# Regularity for quasi-linear parabolic equations with nonhomogeneous
degeneracy or singularity

Yuzhou Fang and Chao Zhang∗ Yuzhou Fang School of Mathematics, Harbin
Institute of Technology, Harbin 150001, China<EMAIL_ADDRESS>Chao Zhang
School of Mathematics and Institute for Advanced Study in Mathematics, Harbin
Institute of Technology, Harbin 150001, China<EMAIL_ADDRESS>
###### Abstract.

We introduce a new class of quasi-linear parabolic equations involving
nonhomogeneous degeneracy or/and singularity

$\partial_{t}u=[|Du|^{q}+a(x,t)|Du|^{s}]\left(\Delta u+(p-2)\left\langle
D^{2}u\frac{Du}{|Du|},\frac{Du}{|Du|}\right\rangle\right),$

where $1<p<\infty$, $-1<q\leq s<\infty$ and $a(x,t)\geq 0$. The motivation to
investigate this model stems not only from the connections to tug-of-war like
stochastic games with noise, but also from the non-standard growth problems of
double phase type. According to different values of $q,s$, such equations
include nonhomogeneous degeneracy or singularity, and may involve these two
features simultaneously. In particular, when $q=p-2$ and $q<s$, it will
encompass the parabolic $p$-Laplacian both in divergence form and in non-
divergence form. We aim to explore the from $L^{\infty}$ to $C^{1,\alpha}$
regularity theory for the aforementioned problem. To be precise, under some
proper assumptions, we use geometrical methods to establish the local Hölder
regularity of spatial gradients of viscosity solutions.

###### Key words and phrases:

Quasi-linear parabolic equation; normalized $p$-Laplacian; nonhomogeneous
degeneracy or/and singularity; viscosity solution; comparison principle;
regularity.

###### 2010 Mathematics Subject Classification:

35B65, 35K65, 35D40, 35K92, 35K67

∗Corresponding author.

##  1\. Introduction

Let $B_{r}\subset\mathbb{R}^{n}$ be a ball with radius $r$ centered at the
origin and $Q_{r}=B_{r}\times(-r^{2},0]$. In this paper, we are concerned with
the interior regularity for viscosity solutions to the following quasi-linear
parabolic equation with nonhomogeneous degeneracy or/and singularity

$\partial_{t}u=[|Du|^{q}+a(x,t)|Du|^{s}]\Delta_{p}^{N}u\quad\textmd{in
}Q_{1},$ (1.1)

where $1<p<\infty$, $-1<q\leq s<\infty$, $0\leq a(x,t)\in C^{1}(Q_{1})$, and
$\Delta_{p}^{N}$ denotes the normalized $p$-Laplace operator defined by

$\Delta_{p}^{N}u=\Delta u+(p-2)\left\langle
D^{2}u\frac{Du}{|Du|},\frac{Du}{|Du|}\right\rangle=\left(\delta_{ij}+(p-2)\frac{u_{i}u_{j}}{|Du|^{2}}\right)u_{ij}.$

Here the summation convention is exploited and $Du$ is the gradient of $u$ in
the spatial variable $x$. Throughout this paper, $u_{i}=\partial_{x_{i}}u$,
$u_{ij}=\partial_{x_{i}x_{j}}u$,
$D_{x,t}u=(\partial_{t}u,\partial_{x_{1}}u,\cdots,\partial_{x_{n}}u)^{T}$.

Over the last decade, a linkage between the stochastic tug-of-war games and
nonlinear equations of $p$-Laplacian type, starting with the pioneering papers
of Peres-Schramm-Sheffield-Wilson [52] and Peres-Sheffield [53], has received
lots of attention. For the parabolic scenario, Manfredi-Parviainen-Rossi [45]
proved that the solutions to

$\partial_{t}u=\Delta_{p}^{N}u$ (1.2)

could be derived as the limits of value functions for tug-of-war games with
noise if the parameter controlling the size of the possible steps tends to
zero. We remark that the normalized $p$-Laplacian can be regarded as the
1-homogeneous version of standard $p$-Laplacian or as a mixture of the
Laplacian and normalized infinity Laplacian,
$\Delta_{\infty}^{N}u=|Du|^{-2}\langle D^{2}uDu,Du\rangle$. The lower
regularity for solutions of the homogeneous or nonhomogeneous elliptic
normalized $p$-Laplace equation was obtained in [44, 55] (see [8, 9, 29] for
the parabolic analogue). The first contribution on the
$C^{1,\alpha}$-regularity for such equations is due to the seminal work of
Jin-Silvestre [36], where they established the interior Hölder gradient
estimates of solutions to (1.2). This result was extended to the inhomogeneous
parabolic normalized $p$-Laplacian in [3]. For the inhomogeneous elliptic
counterpart

$-\Delta_{p}^{N}u=f(x)\quad\textmd{in }\Omega,$

Attouchi-Parviainen-Ruosteenoja [4] showed that the solutions are locally
$C^{1,\alpha}$ regular under the condition that $f(x)\in L^{q}$ with
$q\leq\infty$ possessing a suitably large lower bound; see also [10] for the
case that $f(x)\in L(n,1)$, where $L(n,1)$ denotes the standard Lorentz space.
In addition, the existence of viscosity solutions to (1.2) has been obtained
in [8, 9, 29] by using approximation techniques that are different from the
game-theoretic arguments [45]. Juutinen [37] investigated the asymptotic
behavior for (1.2). For more results on the stochastic tug-of-war game and the
$p$-Laplacian operators, see for instance [40, 42, 46, 54].

On the other hand, equation (1.1) is motivated by the double phase problems as
well. We observe that equation (1.1) is a new model of quasi-linear parabolic
equations featuring a nonhomogeneous degenerate or/and singular term modelled
on the double phase integrand

$H(x,t,\xi):=|\xi|^{p}+a(x,t)|\xi|^{q},\quad\quad a(x,t)\geq 0,\quad 1<p\leq
q.$ (1.3)

In the elliptic case (i.e., the function $a$ is independent of $t$), from a
variational point of view, (1.3) is closely related to the following energy
functional

$u\mapsto\int(|Du|^{p}+a(x)|Du|^{q})\,dx,$ (1.4)

which was originally introduced by Zhikov [57, 58] in the context of
homogenization and Lavrentiev phenomenon. Such functionals can provide useful
models for describing the behaviours of strongly anisotropic materials. More
precisely, considering two diverse materials with hardening exponents $p$ and
$q$ separately, the modulating coefficient $a(\cdot)$ determines the geometry
of the mixture composed of the two, according to whether $x$ belongs to the
zero set $\\{a(x)=0\\}$ or not. These functionals with non-standard growth
conditions

$u\mapsto\int_{\Omega}F(x,u,Du)\,dx,\quad\nu|\xi|^{p}\leq F(x,u,\xi)\leq
L(|\xi|^{q}+1),$

have been a surge of interest. In the autonomous case that energy density
$F(x,u,Du)\equiv F(Du)$, the regularity theory for minima of such functionals
is by now well-known from the prominent works of Marcellini [47, 48, 49]. The
investigation of double phase functional (1.4) has been continued in a series
of nice papers by Colombo, Mingione et al. For instance, the local
$C^{1,\alpha}$-regularity for minimizers of functional (1.4) was obtained in
[11, 19, 20] under some hypotheses that $a(x)$ is Hölder continuous and the
magnitude of the difference $q-p\geq 0$ is suitably small. Whereafter, the
Calderón-Zygmund type estimates for the weak solutions to

$\operatorname{div}(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=\operatorname{div}(|F|^{p-2}F+a(x)|F|^{q-2}F)$

were proved in [14, 21, 25]. See also [26] for the manifold constrained
problem, [16] for the obstacle problem, [31] for the equivalence of weak and
viscosity solutions and [17] for the potential theory. More results can be
found in [7, 18, 30, 43] and reference therein. Very recently, De Filippis
[24] established the quantitative gradient bounds for weak solutions to the
following parabolic double phase equations

$\partial_{t}u-\operatorname{div}(|Du|^{p-2}Du+a(x,t)|Du|^{q-2}Du)=0.$

The Harnack’s type inequality for this equation was also derived in [13].
However, as far as we know, the relevant regularity results regarding the
parabolic double phase models are vary rare in the literature.

Influenced by the aforementioned works, we in the present paper introduce a
new model (1.1) by combining the normalized $p$-Laplacian operator with the
double phase gradient-diffusion. It is worthwhile mentioning that problem
(1.1) exhibits some novel and intriguing characteristics. In the cases that
$0<q\leq s$ and $-1<q\leq s<0$, (1.1) possesses the nonhomogeneous degeneracy
and singularity, respectively. Furthermore, (1.1) has both singularity and
degeneracy when $-1<q<0<s$. In particular, equation (1.1) incorporates
$p$-Laplacian in divergence form together with $p$-Laplacian in non-divergence
form simultaneously provided that $q=p-2$ and $s>q$, that is,

$\partial_{t}u-\operatorname{div}(|Du|^{p-2}Du)-a(x,t)|Du|^{s}\Delta_{p}^{N}u=0.$

Meanwhile, equation (1.1) is a natural extension of canonical quasi-linear
parabolic equations with singularity or degeneracy, whose highly celebrated
prototype is

$\partial_{t}u-|Du|^{q}\Delta_{p}^{N}u=0.$ (1.5)

Imbert-Jin-Silvestre [34] showed the interior $C^{1,\alpha}$-regularity of
viscosity solutions $u$ to (1.5) in $Q_{1}$, which states that

$\|Du\|_{C^{\alpha}(Q_{1/2})}\leq C$

and

$\sup_{\stackrel{{\scriptstyle(x,t),(x,s)\in Q_{1/2}}}{{t\neq
s}}}\frac{|u(x,t)-u(x,s)|}{|t-s|^{\frac{1+\alpha}{2-\alpha q}}}\leq C.$

Later, for the nonhomogeneous analogue,

$\partial_{t}u-|Du|^{q}\Delta_{p}^{N}u=f(x,t),$

the local $C^{1,\alpha}$-regularity of solutions was completed under the
assumption that $f$ is continuous and bounded; see [2] for the degenerate case
$q\geq 0$ and [6] for the singular case $-1<q<0$. Additionally, several extra
aspects of such equations have already been explored as well, such as
existence and uniqueness of solutions [15, 27], the comparison principles [32,
50], Aleksandrov-Bakelman-Pucci type estimate [1], parabolic Harnack’s
inequality [51]. For the related regularity results in the elliptic context,
we refer to [5, 12, 35] and the references therein.

In this work we make use of a unified geometrical method developed by Jin-
Silvestre [36] and Imbert-Jin-Silvestre[34] to study the interior Hölder
continuity for the spatial gradient of solutions to (1.1). The strategies of
the proof concentrate mainly on verifying that the oscillation of gradient is
declining in a shrinking sequence of parabolic cylinders, and then reducing
the iterative step to a dichotomy between two cases: either the gradient $Du$
stays close to a fixed unit vector $e$ for most points $(x,t)$ (in measure),
or it does not, and at last patching these two alternatives together. We shall
first prove the $C^{1,\alpha}$-regularity for solutions to (1.1) with the
strong restriction that $\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ is small (less
than 1). Then, by employing a scaling technique, we infer the $C^{1,\alpha}$
estimates for solutions of (1.1) under the assumption that $a(x,t)\in
C^{1}(Q_{1})$. Due to the presence of coefficient $a(x,t)$ and the fact that
the nonhomogeneous $(q,s)$-growth gradient-diffusion terms are intertwined in
equation (1.1), the theoretical analysis in the current study is radically
much more challenging than the previous ones. The significant distinctions and
difficulties are as follows. First, in order to obtain the improvement of
oscillation for $|Du|$, it is indispensable to incorporate more terms
involving $Da(x,t)$ when we differentiate the regularized equation (3.1) in
$x$-variable. Therefore, in comparison to the proof of Lemma 4.1 in [34], we
need additional elaborate analyses and construct a much more complicated
auxiliary function. Second, the comparison principle (Proposition 3.5) cannot
plainly follow from the known results due to the presence of $a(x,t)$. We have
to meticulously apply the information from the maximum principle for
semicontinuous functions, together with properties such as the local Lipschitz
continuity with respect to the matrix square root. Moreover, the Lipschitz
regularity of viscosity solutions to (1.1) plays a rather crucial role in the
proof. To the best of our knowledge, the comparison principle is new, which is
also of independent interest. Finally, in order to establish the Lipschitz
estimates for equation (3.1) in the spatial variable, we shall employ Ishii-
Lions’ method twice: we first use it to deduce the solutions are Hölder
continuity in $x$-variable, and then we rely on this Hölder regularity and use
the Ishii-Lions’ method again to show the Lipschitz estimates.

The paper is organized as follows. In Section 2, we first recall the
definition of viscosity solutions to (1.1) and then give the main result
derived in this paper. Section 3 contains the Lipschitz continuity in the
spatial variables, the Hölder continuity in the time variable as well as two
important properties of viscosity solutions. Section 4 is devoted to
establishing the Hölder estimates on the spatial gradients of solutions, which
is the most technically challenging part. We complete the proof of comparison
principle (Proposition 3.5) in Section 5. The technical proof of the Lipschitz
continuity in the spatial variables (Lemma 3.1) is postponed to Section 6. At
last, we in Section 7 present the proof of the boundary estimates, Proposition
4.11.

##  2\. Main result

The aim of this paper is to establish the interior Hölder estimates for
spatial gradients of solutions to problem (1.1). To this end, the following
hypotheses will be in force. We first assume that

$1<p<\infty.$ (2.1)

Furthermore, concerning the nonhomogeneous degeneracy or/and singularity term
appearing in (1.1), we shall require that the exponents $q,s$ fulfill

$-1<q\leq s<\infty,$ (2.2)

and that the modulating coefficient $a(\cdot)$ is such that

$0<a^{-}:=\inf_{Q_{1}}a(x,t)\leq a(x,t)\leq a^{+}:=\sup_{Q_{1}}a(x,t)<\infty,$
(2.3)

and

$a(x,t)\in C^{1}(Q_{1})\quad\text{and}\quad
A:=\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}<\infty.$ (2.4)

Throughout this article, the assumptions (2.1) and (2.2) are always supposed
to hold.

In the degenerate case (i.e., $q\geq 0$), the definition of viscosity
solutions is straightforward. Nonetheless, the formulations of defining
viscosity solutions cannot be displayed specifically for the case when
singularity occurs (i.e., $-1<q<0$). Hence we here adopt the same notion of
viscosity solutions as the one utilized in [38] to provide a unified way of
defining solutions in the degenerate and singular conditions. Next let us
recall the definition of viscosity solutions to (1.1).

######  Definition 2.1 (viscosity solution).

A finite almost everywhere and lower semicontinuous function
$u:Q_{1}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is a viscosity supersolution
to (1.1) in $Q_{1}$, if whenever $(x_{0},t_{0})\in Q_{1}$ and $\varphi\in
C^{2}(Q_{1})$ are such that $u-\varphi$ attains a local minimum at
$(x_{0},t_{0})$ and moreover $D\varphi(x,t)\neq 0$ for $x\neq x_{0}$, then we
obtain

$\limsup_{\stackrel{{\scriptstyle(x,t)\rightarrow(x_{0},t_{0})}}{{x\neq
x_{0}}}}\left(\partial_{t}\varphi(x,t)-[|D\varphi(x,t)|^{q}+a(x,t)|D\varphi(x,t)|^{s}]\Delta_{p}^{N}\varphi(x,t)\right)\geq
0.$

A finite almost everywhere and upper semicontinuous function
$u:Q_{1}\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is a viscosity subsolution to
(1.1) in $Q_{1}$, if whenever $(x_{0},t_{0})\in Q_{1}$ and $\varphi\in
C^{2}(Q_{1})$ are such that $u-\varphi$ reaches a local maximum at
$(x_{0},t_{0})$ and moreover $D\varphi(x,t)\neq 0$ for $x\neq x_{0}$, then we
derive

$\liminf_{\stackrel{{\scriptstyle(x,t)\rightarrow(x_{0},t_{0})}}{{x\neq
x_{0}}}}\left(\partial_{t}\varphi(x,t)-[|D\varphi(x,t)|^{q}+a(x,t)|D\varphi(x,t)|^{s}]\Delta_{p}^{N}\varphi(x,t)\right)\leq
0.$

A function $u$ is called a viscosity solution to (1.1) if and only if it is
both viscosity super- and subsolution.

######  Remark 2.2.

When $D\varphi(x_{0},t_{0})\neq 0$, these limits above are explicit,

$\partial_{t}\varphi(x_{0},t_{0})-[|D\varphi(x_{0},t_{0})|^{q}+a(x_{0},t_{0})|D\varphi(x_{0},t_{0})|^{s}]\Delta_{p}^{N}\varphi(x_{0},t_{0})\geq(\leq)0.$

In addition, when $0\leq q\leq s$, the condition that $D\varphi(x,t)\neq 0$
for $x\neq x_{0}$ can be removed actually. For example, if
$D\varphi(x_{0},t_{0})=0$, after careful computations, the supremum limit
turns into

  * (i)

the case that $q=s=0$,

$\begin{cases}\partial_{t}\varphi(x_{0},t_{0})-(1+a(x_{0},t_{0}))\left(\mathrm{tr}(D^{2}\varphi(x_{0},t_{0}))+(p-2)\lambda_{\rm
min}(D^{2}\varphi(x_{0},t_{0}))\right)\geq 0,&\text{{}}p\geq 2,\\\\[5.69054pt]
\partial_{t}\varphi(x_{0},t_{0})-(1+a(x_{0},t_{0}))\left(\mathrm{tr}(D^{2}\varphi(x_{0},t_{0}))+(p-2)\lambda_{\rm
max}(D^{2}\varphi(x_{0},t_{0}))\right)\geq 0,&\text{{}}p\in(1,2).\end{cases}$

  * (ii)

the case that $0=q<s$,

$\begin{cases}\partial_{t}\varphi(x_{0},t_{0})-\left(\mathrm{tr}(D^{2}\varphi(x_{0},t_{0}))+(p-2)\lambda_{\rm
min}(D^{2}\varphi(x_{0},t_{0}))\right)\geq 0,&\text{{when}}\quad p\geq
2,\\\\[5.69054pt]
\partial_{t}\varphi(x_{0},t_{0})-\left(\mathrm{tr}(D^{2}\varphi(x_{0},t_{0}))+(p-2)\lambda_{\rm
max}(D^{2}\varphi(x_{0},t_{0}))\right)\geq 0,&\text{{when}}\quad
p\in(1,2).\end{cases}$

  * (iii)

the case that $0<q\leq s$,

$\partial_{t}\varphi(x_{0},t_{0})\geq 0.$

Here $\mathrm{tr}(N)$ is the trace of matrix $N$, and $\lambda_{\rm min}(N)$
($\lambda_{\rm max}(N)$) denotes the minimum (maximum) eigenvalue of $N$.

The infimal limit in definition can be tackled analogously.

Now we are in position to state our main contribution of this work.

######  Theorem 2.3.

Let the conditions (2.1)–(2.4) be in force. Suppose that $u$ is a bounded
viscosity solution to equation (1.1) in $Q_{1}$. Then there are two constants
$\alpha\in(0,1)$ and $C>0$, both depending upon $n,p,q,s,a^{-},a^{+},A$ and
$\|u\|_{L^{\infty}(Q_{1})}$, such that the following estimates hold

$\|Du\|_{C^{\alpha}(Q_{1/2})}\leq C$

and

$\sup_{\stackrel{{\scriptstyle(x,t),(x,s)\in Q_{1/2}}}{{t\neq
s}}}\frac{|u(x,t)-u(x,s)|}{|t-s|^{\frac{1+\alpha}{2-\alpha q}}}\leq C.$

##  3\. Lower regularity for solutions

In order to circumvent some technical difficulties created by the lack of
smoothness of viscosity solutions to (1.1), we first study the regularized
equation below

$\partial_{t}u=\big{[}(|Du|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|Du|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{u_{i}u_{j}}{|Du|^{2}+\varepsilon^{2}}\right)u_{ij}$
(3.1)

in $Q_{1}$, where $0<\varepsilon<1$. Then we devote to obtaining uniform
estimates with respect to $\varepsilon$ so that we could pass to the limit in
the end.

In this section, we are going to show the Lipschitz continuity in the spatial
variables and the Hölder continuity in the time variable. Now we first present
the Lipschitz estimates independent of $\varepsilon$ on solutions to equation
(3.1). However, the proof of this lemma is rather long and delicate, which was
postponed to Section 6. It is worth mentioning that, from the proof below, we
can easily find that the Lipschitz estimates also hold true for
$\varepsilon=0$. We state this result as follows.

######  Lemma 3.1 (Local Lipschitz estimates in $x$-variable).

Let $\varepsilon\in[0,1)$ and the assumptions (2.1) and (2.2) be in force. Let
$u$ be a smooth solution to (3.1) in $Q_{1}$. Assume that $a(x,t)\geq a^{-}>0$
and $a(x,t)$ is uniformly Lipschitz continuous in $x$-variable, that is, there
exists a constant $C_{\rm lip}>0$, independent of $t$-variable, such that
$|a(x,t)-a(y,t)|\leq C_{\rm lip}|x-y|$. Then for all $r\in(0,\frac{7}{8}]$,
there holds that

$|u(x,t)-u(y,t)|\leq C|x-y|$

for $(x,t),(y,t)\in\overline{Q_{r}}$, where $C>0$ depends on
$n,p,q,s,a^{-},C_{\rm lip}$ and $\|u\|_{L^{\infty}(Q_{1})}$.

