Regularity Theory for Mixed Local and Nonlocal Parabolic p-Laplace Equations

Abstract

We investigate the mixed local and nonlocal parabolic p-Laplace equation

$$\begin{aligned} \partial _t u(x,t)-\Delta _p u(x,t)+\mathcal {L}u(x,t)=0, \end{aligned}$$

where $\Delta _p$ is the usual local p-Laplace operator and $\mathcal {L}$ is the nonlocal p-Laplace type operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi–Nash–Moser iteration, we establish the local boundedness and Hölder continuity of weak solutions for such equations.

1 Introduction

In this paper, we are concerned with the local behaviour of weak solutions to the following mixed problem:

$$\begin{aligned} \partial _t u(x,t)-\Delta _p u(x,t)+{\mathcal {L}}u(x,t)=0 \quad \text {in } Q_T,\quad 1<p<\infty , \end{aligned}$$

(1.1)

where $Q_T:=\Omega \times (0,T)$ with $T>0$ and $\Omega$ is a bounded domain in ${\mathbb {R}}^N$. This kind of evolution equations arises from the Lévy process, image processing etc; see [16] and references therein. The local p-Laplace operator $\Delta _p$ is defined as follows:

$$\begin{aligned} \Delta _p u:={\text {div}}(|\nabla u|^{p-2}\nabla u), \end{aligned}$$

and ${\mathcal {L}}$ is a nonlocal p-Laplace operator given by

$$\begin{aligned} {\mathcal {L}}u(x,t)=\mathrm {P.V.}\int _{{\mathbb {R}}^N}K(x,y,t)|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,\mathrm{d}y, \end{aligned}$$

(1.2)

where the symbol $\mathrm {P.V.}$ stands for the Cauchy principal value. Here, K is a symmetric kernel fulfilling

$$\begin{aligned} K(x,y,t)=K(y,x,t) \end{aligned}$$

and

$$\begin{aligned} \frac{\Lambda ^{-1}}{|x-y|^{N+sp}}\le K(x,y,t)\le \frac{\Lambda }{|x-y|^{N+sp}} \end{aligned}$$

(1.3)

with $\Lambda \ge 1$ and $0<s<1$ for all $x,y\in {\mathbb {R}}^N$ and $t\in (0,T)$.

Before stating our main results, let us mention some known results. For the nonlocal parabolic equations of p-Laplacian type,

$$\begin{aligned} \partial _tu(x,t)+{\mathcal {L}}u(x,t)=0, \end{aligned}$$

(1.4)

the existence and uniqueness of strong solutions were verified by Vázquez [27], where the author studied the long-time behaviours as well. Mazón–Rossi–Toledo [24] established the well-posedness of solutions to Eq. (1.4) together with the asymptotic property. When it comes to regularity theory of this equation, Strömqvist [26] obtained the existence and local boundedness of weak solutions provided $p\ge 2$. Hölder regularity with specific exponents in the case $p\ge 2$ was proved by Brasco–Lindgren–Strömqvist [5]. Furthermore, Ding–Zhang–Zhou [12] showed the local boundedness and Hölder continuity of weak solutions to the nonhomogeneous case under the conditions that $1<p<\infty$ and $2<p<\infty$, respectively. We refer the readers to [6, 18, 22, 28, 29] and references therein for more results.

In the mixed local and nonlocal setting, for the case $p=2$,

$$\begin{aligned} -\Delta u+(-\Delta )^su=0, \end{aligned}$$

(1.5)

Foondun [19] has derived Harnack inequality and interior Hölder estimates for nonnegative solutions, see also [8] for a diverse approach. In addition, the Harnack inequality regarding the parabolic version of (1.5) was established in [2, 7], where, however, the authors only proved such inequality for globally nonnegative solutions. Very recently, Garain–Kinnunen [21] proved a weak Harnack inequality with a tail term for sign changing solutions to the parabolic problem of (1.5). For what concerns maximum principles, interior sobolev regularity along with symmetry results among many other quantitative and qualitative properties for solutions to (1.5), one can see for instance [3, 4, 13,14,15]. In the nonlinear framework (i.e. $p\ne 2$), Garain–Kinnunen [20] developed the local regularity theory for

$$\begin{aligned} -\Delta _pu+\mathrm {P.V.}\int _{{\mathbb {R}}^N}K(x,y)|u(x)-u(y)|^{p-2}(u(x)-u(y))\,\mathrm{d}y=0 \end{aligned}$$

with $K(x,y)\simeq |x-y|^{-(N+sp)}$, involving boundedness, Hölder continuity, Harnack inequality, as well as lower/upper semicontinuity of weak supersolutions/subsolutions. Nonetheless, to the best of our knowledge, there are few results concerning on the mixed local and nonlocal nonlinear parabolic problems. To this end, influenced by the ideas developed in [10, 12, 20], we aim to establish the local boundedness and interior Hölder regularity of weak solutions to Eq. (1.1). It is noteworthy that our results are new even for the case $p=2$.

Before giving the notion of weak solutions to (1.1), let us recall the tail space

$$\begin{aligned} L_\alpha ^q({\mathbb {R}}^{N}):=\left\{ v \in L_{\mathrm{loc}}^q({\mathbb {R}}^{N}): \int _{{\mathbb {R}}^N} \frac{|v(x)|^q}{1+|x|^{N+\alpha }}\,\mathrm{d}x<+\infty \right\} , \quad q>0 \text { and } \alpha >0. \end{aligned}$$

Then, we define the tail appearing in estimates throughout this article,

$$\begin{aligned} \mathrm {Tail}_\infty (v;x_{0},r,I)&=\mathrm {Tail}_{\infty }(v; x_0,r,t_0-T_1,t_0+T_2)\nonumber \\&:=\underset{t \in I}{{\text {ess}}\,\sup }\left( r^p \int _{{\mathbb {R}}^{N} \backslash B_r(x_{0})} \frac{|v(x, t)|^{p-1}}{|x-x_0|^{N+sp}}\,\mathrm{d}x\right) ^{\frac{1}{p-1}}, \end{aligned}$$

(1.6)

where $(x_0, t_0)\in {\mathbb {R}}^N\times (0,T)$ and the interval $I=\left[ t_0-T_1, t_0+T_2\right] \subseteq (0,T)$. This is a parabolic counterpart to the tail introduced in [10]. It is easy to check that $\mathrm {Tail}_\infty (v;x_0,r,I)$ is well-defined for any $v \in L^{\infty }(I; L_{sp}^{p-1}({\mathbb {R}}^{N}))$.

For any $1<p<\infty$ and $0<s<1$, the fractional Sobolev space is defined by

$$\begin{aligned} W^{s,p}(\Omega )=\left\{ u \in L^{p}(\Omega ): \int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{d}x\mathrm{d}y<\infty \right\} \end{aligned}$$

endowed with the norm

$$\begin{aligned} \Vert u\Vert _{W^{s, p}\left( \Omega \right) }=\Vert u\Vert _{L^{p}(\Omega )}+\left( \int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \mathrm{d}x\mathrm{d}y\right) ^{\frac{1}{p}}, \end{aligned}$$

which is a reflexive Banach space, see [11, 25]. From [11, Proposition 2.2], we know that the classical Sobolev space $W^{1,p}(\Omega )$ is continuously embedded in the fractional Sobolev space $W^{s,p}(\Omega )$.

The notion of weak solutions to (1.1) is stated as follows.

Definition 1.1

A function $u\in L^p(I;W_\mathrm{{loc}}^{1,p}(\Omega )) \cap C(I;L_\mathrm{{loc}}^2(\Omega )) \cap L^{\infty }(I;L_{sp}^{p-1}({\mathbb {R}}^N))$ is a local weak subsolution (super-) to (1.1) if for any closed interval $I:=[t_1, t_2] \subseteq (0,T)$, there holds that

$$\begin{aligned}&\int _\Omega u(x,t_2)\varphi (x,t_2)\,\mathrm{d}x-\int _\Omega u(x,t_1)\varphi (x,t_1)\,\mathrm{d}x -\int _{t_1}^{t_2} \int _\Omega u(x,t)\partial _t \varphi (x,t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad +\int _{t_1}^{t_2}\int _\Omega |\nabla u|^{p-2}\nabla u\cdot \nabla \varphi \,\mathrm{d}x\mathrm{d}t +\int _{t_1}^{t_2} {\mathcal {E}}(u,\varphi ,t)\,\mathrm{d}t \le (\ge ) 0, \end{aligned}$$

(1.7)

for every nonnegative test function $\varphi \in L^p (I;W^{1,p}(\Omega ))\cap W^{1,2}(I;L^2(\Omega ))$ with the property that $\varphi$ has spatial support compactly contained in $\Omega$, where

$$\begin{aligned} {\mathcal {E}}(u,\varphi ,t):=\frac{1}{2} \int _{{\mathbb {R}}^N} \int _{{\mathbb {R}}^N}&\Big [|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\\&\times (\varphi (x,t)-\varphi (y,t))K(x,y,t)\Big ]\,\mathrm{d}x\mathrm{d}y. \end{aligned}$$

A function u is a local weak solution to (1.1) if and only if u is a local weak subsolution and supersolution.

