\partial \Omega } h(t, \zeta ) D_\gamma \mathcal H[0,\phi ,0] (T-t, \zeta ) \textrm{d}\zeta \textrm{d}t, \end{aligned}$$
(6.1)

for any \(\phi \in \delta ^\gamma L^\infty ((0,T)\times \Omega )\).

Given this definition, uniqueness is trivial.

Lemma 6.2

Suppose \(u\in L^1(0,T;L^1_\textrm{loc}(\Omega ))\) satisfies

$$\begin{aligned} \int \nolimits _0^T \int \nolimits _\Omega u(t,x) \phi (T-t,x) \textrm{d}x\textrm{d}t=0, \quad \forall \phi \in L^\infty (0,T;L^\infty _c(\Omega )), \end{aligned}$$

then \(u\equiv 0\) in \(\Omega \). In particular, the same implication holds for all \(u\in L^1(0,T;L^1(\Omega ,\delta ^\gamma ))\) with test functions \(\phi \in \delta ^\gamma L^\infty ((0,T)\times \Omega )\).

Proof

For every \(K\Subset \Omega \), choosing \(\phi (t,x)={{\,\textrm{sign}\,}}u(T-t,x) \chi _{K}(x)\) yields \(\int \nolimits _0^T\int \nolimits _K |u| \textrm{d}x\textrm{d}t=0\). \(\square \)

First we check that the solution lies in the correct weighted space.

Lemma 6.3

Given \(h\in L^1((0,T)\times \partial \Omega )\), \(u=\mathcal {H}[0,0,h]\) given by (2.7) lies in \(L^1(0,T;L^1(\Omega ,\delta ^\gamma ))\). Moreover,

$$\begin{aligned} \int \nolimits _0^T\int \nolimits _{\Omega } u(x,t) \delta (x)^\gamma \textrm{d}x\textrm{d}t \le C\int \nolimits _0^T\int \nolimits _{\partial \Omega } h(\tau ,\zeta ) \textrm{d}\zeta \textrm{d}\tau . \end{aligned}$$

Proof

We express

$$\begin{aligned} \int \nolimits _0^T u(t,x)\textrm{d}t&=\int \nolimits _0^T\int \nolimits _0^t\int \nolimits _{\partial \Omega } D_\gamma \mathbb {P}_\alpha (t-\tau ,x,\zeta ) h(\tau ,\zeta ) \textrm{d}\zeta \textrm{d}\tau \textrm{d}t\\&=\int \nolimits _0^T\int \nolimits _{\partial \Omega } \bigg [ \int \nolimits _\tau ^T D_\gamma \mathbb {P}_\alpha (t-\tau ,x,\zeta ) \textrm{d}t \bigg ] \cdot h(\tau ,\zeta ) \textrm{d}\zeta \textrm{d}\tau \\&=\int \nolimits _0^T\int \nolimits _{\partial \Omega } \bigg [ \int \nolimits _{0}^{T-\tau } D_\gamma \mathbb {P}_\alpha (T-\tau -t,x,\zeta ) \textrm{d}t \bigg ] \cdot h(\tau ,\zeta ) \textrm{d}\zeta \textrm{d}\tau \\&=\int \nolimits _0^T\int \nolimits _{\partial \Omega } \bigg [ \int \nolimits _{0}^{T-\tau } D_\gamma \mathbb {P}_\alpha (t,x,\zeta ) \textrm{d}t \bigg ] \cdot h(\tau ,\zeta ) \textrm{d}\zeta \textrm{d}\tau \\&\le \int \nolimits _{\partial \Omega } \bigg [\int \nolimits _0^\infty D_\gamma \mathbb {P}_\alpha (t,x,\zeta ) \textrm{d}t\bigg ] \bigg [\int \nolimits _0^T h(\tau ,\zeta ) \textrm{d}\tau \bigg ] \textrm{d}\zeta . \end{aligned}$$

Using Lemma 5.6, the last t-integral is in \(L^1(\Omega ,\delta ^\gamma )\) (in variable x) and hence the result follows. \(\square \)

Integrating by parts, we see that the only possible solution is precisely (2.7).