######  Remark 3.2.

From the proof of this lemma in Section 6, we can see the explicit
dependencies of the above constant $C$ with

$C:=C(n,p,q,s)\|u\|_{L^{\infty}(Q_{1})}\left[1+\left(\frac{C_{\rm
lip}}{a^{-}}\right)^{2}\right].$

If $a(x,t)$ is supposed to be of class $C^{1}(\overline{Q_{1}})$ in the
previous lemma, then the constant $C_{\rm lip}>0$ can be replaced by
$A:=\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$.

Based on the Lipschitz estimates above and a simple comparison argument, we
can demonstrate that the solutions to (3.1) are Hölder continuous in $t$,
which will be utilized in Lemma 4.8 below.

######  Lemma 3.3 (Local Hölder estimates in $t$-variable).

Suppose that $u$ is a smooth solution of (3.1) in $Q_{1}$ with
$0<\varepsilon<1$. Let $a(x,t)$ satisfy that $|a(x,t)-a(y,t)|\leq C_{\rm
lip}|x-y|$ in $Q_{1}$. Then under the assumptions (2.1)–(2.3), the following
estimates hold:

  * (i)

for $0\leq q\leq s$,

$\sup_{\stackrel{{\scriptstyle(x,t),(x,s)\in Q_{3/4}}}{{t\neq
s}}}\frac{|u(x,t)-u(x,s)|}{|t-s|^{\frac{1}{2}}}\leq C;$

  * (ii)

for $-1<q\leq s<0$ or $-1<q<0\leq s$,

$\sup_{\stackrel{{\scriptstyle(x,t),(x,s)\in Q_{3/4}}}{{t\neq
s}}}\frac{|u(x,t)-u(x,s)|}{|t-s|^{\frac{1}{\beta(1+s)-s}}}\leq C,$

where $\beta=\frac{q+2}{q+1}$ and $C$ depends on $n,p,q,s,a^{-},a^{+},C_{\rm
lip}$ and $\|u\|_{L^{\infty}(Q_{1})}$.

###### Proof.

Let $\beta\geq 2$ be determined later, according to three different scenarios,
i.e., $-1<q\leq s<0$, $-1<q<0\leq s$ and $0\leq q\leq s$. For all
$t_{0}\in\left[-\left(\frac{3}{4}\right)^{2},0\right)$ and $\eta>0$, we now
assert that there are two constants $L_{1},L_{2}>0$ such that

$u(x,t)-u(0,t_{0})\leq\eta+L_{1}(t-t_{0})+L_{2}|x|^{\beta}=:\varphi(x,t)$
(3.2)

for any $\overline{B_{3/4}}\times[t_{0},0]$. We first select $L_{2}\geq
2(\frac{4}{3})^{\beta}\|u\|_{L^{\infty}(Q_{1})}$ such that (3.2) holds for
$x\in\partial B_{3/4}$, and in turn take $L_{2}$ such that (3.2) holds for
$t=t_{0}$. That is to say, we can choose properly such $L_{2}>0$ that (3.2)
does hold on the boundary of $\overline{B_{3/4}}\times[t_{0},0]$. Indeed, due
to $u$ is Lipschitz continuous in the spatial variables, we may take

$L_{2}\geq\frac{\|Du\|^{\beta}_{L^{\infty}(Q_{7/8})}}{\eta^{\beta-1}}$

to guarantee that

$\eta+L_{2}|x|^{\beta}\geq\|Du\|_{L^{\infty}(Q_{7/8})}|x|$

by Young’s inequality, which implies that (3.2) is true for $t=t_{0}$. Here we
note that $\|Du\|_{L^{\infty}(Q_{7/8})}$ is bounded depending on
$n,p,q,s,a^{-},C_{\rm lip}$ and $\|u\|_{L^{\infty}(Q_{1})}$. In the rest of
proof, we fix

$L_{2}=\eta^{1-\beta}\|Du\|^{\beta}_{L^{\infty}(Q_{7/8})}+2\left(\frac{4}{3}\right)^{\beta}\|u\|_{L^{\infty}(Q_{1})}+1.$

Next, we are ready to select $L_{1}$ such that $\varphi(x,t)$ is a
supersolution to certain equations. Inequality (3.2) then follows by the
comparison principle. The remaining proof is completed under three diverse
cases.

Case 1. $-1<q\leq s<0$. We shall show that $\varphi(x,t)$ is a supersolution
to equation (3.1), that is,

$\partial_{t}\varphi-\big{[}(|D\varphi|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|D\varphi|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{\varphi_{i}\varphi_{j}}{|D\varphi|^{2}+\varepsilon^{2}}\right)\varphi_{ij}\geq
0.$ (3.3)

We first calculate

$D\varphi=\beta L_{2}|x|^{\beta-2}x,$ $D^{2}\varphi=\beta
L_{2}|x|^{\beta-2}I+\beta(\beta-2)L_{2}|x|^{\beta-2}\frac{x}{|x|}\otimes\frac{x}{|x|},$

where it is easy to see that $D^{2}\varphi$ is a positive definite matrix and

$\|D^{2}\varphi\|\leq\beta(\beta-1)L_{2}|x|^{\beta-2}.$ (3.4)

Here $\xi\otimes\xi$ is the matrix with entries $\xi_{i}\xi_{j}$ for a vector
$\xi\in\mathbb{R}^{n}$. We next evaluate

$\displaystyle\quad\big{[}(|D\varphi|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|D\varphi|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{\varphi_{i}\varphi_{j}}{|D\varphi|^{2}+\varepsilon^{2}}\right)\varphi_{ij}$
$\displaystyle\leq C(n,p)(|D\varphi|^{q}+a^{+}|D\varphi|^{s})\|D^{2}\varphi\|$
$\displaystyle\leq
C\left(L^{1+q}_{2}|x|^{q(\beta-1)+\beta-2}+L^{1+s}_{2}|x|^{s(\beta-1)+\beta-2}\right)$
$\displaystyle\leq C(L^{1+q}_{2}+L^{1+s}_{2})$

by taking $\beta\geq\frac{q+2}{q+1}(>2)$, where $C$ depends only on
$n,p,q,s,a^{+}$. Thereby, in order to assure (3.3), we need to fix
$L_{1}=C(L^{1+q}_{2}+L^{1+s}_{2})$.

Finally, applying the comparison principle together with the choices of
$L_{1},L_{2}$, we arrive at

$\displaystyle\quad u(0,t)-u(0,t_{0})$ $\displaystyle\leq\eta+L_{1}(t-t_{0})$
$\displaystyle\leq\eta+C\left(\eta^{1-\beta}\|Du\|^{\beta}_{L^{\infty}(Q_{7/8})}+2(4/3)^{\beta}\|u\|_{L^{\infty}(Q_{1})}+1\right)^{1+q}(t-t_{0})$
$\displaystyle\quad+C\left(\eta^{1-\beta}\|Du\|^{\beta}_{L^{\infty}(Q_{7/8})}+2(4/3)^{\beta}\|u\|_{L^{\infty}(Q_{1})}+1\right)^{1+s}(t-t_{0})$
$\displaystyle\leq\eta+C\|Du\|^{\beta(1+q)}_{L^{\infty}(Q_{7/8})}\eta^{(1-\beta)(1+q)}|t-t_{0}|+C\|Du\|^{\beta(1+s)}_{L^{\infty}(Q_{7/8})}\eta^{(1-\beta)(1+s)}|t-t_{0}|$
$\displaystyle\quad+C(\|u\|_{L^{\infty}(Q_{1})}+1)^{1+s}|t-t_{0}|.$

We now pick $\eta=|t-t_{0}|^{\gamma}$ with $0<\gamma<1$ to be fixed later.
Then it follows that

$\displaystyle\quad u(0,t)-u(0,t_{0})$
$\displaystyle\leq|t-t_{0}|^{\gamma}+C\|Du\|^{\beta(1+q)}_{L^{\infty}(Q_{7/8})}|t-t_{0}|^{\gamma(1-\beta)(1+q)+1}$
$\displaystyle\quad+C\|Du\|^{\beta(1+s)}_{L^{\infty}(Q_{7/8})}|t-t_{0}|^{\gamma(1-\beta)(1+s)+1}+C(\|u\|_{L^{\infty}(Q_{1})}+1)^{1+s}|t-t_{0}|.$

Then $\gamma$ can be chosen as

$\begin{cases}\gamma(1-\beta)(1+q)+1-\gamma\geq 0,\\\\[5.69054pt]
\gamma(1-\beta)(1+s)+1-\gamma\geq 0,\\\\[5.69054pt] 0<\gamma<1,\end{cases}$

which leads to

$\gamma\leq\frac{1}{\beta(1+s)-s}(<1)$

by noting that $-1<q\leq s<0$. As has been stated above, we could determine
$\beta=\frac{q+2}{q+1}$ and $\gamma=\frac{1}{\beta(1+s)-s}$. We then get the
desired result for the case that $-1<q\leq s<0$.

Case 2. $-1<q<0\leq s$. Similarly to Case 1, we get

$\displaystyle\quad\big{[}(|D\varphi|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|D\varphi|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{\varphi_{i}\varphi_{j}}{|D\varphi|^{2}+\varepsilon^{2}}\right)\varphi_{ij}$
$\displaystyle\leq
C(n,p)(|D\varphi|^{q}+a^{+}(|D\varphi|^{s}+1))\|D^{2}\varphi\|$
$\displaystyle\leq
C\left(L^{1+q}_{2}|x|^{q(\beta-1)+\beta-2}+L_{2}|x|^{\beta-2}+L^{1+s}_{2}|x|^{s(\beta-1)+\beta-2}\right)$
$\displaystyle\leq C(L^{1+q}_{2}+L_{2}+L^{1+s}_{2})$

by taking $\beta\geq\frac{q+2}{q+1}$, where $C$ depends only on $n,p,q,s$ and
$a^{+}$. Hence we can choose $L_{1}=C(L^{1+q}_{2}+L_{2}+L^{1+s}_{2})$ to
ensure (3.3).

Then through the comparison principle and the choices of $L_{1},L_{2}$, it
yields that

$\displaystyle\quad u(0,t)-u(0,t_{0})$ $\displaystyle\leq\eta+L_{1}(t-t_{0})$
$\displaystyle\leq\eta+C\|Du\|^{\beta(1+q)}_{L^{\infty}(Q_{7/8})}\eta^{(1-\beta)(1+q)}|t-t_{0}|+C\|Du\|^{\beta}_{L^{\infty}(Q_{7/8})}\eta^{1-\beta}|t-t_{0}|$
$\displaystyle\quad+C\|Du\|^{\beta(1+s)}_{L^{\infty}(Q_{7/8})}\eta^{(1-\beta)(1+s)}|t-t_{0}|+C(\|u\|_{L^{\infty}(Q_{1})}+1)^{1+s}|t-t_{0}|.$

Let $\eta=|t-t_{0}|^{\gamma}$. The above display then becomes

$\displaystyle\quad u(0,t)-u(0,t_{0})$
$\displaystyle\leq|t-t_{0}|^{\gamma}+C\|Du\|^{\beta(1+q)}_{L^{\infty}(Q_{7/8})}|t-t_{0}|^{\gamma(1-\beta)(1+q)+1}+C\|Du\|^{\beta}_{L^{\infty}(Q_{7/8})}|t-t_{0}|^{\gamma(1-\beta)+1}$
$\displaystyle\quad+C\|Du\|^{\beta(1+s)}_{L^{\infty}(Q_{7/8})}|t-t_{0}|^{\gamma(1-\beta)(1+s)+1}+C(\|u\|_{L^{\infty}(Q_{1})}+1)^{1+s}|t-t_{0}|.$

We shall pick $0<\gamma<1$ such that

$\begin{cases}\gamma(1-\beta)(1+q)+1-\gamma\geq 0,\\\\[5.69054pt]
\gamma(1-\beta)+1-\gamma\geq 0,\\\\[5.69054pt]
\gamma(1-\beta)(1+s)+1-\gamma\geq 0,\end{cases}$

i.e.,

$\gamma\leq\frac{1}{\beta(1+s)-s}.$

Consequently, we can fix $\beta=\frac{q+2}{q+1}$ and
$\gamma=\frac{1}{\beta(1+s)-s}$. We then finish the proof for the case that
$-1<q<0\leq s$.

Case 3. $0\leq q\leq s$. This time, we can verify that $\varphi(x,t)$ is a
supersolution of a linear parabolic equation with coefficients depending on
$u$. That is,

$\partial_{t}\varphi-\big{[}(|Du|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|Du|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{u_{i}u_{j}}{|Du|^{2}+\varepsilon^{2}}\right)\varphi_{ij}\geq
0.$

Because $q\geq 0$ and $|Du|$ is known to be bounded by the Lipschitz
continuity, we can rewrite this display as

$\partial_{t}\varphi-a_{ij}(x,t)\varphi_{ij}\geq 0,$ (3.5)

where

$|a_{ij}(x,t)|\leq
C(p)((\|Du\|_{L^{\infty}(Q_{7/8})}+\varepsilon)^{q}+a^{+}(\|Du\|_{L^{\infty}(Q_{7/8})}+\varepsilon)^{s}).$
(3.6)

Here the boundedness on $|a_{ij}(x,t)|$ depends on $n,p,q,s,a^{-},a^{+},C_{\rm
lip}$ and $\|u\|_{L^{\infty}(Q_{1})}$ actually. We fix $\beta=2$. Thus by
(3.4) and (3.6), we take

$L_{1}=C(\|Du\|_{L^{\infty}(Q_{7/8})}+1)^{s}L_{2},$

which makes (3.5) hold true. In turn, utilizing the comparison principle
again, we have

$u(0,t)-u(0,t_{0})\leq\eta+C(\|Du\|_{L^{\infty}(Q_{7/8})}+1)^{s}(\eta^{-1}\|Du\|^{2}_{L^{\infty}(Q_{7/8})}+\|u\|_{L^{\infty}(Q_{1})}+1)|t-t_{0}|.$

Taking

$\eta=(\|Du\|_{L^{\infty}(Q_{7/8})}+1)^{\frac{s}{2}+1}|t-t_{0}|^{\frac{1}{2}},$

we derive

$\displaystyle u(0,t)-u(0,t_{0})$
$\displaystyle\leq(\|Du\|_{L^{\infty}(Q_{7/8})}+1)^{\frac{s}{2}+1}|t-t_{0}|^{\frac{1}{2}}$
$\displaystyle\quad+C(\|Du\|_{L^{\infty}(Q_{7/8})}+1)^{s}(\|u\|_{L^{\infty}(Q_{1})}+1)|t-t_{0}|.$

This proof now is finished. ∎

Next we end this section by presenting two important properties of viscosity
solutions, comparison principle and stability, which will be exploited in the
proof of Theorem 4.13 below. However, their proof shall be postponed to
Section 5.

######  Proposition 3.4 (Stability).

Assume that $\\{u_{i}\\}$ is a sequence of viscosity solutions to (3.1) in
$Q_{1}$ with $\varepsilon_{i}\geq 0$ such that $\varepsilon_{i}\rightarrow 0$.
Let $u_{i}$ converge to $u$ locally uniformly in $Q_{1}$. Then we can infer
that $u$ is a viscosity solution to (1.1) in $Q_{1}$.

Once deriving the Lipschitz continuity of solutions to equation (1.1), we can
show the following comparison principle that is interesting by itself.

######  Proposition 3.5 (Comparison principle).

Let the function $a(x,t)>0$ in (1.1) be Lipschitz continuous in time-space
variables. Assume that $u$ and $v$ are a viscosity subsolution and a locally
uniformly Lipschitz continuous viscosity supersolution in $x$-variable to
(1.1) in $Q_{1}$, respectively. If $u\leq v$ on $\partial_{p}Q_{1}$, then
there holds that

$u\leq v\quad\text{in }Q_{1}.$

##  4\. Hölder estimates on the spatial gradients

In this section, we assume that $a(x,t)\in C^{1}(Q_{1})$ and
$A:=\|D_{x,t}a\|_{L^{\infty}(Q_{1})}<\infty$. Since Lemma 3.1 states that the
solutions to (3.1) in $Q_{1}$ have uniform interior Lipschitz estimates in
$x$-variable (which is independent of $\varepsilon\in[0,1)$), we can see that

$\|Du\|_{L^{\infty}(Q_{7/8})}\leq
C(n,p,q,s,a^{-},A,\|u\|_{L^{\infty}(Q_{1})}).$

Here we refer to Lemma 3.1 for the explicit dependencies of parameters. In
what follows, we may suppose that $Du$ is bounded in $Q_{1}$ for convenience,
as we could obtain these conclusions in the preceding section in a larger
domain such as $Q_{2}$.

We are going to establish the Hölder estimates on $Du$ at the origin $(0,0)$,
and then deduce plainly the interior Hölder continuity of $Du$ by standard
translation arguments. The idea of this proof is analogous to that in [34,
36], but there exist many extra delicate difficulties caused by the
coefficient $a(x,t)$ and the $(q,s)$-growth. We will consider the so-called
intrinsic (re-scaled) parabolic cylinder defined as

$Q^{\rho}_{r}=B_{r}\times(-\rho^{-q}r^{2},0]\quad\text{with }r,\rho>0.$

The same family of parabolic cylinders $Q^{\rho}_{r}$ has been utilized in
[28]. If $u$ solves (3.1) in $Q^{\rho}_{r}$ and we denote
$v(x,t)=\frac{1}{\rho r}u(rx,\rho^{-q}r^{2}t)$ with $(x,t)\in Q_{1}$, then it
is easy to check that

$\partial_{t}v=\big{[}(|Dv|^{2}+\overline{\varepsilon}^{2})^{\frac{q}{2}}+\overline{a}(x,t)(|Dv|^{2}+\overline{\varepsilon}^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{v_{i}v_{j}}{|Dv|^{2}+\overline{\varepsilon}^{2}}\right)v_{ij}$
(4.1)

in $Q_{1}$, where

$\overline{a}(x,t)=\rho^{s-q}a(rx,\rho^{-q}r^{2}t)\quad\text{and}\quad\overline{\varepsilon}=\varepsilon\rho^{-1}.$

Particularly, it is noteworthy that if $u$ solves (1.1) in $Q^{\rho}_{r}$,
then $v$ (defined as before) is a solution to

$\partial_{t}v=[|Dv|^{q}+\overline{a}(x,t)|Dv|^{s}]\left(\delta_{ij}+(p-2)\frac{v_{i}v_{j}}{|Dv|^{2}}\right)v_{ij}\quad\text{in
}Q_{1}.$ (4.2)

Hence when we fix $\rho\geq\|Du\|_{L^{\infty}(Q_{1})}+1$, we know that the
solutions of (4.1) or (4.2) satisfy $|Dv|\leq 1$ in $Q_{1}$. Therefore, in the
sequel, we may suppose that the solutions to (3.1) fulfill $|Du|\leq 1$ in
$Q_{1}$. Next, we proceed with considering (3.1) and investigate the Hölder
continuity of gradients of its solutions, from which we can derive the higher
regularity (Theorem 2.3) for the solutions to (1.1) (by sending
$\varepsilon\rightarrow 0$). To this end, we first show the Hölder estimates
on the gradients of solutions to (1.1) under the assumption that
$\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ ($\leq 1$) is small. In turn, by doing
a scaling work, we eventually demonstrate the Hölder regularity for the
gradients of solutions to (1.1) under the condition that
$\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ is finite, that is, $|D_{x,t}a(x,t)|$
exhibits a general bound in $Q_{1}$.

###  4.1. Hölder regularity of spatial gradients in the case that
$\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ is small

We may assume $\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}\leq 1$. Now we are ready
to verify that when the projection of $Du$ onto the direction
$e\in\mathbb{S}^{n-1}$ (i.e., $|e|=1$) is away from 1 in a large portion of
$Q_{1}$, then in a smaller cylinder the inner product $Du\cdot e$ has improved
oscillation.

######  Lemma 4.1.

Let the conditions (2.1) and (2.2) be in force. Assume that $u$ is a smooth
solution of (3.1) with $\varepsilon\in(0,1)$ such that $|Du|\leq 1$ in
$Q_{1}$. For each $l\in(\frac{1}{2},1)$ and $\mu>0$, if $0\leq a(x,t)\in
C^{1}(Q_{1})$ and $\|Da\|_{L^{\infty}(Q_{1})}\leq\kappa$, where
$\kappa\in(0,1]$ is a sufficiently small constant depending on
$n,p,q,s,a^{+},\mu$ and $l$, then there is $\tau_{0}\in(0,\frac{1}{4})$ only
depending on $n,\mu$ and there are $\tau,\delta>0$ depending upon
$n,p,q,s,a^{+},\mu$ and $l$ such that for arbitrary $e\in\mathbb{S}^{n-1}$ if

$|\\{(x,t)\in Q_{1}:Du\cdot e\leq l\\}|>\mu|Q_{1}|,$

one has

$Du\cdot e<1-\delta\quad\text{in }Q^{1-\delta}_{\tau}$

with $Q^{1-\delta}_{\tau}\subset Q_{\tau_{0}}$.

