# Parabolic problem with fractional time derivative with nonlocal and
nonsingular Mittag-Leffler kernel

J.D Djida  ,  A. Atangana  and  I. Area  African Institute for Mathematical
Sciences (AIMS), P.O. Box 608, Limbe Crystal Gardens, South West Region,
Cameroon<EMAIL_ADDRESS>Institute for Groundwater
Studies, Faculty of Natural and Agricultural Sciences, University of the Free
State, 9301, Bloemfontein, South Africa<EMAIL_ADDRESS>Departamento
de Matemática Aplicada II, E.E. Aeronáutica e do Espazo, Universidade de Vigo,
Campus As Lagoas s/n, 32004 Ourense, Spain<EMAIL_ADDRESS>
###### Abstract.

We prove Hölder regularity results for a class of nonlinear parabolic problem
with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular
kernel. Existence of weak solutions via approximating solutions is proved.
Moreover, the Hölder continuity of viscosity solutions is obtained. We get the
similar results as those obtained by Allen (see
https://arxiv.org/abs/1610.10073).

##  1\. Introduction

In this work we prove some Hölder regularity results for viscosity solutions
of integro-differential equations in which the kernels defining the fractional
time operators belongs to a more general class with nonlocal and Mittag-
Leffler nonsingular kernel and the spatial nonlocal operator kernel
corresponds to the fractional Laplacian. One of our aims is to bring into
better alignment this analysis of Hölder with fractional time derivative
involving the Mittag-Leffler function with the results which have been known
[1, 2] where the fractional time derivative in the sense of Caputo were used .

We first mention that we have tried to collect the notation contained herein–
as much as possible– in Section 3.1. We also include a discussion of
background in Section 2.

The specific equation which is of interest is

${\mathcal{L}}u(t,x)-\mathcal{J}u(t,x)=g(t,x),\quad\text{in}\leavevmode\nobreak\
\leavevmode\nobreak\ (-\infty,b)\times\mathbb{R}^{n},$ (1.1)

where the nonlocal spatial operator $\mathcal{J}$ [3, 4] we make is

${\mathcal{J}}u(x)=\int_{\mathbb{R}^{n}}\delta_{h}u(x){\mathcal{K}}(x,h)dh.$
(1.2)

$\delta_{h}u(x)=u(x+h)+u(x-h)-2u(x)$ denote the second order difference
relation. We choose for $\sigma(0,2)$ the kernel

${\mathcal{K}}(x,h)=C(n,\sigma)\left|h\right|^{-n-\sigma},$

which for an appropriate constant $C(n,\sigma)$ gives
$\mathcal{J}=(-\Delta)^{\sigma/2}$.

Instead of considering the class of kernels ${\mathcal{K}}$ as it has been
shown in [5], we consider the the simplest one which allowed to have large
regions where ${\mathcal{K}}(x,h)$ is comparable to
$\left|h\right|^{-n-\alpha}$ from below. This class of kernel that is called
${\mathcal{A}}_{sec}$ (see (3.1)), are treat without assuming any regularity
in the $x$ variable through the assumption that they need only to be above
$\left|h\right|^{-n-\sigma}$ on a possibly set as

$\displaystyle\frac{\lambda}{|h|^{n+\sigma}}\leq{\mathcal{K}}(x,h)\leq\frac{\Lambda}{|h|^{n+\sigma}}\quad
h\in\mathbb{R}^{n}\setminus\\{0\\}.$ (1.3)

Furthermore, for $\alpha\in(0,1)$ for all $t<a$ and for $n\in\mathbb{N}$, we
defined the fractional time-operator ${\mathcal{L}}$ involved in (1.1) as

${\mathcal{L}}u(t)=\mathcal{C}(n,\alpha)\int_{-\infty}^{t}\big{[}u(t)-u(s)\big{]}{\mathcal{T}}(t,s)ds,$
(1.4)

where the kernel ${\mathcal{T}}(t,s)$ belongs to the class of kernels
${\mathcal{T}}_{sec}$ described by

$\displaystyle{\mathcal{T}}_{sec}=\bigg{\\{}{\mathcal{T}}:(-\infty,b)\to\mathbb{R}:{\mathcal{T}}(t,t-s)={\mathcal{T}}(t+s,t),\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}$ (1.5)
$\displaystyle-c\lambda\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha+1)}\leq{\mathcal{T}}(t,s)\leq-c\Lambda\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha+1)}\bigg{\\}}.$

######  Remark 1.1.

This class of kernel contains a number of specific kernels such as the ones
involved in Marchaud derivative [6], Caputo derivative [1, 6, 7, 8] and the
fractional derivative with nonlocal and nonsingular Mittag-Leffler kernel [9]
if we consider the fractional time derivative in the sense of Caputo, defined
as

${}^{C}{\mathcal{D}}_{a}^{\alpha}u(t)=\frac{\alpha}{\Gamma(1-\alpha)}\int_{-\infty}^{t}\big{[}u(t)-u(s)\big{]}(t-s)^{-\alpha-1}ds,$

the kernel $(t-s)^{-\alpha-1}$ belongs the class of kernels
${\mathcal{T}}_{sec}$. We recall that this Caputo derivative has been used in
[1, 10] to prove Hölder regularity of some parabolic problems in the non-
divergence form.

Also, for the fractional time-derivative with nonlocal and nonsingular Mittag-
Leffler kernel [9]

${}^{ABC}{\mathcal{D}}_{a}^{\alpha}u(t):=c_{\alpha}\int_{-\infty}^{t}\big{[}u(t)-u(s)\big{]}(t-s)^{\alpha-1}E_{\alpha,\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}\
ds.$ (1.6)

The kernel
$(t-s)^{\alpha-1}E_{\alpha,\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}$
belongs the class of kernels ${\mathcal{T}}_{sec}$ as well.

Further properties of the fractional time derivative will be given in Section
3.

Besides the mathematical satisfactions of the Caputo fractional derivative,
the attentiveness for the derivative with nonlocal and nonsingular Mittag-
Leffler kernel is based on its properties of portraying the behavior of
orthodox viscoelastic materials, thermal medium, material heterogeneities and
some structure or media with different scales [9]. The nonlocality of the new
kernel allows a good description of the memory within structure and media with
different scale, which cannot be described by classical fractional derivative.
This derivative also takes into account power and exponential decay laws
which, many natural occurrence follow. The one with the Mittag-Leffler
functions allows us to describe phenomena in processes that progress or decay
too slowly to be represented by classical functions like the exponential
function and its successors. The Mittag-Leffler function arises naturally in
the solution of fractional integral equations, and especially in the study of
the fractional generalization of the kinetic equation, random walks, Lévy
flights, and so called super-diffusive transport.

######  Remark 1.2.

There are various definitions—Riemann, Liouville, Caputo, Grunwald-Letnikov,
Marchaud, Weyl, Riesz, Feller, and others— for fractional derivatives and
integrals, (see e.g. [6, 8, 11, 12, 13] and references therein). This
diversity of definitions is due to the fact that fractional operators take
different kernel representations. So instead of studying the parabolic problem
for different kernels, we propose in this note to study the problem for a
large class of kernels ${\mathcal{T}}_{sec}$ satisfying the properties given
by (1.5).

Of interest is the regularity of solutions to these nonlocal equations
involving the fractional time derivative and the nonlocal spacial operator,
some authors as [8, 10, 14, 15] studied the problem of Hölder continuity for
solutions to master equations and Hölder continuity for parabolic equations
with Riemman derivative and the Caputo fractional time derivative and
divergence form nonlocal operator. Furthermore very recently the author in [1]
proves the Hölder continuity of viscosity solutions to (1.1) in the non
divergence form, but using the generalized fractional time derivative of
Marchaud or Caputo type under the appropriate assumptions. They obtained an
estimate which remains uniform as the order of the fractional derivative
$\alpha\to 1$.

With the clear intention to obtain a similar result as in [1, 2] where the
generalized Marchaud or Caputo derivative was used to show the Hölder
continuity of viscosity solutions, we propose to study the same problem, but
now with another kernel (e.g.
${\mathcal{T}}=(t-s)^{\alpha-1}E_{\alpha,\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}$
) where the Caputo kernel can be deduce from and which belongs to the class of
kernels ${\mathcal{T}}_{sec}$.

######  Remark 1.3.

Despite the fact that we obtain similar results as those already presented in
the literature, we consider that this approach with more general kernel is a
useful contribution to the field since up to now (to the best of our
knowledge) most of the kernels of fractional time derivative in the literature
belong to ${\mathcal{T}}_{sec}$. For example if we take the case of Caputo
fractional derivative, it has been proven in [9] that there exists a well
defined function $v$ that equalizes the Caputo fractional time-derivative and
the fractional time derivative with nonlocal and nonsingular Mittag-Leffler
kernel with fractional order $\alpha$

${}^{C}{\mathcal{D}}_{a}^{\alpha}v(t)={}^{ABC}{\mathcal{D}}_{a}^{\alpha}v(t).$

The solution is obtained by applying the Fourier transform in both side. Hence
the result as

$v(t)=v(0)\big{(}1-k_{\alpha}\big{)}t^{\alpha-1}E_{\alpha,\alpha}\big{[}(c+k_{\alpha})t^{\alpha}\big{]}+\frac{cv(0)}{\Gamma(\alpha)}\int_{0}^{t}s^{\alpha-1}\big{(}t-s\big{)}^{\alpha-1}E_{\alpha,\alpha}\big{[}(c+k_{\alpha})s^{\alpha}\big{]}ds,$

where $k_{\alpha}=\frac{B(\alpha)}{1-\alpha}$.