Lemma 6.4

Given \(h\in L^1((0,T)\times \partial \Omega )\), \(u=\mathcal {H}[0,0,h]\) given by (2.7) satisfies (6.1).

Proof

Keeping in mind that

$$\begin{aligned} \begin{aligned}&\mathcal {H}[0,\phi ,0](t,x)\\&\quad =\bigg [\int \nolimits _0^t {\mathcal {P}}_\alpha (t-\tau )\phi (\tau ) \textrm{d}\tau \bigg ](x) =\int \nolimits _0^t\int \nolimits _\Omega \mathbb {P}_\alpha (t-\tau ,x,y) \phi (\tau ,y) \textrm{d}y \textrm{d}\tau , \end{aligned} \end{aligned}$$

we verify that

$$\begin{aligned}&\int \nolimits _0^T\int \nolimits _\Omega \bigg [ \int \nolimits _0^t\int \nolimits _{\partial \Omega } (D_\gamma \mathbb {P}_\alpha )(t-\tau ,x,\zeta ) h(\tau ,\zeta ) \textrm{d}\zeta \textrm{d}\tau \bigg ] \phi (T-t,x) \textrm{d}x\textrm{d}t\\&\quad =\int \nolimits _0^T\int \nolimits _{\partial \Omega } \bigg [ \int \nolimits _\tau ^T\int \nolimits _{\Omega } (D_\gamma \mathbb {P}_\alpha )(t-\tau ,x,\zeta ) \phi (T-t,x) \textrm{d}x\textrm{d}t \bigg ] h(\tau ,\zeta ) \textrm{d}\zeta \textrm{d}\tau \\&\quad =\int \nolimits _0^T\int \nolimits _{\partial \Omega } D_\gamma \bigg [ \int \nolimits _{0}^{T-\tau }\int \nolimits _{\Omega } \mathbb {P}_\alpha (T-t-\tau ,x,\cdot ) \phi (t,x) \textrm{d}x\textrm{d}t \bigg ](\zeta )\, h(\tau ,\zeta ) \textrm{d}\zeta \textrm{d}\tau \\&\quad =\int \nolimits _0^T\int \nolimits _{\partial \Omega } D_\gamma \mathcal {H}[0,\phi ,0](T-\tau ,\zeta ) \,h(\tau ,\zeta ) \textrm{d}\zeta \textrm{d}\tau . \end{aligned}$$

\(\square \)

6.2 Well-posedness when \(u_0, f = 0\) and \(h \ne 0\). Proof of Theorem 2.6 i)

The proof is structured in several steps, using the previous lemmas:

  1. 1.

    Due to compactness from Lemma 5.2, there is at least a convergent subsequence of \(\mathcal H[0,f_j,0]\) in the sense \(L^1 (0,T; L^1 (\Omega , \delta ^\gamma ))\) to some function u.

  2. 2.

    By passing to the limit in (2.9) we observe that u satisfies (6.1).

  3. 3.

    Due to Lemma 6.2, u is the unique \(L^1(0,T; L^1 (\Omega , \delta ^\gamma ))\) solution of (6.1). By uniqueness of the weak limit, we deduce the convergence of the whole sequence \(\mathcal H[0,f_j,0]\).

  4. 4.

    Due to Lemma 6.4, u is given precisely by (2.7).

This completes the proof. \(\square \)

6.3 \(\mathcal H[0,0,h]\) satisfies the singular boundary condition. Proof of Theorem 2.6 ii)

Now we want to see whether (2.11) holds. We observe that (S3) is a form of saying that \(D_\gamma \mathbb S\) uniformly localises at \(t = 0\) on the boundary.

Theorem 6.5

Let \(h \in C(\partial \Omega )\) and assume (S3), in addition to the main assumptions throughout the paper. Then (2.11) holds.