###### Proof.

Set

$a_{ij}(x,t,\eta)=\big{[}(|\eta|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|\eta|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{\eta_{i}\eta_{j}}{|\eta|^{2}+\varepsilon^{2}}\right)$
(4.3)

with $\eta\in\mathbb{R}^{n}$ and

$a_{ij,m}(x,t,\eta):=\frac{\partial a_{ij}(x,t,\eta)}{\partial\eta_{m}},$

where $\eta_{i}$ denotes the $i$-th component of $\eta$. By differentiating
equation (3.1) in $x_{k}$, we get

$\displaystyle\partial_{t}(u_{k})$
$\displaystyle=a_{ij}(x,t,Du)(u_{k})_{ij}+a_{ij,m}(x,t,Du)u_{ij}(u_{k})_{m}$
$\displaystyle\quad+\partial_{k}a(x,t)(|Du|^{2}+\varepsilon^{2})^{\frac{s}{2}}\left(\delta_{ij}+(p-2)\frac{u_{i}u_{j}}{|Du|^{2}+\varepsilon^{2}}\right)u_{ij}.$

In the rest of proof, let

$b_{ij}(Du)=(|Du|^{2}+\varepsilon^{2})^{\frac{s}{2}}\left(\delta_{ij}+(p-2)\frac{u_{i}u_{j}}{|Du|^{2}+\varepsilon^{2}}\right).$

We further have

$\displaystyle\partial_{t}(Du\cdot e-l)$ $\displaystyle=a_{ij}(x,t,Du)(Du\cdot
e-l)_{ij}+a_{ij,m}(x,t,Du)u_{ij}(Du\cdot e-l)_{m}$ $\displaystyle\quad+Da\cdot
eb_{ij}(Du)u_{ij}.$

Let $h=|Du|^{2}$. Then

$\partial_{t}h=2Du\cdot Du_{t},\quad h_{i}=2Du\cdot Du_{i},$
$h_{ij}=2Du_{j}\cdot Du_{i}+2Du\cdot Du_{ij}.$

By direct calculation, it yields that

$\displaystyle\partial_{t}h$
$\displaystyle=a_{ij}(x,t,Du)h_{ij}+a_{ij,m}(x,t,Du)u_{ij}h_{m}+2Da\cdot
Dub_{ij}(Du)u_{ij}$ $\displaystyle\quad-2a_{ij}(x,t,Du)u_{ki}u_{kj}.$

For $\rho=\frac{l}{4}$, define

$w=(Du\cdot e-l+\rho|Du|^{2})_{+}$

with $(f)_{+}:=\max\\{0,f\\}$. In the region $\Omega_{+}:=\\{(x,t)\in
Q_{1}:w>0\\}$, we arrive at

$\displaystyle\partial_{t}w$
$\displaystyle=a_{ij}(x,t,Du)w_{ij}+a_{ij,m}(x,t,Du)u_{ij}w_{m}+Da\cdot(e+2\rho
Du)b_{ij}(Du)u_{ij}$ $\displaystyle\quad-2\rho a_{ij}(x,t,Du)u_{ki}u_{kj}.$
(4.4)

Observe that

$\displaystyle\quad a_{ij,m}(x,t,\eta)$
$\displaystyle=\big{[}q(|\eta|^{2}+\varepsilon^{2})^{\frac{q}{2}-1}\eta_{m}+sa(x,t)(|\eta|^{2}+\varepsilon^{2})^{\frac{s}{2}-1}\eta_{m}\big{]}\left(\delta_{ij}+(p-2)\frac{\eta_{i}\eta_{j}}{|\eta|^{2}+\varepsilon^{2}}\right)$
$\displaystyle\quad+\big{[}(|\eta|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|\eta|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}(p-2)\left(\frac{\delta_{im}\eta_{j}+\delta_{jm}\eta_{i}}{|\eta|^{2}+\varepsilon^{2}}-\frac{2\eta_{i}\eta_{j}\eta_{m}}{(|\eta|^{2}+\varepsilon^{2})^{2}}\right).$

Due to $|Du|>\frac{l}{2}$ in $\Omega_{+}$ and $|Du|\leq 1$, it follows that,
in $\Omega_{+}$,

$|a_{ij,m}(x,t,Du)|\leq\begin{cases}Cl^{-1}&{\text{ if }q\geq
0},\\\\[5.69054pt] Cl^{q-1}&{\text{ if}-1<q<0},\end{cases}$ (4.5)

where $C$ depends only on $p,q,s,a^{+}$. We then calculate

$\displaystyle\quad Da\cdot(e+2\rho Du)b_{ij}(Du)u_{ij}$
$\displaystyle\leq(1+2\rho)H(|Du|^{2}+\varepsilon^{2})^{\frac{s}{2}}\left|\mathrm{tr}(D^{2}u)+(p-2)(|Du|^{2}+\varepsilon^{2})^{-1}\langle
D^{2}uDu,Du\rangle\right|$ $\displaystyle\leq
3(n+|p-2|)H\|D^{2}u\|(|Du|^{2}+\varepsilon^{2})^{\frac{s}{2}}$
$\displaystyle\leq\begin{cases}CH\|D^{2}u\|&{\text{if }s\geq
0},\\\\[5.69054pt] CH\|D^{2}u\|l^{s}&{\text{if }-1<s<0},\end{cases}$ (4.6)

where $H:=\|Da\|_{L^{\infty}(Q_{1})}$ and $C$ depends only on $n,p,s$. We next
estimate the term $2\rho a_{ij}(x,t,Du)u_{ki}u_{kj}$ as

$\displaystyle\quad
2\rho\big{[}(|Du|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|Du|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{u_{i}u_{j}}{|Du|^{2}+\varepsilon^{2}}\right)u_{ki}u_{kj}$
$\displaystyle=2\rho\big{[}(|Du|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|Du|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}\left(\|D^{2}u\|^{2}+(p-2)\frac{|D^{2}uDu|^{2}}{|Du|^{2}+\varepsilon^{2}}\right)$
$\displaystyle\geq
2\min\\{1,p-1\\}\rho(|Du|^{2}+\varepsilon^{2})^{\frac{q}{2}}\|D^{2}u\|^{2}$
$\displaystyle\geq\begin{cases}C\rho l^{q}\|D^{2}u\|^{2}&{\text{if }q\geq
0},\\\\[5.69054pt] C\rho\|D^{2}u\|^{2}&{\text{if }-1<q<0},\end{cases}$ (4.7)

where $C$ depends only on $p,q$. Hence merging these estimates (4.1)–(4.1) and
using Cauchy-Schwarz inequality, for $q\geq 0$, in $\Omega_{+}$ we derive

$\displaystyle\partial_{t}w$ $\displaystyle\leq
a_{ij}(x,t,Du)w_{ij}+Cl^{-1}|Dw|\sum^{n}_{i,j}|u_{ij}|+CH\|D^{2}u\|-Cl^{q+1}\|D^{2}u\|^{2}$
$\displaystyle\leq
a_{ij}(x,t,Du)w_{ij}+\epsilon\|D^{2}u\|^{2}+\frac{C^{2}}{\epsilon
l^{2}}|Dw|^{2}+\epsilon\|D^{2}u\|^{2}+\frac{C^{2}H^{2}}{\epsilon}-Cl^{q+1}\|D^{2}u\|^{2}$
$\displaystyle\leq
a_{ij}(x,t,Du)w_{ij}+Cl^{-q-3}|Dw|^{2}+\widehat{C}l^{-q-1}H^{2},$

by choosing $\epsilon=\frac{1}{2}Cl^{q+1}$. For $-1<q<0\leq s$, in
$\Omega_{+}$ we have

$\displaystyle\partial_{t}w$ $\displaystyle\leq
a_{ij}(x,t,Du)w_{ij}+Cl^{q-1}|Dw|\sum^{n}_{i,j}|u_{ij}|+CH\|D^{2}u\|-C\rho\|D^{2}u\|^{2}$
$\displaystyle\leq
a_{ij}(x,t,Du)w_{ij}+\epsilon\|D^{2}u\|^{2}+\frac{C^{2}}{\epsilon
l^{2(1-q)}}|Dw|^{2}+\epsilon\|D^{2}u\|^{2}+\frac{C^{2}H^{2}}{\epsilon}-C\rho\|D^{2}u\|^{2}$
$\displaystyle\leq
a_{ij}(x,t,Du)w_{ij}+Cl^{2q-3}|Dw|^{2}+\widehat{C}l^{-1}H^{2},$

by letting $\epsilon=\frac{1}{2}C\rho$. Finally, for $-1<s<0$, in $\Omega_{+}$
we get

$\displaystyle\partial_{t}w$ $\displaystyle\leq
a_{ij}(x,t,Du)w_{ij}+Cl^{q-1}|Dw|\sum^{n}_{i,j}|u_{ij}|+CH\|D^{2}u\|l^{s}-C\rho\|D^{2}u\|^{2}$
$\displaystyle\leq
a_{ij}(x,t,Du)w_{ij}+\epsilon\|D^{2}u\|^{2}+\frac{C^{2}}{\epsilon
l^{2(1-q)}}|Dw|^{2}+\epsilon\|D^{2}u\|^{2}+\frac{C^{2}H^{2}}{\epsilon
l^{-2s}}-C\rho\|D^{2}u\|^{2}$ $\displaystyle\leq
a_{ij}(x,t,Du)w_{ij}+Cl^{2q-3}|Dw|^{2}+\widehat{C}l^{2s-1}H^{2},$

by selecting $\epsilon=\frac{1}{2}C\rho$ again. In the previous formulations,
the constants $C$ and $\widehat{C}$ separately depend on $n,p,q,s,a^{+}$ and
$n,p,q,s$. Therefore, we can see that $w$ satisfies in the viscosity sense
that

$\partial_{t}w\leq\tilde{a}_{ij}w_{ij}+C_{1}(l)|Dw|^{2}+C_{2}(l)H^{2},$ (4.8)

where

$\tilde{a}_{ij}(x,t)=\begin{cases}a_{ij}(x,t,Du(x,t))&{\text{if
}(x,t)\in\Omega_{+}},\\\\[5.69054pt]
\delta_{ij}&{\text{elsewhere}}\end{cases}$

and

$C_{1}(l)=\begin{cases}c_{1}l^{-q-3}&{\text{if }q\geq 0},\\\\[5.69054pt]
c_{1}l^{2q-3}&{\text{if }-1<q<0},\end{cases}$

$C_{2}(l)=\begin{cases}c_{2}l^{-q-1}&{\text{if }q\geq 0},\\\\[5.69054pt]
c_{2}l^{-1}&{\text{if }-1<q<0\leq s},\\\\[5.69054pt] c_{2}l^{2s-1}&{\text{if
}-1<s<0}\end{cases}$

with $c_{1}$ depending on $n,p,q,s,a^{+}$ and $c_{2}$ depending only on
$n,p,q,s$. Here we notice that, since $l\in(\frac{1}{2},1)$, the coefficient
$\tilde{a}_{ij}(x,t)$ is uniformly parabolic, i.e., there exist two constants
$0<\lambda\leq\Lambda<\infty$ such that $\lambda
I\leq\tilde{a}_{ij}(x,t)\leq\Lambda I$ for all $(x,t)\in Q_{1}$. Indeed, we
find that $\lambda$ depends only on $p,q$ and $\Lambda$ depends on $p,q,s$ and
$a^{+}$. For simplicity, (4.8) is reformulated as

$\partial_{t}w\leq\tilde{a}_{ij}w_{ij}+C_{1}(l)|Dw|^{2}+\overline{c},$

with $\overline{c}\equiv C_{2}(l)H^{2}$. Set

$W=1-l+\rho+\overline{c},$

and

$U=\frac{1}{\nu}\left(1-e^{\nu(w-\overline{c}t-W)}\right)\quad\text{with
}\nu>0.$

We could determine $\nu>0$, which depends on $n,p,q,s,a^{+}$ and $l$, such
that

$\partial_{t}U\geq\tilde{a}_{ij}U_{ij}\quad\text{in }Q_{1}$

in the viscosity sense. Obviously, $U\geq 0$ in $Q_{1}$.

If $Du\cdot e\leq l$, then it follows from the assumption in the statement
that

$|\\{(x,t)\in Q_{1}:U\geq\nu^{-1}(1-e^{\nu(l-1)})\\}|>\mu|Q_{1}|.$

Thereby, we can infer from Proposition 2.3 in [36] that there are two
constants $\tau_{0},\gamma_{0}>0$ such that

$U\geq\nu^{-1}(1-e^{\nu(l-1)})\gamma_{0}\quad\text{in }Q_{\tau_{0}},$

where $\tau_{0}$ depends only on $n,\mu$ and $\gamma_{0}$ depends on
$n,\mu,p,q,s$ and $a^{+}$. Moreover, since $w-\overline{c}t\leq W$, we could
readily get

$U\leq W-w+\overline{c}t.$

Thus, in $Q_{\tau_{0}}$ we have

$Du\cdot e+\rho|Du|^{2}\leq
1+\rho-\nu^{-1}(1-e^{\nu(l-1)})\gamma_{0}+\overline{c}+\overline{c}t,$

which leads to

$Du\cdot e+\rho(Du\cdot e)^{2}\leq
1+\rho-\nu^{-1}(1-e^{\nu(l-1)})\gamma_{0}+\overline{c}.$

It yields that

$Du\cdot
e\leq\frac{-1+\sqrt{1+4\rho(1+\rho-\nu^{-1}(1-e^{\nu(l-1)})\gamma_{0}+\overline{c})}}{2\rho}\quad\text{in
}Q_{\tau_{0}}.$

If

$H<(C_{2}^{-1}(l)\nu^{-1}(1-e^{\nu(l-1)})\gamma_{0})^{\frac{1}{2}}=:\kappa(\leq
1),$

which implies that $\overline{c}<\nu^{-1}(1-e^{\nu(l-1)})\gamma_{0}$, then
there holds that

$Du\cdot e\leq 1-\delta\quad\text{in }Q_{\tau_{0}},$

where $\delta>0$ depends on $n,p,q,s,a^{+},\mu$ and $l$. Briefly, when the
upper bound on $|Da(x,t)|$ is small enough depending on $n,p,q,s,a^{+},\mu$
and $l$, we arrive at $Du\cdot e\leq 1-\delta$ in $Q_{\tau_{0}}$. Finally, we
select

$\tau=\begin{cases}\tau_{0}(1-\delta)^{\frac{q}{2}}&{\text{if }q\geq
0},\\\\[5.69054pt] \tau_{0}&{\text{if }-1<q<0}\end{cases}$

such that $Q_{\tau}^{1-\delta}\subset Q_{\tau_{0}}$. We now complete the
proof. ∎

######  Remark 4.2.

Observe that the selection of $\tau$ and $\delta$ above implies that

$\tau<(1-\delta)^{\frac{q}{2}}\quad\text{when }q\geq 0.$

In the rest of this work, we shall choose such smaller $\tau$ that

$\tau<(1-\delta)^{1+q}\quad\text{for any }q\geq-1.$ (4.9)

If Lemma 4.1 holds true in all directions $e\in\mathbb{S}^{n-1}$, then it in
effect indicates a reduction in the oscillation of $Du$ in a smaller parabolic
cylinder. This content is stated by the forthcoming corollary.

######  Corollary 4.3.

Let the conditions (2.1) and (2.2) be in force. Suppose that $u$ is a smooth
solution of (3.1) with $\varepsilon\in(0,1)$ such that $|Du|\leq 1$ in
$Q_{1}$. For each $l\in(0,1)$ and $\mu>0$, if $0\leq a(x,t)\in C^{1}(Q_{1})$
and $\|Da\|_{L^{\infty}(Q_{1})}\leq\kappa$, where $\kappa\in(0,1]$ is a small
enough quantity depending on $n,p,q,s,a^{+},\mu$ and $l$, then there exist
$\tau\in(0,\frac{1}{4})$ and $\delta>0$ that both depend upon
$n,p,q,s,a^{+},\mu$ and $l$ such that for all nonnegative integer
$k\leq\log\varepsilon/\log(1-\delta)$ if

$\left|\left\\{(x,t)\in Q_{\tau^{i}}^{(1-\delta)^{i}}:Du\cdot e\leq
l(1-\delta)^{i}\right\\}\right|>\mu\left|Q_{\tau^{i}}^{(1-\delta)^{i}}\right|$
(4.10)

for all $e\in\mathbb{S}^{n-1}$ and $i=0,1,2,\cdots,k$, then one has

$|Du|<(1-\delta)^{i+1}$

in $Q_{\tau^{i+1}}^{(1-\delta)^{i+1}}$ for $i=0,1,2,\cdots,k$.

######  Remark 4.4.

Notice that we could further impose on $\delta$ that $\delta<\frac{1}{2}$ and
$\delta<1-\tau$.

###### Proof.

Argue by induction. If $i=0$, it follows from Lemma 4.1 that $Du\cdot
e<1-\delta$ in $Q_{\tau}^{1-\delta}$ for all $e\in\mathbb{S}^{n-1}$, which
leads to $|Du|<1-\delta$ in $Q_{\tau}^{1-\delta}$. We now suppose that this
claim holds true for $i=0,1,2,\cdots,k-1$. Next, we shall verify it for $i=k$.
Define

$w(x,t)=\frac{1}{\tau^{k}(1-\delta)^{k}}u(\tau^{k}x,\tau^{2k}(1-\delta)^{-kq}t),\quad(x,t)\in
Q_{1}.$

We can readily check that $w$ solves in the viscosity sense

$\displaystyle\partial_{t}w=\big{[}(|Dw|^{2}+\hat{\varepsilon}^{2})^{\frac{q}{2}}+\hat{a}(x,t)(|Dw|^{2}+\hat{\varepsilon}^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{w_{i}w_{j}}{|Dw|^{2}+\hat{\varepsilon}^{2}}\right)w_{ij}$

in $Q_{1}$, where

$\hat{a}(x,t)=(1-\delta)^{k(s-q)}a(\tau^{k}x,\tau^{2k}(1-\delta)^{-qk}t)\quad\text{and}\quad\hat{\varepsilon}^{2}=\frac{\varepsilon^{2}}{(1-\delta)^{2k}}.$

Moreover, there holds that

$|Dw|\leq 1\quad\text{in }Q_{1}$

and

$|\\{(x,t)\in Q_{1}:Dw\cdot e\leq l\\}|>\mu|Q_{1}|\quad\text{for all
}e\in\mathbb{S}^{n-1},$

by the induction assumption. Additionally, note that
$\varepsilon\leq(1-\delta)^{k}$. Applying Lemma 4.1, we obtain

$Dw\cdot e\leq 1-\delta\quad\text{in }Q_{\tau}^{1-\delta}\quad\text{for all
}e\in\mathbb{S}^{n-1},$

which implies that $|Dw|\leq 1-\delta$ in $Q_{\tau}^{1-\delta}$. Rescaling
back, it yields that

$|Du|<(1-\delta)^{k+1}\quad\text{in }Q_{\tau^{k+1}}^{(1-\delta)^{k+1}}.$

We conclude the proof. ∎

######  Remark 4.5.

In order to derive the reduction of oscillation of $|Du|$, we ask that
$\|Da\|_{L^{\infty}(Q_{1})}$ is smaller than $\kappa$ ($\kappa\leq 1$
sufficiently small) in Lemmas 4.1 and 4.3. In fact, we can suppose initially
that $\|D_{x,t}a\|_{L^{\infty}(Q_{1})}\leq\kappa$. These two lemmas still
hold, when $\|Da\|_{L^{\infty}(Q_{1})}\leq\kappa$ is substituted by
$\|D_{x,t}a\|_{L^{\infty}(Q_{1})}\leq\kappa$ in Lemmas 4.1 and 4.3.

If the iteration above can be carried out infinitely, then we will readily
conclude the Hölder continuity of $Du$ at the origin $(0,0)$. Nevertheless,
unless $Du(0,0)=0$, the iteration shall stop unavoidably at some step, that
is, for some nonnegative integer $k$ the condition (4.10) is not true in some
direction $e\in\mathbb{S}^{n-1}$. In this case, we will infer that $u$ can be
approximated by a linear function, then making use of a conclusion on
regularity of small perturbation solutions from [56] to prove the Hölder
regularity for $Du$. Now we first study how the solution is close to a linear
function.

######  Lemma 4.6.

Let the conditions (2.1) and (2.2) be in force. Suppose that $u\in
C(\overline{Q_{1}})$ is a smooth solution to (3.1) with $\varepsilon\in(0,1)$
such that $|Du|\leq M$ in $Q_{1}$. Let $0\leq a(x,t)\leq a^{+}$. If for any
$t\in[-1,0]$ it holds that

$\mathrm{osc}_{B_{1}}u(\cdot,t)\leq L$

with $L$ being a positive constant, then

$\mathrm{osc}_{Q_{1}}u(x,t)\leq\begin{cases}CL&{\text{if }q\geq
0},\\\\[5.69054pt] C(L+L^{1+q}+L^{1+s})&{\text{if }-1<q<0},\end{cases}$

where $C>0$ depends upon $n,p,q,s,a^{+}$ and $M$.

###### Proof.

This proof is similar to that of Lemma 4.4 in [34]. We only give the sketch of
proof here. If $q\geq 0$, for $a_{ij}$ defined in (4.3) we can find that

$|a_{ij}|\leq\overline{\Lambda}:=\big{[}(1+M^{2})^{\frac{q}{2}}+a^{+}(1+M^{2})^{\frac{s}{2}}\big{]}\max\\{1,p-1\\}.$

So this claim can be concluded by the same proof of Lemma 4.3 in [36].

