# Discrete Heat Equation with irregular thermal conductivity and tempered
distributional data

Marianna Chatzakou  Marianna Chatzakou: Department of Mathematics: Analysis,
Logic and Discrete Mathematics Ghent University Ghent Belgium E-mail address
<EMAIL_ADDRESS>,  Aparajita Dasgupta  Aparajita Dasgupta:
Department of Mathematics Indian Institute of Technology, Delhi, Hauz Khas New
Delhi-110016 India E-mail address<EMAIL_ADDRESS>,  Michael
Ruzhansky  Michael Ruzhansky: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium and School of Mathematical
Sciences Queen Mary University of London United Kingdom E-mail address
<EMAIL_ADDRESS>and  Abhilash Tushir  Abhilash Tushir: Department of
Mathematics Indian Institute of Technology, Delhi, Hauz Khas New Delhi-110016
India E-mail address<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider a semi-classical version of the nonhomogeneous heat
equation with singular time-dependent coefficients on the lattice
$\hbar\mathbb{Z}^{n}$. We establish the well-posedeness of such Cauchy
equations in the classical sense when regular coefficients are considered, and
analyse how the notion of very weak solution adapts in such equations when
distributional coefficients are regarded. We prove the well-posedness of both
the classical and the very weak solution in the weighted spaces
$\ell^{2}_{s}(\hbar\mathbb{Z}^{n})$, $s\in\mathbb{R}$, which is enough to
prove the well-posedness in the space of tempered distributions
$\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})$. Notably, when $s=0$, we show that
for $\hbar\rightarrow 0$, the classical (resp. very weak) solution of the heat
equation in the Euclidean setting $\mathbb{R}^{n}$ is recaptured by the
classical (resp. very weak) solution of it in the semi-classical setting
$\hbar\mathbb{Z}^{n}$.

M. Chatzakou is a postdoctoral fellow of the Research Foundation – Flanders
(FWO) under the postdoctoral grant No 12B1223N. A. Dasgupta is supported by
Core Research Grant, RP03890G, Science and Engineering Research Board (SERB),
DST, India. M. Ruzhansky is supported by the EPSRC Grants EP/R003025 by the
FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations
and by the Methusalem programme of the Ghent University Special Research Fund
(BOF) (Grantnumber 01M01021). The last author is supported by the institute
assistantship from Indian Institute of Technology Delhi, India.

###### Contents

  1. 1 Introduction
  2. 2 Main results
  3. 3 Preliminaries
    1. 3.1 Spaces of functions and distributions on the lattice $\hbar\mathbb{Z}^{n}$
    2. 3.2 Space of periodic functions and distributions on the torus $\mathbb{T}_{\hbar}^{n}$
    3. 3.3 Related semi-classical Fourier analysis
    4. 3.4 The discrete fractional Laplacian
  4. 4 Proof of the main results
  5. 5 Semi-classical limit $\hbar\to 0$
  6. 6 Remarks

##  1\. Introduction

The solvability of the heat equation, as well as the inverse problem for the
heat equation where thermal coefficients are constant, time-dependent, or
space-dependent, has been studied extensively in many different settings.

For the case of the heat equation with singular potentials in the space
variable, we refer to the works [CRT22c, ARST21] for an overview of the known
results in different settings. For the case where the time-dependent
coefficients are considered, we refer to the works [Gul02, HLI14, LSST22,
HX17, DMA22] and [TY15], that apply in settings different from the lattice
$\mathbb{Z}^{n}$. In the particular case of the lattice $\mathbb{Z}^{n}$ we
have the work [FNNV96] where constant coefficients are considered. In the
aforesaid setting, the heat equation with constant coefficients reads as the
parabolic Anderson model, see e.g. [Che20] and references therein, and the
asymptotic analysis of its solution is a topic of wide interest in the field
of stochastic analysis.

The setting in this current work is, for a (small) semi-classical parameter
$\hbar>0$, given by

$\hbar\mathbb{Z}^{n}=\\{x\in\mathbb{R}^{n}:x=\hbar
k,~{}k\in\mathbb{Z}^{n}\\},$

and clearly includes the $n$-dimensional lattice $\mathbb{Z}^{n}$ as a special
case. For $\alpha>0$, the discrete fractional Laplacian on
$\hbar\mathbb{Z}^{n}$ denoted by $\left(-\mathcal{L}_{\hbar}\right)^{\alpha}$
is defined by

$\left(-\mathcal{L}_{\hbar}\right)^{\alpha}u(k):=\sum_{j\in\mathbb{Z}^{n}}a_{j}^{(\alpha)}u(k+j\hbar),\quad
k\in\hbar\mathbb{Z}^{n},$ (1.1)

where the expansion coefficient $a_{j}^{(\alpha)}$ is given by

$a_{j}^{(\alpha)}:=\int\limits_{\left[-\frac{1}{2},\frac{1}{2}\right]^{n}}\left[\sum_{l=1}^{n}4\sin^{2}\left(\pi\xi_{l}\right)\right]^{\alpha}e^{-2\pi
ij\cdot\xi}\mathrm{d}\xi.$ (1.2)

For more information on the discrete fractional Laplacian, see Section 3.

For the space variable $k\in\hbar\mathbb{Z}^{n}$ in this setting, we analyse
the semi-classical analogue of the heat equation in the Euclidean setting,
which reads as follows:

$\left\\{\begin{array}[]{l}\partial_{t}u(t,k)+a(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}u(t,k)+b(t)u(t,k)=f(t,k),\quad(t,k)\in(0,T]\times\hbar\mathbb{Z}^{n},\\\
u(0,k)=u_{0}(k),\quad k\in\hbar\mathbb{Z}^{n},\end{array}\right.$ (1.3)

where $a=a(t)\geq 0$ is the thermal conductivity, $b$ is a real-valued bounded
potential and $f$ is the heat source.

The first aim of the current work is to prove the well-posedness of the
classical solution of the heat equation as in (1.3) in the semi-classical
setting $\hbar\mathbb{Z}^{n}$. Let us note that in [DRT22a, DRT22b] the
authors examine the discrete wave equation with time-dependent coefficients
and the discrete Klein-Gordan equation with regular potential and prove that
they are well-posed in $\ell^{2}(\hbar\mathbb{Z}^{n})$. In this work, we are
extending the well-posedness results in [DRT22a, DRT22b] for our consideration
of the Cauchy problem by allowing the initial data to grow polynomially. More
precisely, we investigate the well-posedness of the Cauchy problem (1.3) with
regular/irregular coefficients as well as the heat source and initial Cauchy
data from the space of discrete tempered distributions
$\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})$. A special feature in our approach
in this article is that we also allow the coefficients $a,b$ to be
distributions. Particularly, our analysis allows $a,b$ to have distributional
irregularities; i.e., to have $\delta$-type terms, while also Heaviside
discontinuities e.g. we can take $b(t)=\delta+H(t)$. Taking into account that
the solution $u(t,x)$ might as well have singularities in $t$, this
consideration would lead to foundational mathematical difficulties due to the
problem of the impossibility of multiplying distributions, see Schwartz
[Sch54]. To overcome this problem we employ the theory of very weak solution
as introduced in [GR15] which allows us to recapture the
classical/distributional solution to the Cauchy problem (1.3), provided that
the latter exists.

With the well-posedness of the classical (resp. very weak) solution to the
Cauchy problem (1.3) on the lattice $\hbar\mathbb{Z}^{n}$ at our disposal, two
natural questions arise:

  1. (1)

Can we recapture the classical solution to the heat equation (1.3) when the
space variable lies in the Euclidean space $\mathbb{R}^{n}$ by allowing
$\hbar\rightarrow 0$?

  2. (2)

Can we recapture the very weak solution to (1.3) when the space variable lies
in the Euclidean space $\mathbb{R}^{n}$ by allowing $\hbar\rightarrow 0$?

We will see that both questions are answered in the affirmative, provided
additional Sobolev regularity. In particular, we consider the semi-classical
limits $\hbar\to 0$ for the following cases:

  1. (A)

regular coefficient and Sobolev initial data;

  2. (B)

distributional coefficient and Sobolev initial data,

and prove that in both cases we recover the (classical/very weak) solution in
the Euclidean setting. The idea of ensuring globally convergence for the
solution by adding Sobolev regularity, can be found in the semi-classical
limit theorems in [DRT22a, Theorem 1.2] and [DRT22b, Theorem 1.3].

Note that the Cauchy problem (1.3) is the discrete analogue of heat equation
with time-dependent coefficients in the Euclidean setting $\mathbb{R}^{n}$
given by:

$\left\\{\begin{array}[]{l}\partial_{t}u(t,x)+a(t)(-\mathcal{L})^{\alpha}u(t,x)+b(t)u(t,x)=f(t,x),\quad(t,x)\in(0,T]\times\mathbb{R}^{n},\\\
u(0,x)=u_{0}(x),\quad x\in\mathbb{R}^{n},\end{array}\right.$ (1.4)

where $(-\mathcal{L})^{\alpha}$ is the usual fractional Laplacian
$\left(-\mathcal{L}\right)^{\alpha}$ on $\mathbb{R}^{n}$ defined as a pseudo-
differential operator with symbol
$|2\pi\xi|^{2\alpha}=\left[\sum\limits_{l=1}^{n}(2\pi\xi_{l})^{2}\right]^{\alpha}$
i.e.,

$(-\mathcal{L})^{\alpha}u(x)=\int\limits_{\mathbb{R}^{n}}\left|2\pi\xi\right|^{2\alpha}\widehat{u}(\xi)e^{2\pi
ix\cdot\xi}\mathrm{d}\xi.$ (1.5)

Theorems 1.1 and 1.2 below state the well-posedness results of the Cauchy
problem (1.4) for the above cases in the Euclidean setting $\mathbb{R}^{n}$.

Before presenting the aforementioned results, let us note that, notation-wise,
we write $a\in L_{m}^{\infty}([0,T])$, if $a\in L^{\infty}([0,T])$ is
$m$-times weakly differentiable with $\partial^{j}_{t}a\in L^{\infty}([0,T])$,
for all $j=1,\dots,m$. Let us recall the usual Sobolev space
$H^{m}(\mathbb{R}^{n})$ and its dual $L^{2}_{m}(\mathbb{R}^{n})$ defined as

$f\in H^{m}(\mathbb{R}^{n})\iff(I-\mathcal{L})^{m/2}f\in
L^{2}(\mathbb{R}^{n}),$

and

$g\in L^{2}_{m}(\mathbb{R}^{n})\iff(1+|\xi|^{2})^{m/2}g\in
L^{2}(\mathbb{R}^{n}),$ (1.6)

respectively.

In the sequel, we will be writing $A\lesssim B$ whenever there exists a
constant $C$, independent of the appearing parameters, such that $A\leq CB$.