We now are in a position to state the main contribution of this work. First, we provide the local boundedness of weak solutions in the cases that $p>\frac{2N}{N+2}$ and $1<p\le \frac{2N}{N+2}$. For two real numbers, set

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

Theorem 1.2

(Local boundedness) Let $p> 2N/(N+2)$ and $q:=\max \{p,2\}$. Assume that u is a local weak subsolution to (1.1). Let $(x_0,t_0) \in Q_T$, $R \in (0,1)$ and $Q_R^- \equiv B_R(x_0) \times (t_0-R^p,t_0)$ such that ${\overline{B}}_R(x_0)\subseteq \Omega$ and $[t_0-R^p, t_0] \subseteq (0,T)$. Then it holds that

$$\begin{aligned} \underset{Q_{R/2}^-}{{\text {ess}}\,\sup } u\le \mathrm {Tail}_\infty \left( u_+;x_0,R/2,t_0-R^p, t_0\right) +C\left( {\int \!\!\!\!-}_{Q_R^-} u_+^q\,\mathrm{d}x \mathrm{d}t\right) ^{\frac{p}{N(p\kappa -q)}} \vee 1, \end{aligned}$$

where $\kappa :=1+2/N$ and $C>0$ only depends on N, p, s and $\Lambda$.

In the scenario that $1<p\le 2N/(N+2)$, assuming that the weak subsolution has the following constructions: for $m>\max \left\{ 2,\frac{N(2-p)}{p}\right\}$, there exists a sequence of $\{u_k\}_{k\in {\mathbb {N}}}$ whose components are bounded subsolutions of (1.1) fulfilling

$$\begin{aligned} \Vert u_k\Vert _{L_{\mathrm{loc}}^\infty (0,T;L_{sp}^{p-1}({\mathbb {R}}^N))}\le C \end{aligned}$$

(1.8)

and

$$\begin{aligned} u_k\rightarrow u \quad \text {in } L_{\mathrm{loc}}^m(Q_T) \text { as } k\rightarrow \infty . \end{aligned}$$

(1.9)

Theorem 1.3

(Local boundedness) Let $1<p\le 2N/(N+2)$, $\kappa =1+2/N$ and $m>\max \left\{ 2,\frac{N(2-p)}{p}\right\}$. Suppose that $u\in L_{\mathrm{loc}}^m(Q_T)$ with the properties (1.8) and (1.9) is a local weak subsolution to (1.1). Let $(x_0,t_0) \in Q_T$, $R \in (0,1)$, and $Q_R^- \equiv B_R(x_0) \times (t_0-R^p,t_0)$ such that ${\overline{B}}_R(x_0)\subseteq \Omega$ and $[t_0-R^p, t_0] \subseteq (0,T)$. Then it holds that

$$\begin{aligned} \underset{Q_{R/2}^-}{{\text {ess}}\,\sup }\, u&\le \mathrm {Tail}_\infty \left( u_+;x_0,R/2,t_0-R^p, t_0\right) \\&\quad +C\left( {\int \!\!\!\!-}_{Q_R^-} u_+^m\,\mathrm{d}x \mathrm{d}t\right) ^{\frac{p}{(N+p)(m-2-\beta )}} \vee \left( {\int \!\!\!\!-}_{Q_R^-} u_+^m\,\mathrm{d}x \mathrm{d}t\right) ^{\frac{p}{(N+p)(m-p-\beta )}}, \end{aligned}$$

where $\beta =N(m-p\kappa )/(N+p)$ and $C>0$ only depends on N, p, s, m and $\Lambda$.

Based on the boundedness result (Theorem 1.2), we are able to deduce that the weak solutions are locally Hölder continuous for $p>2$.

Theorem 1.4

(Hölder continuity) Let $p>2$. Assume that u is a local weak solution to (1.1). Let $(x_0,t_0) \in Q_T$, $R \in (0,1)$ and $Q_R \equiv B_R(x_0) \times (t_0-R^p,t_0+R^p)$ such that ${\overline{Q}}_R\subseteq Q_T$. Then there is a constant $\alpha \in (0,p/(p-1))$ such that for every $\rho \in (0,R/2)$,

$$\begin{aligned}&\underset{Q_{\rho , d \rho ^p}}{{\text {ess osc}}} \,u\\&\quad <C\left( \frac{\rho }{R}\right) ^\alpha \left[ \mathrm {Tail}_\infty \left( u;x_0,R/2, t_0-R^p, t_0+R^p\right) +\left( {\int \!\!\!\!-}_{Q_R}|u|^p\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{1}{2}} \vee 1\right] \end{aligned}$$

with some $d\in (0,1)$ and $C\ge 1$ depending on N, p, s and $\Lambda$.

The paper is organized as follows. In Sect. 2, we collect some notations and auxiliary inequalities. Necessary energy estimates are showed in Sect. 3. Sections 4 and 5 are devoted to proving the local boundedness and Hölder regularity of weak solutions, respectively.

2 Preliminaries

In this section, we first give some notations for clarity and then provide some important inequalities to be used later.

2.1 Notation

Let $B_\rho (x)$ be the open ball with radius $\rho$ and centred at $x\in {\mathbb {R}}^N$. We denote the parabolic cylinders by $Q_{\rho ,r}(x, t):=B_\rho (x) \times (t-r,t+r)$, $Q_\rho (x,t):=Q_{\rho ,\rho ^p}(x,t)=B_\rho (x) \times (t-\rho ^p,t+\rho ^p)$ and $Q_\rho ^-(x,t):=Q_{\rho ,\rho ^p}^-(x, t)=B_\rho (x) \times (t-\rho ^p,t)$ with $r,\rho >0$ and $(x,t) \in {\mathbb {R}}^{N} \times (0,T)$. If not important, or clear from the context, we simply write these symbols by $B_\rho =B_\rho (x)$, $Q_{\rho ,r}=Q_{\rho ,r}(x,t)$, $Q_\rho =Q_\rho (x,t)$ and $Q_\rho ^-=Q_\rho ^-(x,t)$. Moreover, for $g\in L^1(V)$, we denote the integral average of g by

$$\begin{aligned} (g)_{V}:={\int \!\!\!\!-}_{V} g(x)\,\mathrm{d}x:=\frac{1}{|V|} \int _{V} g(x)\,\mathrm{d}x. \end{aligned}$$

Define

$$\begin{aligned} J_{p}(a, b)=|a-b|^{p-2}(a-b) \end{aligned}$$

for any $a, b \in {\mathbb {R}}$. We also use the notation

$$\begin{aligned} \mathrm{d}\mu =\mathrm{d}\mu (x,y,t)=K(x,y,t)\,\mathrm{d}x\mathrm{d}y. \end{aligned}$$

It is worth mentioning that the constant C represents a general positive constant which may differ from each other.

Next, we will show several fundamental but very useful Sobolev inequalities. The similar results can be found in [1, 9]. For the sake of readability and completeness, we give the proof of the last two lemmas.

2.2 Sobolev Inequalities

Lemma 2.1

Let $1 \le p$, $\ell \le q<\infty$ satisfy $\frac{N}{p}-\frac{N}{q}\le 1$ and

$$\begin{aligned} \theta \left( 1-\frac{N}{p}+\frac{N}{q}\right) +(1-\theta )\left( \frac{N}{q}-\frac{N}{\ell }\right) =0 \end{aligned}$$

with $\theta \in (0,1)$. Then there exists a constant $C>0$ only depending on $N, p, q, \ell$ such that

$$\begin{aligned} \Vert u\Vert _{L^q(B_1)} \le C\Vert Du\Vert _{L^p(B_1)}^\theta \Vert u\Vert _{L^\ell (B_1)}^{1-\theta } \end{aligned}$$

(2.1)

for all $u \in W^{1,p}(B_1) \cap L^\ell (B_1)$.

Lemma 2.2

Let $1<p<N$. Then for every $u\in W^{1,p}(B_1)$, there holds that

$$\begin{aligned} \Vert u\Vert _{L^{\frac{Np}{N-p}}(B_1)}\le C\Vert u\Vert _{W^{1,p}(B_1)}, \end{aligned}$$

where $C>0$ only depends on N and p.

Lemma 2.3

Let $0<t_1<t_2$ and $p\in (1,\infty )$. Then for every

$$\begin{aligned} u\in L^p\left( t_1,t_2;W^{1,p}(B_r)\right) \cap L^\infty \left( t_1,t_2;L^2(B_r)\right) , \end{aligned}$$

it holds that

$$\begin{aligned}&\int _{t_1}^{t_2} {\int \!\!\!\!-}_{B_r}|u(x,t)|^{p(1+\frac{2}{N})}\,\mathrm{d}x\mathrm{d}t \nonumber \\&\quad \le C\left( r^p \int _{t_1}^{t_2}{\int \!\!\!\!-}_{B_r} |\nabla u(x,t)|^p\,\mathrm{d}x\mathrm{d}t+\int _{t_1}^{t_2} {\int \!\!\!\!-}_{B_r}|u(x,t)|^p\,\mathrm{d}x\mathrm{d}t\right) \nonumber \\&\qquad \times \left( \underset{t_1<t<t_2}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B_r}|u(x, t)|^2\,\mathrm{d}x\right) ^{\frac{p}{N}}, \end{aligned}$$

(2.2)

where $C>0$ only depends on p and N.

Proof

We divide the proof into the following two cases.