To study regularity properties of solutions to equation (1.1), one could in
some sense study the solution $u$ which simultaneously solve the two
inequalities

$\inf_{{\mathcal{T}}\in{\mathcal{T}}_{sec}}\bigg{\\{}{\mathcal{L}}_{{\mathcal{T}}}u(t,x)-{\mathcal{J}}u(t,x)\bigg{\\}}\leq
C\ \text{and}\
\sup_{{\mathcal{T}}\in{\mathcal{T}}_{sec}}\bigg{\\{}{\mathcal{L}}_{{\mathcal{T}}}u(t,x)-{\mathcal{J}}u(t,x)\bigg{\\}}\geq-C\
\textnormal{in}\ (\infty,T)\times\Omega.$

The kernel
${\mathcal{T}}=(t-s)^{\alpha-1}E_{\alpha,\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}$
is chosen in the class of kernels ${\mathcal{T}}_{sec}$ which at least
contains all the kernels involved in the fractional time derivatives in the
literature so that it will be convenient if one wishes to attain further
properties of the extremal operators.

The program of studying existence of solutions and regularity properties of
parabolic problem with fractional nonlocal space-time operators such as (1.1)
was presented in [1, 8, 10], using respectively Riemann fractional derivative
and Caputo fractional derivative. We extend those results to cover the larger
class, ${\mathcal{T}}_{sec}$. Our main results are existence of weak solution
and Hölder regularity estimate.

######  Theorem 1.1 (Existence of weak solutions).

Let $\vartheta$ be a bounded an Lipschitz function on
$(-\infty,b)\times\mathbb{R}^{n}$ and assume that the function $g$ is regular
enough. So for a given smooth bounded initial data $u_{0}>0$, there exists a
weak solution $u$ to the weak formulation

$\displaystyle
c_{\alpha}\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}\int_{-\infty}^{t}\big{[}u(t,x)-u(s,x)\big{]}\big{[}\vartheta(t,x)-\vartheta(s,x)\big{]}{\mathcal{T}}(t,s,x)ds\leavevmode\nobreak\
dt\leavevmode\nobreak\ dx$
$\displaystyle+\int_{a}^{b}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}{\mathcal{K}}(t,x,\xi)\big{[}u(t,x)-u(t,\xi)\big{]}\big{[}\vartheta(t,x)-\vartheta(t,\xi)\big{]}dx\leavevmode\nobreak\
d\xi\leavevmode\nobreak\ dt$
$\displaystyle+c_{\alpha}\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}\int_{-\infty}^{2t-b}u(t,x)\vartheta(t,x){\mathcal{T}}(t,s,x)ds\leavevmode\nobreak\
dt\leavevmode\nobreak\
dx-\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}u(t,x)\leavevmode\nobreak\
{\mathcal{L}}\vartheta(t,x)dt\leavevmode\nobreak\ dx$
$\displaystyle=\int_{-\infty}^{b}\int_{\mathbb{R}^{n}}f(t,x)\vartheta(t,x)dx\leavevmode\nobreak\
dt.$

Next we state the Hölder regularity estimate result.

######  Theorem 1.2 (Hölder Regularity).

Let $\alpha\in(0,1)$, $\sigma\in(\sigma_{0},2)$ and let
${\mathcal{M}}^{\pm}_{{\mathcal{A}}}$ be as defined in (5.2)) and (5.3)).
Assume also that $g\in L^{\infty}(-\infty,b)\times\mathbb{R}^{n}$. There are
positive constants $\beta\in(0,1)$ and $C\geq 1$ depending only on
$n,\lambda,\Lambda,\alpha,\sigma$ such that if $u$ is bounded continuous
viscosity solution in $B_{2}\times[-2,0]$ satisfying

${\mathcal{L}}_{{\mathcal{T}}}u-{\mathcal{M}}^{-}_{{\mathcal{A}}}u\leq\varepsilon_{0}\
\ \text{and}\
{\mathcal{L}}_{{\mathcal{T}}}u-{\mathcal{M}}^{+}_{{\mathcal{A}}}u\geq-\varepsilon_{0},$
(1.7)

then $u$ is Hölder continuous in $B_{1}\times[-1,0]$ and for $(x,t),(y,s)\in
B_{1}\times[-1,0]$ the following estimates holds

$\displaystyle|u(x,t)-u(y,s)|\leq
C(\|u\|_{L^{\infty}}+\varepsilon_{0}^{-1}\|g\|_{L^{\infty}})|x-y|^{\kappa}+|t-s|^{\kappa\alpha/(2\sigma)}.$
(1.8)

Furthermore $C$ remains bounded as $\alpha\to 1$ and $\sigma\to 2$.

######  Remark 1.4.

Notice that in order to get solutions to (1.2), the solution of fractional
differential equations involving the fractional time derivative with nonlocal
and nonsingular Mittag-Leffler kernel is proposed in order to use it as a test
function for of viscosity solution. The case with the Caputo derivative was
well presented by the author in [1] showing that if $|\\{x\in
B_{1}\times(-2,-1):u(x,t)\leq 0\\}|\geq\mu_{1}$, then $u(t)\geq\mu_{2}$ if
$t\in(-1,0)$.

So we omit to prove it in this note in the case for the fractional time
derivative with nonlocal and non singular Mittag-Leffler kernel since the idea
of the proof is similar.

So with the Hölder continuity estimates for ordinary differential equations
involving the Caputo derivative [1], our class of weak solutions will be
considered in the viscosity sense as described in Section 5. This will then
allow us to get the similar result.

The organization of the article is as follows. In Section 2 we review some
background related to 1.2. In Section 3 we collect notation, definitions, and
preliminary results regarding (1.1) and 1.2. Section 4 is dedicated to the
sketch of proof of the existence of weak solutions via approximating
solutions, mainly 1.1. Finally in Section 5 we discuss the pointwise estimates
and put together the remaining pieces of the proof for the and Hölder
Regularity.

##  2\. Background

There are few collection of results related to 1.2 with both space and time
fractional nonlocal operators. We will focus on the type of results which only
depend on the ellipticity constants, $\lambda$ and $\Lambda$, for the spacial
nonlocal operator as well as possibly the order, $\alpha$, $\sigma$ and we
shall try to see if we recover the results of Mark Allen et al. [1, 2, 10] by
using fractional time derivative with nonlocal and nonsingular Mittag-Leffler
kernel.

The problem of regularity of parabolic problem with fractional time derivative
in time are new in the literature. As we can notice in the recent article of
[16], where the authors used the original method of De Giorgi to prove
boundedness of solutions and local Hölder regularity. Similar approach to
prove apriori local Hölder estimates of solutions to the fractional parabolic
type equation was also been used in [2], where they have followed the De
Giorgi method as in [16] but now by taking into account the fractional time
derivative in the sense of Caputo [6]. As an earlier result, the used of the
fractional nature of the derivative made the estimates to not remain uniform
as the fractional order $\alpha\to 1$.

In the same direction, we shall also bring the attention of the reader on the
fact that similar result has been studied by Zacher in [8] but instead of
using Caputo derivative the author used the Riemann-Liouville fractional time
derivative and zero right hand side.

One should notice that these results based on the fractional space and time
where obtained both kernels of these kernel are bounded. In the case of the
fractional spatial nonlocal operator,there are a few interesting distinctions
that are usually made: whether or not ${\mathcal{K}}(x,h)$ is assumed to be
even in $h$; whether or not the corresponding equations are linear; whether or
not a Harnack inequality holds [5].

Regularity results (such as 1.2) as well as the Harnack inequality for linear
equations with operators similar to (1.2) were obtained [5, 17]. Furthermore
in [1] Hölder continuity of viscosity solutions to certain nonlocal parabolic
equations that involve a generalized fractional time derivative of Marchaud or
Caputo type as well is obtained under the assumption that kernel of the
fractional time operator satisfied the symmetry condition
${\mathcal{T}}(t,t-s)={\mathcal{T}}(t+s,t)$. The estimates are uniform as the
order of the operator $\alpha$ approaches $1$, so that the results recover
many of the regularity results for local parabolic problem.

Finally, higher regularity in time type estimates were obtained in [2].

An important class of kernels are those for which the symmetry
${\mathcal{T}}(t,t-s)={\mathcal{T}}(t+s,t)$ is assumed to hold and for which
all the fractional time operators in the literature belongs to the class
$T_{sec}$. This class could be also extended to the non-symmetric case. We
will discuss in this note the case where the kernel of the nonlocal time
operator belongs to a more general class of kernels $T_{sec}$ which contains
must of the kernels known in the literature.

##  3\. Preliminaries

###  3.1. Notation

We first collect some notations which will be used throughout this article.

  * •

${\mathcal{L}}$ \- the nonlocal fractional time-derivative with nonsingular
Mittag-Leffler kernel.

  * •

$\sigma\in(\sigma_{0},2)$ \- denote the order of the nonlocal spatial
operator.

  * •

$\alpha$ \- will always denote the order of the space-time arbitrary order
derivative.

${\mathcal{A}}_{sec}=\bigg{\\{}K:\mathbb{R}^{n}\to\mathbb{R}\ :\
{\mathcal{J}}(-h)={\mathcal{J}}(h),\quad\text{and}\quad\frac{\lambda}{|h|^{n+\sigma}}\leq{\mathcal{J}}(h)\leq\frac{\Lambda}{|h|^{n+\sigma}}\bigg{\\}}$
(3.1) $\displaystyle\delta_{h}u(x)=u(x+h)+u(x-h)-2u(x)$ (3.2)
$\displaystyle{\mathcal{J}}_{{\mathcal{A}}}u(x)$
$\displaystyle=\int_{{\mathbb{R}}^{n}}\delta_{h}u(x){\mathcal{J}}(h)dh$
$\displaystyle\mu(dh)$ $\displaystyle=\left|h\right|^{-n-2\sigma}dh$
$\displaystyle Q_{\varepsilon}(x_{0})$
$\displaystyle=\left\\{x\in\mathbb{R}^{n}\,:\,\left|x-x_{0}\right|_{\infty}<\tfrac{\varepsilon}{2}\right\\}$
$\displaystyle B_{\varepsilon}(x_{0})$
$\displaystyle=\left\\{x\in\mathbb{R}^{n}\,:\,\left|x-x_{0}\right|<\varepsilon\right\\}$

We use $\left|\cdot\right|$ for the absolute value, the Euclidean norm, and
the $n$-dimensional Lebesgue measure at the same time. Throughout this article
$\Omega\subset\mathbb{R}^{n}$ denotes a bounded domain. For cubes and balls
such that $x_{0}=0$ we write $Q_{l}$ instead of $Q_{l}(0)$ and similarly for
$B_{l}$. The following hold:

$B_{1/2}\subset Q_{1}\subset Q_{3}\subset B_{3}\subset B_{2}\,.$

###  3.2. Definitions

We use the definitions and basic properties the of the fractional time
derivative with nonlocal and nonsingular Mittag-Leffler kernel and of
viscosity solutions from [9] and [1], and for Hölder continuity [2, 10, 17,
18].

