.

# Regularity of weak solutions for mixed local and nonlocal double phase
parabolic equations

Bin Shang and Chao Zhang∗ Bin Shang School of Mathematics, Harbin Institute of
Technology, Harbin 150001, P.R. China<EMAIL_ADDRESS>Chao Zhang School
of Mathematics and Institute for Advanced Study in Mathematics, Harbin
Institute of Technology, Harbin 150001, P.R. China<EMAIL_ADDRESS>
###### Abstract.

We study the mixed local and nonlocal double phase parabolic equation

$\displaystyle\partial_{t}u(x,t)-\mathrm{div}(a(x,t)|\nabla u|^{q-2}\nabla
u)+\mathcal{L}u(x,t)=0$

in $Q_{T}=\Omega\times(0,T)$, where $\mathcal{L}$ is the nonlocal $p$-Laplace
type operator. The local boundedness of weak solutions is proved by means of
the De Giorgi-Nash-Moser iteration with the nonnegative coefficient function
$a(x,t)$ being bounded. In addition, when $a(x,t)$ is Hölder continuous, we
discuss the pointwise behavior and lower semicontinuity of weak supersolutions
based on energy estimates and De Giorgi type lemma. Analogously, the
corresponding results are also valid for weak subsolutions.

###### Key words and phrases:

Local boundedness; Semicontinuity representative; Pointwise behavior; Mixed
local and nonlocal parabolic equations; Double phase

###### 2010 Mathematics Subject Classification:

35B45, 35B65, 35D30, 35K55, 35R11

∗Corresponding author.

##  1\. Introduction

In this paper, we investigate some regularity results of the following problem

$\displaystyle\partial_{t}u(x,t)-\mathrm{div}(a(x,t)|\nabla u|^{q-2}\nabla
u)+\mathcal{L}u(x,t)=0\text{ \ \ in }Q_{T},$ (1.1)

where $Q_{T}:=\Omega\times(0,T)$ with $T>0$ and $\Omega$ being a bounded
domain in $\mathbb{R}^{N}$. The fractional $p$-Laplace operator $\mathcal{L}$
is given by

$\displaystyle\mathcal{L}u(x,t)=\mathrm{P.V.}\int_{\mathbb{R}^{N}}K(x,y,t)|u(x,t)-u(y,t)|^{p-2}\left(u(x,t)-u(y,t)\right)\,dy,$

where P.V. stands for the Cauchy principle value and $K(x,y,t)$ is the
symmetric Kernel function satisfies

$\displaystyle\frac{\Lambda^{-1}}{|x-y|^{N+sp}}\leq
K(x,y,t)=K(y,x,t)\leq\frac{\Lambda}{|x-y|^{N+sp}}\ \ \ \forall t\in(0,T),\
\forall x,y\in\mathbb{R}^{N}$

for some $\Lambda\geq 1$. Moreover, we suppose that

$\displaystyle p,q\in(1,\infty),\quad sp<q,\quad s\in(0,1)$ (1.2)

and the coefficient function $a:Q_{T}\rightarrow\mathbb{R}^{+}$ satisfies

$\displaystyle 0\leq a(x,t)\leq\|a\|_{L^{\infty}(Q_{T})}.$ (1.3)

###  1.1. Overview of related literature

Eq. (1.1) we study in this paper is characterized by different type of
operators depending on the size of the coefficient function $a(x,t)$. Observe
that $\eqref{1.1}$ behaves as a purely nonlocal $p$-Laplace equation on
$\\{a(x,t)=0\\}$ while leading to the problem involving different orders of
differentiation on $\\{a(x,t)>0\\}$. For the study on regularity theory of
nonlocal equations, we refer the readers to [2, 37, 18, 7, 32, 1, 27, 31, 20,
13]. The mixed local and nonlocal problem have been widely studied recently
owing to its essential applications in plasma physics [21] and biology [4]. In
the following, we will focus on introducing some known results for the mixed
problems.

For the basic prototype of the mixed local and nonlocal equation

$-\Delta u+(-\Delta)^{s}u=0,$ (1.4)

the Harnack inequality and boundary Harnack principle were derived in [23] and
[9], respectively; see also [10] for the Harnack estimate of globally
nonnegative solutions regarding the parabolic version of (1.4). Moreover, a
weak Harnack inequality with a tail term to the parabolic setting of (1.4) can
be found in [26]. Recently, Biagi-Dipierro-Valdinoci-Vecchi [5, 6] developed
systematic research on mixed problems and established several results
involving radial symmetry, regularity, maximum principle, and other
qualitative properties of weak solutions to (1.4). For the nonlinear scenario
of (1.4)

$\displaystyle-\Delta_{p}u+(-\Delta)^{s}_{p}u=0,$ (1.5)

the regularity theory involving Harnack inequality, local boundedness, and
Hölder continuity along with semicontinuity representative of weak solutions
were discussed by Garain-Kinnunen [25]. When it comes to the parabolic version
of (1.5), Shang-Zhang studied the local Hölder regularity [35] and Harnack
inequality [36] via analytic approach. For what concerns existence,
uniqueness, local boundedness and higher Hölder regularity for weak solutions
to (1.5) with the nonhomogeneous term were provided in [24]. In particular, De
Filippis-Mingione [15] first proposed the functionals with nonstandard growth

$\displaystyle
u\mapsto\int_{\Omega}\left(|Du|^{q}-fu\right)\,dx+\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dxdy,$
(1.6)

and proved the maximal regularity of minimizers to (1.6) with $q>sp$. The case
$q<sp$ can be seen as a natural extension of the above problem and the local
behaviour including the boundedness, Hölder continuity and Harnack inequality
of the minimizers of (1.6) were achieved by Ding-Fang-Zhang [19]. Besides,
Byun-Lee-Song [8] considered the regularity results for minimizers of
functionals modeled after

$\displaystyle
u\mapsto\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dxdy+\int_{\Omega}a(x)|Du|^{q}\,dx,$
(1.7)

where $0\leq a(x)\leq\|a\|_{L^{\infty}(\Omega)}$ and $1<p\leq q$. Functional
(1.7) is linked to the local or nonlocal double phase elliptic problems. We
refer the readers to [12, 11, 14, 3, 22, 16] and references therein for the
regularity results. On the other hand, for the parabolic double phase problem,
the gradient higher integrability, Lipschitz truncation method and the
Calderón-Zygmund type estimates have been provided in [28, 29, 30]. Prasad-
Tewary [34] proved the local boundedness of variational solutions for the
nonlocal double phase parabolic equation by variational method.

As far as we know, there are no results concerning the mixed local and
nonlocal double phase parabolic equation (1.1) for the case $p\neq q$, even
the coefficient function $a(x,t)$ is a constant. To this end, we are
interested in proving the local boundedness by energy estimates and De Giorgi-
Nash-Moser iteration. We would like to mention that our approach to establish
energy estimates differs from [34] since they discussed the variational
solutions that can be proved as parabolic minimizers. Furthermore, based on
energy estimates and De Giorgi type lemma, we take into account the lower
semicontinuity and pointwise behavior of weak supersolutions to (1.1), and the
related results also hold true for subsolutions.

###  1.2. Notations and Definitions

For the sake of simplicity, we give some notations that will be used later.

For radius $\rho>0$, let $B_{\rho}(x)$ be the domain centered at
$x\in\mathbb{R}^{N}$. We denote the forward and backward space-time cylinders
by

$\displaystyle
Q_{\rho,\theta}^{+}(x_{0},t_{0})=B_{\rho}(x_{0})\times(t_{0},t_{0}+\theta)\ \
\text{and}\ \
Q_{\rho,\theta}^{-}(x_{0},t_{0})=B_{\rho}(x_{0})\times(t_{0}-\theta,t_{0}).$

Denote the centered cylinders as

$\displaystyle
Q_{\rho,\theta}(x_{0},t_{0})=Q_{\rho,\theta}^{-}(x_{0},t_{0})\cup
Q_{\rho,\theta}^{+}(x_{0},t_{0}).$

If not important, we will omit $(x_{0},t_{0})$ in the above symbols for
clearly.

For any $a,b\in\mathbb{R}$, define

$\displaystyle J_{p}(a,b)=|a-b|^{p-2}(a-b).$

Set

$\displaystyle d\mu=d\mu(x,y,t)=K(x,y,t)\,dxdy,$

and

$data:=data(N,p,s,q,\Lambda).$

We introduce the tail space as

$\displaystyle L_{\alpha}^{p}(\mathbb{R}^{N}):=\left\\{v\in
L_{\rm{loc}}^{p}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\frac{|v(x)|^{p}}{1+|x|^{N+\alpha}}\,dx<+\infty\right\\},\quad
p>0\text{ and }\alpha>0.$

The tail appearing in estimates is denoted by

$\displaystyle\mathrm{Tail}_{\infty}(v;x_{0},\rho,I)$
$\displaystyle=\mathrm{Tail}_{\infty}(v;x_{0},\rho,t_{0}-\theta,t_{0})$
$\displaystyle:=\operatorname*{ess\sup}_{t\in
I}\left(\rho^{sp}\int_{\mathbb{R}^{N}\backslash
B_{\rho}(x_{0})}\frac{|v(x,t)|^{p-1}}{|x-x_{0}|^{N+sp}}\,dx\right)^{\frac{1}{p-1}}$
(1.8)

with $(x_{0},t_{0})\in\mathbb{R}^{N}\times(0,T)$ and the interval
$I=\left(t_{0}-\theta,t_{0}\right)\subseteq(0,T)$. By the definition, it is
not hard to verify that $\mathrm{Tail}_{\infty}(v;x_{0},\rho,I)<\infty$ for
any $v\in L^{\infty}(I;L_{sp}^{p-1}(\mathbb{R}^{N}))$.

Now, we state the definition of weak solutions to (1.1) as follows.

######  Definition 1.1.

We say that

$u\in L^{q}_{\rm{loc}}(0,T;W_{\rm{loc}}^{1,q}(\Omega))\cap
C_{\rm{loc}}(0,T;L_{\rm{loc}}^{2}(\Omega))\cap
L^{\infty}(0,T;L_{sp}^{p-1}(\mathbb{R}^{N}))$

is a local weak sub(super-)solution to (1.1), if for every closed interval
$[t_{1},t_{2}]\subseteq(0,T)$ and any compact set $K\subset\Omega$, integral
equality

$\displaystyle\int_{K}u(x,t_{2})\varphi(x,t_{2})\,dx-\int_{K}u(x,t_{1})\varphi(x,t_{1})\,dx-\int_{t_{1}}^{t_{2}}\int_{K}u(x,t)\partial_{t}\varphi(x,t)\,dxdt$
$\displaystyle\quad+\int_{t_{1}}^{t_{2}}\int_{K}a(x,t)|\nabla u|^{q-2}\nabla
u\cdot\nabla\varphi\,dxdt+\int_{t_{1}}^{t_{2}}\mathcal{E}(u,\varphi,t)\,dt\leq(\geq)0$
(1.9)

holds for every nonnegative test function $\varphi\in
L^{q}_{\rm{loc}}(0,T;W_{0}^{1,q}(K))\cap W_{\rm{loc}}^{1,2}(0,T;L^{2}(K))$,
where

$\displaystyle\mathcal{E}(u,\varphi,t):=\frac{1}{2}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}$
$\displaystyle\Big{[}|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))$
$\displaystyle\cdot(\varphi(x,t)-\varphi(y,t))K(x,y,t)\Big{]}\,dxdy.$

A function $u$ called a weak solution to (1.1) that is both a weak
supersolution and a weak subsolution.