For the well-posedness of the Cauchy problem (1.4) in the case (A) we have the
following result:

######  Theorem 1.1.

Let $m\in\mathbb{R}$ and $f\in L^{2}([0,T];H^{m}(\mathbb{R}^{n}))$. Assume
that $a\in L_{1}^{\infty}([0,T])$ satisfies
$\inf\limits_{t\in[0,T]}a(t)=a_{0}>0$ and $b\in L^{\infty}([0,T])$. If the
initial Cauchy data $u_{0}\in H^{m}(\mathbb{R}^{n})$, then the Cauchy problem
(1.4) has a unique solution $u\in C([0,T];H^{m}(\mathbb{R}^{n}))$ which
satisfies the estimate

$\|u(t,\cdot)\|_{H^{m}(\mathbb{R}^{n})}^{2}\leq
C_{T,a,b}\left(\|u_{0}\|^{2}_{H^{m}(\mathbb{R}^{n})}+\|f\|^{2}_{L^{2}([0,T];H^{m}(\mathbb{R}^{n})}\right),$
(1.7)

for all $t\in[0,T]$, where the positive constant $C_{T,a,b}$ is given by

$C_{T,a,b}=a_{0}^{-1}\|a\|_{L^{\infty}}e^{a_{0}^{-1}(\|a_{t}\|_{L^{\infty}}+2\|a\|_{L^{\infty}}\|b\|_{L^{\infty}}+\|a\|_{L^{\infty}})T}.$
(1.8)

Similarly, for the well-posedness of the Cauchy problem (1.4) in the case (B)
we have:

######  Theorem 1.2.

Let a and $b$ be distributions with supports included in $[0,T]$ such that
$a\geq a_{0}>0$ for some positive constant $a_{0}$, and let the source term
$f(\cdot,x)$ be a distribution with support included in $[0,T]$, for all
$x\in\mathbb{R}^{n}$. Let $m\in\mathbb{R}$ and $u_{0}\in
H^{m}(\mathbb{R}^{n})$. Then the Cauchy problem (1.4) has a unique very weak
solution $(u_{\varepsilon})_{\varepsilon}\in
L^{2}([0,T];H^{m}(\mathbb{R}^{n}))^{(0,1]}$ of order $m$.

For more details about the very weak solutions for the Cauchy problem (1.4) in
the Euclidean setting, we refer to Section 6.

To give a synopsis of the topic of very weak solution, we refer to the works
[MnRT19, RT17b] on Cauchy problems with singular, time-dependent coefficients
in the Euclidean setting. The concept of very weak solution with space-
dependent coefficients in the general setting of graded Lie groups has been
employed in the works [CRT22a, CRT22b, CRT22c, RT21, RY20] and [Gar21] in the
Euclidean setting.

To summarise, we are able to: use the notion of very weak solution, as
introduced in [GR15] in the setting of hyperbolic Cauchy problems with
distributional coefficients in space; understand distributionally the Cauchy
problem (1.3); and prove its well-posedness. Precisely, the following facts
will be presented in the sequel:

  * •

The Cauchy problem (1.3) is well-posed in the weighted spaces
$\ell^{2}_{s}(\hbar\mathbb{Z}^{n})$, for all $s\in\mathbb{R}$ and admits a
very weak solution even when distributional coefficients $a,b$ are considered.

  * •

The very weak solution is unique in the sense of Definition 2.5.

  * •

If the coefficients $a,b$ are regular enough so that the Cauchy problem (1.3)
has a classical solution, then the very weak solution recaptures the classical
one. This fact indicates that the notion of very weak solution as adapted to
our setting is consistent with the classical one.

  * •

We are able to approximate both the classical and the very weak solution to
the heat equation as in (1.3) in the Euclidean setting $\mathbb{R}^{n}$, by
the corresponding solution in the semi-classical setting
$\hbar\mathbb{Z}^{n}$.

The structure of the paper is as follows: in Section 2 we present our main
results. In Section 3 we discuss the basics of the Fourier analysis in the
case of the lattices $\mathbb{Z}^{n}$ and $\hbar\mathbb{Z}^{n}$ and of the
torus $\mathbb{T}^{n}_{\hbar}$. Additionally to this, the necessary for our
analysis functions of spaces and distributions are recalled. In Section 4 we
provide the proofs of the results on the existence and uniqueness of the very
weak solution of the problem we consider, as well as the consistency of it
with the classical solution. In Section 5 we prove the results on the
approximation of the (classical/very weak) solution on $\mathbb{R}^{n}$, by
the one on $\hbar\mathbb{Z}^{n}$. Finally, in Section 6, we make some remarks
about the very weak solution in the Euclidean setting.

##  2\. Main results

In this section, we will present our main results for the well-posedness of
the Cauchy problem (1.3) with regular/irregular coefficients and the discrete
tempered distributional initial data. We will also present the semi-classical
limit theorems for the classical as well as very weak solution.

First, we consider the Cauchy problem (1.3) with discrete tempered
distributional initial Cauchy data and the regular coefficients $a\in
L_{1}^{\infty}([0,T])$ and $b\in L^{\infty}([0,T])$. In the light of the
relation (3.2), proving the well-posedness in the space of weighted spaces
$\ell^{2}_{s}(\hbar\mathbb{Z}^{n})$ is sufficient to prove the well-posedness
in the space of discrete tempered distributions
$\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})$.

Let us start our exposition of results with the one on the well-posedness
theorem (in the classical sense) for the Cauchy problem (1.3):

######  Theorem 2.1 (Classical solution).

Let $s\in\mathbb{R}$ and $f\in
L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$. Assume that $a\in
L_{1}^{\infty}([0,T])$ satisfies $\inf\limits_{t\in[0,T]}a(t)=a_{0}>0$ and
$b\in L^{\infty}([0,T])$. If for the initial Cauchy data we have
$u_{0}\in\ell^{2}_{s}\left(\hbar\mathbb{\mathbb{Z}}^{n}\right)$, then the
Cauchy problem (1.3) has a unique solution $u\in
C([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$ satisfying the estimate

$\|u(t,\cdot)\|_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}^{2}\leq
C_{T,a,b}(\|u_{0}\|^{2}_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}+\|f\|^{2}_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}),$
(2.1)

for all $t\in[0,T]$ and $\hbar>0$, where the positive constant $C_{T,a,b}$ is
given by

$C_{T,a,b}=a_{0}^{-1}\|a\|_{L^{\infty}}e^{a_{0}^{-1}(\|a_{t}\|_{L^{\infty}}+2\|a\|_{L^{\infty}}\|b\|_{L^{\infty}}+\|a\|_{L^{\infty}})T}.$
(2.2)

Next, we are considering the Cauchy problem (1.3) with discrete tempered
distributional initial Cauchy data and, also allowing the coefficients and the
source term to have singularities in the time variable. As we discussed
earlier in Section 1, in order to deal with such equations where fundamental
mathematical difficulties may arise when adapting the classical approach, we
have to introduce the notion of a very weak solution. To this end, let us
quickly recall the important points for the upcoming analysis:

Let $a\in\mathcal{D}^{\prime}\left(\mathbb{R}\right)$ be a distribution. Using
the Friedrichs-mollifier, i.e., a function $\psi\in
C_{0}^{\infty}\left(\mathbb{R}\right),\psi\geq 0$ and
$\int_{\mathbb{R}}\psi=1$, we are able to construct families of smooth
functions $(a_{\varepsilon})_{\varepsilon}$ by regularizing the distributional
coefficient $a$ as follows:

$a_{\varepsilon}(t):=\left(a*\psi_{\omega(\varepsilon)}\right)\left(t\right),\quad\psi_{\omega(\varepsilon)}(t)=\dfrac{1}{\omega(\varepsilon)}\psi\left(\dfrac{t}{\omega(\varepsilon)}\right),\quad\varepsilon\in(0,1],$
(2.3)

where $\omega(\varepsilon)\geq 0$ and $\omega(\varepsilon)\to 0$ as
$\varepsilon\to 0$. Let us note that since for the purposes of this article,
the distributions $a$ and $b$, as in (1.3), are distributions with domain
$[0,T]$, it is enough to consider that $\text{supp}(\psi)\subseteq K$, with
$K=[0,T]$ throughout this article.

The notions of moderateness and the negligibility for a net of
functions/distributions as follows:

######  Definition 2.2.

(i) A net $\left(a_{\varepsilon}\right)_{\varepsilon}\in
L^{\infty}_{m}(\mathbb{R})^{(0,1]}$ is said to be $L^{\infty}_{m}$-moderate if
for all $K\Subset\mathbb{R}$, there exist $N\in\mathbb{N}_{0}$ and $c>0$ such
that

$\left\|\partial^{k}a_{\varepsilon}\right\|_{L^{\infty}(K)}\leq
c\varepsilon^{-N-k},\quad\text{ for all }k=0,1,\dots,m,$

for all $\varepsilon\in(0,1]$.
(ii) A net $\left(a_{\varepsilon}\right)_{\varepsilon}\in
L^{\infty}_{m}(\mathbb{R})^{(0,1]}$ is said to be $L^{\infty}_{m}$-negligible
if for all $K\Subset\mathbb{R}$ and $q\in\mathbb{N}_{0}$, there exists $c>0$
such that

$\left\|\partial^{k}a_{\varepsilon}\right\|_{L^{\infty}(K)}\leq
c\varepsilon^{q},\quad\text{ for all }k=0,1,\dots,m,$

for all $\varepsilon\in(0,1]$.
(iii) A net $\left(u_{\varepsilon}\right)_{\varepsilon}\in
L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))^{(0,1]}$ is said to be
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-moderate if there exist
$N\in\mathbb{N}_{0}$ and $c>0$ such that

$\|u_{\varepsilon}\|_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}\leq
c\varepsilon^{-N},$

for all $\varepsilon\in(0,1]$.
(iv) A net $\left(u_{\varepsilon}\right)_{\varepsilon}\in
L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))^{(0,1]}$ is said to be
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-negligible if for all
$q\in\mathbb{N}_{0}$ there exists $c>0$ such that

$\|u_{\varepsilon}\|_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}\leq
c\varepsilon^{q},$

for all $\varepsilon\in(0,1]$.

We note that the moderateness requirements are natural in the sense that
distributions are moderately regularized. Moreover, by the structure theorems
for distributions, we have the following inclusion:

$\text{compactly supported distributions
}\mathcal{E}^{\prime}(\mathbb{R})\subset\left\\{L^{2}\text{-moderate
families}\right\\}.$

Therefore, it is possible that the Cauchy problem (1.3) may not have a
solution in the space of compactly supported distributions
$\mathcal{E}^{\prime}(\mathbb{R})$ but it may exist in the space of
$L^{2}$-moderate families in some suitable sense.

The notion of a very weak solution to the Cauchy problem (1.3) can be viewed
as follows:

######  Definition 2.3.

Let $s\in\mathbb{R},f\in L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$, and
$u_{0}\in\ell^{2}_{s}(\hbar\mathbb{Z}^{n}).$ The net
$\left(u_{\varepsilon}\right)_{\varepsilon}\in$
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))^{(0,1]}$ is a very weak
solution of order $s$ of the Cauchy problem (1.3) if there exist

  1. (1)

$L_{1}^{\infty}$-moderate regularisation $a_{\varepsilon}$ of the coefficient
$a$;

  2. (2)

$L^{\infty}$-moderate regularisation $b_{\varepsilon}$ of the coefficient $b$;
and

  3. (3)

$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-moderate regularisation
$f_{\varepsilon}$ of the source term $f$,

such that $\left(u_{\varepsilon}\right)_{\varepsilon}$ solves the regularised
problem

$\left\\{\begin{array}[]{l}\partial_{t}u_{\varepsilon}(t,k)+a_{\varepsilon}(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}u_{\varepsilon}(t,k)+b_{\varepsilon}(t)u_{\varepsilon}(t,k)=f_{\varepsilon}(t,k),\quad
t\in(0,T],\\\ u_{\varepsilon}(0,k)=u_{0}(k),\quad
k\in\hbar\mathbb{Z}^{n},\end{array}\right.$ (2.4)

for all $\varepsilon\in(0,1]$, and is
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-moderate.

It should be noted that Theorem 2.1 provides a unique solution to the
regularised Cauchy problem (1.3) that satisfies estimate (2.1).