###  3.3. The fractional time derivative with nonlocal and nonsingular Mittag-
Leffler kernel

In this section for the convenience of the reader, we recall some definitions
of fractional time derivative with the nonlocal and nonsingular kernel as
stated in [9] and state some of its new properties.

The fractional time derivative with nonlocal and Mittag-Leffler nonsingular
kernel, recently introduced by Atangana and Baleanu and known as the Atangana-
Baleanu derivative is useful in modelling equations arising in porous media.
One formulation of the Atangana-Baleanu derivative is

${}^{ABC}{\mathcal{D}}_{a}^{\alpha}u(t)=\frac{B(\alpha)}{1-\alpha}\int_{a}^{t}E_{\alpha}\big{[}-c\big{(}t-s\big{)}^{\alpha}\big{]}u^{\prime}(s)ds.$
(3.3)

The associate integral of the fractional time derivative with nonlocal and
Mittag-Leffler nonsingular kernel, is defined as

$\prescript{\text{AB}}{a}{I}_{t}^{\alpha}u(t)=\frac{1-\alpha}{B(\alpha)}u(t)+\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{a}^{t}u(y)(t-y)^{\alpha-1}dy.$
(3.4)

In the above formulas $B(\alpha)$ is a constant depending on $\alpha$ such
that

$B(\alpha)=1-\alpha+\frac{\alpha}{\Gamma(\alpha)},\quad
c=-\frac{\alpha}{1-\alpha},$

and $E_{\alpha,\beta}(z)$ is the two-parametric Mittag-Leffler function
defined in terms of a series as the following entire function as

$E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty}\frac{z^{\alpha k}}{\Gamma(\alpha
k+\beta)},\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\quad\beta>0,\leavevmode\nobreak\ \quad z\in\mathbb{C}.$

The others representation of the Atangana-Baleanu fractional time derivative
in the sense of Caputo holds pointwise

${}^{ABC}{\mathcal{D}}_{a}^{\alpha}u(t)=\nu_{\alpha}E_{\alpha}\big{[}-c\big{(}t-a\big{)}^{\alpha}\big{]}\big{[}u(t)-u(a)]\\\
+c_{\alpha}\int_{a}^{t}(t-s)^{\alpha-1}E_{\alpha,\alpha}\big{[}-c\big{(}t-s\big{)}^{\alpha}\big{]}\big{[}u(t)-u(s)\big{]}ds,$
(3.5)

where $\nu_{\alpha}=B(\alpha)(1-\alpha)^{-1}$ and $c_{\alpha}=-c\nu_{\alpha}$.
We set

${\mathcal{T}}(t,s)=(t-s)^{\alpha-1}E_{\alpha,\alpha}\big{[}-c\big{(}t-s\big{)}^{\alpha}\big{]},$
(3.6)

as our bounded kernel in time, satisfying the relation
${\mathcal{T}}(t,t-s)={\mathcal{T}}(t+s,t)$ and

$-c\lambda\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha+1)}\leq{\mathcal{T}}(t,s)\leq-c\Lambda\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha+1)}.$
(3.7)

In this setting, following the idea of [2, 10] we define $u(t)=u(a)$ for
$t<a$,

${\mathcal{L}}u(t):=c_{\alpha}\int_{-\infty}^{t}\big{[}u(t)-u(s)\big{]}{\mathcal{K}}(t,s)\
ds.$ (3.8)

One of the immediate consequence of the formulation (3.8) is that, it allows
to drop out data and it is also useful for viscosity solutions [10].

The notion of viscosity solution and supersolutions for the initial values
problem involving the time derivative with the non singular Mittag-Leffler
kernel in (3.8) is similar to the one of (1.1). For this purpose we state the
following Proposition which shows that (3.8) is well defined.

######  Proposition 3.1.

Let $u$ a continuous bounded function and $w\in\mathcal{C}^{0,\beta}$ with
$\alpha<\beta\leq 1$. If $w\geq(\leq)u$ on $[t_{0}-\varepsilon,t_{0}]$ and
$w(t_{0})=u(t_{0})$, then the integral

$c_{\alpha}\int_{-\infty}^{t_{0}}\big{[}u(t_{0})-u(s)\big{]}{\mathcal{T}}(t,s)\
ds$

is well defined, so that ${\mathcal{L}}u(t_{0})$ is well defined.

###### Proof.

Let assume that $w\geq u$ on $[t_{0}-\varepsilon,t_{0}]$. So it comes from
(3.8)

$\displaystyle
c_{\alpha}\int_{-\infty}^{t_{0}}\big{[}u(t_{0})-u(s)\big{]}{\mathcal{T}}(t,s)\
ds=$ $\displaystyle
c_{\alpha}\int_{-\infty}^{t_{0}-\varepsilon}\big{[}u(t_{0})-u(s)\big{]}{\mathcal{T}}(t,s)\
ds+c_{\alpha}\int_{t_{0}-\varepsilon}^{t_{0}}\big{[}u(t_{0})-u(s)\big{]}{\mathcal{T}}(t,s)\
ds$ $\displaystyle\geq
c_{\alpha}\int_{-\infty}^{t_{0}-\varepsilon}\big{[}u(t_{0})-u(s)\big{]}{\mathcal{T}}(t,s)\
ds+c_{\alpha}\int_{t_{0}-\varepsilon}^{t_{0}}\big{[}w(t_{0})-w(s)\big{]}{\mathcal{T}}(t,s)\
ds$ $\displaystyle\geq
c_{\alpha}\int_{-\infty}^{t_{0}-\varepsilon}\big{[}u(t_{0})-u(s)\big{]}{\mathcal{T}}(t,s)\
ds-
cc_{\alpha}\Lambda\big{\|}w\big{\|}_{\mathcal{C}^{0,\beta}}\int_{t_{0}-\varepsilon}^{t_{0}}\frac{\big{(}t_{0}-s\big{)}^{\alpha+\beta-1}}{\Gamma(1+\alpha)}\bigg{)}ds$
$\displaystyle\geq
c_{\alpha}\int_{-\infty}^{t_{0}-\varepsilon}\big{[}u(t_{0})-u(s)\big{]}{\mathcal{T}}(t,s)ds+cc_{\alpha}\Lambda\frac{\varepsilon^{(\alpha+\beta)}}{\big{(}\alpha+\beta\big{)}\Gamma(1+\alpha)}\big{\|}w\big{\|}_{\mathcal{C}^{0,\beta}}$
$\displaystyle\geq
c_{\alpha}\frac{c\Lambda\varepsilon^{\alpha+\beta}}{\big{(}\alpha+\beta\big{)}\Gamma(1+\alpha)}\bigg{(}2\big{\|}u\big{\|}_{L^{\infty}}+\big{\|}w\big{\|}_{\mathcal{C}^{0,\beta}}\bigg{)}$
$\displaystyle\geq\tilde{A}_{\alpha,\beta}.$

Hence the integral is well defined. ∎

Next we state the following estimate of bound of the fractional time
derivative ${\mathcal{L}}\varrho(t)$ that will be useful for the proof of
Hölder continuity in Section 5.

######  Proposition 3.2.

Let us consider $\varrho(t)=\max\\{2|rt|^{\nu}-1,0\\}$ under the condition
that $\nu<\alpha$, and $r=\min\\{4^{-1},4^{-\alpha/2\sigma}\\}$.

If $t_{1}\leq 0$ then

$-d_{\alpha,\nu}\leq{\mathcal{L}}\varrho(t_{1})\leq 0$

where the constant $d_{\alpha,\nu}$ depends on $\alpha$ and $\nu$.

###### Proof.

From (3.3) and (3.8) the rescaled fractional time derivative take the form

$-\nu\int_{a}^{t}E_{\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}|rs|^{\nu-1}ds\geq-\nu\int_{-\infty}^{t}E_{\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}|rs|^{\nu-1}ds.$

One notice that $|rs|^{\nu-1}$ and the Mittag-Leffler function
$E_{\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}$ are increasing function
of $s$, if $s<0$, so

$\nu\int_{-\infty}^{t}E_{\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}|rs|^{\nu-1}ds$

is an increasing function of $t$.