When $-1<q<0$, we define the comparison functions as follows

$\overline{w}(x,t)=\overline{b}+\Lambda L^{1+q}t+2L|x|^{\beta},$
$\underline{w}(x,t)=\underline{b}-\Lambda L^{1+q}t-2L|x|^{\beta},$

where $\beta=\frac{q+2}{q+1}$ and $\Lambda$ will be determined later. Here
$\overline{b}$ is chosen so that $\overline{w}(\cdot,-1)\geq u(\cdot,-1)$ in
$B_{1}$ and $\overline{w}(\overline{x},-1)=u(\overline{x},-1)$ at some point
$\overline{x}\in\overline{B_{1}}$. Correspondingly, $\underline{b}$ is chosen
so that $\underline{w}(\cdot,-1)\leq u(\cdot,-1)$ in $B_{1}$ and
$\underline{w}(\underline{x},-1)=u(\underline{x},-1)$ at some point
$\underline{x}\in\overline{B_{1}}$. Then $\overline{b}-\underline{b}\leq
L+2\Lambda L^{1+q}$. By direct computations,

$\displaystyle\quad\big{[}(|D\overline{w}|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|D\overline{w}|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{\overline{w}_{i}\overline{w}_{j}}{|D\overline{w}|^{2}+\varepsilon^{2}}\right)\overline{w}_{ij}$
$\displaystyle\leq\big{[}((2L\beta|x|^{\beta-1})^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)((2L\beta|x|^{\beta-1})^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}2pn\beta(\beta-1)L|x|^{\beta-2}$
$\displaystyle\leq(2\beta)^{q+1}(\beta-1)pn\big{[}1+a^{+}(1+(2L\beta)^{2})^{\frac{s-q}{2}}\big{]}L^{q+1}.$

Hence, if we choose

$\Lambda=(2\beta)^{q+1}(\beta-1)pn\big{[}1+a^{+}(1+(2L\beta)^{2})^{\frac{s-q}{2}}\big{]}+1,$

then $\overline{w}$ is a strict supersolution to (3.1). Analogously,
$\underline{w}$ is a strict subsolution. Next, we can prove that
$\overline{w}\geq u\geq\underline{w}$ in $Q_{1}$. For the details, one can see
[34, Lemma 4.4]. Finally we have

$\mathrm{osc}_{Q_{1}}u(x,t)\leq\overline{b}-\underline{b}+4L\leq 2\Lambda
L^{q+1}+5L=:C(L+L^{1+q}+L^{1+s}).$

The proof now is completed. ∎

######  Lemma 4.7.

Let $e\in\mathbb{S}^{n-1}$ and $0<\sigma<\frac{1}{8}$. Let the conditions
(2.1) and (2.2) be in force. Suppose that $u\in C(\overline{Q_{1}})$ is a
smooth solution to (3.1) with $\varepsilon\in(0,1)$ in $Q_{1}$. Let $0\leq
a(x,t)\leq a^{+}$. If for any $t\in[-1,0]$ it holds that

$\mathrm{osc}_{x\in B_{1}}(u(x,t)-x\cdot e)\leq\sigma,$

then one has

$\mathrm{osc}_{(x,t)\in Q_{1}}(u(x,t)-x\cdot e)\leq C\sigma,$

where $C>0$ depends only on $n,p,q,s$ and $a^{+}$.

###### Proof.

Denote

$\overline{w}(x,t)=\overline{b}+x\cdot e+\Lambda\sigma t+2\sigma|x|^{2},$
$\underline{w}(x,t)=\underline{b}+x\cdot e-\Lambda\sigma t-2\sigma|x|^{2},$

with $\Lambda>0$ to be fixed later. Here, the choices of
$\overline{b},\underline{b}$ are the same as that in the proof of Lemma 4.6.
Then we get $\overline{b}-\underline{b}\leq(2\Lambda+1)\sigma$. Owing to
$\sigma<\frac{1}{8}$, by simple calculation, we obtain

$\frac{1}{2}\leq|D\overline{w}(x,t)|,|D\underline{w}(x,t)|\leq\frac{3}{2},\quad\text{for
}(x,t)\in\overline{Q_{1}}.$

Therefore, it follows that

$a_{ij}(x,t,D\overline{w}(x,t))\leq A_{0}I$

and

$a_{ij}(x,t,D\underline{w}(x,t))\leq A_{0}I,$

where the notation $a_{ij}$ is from (4.3), and the constant $A_{0}$ depends on
$p,q,s,a^{+}$.

We next pick $\Lambda=5nA_{0}$ and then show that

$\underline{w}\leq u\leq\overline{w}$

in $Q_{1}$. For the details, we refer to [34, Lemma 4.5]. Finally, we arrive
at

$\mathrm{osc}_{(x,t)\in Q_{1}}(u(x,t)-x\cdot
e)\leq\sup_{Q_{1}}(\overline{w}-x\cdot e)-\inf_{Q_{1}}(\underline{w}-x\cdot
e)\leq\overline{b}-\underline{b}+4\sigma\leq(2\Lambda+5)\sigma.$

We now finish the proof. ∎

Now putting together Lemmas 4.6 and 4.7 with Lemma 3.3, we will conclude the
following result which states that if $Du$ is close to a unit vector in a
large portion of $Q_{1}$, then $u$ is close to some linear function. Since
$\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ is assumed to be smaller than
$\kappa\leq 1$ in Lemmas 4.1 and 4.3 (see Remark 4.5), in the lemma below we
will suppose $\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}\leq 1$ is in force for
simplicity.

######  Lemma 4.8.

Let the assumptions (2.1)–(2.3) be in force and let $\omega$ be a positive
number. Suppose that $|D_{x,t}a(x,t)|\leq 1$ in $Q_{1}$. Assume that $u$ is a
smooth solution of (3.1) with $0<\varepsilon<1$ satisfying $|Du|\leq 1$ in
$Q_{1}$, and that for some $e\in\mathbb{S}^{n-1}$ and two quantities
$\varepsilon_{0},\varepsilon_{1}>0$, there holds that

$|\\{(x,t)\in Q_{1}:|Du-e|>\varepsilon_{0}\\}|\leq\varepsilon_{1}.$

Then if $\varepsilon_{0},\varepsilon_{1}$ are small enough, there is a real
number $d\in\mathbb{R}$ such that

$|u(x,t)-d-e\cdot x|\leq\omega$

for $(x,t)\in Q_{1/2}$. Here both $\varepsilon_{0}$ and $\varepsilon_{1}$
depend upon $n,p,q,s,a^{-},a^{+}$ and $\omega$.

###### Proof.

Set

$f(t):=|\\{x\in B_{1}:|Du(x,t)-e|>\varepsilon_{0}\\}|,$
$D:=\\{t\in(-1,0):f(t)\geq\sqrt{\varepsilon_{1}}\\}.$

We can easily get

$\int^{0}_{-1}f(t)\,dt\leq\varepsilon_{1}\quad\text{and}\quad|D|\leq\sqrt{\varepsilon_{1}}.$

Thereby, it yields that

$|\\{x\in B_{1}:|Du(x,t)-e|>\varepsilon_{0}\\}|\leq\sqrt{\varepsilon_{1}},$
(4.11)

for any $t\in(-1,0]\setminus D$ with $|D|\leq\sqrt{\varepsilon_{1}}$. Applying
(4.11) and Morrey’s inequality to get for all $t\in(-1,0]\setminus D$,

$\displaystyle\quad\mathrm{osc}_{B_{1/2}}(u(x,t)-e\cdot x)$ $\displaystyle\leq
C\|Du-e\|_{L^{2n}(B_{1})}$ $\displaystyle=C\left(\int_{\\{x\in
B_{1}:|Du(x,t)-e|\leq\varepsilon_{0}\\}}+\int_{\\{x\in
B_{1}:|Du(x,t)-e|>\varepsilon_{0}\\}}|Du-e|^{2n}\,dx\right)^{\frac{1}{2n}}$
$\displaystyle\leq C(\varepsilon_{0}+\varepsilon_{1}^{\frac{1}{4n}}),$ (4.12)

where $C$ depends only on $n$.

On the other hand, we can see that

$\mathrm{osc}_{B_{1}}u(\cdot,t)\leq 2$

for $t\in(-1,0]$, due to $|Du|\leq 1$ in $Q_{1}$. Then we employ Lemma 4.6 to
deduce

$\mathrm{osc}_{Q_{1}}u(x,t)\leq C$

with $C$ depending on $n,p,q,s$ and $a^{+}$. Observe that $u(x,t)-u(0,0)$
solves (3.1) as well, and obviously

$\|u(x,t)-u(0,0)\|_{L^{\infty}(Q_{1})}\leq C.$

Via applying Lemma 3.3 to $u(x,t)-u(0,0)$, it follows that

$\sup_{\stackrel{{\scriptstyle(x,t),(x,s)\in Q_{1}}}{{t\neq
s}}}\frac{|u(x,t)-u(x,s)|}{|t-s|^{\gamma}}\leq C,$

where

$\gamma=\begin{cases}\frac{1}{\beta(1+s)-s}&{\text{if }-1<q<0},\\\\[5.69054pt]
\frac{1}{2}&{\text{if }0\leq q}\end{cases}$

with $\beta=\frac{q+2}{q+1}$, and the constant $C$ depends only on
$n,p,q,s,a^{-}$ and $a^{+}$. Thus by $|D|\leq\sqrt{\varepsilon_{1}}$ and
(4.1), we derive

$\mathrm{osc}_{x\in B_{\frac{1}{2}}}(u(x,t)-e\cdot x)\leq
C\big{(}\varepsilon_{0}+\varepsilon_{1}^{\frac{1}{4n}}+\varepsilon_{1}^{\frac{\gamma}{2}}\big{)}$

for any $t\in(-\frac{1}{4},0]$. It follows from Lemma 4.7 that

$\mathrm{osc}_{(x,t)\in Q_{\frac{1}{2}}}(u(x,t)-e\cdot x)\leq
C\big{(}\varepsilon_{0}+\varepsilon_{1}^{\frac{1}{4n}}+\varepsilon_{1}^{\frac{\gamma}{2}}\big{)},$

if $\varepsilon_{0}$ and $\varepsilon_{1}$ are small enough. Consequently, if
$\varepsilon_{0},\varepsilon_{1}$ are sufficiently small, then there is a
constant $d\in\mathbb{R}$ satisfying

$|u(x,t)-d-e\cdot x|\leq\omega$

for any $(x,t)\in Q_{1/2}$. ∎

Taking into account that the framework of equation (1.1) is analogous to that
of

$\partial_{t}u=|Du|^{q}\left(\delta_{ij}+(p-2)\frac{u_{i}u_{j}}{|Du|^{2}}\right)u_{ij},$

we may obtain, with the help of the same tool that is Corollary 1.2 in [56],
the regularity of small perturbation solutions to (3.1) as well.

######  Proposition 4.9.

Let $\beta\in(0,1)$. Suppose that $u$ is a smooth solution to (3.1) in
$Q_{1}$. Then there are $\omega>0$ (small) and $C>0$ (large), both depending
on $n,p,q,s,a^{+},\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ and $\beta$, such
that if a linear function $L(x)$ with $\frac{1}{2}\leq|DL|\leq 2$ fulfills

$\|u(x,t)-L(x)\|_{L^{\infty}(Q_{1})}\leq\omega,$

then

$\|u-L\|_{C^{2,\beta}(Q_{1/2})}\leq C.$

###### Proof.

Note that $L(x)$ is also a solution to (3.1). So we could infer this
conclusion by Corollary 1.2 in [56]. ∎

Once we show that $u$ is close to some linear function, then the Hölder
regularity for $Du$ follows from the above proposition. From Remark 4.5, we
have known that $\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ is less than $\kappa$
($\kappa\leq 1$), so in Proposition 4.9 we may replace
$\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ by $1$ so that $\omega,C$ do not
depend on $\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$.

In the sequel, we shall give a uniformly a priori Hölder estimate on the
solution of (3.1).

######  Theorem 4.10 (A priori Hölder gradient estimate).

Let the assumptions (2.1)–(2.3) be in force and let $\varepsilon\in(0,1)$.
Suppose that $\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}\leq\kappa$, where
$\kappa\in(0,1]$ is a small constant depending on $n,p,q,s,a^{-},a^{+}$.
Assume that $u$ is a smooth solution of (3.1) satisfying $|Du|\leq 1$ in
$Q_{1}$. Then there are two constants $\alpha,C>0$, both of which depend on
$n,p,q,s,a^{-}$ and $a^{+}$, such that the following estimates hold:

$|Du(x,t)-Du(y,s)|\leq C\big{(}|x-y|^{\alpha}+|t-s|^{\frac{\alpha}{2-\alpha
q}}\big{)}$

and

$|u(x,t)-u(x,s)|\leq C|t-s|^{\frac{1+\alpha}{2-\alpha q}}$

for any $(x,t),(y,s),(x,s)\in Q_{\frac{1}{2}}$.

###### Proof.

This proof is similar to that of Theorem 4.8 in [34]. However, for the sake of
completeness and convenience, we give the details of proof here. We first
demonstrate the Hölder continuity of $Du$ at $(0,0)$ and the Hölder continuity
of $u$ in $t$ at (0,0). Then, by standard translation arguments, the interior
Hölder regularity follows.

We choose $\omega$ as the one in Proposition 4.9 with
$\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ replaced by $1$ and
$\beta=\frac{1}{2}$. And then for this $\omega$ we pick two quantities
$\varepsilon_{0},\varepsilon_{1}>0$ small so that Lemma 4.8 holds true. Now we
fix

$l=1-\frac{\varepsilon_{0}^{2}}{2}\quad\text{and}\quad\mu=\frac{\varepsilon_{1}}{|Q_{1}|}.$

Here we observe that if for arbitrary $e\in\mathbb{S}^{n-1}$

$|\\{(x,t)\in Q_{1}:Du\cdot e\leq l\\}|\leq\mu|Q_{1}|,$

then

$|\\{(x,t)\in Q_{1}:|Du-e|>\varepsilon_{0}\\}|\leq\varepsilon_{1}.$

This estimate will be used later.

Let $\tau$ and $\delta$ be two positive constants coming from Corollary 4.3.
By $[\log\varepsilon/\log(1-\delta)]$, we mean the integer part of
$\log\varepsilon/\log(1-\delta)$. Let $k$ be either
$[\log\varepsilon/\log(1-\delta)]$ or the minimum nonnegative integer that
makes condition (4.10) false, whichever is smaller. Then it follows from
Corollary 4.3 that

$|Du(x,t)|\leq(1-\delta)^{m}\quad\text{in }Q_{\tau^{m}}^{(1-\delta)^{m}}$

for $m=0,1,\cdots,k$. When

$(x,t)\in Q_{\tau^{m}}^{(1-\delta)^{m}}\setminus
Q_{\tau^{m+1}}^{(1-\delta)^{m+1}},$

we know that

$|x|\geq\tau^{m+1}\quad\text{or}\quad|t|\geq(1-\delta)^{-(m+1)q}\tau^{2(m+1)}.$

Thus, by taking $\alpha=\frac{\log(1-\delta)}{\log\tau}$, it yields that

$|x|^{\alpha}\geq(1-\delta)^{m+1}\quad\text{or}\quad|t|^{\frac{\alpha}{2-\alpha
q}}\geq(1-\delta)^{(m+1)}.$

Then

$|Du(x,t)|\leq(1-\delta)^{m}\leq
C\left(|x|^{\alpha}+|t|^{\frac{\alpha}{2-\alpha q}}\right)$

with $C=\frac{1}{1-\delta}$ in $Q_{\tau^{m}}^{(1-\delta)^{m}}\setminus
Q_{\tau^{m+1}}^{(1-\delta)^{m+1}}$. Hence, for each $\xi\in\mathbb{R}^{n}$
with $|\xi|\leq(1-\delta)^{k}$,

$|Du(x,t)-\xi|\leq(1-\delta)^{m}+(1-\delta)^{k}\leq
C\left(|x|^{\alpha}+|t|^{\frac{\alpha}{2-\alpha q}}\right)$ (4.13)

in $Q_{1}\setminus Q_{\tau^{k+1}}^{(1-\delta)^{k+1}}$. Observe that if $q\geq
0$, (4.9) implies that $2-\alpha q>0$ and $\frac{\alpha}{2-\alpha
q}<\frac{1}{2}$. For $m=0,1,\cdots,k$, set

$u_{m}(x,t)=\frac{1}{\tau^{m}(1-\delta)^{m}}u(\tau^{m}x,\tau^{2m}(1-\delta)^{-mq}t),\quad(x,t)\in
Q_{1}.$ (4.14)

We can check that $|Du_{m}(x,t)|\leq 1$ in $Q_{1}$ and $u_{m}$ solves

$\displaystyle\partial_{t}u_{m}=\big{[}(|Du_{m}|^{2}+\hat{\varepsilon}^{2})^{\frac{q}{2}}+\hat{a}(x,t)(|Du_{m}|^{2}+\hat{\varepsilon}^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{\partial_{i}u_{m}\partial_{j}u_{m}}{|Du_{m}|^{2}+\hat{\varepsilon}^{2}}\right)\partial_{ij}u_{m},$
(4.15)

where

$\hat{a}(x,t)=(1-\delta)^{m(s-q)}a(\tau^{m}x,\tau^{2m}(1-\delta)^{-mq}t)\quad\text{and}\quad\hat{\varepsilon}=\frac{\varepsilon}{(1-\delta)^{m}}.$

Clearly, $\hat{\varepsilon}^{2}\leq\varepsilon^{2}(1-\delta)^{-2k}\leq 1$.
Noting the framework of this equation, we have, for any $t\in[-1,0]$,

$\mathrm{osc}_{B_{1}}u_{m}(\cdot,t)\leq 2$

and further get

$\mathrm{osc}_{Q_{1}}u_{m}\leq C$

by Lemma 4.6. This indicates

$\mathrm{osc}_{Q_{\tau^{m}}^{(1-\delta)^{m}}}u\leq C\tau^{m}(1-\delta)^{m}.$
(4.16)

In the rest of proof, we let $w(x,t)=u_{k}(x,t)$.

Case 1. $k=[\log\varepsilon/\log(1-\delta)]$. Then we get
$(1-\delta)^{k+1}<\varepsilon\leq(1-\delta)^{k}$, and hence
$\frac{1}{2}<1-\delta<\varepsilon(1-\delta)^{-k}\leq 1$. From this, we find
that, when $m=k$, (4.15) is a uniformly quasilinear parabolic equation with
bounded and smooth coefficients. In view of the standard parabolic quasilinear
equation theory (see, e.g. [41, Theorem 4.4]) together with Schauder
estimates, we may find $\zeta\in\mathbb{R}^{n},|\zeta|\leq 1$ such that in
$Q_{\tau}^{1-\delta}\subset Q_{1/4}$

$|Dw(x,t)-\zeta|\leq C(|x|+|t|^{\frac{1}{2}})\leq
C\big{(}|x|^{\alpha}+|t|^{\frac{\alpha}{2-\alpha q}}\big{)}$

and

$|\partial_{t}w|\leq C,$

where $C>0$ depends on $n,p,q,s$ and $a^{+}$. Here we have utilized the fact
that $\frac{\alpha}{2-\alpha q}\leq\frac{1}{2}$. Rescaling back, we arrive at

$|Du(\tau^{k}x,\tau^{2k}(1-\delta)^{-kq}t)-(1-\delta)^{k}\zeta|\leq
C(1-\delta)^{k}\big{(}|x|^{\alpha}+|t|^{\frac{\alpha}{2-\alpha q}}\big{)}$

for $(x,t)\in Q_{\tau}^{1-\delta}$, which leads to

$\displaystyle|Du(x,t)-(1-\delta)^{k}\zeta|$ $\displaystyle\leq
C(1-\delta)^{k}\big{(}\tau^{-k\alpha}|x|^{\alpha}+(\tau^{-2k}(1-\delta)^{kq})^{\frac{\alpha}{2-\alpha
q}}|t|^{\frac{\alpha}{2-\alpha q}}\big{)}$
$\displaystyle=C\big{(}|x|^{\alpha}+|t|^{\frac{\alpha}{2-\alpha q}}\big{)}$
(4.17)

by $(1-\delta)\tau^{-\alpha}=1$, where $(x,t)\in
Q_{\tau^{k+1}}^{(1-\delta)^{k+1}}$. Similarly,

$|u(x,t)-u(x,0)|\leq C\tau^{-k}(1-\delta)^{k(1+q)}|t|$ (4.18)

with $(x,t)\in Q_{\tau^{k+1}}^{(1-\delta)^{k+1}}$. Therefore, it yields by
(4.13) and (4.1) that for some vector $\xi_{0}\in\mathbb{R}^{n}$,

$|Du(x,t)-\xi_{0}|\leq C\big{(}|x|^{\alpha}+|t|^{\frac{\alpha}{2-\alpha
q}}\big{)}$

in $Q_{1/2}$, where $C>0$ depends on $n,p,q,s,a^{-}$ and $a^{+}$. On the other
hand, from (4.18) we obtain, for $|t|\leq\tau^{2l}(1-\delta)^{-lq}$ with
$l\geq k+1$, that

$|u(0,t)-u(0,0)|\leq C\tau^{l}(1-\delta)^{l},$ (4.19)

where we exploited the fact that $\tau<(1-\delta)^{1+q}$ in (4.9). By means of
(4.16) and (4.19), it follows that