A distribution $a$ is said to be a positive distribution if $\langle
a,\psi\rangle\geq 0$ for all $\psi\in C^{\infty}_{0}(\mathbb{R})$ such that
$\psi\geq 0$. Similarly, a distribution $a$ is said to be a strictly positive
distribution if there exists a positive constant $\alpha$ such that $a-\alpha$
is a positive distribution in the previous sense. In other words,
$a\geq\alpha>0$ in the support of $a$, where $a\geq\alpha$, means that

$\langle a-\alpha,\psi\rangle\geq 0,\quad\text{for all }\psi\in
C_{0}^{\infty}(\mathbb{R}),\psi\geq 0.$ (2.5)

Now we can state the existence theorem for the Cauchy problem (1.3) with
distributional coefficients as follows:

######  Theorem 2.4 (Existence).

Let $a$ and $b$ be distributions with supports contained in $[0,T]$ such that
$a\geq a_{0}>0$ for some positive constant $a_{0}$, and let the source term
$f(\cdot,k)$ be a distribution with support contained in $[0,T]$, for all
$k\in\hbar\mathbb{Z}^{n}$. For $s\in\mathbb{R}$, we assume that the initial
Cauchy data $u_{0}$ satisfies $u_{0}\in\ell^{2}_{s}(\hbar\mathbb{Z}^{n})$.
Then the Cauchy problem (1.3) has a very weak solution
$\left(u_{\varepsilon}\right)_{\varepsilon}\in$
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))^{(0,1]}$ of order s.

Next, we define the uniqueness of the very weak solution obtained in Theorem
2.4 for the Cauchy problem (1.3). This should be regarded as if the family of
very weak solution is not “significantly” affected by the negligible changes
in the approximations of the coefficients $a,b$ and of the source term $f$.
This can be also regarded as a ”stability” property.

Strictly speaking, the notion of the uniqueness of the very weak solution for
the Cauchy problem (1.3) is formulated as follows:

######  Definition 2.5.

We say that the Cauchy problem (1.3) has a
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-unique very weak solution, if

  1. (1)

for all $L^{\infty}_{1}$-moderate nets
$a_{\varepsilon},\tilde{a}_{\varepsilon}$ such that
$(a_{\varepsilon}-\tilde{a}_{\varepsilon})_{\varepsilon}$ is
$L^{\infty}_{1}$-negligible;

  2. (2)

for all $L^{\infty}$-moderate nets $b_{\varepsilon},\tilde{b}_{\varepsilon}$
such that $(b_{\varepsilon}-\tilde{b}_{\varepsilon})_{\varepsilon}$ are
$L^{\infty}$-negligible; and

  3. (3)

for all $L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-moderate nets
$f_{\varepsilon},\tilde{f}_{\varepsilon}$ such that
$(f_{\varepsilon}-\tilde{f}_{\varepsilon})_{\varepsilon}$ is
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-negligible,

the net $(u_{\varepsilon}-\tilde{u}_{\varepsilon})_{\varepsilon}$ is
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-negligible, where
$(u_{\varepsilon})_{\varepsilon}$ and
$(\tilde{u}_{\varepsilon})_{\varepsilon}$ are the families of solution
corresponding to the $\varepsilon$-parametrised problems

$\left\\{\begin{array}[]{l}\partial_{t}u_{\varepsilon}(t,k)+a_{\varepsilon}(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}u_{\varepsilon}(t,k)+b_{\varepsilon}(t)u_{\varepsilon}(t,k)=f_{\varepsilon}(t,k),\quad
t\in(0,T],\\\ u_{\varepsilon}(0,k)=u_{0}(k),\quad
k\in\hbar\mathbb{Z}^{n},\end{array}\right.$ (2.6)

and

$\left\\{\begin{array}[]{l}\partial_{t}\tilde{u}_{\varepsilon}(t,k)+\tilde{a}_{\varepsilon}(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}\tilde{u}_{\varepsilon}(t,k)+\tilde{b}_{\varepsilon}(t)\tilde{u}_{\varepsilon}(t,k)=\tilde{f}_{\varepsilon}(t,k),\quad
t\in(0,T],\\\ \tilde{u}_{\varepsilon}(0,k)=u_{0}(k),\quad
k\in\hbar\mathbb{Z}^{n},\end{array}\right.$ (2.7)

respectively.

######  Remark 2.6.

The Colombeau algebra $\mathcal{G}(\mathbb{R})$ defined as

$\mathcal{G}(\mathbb{R})=\frac{C^{\infty}\text{-moderate nets
}}{C^{\infty}\text{-negligible nets }},$

can also be used to formulate the uniqueness for the Cauchy problem (1.3). For
more details about the Colombeau algebra, we refer to [Obe92]. The uniqueness
of very weak solution in the sense of Colombeau algebra was introduced by
Garetto and the third author in [GR15] and subsequently used in different
settings, see e.g. [RT17a, RT17b].

The following theorem gives the uniqueness of the very weak solution to the
Cauchy problem (1.3) in the sense of Definition 2.5.

######  Theorem 2.7 (Uniqueness).

Let $a$ and $b$ be distributions with supports contained in $[0,T]$ such that
$a\geq a_{0}>0$ for some positive constant $a_{0}$, and let the source term
$f(\cdot,k)$ be a distribution with support contained in $[0,T]$, for all
$k\in\hbar\mathbb{Z}^{n}$. For $s\in\mathbb{R}$, let also
$u_{0}\in\ell^{2}_{s}(\hbar\mathbb{Z}^{n})$. Then the very weak solution of
the Cauchy problem (1.3) is
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-unique.

In the following theorem, we prove the consistency results with the classical
case:

######  Theorem 2.8 (Consistency).

Let $s\in\mathbb{R}^{n}$ and $f\in
L^{2}([0,T],\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$. Assume that $a\in
L_{1}^{\infty}\left([0,T]\right)$ satisfies
$\inf\limits_{t\in[0,T]}a(t)=a_{0}>0$ and $b\in L^{\infty}\left([0,T]\right)$.
Let also $u_{0}\in\ell^{2}_{s}(\hbar\mathbb{Z}^{n})$ for any $s\in\mathbb{R}$.
Then the regularised net $u_{\varepsilon}$ converges, as
$\varepsilon\rightarrow 0$, in
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$ to the classical solution of
the Cauchy problem (1.3).

The following theorem shows that the classical solution of (1.3) in the semi-
classical setting $\hbar\mathbb{Z}^{n}$ obtained in Theorem 2.1 for $s=0$,
recaptures the classical solution in the Euclidean setting $\mathbb{R}^{n}$ as
in Theorem 1.1, provided the latter exists.

######  Theorem 2.9.

Let $\alpha\in(0,1]$ and $s=0$. Let $u$ and $v$ be the classical solution of
the Cauchy problems (1.3) on $\hbar\mathbb{Z}^{n}$ and (1.4) on
$\mathbb{R}^{n}$, respectively. Assume that for the initial Cauchy data we
have $u_{0}\in H^{m}(\mathbb{R}^{n})$ for some $m\geq 4\alpha$. Then we have
the following uniform in $t\in[0,T]$ convergence:

$\|v(t,\cdot)-u(t,\cdot)\|_{\ell^{2}(\hbar\mathbb{Z}^{n})}\rightarrow 0,\text{
as }\hbar\rightarrow 0\,.$ (2.8)

######  Remark 2.10.

Let us point out that the solution $u,v$ as in Theorem 2.9 are regarded as
solution to the Cauchy problems (1.3) and (1.4), respectively, with the same
Cauchy data $u_{0}$, source term $f$ and time-dependent coefficients $a,b$.
The same hypothesis holds true in Theorem 2.11 on the convergence of the nets
of the very weak solution in the semi-classical and Euclidean settings of the
corresponding regularised heat equations.

The analogous to Theorem 2.9 statement in the “very weak sense” reads as
follows:

######  Theorem 2.11.

let $\alpha\in(0,1]$ and $s=0$. Let
$\left(u_{\varepsilon}\right)_{\varepsilon}$ and
$\left(v_{\varepsilon}\right)_{\varepsilon}$ be the very weak solution of the
Cauchy problem (1.3) on $\hbar\mathbb{Z}^{n}$ and (1.4) on $\mathbb{R}^{n}$,
respectively. Assume that for the initial Cauchy data, we have $u_{0}\in
H^{m}(\mathbb{R}^{n})$ for some $m\geq 4\alpha$. Then we have the following
uniform in $t\in[0,T]$ convergence

$\left\|v_{\varepsilon}(t,\cdot)-u_{\varepsilon}(t,\cdot)\right\|_{\ell^{2}(\hbar\mathbb{Z}^{n})}\rightarrow
0\text{ as }\hbar\rightarrow 0,$ (2.9)

and pointwise for $\varepsilon\in(0,1]$.

##  3\. Preliminaries

The Fourier analysis related to the discrete lattice $\mathbb{Z}^{n}$ and the
torus $\mathbb{T}^{n}$ has been developed by Turunen and the third author in
[RT10]. The pseudo-difference operators and the related symbolic calculus on
the weighted sequence space $\ell^{p}_{s}(\mathbb{Z}^{n})$ have been
extensively studied in [BGKR20]. The aim of the section is to recall the
preliminaries and important tools related to the discrete lattice
$\hbar\mathbb{Z}^{n}$ and the torus $\mathbb{T}^{n}_{\hbar}$ that will be
necessary for the analysis that we will follow.

###  3.1. Spaces of functions and distributions on the lattice
$\hbar\mathbb{Z}^{n}$

The Schwartz space $\mathcal{S}(\hbar\mathbb{Z}^{n})$ on the lattice
$\hbar\mathbb{Z}^{n}$ is the space of rapidly decreasing functions
$\varphi:\hbar\mathbb{Z}^{n}\rightarrow\mathbb{C}$; that is, we write
$\varphi\in\mathcal{S}(\hbar\mathbb{Z}^{n})$ if for any $M<\infty$ there
exists a constant $C_{\varphi,M}$ such that

$|\varphi(k)|\leq C_{\varphi,M}(1+|k|)^{-M},\quad\text{ for all
}k\in\hbar\mathbb{Z}^{n},$

where $|k|$ stands for the Euclidean norm of $k\in\hbar\mathbb{Z}^{n}$; i.e.,
for $k=\hbar(k_{1},\cdots,k_{n})$, we have
$|k|=\hbar\left(\sum\limits_{l=1}^{n}k_{l}^{2}\right)^{\frac{1}{2}}$. The
topology on $\mathcal{S}(\hbar\mathbb{Z}^{n})$ is given by the seminorms
$p_{j}$, defined as

$p_{j}(\varphi):=\sup\limits_{k\in\hbar\mathbb{Z}^{n}}(1+|k|)^{j}|\varphi(k)|,\quad\varphi\in\mathcal{S}(\hbar\mathbb{Z}^{n})\text{
and }j\in\mathbb{N}_{0}.$

The topological dual of $\mathcal{S}(\hbar\mathbb{Z}^{n})$ is the space of
tempered distributions defined as

$\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n}):=\mathcal{L}\left(\mathcal{S}(\hbar\mathbb{Z}^{n}),\mathbb{C}\right),$

i.e., $\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})$ is the space of all linear
continuous functionals on $\mathcal{S}(\hbar\mathbb{Z}^{n})$ of the form

$\varphi\mapsto(u,\varphi):=\sum_{k\in\hbar\mathbb{Z}^{n}}u(k)\varphi(k),\quad\varphi\in\mathcal{S}(\hbar\mathbb{Z}^{n}).$

We note that, in contrast to $\mathcal{S}^{\prime}(\mathbb{R}^{n})$, the
tempered distributions in $\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})$ in the
semi-classical setting are pointwise well-defined functions on
$\hbar\mathbb{Z}^{n}$. Furthermore, a tempered distribution
$u:\hbar\mathbb{Z}^{n}\rightarrow\mathbb{C}$ has polynomial growth at
infinity, i.e., there exist positive constants $M$ and $C_{u,M}$ such that

$|u(k)|\leq C_{u,M}(1+|k|)^{M},\quad k\in\hbar\mathbb{Z}^{n}.$

For $s\in\mathbb{R}$, one can extend the usual space
$\ell^{2}(\hbar\mathbb{Z}^{n})$ to the weighted space
$\ell^{2}_{s}(\hbar\mathbb{Z}^{n})$ of order $s$ as follows: for
$f:\hbar\mathbb{Z}^{n}\rightarrow\mathbb{C}$ we write
$f\in\ell^{2}_{s}(\hbar\mathbb{Z}^{n})$ whenever the following norm is finite

$\|f\|_{\ell_{s}^{2}(\hbar\mathbb{Z}^{n})}:=\left(\sum_{k\in\hbar\mathbb{Z}^{n}}(1+|k|)^{2s}|f(k)|^{2}\right)^{\frac{1}{2}}\,.$

Clearly, the weighted $\ell^{2}$-spaces also include
$\ell^{2}(\hbar\mathbb{Z}^{n})$ as a special case when $s=0$.