Let $u$ be a locally essentially bounded below measure function in $Q_{T}$,
the lower semicontinuous regularization of $u$ is given by

$\displaystyle
u_{*}(x,t):=\operatorname*{ess\lim\inf}_{(y,\hat{t})\rightarrow(x,t)}u(y,\hat{t})=\lim_{\rho\rightarrow
0}\operatorname*{ess\inf}_{Q_{\rho,\theta}(x,t)}u\text{ \ \ for every
}(x,t)\in Q_{T}.$

Likewise, for the measurable function $u$ that is essentially bounded above in
$Q_{T}$, we define the upper semicontinuous regularization of $u$ as

$\displaystyle
u^{*}(x,t):=\operatorname*{ess\lim\sup}_{(y,\hat{t})\rightarrow(x,t)}u(y,\hat{t})=\lim_{\rho\rightarrow
0}\operatorname*{ess\sup}_{Q_{\rho,\theta}(x,t)}u\text{ \ \ for every
}(x,t)\in Q_{T}.$

Let $u\in L_{\rm{loc}}^{1}(Q_{T})$, we denote

$\displaystyle\mathcal{F}:=\left\\{(x,t)\in
Q_{T}:|u(x,t)|<\infty,\lim_{\rho\rightarrow
0}{\int-}_{Q_{\rho,\theta}(x,t)}|u(x,t)-u(y,\hat{t})|\,dyd\hat{t}=0\right\\},$

then it follows from the Lebesgue differentiation theorem that
$|\mathcal{F}|=|Q_{T}|$.

###  1.3. Main results

Our results can be stated as follows. The first one we present is the local
boundedness result in the supercritical case. Denote

$\displaystyle a\vee b:=\max\\{a,b\\},\quad a_{+}:=\max\\{a,0\\},\quad
a_{-}:=-\min\\{a,0\\}.$

######  Theorem 1.2 (Local boundedness).

Suppose that (1.2) and (1.3) hold. Let $p>2N/(N+2s)$, $\kappa:=1+2s/N$ and
$q<p\kappa$. Assume that $u$ is a local weak subsolution to (1.1). Let
$(x_{0},t_{0})\in Q_{T}$, $R>0$ and $Q_{R}^{-}\equiv
B_{R}(x_{0})\times(t_{0}-R^{sp},t_{0})$ such that
$\bar{B}_{R}(x_{0})\subseteq\Omega$ and $[t_{0}-R^{sp},t_{0}]\subseteq(0,T)$.
Then for any $\delta=\max\\{2,p,q\\}$, we have

$\displaystyle\operatorname*{ess\sup}_{Q_{R/2}^{-}}u\leq\mathrm{Tail}_{\infty}\left(u_{+};x_{0},R/2,t_{0}-R^{sp},t_{0}\right)+C(2+R^{sp-q})^{\frac{1}{\tau}}\left({\int-}_{Q_{R}^{-}}u_{+}^{\delta}\,dxdt\right)^{\frac{s\delta}{\tau(N\kappa+s\delta)}}\vee
1,$

where $\tau=\min\\{\delta-q,\delta-p,\delta-2\\}$, and the constant $C>0$ only
depends on $\|a\|_{L^{\infty}(Q_{T})}$ and the data.

In the subcritical range $1<p\leq 2N/(N+2s)$, supposing that the weak
solutions can be constructed as: for
$r>\max\left\\{2,\frac{N(2-p)}{sp}\right\\}$, there exists a sequence
$\\{u_{k}\\}_{k\in\mathbb{N}}$ of bounded weak solutions to (1.1) such that

$\displaystyle\|u_{k}\|_{L_{\rm{loc}}^{\infty}(0,T;L_{sp}^{p-1}(\mathbb{R}^{N}))}\leq
C$ (1.10)

and

$\displaystyle u_{k}\rightarrow u\quad\text{in }L_{\rm{loc}}^{r}(Q_{T})\text{
as }k\rightarrow\infty.$ (1.11)

######  Theorem 1.3 (Local boundedness).

Suppose that (1.2) and (1.3) hold. Let $1<p\leq 2N/(N+2s)$, $\kappa=1+2s/N$,
$q<p\kappa$ and $r>\max\left\\{2,\frac{N(2-p)}{sp}\right\\}$. Assume that
$u\in L^{r}_{\rm loc}(Q_{T})$ fulfilling (1.10) and (1.11) is a weak
subsolution to (1.1). Let $(x_{0},t_{0})\in Q_{T}$, and $Q_{R}^{-}\equiv
B_{R}(x_{0})\times(t_{0}-R^{sp},t_{0})$ such that
$\bar{B}_{R}(x_{0})\subseteq\Omega$ and $[t_{0}-R^{sp},t_{0}]\subseteq(0,T)$.
Then we have

$\displaystyle\operatorname*{ess\sup}_{Q_{R/2}^{-}}u\leq$
$\displaystyle\mathrm{Tail}_{\infty}\left(u_{+};x_{0},R/2,t_{0}-R^{sp},t_{0}\right)$
$\displaystyle+C\left(R^{sp-q}+2\right)^{\frac{1}{r-2-\beta}}\left({\int-}_{Q_{R}^{-}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-2-\beta)}}$
$\displaystyle\quad\vee\left(R^{sp-q}+2\right)^{\frac{1}{r-p-\beta}}\left({\int-}_{Q_{R}^{-}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-p-\beta)}}$
$\displaystyle\quad\vee\left(R^{sp-q}+2\right)^{\frac{1}{r-q-\beta}}\left({\int-}_{Q_{R}^{-}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-q-\beta)}},$

where $\beta=N(r-p\kappa)/(N+sp)$, and $C>0$ only depends on
$\|a\|_{L^{\infty}(Q_{T})},r$ and the data.

The following results are regarding the semicontinuous properties of weak
solutions. Here, we assume in addition that

$\displaystyle 0\leq a\in C^{\alpha,\frac{\alpha}{2}}(Q_{T})\text{\ \ and \ \
}q\leq sp+\alpha\text{\ \ for some \ }\alpha\in(0,1].$ (1.12)

The function $a\in C^{\alpha,\frac{\alpha}{2}}(Q_{T})$ means

$\displaystyle|a(x,t)-a(y,t)|\leq[a]_{\alpha,\frac{\alpha}{2}}|x-y|^{\alpha}\text{\
\ and \ \
}|a(x,t)-a(x,s)|\leq[a]_{\alpha,\frac{\alpha}{2}}|t-s|^{\frac{\alpha}{2}}$

for every $(x,y)\in\Omega$ and $(t,s)\in(0,T)$.

######  Theorem 1.4.

Let the assumptions (1.2) and (1.12) hold. Suppose that $u$ is a local weak
solution to (1.1) that is bounded from below in $\mathbb{R}^{N}\times(0,T)$.
Let $\mu^{-}$ be a number such that

$\displaystyle\mu^{-}\leq\operatorname*{ess\inf}_{\mathbb{R}^{N}\times(0,T)}u.$

Then we have $u_{*}(x,t)=u(x,t)$ for all $(x,t)\in\mathcal{F}$. In particular,
$u_{*}$ is a lower semicontinuous representative of $u$ in $Q_{T}$.

######  Remark 1.5.

Based on the same method of proving Theorem 1.4, we can obtain that a
subsolution of (1.1) has an upper semicontinuous representative in $Q_{T}$.

We now present the pointwise behavior of weak supersolutions to (1.1), which
shows that the value of lower semicontinuous representative $u_{*}$ at some
point can be recovered pointwise from the values at previous times.

######  Theorem 1.6.

Let the assumptions (1.2) and (1.12) hold. Suppose that $u$ is a local weak
supersolution to (1.1) that is bounded from below in
$\mathbb{R}^{N}\times(0,T)$. Let $\mu^{-}$ be a number such that

$\displaystyle\mu^{-}\leq\operatorname*{ess\inf}_{\mathbb{R}^{N}\times(0,T)}u.$

For a positive parameter $\eta$, we set the time levels

$\displaystyle\theta=\eta\rho^{q}\text{\ \ \ when
}\operatorname*{\max}_{Q_{\rho,\eta\rho^{sp}}(x_{0},t_{0})}a(x,t)\geq
2[a]_{\alpha,\frac{\alpha}{2}}\rho^{\alpha},$
$\displaystyle\theta=\eta\rho^{sp}\text{\ \ \ when
}\operatorname*{\max}_{Q_{\rho,\eta\rho^{sp}}(x_{0},t_{0})}a(x,t)\leq
2[a]_{\alpha,\frac{\alpha}{2}}\rho^{\alpha}.$

Then, for the lower semicontinuous representative $u_{*}$ given by Theorem
1.4, we have

$\displaystyle u_{*}(x,t)=\inf_{\eta>0}\lim_{\varrho\rightarrow
0}\operatorname*{ess\inf}_{Q^{\prime}_{\rho,\theta}(x,t)}u\text{\ \ for every
}(x,t)\in Q_{T},$

where $Q^{\prime}_{\rho,\theta}(x,t)=B_{\rho}(x)\times(t-2\theta,t-\theta)$.
In particular, there holds

$\displaystyle
u_{*}(x,t)=\operatorname*{ess\lim\inf}_{(y,\hat{t})\rightarrow(x,t),\
\hat{t}<t}u(y,\hat{t})\text{\ \ for every }(x,t)\in Q_{T}.$

###  1.4. Auxiliary lemmas

We introduce several fundamental lemmas. The following one is the useful
Sobolev inequality.

######  Lemma 1.7 (Lemma 2.3, [20]).

Let $0<t_{1}<t_{2}$, $s\in(0,1)$ and $p\in(1,\infty)$. Then for every

$\displaystyle u\in L^{p}\left(t_{1},t_{2};W^{s,p}(B_{\rho})\right)\cap
L^{\infty}\left(t_{1},t_{2};L^{2}(B_{\rho})\right),$

we have

$\displaystyle\quad\int_{t_{1}}^{t_{2}}{\int-}_{B_{\rho}}|u(x,t)|^{p(1+\frac{2s}{N})}\,dxdt$
$\displaystyle\leq
C\left(\rho^{sp}\int_{t_{1}}^{t_{2}}\int_{B\rho}{\int-}_{B_{\rho}}\frac{|u(x,t)-u(y,t)|^{p}}{|x-y|^{N+sp}}\,dxdydt+\int_{t_{1}}^{t_{2}}{\int-}_{B_{\rho}}|u(x,t)|^{p}\,dxdt\right)$
$\displaystyle\quad\times\left(\operatorname*{ess\sup}_{t_{1}<t<t_{2}}{\int-}_{B_{\rho}}|u(x,t)|^{2}\,dx\right)^{\frac{sp}{N}},$
(1.13)

where $C>0$ depends on $s,p$ and $N$.

Next, we give a classical iterative lemma which will be employed to prove the
main results.

######  Lemma 1.8 (Lemma 4.1, [17]).

Let $\\{Y_{j}\\}_{j=0}^{\infty}$ be a sequence of positive numbers, satisfying

$\displaystyle\ Y_{j+1}\leq Kb^{j}Y_{j}^{1+\delta},\quad j=0,1,2,\ldots$

for some constants $K$, $b>1$ and $\delta>0$. If

$\displaystyle Y_{0}\leq K^{-\frac{1}{\delta}}b^{-\frac{1}{\delta^{2}}},$

then $Y_{j}\rightarrow 0$ as $j\rightarrow\infty$.

The article is organized as follows. Section 2 is devoted to proving the
necessary Caccioppoli type inequality. In Section 3, we will consider the
boundedness estimates by using De Giorgi-Nash-Moser iteration. Finally, the
semicontinuity representative and pointwise behavior of weak solutions will be
discussed in Section 4.

##  2\. Energy Estimates

In this section, a Caccioppoli type inequality is established. In general, we
need to begin with time regularization since the time derivative of a weak
solution to (1.1) does not exist in the Sobolev sense. The specific procedure
can be performed as [29, 30], and thus we omit this step here.