Proof

Using (2.7), we write down the ratio

$$\begin{aligned} \frac{\mathcal {H}[0,0,h] (t,x) }{ u^\star (x) }&= \dfrac{ \int \nolimits _0^t \int \nolimits _{\partial \Omega } (D_\gamma \mathbb P_\alpha )(t - \tau ,x,\zeta ) h(\tau , \zeta ) \textrm{d}\zeta \textrm{d}\tau }{ \int \nolimits _{\partial \Omega } D_\gamma \mathbb {G}(\tilde{\zeta },x) \textrm{d}\tilde{\zeta }} \\&= \int \nolimits _0^t \int \nolimits _{\partial \Omega } \frac{ (D_\gamma \mathbb P_\alpha )(t - \tau ,x,\zeta ) }{ \int \nolimits _{\partial \Omega } D_\gamma \mathbb {G}(\tilde{\zeta },x) \textrm{d}\tilde{\zeta }} h(\tau , \zeta ) \textrm{d}\zeta \textrm{d}\tau . \end{aligned}$$

We define

$$\begin{aligned} \Upsilon (t,x,\zeta ) = \frac{ (D_\gamma \mathbb P_\alpha )(t,x,\zeta ) }{ \int \nolimits _{\partial \Omega } D_\gamma \mathbb {G}(\tilde{\zeta },x) \textrm{d}\tilde{\zeta }}. \end{aligned}$$

Then, we notice that

$$\begin{aligned} \int \nolimits _0^t \Upsilon (\sigma ,x,\zeta ) \textrm{d}\sigma= & {} \int \nolimits _0^t \int \nolimits _{0}^{\infty } \alpha \sigma ^{\alpha -1} \tau \Phi _\alpha (\tau ) \dfrac{ D_\gamma \mathbb {S}(\tau \sigma ^{\alpha },x,\zeta ) }{ \int \nolimits _{\partial \Omega } \int \nolimits _0^\infty D_\gamma \mathbb {S}(\tilde{\tau }, \tilde{\zeta },x) \textrm{d}\tilde{\zeta }\textrm{d}\tilde{\tau }} \textrm{d}\tau \textrm{d}\sigma \nonumber \\= & {} \int \nolimits _0^\infty \Phi _\alpha (\tau ) \left( \dfrac{ \int \nolimits _{0}^{\tau t^\alpha } D_\gamma \mathbb {S}(\tau \sigma ^{\alpha },x,\zeta ) \textrm{d}(\tau \sigma ^\alpha )}{ \int \nolimits _{\partial \Omega } \int \nolimits _0^\infty D_\gamma \mathbb {S}(\tilde{\tau }, \tilde{\zeta },x) \textrm{d}\tilde{\zeta }\textrm{d}\tilde{\tau }} \right) \textrm{d}\tau \nonumber \\= & {} \int \nolimits _0^\infty \Phi _\alpha (\tau ) \dfrac{ \int \nolimits _{0}^{\tau t^\alpha } D_\gamma \mathbb {S}(\sigma ,x,\zeta ) \textrm{d}\sigma }{ \int \nolimits _0^\infty \int \nolimits _{\partial \Omega } D_\gamma \mathbb {S}(\tilde{\tau }, \tilde{\zeta },x) \textrm{d}\tilde{\zeta }\textrm{d}\tilde{\tau }} \textrm{d}\tau . \end{aligned}$$
(6.2)

We recover that

$$\begin{aligned} \forall x \in \Omega \text { it holds that } \int \nolimits _0^\infty \int \nolimits _{\partial \Omega } \Upsilon (\sigma ,x,\zeta ) \textrm{d}\zeta \textrm{d}\sigma = 1. \end{aligned}$$
(6.3)

Notice that due to (6.3) and (S3) we have that for any \(t>0\),

$$\begin{aligned} \lim _{x \rightarrow \zeta _0} \int \nolimits _0^t \int \nolimits _{\partial \Omega } \Upsilon (\sigma , x,\zeta ) \textrm{d}\zeta \textrm{d}\sigma = 1. \end{aligned}$$
(6.4)