Observe that the weighted space $\ell^{2}_{s}(\hbar\mathbb{Z}^{n})$ is a
Hilbert space when endowed with the natural inner product:

$(u,v)_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}:=\sum_{k\in\hbar\mathbb{Z}^{n}}(1+|k|)^{2s}u(k)\overline{v(k)}\,,$
(3.1)

where $\overline{v(k)}$ stands for the complex conjugate of $v(k)$.

Let us point out that the structure of the space of tempered distributions
$\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})$, as well as this of the Schwartz
space $\mathcal{S}(\hbar\mathbb{Z}^{n})$, are closely related to the weighted
$\ell^{2}$-spaces. In particular, we have the following useful relations:

$\mathcal{S}(\hbar\mathbb{Z}^{n})=\bigcap_{s\in\mathbb{R}}\ell_{s}^{2}(\hbar\mathbb{Z}^{n})\text{
and
}\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})=\bigcup_{s\in\mathbb{R}}\ell_{s}^{2}(\hbar\mathbb{Z}^{n}).$
(3.2)

###  3.2. Space of periodic functions and distributions on the torus
$\mathbb{T}_{\hbar}^{n}$

Let us now introduce the torus denoted by $\mathbb{T}_{\hbar}^{n}$ that will
be useful for the subsequent analysis, especially when the semi-classical
limit $\hbar\rightarrow 0$ is taken. The torus $\mathbb{T}_{\hbar}^{n}$ can be
realised via the identification

$\mathbb{T}_{\hbar}^{n}=\left[-\frac{1}{2\hbar},\frac{1}{2\hbar}\right]^{n},\quad\hbar>0.$

Consequently, the space $C^{k}(\mathbb{T}_{\hbar}^{n})$ consists of functions
that are $\frac{1}{\hbar}$-periodic and $k$-times continuously differentiable
functions on torus $\mathbb{T}_{\hbar}^{n}$. The space
$C^{\infty}(\mathbb{T}_{\hbar}^{n})$ of test functions on the torus
$\mathbb{T}_{\hbar}^{n}$ can then be defined as

$C^{\infty}(\mathbb{T}_{\hbar}^{n}):=\bigcap\limits_{k=1}^{\infty}C^{k}(\mathbb{T}_{\hbar}^{n}).$

The Fréchet topology on the space of smooth functions
$C^{\infty}(\mathbb{T}_{\hbar}^{n})$ is given by the seminorms $p_{j}$ defined
as

$p_{j}(\psi):=\max\\{\|\partial^{\alpha}\psi\|_{C(\mathbb{T}_{\hbar}^{n})}:|\alpha|\leq
j\\},\quad j\in\mathbb{N}_{0},\alpha\in\mathbb{N}^{n}_{0}.$

The topological dual of $C^{\infty}(\mathbb{T}_{\hbar}^{n})$ is the space of
periodic distributions defined as

$\mathcal{D}^{\prime}(\mathbb{T}_{\hbar}^{n}):=\mathcal{L}\left(C^{\infty}(\mathbb{T}_{\hbar}^{n}),\mathbb{C}\right),$

i.e., it is the space of all linear continuous functionals on
$C^{\infty}(\mathbb{T}_{\hbar}^{n})$ of the form

$\varphi\mapsto\int_{\mathbb{T}_{\hbar}^{n}}\varphi(\xi)\psi(\xi)\mathrm{d}\xi,\quad\psi\in
C^{\infty}(\mathbb{T}_{\hbar}^{n}).$

###  3.3. Related semi-classical Fourier analysis

The Fourier transform operator

$\mathcal{F}_{\hbar\mathbb{Z}^{n}}:\mathcal{S}(\hbar\mathbb{Z}^{n})\to
C^{\infty}(\mathbb{T}_{\hbar}^{n})$

is defined as

$\mathcal{F}_{\hbar\mathbb{Z}^{n}}u(\xi):=\hbar^{n/2}\sum_{k\in\hbar\mathbb{Z}^{n}}u(k)e^{-2\pi
ik\cdot\xi},\quad\xi\in\mathbb{T}_{\hbar}^{n}\,.$

For the inverse Fourier transform operator

$\mathcal{F}^{-1}_{\hbar\mathbb{Z}^{n}}:C^{\infty}(\mathbb{T}_{\hbar}^{n})\to\mathcal{S}(\hbar\mathbb{Z}^{n})$

we have

$\mathcal{F}^{-1}_{\hbar\mathbb{Z}^{n}}v(k):=\hbar^{n/2}\int_{\mathbb{T}_{\hbar}^{n}}v(\xi)e^{2\pi
ik\cdot\xi}\mathrm{d}\xi,\quad k\in\hbar\mathbb{Z}^{n}\,,$

implying that the Fourier inversion formula is given by

$u(k)=\hbar^{n/2}\int_{\mathbb{T}_{\hbar}^{n}}\widehat{u}(\xi)e^{2\pi
ik\cdot\xi}\mathrm{d}\xi,\quad k\in\hbar\mathbb{Z}^{n}.$ (3.3)

The Fourier transform $\mathcal{F}_{\hbar\mathbb{Z}^{n}}$ can be uniquely
extended to the space of tempered distributions
$\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})$ when realised via the
distributional duality

$(\mathcal{F}_{\hbar\mathbb{Z}^{n}}u,\psi):=(u,\iota\circ\mathcal{F}_{\hbar\mathbb{Z}^{n}}^{-1}\psi),$
(3.4)

where $u\in\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})$, $\psi\in
C^{\infty}(\mathbb{T}_{\hbar}^{n}),$ and $(\iota\circ f)(x)=f(-x)$.

Hence for the operator $\mathcal{F}_{\hbar\mathbb{Z}^{n}}$ we can write in
more generality
$\mathcal{F}_{\hbar\mathbb{Z}^{n}}:\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})\to\mathcal{D}^{\prime}(\mathbb{T}_{\hbar}^{n})$,
and consequently we can define the inverse operator
$\mathcal{F}_{\hbar\mathbb{Z}^{n}}^{-1}:\mathcal{D}^{\prime}(\mathbb{T}_{\hbar}^{n})\to\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})$.
With the use of the latter, we can define the periodic Sobolev spaces
$H^{s}(\mathbb{T}_{\hbar}^{n})$, for $s\in\mathbb{R}$ as follows:

$H^{s}(\mathbb{T}_{\hbar}^{n}):=\left\\{u\in\mathcal{D}^{\prime}(\mathbb{T}_{\hbar}^{n}):\|u\|_{H^{s}(\mathbb{T}_{\hbar}^{n})}:=\left(\sum_{k\in\hbar\mathbb{Z}^{n}}(1+|k|)^{2s}\left|\mathcal{F}^{-1}_{\hbar\mathbb{Z}^{n}}u(k)\right|^{2}\right)^{1/2}<\infty\right\\}.$

For any $s\in\mathbb{R}$ the periodic Sobolev space
$H^{s}\left(\mathbb{T}^{n}_{\hbar}\right)$ is a Hilbert space endowed with the
inner product given by

$(u,v)_{H^{s}\left(\mathbb{T}^{n}_{\hbar}\right)}:=\sum_{k\in\hbar\mathbb{Z}^{n}}(1+|k|)^{2s}\mathcal{F}^{-1}_{\hbar\mathbb{Z}^{n}}u(k)\overline{\mathcal{F}^{-1}_{\hbar\mathbb{Z}^{n}}v(k)}.$
(3.5)

Clearly, the periodic Sobolev space also includes
$L^{2}(\mathbb{T}_{\hbar}^{n})$ as a special case when $s=0$.

We have the following relation between the space of test functions
$C^{\infty}(\mathbb{T}_{\hbar}^{n})$ and the space of periodic distributions
$\mathcal{D}^{\prime}(\mathbb{T}_{\hbar}^{n})$ and the periodic Sobolev spaces
$H^{s}(\mathbb{T}_{\hbar}^{n})$:

$C^{\infty}(\mathbb{T}_{\hbar}^{n})=\bigcap_{s\in\mathbb{R}}H^{s}(\mathbb{T}_{\hbar}^{n})\text{
and
}\mathcal{D}^{\prime}(\mathbb{T}_{\hbar}^{n})=\bigcup_{s\in\mathbb{R}}H^{s}(\mathbb{T}_{\hbar}^{n}).$

Combining the inner product (3.1) and (3.5) with the Fourier transform, we
obtain the following Plancherel formula

$\|f\|_{\ell_{s}^{2}(\hbar\mathbb{Z}^{n})}=\left\|(1+|\cdot|)^{s}f(\cdot)\right\|_{\ell^{2}(\hbar\mathbb{Z}^{n})}=\|\widehat{f}\|_{H^{s}(\mathbb{T}_{\hbar}^{n})},\quad
f\in\ell_{s}^{2}(\hbar\mathbb{Z}^{n}).$ (3.6)

###  3.4. The discrete fractional Laplacian

The discrete fractional Laplacian on the lattice $\hbar\mathbb{Z}^{n}$ can be
defined by restricting the usual fractional centered difference operators in
the Euclidean setting $\mathbb{R}^{n}$, see [Ort11, Section 5.4]. For more
details about the fractional difference operators on $\mathbb{R}^{n}$, we
refer to [SKM+93].