Furthermore, if $t\leq-1$, then if follows that

${\mathcal{L}}\varrho\geq-\nu\int_{-\infty}^{t}E_{\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}|rs|^{\nu-1}ds\geq-\nu\int_{-\infty}^{-1}E_{\alpha}\big{[}c\big{(}-1-s\big{)}^{\alpha}\big{]}|rs|^{\nu-1}ds\geq-
d_{\alpha,\nu}.$

Now if $t>-1$, then

${\mathcal{L}}\varrho\geq-\nu\int_{-\infty}^{-1}E_{\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}|rs|^{\nu-1}ds\geq-\nu\int_{-\infty}^{-1}E_{\alpha}\big{[}c\big{(}-1-s\big{)}^{\alpha}\big{]}|rs|^{\nu-1}ds\geq-
d_{\alpha,\nu}.$

∎

Next, in order to get informations over the various time, we are interested on
the solution to the fractional differential equation involving the Atangana-
Baleanu fractional derivative, in the form

${\mathcal{L}}u(t)=-c_{1}\leavevmode\nobreak\ u(t)+c_{0}\leavevmode\nobreak\
h(t),\quad c_{1},\leavevmode\nobreak\ c_{0}<\infty.$ (3.9)

######  Proposition 3.3.

Let $u\in H^{1}(0,b)$, $b>0$, and $h\in\mathcal{C}^{2}$, such that the
Atangana-Baleanu fractional derivative exists. Then, the solution of
differential equation (3.9), for $c_{1}=0$, is given by

$u(t)=\frac{(1-\alpha)c_{0}}{B(\alpha)}h(t)+\frac{\alpha
c_{0}}{B(\alpha)\Gamma(\alpha)}\int_{0}^{t}h(s)(t-s)^{\alpha-1}ds,$ (3.10)

and for $c_{1}\neq 0$ by

$\displaystyle u(t)$ $\displaystyle=\zeta E_{\alpha}\big{[}-\gamma
t^{\alpha}\big{]}u(0)$ (3.11) $\displaystyle+\frac{\alpha
c_{0}\zeta}{B(\alpha)}\int_{0}^{t}\bigg{(}E_{\alpha,\alpha}\big{[}-\gamma(t-s)^{\alpha}\big{]}+\frac{(1-\alpha)}{\alpha}\gamma^{-2\alpha}E_{\alpha,\alpha}\big{[}-\gamma^{-2\alpha}(t-s)^{\alpha}\big{]}\bigg{)}(t-s)^{\alpha-1}h(s)ds,$

with $\gamma=\frac{\alpha c_{1}}{\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}}$ and
$\zeta=\frac{B(\alpha)}{\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}}$.

###### Proof.

For $c_{1}=0$, it is obvious that the result is

$u(t)=\frac{1-\alpha}{B(\alpha)}h(t)+\frac{\alpha c_{0}\leavevmode\nobreak\
}{B(\alpha)\Gamma(\alpha)}\int_{0}^{t}h(s)(t-s)^{\alpha-1}ds.$

We purchase the proof of 3.3 for $c_{1}\neq 0$ by using (3.4) and by applying
the Laplace transform on the equation (3.9). It comes that

$\displaystyle\prescript{\text{AB}}{0}{I}_{t}^{\alpha}\left\\{{}^{ABC}{\mathcal{D}}_{a}^{\alpha}u(t)\right\\}$
$\displaystyle=\prescript{\text{AB}}{0}{I}_{t}^{\alpha}\big{\\{}-c_{1}\leavevmode\nobreak\
u(t)+c_{0}\leavevmode\nobreak\ h(t)\big{\\}}$ (3.12) $\displaystyle u(t)-u(0)$
$\displaystyle=-c_{1}\bigg{\\{}\frac{1-\alpha}{B(\alpha)}u(t)+\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{0}^{t}u(s)(t-s)^{\alpha-1}ds\bigg{\\}}$
$\displaystyle+c_{0}\bigg{\\{}\frac{1-\alpha}{B(\alpha)}h(t)+\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{0}^{t}h(s)(t-s)^{\alpha-1}ds\bigg{\\}}$

If we apply the Laplace transform to both sides of (3.12), it yields

$\bigg{[}\frac{B(\alpha)+(1-\alpha)c_{1}}{B(\alpha)}+\frac{\alpha
c_{1}}{B(\alpha)}\frac{1}{p^{\alpha}}\bigg{]}u(p)=\frac{1}{p}u(0)+\frac{c_{0}(1-\alpha)}{B(\alpha)}h(p)+\frac{\alpha
c_{0}}{B(\alpha)\Gamma(\alpha)}\Gamma(\alpha)p^{-\alpha}h(p)\\\
\frac{1}{B(\alpha)}\bigg{[}\frac{p^{\alpha}\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}+\alpha
c_{1}}{p^{\alpha}}\bigg{]}u(p)=\frac{1}{p}u(0)+\frac{(1-\alpha)c_{0}}{B(\alpha)}h(p)+\frac{\alpha
c_{0}}{B(\alpha)}p^{-\alpha}h(p)\\\
u(p)=\frac{B(\alpha)}{p}\bigg{[}\frac{p^{\alpha}}{p^{\alpha}\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}+\alpha
c_{1}}\bigg{]}u(0)+\bigg{[}\frac{(1-\alpha)c_{0}p^{\alpha}}{p^{\alpha}\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}+\alpha
c_{1}}\bigg{]}h(p)\\\
+B(\alpha)\bigg{[}\frac{p^{\alpha}}{p^{\alpha}\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}+\alpha
c_{1}}\bigg{]}\frac{\alpha c_{0}}{B(\alpha)}p^{-\alpha}h(p)\\\
=\frac{B(\alpha)}{\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}}\bigg{[}\frac{p^{\alpha-1}}{p^{\alpha}+\frac{\alpha
c_{1}}{\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}}}\bigg{]}u(0)\\\
+\frac{(1-\alpha)c_{0}}{\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}}\bigg{[}\frac{p^{\alpha}}{p^{\alpha}+\frac{\alpha
c_{1}}{\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}}}\bigg{]}h(p)\\\
+\frac{\alpha}{\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}}\bigg{[}\frac{c_{0}}{p^{\alpha}+\frac{\alpha
c_{1}}{\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}}}\bigg{]}h(p)$ (3.13)

if we set

$\gamma=\frac{\alpha
c_{1}}{\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\zeta=\frac{B(\alpha)}{\big{(}B(\alpha)+(1-\alpha)c_{1}\big{)}}.$

it comes that

$u(p)=\zeta\leavevmode\nobreak\
\frac{p^{\alpha-1}}{p^{\alpha}+\gamma}\leavevmode\nobreak\
u(0)+\frac{c_{0}\zeta(1-\alpha)}{B(\alpha)}\leavevmode\nobreak\
\frac{p^{\alpha}}{p^{\alpha}+\gamma}\leavevmode\nobreak\ h(p)+\frac{\alpha
c_{0}\zeta}{B(\alpha)}\leavevmode\nobreak\
\frac{1}{p^{\alpha}+\gamma}\leavevmode\nobreak\ h(p)\\\
=\zeta\leavevmode\nobreak\
\frac{p^{\alpha-1}}{p^{\alpha}+\gamma}\leavevmode\nobreak\ u(0)+\frac{\zeta
c_{0}(1-\alpha)}{B(\alpha)}\leavevmode\nobreak\
\frac{1}{1+\big{(}\gamma^{-\alpha}p\big{)}^{-\alpha}}\leavevmode\nobreak\
h(p)+\frac{\alpha c_{0}\zeta}{B(\alpha)}\leavevmode\nobreak\
\frac{1}{p^{\alpha}+\gamma}\leavevmode\nobreak\ h(p)$ (3.14)

with $d=\gamma^{-\alpha}$. We notice that

$\displaystyle\frac{1}{1+\big{(}\gamma^{-\alpha}p\big{)}^{-\alpha}}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{d^{\alpha k-2}t^{\alpha
k-1}}{\Gamma(\alpha
k)}=\frac{d}{dt}E_{\alpha}\big{[}\gamma^{-2\alpha}t^{\alpha}\big{]}=d^{-2\alpha}t^{\alpha-1}E_{\alpha,\alpha}\big{[}\gamma^{-2\alpha}t^{\alpha-1}\big{]}$

Hence by applying the inverse Laplace transform, we the result as

$\displaystyle u(t)$ $\displaystyle=\zeta E_{\alpha}\big{[}-\gamma
t^{\alpha}\big{]}u(0)$ $\displaystyle+\frac{\alpha
c_{0}\zeta}{B(\alpha)}\int_{0}^{t}\bigg{(}E_{\alpha,\alpha}\big{[}-\gamma(t-s)^{\alpha}\big{]}+\frac{(1-\alpha)}{\alpha}\gamma^{-2\alpha}E_{\alpha,\alpha}\big{[}-\gamma^{-2\alpha}(t-s)^{\alpha}\big{]}\bigg{)}(t-s)^{\alpha-1}h(s)ds.$

∎

######  Corollary 3.4.

Let $g:[-2,0]\to\mathbb{R}$ be a solution to
${}^{ABC}{\mathcal{D}}_{a}^{\alpha}g=-c_{1}g+c_{0}h(t)$ with $g(-2)=0$, $h\geq
0$ and $\int_{-2}^{-1}h(t)\geq\mu.$ Then

$g(t)\geq\frac{\alpha}{2}E_{\alpha,\alpha}\big{[}-2c_{1}\big{]}c_{0}\mu,\quad\text{for}\quad-1\leq
t\leq 0.$

###### Proof.

From 3.3 the solution of the differential equation for $g$ can be computed
explicitly

$\displaystyle g(t)$ $\displaystyle=\zeta E_{\alpha}\big{[}-\gamma
t^{\alpha}\big{]}g(-2)$ $\displaystyle+\frac{\alpha
c_{0}\zeta}{B(\alpha)}\int_{-2}^{t}\bigg{(}E_{\alpha,\alpha}\big{[}-\gamma(t-s)^{\alpha}\big{]}+\frac{(1-\alpha)}{\alpha}\gamma^{-2\alpha}E_{\alpha,\alpha}\big{[}-\gamma^{-2\alpha}(t-s)^{\alpha}\big{]}\bigg{)}(t-s)^{\alpha-1}h(s)ds.$

If $c_{o}$ is small enough and $c_{1}$ is large, with the initial condition
$g(-2)=0$, and the fact that $E_{\alpha,\alpha}(t)>0$, we have

$\displaystyle g(t)$ $\displaystyle\geq\frac{\alpha
c_{0}\zeta}{2B(\alpha)}E_{\alpha,\alpha}\big{[}-2\gamma\big{]}\int_{-2}^{t}h(s)ds\geq\frac{\alpha
c_{0}\zeta}{2B(\alpha)}E_{\alpha,\alpha}\big{[}-2c_{1}\big{]}c_{0}\mu$
$\displaystyle\geq\frac{\alpha}{2}E_{\alpha,\alpha}\big{[}-2c_{1}\big{]}c_{0}\mu.$

∎

##  4\. Existence of weak solutions via approximating solutions

In this section we provide details of the proof of existence of a solution to
the weak equation (1.1) via approximating solutions. We follow the idea of [2]
to prove our result. To do this, we shall start with the weak formulation of
the problem, then provide the discretization of the weak formulation that will
enable us later to prove the existence of the unique solution and Hölder
continuity.