We compute the following limit as \(x \rightarrow \zeta _0\),

$$\begin{aligned}&\Bigg | \frac{\mathcal {H}[0,0,h] (t,x) }{ u^\star (x) } - \left( \int \nolimits _0^t \int \nolimits _{\partial \Omega } \Upsilon (t-\sigma , x, \zeta ) \textrm{d}\zeta \textrm{d}\sigma \right) h(t,\zeta _0) \Bigg | \\&\quad \le \int \nolimits _0^t \int \nolimits _{\partial \Omega } \Upsilon (t-\sigma , x, \zeta ) |h(\sigma , \zeta ) - h(t,\zeta _0)| \textrm{d}\zeta \textrm{d}\sigma . \end{aligned}$$

Assume that h is continuous. Now we split the right-hand side into different parts:

  1. 1.

    Close to \((t, \zeta _0)\). We pick \(\delta \) such that ball in \((\sigma , \zeta ) \in B(t, \delta ) \times ( B(\zeta _0, \delta ) \cap \partial \Omega )\) we have \(|h(\sigma , \zeta ) - h(t,\zeta _0)| \le \varepsilon \). Then

    $$\begin{aligned}&\int \nolimits _{t-\delta }^t \int \nolimits _{B(\zeta _0, \delta ) \cap \partial \Omega } \Upsilon (t-\sigma , x, \zeta ) |h(\sigma , \zeta ) - h(t,\zeta _0)| \textrm{d}\zeta \textrm{d}\sigma \\&\quad \le \varepsilon \int \nolimits _0^\infty \int \nolimits _{\partial \Omega } \Upsilon (\sigma , x , \zeta )\textrm{d}\zeta \textrm{d}\sigma = \varepsilon . \end{aligned}$$
  2. 2.

    On \((0,t) \times ( \partial \Omega {\setminus } B(\zeta _0, \delta ) ) \). In this region we use (6.2) to deduce that

    $$\begin{aligned} \int \nolimits _{0}^t \int \nolimits _{\partial \Omega \setminus B(\zeta _0, \delta ) }&\Upsilon (t-\sigma , x, \zeta ) |h(\sigma , \zeta ) - h(t,\zeta _0)| \textrm{d}\zeta \textrm{d}\sigma \\&\le 2 \Vert h\Vert _{L^\infty (\partial \Omega )} \int \nolimits _{\partial \Omega \setminus B(\zeta _0, \delta ) } \int \nolimits _0^\infty \Upsilon (\sigma , x, \zeta ) \textrm{d}\sigma \textrm{d}\zeta \\&= 2 \Vert h\Vert _{L^\infty (\partial \Omega )} \int \nolimits _0^\infty \Phi _\alpha (\tau ) \textrm{d}\tau \dfrac{ \int \nolimits _{\partial \Omega \setminus B(\zeta _0, \delta ) } \int \nolimits _{0}^{\infty } D_\gamma \mathbb {S}(\sigma ,x,\zeta ) \textrm{d}\sigma \textrm{d}\zeta }{ \int \nolimits _0^\infty \int \nolimits _{\partial \Omega } D_\gamma \mathbb {S}(\tilde{\tau }, \tilde{\zeta },x) \textrm{d}\tilde{\zeta }\textrm{d}\tilde{\tau }} \\&= 2 \Vert h\Vert _{L^\infty (\partial \Omega )} \dfrac{ \int \nolimits _{\partial \Omega \setminus B(\zeta _0, \delta ) } D_\gamma \mathbb G(x,\zeta ) \textrm{d}\zeta }{ \int \nolimits _{\partial \Omega } D_\gamma \mathbb G(x, \tilde{\zeta }) \textrm{d}\tilde{\zeta }} \end{aligned}$$

    because of the hypothesis we made above. Taking a smooth non-negative function \(\varphi \) that takes value 1 in \(\partial \Omega {\setminus } B(\zeta _0, \delta )\) and \(\varphi (\zeta _0) = 0\), we use that in [3] the authors prove

    $$\begin{aligned} \lim _{x \rightarrow \zeta _0} \dfrac{ \int \nolimits _{\partial \Omega } D_\gamma \mathbb G(x,\zeta ) \varphi (\zeta ) \textrm{d}\zeta }{ \int \nolimits _{\partial \Omega } D_\gamma \mathbb G(x, \tilde{\zeta }) \textrm{d}\tilde{\zeta }} \rightarrow \varphi (\zeta _0) = 0. \end{aligned}$$
  3. 3.