Rigorously, for a positive $\alpha>0$ and for $u$ being a complex-valued grid
function on $\hbar\mathbb{Z}^{n}$, the discrete fractional Laplacian
$\left(-\mathcal{L}_{\hbar}\right)^{\alpha}$ is defined by

$\left(-\mathcal{L}_{\hbar}\right)^{\alpha}u(k):=\sum_{j\in\mathbb{Z}^{n}}a_{j}^{(\alpha)}u(k+j\hbar),\quad\alpha>0,$
(3.7)

where the generating function $a_{j}^{(\alpha)}$ is given by

$a_{j}^{(\alpha)}:=\int\limits_{\left[-\frac{1}{2},\frac{1}{2}\right]^{n}}\left[\sum_{l=1}^{n}4\sin^{2}\left(\pi\xi_{l}\right)\right]^{\alpha}e^{-2\pi
ij\cdot\xi}\mathrm{d}\xi.$

The Fourier transform and the Fourier inversion formula, allow to verify that

$\sum_{j\in\mathbb{Z}^{n}}a_{j}^{(\alpha)}e^{2\pi
ij\cdot\xi}=\left[\sum_{l=1}^{n}4\sin^{2}\left(\pi\xi_{l}\right)\right]^{\alpha},\quad\xi\in\mathbb{T}_{\hbar}^{n}.$
(3.8)

Further, using the relation (3.8) and the shifting property of the Fourier
transform, we can compute the Fourier transform of the discrete fractional
Laplacian $\left(-\mathcal{L}_{\hbar}\right)^{\alpha}$ as follows:

$\displaystyle\left(\mathcal{F}_{\hbar\mathbb{Z}^{n}}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}u\right)(\xi)$
$\displaystyle=$
$\displaystyle\sum_{k\in\hbar\mathbb{Z}^{n}}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}u(k)e^{-2\pi
ik\cdot\xi}$ (3.9) $\displaystyle=$
$\displaystyle\sum_{k\in\hbar\mathbb{Z}^{n}}\left(\sum_{j\in\mathbb{Z}^{n}}a_{j}^{(\alpha)}u(k+j\hbar)\right)e^{-2\pi
ik\cdot\xi}$ $\displaystyle=$
$\displaystyle\left(\sum_{j\in\mathbb{Z}^{n}}a_{j}^{(\alpha)}e^{2\pi
ij\hbar\cdot\xi}\right)\widehat{u}(\xi)$ $\displaystyle=$
$\displaystyle\left[\sum_{l=1}^{n}4\sin^{2}\left(\pi\hbar\xi_{l}\right)\right]^{\alpha}\widehat{u}(\xi),$

for all $\xi\in\mathbb{T}_{\hbar}^{n},$ and consequently the Fourier transform
of fractional Laplacian is

$\displaystyle\left(\mathcal{F}_{\hbar\mathbb{Z}^{n}}\left(-\mathcal{L}\right)^{\alpha}u\right)(\xi)$
$\displaystyle=$
$\displaystyle\left(\left(-\mathcal{L}\right)^{\alpha}u,e^{2\pi
ik\cdot\xi}\right)$ (3.10) $\displaystyle=$
$\displaystyle\left(u,\left(-\mathcal{L}\right)^{\alpha}e^{2\pi
ik\cdot\xi}\right)$ $\displaystyle=$
$\displaystyle\left(u,|2\pi\xi|^{2\alpha}e^{2\pi ik\cdot\xi}\right)$
$\displaystyle=$ $\displaystyle|2\pi\xi|^{2\alpha}\widehat{u}(\xi),$

for all $\xi\in\mathbb{T}_{\hbar}^{n}$. For more details about the
construction and other properties of discrete fractional Laplacian on
$\hbar\mathbb{Z}^{n}$, see [DRT22b].

The consistency of the formula 3.7 for the discrete fractional Laplacian
$(-\mathcal{L}_{\hbar})^{\alpha}$ with the usual discrete Laplacian introduced
in the monograph [DRT22a], can be understood by explicitly computing the
expansion coefficients $a_{j}^{(\alpha)}$, for $\alpha=1$ and
$j\in\mathbb{Z}^{n}$. Indeed, it is easy to check that

$a_{0}^{(1)}=2n,\quad a_{\pm v_{l}}^{(1)}=-1,\text{ and
}a_{j}^{(1)}=0,\quad\text{for all }j\neq 0,\pm v_{l},$

where $v_{l}$ is the $l^{th}$ basis vector in $\mathbb{Z}^{n}$, having all
zeros except for $1$ at the $l^{th}$ component. This gives

$\displaystyle(-\mathcal{L}_{\hbar})^{1}u(k)$ $\displaystyle=$ $\displaystyle
2nu(k)-\sum\limits_{j=\pm v_{l}}u(k+j\hbar)$ $\displaystyle=$ $\displaystyle
2nu(k)-\sum\limits_{l=1}^{n}\left(u(k+v_{l}\hbar)+u(k-v_{l}\hbar)\right),$

which is usual discrete Laplacian on $\hbar\mathbb{Z}^{n}$.

##  4\. Proof of the main results

In this section, we give the proofs of all the results presented above in
Section 2. Before moving on to proving our first result, let us note that the
proof of Theorem 1.1 in the Euclidean setting $\mathbb{R}^{n}$ follows the
same arguments as the ones that are used in the proof of Theorem 2.1 in the
setting $\hbar\mathbb{Z}^{n}$ except for using the inner product of
$L^{2}_{m}(\mathbb{R}^{n})$ instead of $H^{s}(\mathbb{T}_{\hbar}^{n})$.
Therefore we just prove Theorem 2.1 and the proof of Theorem 1.1 should be
considered verbatim.

###### Proof of Theorem 2.1.

Taking the Fourier transform of the Cauchy problem (1.3) with respect to
$k\in\hbar\mathbb{Z}^{n}$ and using (3.9), we obtain

$\left\\{\begin{array}[]{l}\partial_{t}\widehat{u}(t,\xi)+a(t)\nu^{2}(\xi)\widehat{u}(t,\xi)+b(t)\widehat{u}(t,\xi)=\widehat{f}(t,\xi),\quad\text{
with }(t,\xi)\in[0,T]\times\mathbb{T}_{\hbar}^{n},\\\
\widehat{u}(0,\xi)=\widehat{u}_{0}(\xi),\quad\xi\in\mathbb{T}_{\hbar}^{n},\end{array}\right.$
(4.1)

where

$\nu^{2}(\xi)=\hbar^{-2\alpha}\left[\sum_{l=1}^{n}4\sin^{2}\left(\pi\hbar\xi_{l}\right)\right]^{\alpha}\geq
0.$ (4.2)

Define the energy functional for the Cauchy problem (4.1) by

$E(t,\xi):=(a(t)\widehat{u}(t,\xi),\widehat{u}(t,\xi)),\quad(t,\xi)\in[0,T]\times\mathbb{T}_{\hbar}^{n},$
(4.3)

where $(\cdot,\cdot)$ is the inner product in the Sobolev space
$H^{s}(\mathbb{T}_{\hbar}^{n})$ given by (3.5). It is easy to check that

$\inf_{t\in[0,T]}\\{a(t)\\}(\widehat{u}(t,\xi),\widehat{u}(t,\xi))\leq(a(t)\widehat{u}(t,\xi),\widehat{u}(t,\xi))\leq\sup_{t\in[0,T]}\\{a(t)\\}(\widehat{u}(t,\xi),\widehat{u}(t,\xi)).$
(4.4)

Since $a\in L_{1}^{\infty}([0,T])$, there exist two positive constants $a_{0}$
and $a_{1}$ such that

$\inf_{t\in[0,T]}\\{a(t)\\}=a_{0}\quad\text{ and
}\sup_{t\in[0,T]}\\{a(t)\\}=a_{1}.$ (4.5)

Combining the equations (4.3) and (4.5) together with the inequality (4.4), we
obtain the following bounds for the energy functional

$a_{0}\|\widehat{u}(t,\cdot)\|_{H^{s}}^{2}\leq E(t,\xi)\leq
a_{1}\|\widehat{u}(t,\cdot)\|^{2}_{H^{s}},\quad(t,\xi)\in[0,T]\times\mathbb{T}_{\hbar}^{n}.$
(4.6)

Differentiating the energy functional $E(t,\xi)$ and using (4.1), we obtain

$\displaystyle\partial_{t}E(t,\xi)$ $\displaystyle=$
$\displaystyle(a_{t}(t)\widehat{u}(t,\xi),\widehat{u}(t,\xi))+(a(t)\widehat{u}_{t}(t,\xi),\widehat{u}(t,\xi))+(a(t)\widehat{u}(t,\xi),\widehat{u}_{t}(t,\xi))$
$\displaystyle=$
$\displaystyle(a_{t}(t)\widehat{u}(t,\xi),\widehat{u}(t,\xi))-(a^{2}(t)\nu^{2}(\xi)\widehat{u}(t,\xi),\widehat{u}(t,\xi))-$
$\displaystyle(a(t)b(t)\widehat{u}(t,\xi),\widehat{u}(t,\xi))+(a(t)\widehat{f}(t,\xi),\widehat{u}(t,\xi))-$
$\displaystyle(a(t)\widehat{u}(t,\xi),a(t)\nu^{2}(\xi)\widehat{u}(t,\xi))-(a(t)\widehat{u}(t,\xi),b(t)\widehat{u}(t,\xi))+$
$\displaystyle(a(t)\widehat{u}(t,\xi),\widehat{f}(t,\xi))$ $\displaystyle=$
$\displaystyle
a_{t}(t)(\widehat{u}(t,\xi),\widehat{u}(t,\xi))-2a^{2}(t)(\nu(\xi)\widehat{u}(t,\xi),\nu(\xi)\widehat{u}(t,\xi))-$
$\displaystyle
2a(t)b(t)(\widehat{u}(t,\xi),\widehat{u}(t,\xi))+a(t)(\widehat{f}(t,\xi),\widehat{u}(t,\xi))+$
$\displaystyle a(t)(\widehat{u}(t,\xi),\widehat{f}(t,\xi))$
$\displaystyle\leq$
$\displaystyle\left(|a_{t}(t)|+2|a(t)||b(t)|\right)\|\widehat{u}(t,\cdot)\|^{2}_{H^{s}}+2|a(t)||\operatorname{Re}(\widehat{f}(t,\xi),\widehat{u}(t,\xi))|.$

Using the hypothesis that $a\in L_{1}^{\infty}([0,T])$ and $b\in
L^{\infty}([0,T])$, we obtain

$\displaystyle\partial_{t}E(t,\xi)$ $\displaystyle\leq$
$\displaystyle\left(\|a_{t}\|_{L^{\infty}}+2\|a\|_{L^{\infty}}\|b\|_{L^{\infty}}\right)\|\widehat{u}(t,\cdot)\|^{2}_{H^{s}}+2\|a\|_{L^{\infty}}\|\widehat{f}(t,\cdot)\|_{H^{s}}\|\widehat{u}(t,\cdot)\|_{H^{s}}$
$\displaystyle\leq$
$\displaystyle\left(\|a_{t}\|_{L^{\infty}}+2\|a\|_{L^{\infty}}\|b\|_{L^{\infty}}+\|a\|_{L^{\infty}}\right)\|\widehat{u}(t,\cdot)\|^{2}_{H^{s}}+\|a\|_{L^{\infty}}\|\widehat{f}(t,\cdot)\|^{2}_{H^{s}}.$

If we set
$\kappa_{1}=\|a_{t}\|_{L^{\infty}}+2\|a\|_{L^{\infty}}\|b\|_{L^{\infty}}+\|a\|_{L^{\infty}}$
and $\kappa_{2}=\|a\|_{L^{\infty}}$, then putting together (4.6) and (4), we
obtain

$\partial_{t}E(t,\xi)\leq
a_{0}^{-1}\kappa_{1}E(t,\xi)+\kappa_{2}\|\widehat{f}(t,\cdot)\|^{2}_{H^{s}}.$
(4.8)

Applying the Gronwall’s lemma to the inequality (4.8), we get

$E(t,\xi)\leq
e^{\int_{0}^{t}a_{0}^{-1}\kappa_{1}\mathrm{d}\tau}\left(E(0,\xi)+\int_{0}^{t}\kappa_{2}\|\widehat{f}(\tau,\cdot)\|^{2}_{H^{s}}\mathrm{d}\tau\right),$
(4.9)

for all $(t,\xi)\in[0,T]\times\mathbb{T}_{\hbar}^{n}$. Again combining the
estimates (4.6) and (4.9), we obtain

$\displaystyle a_{0}\|\widehat{u}(t,\cdot)\|^{2}_{H^{s}}\leq E(t,\xi)$
$\displaystyle\leq$ $\displaystyle
e^{\int_{0}^{t}a_{0}^{-1}\kappa_{1}\mathrm{d}\tau}\left(E(0,\xi)+\int_{0}^{t}\kappa_{2}\|\widehat{f}(\tau,\cdot)\|^{2}_{H^{s}}\mathrm{d}\tau\right)$
$\displaystyle\leq$ $\displaystyle
e^{a_{0}^{-1}\kappa_{1}T}\left(a_{1}\|\widehat{u}(0,\cdot)\|^{2}_{H^{s}}+\kappa_{2}\int_{0}^{T}\|\widehat{f}(\tau,\cdot)\|^{2}_{H^{s}}\mathrm{d}\tau\right).$

Further, using the Plancherel formula (3.6), we obtain the required estimate

$\|u(t,\cdot)\|^{2}_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}\leq
C_{T,a,b}\left(\|u_{0}\|^{2}_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}+\|f\|^{2}_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}\right),\quad\text{
for all }t\in[0,T],$

where the constant $C_{T,a,b}$ is given by

$C_{T,a,b}=a_{0}^{-1}\|a\|_{L^{\infty}}e^{a_{0}^{-1}(\|a_{t}\|_{L^{\infty}}+2\|a\|_{L^{\infty}}\|b\|_{L^{\infty}}+\|a\|_{L^{\infty}})T}.$

The uniqueness of the solution follows immediately from the above estimate.
This completes the proof. ∎

Thus, we have obtained the well-posedness for the Cauchy problem (1.3) in the
weighted spaces $\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$ for all $s\in\mathbb{R}$
and consequently we have distributional well-posedness for the Cauchy problem
(1.3) in the space of tempered distribution
$\mathcal{S}^{\prime}(\hbar\mathbb{Z}^{n})$. We will now prove the existence
of the very weak solution to the Cauchy problem (1.3) in the case of
distributional coefficients.