We state the following integration by parts which will be used to prove weak
formulation.

######  Proposition 4.1.

Assume that $\vartheta$ is bounded and Lipschitz function on $(a,b)$, then

$\displaystyle\int_{a}^{b}\vartheta(t){\mathcal{L}}u(t)dt$
$\displaystyle=c_{\alpha}\int_{a}^{b}u(t)\bigg{(}\int_{a}^{t}{\mathcal{T}}(t,s)\big{[}\vartheta(t)-\vartheta(s)\big{]}ds\bigg{)}dt$
$\displaystyle+c_{\alpha}\int_{a}^{b}\int_{a}^{t}{\mathcal{T}}(t,s)\big{[}u(t)-u(s)\big{]}\big{[}\vartheta(t)-\vartheta(s)\big{]}ds\leavevmode\nobreak\
dt$
$\displaystyle-\int_{a}^{b}E_{\alpha}\big{[}c(t-a)^{\alpha}\big{]}\bigg{[}u(t)\vartheta(a)+u(a)\vartheta(t)\bigg{]}\leavevmode\nobreak\
dt.$ $\displaystyle-\int_{a}^{b}u(t){\mathcal{L}}\vartheta(t)dt$

###### Proof.

The proof of the 4.1 follows from the direct computation of the first term
before the equality. ∎

Using the definition of the Atangana-Baleanu fractional derivative in the form
of (3.8), and for $t<a$, we have the following relation

$\displaystyle\int_{-\infty}^{b}\vartheta(t){\mathcal{L}}u(t)dt$
$\displaystyle=c_{\alpha}\int_{-\infty}^{b}\int_{-\infty}^{b}\big{[}u(t)-u(s)\big{]}\big{[}\vartheta(t)-\vartheta(s)\big{]}{\mathcal{T}}(t,s)ds\leavevmode\nobreak\
dt$
$\displaystyle+c_{\alpha}\int_{-\infty}^{b}\int_{-\infty}^{2t-b}u(t)\big{[}\vartheta(t)-\vartheta(s)\big{]}{\mathcal{T}}(t,s)ds\leavevmode\nobreak\
dt$
$\displaystyle-\int_{-\infty}^{b}u(t){\mathcal{L}}\vartheta(t)\leavevmode\nobreak\
dt.$

Now we state the weak formulation of the problem (1.1).

######  Proposition 4.2.

Assume that $\vartheta$ is a bounded and Lipschitz function on $(-\infty,b)$
for any $t<a$ then the following weak formulation of the solutions (1.1) holds

$\displaystyle
c_{\alpha}\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}\int_{-\infty}^{t}\big{[}u(t,x)-u(s,x)\big{]}\big{[}\vartheta(t,x)-\vartheta(s,x)\big{]}{\mathcal{T}}(t,s,x)ds\leavevmode\nobreak\
dt\leavevmode\nobreak\ dx$
$\displaystyle+\int_{a}^{b}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}{\mathcal{J}}(t,x,\xi)\big{[}u(t,x)-u(t,\xi)\big{]}\big{[}\vartheta(t,x)-\vartheta(t,\xi)\big{]}dx\leavevmode\nobreak\
d\xi\leavevmode\nobreak\ dt$
$\displaystyle+c_{\alpha}\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}\int_{-\infty}^{2t-b}u(t,x)\vartheta(t,x){\mathcal{T}}(t,s,x)ds\leavevmode\nobreak\
dt\leavevmode\nobreak\
dx-\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}u(t,x){\mathcal{L}}\vartheta(t,x)dt\leavevmode\nobreak\
dx$
$\displaystyle=\int_{-\infty}^{b}\int_{\mathbb{R}^{n}}f(t,x)\vartheta(t,x)dx\leavevmode\nobreak\
dt.$

###### Proof.

Now to prove the statement of the 4.2 we use the equation (3.5) and we follow
the idea introduced in [10].

Let $\vartheta$ is a bounded and Lipschitz function on $(-\infty,b)$. Then

$\displaystyle\int_{a}^{b}\int_{a}^{t}{\mathcal{T}}(t,s)\big{[}\vartheta(t,x)-\vartheta(s,x)\big{]}ds\leavevmode\nobreak\
dt=\int_{a}^{b}\int_{a}^{t}{\mathcal{T}}(t,s)\vartheta(t,x)-\int_{a}^{b}\int_{a}^{t}{\mathcal{T}}(t,s)\vartheta(s)ds\leavevmode\nobreak\
dt$
$\displaystyle=\int_{a}^{b}\int_{s}^{b}{\mathcal{T}}(t,s)\vartheta(t)-\int_{a}^{b}\int_{a}^{t}{\mathcal{T}}(t,s)\vartheta(s)dt\leavevmode\nobreak\
ds$ (4.1)
$\displaystyle=\int_{a}^{b}\int_{a}^{t}{\mathcal{T}}(t,s)\vartheta(t)-\int_{a}^{b}\int_{t}^{b}{\mathcal{T}}(s,t)\vartheta(s)ds\leavevmode\nobreak\
dt,$

thanks to (1.5) the kernel ${\mathcal{K}}(t,s)$ satisfies that condition,
therefore it can be written as

${\mathcal{T}}(t,t-s)={\mathcal{T}}(t+s,t)=s^{\alpha-1}E_{\alpha,\alpha}\big{[}cs^{\alpha}\big{]}.$
(4.2)

Thus from (4.2), the equation (4) becomes

$\displaystyle\int_{a}^{b}\int_{a}^{t}{\mathcal{T}}(t,s)\big{[}\vartheta(t)-\vartheta(s)\big{]}ds\leavevmode\nobreak\
dt=$
$\displaystyle\int_{a}^{b}\vartheta(t)\bigg{(}\int_{0}^{t-a}s^{\alpha-1}E_{\alpha,\alpha}\big{[}cs^{\alpha}\big{]}ds-\int_{0}^{b-t}s^{\alpha-1}E_{\alpha,\alpha}\big{[}cs^{\alpha}\big{]}ds\bigg{)}dt$
$\displaystyle=\frac{1}{\alpha
c_{\alpha}}\int_{a}^{b}\vartheta(t)\bigg{(}E_{\alpha}\big{[}c(b-t)^{\alpha}\big{]}-E_{\alpha}\big{[}c(t-a)^{\alpha}\big{]}\bigg{)}dt$

By making use of equation (4) and (4.1), it follows that

$\displaystyle
c_{\alpha}\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}\int_{-\infty}^{t}\big{[}u(t,x)-u(s,x)\big{]}\big{[}\vartheta(t,x)-\vartheta(s,x)\big{]}{\mathcal{T}}(t,s,x)ds\leavevmode\nobreak\
dt\leavevmode\nobreak\ dx$ (4.3)
$\displaystyle+\int_{a}^{b}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}{\mathcal{J}}(t,x,\xi)\big{[}u(t,x)-u(t,\xi)\big{]}\big{[}\vartheta(t,x)-\vartheta(t,\xi)\big{]}dx\leavevmode\nobreak\
d\xi\leavevmode\nobreak\ dt$
$\displaystyle+c_{\alpha}\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}\int_{-\infty}^{2t-b}u(t,x)\vartheta(t,x){\mathcal{T}}(t,s,x)ds\leavevmode\nobreak\
dt\leavevmode\nobreak\
dx-\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}u(t,x){\mathcal{L}}\vartheta(t,x)dt\leavevmode\nobreak\
dx$
$\displaystyle=\int_{-\infty}^{b}\int_{\mathbb{R}^{n}}f(t,x)\vartheta(t,x)dx\leavevmode\nobreak\
dt.$

This completes the proof of the (4.2). ∎

###  4.1. Discretization of the problem

In the following, we denote by $\tau=b/\kappa$ the time step which represents
the subdivision of the interval $(a,b)$, where $\kappa\in\mathbb{N}$ denotes
the number of time steps. Also, for $0\leq k\leq\kappa$, $t=k\tau$. So the
discrete form of the Atangana-Baleanu fractional derivative in the sense of
Caputo holds

${\mathcal{L}}u(a+\tau
k)=\tau^{\alpha}c_{\alpha}\sum_{-\infty<i<k}\frac{E_{\alpha,\alpha}\big{[}c\tau^{\alpha}(k-i)^{\alpha}\big{]}\big{[}u(a+\tau
k)-u(a+\tau i)\big{]}}{\big{(}k-i\big{)}^{1-\alpha}}$ (4.4)

Using (4.4) the discrete form of (3.3) takes the form

$\displaystyle\tau^{\alpha}c_{\alpha}\sum_{-\infty<i<k}\frac{E_{\alpha,\alpha}\big{[}c\tau^{\alpha}(k-i)^{\alpha}\big{]}\big{[}u(a+\tau
k)-u(a+\tau i)\big{]}}{\big{(}k-i\big{)}^{1-\alpha}}$
$\displaystyle=\int_{\mathbb{R}^{n}}[u(a+\tau k,\xi)-u(a+\tau k,x)]$
$\displaystyle{\mathcal{J}}(a+\tau k,x,\xi)d\xi+g(a+\tau k,x).$

Next we state the following Lemma

######  Lemma 4.3.

Assume $u(a)=u(a+\varepsilon j)=0$ for $j<0$. Then the discrete integration by
parts type estimate holds

$\displaystyle\sum_{k\leq j}u(a+\tau k){\mathcal{L}}u(a+\tau k)$
$\displaystyle\geq\frac{\tau^{\alpha}}{2}C_{\alpha}\mathop{\sum\sum}_{0\leq
i<k\leq j}\frac{\big{[}u(a+\tau k)-u(a+\tau
i)\big{]}^{2}}{\big{(}k-i\big{)}^{\alpha-1}}E_{\alpha,\alpha}\big{[}c\tau^{\alpha}(k-i)^{\alpha}\big{]}$
(4.5) $\displaystyle+\frac{\tau^{\alpha}}{2}C_{\alpha}\mathop{\sum}_{k\leq
j}\frac{\big{[}u^{2}(a+\tau
k)\big{]}^{2}}{\big{(}j-i\big{)}^{\alpha-1}}E_{\alpha,\alpha}\big{[}c\tau^{\alpha}(j-i)^{\alpha}\big{]}$
.