######  Lemma 2.1 (Caccioppoli-type inequality).

Let $p,q>1$ and every cylinder $Q^{-}_{\rho,\theta}\subset Q_{T}$. Suppose
that $u$ is a local weak subsolution to (1.1). There exists constants $C>0$
depending on the data such that for every $k\in\mathbb{R}$, there holds

$\displaystyle\quad\operatorname*{ess\sup}_{t_{0}-\theta<t<t_{0}}\int_{B_{\rho}}w_{+}^{2}(x,t)\psi^{m}(x,t)\,dx+\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)|\nabla
w_{+}(x,t)|^{q}\psi^{m}(x,t)\,dxdt$
$\displaystyle\quad+\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}\int_{B_{\rho}}(\min\\{\psi(x,t),\psi(y,t)\\})^{m}|w_{+}(x,t)-w_{+}(y,t)|^{p}\,d\mu
dt$ $\displaystyle\leq
C\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}w_{+}^{2}(x,t)|\partial_{t}\psi^{m}|\,dxdt+C\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)w_{+}^{q}(x,t)|\nabla\psi(x,t)|^{q}\,dxdt$
$\displaystyle\quad+C\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}\int_{B_{\rho}}(\max\\{w_{+}(x,t),w_{+}(y,t)\\})^{p}\left|\psi^{\frac{m}{p}}(x,t)-\psi^{\frac{m}{p}}(y,t)\right|^{p}\,d\mu
dt$ $\displaystyle\quad+C\mathop{\mathrm{ess}\sup}_{\stackrel{{\scriptstyle
t_{0}-\theta<t<t_{0}}}{{x\in\mathrm{supp}\,\psi(\cdot,t)}}}\int_{\mathbb{R}^{N}\backslash
B_{\rho}}\frac{w_{+}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}w_{+}(x,t)\psi^{m}(x,t)\,dxdt,$
(2.1)

where $w_{+}(x,t):=\left(u(x,t)-k\right)_{+}$, $m=\max\\{p,q\\}$ and
$\psi(x,t)\in[0,1]$ is a piecewise smooth cutoff function vanishing on
$\partial B_{\rho}(x_{0})\times(t_{0}-\theta,t_{0})$.

###### Proof.

Denote $w_{+}(x,t):=\left(u(x,t)-k\right)_{+}$ and $m=\max\\{p,q\\}$. Choose
the piecewise smooth cutoff function $\psi(\cdot,t)\in[0,1]$ that is compactly
supported in $B_{\rho}$ for any $(t_{0}-\theta,t_{0})$. Testing the weak
formulation (1.1) by $\varphi(x,t):=w_{+}(x,t)\psi^{m}(x,t)$, we have

$\displaystyle 0$
$\displaystyle\geq\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}\partial_{t}u(x,t)w_{+}(x,t)\psi^{m}(x,t)\,dxdt$
$\displaystyle\quad+\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)|\nabla
u(x,t)|^{q-2}\nabla u(x,t)\nabla(w_{+}(x,t)\psi^{m}(x,t))\,dxdt$
$\displaystyle\quad+\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}\int_{B_{\rho}}J_{p}(w(x,t),w(y,t))\left(w_{+}(x,t)\psi^{m}(x,t)-w_{+}(y,t)\psi^{m}(y,t)\right)\,d\mu
dt$
$\displaystyle\quad+2\int_{t_{0}-\theta}^{t_{0}}\int_{\mathbb{R}^{N}\backslash
B_{\rho}}\int_{B_{\rho}}J_{p}\left(w(x,t),w(y,t)\right)w_{+}(x,t)\psi^{m}(x,t)\,d\mu
dt$ $\displaystyle=:I_{1}+I_{2}+I_{3}+I_{4}.$

Then we estimate $I_{1},I_{2},I_{3}$ and $I_{4}$ separately. First, we
evaluate

$\displaystyle I_{1}$
$\displaystyle=\frac{1}{2}\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}\partial_{t}w_{+}^{2}(x,t)\psi^{m}(x,t)\,dxdt$
$\displaystyle=\frac{1}{2}\int_{B_{\rho}}w_{+}^{2}(x,t)\psi^{m}(x,t)\,dx\bigg{|}_{t_{0}-\theta}^{t_{0}}-\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}w_{+}^{2}(x,t)\partial_{t}\psi^{m}\,dxdt.$
(2.2)

By Young’s inequality, the term $I_{2}$ can be treated as

$\displaystyle I_{2}$
$\displaystyle=\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)|\nabla
w_{+}(x,t)|^{q-2}\nabla w_{+}(x,t)\nabla(w_{+}(x,t)\psi^{m}(x,t))\,dxdt$
$\displaystyle\geq\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)|\nabla
w_{+}(x,t)|^{q}\psi^{m}(x,t)\,dxdt$
$\displaystyle\quad-\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)m\psi^{m-1}|\nabla\psi(x,t)||\nabla
w_{+}|^{q-1}w_{+}(x,t)\,dxdt$
$\displaystyle\geq\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)|\nabla
w_{+}|^{q}\psi^{m}(x,t)\,dxdt-\varepsilon\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)\psi^{\frac{q(m-1)}{q-1}}|\nabla
w_{+}|^{q}\,dxdt$
$\displaystyle\quad-C(p,q,\varepsilon)\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)|\nabla\psi(x,t)|^{q}w_{+}^{q}(x,t)\,dxdt$
$\displaystyle\geq\frac{1}{2}\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)|\nabla
w_{+}|^{q}\psi^{m}\,dxdt-C(p,q)\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)|\nabla\psi|^{q}w_{+}^{q}(x,t)\,dxdt,$
(2.3)

where we used the property $0\leq\psi(x,t)\leq 1$ in the last line. For the
term $I_{3}$, we utilize the idea from Lemma 4.1 in [15] to obtain the
pointwise estimates. We consider the case $u(x,t)\geq u(y,t)$ with some
$t\in(0,T)$, while case $u(x,t)\leq u(y,t)$ can be verified by exchanging the
role of $x$ and $y$. Denote

$\displaystyle\Gamma(x,y,t):=J_{p}(w(x,t),w(y,t))\left(w_{+}(x,t)\psi^{m}(x,t)-w_{+}(y,t)\psi^{m}(y,t)\right).$

Since $u(x,t)\geq u(y,t)$, it is not hard to check that
$\Gamma_{1}(x,y,t)+\Gamma_{2}(x,y,t)\leq\Gamma(x,y,t)$, where

$\displaystyle\Gamma_{1}(x,y,t):=|w_{+}(x,t)-w_{+}(y,t)|^{p}\psi^{m}(x,t)$

and

$\displaystyle\Gamma_{2}(x,y,t):=J_{p}(w_{+}(x,t),w_{+}(y,t))(\psi^{m}(x,t)-\psi^{m}(y,t))w_{+}(y,t).$

Moreover, we assume that $\psi(x,t)\geq\psi(y,t)$. By Mean Value Theorem with
respect to space variable, we have

$\displaystyle|\Gamma_{2}(x,y,t)|\leq
C\psi^{\frac{m(p-1)}{p}}(x,t)\left|\psi^{\frac{m}{p}}(x,t)-\psi^{\frac{m}{p}}(y,t)\right||w_{+}(x,t)-w_{+}(y,t)|^{p-1}w_{+}(y,t).$

To estimate $\Gamma_{1}$, we employ Young’s inequality to yield

$\displaystyle\Gamma_{1}(x,y,t)\leq
C\Gamma(x,y,t)+C\left|\psi^{\frac{m}{p}}(x,t)-\psi^{\frac{m}{p}}(y,t)\right|^{p}w_{+}^{p}(y,t).$

Thus, we conclude that

$\displaystyle\quad|w_{+}(x,t)-w_{+}(y,t)|^{p}\psi^{m}(x,t)$
$\displaystyle\leq
C\Gamma(x,y,t)+C(\max\\{w_{+}(x,t),w_{+}(y,t)\\})^{p}\left|\psi^{\frac{m}{p}}(x,t)-\psi^{\frac{m}{p}}(y,t)\right|^{p},$
(2.4)

where constants $C$ only depend on $p$ and $q$. Note that the inequality (2)
apparently holds when $\psi(x,t)\leq\psi(y,t)$. Then, it follows from (2) that

$\displaystyle I_{3}$
$\displaystyle\geq\frac{1}{C}\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}\int_{B_{\rho}}(\min\\{\psi(x,t),\psi(y,t)\\})^{m}|w_{+}(x,t)-w_{+}(y,t)|^{p}\,d\mu
dt$
$\displaystyle\quad-C\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}\int_{B_{\rho}}(\max\\{w_{+}(x,t),w_{+}(y,t)\\})^{p}\left|\psi^{\frac{m}{p}}(x,t)-\psi^{\frac{m}{p}}(y,t)\right|^{p}\,d\mu
dt.$ (2.5)

Finally, we consider the term $I_{4}$. When $w(x,t)>0$, we have

$\displaystyle|w(x,t)-w(y,t)|^{p-2}(w(x,t)-w(y,t))w_{+}(x,t)$
$\displaystyle\geq$ $\displaystyle-(w(y,t)-w(x,t))_{+}^{p-1}w_{+}(x,t)$
$\displaystyle\geq$ $\displaystyle-w_{+}^{p-1}(y,t)w_{+}(x,t).$

When $w(x,t)\leq 0$, the above inequality still holds due to the both sides
are zero. Thus we derive that

$\displaystyle I_{4}\geq$
$\displaystyle-C\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}\int_{\mathbb{R}^{N}\backslash
B_{\rho}}w_{+}^{p-1}(y,t)w_{+}(x,t)\psi^{m}(x,t)\,d\mu dt$ $\displaystyle\geq$
$\displaystyle-C\mathop{\mathrm{ess}\sup}_{\stackrel{{\scriptstyle
t_{0}-\theta<t<t_{0}}}{{x\in\mathrm{supp}\,\psi(\cdot,t)}}}\int_{\mathbb{R}^{N}\backslash
B_{\rho}}\frac{w_{+}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}w_{+}(x,t)\psi^{m}(x,t)\,dxdt.$
(2.6)

Putting the estimates (2), (2), (2) and (2) together, we conclude that

$\displaystyle\quad\int_{B_{\rho}}w_{+}^{2}(x,t_{0})\psi^{m}(x,t)\,dx+\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)|\nabla
w_{+}(x,t)|^{q}\psi^{m}(x,t)\,dxdt$
$\displaystyle\quad+\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}\int_{B_{\rho}}(\min\\{\psi(x,t),\psi(y,t)\\})^{m}|w_{+}(x,t)-w_{+}(y,t)|^{p}\,d\mu
dt$ $\displaystyle\leq
C\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}a(x,t)|\nabla\psi(x,t)|^{q}w_{+}^{q}(x,t)\,dxdt$
$\displaystyle\quad+C\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}\int_{B_{\rho}}\max\left\\{w_{+}(x,t),w_{+}(y,t)\right\\}^{p}\left|\psi^{\frac{m}{p}}(x,t)-\psi^{\frac{m}{p}}(y,t)\right|^{p}\,d\mu
dt$ $\displaystyle\quad+C\mathop{\mathrm{ess}\sup}_{\stackrel{{\scriptstyle
t_{0}-\theta<t<t_{0}}}{{x\in\mathrm{supp}\,\psi(\cdot,t)}}}\int_{\mathbb{R}^{N}\backslash
B_{\rho}}\frac{w_{+}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}w_{+}(x,t)\psi^{m}(x,t)\,dxdt$
$\displaystyle\quad+C\int_{t_{0}-\theta}^{t_{0}}\int_{B_{\rho}}w_{+}^{2}(x,t)|\partial_{t}\psi^{m}|\,dxdt,$

which leads to the desired result. ∎

##  3\. Local boundedness

In this section, we aim at investigating the local boundedness of weak
solutions. Before doing this, we give some notations as below.