###### Proof of Theorem 2.4.

Consider the regularised Cauchy problem

$\left\\{\begin{array}[]{l}\partial_{t}u_{\varepsilon}(t,k)+a_{\varepsilon}(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}u_{\varepsilon}(t,k)+b_{\varepsilon}(t)u_{\varepsilon}(t,k)=f_{\varepsilon}(t,k),\quad
t\in(0,T],\\\ u_{\varepsilon}(0,k)=u_{0}(k),\quad
k\in\hbar\mathbb{Z}^{n},\end{array}\right.$ (4.10)

where the coefficient $a_{\varepsilon}$ is $L^{\infty}_{1}$-moderate,
$b_{\varepsilon}$ is $L^{\infty}$-moderate and the source term
$f_{\varepsilon}$ is $L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n})$-moderate
regularisation of the coefficients $a,b$ and the source term $f$,
respectively. Taking the Fourier transform with respect to
$k\in\hbar\mathbb{Z}^{n}$, we obtain

$\left\\{\begin{array}[]{l}\partial_{t}\widehat{u}_{\varepsilon}(t,\xi)+a_{\varepsilon}(t)\nu^{2}(\xi)\widehat{u}_{\varepsilon}(t,\xi)+b_{\varepsilon}(t)\widehat{u}_{\varepsilon}(t,\xi)=\widehat{f}_{\varepsilon}(t,\xi),\quad(t,\xi)\in(0,T]\times\mathbb{T}_{\hbar}^{n},\\\
\widehat{u}_{\varepsilon}(0,\xi)=\widehat{u}_{0}(\xi),\quad
k\in\mathbb{T}_{\hbar}^{n},\end{array}\right.$

where $\nu^{2}(\xi)$ is given by (4.2). Define the energy functional for the
Cauchy problem (4.10) by

$E_{\varepsilon}(t,\xi):=(a_{\varepsilon}(t)\widehat{u}_{\varepsilon}(t,\xi),\widehat{u}_{\varepsilon}(t,\xi)),\quad(t,\xi)\in[0,T]\times\mathbb{T}_{\hbar}^{n},$
(4.11)

where $(\cdot,\cdot)$ is the inner product in the Sobolev space
$H^{s}(\mathbb{T}_{\hbar}^{n})$. It is easy to check that

$\inf_{t\in[0,T]}\\{a_{\varepsilon}(t)\\}(\widehat{u}_{\varepsilon}(t,\xi),\widehat{u}_{\varepsilon}(t,\xi))\leq(a_{\varepsilon}(t)\widehat{u}_{\varepsilon}(t,\xi),\widehat{u}_{\varepsilon}(t,\xi))\leq\sup_{t\in[0,T]}\\{a_{\varepsilon}(t)\\}(\widehat{u}_{\varepsilon}(t,\xi),\widehat{u}_{\varepsilon}(t,\xi)).$

Since $a$ and $b$ are distributions, by the structure theorem for compactly
supported distributions, there exist $L_{1},L_{2}\in\mathbb{N}$ and
$c_{1},c_{2}>0$ such that

$\left|\partial_{t}^{k}a_{\varepsilon}(t)\right|\leq
c_{1}\omega(\varepsilon)^{-L_{1}-k}\text{ and
}\left|\partial_{t}^{k}b_{\varepsilon}(t)\right|\leq
c_{2}\omega(\varepsilon)^{-L_{2}-k},\quad k\in\mathbb{N}_{0},$ (4.12)

for all $t\in[0,T]$, where $\omega(\varepsilon)$ is given by (2.3). Since
$a\geq a_{0}>0$, we can write

$a_{\varepsilon}(t)=\left(a*\psi_{\omega(\varepsilon)}\right)(t)=\langle
a,\tau_{t}\tilde{\psi}_{\omega(\varepsilon)}\rangle\geq\tilde{a}_{0}>0,$
(4.13)

where $\tilde{\psi}(x)=\psi(-x),x\in\mathbb{R}$ and
$\tau_{t}\psi(\xi)=\psi(\xi-t),\xi\in\mathbb{R}$.

Now applying Theorem 2.1 to the Cauchy problem (4.10), we have the following
estimate

$\|u_{\varepsilon}(t,\cdot)\|^{2}_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}\leq
C_{T,a_{\varepsilon},b_{\varepsilon}}\left(\|u_{0}\|^{2}_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}+\|f_{\varepsilon}\|^{2}_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}\right),\quad\text{
for all }t\in[0,T],$ (4.14)

where the constant $C_{T,a_{\varepsilon},b_{\varepsilon}}$ is given by

$C_{T,a_{\varepsilon},b_{\varepsilon}}=\tilde{a}_{0}^{-1}\|a_{\varepsilon}\|_{L^{\infty}}e^{\tilde{a}_{0}^{-1}(\|\partial_{t}a_{\varepsilon}\|_{L^{\infty}}+2\|a_{\varepsilon}\|_{L^{\infty}}\|b_{\varepsilon}\|_{L^{\infty}}+\|a_{\varepsilon}\|_{L^{\infty}})T}.$
(4.15)

Combining the estimates (4.12) and (4.14) with (4.15) we get

$\displaystyle\|u_{\varepsilon}(t,\cdot)\|^{2}_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}$
$\displaystyle\leq$
$\displaystyle\tilde{a}_{0}^{-1}c_{1}\omega(\varepsilon)^{-L_{1}}e^{\left(c_{1}\omega(\varepsilon)^{-L_{1}-1}+2c_{1}c_{2}\omega(\varepsilon)^{-L_{1}-L_{2}}+c_{1}\omega(\varepsilon)^{-L_{1}}\right)T}\times$
$\displaystyle\left(\|u_{0}\|^{2}_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}+\|f_{\varepsilon}\|^{2}_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}\right)$
$\displaystyle\leq$
$\displaystyle\tilde{a}_{0}^{-1}e^{K_{T}\left(\omega(\varepsilon)^{-L_{1}-1}+\omega(\varepsilon)^{-L_{1}-L_{2}}+\omega(\varepsilon)^{-L_{1}}\right)T}\times$
$\displaystyle\left(\|u_{0}\|^{2}_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}+\|f_{\varepsilon}\|^{2}_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}\right),$

where $K_{T}=\tilde{a}_{0}^{-1}c_{1}T\max\\{1,2c_{2},T^{-1}\\}$. Putting
$\omega(\varepsilon)\sim|\log(\varepsilon)|^{-1}$, we obtain

$\|u_{\varepsilon}(t,\cdot)\|_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}^{2}\lesssim\varepsilon^{-3L_{1}-L_{2}-1}\left(\|u_{0}\|_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}^{2}+\|f_{\varepsilon}\|_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}^{2}\right),$
(4.16)

for all $t\in[0,T]$ and $\varepsilon\in(0,1]$.

Since $(f_{\varepsilon})_{\varepsilon}$ is
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-moderate regularisation of
$f$, there exist positive constants $L_{3}$ and $c$ such that

$\|f_{\varepsilon}\|_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}\leq
c\varepsilon^{-L_{3}}.$ (4.17)

Now by integrating the estimate (4.16) with respect to $t\in[0,T]$ and then
using (4.17), we obtain

$\|u_{\varepsilon}\|_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}\lesssim\varepsilon^{-3L_{1}-L_{2}-L_{3}-1}.$
(4.18)

Thus we conclude that $(u_{\varepsilon})_{\varepsilon}$ is
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-moderate. This completes the
proof. ∎

Therefore, we have established the existence of very weak solution for the
Cauchy problem (1.3) with irregular coefficients. We will now prove the
uniqueness of the very weak solution of it in the sense of Definition 2.5.

###### Proof of Theorem 2.7.

Let $(u_{\varepsilon})_{\varepsilon}$ and
$(\tilde{u}_{\varepsilon})_{\varepsilon}$ be the families of solution
corresponding to the Cauchy problems (2.6) and (2.7), respectively. Denoting
$w_{\varepsilon}:=u_{\varepsilon}-\tilde{u}_{\varepsilon}$, we get

$\left\\{\begin{array}[]{l}\partial_{t}w_{\varepsilon}(t,k)+a_{\varepsilon}(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}w_{\varepsilon}(t,k)+b_{\varepsilon}(t)w_{\varepsilon}(t,k)=g_{\varepsilon}(t,k),\quad
t\in(0,T],\\\ w_{\varepsilon}(0,k)=0,\quad
k\in\hbar\mathbb{Z}^{n},\end{array}\right.$ (4.19)

where

$g_{\varepsilon}(t,k):=(\tilde{a}_{\varepsilon}-a_{\varepsilon})(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}\tilde{u}_{\varepsilon}(t,k)+(\tilde{b}_{\varepsilon}-b_{\varepsilon})(t)\tilde{u}_{\varepsilon}(t,k)+(f_{\varepsilon}-\tilde{f}_{\varepsilon})(t,k).$
(4.20)

Since the nets $(\tilde{a}_{\varepsilon}-a_{\varepsilon})_{\varepsilon}$,
$(\tilde{b}_{\varepsilon}-b_{\varepsilon})_{\varepsilon}$ and
$(f_{\varepsilon}-\tilde{f}_{\varepsilon})_{\varepsilon}$ are
$L^{\infty}_{1}$-negligible, $L^{\infty}$-negligible and
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-negligible, respectively, it
follows that $g_{\varepsilon}$ is
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-negligible.