###### Proof.

The proof of this Lemma follows from the direct computation of the discrete
integration by parts type estimate. ∎

Now to get the discrete form of the weak formulation 4.2 we use the
integration by parts type estimate given by 4.3

$\displaystyle c_{\alpha}\mathop{\sum\sum}_{0\leq i<k\leq
j}\int_{\tau(k-1)}^{\tau k}\int_{\tau(i-1)}^{\tau i}\big{[}u_{\tau}(\tau
k)-u_{\tau}(\tau i)\big{]}\big{[}\vartheta(\tau k)-\vartheta(\tau
i)\big{]}\frac{E_{\alpha,\alpha}\big{[}c\tau^{\alpha}(k-i)^{\alpha}\big{]}}{\big{(}k-i\big{)}^{1-\alpha}}.$
(4.6) $\displaystyle+\int_{\tau(k-1)}^{\tau
k}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}{\mathcal{J}}(t,x,\xi)\big{[}u(\tau
k,\xi)-u(\tau k,x)\big{]}\big{[}\vartheta(\tau k,x)-\vartheta(\tau
k,\xi)\big{]}$
$\displaystyle+c_{\alpha}\int_{\mathbb{R}^{n}}\tau\mathop{\sum\sum}_{0\leq
i<k\leq j}\frac{u_{\tau}(\tau k)\vartheta(\tau
k)}{(\tau(k-i))^{1-\alpha}}E_{\alpha,\alpha}\big{[}c\tau^{\alpha}(k-i)^{\alpha}\big{]}-\int_{\mathbb{R}^{n}}\tau\sum_{0<k\leq
j}u_{\tau}(\tau k){\mathcal{L}}\vartheta(\tau k)$
$\displaystyle=\int_{\tau(k-1)}^{\tau k}\int_{\mathbb{R}^{n}}g(\tau
k,x)\vartheta(\tau k,x).$

Now we are ready to prove the 1.1 on existence of weak solutions following the
idea of the authors in [10] by using the approximating method. For this
purpose, we write the operators in (4.3) and (4.5) as $\mathcal{H}$ and
$\mathcal{H}_{\tau}$ respectively.

###### Proof.

For $\vartheta$ be a bounded an Lipschitz function on
$(-\infty,b)\times\mathbb{R}^{n}$. There exists a sequence of solutions
$u_{\tau}$ to (4.3) with $\tau\to 0$, such that

$u_{\tau}\to u\in L^{p}\big{(}(-\infty,b)\times\mathbb{R}^{n}\big{)},$

with $p$ as defined in [2] as

$p=2\bigg{(}\frac{\alpha n+\beta}{\alpha n+(1-\alpha)\beta}\bigg{)}.$

For $\tau(k-1)<t\leq\tau k$, we let $\mathcal{B}_{\tau}$ be the bilinear form
associated with $K_{\tau}$. Our aim is to show that for $\vartheta$ be a
bounded and Lipschitz function on $(-\infty,b)\times\mathbb{R}^{{}^{n}}$,

$\mathcal{H}(u,\vartheta)+\mathcal{H}_{\tau}(u_{\tau},\vartheta)\to 0.$

The first part of the proof where we shall consider the fractional Laplacian,
we mean

$\lim_{\tau\to 0}\int_{a}^{b}\mathcal{B}_{\tau}(u_{\tau},\vartheta)\
dt=\lim_{\tau\to 0}\tau\sum_{0<k\leq j}\mathcal{B}(u_{\tau}(\tau
k,x),\vartheta(\tau k,x))$

has been prove by the authors in [2]. So next we shall focus on the pieces in
time. To do this we start by showing that

$\displaystyle\lim_{\tau\to
0}\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}\int_{-\infty}^{t}\big{[}u_{\tau}(t,x)-u_{\tau}(s,x)\big{]}\big{[}\vartheta(t,x)-\vartheta(s,x)\big{]}\frac{E_{\alpha,\alpha}\big{[}c(t-s)^{\alpha}\big{]}}{(t-s)^{1-\alpha}}ds\leavevmode\nobreak\
dt$ $\displaystyle\quad=\lim_{\tau\to 0}\mathop{\sum\sum}_{0\leq i<k\leq
j}\int_{\tau(k-1)}^{\tau k}\int_{\tau(i-1)}^{\tau i}\big{[}u_{\tau}(\tau
k)-u_{\tau}(\tau i)\big{]}\big{[}\vartheta(\tau k)-\vartheta(\tau
i)\big{]}\frac{E_{\alpha,\alpha}\big{[}c\tau^{\alpha}(k-i)^{\alpha}\big{]}}{\big{(}\tau(k-i)\big{)}^{1-\alpha}}.$

In order to achieve our goal, since $\vartheta$ is a bounded and Lipschitz
function, then $\vartheta_{\tau}(t)\to\vartheta(t)$ and $u_{\tau}\to u\in
L^{p}\big{(}(-\infty,b)\times\mathbb{R}^{n}\big{)}$ we have that

$\lim_{\tau\to
0}\left|\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}\int_{\infty}^{t}\frac{E_{\alpha,\alpha}\big{[}c(t-s)^{\alpha}\big{]}}{(t-s)^{1-\alpha}}\big{[}u_{\tau}(t)-u_{\tau}(s)\big{]}\left[(\vartheta(t)-\vartheta(s))-(\vartheta_{\tau}(t)-\vartheta_{\tau}(s))\right]\
ds\ dt\right|\to 0.$

We show also that

$\displaystyle\lim_{\tau\to 0}\mathop{\sum\sum}_{0\leq i<k\leq
j}(u_{\tau}(\tau k)-u_{\tau}(\tau i))(\vartheta(\tau k)-\vartheta(\tau i))$
$\displaystyle\times\int_{\tau(k-1)}^{\tau k}\int_{\tau(i-1)}^{\tau
i}\bigg{(}\frac{E_{\alpha,\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}}{(t-s)^{1-\alpha}}-\frac{E_{\alpha,\alpha}\big{[}c\tau^{\alpha}\big{(}k-i\big{)}^{\alpha}\big{]}}{\big{(}\tau(k-i)\big{)}^{1-\alpha}}\bigg{)}=0.$

To do this we break up the integral over the sets $(t-s)\leq\tau^{1/2}$ and
$(t-s)>\tau^{1/2}$. Then with the relation (1.5)

$\displaystyle 0$ $\displaystyle=\lim_{\tau\to
0}\int_{\mathbb{R}^{n}}\mathop{\int\int}_{t-s\leq\tau^{1/2}}\frac{|(u_{\tau}(t)-u_{\tau}(s)(\vartheta_{\tau}(t)-\vartheta_{\tau}(s))|}{(t-s)^{1-\alpha}}E_{\alpha,\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}$
$\displaystyle\geq\lim_{\tau\to
0}\int_{\mathbb{R}^{n}}\mathop{\sum\sum}_{\tau(k-i)\leq\tau^{1/2}}\bigg{|}(u_{\tau}(\tau
k)-u_{\tau}(\tau i))(\vartheta(\tau k)-\vartheta(\tau
i))\bigg{|}\int_{\tau(k-1)}^{\tau k}\int_{\tau(i-1)}^{\tau
i}\frac{E_{\alpha,\alpha}\big{[}c\tau^{\alpha}\big{(}k-i\big{)}^{\alpha}\big{]}}{(\tau(k-i))^{1-\alpha}}$

For $t-s>\tau^{1/2}$, and $\tau(i-1)\leq s\leq\tau i$ and $\tau(k-1)\leq
t\leq\tau k$, we can compute the estimate

$\displaystyle\big{|}(t-s)^{\alpha-1}E_{\alpha,\alpha}\big{[}c\big{(}t-s\big{)}^{\alpha}\big{]}-\big{(}\tau(k-i)\big{)}^{\alpha-1}\big{|}$
$\displaystyle\leq\big{(}\tau^{1/2}-\tau\big{)}^{\alpha-1}$
$\displaystyle\times
E_{\alpha,\alpha}\big{[}c\big{(}\tau^{1/2}-\tau\big{)}^{\alpha}\big{]}-\big{(}\tau^{1/2}\big{)}^{\alpha-1}E_{\alpha,\alpha}\big{[}c\big{(}\tau^{1/2}\big{)}^{\alpha}\big{]}$
$\displaystyle\leq T_{\alpha,\tau}$

Hence the result follows for $\tau\to 0$. Next we consider the following

$\int_{\mathbb{R}^{n}}\int_{-\infty}^{b}u_{\tau}(t){\mathcal{L}}\vartheta(t)\
dt\ dx-\int_{\mathbb{R}^{n}}\tau\sum_{0<k\leq j}u_{\tau}(\tau
k){\mathcal{L}}\vartheta(\tau k).$

Similarly as in the previous case, since $\vartheta$ is a bounded Lipschitz
function and $u_{\tau}\to u\in
L^{p}\big{(}(-\infty,b)\times\mathbb{R}^{n}\big{)}$, and from (4.5) one show
that this term also goes to zero.

The remaining pieces in time are handled in the similar manner. Thus the
Theorem is proved. ∎

##  5\. Pointwise Estimates and Hölder Regularity

This section contains the some auxiliaries results, which are the key to prove
1.2. The proof of 5.6 uses the main contributions of this note. Once 5.6 is
established, a-priori Hölder regularity estimates follow by the classical
method of diminishing oscillation given by 5.7.

Before going into this, we first collect the ingredients that will be useful.
As one feature, we underline the viscosity solution. One of the useful
property of viscosity solution is that viscosity subsolutions themselves can
be used to evaluate their corresponding equation classically at all of the
points where where the subsolution can be touched from above by a smooth test
function.