Let $(x_{0},t_{0})\in Q_{T}$, $\rho>0$ and $Q_{\rho}^{-}\equiv
B_{\rho}(x_{0})\times(t_{0}-\theta,t_{0})$ satisfy
$\bar{B}_{\rho}(x_{0})\subseteq\Omega$ and
$\left[t_{0}-\theta,t_{0}\right]\subseteq(0,T)$. For $\sigma\in[1/2,1)$, we
take decreasing sequences

$\displaystyle\rho_{0}:=\rho,\quad\rho_{j}:=\sigma\rho+2^{-j}(1-\sigma)\rho,\quad\tilde{\rho}_{j}:=\frac{\rho_{j}+\rho_{j+1}}{2},\quad
j=0,1,2,\ldots$

and

$\displaystyle\theta_{0}:=\theta,\quad\theta_{j}:=\sigma\theta+2^{-j}(1-\sigma)\theta,\quad\tilde{\theta}_{j}:=\frac{\theta_{j}+\theta_{j+1}}{2},\quad
j=0,1,2,\ldots.$

Set the cylinders

$\displaystyle Q_{j}^{-}:=B_{j}\times
I_{j}:=B_{\rho_{j}}(x_{0})\times\left(t_{0}-\theta_{j},t_{0}\right),\quad
j=0,1,2,\ldots,$

and

$\displaystyle\tilde{Q}_{j}^{-}:=\tilde{B}_{j}\times\tilde{I}_{j}:=B_{\tilde{\rho}_{j}}(x_{0})\times\left(t_{0}-\tilde{\theta}_{j},t_{0}\right),\quad
j=0,1,2,\ldots.$

Denote the sequences of increasing levels

$\displaystyle
k_{j}:=\left(1-2^{-j}\right)\tilde{k},\quad\tilde{k}_{j}:=\frac{k_{j+1}+k_{j}}{2},\quad
j=0,1,2,\ldots$

with

$\displaystyle\tilde{k}\geq\frac{\mathrm{Tail}_{\infty}\left(u_{+};x_{0},\sigma\rho,t_{0}-\theta,t_{0}\right)}{2}.$

Let

$\displaystyle
w_{j}:=(u-k_{j})_{+},\quad\tilde{w}_{j}:=\left(u-\tilde{k}_{j}\right)_{+},\quad
j=0,1,2,\ldots.$

The following recursive estimate is obtained by utilizing the Caccioppoli type
inequality in a suitable cylinder.

######  Lemma 3.1.

Let the assumptions (1.2) and (1.3) hold. Suppose that $u$ is a local weak
subsolutions to (1.1). Let $(x_{0},t_{0})\in Q_{T}$, $\rho>0$ and
$Q_{\rho}^{-}=B_{\rho}(x_{0})\times(t_{0}-\theta,t_{0})$ such that
$\bar{B}_{\rho}(x_{0})\subseteq\Omega$ and
$\left[t_{0}-\theta,t_{0}\right]\subseteq(0,T)$. Let $\delta>0$ be a parameter
such that $\delta\geq\max\\{p,q,2\\}$. Then it holds for all $j\in\mathbb{N}$
that

$\displaystyle\quad\operatorname*{ess\sup}_{t\in
I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{2}(x,t)\,dx+\int_{I_{j+1}}{\int-}_{B_{j+1}}a(x,t)|\nabla\tilde{w}_{j}(x,t)|^{q}\,dxdt$
$\displaystyle\quad+\int_{I_{j+1}}\int_{B_{j+1}}{\int-}_{B_{j+1}}\frac{|\tilde{w}_{j}(x,t)-\tilde{w}_{j}(y,t)|^{p}}{|x-y|^{N+sp}}\,dxdydt$
$\displaystyle\leq
C\left[\frac{1}{\sigma^{sp}(1-\sigma)^{N+sp}}+\frac{1}{(1-\sigma)^{m}}\right]$
$\displaystyle\quad\times\left[\frac{2^{\delta
j}}{\rho^{q}\tilde{k}^{\delta-q}}+\frac{2^{(\delta+sp-2)j}}{\theta\tilde{k}^{\delta-2}}+\frac{2^{(\delta+m-p)j}}{\rho^{sp}\tilde{k}^{\delta-p}}+\frac{2^{(\delta+N+m-1)j}}{\rho^{sp}\tilde{k}^{\delta-p}}\right]\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{\delta}(x,t)\,dxdt,$
(3.1)

where $m=\max\\{p,q\\}$, the constant $C>0$ only depends on the data, $\delta$
and $\|a\|_{L^{\infty}(Q_{T})}$.

###### Proof.

For every $0<\tau<\delta$, we can check that

$\displaystyle\tilde{w}_{j}^{\tau}(x,t)\leq\frac{C2^{(\delta-\tau)j}}{\tilde{k}^{\delta-\tau}}w_{j}^{\delta}(x,t)\quad\text{
in }Q_{T}.$ (3.2)

Choose the cutoff functions $\psi_{j}\in C_{0}^{\infty}(Q_{j})$ such that

$\displaystyle 0\leq\psi_{j}\leq
1,\quad\left|\nabla\psi_{j}\right|\leq\frac{C2^{j}}{(1-\sigma)\rho}\text{ in
}\tilde{B}_{j},\quad\left|\partial_{t}\psi_{j}\right|\leq\frac{C2^{spj}}{(1-\sigma)^{sp}\theta}\text{
in }\tilde{I}_{j},\quad\psi_{j}\equiv 1\text{ in }Q_{j+1}.$

An application of Lemma 2.1 with $\rho=\rho_{j}$, $\theta=\theta_{j}$ provides
that

$\displaystyle\quad\operatorname*{ess\sup}_{t\in
I_{j+1}}\int_{B_{j+1}}\tilde{w}_{j}^{2}(x,t)\,dx+\int_{I_{j+1}}\int_{B_{j+1}}a(x,t)|\nabla\tilde{w}_{j}(x,t)|^{q}\,dxdt$
$\displaystyle\quad+\int_{I_{j+1}}\int_{B_{j+1}}\int_{B_{j+1}}\frac{\left|\tilde{w}_{j}(x,t)-\tilde{w}_{j}(y,t)\right|^{p}}{|x-y|^{N+sp}}\,dxdydt$
$\displaystyle\leq
C\int_{I_{j}}\int_{B_{j}}a(x,t)|\nabla\psi_{j}(x)|^{q}\tilde{w}_{j}^{q}(x,t)\,dxdt$
$\displaystyle\quad+C\int_{I_{j}}\int_{B_{j}}\int_{B_{j}}(\max\\{\tilde{w}_{j}(x,t),\tilde{w}_{j}(y,t)\\})^{p}\left|\psi_{j}^{\frac{m}{p}}(x,t)-\psi_{j}^{\frac{m}{p}}(y,t)\right|^{p}\,d\mu
dt$ $\displaystyle\quad+C\mathop{\mathrm{ess}\sup}_{\stackrel{{\scriptstyle
t\in
I_{j}}}{{x\in\mathrm{supp}\,\psi_{j}(\cdot,t)}}}\int_{\mathbb{R}^{N}\backslash
B_{j}}\frac{\tilde{w}_{j}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\int_{I_{j}}\int_{B_{j}}\tilde{w}_{j}(x,t)\psi_{j}^{m}(x,t)\,dxdt$
$\displaystyle\quad+C\int_{I_{j}}\int_{B_{j}}\tilde{w}_{j}^{2}(x,t)\left|\partial_{t}\psi^{m}\right|\,dxdt$
$\displaystyle=:I_{1}+I_{2}+I_{3}+I_{4}.$ (3.3)

By the definition of $\psi_{j}$ and (3.2), we have

$\displaystyle I_{1}$
$\displaystyle\leq\frac{C2^{qj}\|a\|_{L^{\infty}(Q_{T})}}{(1-\sigma)^{q}\rho^{q}}\int_{I_{j}}\int_{B_{j}}\tilde{w}_{j}^{q}(x,t)\,dxdt$
$\displaystyle\leq\frac{C2^{\delta
j}}{\tilde{k}^{\delta-q}(1-\sigma)^{q}\rho^{q}}\int_{I_{j}}\int_{B_{j}}w_{j}^{\delta}(x,t)\,dxdt$
(3.4)

and

$\displaystyle I_{2}$
$\displaystyle\leq\frac{C2^{(m+\delta-p)j}}{\tilde{k}^{\delta-p}(1-\sigma)^{m}\rho^{sp}}\int_{I_{j}}\int_{B_{j}}w_{j}^{\delta}(x,t)\,dxdt.$
(3.5)

Observe that

$\displaystyle\frac{|y-x_{0}|}{|y-x|}\leq 1+\frac{|x-x_{0}|}{|x-y|}\leq
1+\frac{\tilde{\rho}_{j}}{\rho_{j}-\tilde{\rho}_{j}}\leq
4+\frac{2^{j+2}\sigma}{1-\sigma},$

then we obtain

$\displaystyle\mathop{\mathrm{ess}\sup}_{\stackrel{{\scriptstyle t\in
I_{j}}}{{x\in\mathrm{supp}\ \psi_{j}(\cdot,t)}}}\int_{\mathbb{R}^{N}\backslash
B_{j}}\frac{\tilde{w}_{j}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy$ $\displaystyle\leq$
$\displaystyle\frac{C2^{(N+sp)j}}{(1-\sigma)^{N+sp}}\operatorname*{ess\sup}_{t\in
I_{j}}\int_{\mathbb{R}^{N}\backslash
B_{j}}\frac{\tilde{w}_{j}^{p-1}(y,t)}{|x_{0}-y|^{N+sp}}\,dy$
$\displaystyle\leq$
$\displaystyle\frac{C2^{(N+sp)j}}{(1-\sigma)^{N+sp}}\operatorname*{ess\sup}_{t\in
I_{j}}\int_{\mathbb{R}^{N}\backslash
B_{\sigma\rho}}\frac{\tilde{w}_{0}^{p-1}(y,t)}{|x_{0}-y|^{N+sp}}\,dy$
$\displaystyle\leq$
$\displaystyle\frac{C2^{(N+sp)j}}{\rho^{sp}\sigma^{sp}(1-\sigma)^{N+sp}}[\mathrm{Tail}_{\infty}(u_{+};x_{0},\sigma\rho,t_{0}-\theta,t_{0})]^{p-1},$

where we used formula (1.2). Recalling the choice of $\tilde{k}$, it can be
derived from (3.2) that

$\displaystyle
I_{3}\leq\frac{2^{(N+sp+\delta-1)j}}{(1-\sigma)^{N+sp}\rho^{sp}\sigma^{sp}\tilde{k}^{\delta-p}}\int_{I_{j}}\int_{B_{j}}w_{j}^{\delta}(x,t)\,dxdt.$
(3.6)

Moreover, the term $I_{4}$ can be estimated still by (3.2) as

$\displaystyle I_{4}$
$\displaystyle\leq\frac{C2^{spj}}{(1-\sigma)^{sp}\theta}\int_{I_{j}}\int_{B_{j}}w_{j}^{2}(x,t)\,dxdt$
$\displaystyle\leq\frac{C2^{(sp+\delta-2)j}}{\tilde{k}^{\delta-2}(1-\sigma)^{sp}\theta}\int_{I_{j}}\int_{B_{j}}w_{j}^{\delta}(x,t)\,dxdt.$
(3.7)