$|u(0,t)-u(0,0)|\leq C|t|^{\beta}$

for every $t\in(-\frac{1}{4},0]$, where $\beta=\frac{1+\alpha}{2-\alpha q}$.
It is easy to see that $\beta>\frac{1}{2}$ if $q>-2$. Indeed, if
$t\in(-\frac{1}{4},0)$, there always exists
$\tilde{l}\in\\{0,1,\cdots,k,k+1,\cdots,l,\cdots\\}$ such that

$t\in\big{(}-\tau^{2\tilde{l}}(1-\delta)^{-q\tilde{l}},-\tau^{2(\tilde{l}+1)}(1-\delta)^{-q(\tilde{l}+1)}\big{]}.$

Via (4.16) and (4.19), when we select $\beta$ verifying

$\tau(1-\delta)=(\tau^{2}(1-\delta)^{-q})^{\beta},$

then we get

$|u(0,t)-u(0,0)|\leq
C\tau^{\tilde{l}}(1-\delta)^{\tilde{l}}=\frac{C}{(\tau^{2}(1-\delta)^{-q})^{\beta}}(\tau^{2}(1-\delta)^{-q})^{\beta(\tilde{l}+1)}\leq\frac{C}{\tau(1-\delta)}|t|^{\beta}.$

Case 2. $k<[\log\varepsilon/\log(1-\delta)]$. Then for some
$e\in\mathbb{R}^{n}$,

$\left|\left\\{(x,t)\in Q_{\tau^{k}}^{(1-\delta)^{k}}:Du\cdot e\leq
l(1-\delta)^{k}\right\\}\right|\leq\mu\left|Q_{\tau^{k}}^{(1-\delta)^{k}}\right|.$
(4.20)

Additionally,

$|Du|<(1-\delta)^{l}\quad\text{in }Q_{\tau^{l}}^{(1-\delta)^{l}}\quad\text{for
all }l=0,1,\cdots,k.$

We can easily see that $w$ ($w=u_{k}$) satisfies $|Dw|\leq 1$ and solves
equation (4.15) in $Q_{1}$. By virtue of (4.20) and the selections of $l$ and
$\mu$, we get

$|\\{(x,t)\in Q_{1}:|Du-e|>\varepsilon_{0}\\}|\leq\varepsilon_{1}.$

It thus follows from Lemma 4.8 that there is $d\in\mathbb{R}$ satisfying

$|w(x,t)-d-e\cdot x|\leq\omega\quad\text{for all }(x,t)\in Q_{1/2}.$

By Proposition 4.9, there is $b\in\mathbb{R}^{n}$ fulfilling

$|Dw-b|\leq C(|x|+\sqrt{|t|})$

and

$|\partial_{t}w|\leq C$

in $Q_{\tau}^{1-\delta}\subset Q_{1/4}$. As in Case $1$, we also arrive at

$|Du(x,t)-\xi_{1}|\leq C\big{(}|x|^{\alpha}+|t|^{\frac{\alpha}{2-\alpha
q}}\big{)}\quad\text{in }Q_{1/2}$

for some vector $\xi_{1}\in\mathbb{R}^{n}$ with $|\xi_{1}|\leq 1$, and

$|u(0,t)-u(0,0)|\leq C|t|^{\beta}\quad\text{for
}t\in\left(-\frac{1}{4},0\right],$

where $C>0$ depends on $n,p,q,s,a^{-}$ and $a^{+}$.

In summary, we have showed that there are $\alpha,C>0$, depending on
$n,p,q,s,a^{-}$ and $a^{+}$, as well as $\xi\in\mathbb{R}^{n}$ with $|\xi|\leq
1$, such that

$|Du(x,t)-\xi|\leq C\big{(}|x|^{\alpha}+|t|^{\frac{\alpha}{2-\alpha
q}}\big{)}\quad\text{in }Q_{1/2}$

and

$|u(0,t)-u(0,0)|\leq C|t|^{\beta}\quad\text{for
}t\in\left(-\frac{1}{4},0\right].$

Finally the claim follows by using the standard translation arguments. ∎

Next we are going to use the solution of equation (3.1) to approximate the
solution of (1.1). Before that, we need some crucial results on viscosity
solutions such as the boundary estimates, apart from the known comparison
principle and stability. We prescribe that the assumptions (2.1) and (2.2)
hold in the following two conclusions. Here two notations are introduced, for
two real numbers $a$ and $b$, $a\vee b=\max\\{a,b\\}$ and $a\wedge
b=\min\\{a,b\\}$.

######  Proposition 4.11 (Boundary estimates).

Assume that $u\in C(\overline{Q_{1}})$ is a solution to (3.1) with
$\varepsilon\in(0,1)$ and $a(x,t)$ satisfying (2.3) and
$\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}\leq 1$, and that
$\varphi:=u\mid_{\partial_{p}Q_{1}}$ possesses a modulus of continuity denoted
by $\rho$. Then there is another modulus of continuity $\rho^{*}$, which
depends on $n,p,q,s,a^{-},a^{+},\rho$ and
$\|\varphi\|_{L^{\infty}(\partial_{p}Q_{1})}$, such that

$|u(x,t)-u(y,s)|\leq\rho^{*}(|x-y|\vee\sqrt{|t-s|})$

for every $(x,t),(y,s)\in\overline{Q_{1}}$.

We will prove the aformentioned proposition in Section 7. The last ingredient
to be applied in the approximation step is the next lemma, which follows
directly via the classical quasi-linear equation theory (see [41, Theorem
4.4]) and the Schauder estimates.

######  Lemma 4.12.

Let $g\in C(\partial_{p}Q_{1})$. Let $0\leq a(x,t)\in
C^{1}(\overline{Q_{1}})$. For $\varepsilon>0$, there is a unique smooth
solution $u^{\varepsilon}\in C(\overline{Q_{1}})$ of (3.1) satisfying
$u^{\varepsilon}=g$ on $\partial_{p}Q_{1}$.

With Propositions 3.4, 3.5, 4.11 and Lemma 4.12 in hand, we now are in a
position to establish a crucial intermediate result under the assumption that
$\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ ($\leq 1$) is small, through letting
$\varepsilon\rightarrow 0$ in the a priori Hölder estimate in Theorem 4.10.

######  Theorem 4.13.

Let the assumptions (2.1)–(2.3) be in force. Let
$\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}\leq\kappa$, where $\kappa\in(0,1]$ is a
small constant depending on $n,p,q,s,a^{-},a^{+}$. Assume that $u$ is a
bounded viscosity solution to (1.1) in $Q_{1}$. There are two constants
$\alpha\in(0,1),C>0$, both of which depend on $n,p,q,s,a^{-},a^{+}$ and
$\|u\|_{L^{\infty}(Q_{1})}$, such that the following estimates hold:

$|Du(x,t)-Du(y,s)|\leq C\big{(}|x-y|^{\alpha}+|t-s|^{\frac{\alpha}{2-\alpha
q}}\big{)}$

and

$|u(x,t)-u(x,s)|\leq C|t-s|^{\frac{1+\alpha}{2-\alpha q}}$

for any $(x,t),(y,s),(x,s)\in Q_{\frac{1}{2}}$.

###### Proof.

Given Theorem 4.10, Propositions 3.4, 3.5, 4.11 and Lemma 4.12, the proof of
this theorem is identical to that of [36, Theorem 1]. ∎

###  4.2. Hölder regularity of spatial gradients in the case that
$\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}$ is finite

In this subsection, we prove the Hölder estimates on the gradients of
solutions to equation (1.1) under the assumption that $D_{x,t}a(x,t)$ has a
general bound instead of a small bound (less than 1). Now define

$\hat{u}(x,t)=\frac{1}{\epsilon}u(\epsilon
x,\epsilon^{2}t),\quad\hat{a}(x,t)=a(\epsilon x,\epsilon^{2}t)$

with $0<\epsilon<1$. For simplicity, let $(y,s):=(\epsilon x,\epsilon^{2}t)$,
then

$\displaystyle\partial_{t}\hat{u}(x,t)=\epsilon\partial_{s}u(\epsilon
x,\epsilon^{2}t),$
$\displaystyle\partial_{x_{i}}\hat{u}(x,t)=\partial_{y_{i}}u(\epsilon
x,\epsilon^{2}t),$
$\displaystyle\partial_{x_{i}x_{j}}\hat{u}(x,t)=\epsilon\partial_{y_{i}y_{j}}u(\epsilon
x,\epsilon^{2}t).$

Therefore if $u$ is a solution to (1.1) in $Q_{1}$, then we can easily check
that $\hat{u}$ solves (in the viscosity sense)

$\partial_{t}\hat{u}=[|D\hat{u}|^{q}+\hat{a}(x,t)|D\hat{u}|^{s}]\left(\delta_{ij}+(p-2)\frac{\hat{u}_{i}\hat{u}_{j}}{|D\hat{u}|^{2}}\right)\hat{u}_{ij}$
(4.21)

in $Q_{\epsilon^{-1}}$ and moreover

$\|D_{x,t}\hat{a}\|_{L^{\infty}(Q_{\epsilon^{-1}})}\leq\epsilon\|D_{x,t}a\|_{L^{\infty}(Q_{1})}<\kappa$

by selecting

$\epsilon\leq\frac{\kappa}{\|D_{x,t}a\|_{L^{\infty}(Q_{1})}+1}.$

By the dependencies of $\kappa$ (see Theorem 4.13), we know that $\epsilon$
depends only on $n,p,q,s,a^{-},a^{+}$ and $\|D_{x,t}a\|_{L^{\infty}(Q_{1})}$.
Notice that the framework of equation (4.21) is the same as that of (1.1)
(with $\|D_{x,t}a\|_{L^{\infty}(Q_{1})}$ being small). Thus this allows us to
make use of these results obtained above to demonstrate the interior Hölder
continuity of gradients of the solutions to (4.21) and interior Hölder
continuity of solutions in the time variable. In turn, by rescaling back, we
can derive the local $C^{1,\alpha}$ regularity of solutions, $u$, to (1.1)
under the condition that $\|D_{x,t}a(x,t)\|_{L^{\infty}(Q_{1})}\leq A$.

As has been stated above, we now conclude the proof of Theorem 2.3.

##  5\. Comparison principle and stability

In this part, we are ready to show the comparison principle and stability
property for the viscosity solution. When proving comparison principle, we
will make use of Ishii-Lions’ method. Here we consider these two properties in
a more general domain. Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$.
We denote a general parabolic cylinder by $\Omega_{T}:=\Omega\times[0,T)$, and
$\partial_{p}\Omega_{T}$ stands for its parabolic boundary.

For the convenience of readers, here we repeat the statement before proceeding
with the proof. Let $Sym(n)$ stand for the set of all symmetric $n\times n$
real matrices.

######  Proposition 5.1.

Let the function $a(x,t)>0$ be Lipschitz continuous in time-space variables.
Assume that $u$ and $v$ are a viscosity subsolution and a locally uniformly
Lipschitz continuous viscosity supersolution in $x$-variable to (1.1) in
$\Omega_{T}$, respectively. If $u\leq v$ on $\partial_{p}\Omega_{T}$, then

$u\leq v\quad\text{in }\Omega_{T}.$

###### Proof.

For simplicity, we can first suppose that $v$ is a strict supersolution, that
is,

$\partial_{t}v-[|Dv|^{q}+a(x,t)|Dv|^{s}]\left(\Delta v+(p-2)\left\langle
D^{2}v\frac{Dv}{|Dv|},\frac{Dv}{|Dv|}\right\rangle\right)>0$

in the viscosity sense by considering $w:=v+\frac{\varepsilon}{T-t}$ instead.
Indeed, we let $\psi\in C^{2}(\Omega_{T})$, with $D\psi(x,t)\neq 0$ for $x\neq
x_{0}$, be such that $w-\psi$ attains a local minimum at
$(x_{0},t_{0})\in\Omega_{T}$, then so does $v-\varphi$ by denoting
$\varphi(x,t):=\psi(x,t)-\frac{\varepsilon}{T-t}$. Since $v$ is a viscosity
supersolution, then it yields that

$\displaystyle 0\leq$
$\displaystyle\limsup_{\stackrel{{\scriptstyle(x,t)\rightarrow(x_{0},t_{0})}}{{x\neq
x_{0}}}}\left(\partial_{t}\varphi(x,t)-[|D\varphi(x,t)|^{q}+a(x,t)|D\varphi(x,t)|^{s}]\Delta_{p}^{N}\varphi(x,t)\right)$
$\displaystyle\leq$
$\displaystyle-\frac{\varepsilon}{(T-t_{0})^{2}}+\limsup_{\stackrel{{\scriptstyle(x,t)\rightarrow(x_{0},t_{0})}}{{x\neq
x_{0}}}}\left(\partial_{t}\psi(x,t)-[|D\psi(x,t)|^{q}+a(x,t)|D\psi(x,t)|^{s}]\Delta_{p}^{N}\psi(x,t)\right),$

and further

$0<\limsup_{\stackrel{{\scriptstyle(x,t)\rightarrow(x_{0},t_{0})}}{{x\neq
x_{0}}}}\left(\partial_{t}\psi(x,t)-[|D\psi(x,t)|^{q}+a(x,t)|D\psi(x,t)|^{s}]\Delta_{p}^{N}\psi(x,t)\right),$

which implies that $w$ is a strict viscosity solution by Definition 2.1.

In order to show this assertion, we argue by contradiction. If the conclusion
does not hold, then we may find some point
$(\hat{x},\hat{t})\in\Omega\times(0,T)$ such that

$\omega_{0}:=u(\hat{x},\hat{t})-v(\hat{x},\hat{t})=\sup_{\Omega_{T}}(u-v)>0.$

Now define

$\Theta_{j}(x,y,t,s):=u(x,t)-v(y,s)-\Psi_{j}(x,y,t,s),$

where $\Psi_{j}(x,y,t,s)=\frac{j}{l}|x-y|^{l}+\frac{j}{2}(t-s)^{2}$ with

$l>\max\left\\{2,\frac{q+2}{q+1},\frac{s+2}{s+1}\right\\}.$

We denote by $(x_{j},y_{j},t_{j},s_{j})$ the maximum point of $\Theta_{j}$ in
$\overline{\Omega}\times\overline{\Omega}\times[0,T)\times[0,T)$. It is easy
to know that
$(x_{j},y_{j},t_{j},s_{j})\in\Omega\times\Omega\times(0,T)\times(0,T)$ (for
$j$ large enough) and
$(x_{j},y_{j},t_{j},s_{j})\rightarrow(\hat{x},\hat{x},\hat{t},\hat{t})$ as
$j\rightarrow\infty$ by Lemma 7.2 in [23]. In the rest of proof, we shall
distinguish between two scenarios that $x_{j}=y_{j}$ and $x_{j}\neq y_{j}$.

Case 1. $x_{j}=y_{j}$. Observe that, by the choice of
$(x_{j},y_{j},t_{j},s_{j})$,

$u(x_{j},t_{j})-v(y_{j},s_{j})-\Psi_{j}(x_{j},y_{j},t_{j},s_{j})\geq
u(x_{j},t_{j})-v(y,s)-\Psi_{j}(x_{j},y,t_{j},s).$

Let

$\phi(y,s):=-\Psi_{j}(x_{j},y,t_{j},s)+\Psi_{j}(x_{j},y_{j},t_{j},s_{j})+v(y_{j},s_{j}).$

Clearly, $v(y,s)-\phi(y,s)$ has a local minimum at $(y_{j},s_{j})$. We first
evaluate

$\partial_{s}\phi=j(t_{j}-s),\quad D\phi=j|x_{j}-y|^{l-2}(x_{j}-y)$

and

$D^{2}\phi=j|x_{j}-y|^{l-2}I+j(l-2)|x_{j}-y|^{l-2}\frac{x_{j}-y}{|x_{j}-y|}\otimes\frac{x_{j}-y}{|x_{j}-y|},$

where $\xi\otimes\xi$ is the matrix with entries $\xi_{i}\xi_{j}$ for a vector
$\xi\in\mathbb{R}^{n}$. Obviously, $D\phi(y,s)\neq 0$ for $y\neq
x_{j}(=y_{j})$. Owing to $v$ being a strict supersolution, we have

$0<\limsup_{\stackrel{{\scriptstyle(y,s)\rightarrow(y_{j},s_{j})}}{{y\neq
y_{j}}}}\left(\partial_{s}\phi(y,s)-[|D\phi(y,s)|^{q}+a(y,s)|D\phi(y,s)|^{s}]\Delta_{p}^{N}\phi(y,s)\right).$
(5.1)

Next we carefully compute

$\displaystyle\quad\left\langle
D^{2}\phi\frac{D\phi}{|D\phi|},\frac{D\phi}{|D\phi|}\right\rangle$
$\displaystyle=\left\langle\left(j|x_{j}-y|^{l-2}I+j(l-2)|x_{j}-y|^{l-2}\frac{x_{j}-y}{|x_{j}-y|}\otimes\frac{x_{j}-y}{|x_{j}-y|}\right)\frac{x_{j}-y}{|x_{j}-y|},\frac{x_{j}-y}{|x_{j}-y|}\right\rangle$
$\displaystyle=j(l-1)|x_{j}-y|^{l-2}$

and then

$\displaystyle[|D\phi|^{q}+a(y,s)|D\phi|^{s}]\left(\mathrm{tr}(D^{2}\phi)+(p-2)\left\langle
D^{2}\phi\frac{D\phi}{|D\phi|},\frac{D\phi}{|D\phi|}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left[(j|x_{j}-y|^{l-1})^{q}+a(y,s)(j|x_{j}-y|^{l-1})^{s}\right]\left(j((n+l-2)+(p-2)(l-1))|x_{j}-y|^{l-2}\right)$
$\displaystyle=$
$\displaystyle\left(n+(p-2)(l-1)+l-2\right)\left[j^{q+1}|x_{j}-y|^{q(l-1)+l-2}+a(y,s)j^{s+1}|x_{j}-y|^{s(l-1)+l-2}\right],$

where the powers of $|x_{j}-y|$, $q(l-1)+l-2$ and $s(l-1)+l-2$, are positive,
by the definition of $l$. From the above estimate, (5.1) turns into

$j(t_{j}-s_{j})>0.$

On the other hand, we can see that

$\psi(x,t):=\Psi_{j}(x,y_{j},t,s_{j})-\Psi_{j}(x_{j},y_{j},t_{j},s_{j})+u(x_{j},t_{j})$

is a good testing function with respect to $u$ at $(x_{j},t_{j})$. In a
similar way, we will readily get

$\displaystyle j(t_{j}-s_{j})$ $\displaystyle=$
$\displaystyle\liminf_{\stackrel{{\scriptstyle(x,t)\rightarrow(x_{j},t_{j})}}{{x\neq
x_{j}}}}\left(\partial_{t}\psi(x,t)-[|D\psi(x,t)|^{q}+a(x,t)|D\psi(x,t)|^{s}]\Delta_{p}^{N}\psi(x,t)\right)\leq
0.$ (5.2)

Combining (5.1) and (5), we have

$0<j(t_{j}-s_{j})-j(t_{j}-s_{j})=0,$

which is a contradiction.

Case 2. $x_{j}\neq y_{j}$. In this case, we shall employ the definition with
jets. Applying theorem of sums (see [23]), for each $\mu>0$, there exist
$X_{j},Y_{j}\in Sym(n)$ such that

$(\partial_{t}\Psi_{j},D_{x}\Psi_{j},X_{j})\in\overline{\mathcal{P}}^{2,+}u(x_{j},t_{j}),$
(5.3)
$(-\partial_{s}\Psi_{j},-D_{y}\Psi_{j},Y_{j})\in\overline{\mathcal{P}}^{2,-}v(y_{j},s_{j})$
(5.4)

and

$\left(\begin{array}[]{cc}X_{j}&\\\ &-Y_{j}\end{array}\right)\leq
D^{2}\Psi_{j}+\frac{1}{\mu}(D^{2}\Psi_{j})^{2},$

where all the derivatives are evaluated at $(x_{j},y_{j},t_{j},s_{j})$ and

$D^{2}\Psi_{j}=\left(\begin{array}[]{cc}D_{xx}\Psi_{j}&D_{xy}\Psi_{j}\\\
D_{yx}\Psi_{j}&D_{yy}\Psi_{j}\end{array}\right)=:\left(\begin{array}[]{cc}B&-B\\\
-B&B\end{array}\right)$

with

$B:=j|x_{j}-y_{j}|^{l-2}I+(l-2)j|x_{j}-y_{j}|^{l-4}(x_{j}-y_{j})\otimes(x_{j}-y_{j}).$

So we get

$\left(\begin{array}[]{cc}X_{j}&\\\
&-Y_{j}\end{array}\right)\leq\left(\begin{array}[]{cc}B&-B\\\
-B&B\end{array}\right)+\frac{2}{\mu}\left(\begin{array}[]{cc}B^{2}&-B^{2}\\\
-B^{2}&B^{2}\end{array}\right),$ (5.5)

with

$B^{2}:=j^{2}|x_{j}-y_{j}|^{2l-4}I+l(l-2)j^{2}|x_{j}-y_{j}|^{2l-6}(x_{j}-y_{j})\otimes(x_{j}-y_{j}).$

We plainly derive $X_{j}\leq Y_{j}$, i.e.,
$\langle(X_{j}-Y_{j})\xi,\xi\rangle\leq 0$ for all $\xi\in\mathbb{R}^{n}$. In
the sequel, we choose $\mu=j$ in (5.5). Now from (5.5), we can get a more
accurate estimate on $X_{j}-Y_{j}$ as follows:

$X_{j}\xi\cdot\xi-Y_{j}\eta\cdot\eta\leq
j\left[(l-1)|x_{j}-y_{j}|^{l-2}+2(l-1)^{2}|x_{j}-y_{j}|^{2(l-2)}\right]|\xi-\eta|^{2}$
(5.6)

for any $\xi,\eta\in\mathbb{R}^{n}$.