Case 1 $1<p<N$. Applying Lemma 2.2 and Hölder inequality, we infer that

$$\begin{aligned}&\int _{t_1}^{t_2} {\int \!\!\!\!-}_{B_1}|u(x,t)|^{p(1+\frac{2}{N})}\, \mathrm{d}x\mathrm{d}t\nonumber \\&\quad =\int _{t_1}^{t_2} {\int \!\!\!\!-}_{B_1}|u(x,t)|^{\frac{2p}{N}}|u(x,t)|^p\, \mathrm{d}x\mathrm{d}t \nonumber \\&\quad \le \int _{t_1}^{t_2}\left( {\int \!\!\!\!-}_{B_1}|u(x,t)|^{2}\, \mathrm{d}x\right) ^{\frac{ p}{N}}\left( {\int \!\!\!\!-}_{B_1}|u(x,t)|^{\frac{pN}{N-p}}\, \mathrm{d}x\right) ^{\frac{N-p}{N}}\, \mathrm{d}t\nonumber \\&\quad \le C\left( \underset{t_1<t<t_2}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B_1}|u(x,t)|^{2}\, \mathrm{d} x\right) ^{\frac{p}{N}} \int _{t_1}^{t_2} {\int \!\!\!\!-}_{B_1}|u(x,t)|^p+|\nabla u(x,t)|^p\,\mathrm{d}x \mathrm{d}t. \end{aligned}$$

By the scaling argument, we get

$$\begin{aligned}&\int _{t_1}^{t_2} {\int \!\!\!\!-}_{B_r}|u(x,t)|^{p(1+\frac{2}{N})}\,\mathrm{d}x\mathrm{d}t \nonumber \\&\quad \le C\left( r^p \int _{t_1}^{t_2}{\int \!\!\!\!-}_{B_r} |\nabla u(x,t)|^p\,\mathrm{d}x\mathrm{d}t+\int _{t_1}^{t_2} {\int \!\!\!\!-}_{B_r}|u(x,t)|^p\,\mathrm{d}x\mathrm{d}t\right) \nonumber \\&\qquad \times \left( \underset{t_1<t<t_2}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B_r}|u(x, t)|^2\,\mathrm{d}x\right) ^{\frac{p}{N}}. \end{aligned}$$

Case 2: $p\ge N$. We can easily find that

$$\begin{aligned} 1\ge \frac{N}{p}-\frac{N}{p(1+\frac{2}{N})}, \quad p\left( 1+\frac{2}{N}\right) >2 \end{aligned}$$

and

$$\begin{aligned} \theta \left( 1-\frac{N}{p}+\frac{N}{p\left( 1+\frac{2}{N}\right) }\right) +(1-\theta )\left( \frac{N}{p\left( 1+\frac{2}{N}\right) }-\frac{N}{2}\right) =0 \end{aligned}$$

with $\theta =\frac{N}{N+2}$. Thus by Lemma 2.1, it follows that

$$\begin{aligned} \Vert u\Vert _{L^{p\left( 1+\frac{2}{N}\right) }(B_1)} \le C\Vert Du\Vert _{L^p(B_1)}^{\frac{N}{N+2}} \Vert u\Vert _{L^2(B_1)}^{\frac{2}{N+2}} \end{aligned}$$

for any $t\in (t_1,t_2)$. Using the rescaling argument, we have

$$\begin{aligned} {\int \!\!\!\!-}_{B_r}|u(x,t)|^{p(1+\frac{2}{N})}\,\mathrm{d}x \le C r^p {\int \!\!\!\!-}_{B_r}|\nabla u(x,t)|^p\,\mathrm{d}x \left( {\int \!\!\!\!-}_{B_r}|u(x,t)|^2\,\mathrm{d}x\right) ^{\frac{p}{N}} \end{aligned}$$

for any $t\in (t_1,t_2)$. Integrating the above inequality over $(t_1,t_2)$, we get

$$\begin{aligned}&\int _{t_1}^{t_2}{\int \!\!\!\!-}_{B_r}|u(x,t)|^{p(1+\frac{2}{N})}\,\mathrm{d}x\mathrm{d}t\\&\quad \le C r^p \int _{t_1}^{t_2}{\int \!\!\!\!-}_{B_r}|\nabla u(x,t)|^p\,\mathrm{d}x \left( {\int \!\!\!\!-}_{B_r}|u(x,t)|^2\,\mathrm{d}x\right) ^{\frac{p}{N}}\,\mathrm{d}t\\&\quad \le C r^p \int _{t_1}^{t_2}{\int \!\!\!\!-}_{B_r}|\nabla u(x,t)|^p\,\mathrm{d}x\mathrm{d}t \left( \underset{t_1<t<t_2}{{\text {ess}}\,\sup }{\int \!\!\!\!-}_{B_r}|u(x,t)|^2\,\mathrm{d}x\right) ^{\frac{p}{N}}. \end{aligned}$$

$\square$

Lemma 2.4

Let $0<t_1<t_2$ and $p\in (1,\infty )$. Then for every

$$\begin{aligned} u\in L^p\left( t_1,t_2;W^{1,p}(B_r)\right) \cap L^\infty \left( t_1,t_2;L^p(B_r)\right) , \end{aligned}$$

it holds that

$$\begin{aligned}&\int _{t_1}^{t_2} {\int \!\!\!\!-}_{B_r}|u(x,t)|^{p(1+\frac{p}{N})}\,\mathrm{d}x\mathrm{d}t \nonumber \\&\quad \le Cr^p \int _{t_1}^{t_2}{\int \!\!\!\!-}_{B_r} |\nabla u(x,t)|^p\,\mathrm{d}x\mathrm{d}t\left( \underset{t_1<t<t_2}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B_r}|u(x, t)|^p\,\mathrm{d}x\right) ^{\frac{p}{N}}, \end{aligned}$$

(2.3)

where $C>0$ only depends on p and N.

Proof

It follows from Lemma 2.1 that

$$\begin{aligned} \Vert u\Vert _{L^{p\left( 1+\frac{p}{N}\right) }(B_1)} \le C\Vert Du\Vert _{L^p(B_1)}^{\frac{N}{N+p}} \Vert u\Vert _{L^p(B_1)}^{\frac{p}{N+p}} \end{aligned}$$

for all $t\in (t_1,t_2)$, where $C>0$ only depends on p and N. Using the rescaling argument, we have

$$\begin{aligned} {\int \!\!\!\!-}_{B_r}|u(x,t)|^{p(1+\frac{p}{N})}\,\mathrm{d}x \le C r^p {\int \!\!\!\!-}_{B_r}|\nabla u(x,t)|^p\,\mathrm{d}x \left( {\int \!\!\!\!-}_{B_r}|u(x,t)|^p\,\mathrm{d}x\right) ^{\frac{p}{N}}. \end{aligned}$$

Integrating the above inequality over $(t_1,t_2)$, we get

$$\begin{aligned}&\int _{t_1}^{t_2}{\int \!\!\!\!-}_{B_r}|u(x,t)|^{p(1+\frac{p}{N})}\,\mathrm{d}x\mathrm{d}t\\&\quad \le C r^p \int _{t_1}^{t_2}{\int \!\!\!\!-}_{B_r}|\nabla u(x,t)|^p\,\mathrm{d}x \left( {\int \!\!\!\!-}_{B_r}|u(x,t)|^p\,\mathrm{d}x\right) ^{\frac{p}{N}}\,\mathrm{d}t\\&\quad \le C r^p \int _{t_1}^{t_2}{\int \!\!\!\!-}_{B_r}|\nabla u(x,t)|^p\,\mathrm{d}x\mathrm{d}t \left( \underset{t_1<t<t_2}{{\text {ess}}\,\sup }{\int \!\!\!\!-}_{B_r}|u(x,t)|^p\,\mathrm{d}x\right) ^{\frac{p}{N}}. \end{aligned}$$

$\square$

3 Energy Estimates

In this section, we will establish the Caccioppoli inequality and Logarithmic form inequality for Eq. (1.1). The first step of the proof should be the regularization procedure with respect to the time variable, which can be performed by straightforward adaptation of standard reasonings as used in [5, 12, 23]. We omit this step here.