We conclude from (3)–(3) that

$\displaystyle\operatorname*{ess\,\sup}_{t\in
I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{2}(x,t)\,dx+\int_{I_{j+1}}{\int-}_{B_{j+1}}a(x,t)|\nabla\tilde{w}_{j}(x,t)|^{q}\,dxdt$
$\displaystyle+\int_{I_{j+1}}\int_{B_{j+1}}{\int-}_{B_{j+1}}\frac{|\tilde{w}_{j}(x,t)-\tilde{w}_{j}(y,t)|^{p}}{|x-y|^{N+sp}}\,dxdydt$
$\displaystyle\leq$ $\displaystyle
C\left[\frac{1}{(1-\sigma)^{q}}+\frac{1}{(1-\sigma)^{m}}+\frac{1}{(1-\sigma)^{sp}}+\frac{1}{(1-\sigma)^{N+sp}\sigma^{sp}}\right]$
$\displaystyle\times\left[\frac{2^{\delta
j}}{\rho^{q}\tilde{k}^{\delta-q}}+\frac{2^{(m+\delta-p)j}}{\rho^{sp}\tilde{k}^{\delta-p}}+\frac{2^{(N+sp+\delta-1)j}}{\rho^{sp}\tilde{k}^{\delta-p}}+\frac{2^{(sp+\delta-2)j}}{\theta\tilde{k}^{\delta-2}}\right]\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{\delta}(x,t)\,dxdt,$

as desired. ∎

######  Lemma 3.2.

Let the assumptions (1.2) and (1.3) hold, $p>2N/(N+2s),\kappa=1+2s/N$ and
$q<p\kappa$. Suppose that $u$ is a local weak subsolution to (1.1). Let
$(x_{0},t_{0})\in Q_{T}$, $\rho>0$ and
$Q_{\rho}^{-}=B_{\rho}(x_{0})\times(t_{0}-\theta,t_{0})$ satisfy
$\bar{B}_{\rho}(x_{0})\subseteq\Omega$ and
$\left[t_{0}-\theta,t_{0}\right]\subseteq(0,T)$. Let $\delta$ be a parameter
such that $\max\\{p,q,2\\}\leq\delta\leq\kappa p$. Then we infer for all
$j\in\mathbb{N}$ that

$\displaystyle\frac{1}{\rho^{sp}}\int_{I_{j+1}}{\int-}_{B_{j+1}}w_{j+1}^{\delta}(x,t)dxdt\leq
C2^{bj}\tilde{B}\left(\frac{A_{k}}{\rho^{sp}}\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{\delta}dxdt\right)^{1+\frac{s\delta}{N\kappa}},$
(3.8)

where $C>0$ depends on $\|a\|_{L^{\infty}(Q_{T})},\delta$, the data and

$\displaystyle b=\left(1+\frac{sp}{N}\right)\left(N+m+\delta\right),\quad
A_{k}:=\frac{1}{\tilde{k}^{\delta-p}}+\frac{\rho^{sp-q}}{\tilde{k}^{\delta-q}}+\frac{\rho^{sp}}{\theta\tilde{k}^{\delta-2}},$
$\displaystyle\tilde{B}=\left[\frac{1}{\sigma^{ps}(1-\sigma)^{N+ps}}+\frac{1}{(1-\sigma)^{m}}\right]^{\left(1+\frac{sp}{N}\right)\frac{\delta}{p\kappa}}.$

###### Proof.

Since $\delta\leq p\kappa$, we derive from Hölder inequality that

$\displaystyle\quad\int_{I_{j+1}}{\int-}_{B_{j+1}}w_{j+1}^{\delta}(x,t)\,dxdt$
$\displaystyle\leq\int_{I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{\delta}(x,t)\,dxdt$
$\displaystyle\leq\left(\int_{I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{p\kappa}(x,t)\,dxdt\right)^{\frac{\delta}{p\kappa}}\left(\int_{I_{j+1}}{\int-}_{B_{j+1}}\chi_{\\{u\geq\tilde{k}_{j}\\}}(x,t)\,dxdt\right)^{1-\frac{\delta}{p\kappa}}.$
(3.9)

In view of (3.2), we can directly get

$\displaystyle\int_{I_{j+1}}{\int-}_{B_{j+1}}\chi_{\\{u\geq\tilde{k}_{j}\\}}(x,t)\,dxdt\leq\frac{C2^{\delta
j}}{\tilde{k}^{\delta}}\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{\delta}(x,t)\,dxdt$
(3.10)

and

$\displaystyle\int_{I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{p}(x,t)\,dxdt\leq\frac{C2^{(\delta-p)j}}{\tilde{k}^{\delta-p}}\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{\delta}(x,t)\,dxdt.$
(3.11)

Taking into account (3.11), Lemma 1.7 and Lemma 3.1, there holds that

$\displaystyle\quad\int_{I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{p\kappa}(x,t)\,dxdt$
$\displaystyle\leq
C\rho^{sp}\left(\int_{I_{j+1}}{\int-}_{B_{j+1}}{\int-}_{B_{j+1}}\frac{|\tilde{w}_{j}(x,t)-\tilde{w}_{j}(y,t)|^{p}}{|x-y|^{N+sp}}\,dxdydt+\frac{1}{\rho^{sp}}\int_{I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{p}(x,t)\,dxdt\right)$
$\displaystyle\quad\times\left(\operatorname*{ess\sup}_{t\in
I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{2}(x,t)\,dx\right)^{\frac{sp}{N}}$
$\displaystyle\leq C\rho^{sp}\left[\frac{2^{\delta
j}}{\rho^{q}\tilde{k}^{\delta-q}}+\frac{2^{(sp+\delta-2)j}}{\theta\tilde{k}^{\delta-2}}+\frac{2^{(N+m+\delta-1)}}{\rho^{sp}\tilde{k}^{\delta-p}}\right]^{1+\frac{sp}{N}}\times\left(B\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{\delta}\,dxdt\right)^{1+\frac{sp}{N}}$
$\displaystyle=C\rho^{sp}\left[\frac{2^{\delta
j}\rho^{sp-q}}{\tilde{k}^{\delta-q}}+\frac{2^{(sp+\delta-2)j}\rho^{sp}}{\theta\tilde{k}^{\delta-2}}+\frac{2^{(N+m+\delta-1)}}{\tilde{k}^{\delta-p}}\right]^{1+\frac{sp}{N}}\times\left(\frac{B}{\rho^{sp}}\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{\delta}\,dxdt\right)^{1+\frac{sp}{N}}$
$\displaystyle\leq
C2^{bj}\rho^{sp}\left(\frac{BA_{k}}{\rho^{sp}}\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{\delta}\,dxdt\right)^{1+\frac{sp}{N}},$
(3.12)

where

$\displaystyle b=\left(1+\frac{sp}{N}\right)\left(N+m+\delta\right),\quad
A_{k}:=\frac{1}{\tilde{k}^{\delta-p}}+\frac{\rho^{sp-q}}{\tilde{k}^{\delta-q}}+\frac{\rho^{sp}}{\theta\tilde{k}^{\delta-2}},$
$\displaystyle
B=\frac{1}{\sigma^{sp}(1-\sigma)^{N+sp}}+\frac{1}{(1-\sigma)^{m}}.$

Substituting (3.10) and (3) into (3), we yield the claim (3.8) with
$\tilde{B}=B^{\left(1+\frac{sp}{N}\right)\frac{\delta}{p\kappa}}$. ∎

######  Lemma 3.3.

Let the assumptions (1.4) and (1.5) hold, $p\leq 2N/(N+2s)$, $\kappa=1+2s/N$,
$q<p\kappa$ and $r>\max\left\\{2,\frac{N(2-p)}{sp}\right\\}$. Suppose that
$u\in L_{\rm{loc}}^{\infty}(Q_{T})$ be a local weak subsolution to (1.1). Let
$(x_{0},t_{0})\in Q_{T}$, $\rho>0$ and
$Q_{\rho}^{-}=B_{\rho}(x_{0})\times(t_{0}-\theta,t_{0})$ such that
$\bar{B}_{\rho}(x_{0})\subseteq\Omega$ and
$\left[t_{0}-\theta,t_{0}\right]\subseteq(0,T)$. For any $j\in\mathbb{N}$, we
have

$\displaystyle\frac{1}{\rho^{sp}}\int_{I_{j+1}}{\int-}_{B_{j+1}}w_{j+1}^{r}(x,t)\,dxdt$
$\displaystyle\leq$ $\displaystyle
C2^{b^{\prime}j}\left\|\tilde{w}_{j}\right\|_{L^{\infty}\left(Q_{j+1}^{-}\right)}^{r-p\kappa}\left(\frac{A^{\prime}_{k}B}{\rho^{sp}}\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{r}\,dxdt\right)^{1+\frac{sp}{N}},$
(3.13)

where $C>0$ only depends on $r,\|a\|_{L^{\infty}(Q_{T})}$, the data and

$\displaystyle b^{\prime}:=\left(1+\frac{sp}{N}\right)\left(N+m+r\right),\
A^{\prime}_{k}:=\frac{1}{\tilde{k}^{r-p}}+\frac{\rho^{sp-q}}{\tilde{k}^{r-q}}+\frac{\rho^{sp}}{\theta\tilde{k}^{r-2}},$
$\displaystyle
B=\frac{1}{\sigma^{sp}(1-\sigma)^{N+sp}}+\frac{1}{(1-\sigma)^{m}}.$

###### Proof.

Thanks to the assumptions, we can see

$\displaystyle\int_{I_{j+1}}{\int-}_{B_{j+1}}w_{j+1}^{r}(x,t)\,dxdt$
$\displaystyle\leq\int_{I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{r}(x,t)\,dxdt$
$\displaystyle\leq\left\|\tilde{w}_{j}\right\|_{L^{\infty}\left(Q_{j+1}^{-}\right)}^{r-p\kappa}\int_{I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{p\kappa}(x,t)\,dxdt$
(3.14)

with $\kappa=1+2s/N$. Similar to the proof of Lemma 3.2 with $\delta=r$, it
gives that

$\displaystyle\quad\int_{I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{p\kappa}(x,t)\,dxdt$
$\displaystyle\leq
C\rho^{sp}\left(\int_{I_{j+1}}{\int-}_{B_{j+1}}{\int-}_{B_{j+1}}\frac{|\tilde{w}_{j}(x,t)-\tilde{w}_{j}(y,t)|^{p}}{|x-y|^{N+sp}}\,dxdydt+\frac{1}{\rho^{sp}}\int_{I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{p}(x,t)\,dxdt\right)$
$\displaystyle\quad\times\left(\operatorname*{ess\,\sup}_{t\in
I_{j+1}}{\int-}_{B_{j+1}}\tilde{w}_{j}^{2}(x,t)\,dx\right)^{\frac{sp}{N}}$
$\displaystyle\leq
C2^{b^{\prime}j}\rho^{sp}\left(\frac{A^{\prime}_{k}B}{\rho^{sp}}\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{r}\,dxdt\right)^{1+\frac{sp}{N}},$
(3.15)

where $b^{\prime},A^{\prime}_{k},B$ are defined as in the statement of claim.
Thus, we get the conclusion by merging inequalities (3) and (3). ∎

######  Remark 3.4.

Due to the fact that $\sigma\in[1/2,1)$, we can rewrite (3.3) as

$\displaystyle\quad\frac{1}{\rho^{sp}}\int_{I_{j+1}}{\int-}_{B_{j+1}}w_{j+1}^{r}(x,t)\,dxdt$
$\displaystyle\leq
C2^{b^{\prime}j}\left\|\tilde{w}_{j}\right\|_{L^{\infty}\left(Q_{j+1}^{-}\right)}^{r-p\kappa}\frac{1}{(1-\sigma)^{\frac{(N+sp)(N+m)}{N}}}\left(\frac{A^{\prime}_{k}}{\rho^{sp}}\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{r}\,dxdt\right)^{1+\frac{sp}{N}}.$

Now we are beginning to prove the boundedness results.