    Lastly, the region \((0,t-\delta ) \times ( B(\zeta _0, \delta ) \cap \partial \Omega ) \).

    $$\begin{aligned}&\int \nolimits _{0}^{t-\delta } \int \nolimits _{\partial \Omega \setminus B(\zeta _0, \delta ) } \Upsilon (t-\sigma , x, \zeta ) |h(\sigma , \zeta ) - h(t,\zeta _0)| \textrm{d}\zeta \textrm{d}\sigma \\&\quad \le 2 \Vert h \Vert _{L^\infty } \int \nolimits _{0}^{t-\delta } \int \nolimits _{\partial \Omega } \Upsilon (t-\sigma , x, \zeta ) \textrm{d}\sigma \textrm{d}\zeta \\&\quad \le 2 \Vert h \Vert _{L^\infty } \int \nolimits _{\delta }^{t} \int \nolimits _{\partial \Omega } \Upsilon (\sigma , x, \zeta ) \textrm{d}\sigma \textrm{d}\zeta . \end{aligned}$$

    Now we notice that

    $$\begin{aligned}&\int \nolimits _{\delta }^{t} \int \nolimits _{\partial \Omega } \Upsilon (\sigma , x, \zeta ) \textrm{d}\sigma \textrm{d}\zeta \\&\quad = \int \nolimits _{0}^{t } \int \nolimits _{\partial \Omega } \Upsilon (\sigma , x, \zeta ) \textrm{d}\sigma \textrm{d}\zeta - \int \nolimits _{0}^{\delta } \int \nolimits _{\partial \Omega } \Upsilon (\sigma , x, \zeta ) \textrm{d}\sigma \textrm{d}\zeta \\&\quad =\int \nolimits _0^\infty \Phi _\alpha (\tau ) \dfrac{ \int \nolimits _{\tau \delta ^\alpha }^{\tau t^\alpha } \int \nolimits _{\partial \Omega } D_\gamma \mathbb {S}(\sigma ,x,\zeta ) \textrm{d}\zeta \textrm{d}\sigma }{ \int \nolimits _0^\infty \int \nolimits _{\partial \Omega } D_\gamma \mathbb {S}(\tilde{\tau }, \tilde{\zeta },x) \textrm{d}\tilde{\zeta }\textrm{d}\tilde{\tau }} \textrm{d}\tau \textrm{d}\sigma \\&\quad \le \int \nolimits _0^\infty \Phi _\alpha (\tau ) \dfrac{ \int \nolimits _{\tau \delta ^\alpha }^{\infty } \int \nolimits _{\partial \Omega } D_\gamma \mathbb {S}(\sigma ,x,\zeta ) \textrm{d}\zeta \textrm{d}\sigma }{ \int \nolimits _0^\infty \int \nolimits _{\partial \Omega } D_\gamma \mathbb {S}(\tilde{\tau }, \tilde{\zeta },x) \textrm{d}\tilde{\zeta }\textrm{d}\tilde{\tau }} \textrm{d}\tau \textrm{d}\sigma . \end{aligned}$$

    As \(x \rightarrow \zeta _0\) this converges to 0 due to (S3).

We have proved that, for any \(\varepsilon > 0\) we have

$$\begin{aligned} \lim _{x \rightarrow \zeta _0} \Bigg | \frac{\mathcal {H}[0,0,h] (t,x) }{ u^\star (x) } - \left( \int \nolimits _0^t \int \nolimits _{\partial \Omega } \Upsilon (t-\sigma , x, \zeta ) \textrm{d}\zeta \textrm{d}\sigma \right) h(t,\zeta _0) \Bigg | \le \varepsilon . \end{aligned}$$

Recalling (6.4) the proof is finished. \(\square \)