Lemma 3.1

(Caccioppoli-type inequality) Let $p>1$ and u be a local subsolution to (1.1). Let $B_r \equiv B_r(x_0)$ satisfy ${\overline{B}}_r \subseteq \Omega$ and $0<\tau _1<\tau _2$, $\ell >0$ satisfy $\left[ \tau _1-\ell , \tau _2\right] \subseteq (0,T).$ For any nonnegative functions $\psi \in C_0^\infty (B_r)$ and $\eta \in C^\infty ({\mathbb {R}})$ such that $\eta (t) \equiv 0$ if $t \le \tau _1-\ell$ and $\eta (t) \equiv 1$ if $t \ge \tau _1$, it holds that

$$\begin{aligned}&\int _{\tau _1-\ell }^{\tau _2} \int _{B_r} |\nabla w_+(x,t)|^p\psi ^p(x)\eta ^2(t)\,\mathrm{d}x\mathrm{d}t+\underset{\tau _1<t<\tau _2}{{\text {ess}}\,\sup }\int _{B_r} w_+^2(x,t) \psi ^p(x)\,\mathrm{d}x\nonumber \\&\qquad +\int _{\tau _1-\ell }^{\tau _2} \int _{B_r} \int _{B_r}|w_+(x,t)\psi (x)-w_+(y,t)\psi (y)|^p \eta ^2(t)\, \mathrm{d}\mu \mathrm{d}t\nonumber \\&\quad \le C \int _{\tau _1-\ell }^{\tau _2} \int _{B_r} |\nabla \psi (x)|^p w_+^p(x,t)\eta ^2(t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\qquad + C \int _{\tau _1-\ell }^{\tau _2} \int _{B_r} \int _{B_r} (\max \{w_+(x,t), w_+(y,t)\})^p|\psi (x)-\psi (y)|^p \eta ^2(t)\, \mathrm{d}\mu \mathrm{d}t\nonumber \\&\qquad +C \underset{{\mathop {x \in \mathrm {supp}\,\psi }\limits ^{\tau _1-\ell<t<\tau _2}}}{\mathrm {ess}\,\sup } \int _{{\mathbb {R}}^N \backslash B_r} \frac{w_+^{p-1}(y,t)}{|x-y|^{N+sp}}\,\mathrm{d}y \int _{\tau _1-\ell }^{\tau _2} \int _{B_r} w_+(x,t) \psi ^p(x) \eta ^2(t)\, \mathrm{d}x\mathrm{d}t\nonumber \\&\qquad +C \int _{\tau _1-\ell }^{\tau _2} \int _{B_r} w_+^2(x,t)\psi ^p(x)\eta (t)|\partial _t \eta (t)|\,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

(3.1)

where $w_+(x,t):=\left( u(x,t)-k\right) _+$ and $C>0$ depends on $N,p,s,\Lambda$.

Proof

Taking $\varphi (x,t):=w_+(x,t)\psi ^p(x)\eta ^2(t)=\left( u(x,t)-k\right) _+\psi ^p(x)\eta ^2(t)$ as a test function in (1.7), for $s\in \left[ \tau _1,\tau _2\right]$, we can obtain

$$\begin{aligned} 0&\ge \int _{\tau _1-\ell }^{s}\int _{B_r}\partial _t u(x,t) w_+(x,t)\psi ^p(x)\eta ^2(t)\,\mathrm{d}x\mathrm{d}t\\&\quad +\int _{\tau _1-\ell }^{s}\int _{B_r}|\nabla u(x,t)|^{p-2}\nabla u(x,t)\cdot \nabla (w_+(x,t)\psi ^p(x)\eta ^2(t))\,\mathrm{d}x\mathrm{d}t\\&\quad +\frac{1}{2}\int _{\tau _1-\ell }^{s}\int _{B_r}\int _{B_r}J_p(w(x,t),w(y,t))\left( w_+(x,t)\psi ^p(x)\eta ^2(t)\right. \\&\quad \left. -w_+(y,t)\psi ^p(y)\eta ^2(t)\right) \,\mathrm{d}\mu \mathrm{d}t\\&\quad +\int _{\tau _1-\ell }^{s}\int _{{\mathbb {R}}^N\backslash B_r}\int _{B_r}J_p\left( w(x,t),w(y,t)\right) w_+(x,t)\psi ^p(x)\eta ^2(t)\,\mathrm{d}\mu \mathrm{d}t\\&=:I_1+I_2+\frac{1}{2}I_3+I_4. \end{aligned}$$

Then we are going to estimate $I_1, I_2, I_3$ and $I_4$. First, we evaluate

$$\begin{aligned} I_1&=\frac{1}{2}\int _{\tau _1-\ell }^{s}\int _{B_r}\partial _t w_+^2(x,t)\psi ^p(x)\eta ^2(t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&=\frac{1}{2}\int _{B_r}w_+^2(x,t)\psi ^p(x)\eta ^2(t)\mathrm{d}x\bigg |_{\tau _1-\ell }^s\nonumber \\&\quad -\int _{\tau _1-\ell }^{s}\int _{B_r}w_+^2(x,t)\psi ^p(x)\eta (t)\partial _t\eta (t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&=\frac{1}{2}\int _{B_r}w_+^2(x,s)\psi ^p(x)\,\mathrm{d}x-\int _{\tau _1-\ell }^{s}\int _{B_r}w_+^2(x,t)\psi ^p(x)\eta (t)\partial _t\eta (t)\,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

(3.2)

where in the last line we note that $\eta (\tau _1-\ell )=0$ and $\eta (s)=1$ when $s\ge \tau _1$. We proceed with estimating $I_2$. By Young’s inequality with $\varepsilon$, it yields that

$$\begin{aligned} I_2&=\int _{\tau _1-\ell }^{s}\int _{B_r}\psi ^p\eta ^2|\nabla u|^{p-2}\nabla u\cdot \nabla w_++p\psi ^{p-1}\eta ^2w_+|\nabla u|^{p-2}\nabla u\cdot \nabla \psi \,\mathrm{d}x\mathrm{d}t\nonumber \\&\ge \int _{\tau _1-\ell }^{s}\int _{B_r}|\nabla w_+|^{p}\psi ^p\eta ^2\,\mathrm{d}x\mathrm{d}t-\int _{\tau _1-\ell }^{s}\int _{B_r}p\psi ^{p-1}\eta ^2 w_+|\nabla u|^{p-1}|\nabla \psi |\,\mathrm{d}x\mathrm{d}t\nonumber \\&\ge \int _{\tau _1-\ell }^{s}\int _{B_r}|\nabla w_+|^{p}\psi ^p\eta ^2\,\mathrm{d}x\mathrm{d}t-\varepsilon \int _{\tau _1-\ell }^{s}\int _{B_r}\psi ^{p}\eta ^2|\nabla w_+|^p\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad -C(p,\varepsilon )\int _{\tau _1-\ell }^{s}\int _{B_r}|\nabla \psi |^p\eta ^2 w_+^p\,\mathrm{d}x\mathrm{d}t\nonumber \\&=\frac{1}{2}\int _{\tau _1-\ell }^{s}\int _{B_r}|\nabla w_+|^{p}\psi ^p\eta ^2\,\mathrm{d}x\mathrm{d}t-C(p)\int _{\tau _1-\ell }^{s}\int _{B_r}|\nabla \psi |^p\eta ^2 w_+^p\,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

(3.3)

with $\varepsilon =\frac{1}{2}$. As the same proof of Lemma 3.3 in [12], we have

$$\begin{aligned} I_3&\ge \frac{1}{2^{p+1}}\int _{\tau _1-\ell }^{s}\int _{B_r}\int _{B_r}|w_+(x,t)\psi (x)-w_+(y,t)\psi (y)|^p\eta ^2(t)\,\mathrm{d}\mu \mathrm{d}t\nonumber \\&\quad -C\int _{\tau _1-\ell }^{s}\int _{B_r}\int _{B_r}(\max \{w_+(x,t),w_+(y,t)\})^p|\psi (x)-\psi (y)|^p\eta ^2(t)\,\mathrm{d}\mu \mathrm{d}t \end{aligned}$$

(3.4)

and

$$\begin{aligned} I_4\ge -C \underset{{\mathop {x \in \mathrm {supp}\,\psi }\limits ^{\tau _1-\ell<t<s}}}{\mathrm {ess}\,\sup }\int _{{\mathbb {R}}^N \backslash B_r}\frac{w_+^{p-1}(y,t)}{|x-y|^{N+sp}}\,\mathrm{d}y\int _{\tau _1-\ell }^{s}\int _{B_r}w_+(x,t)\psi ^p(x)\eta ^2(t)\,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

(3.5)

Merging the estimates on (3.2)–(3.5), we get

$$\begin{aligned}&\int _{\tau _1-\ell }^{s} \int _{B_r} |\nabla w_+(x,t)|^p\psi ^p(x)\eta ^2(t)\,\mathrm{d}x\mathrm{d}t+\int _{B_r} w_+^2(x,s) \psi ^p(x)\,\mathrm{d}x\nonumber \\&\qquad +\int _{\tau _1-\ell }^{s} \int _{B_r} \int _{B_r}|w_+(x,t)\psi (x)-w_+(y,t)\psi (y)|^p \eta ^2(t)\, \mathrm{d}\mu \mathrm{d}t\nonumber \\&\quad \le C \int _{\tau _1-\ell }^{\tau _2} \int _{B_r} |\nabla \psi (x)|^p w_+^p(x,t)\eta ^2(t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\qquad + C \int _{\tau _1-\ell }^{\tau _2} \int _{B_r} \int _{B_r} \max \left\{ w_+(x,t), w_+(y,t)\right\} ^p|\psi (x)-\psi (y)|^p \eta ^2(t)\, \mathrm{d}\mu \mathrm{d}t\nonumber \\&\qquad +C \underset{{\mathop {x \in \mathrm {supp}\, \psi }\limits ^{\tau _1-\ell<t<\tau _2}}}{\mathrm {ess}\,\sup } \int _{{\mathbb {R}}^N \backslash B_r} \frac{w_+^{p-1}(y,t)}{|x-y|^{N+sp}}\,\mathrm{d}y \int _{\tau _1-\ell }^{\tau _2} \int _{B_r} w_+(x,t) \psi ^p(x) \eta ^2(t)\, \mathrm{d}x\mathrm{d}t\nonumber \\&\qquad +C \int _{\tau _1-\ell }^{\tau _2} \int _{B_r} w_+^2(x,t)\psi ^p(x)\eta (t)|\partial _t \eta (t)|\,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

which leads to the desired result. $\square$

From the forthcoming lemma, one can interpret the reason why we only derive the Hölder continuity of weak solutions in the case $p>2$. For the subquadratic case, we at present cannot show a similar logarithmic estimate.