######  Proof of Theorem 1.2.

Let $\rho=R$, $\sigma=\frac{1}{2}$, then $\rho_{j}=\frac{R}{2}+2^{-j-1}R$. For
$\delta=\max\\{2,p,q\\}$, set

$\displaystyle
Y_{j}=\frac{1}{\rho^{sp}}\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{\delta}\,dxdt,\quad
j=0,1,2,\ldots.$

It follows from Lemma 3.2 that

$\displaystyle\frac{1}{\rho^{sp}}\int_{I_{j+1}}{\int-}_{B_{j+1}}w_{j+1}^{\delta}(x,t)\,dxdt\leq
C2^{bj}\left(\frac{A_{k}}{\rho^{sp}}\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{\delta}\,dxdt\right)^{1+\frac{s\delta}{N\kappa}}$
(3.16)

with $b=\left(1+\frac{sp}{N}\right)\left(N+m+\delta\right),m=\max\\{p,q\\}$
and $\kappa:=1+2s/N$. If assuming $\tilde{k}\geq 1$ and taking
$\tau=\min\\{\delta-q,\delta-p,\delta-2\\}$, it shows that

$\displaystyle
A_{k}=\frac{1}{\tilde{k}^{\delta-p}}+\frac{\rho^{sp-q}}{\tilde{k}^{\delta-q}}+\frac{\rho^{sp}}{\theta\tilde{k}^{\delta-2}}\leq\frac{A}{\tilde{k}^{\tau}},$
(3.17)

where

$\displaystyle A=1+\rho^{sp-q}+\frac{\rho^{sp}}{\theta}.$

By using (3.16) and (3.17), we have

$\displaystyle Y_{j+1}\leq
C2^{bj}\left(\frac{A}{\tilde{k}^{\tau}}Y_{j}\right)^{1+\frac{s\delta}{N\kappa}}.$

Choose $\theta=\rho^{sp}$. Let $\tilde{k}$ be taken to satisfy

$\displaystyle\tilde{k}\geq\max\bigg{\\{}\mathrm{Tail}_{\infty}\left(u_{+};x_{0},R/2,t_{0}-R^{sp},t_{0}\right),CA^{\frac{1}{\tau}}\left({\int-}^{t_{0}}_{t_{0}-R^{sp}}{\int-}_{B_{R}}u_{+}^{\delta}\,dxdt\right)^{\frac{s\delta}{\tau(N\kappa+s\delta)}}\vee
1\bigg{\\}},$

where $C$ only depends on the data and $\|a\|_{L^{\infty}(Q_{T})}$. Applying
Lemma 1.8, we obtain $\lim_{j\rightarrow\infty}Y_{j}=0$, which implies that

$\displaystyle\operatorname*{ess\,\sup}_{Q^{-}_{R/2,R^{sp}/2}}u\leq\mathrm{Tail}_{\infty}\left(u_{+};x_{0},R/2,t_{0}-R^{sp},t_{0}\right)+CA^{\frac{1}{\tau}}\left({\int-}^{t_{0}}_{t_{0}-R^{sp}}{\int-}_{B_{R}}u_{+}^{\delta}\,dxdt\right)^{\frac{s\delta}{\tau(N\kappa+s\delta)}}\vee
1,$

as intended. ∎

######  Proof of Theorem 1.3.

Let the conditions in Theorem 1.3 hold. By means of the appropriate
approximation method, we may suppose that $u$ is qualitatively locally bounded
based on (1.10) and (1.11). In fact, since the approximation subsolutions
$u_{k}$ are bounded, the following estimate still holds if we substitute $u$
by $u_{k}$, and combining with (1.10) and (1.11) gives a $k$-independent bound
on $u_{k}$ in $L^{\infty}$. Therefore, we obtain $u$ is qualitatively locally
bounded by the a.e. convergence of $u_{k}$.

Let $\theta=R^{sp}$, $\theta_{n}=R_{n}^{sp}$, $R_{0}=R/2$ and
$R_{n}=R/2+\sum_{i=1}^{n}2^{-i-1}R$ for $n\in\mathbb{N}^{+}$, and
$Q_{n}^{-}:=B_{R_{n}}(x_{0})\times\left(t_{0}-\theta_{n},t_{0}\right)=B_{R_{n}}(x_{0})\times\left(t_{0}-R_{n}^{sp},t_{0}\right)$.
Denote

$\displaystyle M_{n}=\operatorname{ess}\sup_{Q_{n}^{-}}u_{+},\quad
n=0,1,2,3,\ldots.$

Taking $\rho=R_{n+1}$ and $\sigma\rho=R_{n}$, then

$\displaystyle\sigma=\frac{1/2+\sum_{i=1}^{n}2^{-i-1}}{1/2+\sum_{i=1}^{n+1}2^{-i-1}}\geq\frac{1}{2}.$

Set

$\displaystyle
Y_{j}=\frac{1}{\rho^{sp}}\int_{I_{j}}{\int-}_{B_{j}}\left(u-k_{j}\right)_{+}^{r}\,dxdt,\quad
j=0,1,2,\ldots.$

In light of Lemma 3.3, we have

$\displaystyle Y_{j+1}\leq$ $\displaystyle
C2^{b^{\prime}j}\left(2+R^{sp-q}\right)^{1+\frac{sp}{N}}\left\|u_{+}\right\|_{L^{\infty}\left(Q_{n+1}^{-}\right)}^{r-p\kappa}\frac{1}{(1-\sigma)^{\frac{(N+sp)(N+m)}{N}}}$
$\displaystyle\times\left(\frac{1}{\tilde{k}^{r-2}}+\frac{1}{\tilde{k}^{r-p}}+\frac{1}{\tilde{k}^{r-q}}\right)^{1+\frac{sp}{N}}\left(\int_{I_{j}}{\int-}_{B_{j}}w_{j}^{r}\,dxdt\right)^{1+\frac{sp}{N}}$
$\displaystyle\leq$ $\displaystyle
C2^{b^{\prime}j+dn}A^{1+\frac{sp}{N}}M_{n+1}^{r-p\kappa}\left(\frac{1}{\tilde{k}^{r-2}}+\frac{1}{\tilde{k}^{r-p}}+\frac{1}{\tilde{k}^{r-q}}\right)^{1+\frac{sp}{N}}Y_{j}^{1+\frac{sp}{N}}$

with $b^{\prime}:=\left(1+\frac{sp}{N}\right)\left(N+m+r\right)$,
$d=(N+sp)(N+m)/N$ and $A=2+R^{sp-q}$. Then from Lemma 1.8 we get
$\lim_{j\rightarrow\infty}Y_{j}=0$ if

$\displaystyle Y_{0}\leq
C2^{-\frac{dnN}{sp}-\frac{b^{\prime}N^{2}}{(sp)^{2}}}A^{-\frac{N+sp}{sp}}M_{n+1}^{-\frac{N(r-p\kappa)}{sp}}\left(\frac{1}{\tilde{k}^{r-2}}+\frac{1}{\tilde{k}^{r-p}}+\frac{1}{\tilde{k}^{r-q}}\right)^{-\frac{N+sp}{sp}},$
(3.18)

that is

$\displaystyle
C2^{\frac{dnN}{N+sp}}AM_{n+1}^{\frac{N(r-p\kappa)}{N+sp}}\left({\int-}_{t_{0}-R_{n+1}^{sp}}^{t_{0}}{\int-}_{B_{R_{n+1}}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{N+sp}}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{\tilde{k}^{r-p}}+\frac{1}{\tilde{k}^{r-q}}+\frac{1}{\tilde{k}^{r-2}}\right)^{-1}.$

To ensure the inequality (3.18) holds true, we can take

$\displaystyle\tilde{k}$
$\displaystyle=C2^{\frac{dnN}{(N+sp)(r-p)}}A^{\frac{1}{r-p}}M_{n+1}^{\frac{N(r-p\kappa)}{(N+sp)(r-p)}}\left({\int-}_{t_{0}-R_{n+1}^{sp}}^{t_{0}}{\int-}_{B_{R_{n+1}}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-p)}}$
$\displaystyle\quad+C2^{\frac{dnN}{(N+sp)(r-q)}}A^{\frac{1}{r-q}}M_{n+1}^{\frac{N(r-p\kappa)}{(N+sp)(r-q)}}\left({\int-}_{t_{0}-R_{n+1}^{sp}}^{t_{0}}{\int-}_{B_{R_{n+1}}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-q)}}$
$\displaystyle\quad+C2^{\frac{dnN}{(N+sp)(r-2)}}A^{\frac{1}{r-2}}M_{n+1}^{\frac{N(r-p\kappa)}{(N+sp)(r-2)}}\left({\int-}_{t_{0}-R_{n+1}^{sp}}^{t_{0}}{\int-}_{B_{R_{n+1}}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-2)}}$
$\displaystyle\quad+\frac{\mathrm{Tail}_{\infty}\left(u_{+};x_{0},R_{n},t_{0}-R_{n+1}^{sp},t_{0}\right)}{2},$

where constants $C$ depend on the data, $r$ and $\|a\|_{L^{\infty}(Q_{T})}$.
Thus by Lemma 1.8 we obtain

$\displaystyle M_{n}$
$\displaystyle=\operatorname*{ess\sup}_{Q_{R_{n}}^{-}}u^{+}$
$\displaystyle\leq
C2^{\frac{dnN}{(N+sp)(r-p)}}A^{\frac{1}{r-p}}M_{n+1}^{\frac{N(r-p\kappa)}{(N+sp)(r-p)}}\left({\int-}_{t_{0}-R_{n+1}^{sp}}^{t_{0}}{\int-}_{B_{R_{n+1}}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-p)}}$
$\displaystyle\quad+C2^{\frac{dnN}{(N+sp)(r-q)}}A^{\frac{1}{r-q}}M_{n+1}^{\frac{N(r-p\kappa)}{(N+sp)(r-q)}}\left({\int-}_{t_{0}-R_{n+1}^{sp}}^{t_{0}}{\int-}_{B_{R_{n+1}}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-q)}}$
$\displaystyle\quad+C2^{\frac{dnN}{(N+sp)(r-2)}}A^{\frac{1}{r-2}}M_{n+1}^{\frac{N(r-p\kappa)}{(N+sp)(r-2)}}\left({\int-}_{t_{0}-R_{n+1}^{sp}}^{t_{0}}{\int-}_{B_{R_{n+1}}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-2)}}$
$\displaystyle\quad+\frac{\mathrm{Tail}_{\infty}\left(u_{+};x_{0},R_{n},t_{0}-R_{n+1}^{sp},t_{0}\right)}{2}.$
(3.19)

In view of the assumption $r>\max\left\\{2,\frac{N(2-p)}{sp}\right\\}$, which
indicates that

$\displaystyle
0<\frac{N(r-p\kappa)}{(N+sp)(r-p)},\frac{N(r-p\kappa)}{(N+sp)(r-q)},\frac{N(r-p\kappa)}{(N+sp)(r-2)}<1.$

Applying Young’s inequality to (3), we get

$\displaystyle M_{n}$ $\displaystyle\leq\varepsilon
M_{n+1}+C2^{\frac{dnN}{(N+sp)(r-p-\beta)}}\varepsilon^{-\frac{\beta}{r-p-\beta}}A^{\frac{1}{r-p-\beta}}\left({\int-}_{t_{0}-R^{sp}}^{t_{0}}{\int-}_{B_{R}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-p-\beta)}}$
$\displaystyle\quad+C2^{\frac{dnN}{(N+sp)(r-q-\beta)}}\varepsilon^{-\frac{\beta}{r-q-\beta}}A^{\frac{1}{r-q-\beta}}\left({\int-}_{t_{0}-R^{sp}}^{t_{0}}{\int-}_{B_{R}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-q-\beta)}}$
$\displaystyle\quad+C2^{\frac{dnN}{(N+sp)(r-2-\beta)}}\varepsilon^{-\frac{\beta}{r-2-\beta}}A^{\frac{1}{r-2-\beta}}\left({\int-}_{t_{0}-R^{sp}}^{t_{0}}{\int-}_{B_{R}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-2-\beta)}}$
$\displaystyle\quad+\frac{\mathrm{Tail}_{\infty}\left(u_{+};x_{0},R/2,t_{0}-R^{sp},t_{0}\right)}{2}$

with $\beta=(r-p\kappa)N/(sp+N)$. By the induction argument, we deduce that

$\displaystyle M_{0}$ $\displaystyle\leq\varepsilon^{n+1}M_{n+1}$
$\displaystyle\quad+C\varepsilon^{-\frac{\beta}{r-p-\beta}}A^{\frac{1}{r-p-\beta}}\left({\int-}_{t_{0}-R^{sp}}^{t_{0}}{\int-}_{B_{R}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-p-\beta)}}\sum_{i=0}^{n}\left(2^{{\frac{dN}{(N+sp)(r-p-\beta)}}}\varepsilon\right)^{i}$
$\displaystyle\quad+C\varepsilon^{-\frac{\beta}{r-q-\beta}}A^{\frac{1}{r-q-\beta}}\left({\int-}_{t_{0}-R^{sp}}^{t_{0}}{\int-}_{B_{R}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-q-\beta)}}\sum_{i=0}^{n}\left(2^{{\frac{dN}{(N+sp)(r-q-\beta)}}}\varepsilon\right)^{i}$
$\displaystyle\quad+C\varepsilon^{-\frac{\beta}{r-2-\beta}}A^{\frac{1}{r-2-\beta}}\left({\int-}_{t_{0}-R^{sp}}^{t_{0}}{\int-}_{B_{R}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-2-\beta)}}\sum_{i=0}^{n}\left(2^{{\frac{dN}{(N+sp)(r-2-\beta)}}}\varepsilon\right)^{i}$
$\displaystyle\quad+\frac{\mathrm{Tail}_{\infty}\left(u_{+};x_{0},R/2,t_{0}-R^{sp},t_{0}\right)}{2}\sum_{i=0}^{n}\varepsilon^{i},\quad
n=0,1,2,\ldots.$ (3.20)