Taking the Fourier transform of the Cauchy problem (4.19) with respect to
$k\in\hbar\mathbb{Z}^{n}$, we obtain

$\left\\{\begin{array}[]{l}\partial_{t}\widehat{w}_{\varepsilon}(t,\xi)+a_{\varepsilon}(t)\nu^{2}(\xi)\widehat{w}_{\varepsilon}(t,\xi)+b_{\varepsilon}(t)\widehat{w}_{\varepsilon}(t,\xi)=\widehat{g}_{\varepsilon}(t,\xi),\quad(t,\xi)\in(0,T]\times\mathbb{T}_{\hbar}^{n},\\\
\widehat{w}_{\varepsilon}(0,\xi)=0,\quad\xi\in\mathbb{T}_{\hbar}^{n},\end{array}\right.$
(4.21)

where $\nu^{2}(\xi)$ is given by (4.2). The energy functional for the Cauchy
problem (4.21) is given by

$E_{\varepsilon}(t,\xi):=(a_{\varepsilon}(t)\widehat{w}_{\varepsilon}(t,\xi),\widehat{w}_{\varepsilon}(t,\xi)),\quad(t,\xi)\in[0,T]\times\mathbb{T}_{\hbar}^{n},$
(4.22)

where $(\cdot,\cdot)$ is the inner product in the Sobolev space
$H^{s}(\mathbb{T}_{\hbar}^{n})$. Then using (4.12), we have the following
energy bounds

$\tilde{a}_{0}\|\widehat{w}_{\varepsilon}(t,\cdot)\|^{2}_{H^{s}}\leq
E_{\varepsilon}(t,\xi)\leq
c_{1}\omega(\varepsilon)^{-L_{1}}\|\widehat{w}_{\varepsilon}(t,\cdot)\|^{2}_{H^{s}},\quad(t,\xi)\in[0,T]\times\mathbb{T}_{\hbar}^{n}.$
(4.23)

Further we can calculate

$\partial_{t}E_{\varepsilon}(t,\xi)\leq\left(|a^{\prime}_{\varepsilon}(t)|+2|a_{\varepsilon}(t)||b_{\varepsilon}(t)|+|a_{\varepsilon}(t)|\right)\|\widehat{w}_{\varepsilon}(t,\cdot)\|^{2}_{H^{s}}+|a_{\varepsilon}(t)|\|\widehat{g}_{\varepsilon}(t,\cdot)\|^{2}_{H^{s}}.$
(4.24)

Combining the estimates (4.12), (4.23) and (4.24), we obtain

$\partial_{t}E_{\varepsilon}(t,\xi)\leq\kappa_{1}\left(\omega(\varepsilon)^{-L_{1}-1}+\omega(\varepsilon)^{-L_{1}-L_{2}}+\omega(\varepsilon)^{-L_{1}}\right)E_{\varepsilon}(t,\xi)+c_{1}\omega(\varepsilon)^{-L_{1}}\|\widehat{g}_{\varepsilon}(t,\cdot)\|^{2}_{H^{s}},$
(4.25)

where $\kappa_{1}=\tilde{a}_{0}^{-1}c_{1}\max\\{1,2c_{2}\\}$. Applying the
Gronwall’s lemma to the inequality (4.25), we obtain

$E_{\varepsilon}(t,\xi)\leq\\\
e^{\int_{0}^{t}\kappa_{1}\left(\omega(\varepsilon)^{-L_{1}-1}+\omega(\varepsilon)^{-L_{1}-L_{2}}+\omega(\varepsilon)^{-L_{1}}\right)\mathrm{d}\tau}\left(E_{\varepsilon}(0,\xi)+c_{1}\omega(\varepsilon)^{-L_{1}}\int_{0}^{t}\|\widehat{g}_{\varepsilon}(\tau,\cdot)\|^{2}_{H^{s}}\mathrm{d}\tau\right).$
(4.26)

Putting together (4.23) and (4.26), and then using the fact that
$\widehat{w}_{\varepsilon}(0,\xi)\equiv 0$ for all $\varepsilon\in(0,1]$, we
get

$\displaystyle\|\widehat{w}_{\varepsilon}(t,\cdot)\|^{2}$ $\displaystyle\leq$
$\displaystyle\tilde{a}_{0}^{-1}c_{1}\omega(\varepsilon)^{-L_{1}}e^{\kappa_{1}\left(\omega(\varepsilon)^{-L_{1}-1}+\omega(\varepsilon)^{-L_{1}-L_{2}}+\omega(\varepsilon)^{-L_{1}}\right)T}\int_{0}^{T}\|\widehat{g}_{\varepsilon}(\tau,\cdot)\|^{2}_{H^{s}}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\tilde{a}_{0}^{-1}c_{1}e^{\kappa_{T}\left(\omega(\varepsilon)^{-L_{1}-1}+\omega(\varepsilon)^{-L_{1}-L_{2}}+\omega(\varepsilon)^{-L_{1}}\right)}\int_{0}^{T}\|\widehat{g}_{\varepsilon}(\tau,\cdot)\|^{2}_{H^{s}}\mathrm{d}\tau,$

where $\kappa_{T}=c_{1}T\max\\{1,2c_{2},T^{-1}\\}$. Putting
$\omega(\varepsilon)\sim|\log(\varepsilon)|^{-1}$ and using the Plancherel
formula (3.6), we obtain

$\|w_{\varepsilon}(t,\cdot)\|_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}^{2}\lesssim\varepsilon^{-3L_{1}-L_{2}-1}\|g_{\varepsilon}\|_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}^{2},$

for all $t\in[0,T]$. Since $g_{\varepsilon}$ is
$L^{2}\left([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n})\right)$-negligible, we
obtain

$\left\|w_{\varepsilon}(t,\cdot)\right\|_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}^{2}\lesssim\varepsilon^{-3L_{1}-L_{2}-1}\varepsilon^{3L_{1}+L_{2}+1+q}=\varepsilon^{q},\quad\text{
for all }q\in\mathbb{N}_{0},$

for all $t\in[0,T]$. Now by integrating the above estimate with respect to
$t\in[0,T]$, we get

$\left\|w_{\varepsilon}\right\|_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}\lesssim\varepsilon^{q},\quad\text{
for all }q\in\mathbb{N}_{0}.$

Thus $\left(u_{\varepsilon}-\tilde{u}_{\varepsilon}\right)_{\varepsilon}$ is
$L^{2}\left([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n})\right)$-negligible. This
completes the proof. ∎

Next we prove that the very weak solution obtained in Theorem 2.4 is
consistent with the classical solution obtained in Theorem 2.1.

###### Proof of Theorem 2.8.

Let $\tilde{u}$ be the classical solution given by Theorem 2.1, that is,
$\tilde{u}$ satisfies the Cauchy problem

$\left\\{\begin{array}[]{l}\partial_{t}\tilde{u}(t,k)+a(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}\tilde{u}(t,k)+b(t)\tilde{u}(t,k)=f(t,k),\quad(t,k)\in(0,T]\times\hbar\mathbb{Z}^{n},\\\
\tilde{u}(0,k)=u_{0}(k),\quad k\in\hbar\mathbb{Z}^{n},\end{array}\right.$
(4.27)

and let $(u_{\varepsilon})_{\varepsilon}$ be the very weak solution obtained
by Theorem 2.4, that is, $(u_{\varepsilon})_{\varepsilon}$ satisfies the
regularised Cauchy problem

$\left\\{\begin{array}[]{l}\partial_{t}u_{\varepsilon}(t,k)+a_{\varepsilon}(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}u_{\varepsilon}(t,k)+b_{\varepsilon}(t)u_{\varepsilon}(t,k)=f_{\varepsilon}(t,k),\quad
t\in(0,T],\\\ u_{\varepsilon}(0,k)=u_{0}(k),\quad
k\in\hbar\mathbb{Z}^{n}.\end{array}\right.$ (4.28)

Note that by the hypothesis the nets
$\left(a_{\varepsilon}-a\right)_{\varepsilon},\left(b_{\varepsilon}-b\right)_{\varepsilon}$
and $\left(f_{\varepsilon}-f\right)_{\varepsilon}$ are converging to $0$
uniformly. Using (4.27) and (4.28), we have

$\left\\{\begin{array}[]{l}\partial_{t}\tilde{u}(t,k)+a_{\varepsilon}(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}\tilde{u}(t,k)+b_{\varepsilon}(t)\tilde{u}(t,k)=f_{\varepsilon}(t,k)+g_{\varepsilon}(t,k),\quad
t\in(0,T],\\\ \tilde{u}(0,k)=u_{0}(k),\quad
k\in\hbar\mathbb{Z}^{n},\end{array}\right.$ (4.29)

where

$g_{\varepsilon}(t,k):=\left(a_{\varepsilon}-a\right)(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}\tilde{u}(t,k)+\left(b_{\varepsilon}-b\right)(t)\tilde{u}(t,k)+\left(f-f_{\varepsilon}\right)(t,k),$

$g_{\varepsilon}\in L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$ and
$g_{\varepsilon}\to 0$ in $L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$ as
$\varepsilon\rightarrow 0$.

Combining the Cauchy problem (4.28) and (4.29), we deduce that the net
$w_{\varepsilon}:=\left(\tilde{u}-u_{\varepsilon}\right)$ solves the Cauchy
problem

$\left\\{\begin{array}[]{l}\partial_{t}w_{\varepsilon}(t,k)+a_{\varepsilon}(t)\hbar^{-2\alpha}\left(-\mathcal{L}_{\hbar}\right)^{\alpha}w_{\varepsilon}(t,k)+b_{\varepsilon}(t)w_{\varepsilon}(t,k)=g_{\varepsilon}(t,k),\quad
t\in(0,T],\\\ w_{\varepsilon}(0,k)=0,\quad
k\in\hbar\mathbb{Z}^{n}.\end{array}\right.$ (4.30)

Taking the Fourier transform of the Cauchy problem (4.30) with respect to
$k\in\hbar\mathbb{Z}^{n}$, we obtain

$\left\\{\begin{array}[]{l}\partial_{t}\widehat{w}_{\varepsilon}(t,\xi)+a_{\varepsilon}(t)\nu^{2}(\xi)\widehat{w}_{\varepsilon}(t,\xi)+b_{\varepsilon}(t)\widehat{w}_{\varepsilon}(t,\xi)=\widehat{g}_{\varepsilon}(t,\xi),\quad(t,\xi)\in(0,T]\times\mathbb{T}_{\hbar}^{n},\\\
\widehat{w}_{\varepsilon}(0,\xi)=0,\quad\xi\in\mathbb{T}_{\hbar}^{n},\end{array}\right.$
(4.31)

where $\nu^{2}(\xi)$ is given by (4.2). Define the energy functional for the
Cauchy problem (4.31) by

$E_{\varepsilon}(t,\xi):=(a_{\varepsilon}(t)\widehat{w}_{\varepsilon}(t,\xi),\widehat{w}_{\varepsilon}(t,\xi)),\quad(t,\xi)\in[0,T]\times\mathbb{T}_{\hbar}^{n},$

where $(\cdot,\cdot)$ is the inner product in the Sobolev space
$H^{s}(\mathbb{T}_{\hbar}^{n})$.

Since the coefficients are sufficiently regular, following the lines of
Theorem 2.1, the next energy estimate holds true

$\partial_{t}E_{\varepsilon}(t,\xi)\leq\kappa_{1}E_{\varepsilon}(t,\xi)+\kappa_{2}\left|\widehat{g}_{\varepsilon}(t,\xi)\right|^{2},$

for some positive constants $\kappa_{1}$ and $\kappa_{2}$. Then using the
Gronwall’s lemma and the energy bounds similar to (4.6) alongwith the
Plancherel formula (3.6), we obtain the following estimate

$\|w_{\varepsilon}(t,\cdot)\|^{2}_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}\lesssim\|w_{\varepsilon}(0,\cdot)\|_{\ell^{2}_{s}(\hbar\mathbb{Z}^{n})}^{2}+\|g_{\varepsilon}\|_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}^{2},$

\+ for all $t\in[0,T]$. Now by integrating the above estimate with respect to
$t\in[0,T]$ and using the fact that $w_{\varepsilon}(0,k)\equiv 0$ for all
$\varepsilon\in(0,1]$, we get

$\|w_{\varepsilon}\|_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}^{2}\lesssim\|g_{\varepsilon}\|_{L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))}^{2}.$

Since $g_{\varepsilon}\rightarrow 0$ in
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$, we have

$w_{\varepsilon}\to 0\text{ in
}L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n})),\quad\varepsilon\to 0,$

implying also that

$u_{\varepsilon}\to\tilde{u}\text{ in
}L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n})),\quad\varepsilon\to 0.$

Furthermore, the limit is the same for every representation of $u$, since they
will differ from $\left(u_{\varepsilon}\right)_{\varepsilon}$ by a
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-negligible net. This
completes the proof. ∎

##  5\. Semi-classical limit $\hbar\to 0$

In this section, we will prove the semi-classical limit theorems for the
classical solution as well as for the very weak solution.