Next we give some notations that will be utilized later. Denote

$F_{1}(\xi,M):=|\xi|^{q}\left(\mathrm{tr}M+(p-2)\left\langle
M\frac{\xi}{|\xi|},\frac{\xi}{|\xi|}\right\rangle\right),$ $\displaystyle
F_{2}(x,t,\xi,M)$
$\displaystyle:=a(x,t)|\xi|^{s}\left(\mathrm{tr}M+(p-2)\left\langle
M\frac{\xi}{|\xi|},\frac{\xi}{|\xi|}\right\rangle\right)$
$\displaystyle=\mathrm{tr}(A(x,t,\xi)M),$

where

$A(x,t,\xi):=a(x,t)|\xi|^{s}\left(I+(p-2)\frac{\xi}{|\xi|}\otimes\frac{\xi}{|\xi|}\right)$

with $(x,t)\in\Omega_{T},M\in Sym(n)$.

Let

$\eta_{j}:=D_{x}\Psi_{j}=-D_{y}\Psi_{j}=j|x_{j}-y_{j}|^{l-2}(x_{j}-y_{j}).$

It is essential that $\eta_{j}$ is nonzero, which allows us to exploit jets.
Because $u$ is a subsolution and $v$ is a strict supersolution, we arrive at

$-\partial_{s}\Psi_{j}-F_{1}(\eta_{j},Y_{j})-F_{2}(y_{j},s_{j},\eta_{j},Y_{j})>0$

and

$\partial_{t}\Psi_{j}-F_{1}(\eta_{j},X_{j})-F_{2}(x_{j},t_{j},\eta_{j},X_{j})\leq
0$

by (5.3) and (5.4). Subtracting these two inequalities above, we get

$\displaystyle 0$
$\displaystyle<-\partial_{s}\Psi_{j}-\partial_{t}\Psi_{j}+F_{1}(\eta_{j},X_{j})-F_{1}(\eta_{j},Y_{j})+F_{2}(x_{j},t_{j},\eta_{j},X_{j})-F_{2}(y_{j},s_{j},\eta_{j},Y_{j})$
$\displaystyle=:J_{1}+J_{2}+J_{3}.$ (5.7)

First, notice that

$-\partial_{s}\Psi_{j}=j(t_{j}-s_{j})=\partial_{t}\Psi_{j},$

then we get

$J_{1}=0.$

Second, through the increasing monotonicity of $F_{1}(\xi,M)$ with respect to
the second variable $M$, and applying $X_{j}\leq Y_{j}$, we obtain

$J_{2}=F_{1}(\eta_{j},X_{j})-F_{1}(\eta_{j},Y_{j})\leq 0.$

In turn, we are going to estimate the third term $J_{3}$, which is the most
delicate part of the proof. Since $a(x,t)>0$, the matrix $A(x,t,\xi)$ is
positive definite so that it has matrix square root denoted by
$A^{\frac{1}{2}}(x,t,\xi)$. By $A^{\frac{1}{2}}_{k}(x,t,\xi)$, we mean the
$k$-th column of $A^{\frac{1}{2}}(x,t,\xi)$. Then it yields that

$\displaystyle J_{3}$
$\displaystyle=\mathrm{tr}(A(x_{j},t_{j},\eta_{j})X_{j})-\mathrm{tr}(A(y_{j},s_{j},\eta_{j})Y_{j})$
$\displaystyle=\sum^{n}_{k=1}X_{j}A^{\frac{1}{2}}_{k}(x_{j},t_{j},\eta_{j})\cdot
A^{\frac{1}{2}}_{k}(x_{j},t_{j},\eta_{j})-\sum^{n}_{k=1}Y_{j}A^{\frac{1}{2}}_{k}(y_{j},s_{j},\eta_{j})\cdot
A^{\frac{1}{2}}_{k}(y_{j},s_{j},\eta_{j})$ $\displaystyle\leq
Cj|x_{j}-y_{j}|^{l-2}\|A^{\frac{1}{2}}(x_{j},t_{j},\eta_{j})-A^{\frac{1}{2}}(y_{j},s_{j},\eta_{j})\|^{2}_{2}$
$\displaystyle\leq\frac{Cj|x_{j}-y_{j}|^{l-2}}{(\lambda_{\rm
min}(A^{\frac{1}{2}}(x_{j},t_{j},\eta_{j}))+\lambda_{\rm
min}(A^{\frac{1}{2}}(y_{j},s_{j},\eta_{j})))^{2}}\|A(x_{j},t_{j},\eta_{j})-A(y_{j},s_{j},\eta_{j})\|^{2}_{2},$
(5.8)

where the penultimate inequality is derived by (5.6) and the last inequality
is obtained from the local Lipschitz continuity of $M\mapsto M^{\frac{1}{2}}$
(see [33, page 410]). Here $\lambda_{\rm min}(M)$ stands for the smallest
eigenvalue of a symmetric $n\times n$ matrix $M$. Let us mention that the
inequality similar to (5) can be found in [39, page 1484].

We proceed with evaluating

$\displaystyle\quad\|A(x_{j},t_{j},\eta_{j})-A(y_{j},s_{j},\eta_{j})\|_{2}$
$\displaystyle=\left\|(a(x_{j},t_{j})-a(y_{j},s_{j}))|\eta_{j}|^{s}\left(I+(p-2)\frac{\eta_{j}}{|\eta_{j}|}\otimes\frac{\eta_{j}}{|\eta_{j}|}\right)\right\|_{2}$
$\displaystyle\leq|\eta_{j}|^{s}|a(x_{j},t_{j})-a(y_{j},s_{j})|(\sqrt{n}+|p-2|).$
(5.9)

In addition,

$\begin{split}&\lambda_{\rm
min}(A^{\frac{1}{2}}(x_{j},t_{j},\eta_{j}))=\lambda^{\frac{1}{2}}_{\rm
min}(A(x_{j},t_{j},\eta_{j}))\geq\min\\{1,\sqrt{p-1}\\}|\eta_{j}|^{\frac{s}{2}}\sqrt{a(x_{j},t_{j})},\\\
&\lambda_{\rm
min}(A^{\frac{1}{2}}(y_{j},s_{j},\eta_{j}))\geq\min\\{1,\sqrt{p-1}\\}|\eta_{j}|^{\frac{s}{2}}\sqrt{a(y_{j},s_{j})}.\end{split}$
(5.10)

Merging (5.10), (5) with (5), we finally have

$\displaystyle J_{3}$
$\displaystyle\leq\frac{Cj|x_{j}-y_{j}|^{l-2}(\sqrt{n}+|p-2|)^{2}}{\min\\{1,p-1\\}|\eta_{j}|^{s}(\sqrt{a(x_{j},t_{j})}+\sqrt{a(y_{j},s_{j})})^{2}}|\eta_{j}|^{2s}|a(x_{j},t_{j})-a(y_{j},s_{j})|^{2}$
$\displaystyle\leq
C\left(\sqrt{a(x_{j},t_{j})}+\sqrt{a(y_{j},s_{j})}\right)^{-2}j|x_{j}-y_{j}|^{l-2}|\eta_{j}|^{s}(|x_{j}-y_{j}|^{2}+|t_{j}-s_{j}|^{2}),$

where we have used the assumption that $a(x,t)$ is Lipschitz continuous in
$\Omega_{T}$, which implies that $|a(x,t)-a(y,s)|\leq
C\sqrt{|x-y|^{2}+|t-s|^{2}}$. Thus (5) becomes

$0<C\left(\sqrt{a(x_{j},t_{j})}+\sqrt{a(y_{j},s_{j})}\right)^{-2}j|x_{j}-y_{j}|^{l-2}|\eta_{j}|^{s}(|x_{j}-y_{j}|^{2}+|t_{j}-s_{j}|^{2})=:H_{j}.$

We now verify that $H_{j}$ tends to 0 as $j\rightarrow\infty$, which leads to
a contradiction. Next, we split the proof into two cases.

Observe that

$\begin{split}u(x_{j},t_{j})-v(x_{j},t_{j})&\leq\max_{\overline{\Omega}\times[0,T)}\\{u(x,t)-v(x,t)\\}\\\
&\leq
u(x_{j},t_{j})-v(y_{j},s_{j})-\frac{j}{l}|x_{j}-y_{j}|^{l}-\frac{j}{2}(t_{j}-s_{j})^{2}.\end{split}$

This leads to

$\displaystyle\frac{j}{l}|x_{j}-y_{j}|^{l}+\frac{j}{2}(t_{j}-s_{j})^{2}$
$\displaystyle\leq v(x_{j},t_{j})-v(y_{j},s_{j})$ $\displaystyle\rightarrow
v(\hat{x},\hat{t})-v(\hat{x},\hat{t})=0,$ (5.11)

as $j\rightarrow\infty$, where we have utilized the fact that $v$ is
continuous in $\Omega_{T}$. On the other hand,

$\displaystyle\quad
u(x_{j},t_{j})-v(y_{j},s_{j})-\frac{j}{l}|x_{j}-y_{j}|^{l}-\frac{j}{2}(t_{j}-s_{j})^{2}$
$\displaystyle=\max_{\overline{\Omega}\times\overline{\Omega}\times[0,T)\times[0,T)}\left\\{u(x,t)-v(y,s)-\frac{j}{l}|x-y|^{l}-\frac{j}{2}(t-s)^{2}\right\\}$
$\displaystyle\geq
u(x_{j},t_{j})-v(x_{j},s_{j})-\frac{j}{l}|x_{j}-x_{j}|^{l}-\frac{j}{2}(t_{j}-s_{j})^{2},$

i.e.,

$v(x_{j},s_{j})-v(y_{j},s_{j})\geq\frac{j}{l}|x_{j}-y_{j}|^{l}.$

By virtue of the uniform Lipschitz continuity of $v$ in the spatial variables,
we have

$\frac{j}{l}|x_{j}-y_{j}|^{l}\leq C|x_{j}-y_{j}|$

and further

$\frac{j}{l}|x_{j}-y_{j}|^{l-1}\leq C.$

Hence it follows that

$|\eta_{j}|\leq Cl.$

If $s\geq 0$, then it follows from (5), $l>2$ and $|\eta_{j}|\leq Cl$ that

$H_{j}=C\left(\sqrt{a(x_{j},t_{j})}+\sqrt{a(y_{j},s_{j})}\right)^{-2}|\eta_{j}|^{s}(j|x_{j}-y_{j}|^{l}+j|t_{j}-s_{j}|^{2}|x_{j}-y_{j}|^{l-2})\rightarrow
0$

by sending $j\rightarrow\infty$.

If $-1<s<0$, we arrive at

$\displaystyle H_{j}$
$\displaystyle=C\left(\sqrt{a(x_{j},t_{j})}+\sqrt{a(y_{j},s_{j})}\right)^{-2}|\eta_{j}|^{s+1}\frac{j|x_{j}-y_{j}|^{l-2}}{j|x_{j}-y_{j}|^{l-1}}(|x_{j}-y_{j}|^{2}+|t_{j}-s_{j}|^{2})$
$\displaystyle=C\left(\sqrt{a(x_{j},t_{j})}+\sqrt{a(y_{j},s_{j})}\right)^{-2}(|\eta_{j}|^{s+1}|x_{j}-y_{j}|+|\eta_{j}|^{s+1}|t_{j}-s_{j}|^{2}|x_{j}-y_{j}|^{-1}).$

Due to $|\eta_{j}|$ is bounded and $s+1>0$, it yields that

$|\eta_{j}|^{s+1}|x_{j}-y_{j}|\rightarrow 0$

by $j\rightarrow\infty$, as $x_{j},y_{j}\rightarrow\hat{x}$. Furthermore,
using (5), we can justify the following limit,

$\displaystyle|\eta_{j}|^{s+1}|t_{j}-s_{j}|^{2}|x_{j}-y_{j}|^{-1}$
$\displaystyle=\left(j|x_{j}-y_{j}|^{l-1}|t_{j}-s_{j}|^{\frac{2}{s+1}}|x_{j}-y_{j}|^{\frac{-1}{s+1}}\right)^{s+1}$
$\displaystyle=\left(j|t_{j}-s_{j}|^{\frac{2}{s+1}}|x_{j}-y_{j}|^{l-1-\frac{1}{s+1}}\right)^{s+1}$
$\displaystyle\rightarrow 0$

as $j\rightarrow\infty$, where we need to notice that $\frac{2}{s+1}>2$ by
$-1<s<0$, together with $l-1-\frac{1}{s+1}>0$ by $l>\frac{s+2}{s+1}$. In
conclusion, we have proved that $H_{j}$ converge to 0 as $j\rightarrow\infty$.
The proof now is completed. ∎

We now conclude this section with stability properties of viscosity solution.

######  Proposition 5.2.

Let $\\{u_{i}\\}$ be a sequence of viscosity solutions to (3.1) in
$\Omega_{T}$ with $\varepsilon_{i}\geq 0$ such that
$\varepsilon_{i}\rightarrow 0$. Suppose that $u_{i}$ converges to $u$ locally
uniformly in $\Omega_{T}$. Then $u$ is a solution to (1.1) in $\Omega_{T}$.

###### Proof.

We only prove that $u$ is a viscosity supersolution to (1.1). The case of
subsolution then follows in a similar way. Let $\varphi\in C^{2}(Q_{1})$ be
such that $u-\varphi$ reaches a local minimum at $(x_{0},t_{0})\in\Omega_{T}$
and moreover $D\varphi(x,t)\neq 0$ for $x\neq x_{0}$. Taking into account that
$u_{i}$ converges to $u$ locally uniformly, we can find a sequence
$\\{(x_{i},t_{i})\\}\subset\Omega_{T}$ satisfying
$(x_{i},t_{i})\rightarrow(x_{0},t_{0})$ as $i\rightarrow\infty$, such that
$u_{i}-\varphi$ attains a local minimum at $(x_{i},t_{i})$. Since $u_{i}$ is a
viscosity supersolution to (3.1), we have

$\displaystyle 0$
$\displaystyle\leq\partial_{t}\varphi(x_{i},t_{i})-\left[(|D\varphi(x_{i},t_{i})|^{2}+\varepsilon_{i}^{2})^{\frac{q}{2}}+a(x_{i},t_{i})(|D\varphi(x_{i},t_{i})|^{2}+\varepsilon_{i}^{2})^{\frac{s}{2}}\right]$
$\displaystyle\quad\cdot\left(\mathrm{tr}D^{2}\varphi(x_{i},t_{i})+(p-2)\frac{D^{2}\varphi(x_{i},t_{i})D\varphi(x_{i},t_{i})\cdot
D\varphi(x_{i},t_{i})}{|D\varphi(x_{i},t_{i})|^{2}+\varepsilon_{i}^{2}}\right).$

Furthermore, on account of $(x_{i},t_{i})\rightarrow(x_{0},t_{0})$, we hence
conclude that

$0\leq\limsup_{\stackrel{{\scriptstyle(x,t)\rightarrow(x_{0},t_{0})}}{{x\neq
x_{0}}}}\left(\partial_{t}\varphi-[|D\varphi|^{q}+a(x,t)|D\varphi|^{s}]\left(\mathrm{tr}D^{2}\varphi+(p-2)\left\langle
D^{2}\varphi\frac{D\varphi}{|D\varphi|},\frac{D\varphi}{|D\varphi|}\right\rangle\right)\right),$

which implies that $u$ is a viscosity supersolution to (1.1). ∎

##  6\. Proof of Lipschitz continuity of solutions

This section is devoted to showing the Lipschitz continuity of solutions to
(3.1) with $\varepsilon\in[0,1)$, that is Lemma 3.1. Our proof follows roughly
the similar lines as the one in [2]. We divide the proof of Lemma 3.1 into two
steps. In the first step, we will make use of Ishii-Lions’ method to infer the
Hölder continuity of solutions in spatial variables. Subsequently, the Hölder
continuity shall be improved into the Lipschitz continuity by employing again
the Ishii-Lions’ method in the second step.

To begin with, we prove the $C^{0,\gamma}$ estimates on solutions with respect
to $x$-variable for all $\gamma\in(0,1)$.

######  Lemma 6.1 (Local Hölder estimates).

Let the conditions (2.1) and (2.2) be in force. Let $u$ be a bounded viscosity
solution to (3.1) with $\varepsilon\in[0,1)$ in $Q_{1}$. Assume that
$a(x,t)\geq a^{-}>0$ and $a(x,t)$ is uniformly Lipschitz continuous in
$x$-variable. Then for any $\gamma\in(0,1)$, there is a positive constant $C$
depending only on $n,p,q,s,\gamma$, such that

$|u(x,t)-u(y,t)|\leq C\|u\|_{L^{\infty}(Q_{1})}\left(1+\frac{C_{\rm
lip}}{a^{-}}\right)|x-y|^{\gamma}$

for all $x,y\in B_{\frac{15}{16}}$ and
$t\in\left(-\left(\frac{15}{16}\right)^{2},0\right]$. Here $C_{\rm lip}$ is
the same as the one in Lemma 3.1.

###### Proof.

Fix $x_{0},y_{0}\in B_{\frac{15}{16}}$ and
$t_{0}\in\left(-\left(\frac{15}{16}\right)^{2},0\right)$. We are ready to
prove that there exist two suitable constants $L_{1},L_{2}>0$ such that

$L:=\sup_{(x,t),(y,t)\in\overline{Q_{\frac{15}{16}}}}(u(x,t)-u(y,t)-L_{1}\phi(|x-y|)-\Psi(x,y,t))\leq
0,$

where

$\Psi(x,y,t)=\frac{L_{2}}{2}|x-x_{0}|^{2}+\frac{L_{2}}{2}|y-y_{0}|^{2}+\frac{L_{2}}{2}|t-t_{0}|^{2}$

with $\phi(r):=r^{\gamma}$. Thriving for a contradiction. We suppose that
$L>0$ and
$(\overline{x},\overline{y},\overline{t})\in\overline{B_{\frac{15}{16}}}\times\overline{B_{\frac{15}{16}}}\times\left[-\left(\frac{15}{16}\right)^{2},0\right]$
denotes a point reaching the maximum. By $L>0$, we know that
$\overline{x}\neq\overline{y}$. Choosing

$L_{2}\geq\frac{32\|u\|_{L^{\infty}(Q_{1})}}{(\min\\{\mathrm{dist}((x_{0},t_{0}),\partial
Q_{15/16}),\mathrm{dist}((y_{0},t_{0}),\partial Q_{15/16})\\})^{2}},$

we get

$|\overline{x}-x_{0}|+|\overline{t}-t_{0}|\leq
2\sqrt{\frac{2\|u\|_{L^{\infty}(Q_{1})}}{L_{2}}}\leq\frac{\mathrm{dist}((x_{0},t_{0}),\partial
Q_{15/16})}{2}$

and

$|\overline{y}-y_{0}|+|\overline{t}-t_{0}|\leq\frac{(\mathrm{dist}(y_{0},t_{0}),\partial
Q_{15/16})}{2},$

so that $\overline{x},\overline{y}\in B_{15/16}$ and
$\overline{t}\in\left(-\left(\frac{15}{16}\right)^{2},0\right)$. In addition,
if $L_{1}$ is large enough, we then find that

$|\overline{x}-\overline{y}|\leq\left(\frac{2\|u\|_{L^{\infty}(Q_{1})}}{L_{1}}\right)^{\frac{1}{\gamma}}$

is sufficiently small, which is crucial and will be used later.