Observe that the sum on the right-hand side of (3) can be revised by a
convergent series if we choose

$\displaystyle\varepsilon=2^{-\left[{\frac{dN}{(N+sp)(r-2-\beta)}}+1\right]}.$

To complete the proof, we let $n\rightarrow\infty$ and arrive at

$\displaystyle\operatorname*{ess\sup}_{Q_{R/2,R^{sp}/2}^{-}}u$
$\displaystyle\leq\mathrm{Tail}_{\infty}\left(u_{+};x_{0},R/2,t_{0}-R^{sp},t_{0}\right)$
$\displaystyle\quad+C\left(R^{sp-q}+2\right)^{\frac{1}{r-2-\beta}}\left({\int-}_{t_{0}-R^{sp}}^{t_{0}}{\int-}_{B_{R}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-2-\beta)}}$
$\displaystyle\quad\quad\vee\left(R^{sp-q}+2\right)^{\frac{1}{r-p-\beta}}\left({\int-}_{t_{0}-R^{sp}}^{t_{0}}{\int-}_{B_{R}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-p-\beta)}}$
$\displaystyle\quad\quad\vee\left(R^{sp-q}+2\right)^{\frac{1}{r-q-\beta}}\left({\int-}_{t_{0}-R^{sp}}^{t_{0}}{\int-}_{B_{R}}u_{+}^{r}\,dxdt\right)^{\frac{sp}{(N+sp)(r-q-\beta)}},$

where $C>0$ depends only on the data, $r$ and $\|a\|_{L^{\infty}(Q_{T})}$. ∎

##  4\. Lower semicontinuity and pointwise behavior

In this section, we study the lower semicontinuity and pointwise behavior of
weak supersolutions to (1.1). We begin with introducing the measure theoretic
property of weak solutions. Then we establish the De Giorgi type lemma 4.2,
which is a key ingredient to obtain the Theorem 1.4.

What follows is the property $(\mathcal{D})$ of weak solutions that can be
found in [33]. Assume that $u$ is a measurable function, locally bounded from
below in $Q_{T}$. Let the parameters $0<b,c<1,L>0$ and the cylinders
$Q_{\rho,\theta}(x_{0},t_{0})\subset Q_{T}$. Let the number $\mu^{-}$ satisfy

$\displaystyle\mu^{-}\leq\operatorname*{ess\inf}_{Q_{\rho,\theta}(x_{0},t_{0})}u.$

We mean the function $u$ satisfies the property $(\mathcal{D})$ if there
exists a constant $\nu\in(0,1)$ only depending on $b,L,\mu^{-}$ and other
data, but independent of $\rho$ such that if

$\displaystyle|[u\leq\mu^{-}+L]\cap
Q_{\rho,\theta}(x_{0},t_{0})|\leq\nu|Q_{\rho,\theta}(x_{0},t_{0})|,$

then

$\displaystyle u\geq\mu^{-}+bL\text{ \ \ a.e.\ in\
}Q_{c\rho,c\theta}(x_{0},t_{0}).$

Similar to the Theorem 2.1 in [33], we present the following result.

######  Theorem 4.1.

Let $u$ be a locally essentially bounded below measurable function in $Q_{T}$
that satisfies the property $(\mathcal{D})$, then we have $u_{*}(x,t)=u(x,t)$
for every $(x,t)\in\mathcal{F}$. In particular, $u_{*}$ is a lower
semiconutinuous representative of $u$ in $Q_{T}$.

Next, we devote to verifying the weak solutions of (1.1) fulfill the property
$(\mathcal{D})$, which together with Theorem 4.1 leads to the conclusion of
Theorem 1.4.

######  Lemma 4.2 (De Giorgi type lemma).

Let the assumptions (1.2) and (1.12) hold. Set $M:=\sup_{Q_{T}}|u|$ and fix
the radius $\rho\in(0,1]$, the constants $L\in(0,M)$, $b\in(0,1]$. Assume that
$u$ is a local weak supersolution to (1.1). For a positive parameter $\eta$,
we set the time levels

$\displaystyle\theta=\eta\rho^{q}\text{\ \ \ when
}\operatorname*{\max}_{Q_{\rho,\eta\rho^{sp}}(x_{0},t_{0})}a(x,t)\geq
2[a]_{\alpha,\frac{\alpha}{2}}\rho^{\alpha},$
$\displaystyle\theta=\eta\rho^{sp}\text{\ \ \ when
}\operatorname*{\max}_{Q_{\rho,\eta\rho^{sp}}(x_{0},t_{0})}a(x,t)\leq
2[a]_{\alpha,\frac{\alpha}{2}}\rho^{\alpha}.$ (4.1)

Let $Q_{\rho,\theta}^{-}(x_{0},t_{0})\subset Q_{T}$, and the number $\mu^{-}$
satisfy

$\displaystyle\mu^{-}\leq\operatorname*{ess\inf}_{\mathbb{R}^{N}\times(0,T)}u.$
(4.2)

There exists a constant $\nu\in(0,1)$ depending on the data,
$M,L,b,\eta,\mu^{-}$ and $[a]_{\alpha,\frac{\alpha}{2}}$ such that if

$\displaystyle|[u\leq\mu^{-}+L]\cap
Q^{-}_{\rho,\theta}(x_{0},t_{0})|\leq\nu|Q^{-}_{\rho,\theta}(x_{0},t_{0})|,$

then

$\displaystyle u\geq\mu^{-}+bL\text{\ a.e. in
}Q_{\frac{\rho}{2},\frac{\theta}{2}}(x_{0},t_{0}).$

###### Proof.

We may assume $(x_{0},t_{0})=(0,0)$ without loss of generality. Let

$\displaystyle k_{j}=\mu^{-}+bL+\frac{(1-b)L}{2^{j}},\ w_{j}=(k_{j}-u)_{+},$
$\displaystyle\rho_{j}=\rho+\frac{\rho}{2^{j}},\
\tilde{\rho}_{j}=\frac{\rho_{j}+\rho_{j+1}}{2},\
\hat{\rho}_{j}=\frac{3\rho_{j}+\rho_{j+1}}{4},$
$\displaystyle\theta_{j}=\theta+\frac{\theta}{2^{j}},\
\tilde{\theta}=\frac{\theta_{j}+\theta_{j+1}}{2},\
\hat{\theta_{j}}=\frac{3\theta_{j}+\theta_{j+1}}{4},$ $\displaystyle
B_{j}=B_{\rho_{j}},\ \tilde{B}_{j}=\tilde{B}_{\rho_{j}},\
\hat{B}_{j}=\hat{B}_{r_{j}},$ $\displaystyle
Q_{j}=B_{j}\times(-\theta_{j},0),\
\tilde{Q}_{j}=\tilde{B}_{j}\times(-\tilde{\theta}_{j},0),\
\hat{Q}_{j}=\hat{B}_{j}\times(-\hat{\theta}_{j},0).$

Obviously, there holds

$\displaystyle Q_{j+1}\subset\tilde{Q}_{j}\subset\hat{Q}_{j}\subset Q_{j}.$

Choose the cutoff functions $\zeta\in C_{0}^{\infty}(Q_{j})$ vanishing outside
$\hat{Q}_{j}$, and satisfy

$\displaystyle\zeta=1\text{\ in }\tilde{Q}_{j},\
|\nabla\zeta|\leq\frac{2^{j}}{\rho},\quad|\partial_{t}\zeta|\leq\frac{2^{spj}}{\theta}.$

Let

$\bar{a}_{\rho}:=\operatorname*{\max}_{Q_{\rho,\eta\rho^{sp}}(x_{0},t_{0})}a(x,t).$

We distinguish two cases: $\bar{a}_{\rho}\leq
2[a]_{\alpha,\frac{\alpha}{2}}\rho^{\alpha}$ and $\bar{a}_{\rho}\geq
2[a]_{\alpha,\frac{\alpha}{2}}\rho^{\alpha}$. For the first case, we utilize
the Caccioppoli type inequality (2.1) in $Q_{j}$ along with the property
$\zeta(x,t)=1$ in $\tilde{Q}_{j}$, which gives that

$\displaystyle\operatorname*{ess\sup}_{-\tilde{\theta}_{j}<t<0}\int_{\tilde{B}_{j}}w_{j}^{2}(x,t)\,dx+\int_{-\tilde{\theta}_{j}}^{0}\int_{\tilde{B}_{j}}\int_{\tilde{B}_{j}}\frac{\left|w_{j}(x,t)-w_{j}(y,t)\right|^{p}}{|x-y|^{N+sp}}\,dxdydt$
$\displaystyle\leq$ $\displaystyle
C\int_{-\theta_{j}}^{0}\int_{B_{j}}w_{j}^{2}(x,t)|\partial_{t}\zeta^{m}|\,dxdt+C\int_{-\theta_{j}}^{0}\int_{B_{j}}a(x,t)w_{j}^{q}(x,t)|\nabla\zeta(x,t)|^{q}\,dxdt$
$\displaystyle+C\int_{-\theta_{j}}^{0}\int_{B_{j}}\int_{B_{j}}(\max\\{w_{j}(x,t),w_{j}(y,t)\\})^{p}\left|\zeta^{\frac{m}{p}}(x,t)-\zeta^{\frac{m}{p}}(y,t)\right|^{p}\,d\mu
dt$
$\displaystyle+C\mathop{\mathrm{ess}\,\sup}_{\stackrel{{\scriptstyle-\theta_{j}<t<0}}{{x\in\hat{B}_{j}}}}\int_{\mathbb{R}^{N}\backslash
B_{j}}\frac{w_{j}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\int_{-\theta_{j}}^{0}\int_{B_{j}}w_{j}(x,t)\zeta^{m}(x,t)\,dxdt,$
(4.3)

where $m=\max\\{p,q\\}$. Denote $A_{j}:=[u<k_{j}]\cap Q_{j}$, we estimate

$\displaystyle\int_{-\theta_{j}}^{0}\int_{B_{j}}w_{j}^{2}(x,t)|\partial_{t}\zeta^{m}|\,dxdt\leq
C\frac{2^{spj}}{\theta}L^{2}|A_{j}|.$ (4.4)

By the condition (1.12), it shows that

$\displaystyle\int_{-\theta_{j}}^{0}\int_{B_{j}}a(x,t)w_{j}^{q}(x,t)|\nabla\zeta(x,t)|^{q}\,dxdt$
$\displaystyle\leq C\frac{2^{jq}}{\rho^{q}}\bar{a}_{\rho}L^{q}|A_{j}|$
$\displaystyle\leq C2^{jq}\rho^{\alpha-q}M^{q-p}L^{p}|A_{j}|$
$\displaystyle\leq C\frac{2^{jq}}{\rho^{sp}}L^{p}|A_{j}|,$ (4.5)

where $C$ depends on the data, $[a]_{\alpha,\frac{\alpha}{2}}$ and $M$.
Moreover, we have