Lemma 3.2

(Logarithmic estimates) Let $p>2$ and u be a local weak solution to (1.1). Let $B_r\equiv B_r(x_0)$ and $(x_0,t_0)\in Q_{T}, T_0>0$, $0<r\le R/2$. We also denote ${\tilde{Q}} \equiv B_R(x_0) \times (t_0-2T_0, t_0+2T_0)$ such that ${\overline{B}}_{R}(x_0) \subseteq \Omega$ and $\left[ t_0-2T_0,t_0+2T_0\right] \subseteq (0,T)$. If $u\in L^\infty ({\tilde{Q}})$ and $u\ge 0$ in ${\tilde{Q}}$, then for any $d>0$, it holds that

$$\begin{aligned}&\int _{t_0-T_0}^{t_0+T_0} \int _{B_r}|\nabla \log \left( u(x,t)+d\right) |^p\,\mathrm{d}x\mathrm{d}t\nonumber \\&\qquad +\int _{t_0-T_0}^{t_0+T_0} \int _{B_r}\int _{B_r} \left| \log \left( \frac{u(x,t)+d}{u(y,t)+d}\right) \right| ^p\,\mathrm{d}\mu \mathrm{d}t\nonumber \\&\quad \le C T_0 r^Nd^{1-p}R^{-p}\left[ \mathrm {Tail}_\infty (u;x_0,R,t_0-2 T_0,t_0+2 T_0)\right] ^{p-1}\nonumber \\&\qquad +Cr^Nd^{2-p}+CT_0r^{N-sp}+CT_0r^{N-p}, \end{aligned}$$

(3.6)

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

Proof

Let $\psi \in C_0^\infty (B_{3r/2})$ and $\eta \in C_0^\infty (t_0-2T_0,t_0+2T_0)$ fulfil

$$\begin{aligned} 0\le \psi \le 1,\quad |\nabla \psi |<Cr^{-1} \text { in } B_{2r},\quad \psi \equiv 1 \text { in } B_r, \end{aligned}$$

and

$$\begin{aligned} 0\le \eta \le 1,\quad |\partial _{t}\eta |<CT_0^{-1} \text { in } (t_0-2T_0,t_0+2T_0),\quad \eta \equiv 1\text { in } (t_0-T_0,t_0+T_0). \end{aligned}$$

Choosing $\varphi (x,t):=(u(x,t)+d)^{1-p} \psi ^p(x) \eta ^2(t)$ to test the weak formulation of (1.1), we get

$$\begin{aligned} 0&=-\int _{t_0-2T_0}^{t_0+2T_0} \int _{B_{2r}} \partial _{t}\left( (u(x,t)+d)^{1-p} \psi ^p(x) \eta ^2(t)\right) u(x,t)\,\mathrm{d}x\mathrm{d}t\\&\quad +\int _{t_0-2T_0}^{t_0+2T_0} \int _{B_{2r}}|\nabla u|^{p-2}\nabla u\cdot \nabla \left( (u(x,t)+d)^{1-p} \psi ^p(x) \eta ^2(t)\right) \,\mathrm{d}x\mathrm{d}t\\&\quad +\frac{1}{2}\int _{t_0-2T_0}^{t_0+2T_0} \int _{B_{2r}}\int _{B_{2r}} J_p(u(x,t),u(y,t))\left[ \frac{\psi ^p(x)}{(u(x,t)+d)^{p-1}}\right. \\&\quad \left. -\frac{\psi ^p(y)}{(u(y, t)+d)^{p-1}}\right] \eta ^2(t)\, \mathrm{d}\mu \mathrm{d}t\\&\quad +\int _{t_0-2T_0}^{t_0+2T_0}\int _{{\mathbb {R}}^{N} \backslash B_{2r}}\int _{B_{2r}}J_p(u(x,t),u(y,t))\frac{\psi ^p(x)}{(u(x,t)+d)^{p-1}}\eta ^2(t)\, \mathrm{d}\mu \mathrm{d}t\\&=:I_1+I_2+\frac{1}{2}I_3+I_4. \end{aligned}$$

It follows from the proof of [12, Lemma 3.5] that

$$\begin{aligned} I_1&=\int _{t_0-2T_0}^{t_0+2T_0} \int _{B_{2r}} \left( (u(x,t)+d)^{1-p} \psi ^p(x) \eta ^2(t)\right) \partial _{t}u(x,t)\,\mathrm{d}x\mathrm{d}t\le Cr^N d^{2-p}, \end{aligned}$$

(3.7)

$$\begin{aligned} I_3&\le -C \int _{t_0-T_0}^{t_0+T_0} \int _{B_r} \int _{B_r}\left| \log \left( \frac{u(x,t)+d}{u(y,t)+d}\right) \right| ^p\,\mathrm{d}x\mathrm{d}y\mathrm{d}t+C T_0 r^{N-sp}, \end{aligned}$$

(3.8)

and

$$\begin{aligned} I_4 \le C T_0 r^{N-sp}+C T_0 r^N R^{-p}d^{1-p}\left[ \mathrm {Tail}_\infty \left( u;x_0,R,t_0-2T_0,t_0+2T_0\right) \right] ^{p-1}. \end{aligned}$$

(3.9)

By Young’s inequality with $\varepsilon$, we estimate the integral $I_2$ as

$$\begin{aligned} I_2&=-(p-1)\int _{t_0-2T_0}^{t_0+2T_0} \int _{B_{2r}}(u(x,t)+d)^{-p}\psi ^p(x)|\nabla u(x,t)|^p\eta ^2(t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad +p\int _{t_0-2T_0}^{t_0+2T_0} \int _{B_{2r}}(u(x,t)+d)^{1-p}\psi ^{p-1}(x)|\nabla u(x,t)|^{p-2}\nabla u(x,t)\nonumber \\&\quad \cdot \nabla \psi (x)\eta ^2(t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\le -(p-1)\int _{t_0-2T_0}^{t_0+2T_0} \int _{B_{2r}}(u(x,t)+d)^{-p}\psi ^p(x)|\nabla (u(x,t)+d)|^p\eta ^2(t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad +\varepsilon \int _{t_0-2T_0}^{t_0+2T_0} \int _{B_{2r}}(u(x,t)+d)^{-p}\psi ^p(x)|\nabla (u(x,t)+d)|^p\eta ^2(t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad +C(\varepsilon )\int _{t_0-2T_0}^{t_0+2T_0} \int _{B_{2r}}|\nabla \psi (x)|^p\eta ^2(t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\le -C(p)\int _{t_0-2T_0}^{t_0+2T_0} \int _{B_r}(u(x,t)+d)^{-p}|\nabla (u(x,t)+d)|^p\eta ^2(t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad +C(N,p)T_0 r^{N-p}\nonumber \\&\le -C(p)\int _{t_0-T_0}^{t_0+T_0} \int _{B_r}|\nabla \log (u(x,t)+d)|^p\,\mathrm{d}x\mathrm{d}t+C(N,p)T_0 r^{N-p} \end{aligned}$$

(3.10)

with $\varepsilon$ satisfying that $\varepsilon <p-1$. Combining with (3.7)–(3.10), we can obtain the Logarithmic estimates. $\square$

Next, we will give a corollary of Lemma 3.2, which plays a crucial role in obtaining the Hölder continuity.

Corollary 3.3

Let $p>2$ and u be a local weak solution to (1.1). Let $B_r\equiv B_r(x_0)$ and $(x_0,t_0)\in Q_{T}, T_0>0$, $0<r\le R/2$. We also denote ${\tilde{Q}} \equiv B_R(x_0) \times (t_0-2T_0, t_0+2T_0)$ such that ${\overline{B}}_{R}(x_0) \subseteq \Omega$ and $\left[ t_0-2T_0,t_0+2T_0\right] \subseteq (0,T)$. Suppose that $u\in L^\infty ({\tilde{Q}})$ and $u\ge 0$ in ${\tilde{Q}}$. Let $a,d>0$, $b>1$ and define

$$\begin{aligned} v:=\min \big \{(\log (a+d)-\log (u+d))_+, \log b\big \}. \end{aligned}$$

Then it holds that

$$\begin{aligned}&\int _{t_0-T_0}^{t_0+T_0}{\int \!\!\!\!-}_{B_r}\left| v(x,t)-(v)_{B_r}(t)\right| ^p\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad \le C T_0 d^{1-p}\left( \frac{r}{R}\right) ^p\left[ \mathrm {Tail}_\infty \left( u;x_0, R,t_0-2 T_0,t_0+2 T_0\right) \right] ^{p-1}\nonumber \\&\qquad +CT_0+Cd^{2-p}r^p+CT_0r^{p-sp}, \end{aligned}$$

(3.11)