###### Proof of Theorem 2.9.

Consider two Cauchy problems:

$\left\\{\begin{array}[]{l}\partial_{t}u(t,k)+a(t)\hbar^{-2\alpha}(-\mathcal{L}_{\hbar})^{\alpha}u(t,k)+b(t)u(t,k)=f(t,k),\quad(t,k)\in(0,T]\times\hbar\mathbb{Z}^{n},\\\
u(0,k)=u_{0}(k),\quad k\in\hbar\mathbb{Z}^{n},\end{array}\right.$ (5.1)

and

$\left\\{\begin{array}[]{ll}\partial_{t}v(t,x)+a(t)(-\mathcal{L})^{\alpha}v(t,x)+b(t)v(t,x)=f(t,x),\quad(t,x)\in(0,T]\times\mathbb{R}^{n},\\\
v(0,x)=u_{0}(x),\quad x\in\mathbb{R}^{n},\end{array}\right.$ (5.2)

where $\left(-\mathcal{L}\right)^{\alpha}$ is the usual fractional Laplacian
on $\mathbb{R}^{n}$ given by (1.5). We have assumed that $f$ and $u_{0}$ in
(5.1) are the restrictions in $\hbar\mathbb{Z}^{n}$ of the corresponding ones
in (5.2) defined on $\mathbb{R}^{n}$. From the equations (5.1) and (5.2),
denoting $w:=u-v$, we get

$\left\\{\begin{array}[]{ll}\partial_{t}w(t,k)+a(t)\hbar^{-2\alpha}(-\mathcal{L}_{\hbar})^{\alpha}w(t,k)+b(t)w(t,k)=a(t)\left((-\mathcal{L})^{\alpha}-\hbar^{-2\alpha}(-\mathcal{L}_{\hbar})^{\alpha}\right)v(t,k),\\\
w(0,k)=0,\quad k\in\hbar\mathbb{Z}^{n}.\end{array}\right.$ (5.3)

Since $w_{0}=0$, applying Theorem 2.1 with s = 0 for the Cauchy problem (5.3)
and using the estimate (2.1), we obtain

$\displaystyle\left\|w(t,\cdot)\right\|^{2}_{\ell^{2}(\hbar\mathbb{Z}^{n})}$
$\displaystyle\leq$ $\displaystyle
C_{T,a,b}\left\|a\left((-\mathcal{L})^{\alpha}-\hbar^{-2\alpha}(-\mathcal{L}_{\hbar})^{\alpha}\right)v\right\|^{2}_{L^{2}([0,T];\ell^{2}(\hbar\mathbb{Z}^{n}))}$
$\displaystyle\leq$ $\displaystyle
C_{T,a,b}\left\|a\right\|^{2}_{L^{\infty}([0,T])}\left\|\left((-\mathcal{L})^{\alpha}-\hbar^{-2\alpha}(-\mathcal{L}_{\hbar})^{\alpha}\right)v\right\|^{2}_{L^{2}([0,T];\ell^{2}(\hbar\mathbb{Z}^{n}))},$

for all $t\in[0,T]$, where the constant $C_{T,a,b}$ is given by

$C_{T,a,b}=a_{0}^{-1}\|a\|_{L^{\infty}}e^{a_{0}^{-1}(\|a_{t}\|_{L^{\infty}}+2\|a\|_{L^{\infty}}\|b\|_{L^{\infty}}+\|a\|_{L^{\infty}})T}.$

Now we will estimate the term
$\left\|\left((-\mathcal{L})^{\alpha}-\hbar^{-2\alpha}(-\mathcal{L}_{\hbar})^{\alpha}\right)v\right\|^{2}_{L^{2}([0,T];\ell^{2}(\hbar\mathbb{Z}^{n}))}.$
Using the Plancherel formula (3.6) for $s=0$, (3.9) and (3.10), we have

$\left\|\left((-\mathcal{L})^{\alpha}-\hbar^{-2\alpha}(-\mathcal{L}_{\hbar})^{\alpha}\right)v(t,\cdot)\right\|_{\ell^{2}(\hbar\mathbb{Z}^{n})}=\\\
\left\|\left(\left[\sum_{l=1}^{n}(2\pi\xi_{l})^{2}\right]^{\alpha}-\hbar^{-2\alpha}\left[\sum_{l=1}^{n}4\sin^{2}\left(\pi\hbar\xi_{l}\right)\right]^{\alpha}\right)\widehat{v}(t,\cdot)\right\|_{L^{2}(\mathbb{T}_{\hbar}^{n})}.$
(5.5)

Since $|\cdot|^{\alpha}:\mathbb{R}\to\mathbb{R}$ is $\alpha$-Hölder continuous
for $0<\alpha\leq 1$, we have the following inequality

$\left||x|^{2\alpha}-|y|^{2\alpha}\right|\lesssim\left||x|^{2}-|y|^{2}\right|^{\alpha},\quad
x,y\in\mathbb{R}^{n},$

and $|x|=\sqrt{x_{1}^{2}+\dots+x_{n}^{2}}$. Now, using the above inequality
and the Taylor expansion for $\sin^{2}(\pi\hbar\xi_{l})$, we get

$\displaystyle\left|\left[\sum_{l=1}^{n}(2\pi\xi_{l})^{2}\right]^{\alpha}-\hbar^{-2\alpha}\left[\sum_{l=1}^{n}4\sin^{2}\left(\pi\hbar\xi_{l}\right)\right]^{\alpha}\right|$
(5.6) $\displaystyle\lesssim$
$\displaystyle\left|\sum\limits_{l=1}^{n}4\pi^{2}\xi^{2}_{l}-\hbar^{-2}\sum\limits_{l=1}^{n}4\sin^{2}(\pi\hbar\xi_{l})\right|^{\alpha}$
$\displaystyle=$
$\displaystyle\left|\sum\limits_{l=1}^{n}\left[4\pi^{2}\xi^{2}_{l}-\hbar^{-2}4\left(\pi^{2}\hbar^{2}\xi_{l}^{2}-\frac{\pi^{4}}{3}\hbar^{4}\xi_{l}^{4}\cos\left(2\theta_{l}\right)\right)\right]\right|^{\alpha}$
$\displaystyle=$
$\displaystyle\left(\frac{4\pi^{4}\hbar^{2}}{3}\right)^{\alpha}\left|\sum\limits_{l=1}^{n}\xi_{l}^{4}\cos(2\theta_{l})\right|^{\alpha}$
$\displaystyle\lesssim$
$\displaystyle\hbar^{2\alpha}\left[\sum\limits_{l=1}^{n}\xi^{4}_{l}\right]^{\alpha}$
$\displaystyle\lesssim$ $\displaystyle\hbar^{2\alpha}|\xi|^{4\alpha},$

where
$|\xi|^{4\alpha}=\left[\sum\limits_{l=1}^{n}\xi^{2}_{l}\right]^{2\alpha}$ and
$\theta_{l}\in(0,\pi\hbar\xi_{l})$ or $(\pi\hbar\xi_{l},0)$ depending on the
sign of $\xi_{l}$. Now, combining the estimate (5.6) with (5.5), we get

$\displaystyle\left\|\left((-\mathcal{L})^{\alpha}-\hbar^{-2\alpha}(-\mathcal{L}_{\hbar})^{\alpha}\right)v(t,\cdot)\right\|_{\ell^{2}(\hbar\mathbb{Z}^{n})}$
$\displaystyle\lesssim$
$\displaystyle\hbar^{2\alpha}\|(1+|\cdot|^{2})^{2\alpha}\widehat{v}(t,\cdot)\|_{L^{2}(\mathbb{T}_{\hbar}^{n})}$
(5.7) $\displaystyle\lesssim$
$\displaystyle\hbar^{2\alpha}\|(1+|\cdot|^{2})^{m/2}\widehat{v}(t,\cdot)\|_{L^{2}(\mathbb{T}_{\hbar}^{n})},$

whenever $m\geq 4\alpha$. Since $u_{0}\in H^{m}(\mathbb{R}^{n})$ with $m\geq
4\alpha$, using Theorem 1.1, we have $v\in H^{m}(\mathbb{R}^{n})$ with $m\geq
4\alpha$. Therefore, using (5) and (5.7), we deduce that
$\left\|w(t,\cdot)\right\|_{\ell^{2}(\hbar\mathbb{Z}^{n})}\to 0$ as $\hbar\to
0.$ Hence we have

$\left\|v(t,\cdot)-u(t,\cdot)\right\|_{\ell^{2}(\hbar\mathbb{Z}^{n})}\to
0\text{ as }\hbar\to 0,\text{ for all }t\in[0,T].$

This finishes the proof of Theorem 2.9. ∎

###### Proof of Theorem 2.11.

Without making any significant changes to the proof of Theorem 2.9, we can
prove Theorem 2.11. ∎

##  6\. Remarks

In this section we make a few remarks related to the very weak solution for
the Cauchy problem (1.4) in the Euclidean setting:

  1. (1)

For a net $u_{\varepsilon}=u_{\varepsilon}(t,x)$, where $x\in\mathbb{R}^{n}$
is a variable in the Euclidean space, the definitions of moderateness and
negligibility are adapted accordingly from Definition (2.2); i.e., the net
$(u_{\varepsilon})_{\varepsilon}\in{L^{2}([0,T];H^{m}(\mathbb{R}^{n}))}$ is
$L^{2}([0,T];H^{m}(\mathbb{R}^{n}))$-moderate if there exist
$N\in\mathbb{N}_{0}$ and $c>0$ such that

$\|u_{\varepsilon}\|_{L^{2}([0,T];H^{m}(\mathbb{R}^{n}))}\leq
c\varepsilon^{-N},$

for all $\varepsilon\in(0,1]$ and is
$L^{2}([0,T];H^{m}(\mathbb{R}^{n}))$-negligible if for all
$q\in\mathbb{N}_{0}$ there exists $c>0$ such that

$\|u_{\varepsilon}\|_{L^{2}([0,T];H^{m}(\mathbb{R}^{n}))}\leq
c\varepsilon^{q},$

for all $\varepsilon\in(0,1]$.

  2. (2)

The notion of a very weak solution for the Cauchy problem (1.4) can be adapted
from Definition 2.3 by simply replacing the
$L^{2}([0,T];\ell^{2}_{s}(\hbar\mathbb{Z}^{n}))$-moderate regularisation by
the $L^{2}([0,T];H^{m}(\mathbb{R}^{n}))$-moderate regularisation for the
source term $f$ and the net $(u_{\varepsilon})_{\varepsilon}.$

  3. (3)

The proof of Theorem 1.2 will be similar to the proof of Theorem 2.4 except
for using the inner product of $L^{2}_{m}(\mathbb{R}^{n})$ instead of
$H^{s}(\mathbb{T}_{\hbar}^{n})$ in (4.11).

  4. (4)

The notion of the uniqueness of very weak solution for the Cauchy problem
(1.4) can also be formulated by making similar modifications to Definition
2.5.

  5. (5)

If the coefficients $a,b$ are regular, the very weak solution obtained in
Theorem 1.2 recaptures the classical solution obtained in Theorem 1.1 in the
limit $L^{2}([0,T];H^{m}(\mathbb{R}^{n}))$ as $\varepsilon\to 0$. More
precisely, we have the consistency result similar to Theorem 2.8 with the same
modifications as above.

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