By Jensen-Ishii’s lemma (see [23, Theorem 8.3]), there are

$(\sigma+L_{2}(\overline{t}-t_{0}),\eta_{1},X+L_{2}I)\in\overline{\mathcal{P}}^{2,+}u(\overline{x},\overline{t}),$
$(\sigma,\eta_{2},Y-L_{2}I)\in\overline{\mathcal{P}}^{2,-}u(\overline{y},\overline{t}),$

where

$\displaystyle\eta_{1}=L_{1}\phi^{\prime}(|\overline{x}-\overline{y}|)\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}+L_{2}(\overline{x}-x_{0}),$
$\displaystyle\eta_{2}=L_{1}\phi^{\prime}(|\overline{x}-\overline{y}|)\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}-L_{2}(\overline{y}-y_{0}).$

By choosing $L_{1}\geq C(\gamma)L_{2}$ large enough, there holds that

$\frac{L_{1}}{2}\gamma|\overline{x}-\overline{y}|^{\gamma-1}\leq|\eta_{1}|,|\eta_{2}|\leq
2L_{1}\gamma|\overline{x}-\overline{y}|^{\gamma-1}.$ (6.1)

By means of Jensen-Ishii’s lemma [22, Theorem 12.2], we could take $X,Y\in
Sym(n)$ such that for any $\tau>0$ satisfying $\tau Z<I$, it holds that

$-\frac{2}{\tau}\left(\begin{array}[]{cc}I&\\\
&I\end{array}\right)\leq\left(\begin{array}[]{cc}X&\\\
&-Y\end{array}\right)\leq\left(\begin{array}[]{cc}Z^{\tau}&-Z^{\tau}\\\
-Z^{\tau}&Z^{\tau}\end{array}\right),$ (6.2)

where

$\displaystyle Z$
$\displaystyle=L_{1}\phi^{\prime\prime}(|\overline{x}-\overline{y}|)\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}\otimes\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}+\frac{L_{1}\phi^{\prime}(|\overline{x}-\overline{y}|)}{|\overline{x}-\overline{y}|}\left(I-\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}\otimes\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}\right)$
$\displaystyle=L_{1}\gamma|\overline{x}-\overline{y}|^{\gamma-2}\left(I+(\gamma-2)\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}\otimes\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}\right)$

and

$Z^{\tau}=(I-\tau Z)^{-1}Z.$

Here $(I-\tau Z)^{-1}$ stands for the inverse of the matrix $I-\tau Z$. We now
pick $\tau=\frac{1}{2L_{1}\gamma|\overline{x}-\overline{y}|^{\gamma-2}}$ such
that

$Z^{\tau}=2L_{1}\gamma|\overline{x}-\overline{y}|^{\gamma-2}\left(I-2\frac{2-\gamma}{3-\gamma}\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}\otimes\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}\right).$

Furthermore, for
$\xi=\frac{\overline{x}-\overline{y}}{|\overline{x}-\overline{y}|}$ we get

$\langle
Z^{\tau}\xi,\xi\rangle=2\gamma\frac{\gamma-1}{3-\gamma}L_{1}|\overline{x}-\overline{y}|^{\gamma-2}<0.$
(6.3)

It follows from (6.2) that

$X\leq Y$

and

$\|X\|,\|Y\|\leq 4\gamma L_{1}|\overline{x}-\overline{y}|^{\gamma-2}.$ (6.4)

We next introduce a notation. Let

$A^{\varepsilon}(\eta)=I+(p-2)\frac{\eta}{(|\eta|^{2}+\varepsilon^{2})^{\frac{1}{2}}}\otimes\frac{\eta}{(|\eta|^{2}+\varepsilon^{2})^{\frac{1}{2}}}.$

It is easy to recognize that the eigenvalues of $A^{\varepsilon}(\eta)$ belong
to $\left(\min\\{1,p-1\\},\max\\{1,p-1\\}\right)$. Since $u$ is a viscosity
solution of (3.1), we will obtain the following viscosity inequalities

$\sigma+L_{2}(\overline{t}-t_{0})-\left[(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(\overline{x},\overline{t})(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{s}{2}}\right]\mathrm{tr}(A^{\varepsilon}(\eta_{1})(X+L_{2}I))\leq
0$

and

$\sigma-\left[(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(\overline{y},\overline{t})(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{s}{2}}\right]\mathrm{tr}(A^{\varepsilon}(\eta_{2})(Y-L_{2}I))\geq
0.$

Thus

$\displaystyle\quad L_{2}(\overline{t}-t_{0})$
$\displaystyle\leq(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{1})(X+L_{2}I))-(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{2})(Y-L_{2}I))$
$\displaystyle\quad+a(\overline{x},\overline{t})\left[(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{s}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{1})(X+L_{2}I))-(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{s}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{2})(Y-L_{2}I))\right]$
$\displaystyle\quad+(a(\overline{x},\overline{t})-a(\overline{y},\overline{t}))(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{s}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{2})(Y-L_{2}I))$
$\displaystyle=:J_{1}+J_{2}+J_{3}.$ (6.5)

We first evaluate $J_{3}$ as

$\displaystyle J_{3}$
$\displaystyle=(a(\overline{x},\overline{t})-a(\overline{y},\overline{t}))(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{s}{2}}[\mathrm{tr}(A^{\varepsilon}(\eta_{2})Y)-L_{2}\mathrm{tr}(A^{\varepsilon}(\eta_{2}))]$
$\displaystyle\leq C_{\rm
lip}|\overline{x}-\overline{y}|C_{0}(s)\gamma^{s}L^{s}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)s}[n\|A^{\varepsilon}(\eta_{2})\|\|Y\|+L_{2}(n+|p-2|)]$
$\displaystyle\leq C_{\rm
lip}C_{0}(s)\gamma^{s}L^{s}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)s+1}[4\gamma
n\max\\{1,p-1\\}L_{1}|\overline{x}-\overline{y}|^{\gamma-2}+L_{2}(n+|p-2|)],$

where we have used the inequalities (6.1) and (6.4) and the fact that $a(x,t)$
is uniformly Lipschitz continuous in $x$-variable.

We rewrite $J_{1}$ as

$\displaystyle J_{1}$
$\displaystyle=(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{1})(X-Y))+(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}((A^{\varepsilon}(\eta_{1})-A^{\varepsilon}(\eta_{2}))Y)$
$\displaystyle\quad+[(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}-(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{q}{2}}]\mathrm{tr}(A^{\varepsilon}(\eta_{2})Y)$
$\displaystyle\quad+L_{2}\left[(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{1}))+(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{2}))\right]$
$\displaystyle=:J_{1,1}+J_{1,2}+J_{1,3}+J_{1,4}.$

By (6.1), it is easy to get

$J_{1,4}\leq
2n\gamma^{q}C(q)\max\\{1,p-1\\}L_{2}L^{q}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)q}.$

In view of (6.2), we know that all the eigenvalues of $X-Y$ are non-positive
and at least one eigenvalue denoted by $\overline{\lambda}(X-Y)$ is smaller
than
$8\gamma\frac{\gamma-1}{3-\gamma}L_{1}|\overline{x}-\overline{y}|^{\gamma-2}$.
Hence,

$\displaystyle J_{1,1}$
$\displaystyle\leq(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\sum^{n}_{i=1}\lambda_{i}(A^{\varepsilon}(\eta_{1}))\lambda_{i}(X-Y)$
$\displaystyle\leq(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\min\\{1,p-1\\}\overline{\lambda}(X-Y)$
$\displaystyle\leq C_{1}(q)(\gamma
L_{1}|\overline{x}-\overline{y}|^{\gamma-1})^{q}\min\\{1,p-1\\}8\gamma\frac{\gamma-1}{3-\gamma}L_{1}|\overline{x}-\overline{y}|^{\gamma-2}$
$\displaystyle=8\gamma^{1+q}C_{1}(q)\min\\{1,p-1\\}\frac{\gamma-1}{3-\gamma}L^{1+q}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)(q+1)-1}.$

To estimate $J_{1,2}$, we first note that

$\displaystyle\quad\|A^{\varepsilon}(\eta_{1})-A^{\varepsilon}(\eta_{2})\|$
$\displaystyle=2|p-2|\left|\frac{\eta_{1}}{(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{1}{2}}}-\frac{\eta_{2}}{(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{1}{2}}}\right|$
$\displaystyle\leq
2|p-2|\left(\left|\frac{\eta_{1}}{|\eta_{1}|}-\frac{\eta_{2}}{|\eta_{2}|}\right|\frac{|\eta_{1}|}{(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{1}{2}}}+\left|\frac{|\eta_{1}|}{(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{1}{2}}}-\frac{|\eta_{2}|}{(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{1}{2}}}\right|\right)$
$\displaystyle\leq
4|p-2|\max\left\\{\frac{|\eta_{1}-\eta_{2}|}{|\eta_{1}|},\frac{|\eta_{1}-\eta_{2}|}{|\eta_{2}|}\right\\}$
$\displaystyle\leq\frac{32|p-2|L_{2}}{\gamma
L_{1}|\overline{x}-\overline{y}|^{\gamma-1}},$

where $|\eta_{1}-\eta_{2}|\leq 4L_{2}$. Thus via (6.1) and (6.4) we arrive at

$\displaystyle J_{1,2}$
$\displaystyle\leq(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}n\|A^{\varepsilon}(\eta_{1})-A^{\varepsilon}(\eta_{2})\|\|Y\|$
$\displaystyle\leq C_{2}(q)(\gamma
L_{1}|\overline{x}-\overline{y}|^{\gamma-1})^{q}n\frac{32|p-2|L_{2}}{\gamma
L_{1}|\overline{x}-\overline{y}|^{\gamma-1}}4\gamma
L_{1}|\overline{x}-\overline{y}|^{\gamma-2}$
$\displaystyle=128nC_{2}(q)\gamma^{q}|p-2|L_{2}L_{1}^{q}|\overline{x}-\overline{y}|^{(\gamma-1)q-1}.$

We finally estimate $J_{1,3}$. Applying the mean value theorem and (6.1), we
evaluate

$\displaystyle\quad|(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}-(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{q}{2}}|$
$\displaystyle=\frac{q}{2}\zeta^{\frac{q}{2}-1}||\eta_{1}|^{2}-|\eta_{2}|^{2}|$
$\displaystyle\leq C_{3}(q)(\gamma
L_{1}|\overline{x}-\overline{y}|^{\gamma-1})^{q-2}(\gamma
L_{1}|\overline{x}-\overline{y}|^{\gamma-1})||\eta_{1}|-|\eta_{2}||$
$\displaystyle\leq
C_{3}(q)\gamma^{q-1}L_{1}^{q-1}|\overline{x}-\overline{y}|^{(q-1)(\gamma-1)}|\eta_{1}-\eta_{2}|$
$\displaystyle\leq
4C_{3}(q)\gamma^{q-1}L_{2}L_{1}^{q-1}|\overline{x}-\overline{y}|^{(q-1)(\gamma-1)},$

where $\zeta$ is between $|\eta_{1}|^{2}+\varepsilon^{2}$ and
$|\eta_{2}|^{2}+\varepsilon^{2}$. Then it follows from (6.4) that

$\displaystyle J_{1,3}$ $\displaystyle\leq
4C_{3}(q)\gamma^{q-1}L_{2}L_{1}^{q-1}|\overline{x}-\overline{y}|^{(q-1)(\gamma-1)}n\|Y\|\|A^{\varepsilon}(\eta_{2})\|$
$\displaystyle\leq
16nC_{3}(q)\max\\{1,p-1\\}\gamma^{q}L_{2}L_{1}^{q}|\overline{x}-\overline{y}|^{q(\gamma-1)-1}.$

Combining the estimates on $J_{1,1}$, $J_{1,2}$, $J_{1,3}$ and $J_{1,4}$, we
derive

$\displaystyle J_{1}\leq$
$\displaystyle-8\gamma^{1+q}C_{1}(q)\min\\{1,p-1\\}\frac{1-\gamma}{3-\gamma}L^{1+q}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)(q+1)-1}$
$\displaystyle+128nC_{2}(q)\gamma^{q}|p-2|L_{2}L_{1}^{q}|\overline{x}-\overline{y}|^{(\gamma-1)q-1}$
$\displaystyle+16nC_{3}(q)\max\\{1,p-1\\}\gamma^{q}L_{2}L_{1}^{q}|\overline{x}-\overline{y}|^{q(\gamma-1)-1}$
$\displaystyle+2n\gamma^{q}C(q)\max\\{1,p-1\\}L_{2}L^{q}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)q}.$

Analogously, we can arrive at

$\displaystyle J_{2}$ $\displaystyle\leq
a(\overline{x},\overline{t})\big{[}-8\gamma^{1+s}C_{1}(s)\min\\{1,p-1\\}\frac{1-\gamma}{3-\gamma}L^{1+s}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)(s+1)-1}$
$\displaystyle\quad+128nC_{2}(s)\gamma^{s}|p-2|L_{2}L_{1}^{s}|\overline{x}-\overline{y}|^{(\gamma-1)s-1}$
$\displaystyle\quad+16nC_{3}(s)\max\\{1,p-1\\}\gamma^{s}L_{2}L_{1}^{s}|\overline{x}-\overline{y}|^{(\gamma-1)s-1}$
$\displaystyle\quad+2n\gamma^{s}C(s)\max\\{1,p-1\\}L_{2}L^{s}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)s}\big{]}.$

Because the constant coefficients in $J_{1},J_{2},J_{3}$ are too long, we
simply denote them by $C$, possibly varying from line to line. And moreover
relevant dependencies on parameters will be emphasised using parentheses.
Consequently, (6) becomes

$\displaystyle 0$ $\displaystyle\leq
L_{2}+\big{[}-8C(p,q,\gamma)L^{1+q}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)(q+1)-1}+C(n,p,q,\gamma)L_{2}L_{1}^{q}|\overline{x}-\overline{y}|^{(\gamma-1)q-1}$
$\displaystyle\quad+C(n,p,q,\gamma)L_{2}L_{1}^{q}|\overline{x}-\overline{y}|^{q(\gamma-1)-1}+C(n,p,q,\gamma)L_{2}L^{q}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)q}\big{]}$
$\displaystyle\quad+a(\overline{x},\overline{t})\big{[}-8C(p,s,\gamma)L^{1+s}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)(s+1)-1}+C(n,p,s,\gamma)L_{2}L_{1}^{s}|\overline{x}-\overline{y}|^{(\gamma-1)s-1}$
$\displaystyle\quad+C(n,p,s,\gamma)L_{2}L_{1}^{s}|\overline{x}-\overline{y}|^{(\gamma-1)s-1}+C(n,p,s,\gamma)L_{2}L^{s}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)s}\big{]}$
$\displaystyle\quad+C_{\rm
lip}\big{[}C(n,p,s,\gamma)L_{1}^{s+1}|\overline{x}-\overline{y}|^{(\gamma-1)(s+1)}+C(n,p,s,\gamma)L_{2}L_{1}^{s}|\overline{x}-\overline{y}|^{(\gamma-1)s}\big{]}.$

We rearrange the previous display as

$\displaystyle 0$
$\displaystyle\leq\big{[}-8C(p,q,\gamma)L^{1+q}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)(q+1)-1}+C(n,p,q,\gamma)L_{2}L_{1}^{q}|\overline{x}-\overline{y}|^{(\gamma-1)q-1}\big{]}$
$\displaystyle\quad+a(\overline{x},\overline{t})\big{[}-8C(p,s,\gamma)L^{1+s}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)(s+1)-1}+C(n,p,s,\gamma)L_{2}L_{1}^{s}|\overline{x}-\overline{y}|^{(\gamma-1)s-1}$
$\displaystyle\quad+a(\overline{x},\overline{t})^{-1}C_{\rm
lip}C(n,p,s,\gamma)L_{1}^{s+1}|\overline{x}-\overline{y}|^{(\gamma-1)(s+1)}$
$\displaystyle\quad+a(\overline{x},\overline{t})^{-1}C_{\rm
lip}C(n,p,s,\gamma)L_{2}L_{1}^{s}|\overline{x}-\overline{y}|^{(\gamma-1)s-1}\big{]}.$

Therefore, we can select $L_{1}$ large enough such that

$\begin{cases}L_{1}|\overline{x}-\overline{y}|^{\gamma-1}\geq
C(n,p,q,\gamma)L_{2},\\\\[5.69054pt] |\overline{x}-\overline{y}|^{-1}\geq
C(n,p,s,\gamma)(a^{-})^{-1}C_{\rm lip},\\\\[5.69054pt]
L_{1}|\overline{x}-\overline{y}|^{\gamma-1}\geq
C(n,p,s,\gamma)L_{2},\\\\[5.69054pt]
L_{1}|\overline{x}-\overline{y}|^{\gamma-1}\geq
C(n,p,s,\gamma)(a^{-})^{-1}C_{\rm lip}L_{2}.\end{cases}$

Thanks to
$|\overline{x}-\overline{y}|\leq\left(\frac{2\|u\|_{L^{\infty}(Q_{1})}}{L_{1}}\right)^{\frac{1}{\gamma}}$,
then we require that

$\left(\frac{2\|u\|_{L^{\infty}(Q_{1})}}{L_{1}}\right)^{-\frac{1}{\gamma}}\geq
C(n,p,s,\gamma)(a^{-})^{-1}C_{\rm lip}$

so that $|\overline{x}-\overline{y}|^{-1}\geq
C(n,p,s,\gamma)(a^{-})^{-1}C_{\rm lip}$ holds true. As a consequence, we can
fix

$L_{1}=C(n,p,q,s,\gamma)\left(\|u\|_{L^{\infty}(Q_{1})}+(a^{-})^{-1}C_{\rm
lip}\|u\|_{L^{\infty}(Q_{1})}\right),$

then derive that

$0\leq-7C(p,q,\gamma)L^{1+q}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)(q+1)-1}-a^{-}5C(p,s,\gamma)L^{1+s}_{1}|\overline{x}-\overline{y}|^{(\gamma-1)(s+1)-1}.$

That is a contradiction. Finally, we obtain the desired result by the
dependence of $L_{1}$. ∎

Based on Lemma 6.1, we can demonstrate the Lipschitz continuity (Lemma 3.1) of
solutions to (3.1) in the spatial variables via applying Ishii-Lions methods
again.

Proof of Lemma 3.1. Fix $r=\frac{7}{8}$ and $x_{0},y_{0}\in B_{r}$,
$t_{0}\in(-r^{2},0)$. We are going to show that there exist two suitable
constants $M_{1},M_{2}>0$ such that

$M_{0}:=\sup_{(x,t),(y,t)\in\overline{Q_{\frac{7}{8}}}}(u(x,t)-u(y,t)-M_{1}\varphi(|x-y|)-\Phi(x,y,t))\leq
0,$

where

$\Phi(x,y,t)=\frac{M_{2}}{2}|x-x_{0}|^{2}+\frac{M_{2}}{2}|y-y_{0}|^{2}+\frac{M_{2}}{2}|t-t_{0}|^{2}$

and

$\varphi(r):=\begin{cases}r-\kappa_{0}r^{\nu}&{0\leq r\leq
r_{1}:=\left(\frac{1}{4\nu\kappa_{0}}\right)^{\frac{1}{\nu-1}}},\\\\[5.69054pt]
\varphi(r_{1})&{r>r_{1}}\end{cases}$

with $1<\nu<2$ and $0<\kappa_{0}<1$ such that $2<r_{1}$. Observe that, for
$r\in(0,r_{1})$,

$\begin{cases}\varphi^{\prime}(r)=1-\nu\kappa_{0}r^{\nu-1},\\\\[5.69054pt]
\varphi^{\prime\prime}(r)=-\nu(\nu-1)\kappa_{0}r^{\nu-2},\end{cases}$

and then $\frac{3}{4}\leq\varphi^{\prime}(r)\leq 1$ and
$\varphi^{\prime\prime}(r)<0$ when $r\in(0,2]$. We now argue by contradiction.
Assume that $M_{0}>0$ and
$(\hat{x},\hat{y},\hat{t})\in\overline{B_{r}}\times\overline{B_{r}}\times[-r^{2},0]$
represents a point attaining the maximum. As in the proof of Lemma 6.1, we
recognize that $\hat{x}\neq\hat{y}$ and $\hat{x},\hat{y}\in B_{r}$,
$\hat{t}\in(-r^{2},0)$ for $M_{2}\geq C\|u\|_{L^{\infty}(Q_{1})}$.
Furthermore, we know from Lemma 6.1 that $u$ is locally Hölder continuous in
$x$-variable and for some $\gamma\in(0,1)$, it holds that

$|u(x,t)-u(y,t)|\leq C\|u\|_{L^{\infty}(Q_{1})}\left[1+(a^{-})^{-1}C_{\rm
lip}\right]|x-y|^{\gamma}.$

In the rest of proof, for shortness we denote

$K:=C\|u\|_{L^{\infty}(Q_{1})}\left[1+(a^{-})^{-1}C_{\rm lip}\right].$

Employing the above inequality and choosing $2M_{2}\leq K$, we get

$M_{2}|\hat{y}-y_{0}|,M_{2}|\hat{x}-x_{0}|\leq
K|\hat{x}-\hat{y}|^{\frac{\gamma}{2}}.$ (6.6)

In addition, it follows from $M_{0}>0$ that

$M_{1}(|\hat{x}-\hat{y}|-\kappa_{0}|\hat{x}-\hat{y}|^{\nu})\leq
2\|u\|_{L^{\infty}(Q_{1})},$

i.e.,

$M_{1}|\hat{x}-\hat{y}|(1-\kappa_{0}|\hat{x}-\hat{y}|^{\nu-1})\leq
2\|u\|_{L^{\infty}(Q_{1})}.$

Thus we can fix $0<\kappa_{0}<1$ such that $\frac{1}{2}\leq
1-\kappa_{0}|\hat{x}-\hat{y}|^{\nu-1}$, from which we notice that