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

Proof

By the Poincaré inequality (see for example [17, p. 276]), it yields that

$$\begin{aligned} {\int \!\!\!\!-}_{B_r}\left| v(x,t)-(v)_{B_r}(t)\right| ^p\,\mathrm{d}x\le Cr^{p-N}\int _{B_r}|\nabla v(x,t)|^p\,\mathrm{d}x \end{aligned}$$

for any $t\in \left( t_0-T_0,t_0+T_0\right)$. Integrating the above inequality over $\left( t_0-T_0,t_0+T_0\right)$ leads to

$$\begin{aligned} \int _{t_0-T_0}^{t_0+T_0}{\int \!\!\!\!-}_{B_r}\left| v(x,t)-(v)_{B_r}(t)\right| ^p\,\mathrm{d}x\mathrm{d}t\le Cr^{p-N}\int _{t_0-T_0}^{t_0+T_0}\int _{B_r}|\nabla v(x,t)|^p\,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

(3.12)

where $C>0$ depends only on N, p. Observing that v is a truncation function of the sum of a constant and $\log (u+d)$, which gives that

$$\begin{aligned} \int _{t_0-T_0}^{t_0+T_0}\int _{B_r}|\nabla v(x,t)|^p\,\mathrm{d}x\mathrm{d}t\le \int _{t_0-T_0}^{t_0+T_0}\int _{B_r}|\nabla \log (u(x,t)+d)|^p\,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

(3.13)

We can get the results from Lemma 3.2 along with (3.12) and (3.13). $\square$

4 Local Boundedness

In this part, we are ready to study the local boundedness of weak solutions. To this end, we first introduce some notations. For $\sigma \in [1/2,1)$, set

$$\begin{aligned} r_0:=r, \quad r_j:=\sigma r+2^{-j}(1-\sigma ) r, \quad {\tilde{r}}_j:=\frac{r_j+r_{j+1}}{2}, \quad j=0,1,2, \ldots \end{aligned}$$

and

$$\begin{aligned} Q_j^-&{:}{=}B_j \times \Gamma _j:=B_{r_j}(x_0) \times \left( t_0-r_j^p,t_0\right) , \quad j=0,1,2, \ldots , \\ {\tilde{Q}}_j^-&{:}{=}{\tilde{B}}_j\times {\tilde{\Gamma }}_j:=B_{{\tilde{r}}_j}(x_{0}) \times \left( t_0-{\tilde{r}}_j^p,t_0\right) , \quad j=0,1,2, \ldots . \end{aligned}$$

Denote

$$\begin{aligned} k_j:=\left( 1-2^{-j}\right) {\tilde{k}}, \quad {\tilde{k}}_j:=\frac{k_{j+1}+k_j}{2}, \quad j=0,1,2, \ldots \end{aligned}$$

with

$$\begin{aligned} {\tilde{k}} \ge \frac{\mathrm {Tail}_\infty \left( u_+;x_0,\sigma r,t_0-r^p,t_0\right) }{2}. \end{aligned}$$

Let

$$\begin{aligned} w_j:=(u-k_j)_+, \quad {\tilde{w}}_j:=\left( u-{\tilde{k}}_j\right) _+, \quad j=0,1,2, \ldots . \end{aligned}$$

We now provide a Caccioppoli-type inequality in a special cylinder, which leads to the recursive inequalities (see Lemmas 4.2 and 4.3).

Lemma 4.1

Let $p>1$ and u be a local weak subsolution to (1.1). Let $(x_0, t_0) \in Q_T$, $0<r<1$ and $Q_r^-=B_r(x_0) \times (t_0-r^p,t_0)$ such that ${\overline{B}}_{r}(x_0) \subseteq \Omega$ and $\left[ t_0-r^p,t_0\right] \subseteq (0,T)$. Suppose q is a parameter satisfying $q\ge \max \{p,2\}$, it holds that

$$\begin{aligned}&\int _{\Gamma _{j+1}}{\int \!\!\!\!-}_{B_{j+1}}|\nabla {\tilde{w}}_j(x,t)|^p\,\mathrm{d}x\mathrm{d}t+\underset{t\in \Gamma _{j+1}}{{\text {ess}}\,\sup }{\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^2(x,t)\, \mathrm{d}x\nonumber \\&\qquad +\int _{\Gamma _{j+1}} \int _{B_{j+1}} {\int \!\!\!\!-}_{B_{j+1}} \frac{|{\tilde{w}}_j(x,t)-{\tilde{w}}_j(y,t)|^p}{|x-y|^{N+sp}}\,\mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad \le \frac{C}{r^p}\left( \frac{1}{\sigma ^p(1-\sigma )^{N+s p}}+\frac{1}{(1-\sigma )^p}\right) \left( \frac{2^{(p+q-2) j}}{{\tilde{k}}^{q-2}}+\frac{2^{(N+sp+q-1)j}}{{\tilde{k}}^{q-p}}\right) \nonumber \\&\qquad \int _{\Gamma _j} {\int \!\!\!\!-}_{B_j} w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

(4.1)

where $C>0$ only depends on $N,p,s,\Lambda$ and q.

Proof

First we give a trivial but very useful inequality

$$\begin{aligned} {\tilde{w}}_j^{\tau }(x,t) \le \frac{C 2^{(q-\tau )j}}{{\tilde{k}}^{q-\tau }} w_j^q(x,t) \quad \text{ in } Q_T, \end{aligned}$$

(4.2)

where $0\le \tau < q$. We take the cut-off functions $\psi _j \in C_0^\infty ({\tilde{B}}_j)$ and $\eta _j \in C_0^\infty ({\tilde{\Gamma }}_j)$ such that

$$\begin{aligned} 0 \le \psi _j \le 1, \quad \left| \nabla \psi _j\right| \le \frac{C 2^j}{(1-\sigma ) r} \text{ in } {\tilde{B}}_j, \quad \psi _j \equiv 1 \text{ in } B_{j+1} \end{aligned}$$

and

$$\begin{aligned} 0 \le \eta _j \le 1, \quad \left| \partial _t \eta _j\right| \le \frac{C 2^{pj}}{(1-\sigma )^pr^p} \text{ in } {\tilde{\Gamma }}_j, \quad \eta _j \equiv 1 \text{ in } \Gamma _{j+1}. \end{aligned}$$

Let $r=r_j$, $\tau _2=t_0$, $\tau _1=t_0-r_{j+1}^p$ and $\ell ={\tilde{r}}_j^p-r_{j+1}^p$ in Lemma 3.1. Then we arrive at

$$\begin{aligned}&\int _{{\tilde{\Gamma }}_j} \int _{B_j}|\nabla {\tilde{w}}_j(x,t)|^p\psi _j^p(x)\eta _j^2(t)\,\mathrm{d}x\mathrm{d}t+\underset{t\in \Gamma _{j+1}}{{\text {ess}}\,\sup }\int _{B_j} {\tilde{w}}_j^2(x,t)\psi _j^p(x)\, \mathrm{d}x\nonumber \\&\qquad +\int _{{\tilde{\Gamma }}_j} \int _{B_j} \int _{B_j} \left| {\tilde{w}}_j(x,t) \psi _j(x)-{\tilde{w}}_j(y,t) \psi _j(y)\right| ^p\eta _j^2(t)\, \mathrm{d}\mu \mathrm{d}t\nonumber \\&\quad \le C\int _{{\tilde{\Gamma }}_j} \int _{B_j}|\nabla \psi _j(x)|^p {\tilde{w}}_j^p(x,t)\eta _j^2(t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\qquad +C\int _{{\tilde{\Gamma }}_j} \int _{B_j} \int _{B_j}(\max \{{\tilde{w}}_j(x,t), {\tilde{w}}_j(y,t)\})^p\left| \psi _j(x)-\psi _j(y)\right| ^p \eta _j^2(t)\,\mathrm{d}\mu \mathrm{d}t\nonumber \\&\qquad +C\underset{{\mathop {x \in \mathrm {supp}\, \psi _j}\limits ^{t\in {\tilde{\Gamma }}_j}}}{\mathrm {ess}\,\sup }\int _{{\mathbb {R}}^N\backslash B_j} \frac{{\tilde{w}}_j^{p-1}(y,t)}{|x-y|^{N+s p}}\,\mathrm{d}y\int _{{\tilde{\Gamma }}_j} \int _{B_j}{\tilde{w}}_j(x,t)\psi _j^p(x) \eta _j^2(t)\,\mathrm{d} x\mathrm{d}t\nonumber \\&\qquad +C\int _{{\tilde{\Gamma }}_j} \int _{B_j} {\tilde{w}}_j^2(x,t)\psi _j^p(x) \eta _j(t)\left| \partial _t \eta _j(t)\right| \,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad =: I_1+I_2+I_3+I_4. \end{aligned}$$

(4.3)

Using (4.2) and the definition of $\psi _j$, we estimate $I_1$ as

$$\begin{aligned} I_1&\le \frac{C 2^{pj}}{(1-\sigma )^{p} r^p} \int _{{\Gamma }_j} \int _{B_j} {\tilde{w}}_j^p(x,t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\le \frac{C 2^{qj}}{{\tilde{k}}^{q-p}(1-\sigma )^{p} r^p} \int _{{\Gamma }_j} \int _{B_j} w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

(4.4)

Analogous to the proof of Lemma 4.1 in [12], we have

$$\begin{aligned} I_2&\le \frac{C 2^{qj}}{{\tilde{k}}^{q-p}(1-\sigma )^p r^{sp}} \int _{\Gamma _j} \int _{B_j} w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

(4.5)

$$\begin{aligned} I_3&\le \frac{C 2^{(N+sp+q-1)j}}{{\tilde{k}}^{q-p} \sigma ^p(1-\sigma )^{N+sp}r^p} \int _{\Gamma _j} \int _{B_j} w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t \end{aligned}$$