$\displaystyle\int_{-\theta_{j}}^{0}\int_{B_{j}}\int_{B_{j}}(\max\\{w_{j}(x,t),w_{j}(y,t)\\})^{p}\left|\zeta^{\frac{m}{p}}(x,t)-\zeta^{\frac{m}{p}}(y,t)\right|^{p}\,d\mu
dt\leq C\frac{2^{jm}}{\rho^{sp}}L^{p}|A_{j}|.$ (4.6)

For $|y|\geq\rho_{j}$ and $|x|\leq\hat{\rho}_{j}$, we can get

$\displaystyle\frac{|y-x|}{|y|}\geq
1-\frac{\hat{\rho}_{j}}{\rho_{j}}=\frac{1}{4}\left(\frac{\rho_{j}-\rho_{j+1}}{\rho_{j}}\right)\geq\frac{1}{2^{j+4}},$

which along with (4.2) gives that

$\displaystyle\mathop{\mathrm{ess}\sup}_{\stackrel{{\scriptstyle-\theta_{j}<t<0}}{{x\in\hat{B}_{j}}}}\int_{\mathbb{R}^{N}\backslash
B_{j}}\frac{w_{j}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\int_{-\theta_{j}}^{0}\int_{B_{j}}w_{j}(x,t)\zeta^{m}(x,t)\,dxdt\leq
C\frac{2^{j(N+sp)}}{\rho^{sp}}L^{p}|A_{j}|.$ (4.7)

We deduce from (4)–(4.7) that

$\displaystyle\operatorname*{ess\sup}_{-\tilde{\theta}_{j}<t<0}\int_{\tilde{B}_{j}}w_{j}^{2}(x,t)\,dx+\int_{-\tilde{\theta}_{j}}^{0}\int_{\tilde{B}_{j}}\int_{\tilde{B}_{j}}\frac{\left|w_{j}(x,t)-w_{j}(y,t)\right|^{p}}{|x-y|^{N+sp}}\,dxdydt$
$\displaystyle\leq$ $\displaystyle
C2^{j(N+2m)}\left(\frac{L^{2}}{\theta}+\frac{L^{p}}{\rho^{sp}}\right)|A_{j}|,$

where $C$ depends on the data, $[a]_{\alpha,\frac{\alpha}{2}}$, and $M$.
Employing the Sobolev embedding Lemma 1.7 and recalling the fact $\zeta=1$ in
$\tilde{Q}_{j}$ yields that

$\displaystyle\iint_{\tilde{Q}_{j}}(w_{j}\zeta)^{p(1+\frac{2s}{N})}\,dxdt=\iint_{\tilde{Q}_{j}}w_{j}^{p(1+\frac{2s}{N})}\,dxdt$
$\displaystyle\leq$
$\displaystyle\int_{-\tilde{\theta}_{j}}^{0}\int_{\tilde{B}_{j}}\int_{\tilde{B}_{j}}\frac{\left|w_{j}(x,t)-w_{j}(y,t)\right|^{p}}{|x-y|^{N+sp}}\,dxdydt\times\left(\operatorname*{ess\sup}_{-\tilde{\theta}_{j}<t<0}\int_{\tilde{B}_{j}}w_{j}^{2}(x,t)\,dx\right)^{\frac{sp}{N}}$
$\displaystyle\leq$ $\displaystyle
C\left[2^{j(N+2m)}\left(\frac{L^{2}}{\theta}+\frac{L^{p}}{\rho^{sp}}\right)\right]^{\frac{N+sp}{N}}|A_{j}|^{\frac{N+sp}{N}}.$
(4.8)

On the other hand,

$\displaystyle\iint_{\tilde{Q}_{j}}(w_{j}\zeta)^{p(1+\frac{2s}{N})}\,dxdt$
$\displaystyle\geq\iint_{\tilde{Q}_{j}}w_{j}^{p(1+\frac{2s}{N})}\chi_{[u<k_{j+1}]}\,dxdt$
$\displaystyle\geq\iint_{Q_{j+1}\cap[u<k_{j+1}]}(k_{j}-k_{j+1})^{p(1+\frac{2s}{N})}\,dxdt$
$\displaystyle\geq\left[\frac{(1-b)L}{2^{j+1}}\right]^{p\frac{N+2s}{N}}|Q_{j+1}\cap[u<k_{j+1}]|.$
(4.9)

Set

$\displaystyle Y_{j}=\frac{|[u<k_{j}]\cap Q_{j}|}{|Q_{j}|},$

then it follows from (4) and (4) that

$\displaystyle
Y_{j+1}\leq\frac{Ch^{j}}{(1-b)^{p\frac{N+2s}{N}}}\left(\frac{\theta}{L^{2-p}}\right)^{\frac{sp}{N}}\left(\frac{1}{\rho^{sp}}+\frac{L^{2-p}}{\theta}\right)^{\frac{N+sp}{N}}\rho^{sp}|Y_{j}|^{\frac{N+sp}{N}},$

where $h=h(N,s,p,q)>1$. Since $\theta=\eta\rho^{sp}$, we can obtain
$Y_{j}\rightarrow 0$ as $j\rightarrow\infty$ by Lemma 1.8 provided

$\displaystyle Y_{0}\leq
C^{-\frac{N}{sp}}h^{-\left(\frac{N}{sp}\right)^{2}}(1-b)^{N+2}\frac{L^{2-p}}{\eta}\left(1+\frac{L^{2-p}}{\eta}\right)^{-\frac{N+sp}{sp}}:=\nu.$

For the case $\bar{a}_{\rho}\geq 2[a]_{\alpha,\frac{\alpha}{2}}\rho^{\alpha}$,
we can check that $\frac{2}{3}\bar{a}_{\rho}\leq a(x,t)\leq 2\bar{a}_{\rho}$
in $Q_{\rho,\eta\rho^{q}}(x_{0},t_{0})$ which leads to the nonstandard growth
problem. In this case, we can use the method of proving Lemma 3.2 in our
previous paper [36] to get this result. Similar to the proof above, employ
Caccioppoli inequality (2.1) in $Q_{j}$ and Lemma 2.1 in [36], we can arrive
at the conclusion by choosing

$\displaystyle\nu=C^{-\frac{N}{q}}\tilde{h}^{-\left(\frac{N}{q}\right)^{2}}(1-b)^{N+2}\frac{L^{2-q}}{\eta}\left(1+\frac{L^{2-q}}{\eta}+L^{p-q}\right)^{-\frac{N+q}{q}},$

where $\tilde{h}=\tilde{h}(N,s,p,q)>1$. ∎

We now prepare to prove Theorem 1.6. From the definition of $u_{*}$, we can
obtain

$\displaystyle
u_{*}(x,t)=\operatorname*{ess\lim\inf}_{(y,\hat{t})\rightarrow(x,t)}u(y,\hat{t})\leq\inf_{\eta>0}\lim_{\varrho\rightarrow
0}\operatorname*{ess\inf}_{Q^{\prime}_{\rho,\theta}(x,t)}u\text{\ \ for every
}(x,t)\in Q_{T}.$ (4.10)

For the reverse inequality of (4.10), we need prove a De Giorgi lemma over the
forward cylinders with initial data as below.

######  Lemma 4.3.

Let the assumptions (1.2) and (1.12) hold. Set $M:=\sup_{Q_{T}}|u|$ and fix
the radius $\rho\in(0,1]$, the constants $L\in(0,M)$, $b\in(0,1]$. Assume that
$u$ is a local weak supersolution to (1.1). Set the same time levels as (4.2).
Let the number $\mu^{-}$ satisfy

$\displaystyle\mu^{-}\leq\operatorname*{ess\inf}_{\mathbb{R}^{N}\times(0,T)}u.$

There exists a constant $\eta>0$ depending on the data, $M,L,b,\mu^{-}$ and
$[a]_{\alpha,\frac{\alpha}{2}}$ such that if

$\displaystyle u(\cdot,t_{0})\geq\mu^{-}+L\text{\ a.e. in }B_{\rho}(x_{0}),$

then

$\displaystyle u\geq\mu^{-}+bL\text{\ a.e. in
}Q_{\frac{\rho}{2},\frac{\theta}{2}}(x_{0},t_{0}).$

###### Proof.

We may assume $(x_{0},t_{0})=(0,0)$. Let
$k_{j},\rho_{j},\tilde{\rho}_{j},\hat{\rho}_{j},\theta_{j},\tilde{\theta}_{j},\hat{\theta}_{j},B_{j},\tilde{B}_{j},\hat{B}_{j}$
be as in Lemma 4.2. Define the forward cylinders

$\displaystyle Q_{j}=B_{j}\times(0,\theta_{j}),\
\tilde{Q}_{j}=\tilde{B}_{j}\times(0,\tilde{\theta_{j}}),\
\hat{Q}_{j}=\hat{B}_{j}\times(0,\hat{\theta}_{j}).$

Choose the piecewise cutoff function $\zeta(x,t)=\zeta(x)$ vanishing outside
$\hat{B}_{j}$ such that

$\displaystyle 0\leq\zeta\leq 1,\quad\zeta(x)=1\text{\ in
}\tilde{B}_{j},\quad|\nabla\zeta|\leq\frac{2^{j}}{\rho}.$

When $\bar{a}_{\rho}\leq 2[a]_{\alpha,\frac{\alpha}{2}}\rho^{\alpha}$, a
combination of energy estimates (2.1) in the cylinder $\tilde{Q}_{j}$ and the
fact $u(\cdot,0)\geq\mu^{-}+L$ yield that

$\displaystyle\operatorname*{ess\sup}_{0<t<\theta_{j}}\int_{\tilde{B}_{j}}w_{j}^{2}(x,t)\,dx+\int_{0}^{\theta_{j}}\int_{\tilde{B}_{j}}\int_{\tilde{B}_{j}}\frac{\left|w_{j}(x,t)-w_{j}(y,t)\right|^{p}}{|x-y|^{N+sp}}\,dxdydt$
$\displaystyle\leq$ $\displaystyle
C2^{j(N+2m)}\frac{L^{p}}{\rho^{sp}}|A_{j}|,$

where $A_{j}:=[u<k_{j}]\cap Q_{j}$. Let $Y_{j}=\frac{|A_{j}|}{|Q_{j}|}$.
Following the same steps as proving Lemma 4.2, we obtain

$\displaystyle
Y_{j+1}\leq\frac{Ch^{j}}{(1-b)^{p\frac{N+2s}{N}}}\left(\frac{\eta}{L^{2-p}}\right)^{\frac{sp}{N}}|Y_{j}|^{\frac{N+sp}{N}}$

with $h=h(N,p,s,q)>1$. By Lemma 1.8, it shows that if

$\displaystyle Y_{0}\leq
C^{-\frac{N}{sp}}h^{-\frac{N^{2}}{s^{2}p^{2}}}(1-b)^{N+2}\frac{L^{2-p}}{\eta},$

then $Y_{j}\rightarrow 0$. Thus, we can arrive at the claim by choosing

$\displaystyle\eta=C^{-\frac{N}{sp}}(1-b)^{N+2s}L^{2-p}.$

Proceeding similarly as above, we may choose

$\eta=C^{-\frac{N}{q}}(1-b)^{N+2}L^{2-q}(1+L^{p-q})^{-\frac{N+q}{q}}$

in the case $\bar{a}_{\rho}\geq 2[a]_{\alpha,\frac{\alpha}{2}}\rho^{\alpha}$
to finish the proof. ∎

The rest of the proof of the inverse inequality of (4.10) is based on Lemma
4.3. Since it is similar to [Theorem 3.1, [33]], we omit the proof here. The
proof of Theorem 1.6 is finally complete.

## Acknowledgment

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

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