$|\hat{x}-\hat{y}|\leq\frac{4\|u\|_{L^{\infty}(Q_{1})}}{M_{1}}.$

By Jensen-Ishii’s lemma, we have

$(\sigma+M_{2}(\hat{t}-t_{0}),\eta_{1},X+M_{2}I)\in\overline{\mathcal{P}}^{2,+}u(\hat{x},\hat{t}),$
$(\sigma,\eta_{2},Y-M_{2}I)\in\overline{\mathcal{P}}^{2,-}u(\hat{y},\hat{t}),$

and for any $\tau>0$,

$-(\tau+2\|Z\|)\left(\begin{array}[]{cc}I&\\\
&I\end{array}\right)\leq\left(\begin{array}[]{cc}X&\\\
&-Y\end{array}\right)\leq\left(\begin{array}[]{cc}Z&-Z\\\
-Z&Z\end{array}\right)+\frac{2}{\tau}\left(\begin{array}[]{cc}Z^{2}&-Z^{2}\\\
-Z^{2}&Z^{2}\end{array}\right),$ (6.7)

where

$\displaystyle\eta_{1}=M_{1}\varphi^{\prime}(|\hat{x}-\hat{y}|)\frac{\hat{x}-\hat{y}}{|\hat{x}-\hat{y}|}+M_{2}(\hat{x}-x_{0}),$
$\displaystyle\eta_{2}=M_{1}\varphi^{\prime}(|\hat{x}-\hat{y}|)\frac{\hat{x}-\hat{y}}{|\hat{x}-\hat{y}|}-M_{2}(\hat{y}-y_{0})$

and

$Z=M_{1}\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)\frac{\hat{x}-\hat{y}}{|\hat{x}-\hat{y}|}\otimes\frac{\hat{x}-\hat{y}}{|\hat{x}-\hat{y}|}+M_{1}\frac{\varphi^{\prime}(|\hat{x}-\hat{y}|)}{|\hat{x}-\hat{y}|}\left(I-\frac{\hat{x}-\hat{y}}{|\hat{x}-\hat{y}|}\otimes\frac{\hat{x}-\hat{y}}{|\hat{x}-\hat{y}|}\right).$

Direct computations give that

$\frac{M_{1}}{2}\leq|\eta_{1}|,|\eta_{2}|\leq 2M_{1},\quad\text{if }M_{1}\geq
4K$ (6.8)

and

$\|Z\|\leq
M_{1}\frac{\varphi^{\prime}(|\hat{x}-\hat{y}|)}{|\hat{x}-\hat{y}|},$ (6.9)
$\|Z^{2}\|\leq
M_{1}^{2}\left(|\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)|+\frac{\varphi^{\prime}(|\hat{x}-\hat{y}|)}{|\hat{x}-\hat{y}|}\right)^{2}.$
(6.10)

According to (6.7), we infer that $X-Y\leq 0$. Besides, by taking

$\tau=4M_{1}\left(|\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)|+\frac{\varphi^{\prime}(|\hat{x}-\hat{y}|)}{|\hat{x}-\hat{y}|}\right)$

and applying (6.7) to the vector $(\xi,-\xi)$ with
$\xi=\frac{\hat{x}-\hat{y}}{|\hat{x}-\hat{y}|}$, after careful manipulation,
we derive that

$\langle(X-Y)\xi,\xi\rangle\leq 4\left(\langle
Z\xi,\xi\rangle+\frac{2}{\tau}\langle Z^{2}\xi,\xi\rangle\right)\leq
2M_{1}\varphi^{\prime\prime}(|\hat{x}-\hat{y}|),$ (6.11)

which indicates that at least one eigenvalue of $X-Y$ denoted by
$\overline{\lambda}(X-Y)$ is smaller than
$2M_{1}\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)<0$. Writing the viscosity
inequalities and adding them, we arrive at

$\displaystyle\quad M_{2}(\hat{t}-t_{0})$
$\displaystyle\leq(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{1})(X+M_{2}I))-(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{2})(Y-M_{2}I))$
$\displaystyle\quad+a(\hat{x},\hat{t})\left[(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{s}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{1})(X+M_{2}I))-(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{s}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{2})(Y-M_{2}I))\right]$
$\displaystyle\quad+(a(\hat{x},\hat{t})-a(\hat{y},\hat{t}))(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{s}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{2})(Y-M_{2}I))$
$\displaystyle=:J_{1}+J_{2}+J_{3}.$ (6.12)

Putting together (6.7), (6.9) and (6.10), we deduce that

$\displaystyle\|Y\|$ $\displaystyle\leq 2|\langle
Z\overline{\xi},\overline{\xi}\rangle|+\frac{4}{\tau}|\langle
Z^{2}\overline{\xi},\overline{\xi}\rangle|$ $\displaystyle\leq
4M_{1}\left(|\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)|+\frac{\varphi^{\prime}(|\hat{x}-\hat{y}|)}{|\hat{x}-\hat{y}|}\right),$
(6.13)

where $\overline{\xi}$ is a unit vector. In what follows, we estimate the
terms $J_{1},J_{2},J_{3}$ in a similar way to that in proof of Lemma 6.1. So
we write it briefly. It yields by (6.8) and (6.11) that

$(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{1})(X-Y))\leq
2\min\\{1,1-p\\}C_{1}(q)M_{1}^{1+q}\varphi^{\prime\prime}(|\hat{x}-\hat{y}|).$
(6.14)

In view of (6.6), (6.8), (6), we have

$\displaystyle\quad(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}((A^{\varepsilon}(\eta_{1})-A^{\varepsilon}(\eta_{2}))Y)$
$\displaystyle\leq
64nC_{2}(q)|p-2|KM^{q}_{1}\left(\varphi^{\prime}(|\hat{x}-\hat{y}|)|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}+|\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)|\right).$
(6.15)

By the mean value theorem together with the inequalities (6.6) and (6.8), it
follows that

$|(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}-(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{q}{2}}|\leq
C_{3}(q)KM^{q-1}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}},$

which leads to

$\displaystyle\quad|(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}-(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{q}{2}}||\mathrm{tr}(A^{\varepsilon}(\eta_{2})Y)|$
$\displaystyle\leq
nC_{3}(q)\max\\{1,p-1\\}KM^{q}_{1}\left(|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}+|\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)|\right).$
(6.16)

We evaluate

$\begin{split}&\quad
M_{2}\left[(|\eta_{1}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{1}))+(|\eta_{2}|^{2}+\varepsilon^{2})^{\frac{q}{2}}\mathrm{tr}(A^{\varepsilon}(\eta_{2}))\right]\\\
&\leq 2^{q+1}n\max\\{1,p-1\\}M_{2}M^{q}_{1}.\end{split}$ (6.17)

Merging the estimates (6.14)–(6.17), we obtain

$\displaystyle J_{1}$ $\displaystyle\leq
2C_{1}(q)\min\\{1,p-1\\}M^{1+q}_{1}\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)$
$\displaystyle\quad+64nC_{2}(q)|p-2|KM_{1}^{q}\left(|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}+|\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)|\right)$
$\displaystyle\quad+nC_{3}(q)\max\\{1,p-1\\}KM_{1}^{q}\left(|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}+|\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)|\right)$
$\displaystyle\quad+2^{q+1}n\max\\{1,p-1\\}M_{2}M^{q}_{1}.$

Likewise, we can arrive at

$\displaystyle J_{2}$ $\displaystyle\leq
a(\hat{x},\hat{t})\big{[}2C_{1}(s)\min\\{1,p-1\\}M^{1+s}_{1}\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)$
$\displaystyle\quad+64nC_{2}(s)|p-2|KM_{1}^{s}(|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}+|\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)|)$
$\displaystyle\quad+nC_{3}(s)\max\\{1,p-1\\}KM_{1}^{s}(|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}+|\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)|)$
$\displaystyle\quad+2^{s+1}n\max\\{1,p-1\\}M_{2}M^{s}_{1}\big{]}.$

Finally, $J_{3}$ could be evaluated as

$J_{3}\leq 4nC_{4}(s)\max\\{1,p-1\\}C_{\rm
lip}M^{s}_{1}(M_{1}(1+|\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)||\hat{x}-\hat{y}|)+M_{2}),$

where we have used the fact that $|a(\hat{x},\hat{t})-a(\hat{y},\hat{t})|\leq
C_{\rm lip}|\hat{x}-\hat{y}|$. Observe that
$\varphi^{\prime\prime}(|\hat{x}-\hat{y}|)=-\nu(\nu-1)\kappa_{0}|\hat{x}-\hat{y}|^{\nu-2}$.
Because the constant coefficients in $J_{1},J_{2},J_{3}$ are too long, we
simply denote by $C$ as before. And moreover relevant dependencies on
parameters will be emphasised using parentheses. Next, we take
$\nu=\frac{\gamma}{2}+1$. Therefore, after rearrangement inequality (6) turns
into

$\displaystyle 0$
$\displaystyle\leq\big{[}-2C(p,q)M^{1+q}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}+C(n,p,q)KM^{q}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}+C(n,p,q)M_{2}M_{1}^{q}\big{]}$
$\displaystyle\quad+a(\hat{x},\hat{t})\big{[}-2C(p,s)M^{1+s}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}+C(n,p,s)KM_{1}^{s}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}+C(n,p,s)M_{2}M^{s}_{1}$
$\displaystyle\quad+a(\hat{x},\hat{t})^{-1}C(n,p,s)C_{\rm
lip}M_{1}^{1+s}+a(\hat{x},\hat{t})^{-1}C(n,p,s)C_{\rm
lip}M_{2}M_{1}^{s}\big{]}.$ (6.18)

To reach a contradiction, we shall choose $M_{1}$ large so that

$\begin{cases}\frac{1}{2}C(p,q)M^{1+q}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}\geq
C(n,p,q)KM_{1}^{q}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1},\\\\[5.69054pt]
\frac{1}{2}C(p,q)M^{1+q}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}\geq
C(n,p,q)M_{2}M_{1}^{q},\\\\[5.69054pt]
\frac{1}{4}C(p,s)M^{1+s}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}\geq
C(n,p,s)(a^{-})^{-1}C_{\rm lip}M^{1+s}_{1},\\\\[5.69054pt]
\frac{1}{4}C(p,s)M^{1+s}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}\geq
C(n,p,s)KM_{1}^{s}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1},\\\\[5.69054pt]
\frac{1}{4}C(p,s)M^{1+s}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}\geq
C(n,p,s)M_{2}M^{s}_{1},\\\\[5.69054pt]
\frac{1}{4}C(p,s)M^{1+s}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}\geq
C(n,p,s)(a^{-})^{-1}C_{\rm lip}M_{2}M_{1}^{s}.\end{cases}$

Utilizing the known fact that
$|\hat{x}-\hat{y}|\leq\frac{4\|u\|_{L^{\infty}(Q_{1})}}{M_{1}}$, we can take

$M_{1}\geq C(n,p,q,s)\left(\|u\|_{L^{\infty}(Q_{1})}\left(1+(a^{-})^{-1}C_{\rm
lip}+[(a^{-})^{-1}C_{\rm lip}]^{\frac{2}{2-\gamma}}\right)+K\right)$

to satisfy the above requirement. After manipulation, we can eventually fix

$M_{1}=C(n,p,q,s)\|u\|_{L^{\infty}(Q_{1})}\left(1+[(a^{-})^{-1}C_{\rm
lip}]^{2}\right)$

such that display (6) becomes

$0\leq-C(p,q)M^{1+q}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}-a(\hat{x},\hat{t})C(p,s)M^{1+s}_{1}|\hat{x}-\hat{y}|^{\frac{\gamma}{2}-1}.$

That is a contradiction. Therefore, we obtain the desired result by the
dependence of $M_{1}$.

##  7\. The proof of Proposition 4.11

In this section, we will give the proof of the boundary estimates in
Proposition 4.11. For the sake of convenience, denote

$F_{\varepsilon}(x,t,Du,D^{2}u)=\big{[}(|Du|^{2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(|Du|^{2}+\varepsilon^{2})^{\frac{s}{2}}\big{]}\left(\delta_{ij}+(p-2)\frac{u_{i}u_{j}}{|Du|^{2}+\varepsilon^{2}}\right)u_{ij}.$

In what follows, we assume that the conditions (2.1)–(2.3) are in force.

######  Lemma 7.1.

For each $y\in\partial B_{1}$, there is a function $W_{y}(x)\in
C(\overline{B_{1}})$ fulfilling $W_{y}(y)=0$ and $W_{y}(x)>0$ in
$\overline{B_{1}}\setminus\\{y\\}$, and for all $t\in[-1,0]$

$F_{\varepsilon}(x,t,DW_{y},D^{2}W_{y})\leq-1\quad\text{in }B_{1}.$

###### Proof.

Let $y\in\partial B_{1}$, $f(r)=\sqrt{(r-1)_{+}}$ and $g_{y}(x)=f(|x-2y|)$.
Here $(r-1)_{+}:=\max\\{r-1,0\\}$. By direct calculation, then for $x\in
B_{1}$ and $t\in[-1,0]$, we derive

$\displaystyle\quad F_{\varepsilon}(x,t,Dg_{y},D^{2}g_{y})$
$\displaystyle=\left[(f^{\prime
2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(f^{\prime
2}+\varepsilon^{2})^{\frac{s}{2}}\right]\left[\left(1+(p-2)\frac{f^{\prime
2}}{f^{\prime
2}+\varepsilon^{2}}\right)f^{\prime\prime}+\frac{n-1}{|x-2y|}f^{\prime}\right]$
$\displaystyle\leq\left[(f^{\prime
2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(f^{\prime
2}+\varepsilon^{2})^{\frac{s}{2}}\right](r-1)^{-\frac{1}{2}}\left(\frac{n-1}{2}-\frac{\min\\{1,p-1\\}}{4}(r-1)^{-1}\right)$
$\displaystyle\leq\left(\frac{(r-1)^{-1}}{4}+\varepsilon^{2}\right)^{\frac{q}{2}}(r-1)^{-\frac{1}{2}}\left(\frac{n-1}{2}-\frac{\min\\{1,p-1\\}}{4}(r-1)^{-1}\right),$

if we choose $r>1$ sufficiently close to 1, where $r:=|x-2y|$. Therefore,
there exists $\delta>0$, that depends only on $n,p,q$, such that for $x\in
B_{1}\cap B_{1+\delta}(2y)$ and $t\in[-1,0]$ it holds that

$F_{\varepsilon}(x,t,Dg_{y},D^{2}g_{y})\leq-1.$

Set

$h_{y}(x)=b\left(2^{\sigma}-\frac{1}{|x-2y|^{\sigma}}\right)$

with

$\sigma=\frac{2n}{\min\\{1,p-1\\}}+2\quad\text{and}\quad b>0.$

Clearly, $h_{y}(x)\geq b(2^{\sigma}-1)$ in $B_{1}$. Also, for $x\in B_{1}$ and
$t\in[-1,0]$, using the choice of $\sigma$ and $1<r<3$, we get

$\displaystyle\quad F_{\varepsilon}(x,t,Dh_{y},D^{2}h_{y})$
$\displaystyle=b\left[(b^{2}\sigma^{2}r^{-2\sigma-2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(b^{2}\sigma^{2}r^{-2\sigma-2}+\varepsilon^{2})^{\frac{s}{2}}\right]$
$\displaystyle\quad\cdot\left[\left(1+\frac{(p-2)\sigma^{2}}{\sigma^{2}+\varepsilon^{2}b^{-2}r^{2\sigma+2}}\right)\sigma(-\sigma-1)r^{-\sigma-2}+(n-1)\sigma
r^{-\sigma-2}\right]$ $\displaystyle\leq
b\left[(b^{2}\sigma^{2}r^{-2\sigma-2}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(b^{2}\sigma^{2}r^{-2\sigma-2}+\varepsilon^{2})^{\frac{s}{2}}\right]\left(-\frac{1}{2}\sigma
r^{-\sigma-2}\right)$ $\displaystyle\leq-\frac{b}{2}\sigma
r^{-\sigma-2}(b^{2}\sigma^{2}r^{-2\sigma-2}+\varepsilon^{2})^{\frac{q}{2}}$
$\displaystyle\leq\begin{cases}-\frac{b^{1+q}}{2}3^{-\sigma-2-q(\sigma+1)}\sigma^{1+q}&{\text{if
}q\geq 0},\\\\[5.69054pt]
-\frac{b}{2}3^{-\sigma-2}(1+\sigma^{2})^{\frac{q}{2}}\sigma&{\text{if
}-1<q<0}.\end{cases}$

Then we take $b<1$ satisfying

$b\left(2^{\sigma}-\frac{1}{|1+\delta|^{\sigma}}\right)=\sqrt{\frac{\delta}{2}}.$

Thanks to $g_{y}(y)=0$ and $h_{y}(y)>0$, the function

$W_{y}(x)=\begin{cases}h_{y}(x)&{\text{if }x\in\overline{B_{1}},|x-2y|\geq
1+\delta},\\\\[5.69054pt] \min\\{g_{y}(x),h_{y}(x)\\}&{\text{if
}x\in\overline{B_{1}},|x-2y|\leq 1+\delta}\end{cases}$

consists with $g_{y}$ in a neighborhood of $y$, and, by the selection of $b$,
consists with $h_{y}$ when $x\in\overline{B_{1}}$ and $|x-2y|\geq
1+\tilde{\delta}$ for some $\tilde{\delta}\in(0,\delta)$. Furthermore,

$F_{\varepsilon}(x,t,DW_{y},D^{2}W_{y})\leq-C$

for $x\in B_{1}$ and $t\in[-1,0]$, where $C>0$ depends only upon $n,p$ and
$q$. We conclude this proof through multiplying a large positive constant to
$W_{y}$. ∎

######  Lemma 7.2.

For each $(y,\tau)\in\partial_{p}Q_{1}$, there is $W_{y,\tau}\in
C(\overline{Q_{1}})$ satisfying $W_{y,\tau}(y,\tau)=0$, $W_{y,\tau}>0$ in
$\overline{Q_{1}}\setminus\\{(y,\tau)\\}$ as well as

$\partial_{t}W_{y,\tau}-F_{\varepsilon}(x,t,DW_{y,\tau},D^{2}W_{y,\tau})\geq
1\quad\text{in }Q_{1}.$

###### Proof.

For $\tau\geq-1$ and $y\in\partial B_{1}$, we can construct

$W_{y,\tau}(x,t)=\frac{(t-\tau)^{2}}{2}+2W_{y}(x),$

which is a desired function apparently. Here $W_{y}(x)$ comes from Lemma 7.1.
If $\tau=-1$ and $y\in B_{1}$, define

$W_{y,\tau}(x,t)=B(t+1)+|x-y|^{\iota}$

with

$\iota=\max\left\\{2,\frac{q+2}{q+1},\frac{s+2}{s+1}\right\\}.$

By virtue of the definition of $\iota$, we now evaluate

$\displaystyle\quad\partial_{t}W_{y,\tau}-F_{\varepsilon}(x,t,DW_{y,\tau},D^{2}W_{y,\tau})$
$\displaystyle=B-\left[(\iota^{2}|x-y|^{2(\iota-1)}+\varepsilon^{2})^{\frac{q}{2}}+a(x,t)(\iota^{2}|x-y|^{2(\iota-1)}+\varepsilon^{2})^{\frac{s}{2}}\right]$
$\displaystyle\quad\cdot\left((n-1)\iota+\iota(\iota-1)+(p-2)\frac{\iota^{2}|x-y|^{2(\iota-1)}}{\iota^{2}|x-y|^{2(\iota-1)}+\varepsilon^{2}}\iota(\iota-1)\right)|x-y|^{\iota-2}$
$\displaystyle\geq
B-\left[(\iota^{2}|x-y|^{2(\iota-1)}+\varepsilon^{2})^{\frac{q}{2}}+a^{+}(\iota^{2}|x-y|^{2(\iota-1)}+\varepsilon^{2})^{\frac{s}{2}}\right]\iota(n-1+p(\iota-1))|x-y|^{\iota-2}$
$\displaystyle\geq\begin{cases}B-\big{[}(16+1)^{\frac{q}{2}}+a^{+}(16+1)^{\frac{s}{2}}\big{]}2(n-1+p)&{\text{if
}q\geq 0},\\\\[5.69054pt]
B-\big{[}\iota^{q}|x-y|^{q(\iota-1)+\iota-2}+a^{+}\iota^{s}|x-y|^{s(\iota-1)+\iota-2}\big{]}\iota(n-1+p(\iota-1))&{\text{if
}-1<s<0},\\\\[5.69054pt]
B-\big{[}\iota^{q}|x-y|^{q(\iota-1)+\iota-2}+a^{+}(\iota^{2}2^{2(\iota-1)}+1)^{\frac{s}{2}}\big{]}\iota(n-1+p(\iota-1))&{{}\text{others}},\end{cases}$

which leads to

$\partial_{t}W_{y,\tau}-F_{\varepsilon}(x,t,DW_{y,\tau},D^{2}W_{y,\tau})\geq
B-C,$

where $C>0$ depends on $n,p,q,s$ and $a^{+}$. From that, we can see that
$W_{y,\tau}$ will be a desired function as well, if we pick $B=C+1$. ∎

Recall that $a\vee b=\max\\{a,b\\}$ and $a\wedge b=\min\\{a,b\\}$ with $a$ and
$b$ being two real numbers. With Lemma 7.2 in hand, we can deduce the
following result. The proof is the same as that of Theorem A.3 in [34], so we
omit it here.

######  Corollary 7.3.

Suppose that $u\in C(\overline{Q_{1}})$ is a solution to (3.1) with
$\varepsilon\in(0,1)$. Let $\varphi:=u\mid_{\partial_{p}Q_{1}}$ and $\rho$ be
a modulus of continuity of $\varphi$. Then there is another modulus of
continuity $\tilde{\rho}$, which depends on $n,p,q,s,a^{+}$ and $\rho$, such
that, for any $(x,t)\in\overline{Q_{1}}$ and $(y,s)\in\partial_{p}Q_{1}$,

$|u(x,t)-u(y,s)|\leq\tilde{\rho}(|x-y|\vee\sqrt{|t-s|})$

holds true.

As a consequence, merging Lemmas 3.1 and 3.3 with Corollary 7.3, we could
arrive at the boundary estimates (Proposition 4.11) by following the proof of
Proposition 2.5 in [36]. We omit the detailed proof.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(No. 12071098).

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