(4.6)

and

$$\begin{aligned} I_4\le \frac{C 2^{(p+q-2)j}}{{\tilde{k}}^{q-2}(1-\sigma )^p r^p } \int _{\Gamma _j} \int _{B_j} w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

(4.7)

By virtue of the fact that $\psi _j\equiv 1$ in $B_{j+1}$, $\eta _j \equiv 1$ in $\Gamma _{j+1}$ and (1.3), merging inequalities (4.3)–(4.7), we get

$$\begin{aligned}&\int _{\Gamma _{j+1}}{\int \!\!\!\!-}_{B_{j+1}}|\nabla {\tilde{w}}_j(x,t)|^p\,\mathrm{d}x\mathrm{d}t+\underset{t\in \Gamma _{j+1}}{{\text {ess}}\,\sup }{\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^2(x,t)\, \mathrm{d}x\\&\qquad +\int _{\Gamma _{j+1}} \int _{B_{j+1}} {\int \!\!\!\!-}_{B_{j+1}} \frac{|{\tilde{w}}_j(x,t)-{\tilde{w}}_j(y,t)|^p}{|x-y|^{N+sp}}\,\mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad \le \frac{C}{r^p(1-\sigma )^p}\left( \frac{2^{qj}}{r^{(s-1)p}{\tilde{k}}^{q-p}}+\frac{2^{(N+sp+q-1)j}}{\sigma ^p(1-\sigma )^{N+p(s- 1)}{\tilde{k}}^{q-p}}+\frac{2^{qj}}{{\tilde{k}}^{q-p}}+\frac{2^{(p+q-2)j}}{{\tilde{k}}^{q-2}}\right) \nonumber \\&\qquad \times \int _{\Gamma _j} {\int \!\!\!\!-}_{B_j} w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

After rearrangement, we get the desired result. $\square$

The following two lemmas are the consequences of Lemmas 2.3 and 4.1 .

Lemma 4.2

Let $p>2N/(N+2)$ and $\max \{p,2\}\le q<p(N+2)/N$. Let $(x_0,t_0) \in Q_T$, $0<r<1$ and $Q_r^-=B_r(x_0) \times (t_0-r^p,t_0)$ such that ${\overline{B}}_{r}(x_0) \subseteq \Omega$ and $\left[ t_0-r^p,t_0\right] \subseteq (0,T)$. Then for a local subsolution u to (1.1), we infer that

$$\begin{aligned}&\int _{\Gamma _{j+1}}{\int \!\!\!\!-}_{B_{j+1}}w_{j+1}^q(x,t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad \le \frac{C 2^{bj}}{r^{\frac{pq}{\kappa N}}}\left( \frac{1}{\sigma ^{\frac{q(N+p)}{\kappa N}}(1-\sigma )^{\frac{q(N+p)(N+sp)}{p \kappa N}}}+\frac{1}{(1-\sigma )^{\frac{q(N+p)}{\kappa N}}}\right) \nonumber \\&\qquad \times \left( \frac{1}{{\tilde{k}}^{\frac{q}{\kappa }(\frac{q}{N}+1-\frac{2}{p})}}+\frac{1}{{\tilde{k}}^{\frac{q}{\kappa }(\frac{q}{N}+\frac{2}{N}-\frac{p}{N})}}\right) \left( \int _{\Gamma _j} {\int \!\!\!\!-}_{B_j}w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t\right) ^{1+\frac{q}{\kappa N}} \end{aligned}$$

(4.8)

for $j\in {\mathbb {N}}$, where $b:=(1+p/N)(N+p+q)$, $\kappa :=1+2/N$ and $C>0$ only depends on $N,p,s,\Lambda$ and q.

Proof

Since $q<p\kappa$, it follows from Hölder inequality that

$$\begin{aligned}&\int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} w_{j+1}^q(x,t)\,\mathrm{d}x\mathrm{d}t \nonumber \\&\quad \le \int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^q(x,t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad \le \left( \int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^{p \kappa }(x,t)\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{q}{p\kappa }}\left( \int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} \chi _{\{u \ge {\tilde{k}}_j\}}(x,t)\,\mathrm{d}x\mathrm{d}t\right) ^{1-\frac{q}{p \kappa }}. \end{aligned}$$

(4.9)

From (4.2), we can get

$$\begin{aligned} \int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} \chi _{\{u \ge {\tilde{k}}_j\}}(x,t)\, \mathrm{d}x\mathrm{d}t \le \frac{C 2^{qj}}{{\tilde{k}}^q} \int _{\Gamma _j} {\int \!\!\!\!-}_{B_j} w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t \end{aligned}$$

(4.10)

and

$$\begin{aligned} \int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^p(x,t)\,\mathrm{d}x\mathrm{d}t \le \frac{C 2^{(q-p)j}}{{\tilde{k}}^{q-p}} \int _{\Gamma _j} {\int \!\!\!\!-}_{B_j} w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

(4.11)

Now by (4.11), Lemmas 2.3 and 4.1 we estimate

$$\begin{aligned}&\int _{\Gamma _{j+1}}{\int \!\!\!\!-}_{B_{j+1}}{\tilde{w}}_j^{p\kappa }(x,t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad \le C \left( r^p \int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} |\nabla {\tilde{w}}_j(x,t)|^p\,\mathrm{d}x\mathrm{d}t +\int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^p(x,t)\,\mathrm{d}x\mathrm{d}t\right) \nonumber \\&\qquad \times \left( \underset{t\in \Gamma _{j+1}}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^2(x,t)\,\mathrm{d}x\right) ^{\frac{p}{N}}\nonumber \\&\quad \le Cr^{-\frac{p^2}{N}}\left( \frac{1}{\sigma ^p(1-\sigma )^{N+s p}}+\frac{1}{(1-\sigma )^p}\right) ^\frac{p}{N} \left( \frac{2^{(p+q-2) j}}{{\tilde{k}}^{q-2}}+\frac{2^{(N+sp+q-1)j}}{{\tilde{k}}^{q-p}}\right) ^\frac{p}{N} \nonumber \\&\qquad \times \left[ \left( \frac{1}{\sigma ^p(1-\sigma )^{N+s p}}+\frac{1}{(1-\sigma )^p}\right) \left( \frac{2^{(p+q-2) j}}{{\tilde{k}}^{q-2}}+\frac{2^{(N+sp+q-1)j}}{{\tilde{k}}^{q-p}}\right) +\frac{2^{(q-p)j}}{{\tilde{k}}^{q-p}}\right] \nonumber \\&\qquad \times \left( \int _{\Gamma _j} {\int \!\!\!\!-}_{B_j} w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t\right) ^{1+\frac{p}{N}} \nonumber \\&\quad \le \frac{C 2^{bj}}{r^{\frac{p^2}{N}}}\left( \frac{1}{\sigma ^{\frac{p(N+p)}{N}}(1-\sigma )^{\frac{(N+p)(N+sp)}{N}}}+\frac{1}{(1-\sigma )^{\frac{p(N+p)}{N}}}\right) \nonumber \\&\qquad \times \left( \frac{1}{{\tilde{k}}^{q-2}}+\frac{1}{{\tilde{k}}^{q-p}}\right) ^{1+\frac{p}{N}}\left( \int _{\Gamma _j} {\int \!\!\!\!-}_{B_j} w_j^q(x,t)\,\mathrm{d}x\mathrm{d}t\right) ^{1+\frac{p}{N}} \end{aligned}$$

(4.12)

with $b=(1+p/N)(N+p+q)$. Combining (4.9), (4.10) and (4.12), we get the desired result. $\square$

Lemma 4.3

Let $1<p\le 2N/(N+2)$ and $m>\max \left\{ 2,\frac{N(2-p)}{p}\right\}$. Let $(x_0,t_0) \in Q_T$, $0<r<1$ and $Q_r^-=B_r(x_0) \times (t_0-r^p,t_0)$ such that ${\overline{B}}_{r}(x_0) \subseteq \Omega$ and $\left[ t_0-r^p,t_0\right] \subseteq (0,T)$. Suppose that $u\in L_{\mathrm{loc}}^\infty (Q_T)$ is a local weak subsolution to (1.1). Then for any $j\in {\mathbb {N}}$, we have

$$\begin{aligned}&\int _{\Gamma _{j+1}}{\int \!\!\!\!-}_{B_{j+1}}w_{j+1}^m(x,t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad \le \frac{C 2^{bj}}{r^{\frac{p^2}{N}}}\left( \frac{1}{\sigma ^\frac{p(N+p)}{N}(1-\sigma )^{\frac{(N+p)(N+sp)}{N}}}+\frac{1}{(1-\sigma )^{\frac{p(N+p)}{N}}}\right) \left( \frac{1}{{\tilde{k}}^{m-p}}+\frac{1}{{\tilde{k}}^{m-2}}\right) ^{1+\frac{p}{N}}\nonumber \\&\qquad \times \left\| {\tilde{w}}_j\right\| _{L^\infty \left( Q_{j+1}^-\right) }^{m-p\kappa }\left( \int _{\Gamma _j} {\int \!\!\!\!-}_{B_j}w_j^m(x,t)\,\mathrm{d}x\mathrm{d}t\right) ^{1+\frac{p}{N}}, \end{aligned}$$

(4.13)

where $b:=(1+p/N)(N+p+m)$, $\kappa :=1+2/N$ and $C>0$ only depends on N, p, s, m and $\Lambda$.