We need the following property

${\mathcal{L}}u(t,x)-M^{+}_{{\mathcal{A}}}u(t,x)\leq g(t,x)\ \quad\text{in}\
B_{1}\times[-1,0].$ (5.1)

Next we state the following proposition to clarify that $u$ is a solution on
(5.1) and (1.1) in the viscosity sense by also making reference to the 3.1

######  Proposition 5.1.

Let $u$ be a continuous bounded function on $(-\infty,b)$ and assume that for
some $t\in(-\infty,b)$ there is a Lipschitz function touching $u$ by above at
$t$. Then

$\int_{-\infty}^{t}\big{[}u(t,x)-u(s,x)\big{]}{\mathcal{T}}(t,s)\ ds\geq
g(t,x)$

if and only if ${\mathcal{L}}u(t,x)\geq g(t,x)$ in the viscosity sense.

###### Proof.

The proof is standard and is based on the proof of Proposition 2.3 in [1]. ∎

######  Proposition 5.2.

Assume $u$ solves (5.1) in the viscosity sense. $\phi\geq u$ defined on the
cylinder $\mathcal{C}:=[t_{0}-\varepsilon,t_{0}]\times B_{\varepsilon}(x_{0})$
has a global maximum and touches $u$ from above at
$(x_{0},t_{0})\in\mathcal{C}$ and we define $v$ as

$v(x,t):=\begin{cases}\phi(x,t)&\text{if }(x,t)\in\mathcal{C}\\\
u(x,t)&\text{otherwise},\\\ \end{cases}$

then $v$ is solution to (5.1) at $(x_{0},t_{0})$ or

${\mathcal{L}}v(t_{0},x_{0})-\mathcal{J}v(t_{0},x_{0})\leq g(t_{0},x_{0}).$

So the solution is both a subsolution and supersolution.

The proof is straightforward.

Before we state the point evaluation proposition, we recall the comparison
principle for which the proof is similar as in [1].

######  Lemma 5.3.

(Comparison Principle). Let $u$ be bounded and upper simi-continuous and $w$
be bounded and lower semi-continuous $(-\infty,b_{2})$. Let $g$ be a
continuous function such that ${\mathcal{L}}u\leq g\leq{\mathcal{L}}w$ on
$(b_{1},b_{2}]$, with $u\leq w$ on $(-\infty,b_{1})$. Then $u\leq w$ on
$(-\infty,b_{2})$, and if $u(t_{0})=w(t_{0})$ for some
$t_{0}\in(b_{1},b_{2}]$, then $u(t)=w(t)$ for all $t\leq t_{0}$.

######  Lemma 5.4.

Let $\phi(t)$ be continuous $(-\infty,b_{1})$. Let $g$ be a continuous
function on $[b_{1},b_{2}]$. There exists a unique viscosity solution $u$ to ,
then

$\begin{cases}{\mathcal{L}}u(t)&=g(t)\qquad\text{for}\quad\quad\leavevmode\nobreak\
t\in[b_{1},b_{2}]\\\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ u(t)&=\phi(t)\qquad\text{if}\quad\quad t\leq
b_{1}\end{cases}$

on $(b_{1},b_{2}]$.

We start by recalling the definitions of Pucci’s extremal operators as defined
in [5] for the spatial operator and next for the fractional-time derivative
operator.

######  Lemma 5.5 (Extremal Formula).

Assume $u\in C^{1,1}(-\infty,b)\bigcap L^{\infty}(\mathbb{R}^{n})$. Then we
have the following elliptic spacial operator

${\mathcal{M}}^{+}_{{\mathcal{A}}}u(t,x)=\int_{\mathbb{R}^{n}}\bigg{(}\Lambda\big{(}\delta_{h}u(t,x)\big{)}_{+}-\lambda\big{(}\delta_{h}u(t,x)\big{)}_{-}\bigg{)}\mu(dh),$
(5.2)
${\mathcal{M}}^{-}_{{\mathcal{A}}}u(t,x)=\int_{\mathbb{R}^{n}}\bigg{(}\lambda\big{(}\delta_{h}u(t,x)\big{)}_{+}-\Lambda\big{(}\delta_{h}u(t,x)\big{)}_{-}\bigg{)}\mu(dh),$
(5.3)

and for the fractional-time derivative operator as

${\mathcal{M}}_{{\mathcal{T}}_{sec}}^{+}u(t,x):=\mathcal{C}(n,\alpha)\int_{-\infty}^{t}\bigg{[}\Lambda\big{(}u(t,x)-u(s,x)\big{)}_{+}-\lambda\big{(}u(t,x)-u(s,x)\big{)}_{-}\bigg{]}{\mathcal{T}}_{sec}$
(5.4)
${\mathcal{M}}_{{\mathcal{T}}_{sec}}^{-}u(t,x):=\mathcal{C}(n,\alpha)\int_{-\infty}^{t}\bigg{[}\lambda\big{(}u(t,x)-u(s,x)\big{)}_{+}-\Lambda\big{(}u(t,x)-u(s,x)\big{)}_{-}\bigg{]}{\mathcal{T}}_{sec}$
(5.5)

Next, in order to prove the Hölder continuity, we use essentially the same
ideas as the proof in [10, 17].

######  Proposition 5.6.

(Point Estimate). Let $u\leq 1$ in
$(\mathbb{R}^{n}\times[-2,0])\cup(B_{1}\times[-\infty,0])$ and assume it
satisfies the following inequality in the viscosity sense in
$B_{2}\times[-2,0]$

${\mathcal{L}}u-{\mathcal{M}}^{+}u\leq\varepsilon_{0}.$

Assume also that $\big{|}\big{\\{}u(x,t)\leq
0\big{\\}}\cap(B_{1}\times[-2,-1])\big{|}\geq\mu>0$.

Then if $\varepsilon_{0}$ is small enough there exists $\theta>0$ such that
$u\leq(1-\theta)$ in $B_{1}\times[-2,0]$. The maximal value of $\theta$ as
well as $\varepsilon_{0}$ depends on $\alpha,\lambda,\Lambda,n$ and $\sigma$,
but remain uniform as $\alpha\to 1$.

###### Proof.

We consider the differential equation

$\begin{cases}{\mathcal{L}}g(t)=c_{0}\big{|}\big{\\{}x\in B_{1}:u(x,t)\leq
0\big{\\}}\big{|}-c_{1}g(t).\\\ g(-2)=0,\quad\text{for}\leavevmode\nobreak\
\leavevmode\nobreak\ t\leq-2\end{cases}$ (5.6)

From 3.3 this ordinary differential equation can be computed explicitly and
from 3.4 we have that

$g(t)\geq\frac{\alpha
c_{0}\mu}{2}E_{\alpha,\alpha}\big{[}-2c_{1}\big{]},\qquad\text{for}-1\leq
t\leq 0.$

In the following we will show that if $c_{0}$ is small and $c_{1}$ is large,
then $u\leq 1-g(t)+\varepsilon_{0}c_{\alpha}2^{\alpha}$ in
$B_{1}\times[-1,0]$. The constant $c_{\alpha}$ is chosen such that
${\mathcal{L}}c_{\alpha}(2+t)^{\alpha}_{+}=1$ for $t\geq-2$.

Since for $t\in[-1,0]$

$\displaystyle g(t)$
$\displaystyle\geq\frac{\alpha}{2}E_{\alpha,\alpha}\big{[}-2c_{1}\big{]}c_{0}\big{|}\big{\\{}x:u(x,t)\leq
0\big{\\}}\cap B_{1}\times[-2,-1]\big{|}$
$\displaystyle\geq\frac{\alpha}{2}E_{\alpha,\alpha}\big{[}-2c_{1}\big{]}c_{0}\mu,$

we set $\theta=\frac{\alpha
c_{0}\mu}{4}E_{\alpha,\alpha}\big{[}-2c_{1}\big{]}$ for $\varepsilon_{0}$
small and finish the proof of the Lemma.

Next as in [10], let $\beta:\mathbb{R}\to\mathbb{R}$ be a fixed smooth non
increasing function such that $\beta(x)=0$ if $x\geq 2$. Let
$\eta(x,t)=\beta(|x|)$. As a function of $x$, $\eta(x,t)$ looks like a bump
function for every fixed $t$. The main strategy of the proof is to show that
the function $u(x,t)$ stays below
$1-g(t)\eta(x,t)+\varepsilon_{0}c_{\alpha}(2+t)_{+}^{\alpha}$.

In order to arrive to a contradiction, we assume that
$\eta(x,t)>1-g(t)+\varepsilon_{0}c_{\alpha}(2+t)_{+}^{\alpha}$ for some point
$(x,t)\in B_{1}\times[-1,0]$. We then look at the maximum of the function

$\tilde{\eta}(x,t)=u(x,t)+g(t)\eta(x,t)-\varepsilon_{0}c_{\alpha}(2+t)_{+}^{\alpha}.$

Assuming that there is one point $(x_{0},t_{0})$ in $B_{1}\times[-1,0)$ where
$\tilde{\eta}(x,t)>1$, $\tilde{\eta}$ must be larger that $1$ at the point
that realize the maximum of $\tilde{\eta}$. We mean by that

$\tilde{\eta}(x_{0},t_{0})=\max_{\mathbb{R}^{n}\times(-\infty,0]}\tilde{\eta}(x,t).$

Since $\tilde{\eta}(x_{0},t_{0})>1$, the point $(x_{0},t_{0})$ must belong to
the compact support $\eta$. Hence $|x|<2$. Now on the remain of the proof is
exactly as in [10].

Next we call $\varphi(x,t)$ the function such that

$\varphi(x,t):=\tilde{\eta}(x_{0},t_{0})-g(t)\eta(x,t)+c_{\alpha}(2+t)_{+}^{\alpha}$

$\varphi$ touches $u$ from above at the point $(x_{0},t_{0})$. We define

$v(x,t):=\begin{cases}\varphi(x,t)\quad\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ x\in B_{r}\\\ u(x,t)\quad\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ x\notin B_{r}.\end{cases}$

Then at the point that $\tilde{\eta}$ realise it maximum

${\mathcal{L}}v(x_{0},t_{0})-{\mathcal{M}}^{+}v(x_{0},t_{0})\leq\varepsilon_{0}.$
(5.7)

So one can have

${\mathcal{L}}v(x_{0},t_{0})-{\mathcal{L}}g(t_{0})\eta(x_{0})+\varepsilon_{0}.$
(5.8)

Hence for $G:=\big{\\{}x\in B_{1}\big{|}u(x,t_{0})\leq o\big{\\}}$, the
following bound is obtain

${\mathcal{M}}^{+}v(x_{0},t_{0})\leq-g(t_{0}){\mathcal{M}}^{-}\eta(x_{0},t_{0})-c_{0}\big{|}G\setminus
B_{r}\big{|}$ (5.9)

Now if we insert the relation (5.9), (5.7), (5.8) into (5.6), we obtain

$-\leavevmode\nobreak\
{\mathcal{L}}g(t_{0})\eta(x_{0},t_{0})+\varepsilon_{0}+c_{0}\big{|}G\setminus
B_{r}\big{|}\leq\varepsilon_{0},$

or in its explicit form

$\bigg{(}-c_{0}\big{|}\big{\\{}x\in B_{1}:u(x,t)\leq
0\big{\\}}\big{|}+c_{1}g(t)\bigg{)}\eta(x_{0})+\varepsilon_{0}+c_{0}\big{|}G\setminus
B_{r}\big{|}\leq\varepsilon_{0}.$

But notice that for any $c_{1}>0$ this contradicts (5.6).

We now analyse the case where $\eta(x_{0},t_{0})>\beta_{1}$. As we said
previously that $\eta$ is a smooth compactly supported function, then there
exist some constant $\tilde{C}$ such that
$\big{|}{\mathcal{M}}^{-}\eta\big{|}\leq\tilde{C}$. Then we have the bound

${\mathcal{M}}^{+}v(x_{0},t_{0})\leq-\tilde{C}g(t_{0})+c_{0}\big{|}G\setminus
B_{r}\big{|}.$ (5.10)

As in the previous case, we insert (5.10) in (5.7), (5.8) into (5.6), we
obtain

$\bigg{(}-c_{0}\big{|}\big{\\{}x\in B_{1}:u(x,t)\leq
0\big{\\}}\big{|}+c_{1}g(t)\bigg{)}\eta(x_{0})+\varepsilon_{0}-\tilde{C}g(t_{0})+c_{0}\big{|}G\setminus
B_{r}\big{|}\leq\varepsilon_{0}.$

We recall that $\eta(x_{0},t_{0})>0$ and by letting $r\to 0$, we obtain

$c_{0}\big{(}1-\eta(x_{0})\big{)}|G|+\big{(}c_{1}\eta(x_{0})-\tilde{C}\big{)}g(t_{0})\leq
0$

Which can be written in the form

$c_{0}\big{(}1-\eta(x_{0})\big{)}|G|+\big{(}c_{1}\beta_{1}-\tilde{C}\big{)}g(t_{0})\leq
0.$

Choosing $c_{1}$ large enough, we arrived to a contradiction. This ends the
proof. ∎

In the following we shall give an approach of the proof of the Hölder
continuity. For this purpose, we state and prove the so called growth Lemma,
which say that if a solution of the equation (1.1) in the unit cylinder
$Q_{1}=B_{1}\times[-1,0]$ has oscillation one then its oscillation in a
smaller cylinder $Q_{r}$ is less that a fixed constant $(1-\theta)$ [10, 17].

######  Lemma 5.7.

(Diminish of Oscillation). Let $u$ be a bounded continuous function which
satisfies (1.1) or the following two inequalities in the viscosity sense in
$Q_{1}$.

$\displaystyle{\mathcal{L}}u-{\mathcal{M}}^{+}u\leq\varepsilon_{0}/2$
$\displaystyle{\mathcal{L}}u-{\mathcal{M}}^{+}u\geq-\varepsilon_{0}/2.$

Then there are universal constants $\theta>0$ and $\nu>0$, depending on
$n,\sigma,\Lambda,\lambda$ and $\alpha$, such that if

$\displaystyle\big{\|}g\big{\|}_{L^{\infty}(Q)}$
$\displaystyle\leq\varepsilon_{0},\qquad|u|\leq 1\leavevmode\nobreak\
\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\ \leavevmode\nobreak\ Q_{1}$
$\displaystyle|u(x,t)|$ $\displaystyle\leq 2|rx|^{\nu}-1\leavevmode\nobreak\
\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\ \leavevmode\nobreak\
\big{(}\mathbb{R}^{n}\setminus B_{1}\big{)}\times[-1,0]$
$\displaystyle|u(x,t)|$ $\displaystyle\leq 2|rt|^{\nu}-1\leavevmode\nobreak\
\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\ \leavevmode\nobreak\
B_{1}\times[-\infty,-1]$

with $r=\min\\{4^{-1},4^{-\alpha/2\sigma}\\}$, then

$\text{osc}_{Q_{r}}u\leq(1-\theta)$

###### Proof.

With the 3.2 in hand the reader can finish the proof by following the idea of
the authors in [10, 17]. ∎

Having the diminish of oscillation Lemma in hand, where the proof is similar
to the one proposed by the author Mark Allen in [1], we are now going to prove
one of our main results about Hölder continuity. The result requires the
function to solve the equation only in a cylinder in order to have Hölder
continuity in a smaller cylinder. To do this, we renormalized the function
$u$.

If $u$ satisfies (1.1), then the rescaled function
$\varrho(x,t)=u(rx,r^{2\sigma/\alpha}t)$ satisfies also (1.1), with
$r\in(0,1)$ such that $r=\min\\{4^{-1},4^{-\alpha/2\sigma}\\}$. This means
that the structure of the space is not modified if we make a parabolic
dilation to make the profile a little bit in the future. We then define the
parabolic cylinders in terms of the scaling of the equation by

$Q_{r}:=B_{r}\times[-r^{2\sigma/\alpha},0].$

Now we state the result on Hölder continuity.

######  1.2.

The proof is the adaptation of the proof in [1, 17]. We prove
$\mathcal{C}^{\alpha}$ estimate of (1.8) by proving a $\mathcal{C}^{\alpha}$
estimate for $\varrho(x,t)$ at the point $(x_{0},t_{0})\in Q_{1}$. For any
point $(x_{0},t_{0})$ we consider the normalized function

$\varrho(x,t)=\frac{u(x_{0}+x,t+t_{0})}{\|u\|_{L^{\infty}}+\varepsilon_{0}^{-1}\|g\|_{L^{\infty}}},$

where $\varepsilon_{0}$ is a constant from 5.7, one can show that
$\text{osc}_{\mathbb{R}\times[-1,0]}\rho\leq 1$ and in $B_{2}\times[-1,0]$,
$\varrho$ is a solution to

$\displaystyle{\mathcal{L}}\varrho-{\mathcal{M}}_{\sigma}^{+}\varrho$
$\displaystyle\leq\varepsilon_{0}$
$\displaystyle{\mathcal{L}}\varrho-{\mathcal{M}}_{\sigma}^{-}\varrho$
$\displaystyle\geq-\varepsilon_{0}.$

Let $r\in(0,1)$, such that $r=\min\\{4^{-1},4^{-\alpha/2\sigma}\\}$ . To prove
then the Hölder estimate, we prove by induction the following decay of the
oscillation in cylinders for some $\kappa$ [14]

$\text{osc}_{Q_{r_{k}}}\varrho\leq 2r^{\kappa
k}\quad\text{for}\leavevmode\nobreak\ k=0,1,2,\dots.$ (5.11)

We construct two sequences $S_{k}\leq\varrho\leq T_{k}$ in $Q_{r_{k}}$,
$T_{k}-S_{k}=2r^{\kappa k}$ with $S_{k}$ nondecreasing and $T_{k}$
nonincreasing. Indeed. for $k=0$, by hypothesis it is true that
$\text{osc}_{Q_{r_{k}}}\varrho=1$. We assume that the sequences holds until
certain value of $k$.

We scale once more by considering

$z(x,t)=(\varrho(r^{k}x,r^{2k\sigma/\alpha}t)-(S_{k}+T_{k})/2)r^{-\kappa k}.$

Then we have

$\displaystyle|z|\leq 1$ $\displaystyle\quad\text{in }\quad Q_{1}$
$\displaystyle|z|\leq 2r^{-\kappa k}-1$ $\displaystyle\quad\text{in }\quad
Q_{r^{-k}}.$

and so

$\displaystyle|z(x,t)|\leq 2|x|^{\nu}-1$ $\displaystyle\quad\text{for
}\quad(x,t)\in B_{1}^{c}\times[-1,0]$ $\displaystyle|z(x,t)|\leq 2|t|^{\nu}-1$
$\displaystyle\quad\text{for }\quad(x,t)\in B_{1}\times(-\infty,-1).$

For $\kappa<2\sigma$, $z$ has right hand side bounded

$\big{\|}r^{k(\kappa-2\sigma)}g\big{\|}_{L^{\infty}}\leq\varepsilon_{0}r^{k(\kappa-2\sigma)}<\varepsilon_{0}$

If $\kappa$ small enough we can apply 5.7 to obtain

$\text{osc}_{Q_{r}}z\leq 1-\theta.$

Furthermore if we choose $\kappa$ smaller than the one in 5.7 and also so that
$r^{\kappa}\geq 1-\theta$, we have that

$\text{osc}_{Q_{r^{k}}}z\leq r^{\kappa}$

which simply means $\text{osc}_{Q_{r^{k+1}}}z\leq r^{\kappa(k+1)}$ so we can
find two sequences $S_{k+1}$ and $T_{k+1}$. This finishes the proof by
induction. ∎

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