Proof

Based on the assumptions, we have

$$\begin{aligned} \int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} w_{j+1}^m(x,t)\,\mathrm{d}x\mathrm{d}t&\le \int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^m(x,t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\le \left\| {\tilde{w}}_j\right\| _{L^\infty \left( Q_{j+1}^-\right) }^{m-p\kappa } \int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^{p\kappa }(x,t)\,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

(4.14)

By utilizing Lemma 2.3, Lemma 4.1 with $q=m$ and the inequality (4.11) with $q=m$, it yields that

$$\begin{aligned}&\int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^{p\kappa }(x,t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad \le C \left( r^p \int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} |\nabla {\tilde{w}}_j(x,t)|^p\,\mathrm{d}x\mathrm{d}t +\int _{\Gamma _{j+1}} {\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^p(x,t)\,\mathrm{d}x\mathrm{d}t\right) \nonumber \\&\qquad \times \left( \underset{t\in \Gamma _{j+1}}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B_{j+1}} {\tilde{w}}_j^2(x,t)\,\mathrm{d}x\right) ^{\frac{p}{N}}\nonumber \\&\quad \le \frac{C 2^{bj}}{r^{\frac{p^2}{N}}}\left[ \frac{1}{\sigma ^{\frac{p(N+p)}{N}}(1-\sigma )^{\frac{(N+p)(N+sp)}{N}}}+\frac{1}{(1-\sigma )^{\frac{p(N+p)}{N}}}\right] \left( \frac{1}{{\tilde{k}}^{m-p}}+\frac{1}{{\tilde{k}}^{m-2}}\right) ^{1+\frac{p}{N}}\nonumber \\&\qquad \times \left( \int _{\Gamma _j} {\int \!\!\!\!-}_{B_j}w_j^m(x,t)\,\mathrm{d}x\mathrm{d}t\right) ^{1+\frac{p}{N}} \end{aligned}$$

(4.15)

with $b:=(1+p/N)(N+p+m)$. Thus combining (4.14) and (4.15), we get the desired inequality (4.13). $\square$

Remark 4.4

In Lemma 4.3, the quantity $\sigma ^{\frac{p(N+p)}{N}}$ can be removed, since $\sigma \in [1/2,1)$. In addition,

$$\begin{aligned} \max \left\{ (1-\sigma )^{\frac{(N+p)(N+sp)}{N}},(1-\sigma )^{\frac{p(N+p)}{N}}\right\} \ge (1-\sigma )^{\frac{(N+p)^2}{N}}. \end{aligned}$$

Thus, we can get

$$\begin{aligned}&\int _{\Gamma _{j+1}}{\int \!\!\!\!-}_{B_{j+1}}w_{j+1}^m(x,t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad \le \frac{C 2^{bj}}{r^{\frac{p^2}{N}}(1-\sigma )^{\frac{(N+p)^2}{N}}}\left( \frac{1}{{\tilde{k}}^{m-p}}+\frac{1}{{\tilde{k}}^{m-2}}\right) ^{1+\frac{p}{N}}\nonumber \\&\qquad \times \left\| {\tilde{w}}_j\right\| _{L^\infty \left( Q_{j+1}^-\right) }^{m-p\kappa }\left( \int _{\Gamma _j} {\int \!\!\!\!-}_{B_j}w_j^m(x,t)\,\mathrm{d}x\mathrm{d}t\right) ^{1+\frac{p}{N}}. \end{aligned}$$

Next, we introduce an analysis lemma which will be used later.

Lemma 4.5

[9, Lemma 4.1] Let $\{Y_j\}_{j=0}^\infty$ be a sequence of positive numbers such that

$$\begin{aligned} Y_0\le K^{-\frac{1}{\delta }}b^{-\frac{1}{\delta ^2}}\ \text { and }\ Y_{j+1}\le Kb^jY_j^{1+\delta },\quad j=0,1,2, \ldots \end{aligned}$$

for some constants K, $b>1$ and $\delta >0$. Then we have $\lim _{j\rightarrow \infty }Y_j=0$.

Finally, we end this section by proving the results of local boundedness.

Proof of Theorem 1.2

Let $r=R$, $\sigma =\frac{1}{2}$, then $r_j=\frac{R}{2}+2^{-j-1}R$. Fix $q=\max \{2,p\}$. We denote

$$\begin{aligned} Y_j=\int _{\Gamma _j} {\int \!\!\!\!-}_{B_j}\left( u-k_j\right) _+^q\,\mathrm{d}x\mathrm{d}t, \quad j=0,1,2, \ldots . \end{aligned}$$

Supposing ${\tilde{k}}\ge 1$ and recalling $r<1$, we derive from Lemma 4.2 that

$$\begin{aligned} \frac{Y_{j+1}}{r^p}&\le \frac{C 2^{bj} Y_j^{1+\frac{q}{N\kappa }}}{r^{p\left( 1+\frac{q}{N \kappa }\right) }{\tilde{k}}^{\frac{q}{\kappa }\left( \frac{q}{N}+\frac{2}{N}-\frac{p}{N}\right) }}+\frac{C 2^{bj} Y_j^{1+\frac{q}{N\kappa }}}{r^{p\left( 1+\frac{q}{N \kappa }\right) }{\tilde{k}}^{\frac{q}{\kappa }\left( \frac{q}{N}+1-\frac{2}{p}\right) }}\nonumber \\&\le \frac{C 2^{bj}}{{\tilde{k}}^{q\left( 1-\frac{q}{p\kappa }\right) }}\left( \frac{Y_j}{r^p}\right) ^{1+\frac{q}{N\kappa }}, \end{aligned}$$

(4.16)

where $b:=(1+p/N)(N+p+q)$, $\kappa :=1+2/N$ and $C>0$ only depends on $N,p,s,\Lambda$. For any $j\in {\mathbb {N}}$, define $W_j=Y_j/r^p$. Thus we get

$$\begin{aligned} W_{j+1}\le \frac{C 2^{bj}}{{\tilde{k}}^{q\left( 1-\frac{q}{p\kappa }\right) }}W_j^{1+\frac{q}{N\kappa }}, \end{aligned}$$

where ${\tilde{k}}$ is such that

$$\begin{aligned} {\tilde{k}} \ge \max \left\{ \mathrm {Tail}_\infty \left( u_+;x_{0},R/2,t_0-R^p, t_0\right) , C\left( {\int \!\!\!\!-}^{t_0}_{t_0-R^p} {\int \!\!\!\!-}_{B_R} u_+^q\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{p}{N(p\kappa -q)}} \vee 1\right\} \end{aligned}$$

with C only depending on N, p, s and $\Lambda$. Thus along with Lemma 4.5 we can deduce that $\lim _{j\rightarrow \infty }W_j=0$. Consequently, we get the desired result

$$\begin{aligned} \underset{Q_{R/2}^-}{{\text {ess}}\,\sup }\, u \le \mathrm {Tail}_\infty \left( u_+;x_0,R/2,t_0-R^p, t_0\right) +C\left( {\int \!\!\!\!-}_{Q_R^-} u_+^q\,\mathrm{d}x \mathrm{d}t\right) ^{\frac{p}{N(p\kappa -q)}} \vee 1. \end{aligned}$$

We now finish the proof. $\square$

Proof of Theorem 1.3

Under the hypotheses (1.8) and (1.9), we may suppose, by a proper approximation procedure that u is locally bounded in advance. Indeed, observe that the approximation subsolutions $u_k$ are bounded (i.e. ${\text {ess}} \sup _{Q_n^-} (u_k)_+<\infty$). Therefore, we could substitute u by $u_k$ to perform the proof below. In other words, the estimate (4.20) below still holds true for u replaced by $u_k$, which along with (1.8) and (1.9) leads to a k-independent bound on $u_k$ in $L^\infty$. We eventually find that u is qualitatively locally bounded via the a.e. convergence of $u_k$. That is, ${\text {ess}} \sup _{Q_n^-} u_+$ is finite for $n=0,1,2,3,\ldots$.

Let $R_0=R/2$ and $R_n=R/2+\sum _{i=1}^n 2^{-i-1}R$ for $n\in {\mathbb {N}}^+$, and $Q_n^-=B_{R_n}(x_0) \times \left( t_0-R_n^p, t_0\right)$. Set

$$\begin{aligned} M_n={\text {ess}} \sup _{Q_n^-} u_+, \quad n=0,1,2,3, \ldots . \end{aligned}$$

Choosing $r=R_{n+1}$ and $\sigma r=R_{n}$, then

$$\begin{aligned} \sigma =\frac{1 / 2+\sum _{i=1}^{n} 2^{-i-1}}{1 / 2+\sum _{i=1}^{n+1} 2^{-i-1}} \ge \frac{1}{2}. \end{aligned}$$

We denote

$$\begin{aligned} Y_j=\int _{\Gamma _j} {\int \!\!\!\!-}_{B_j}\left( u-k_j\right) _+^m\,\mathrm{d}x\mathrm{d}t, \quad j=0,1,2, \ldots . \end{aligned}$$

Due to Lemma 4.3, we obtain

$$\begin{aligned} Y_{j+1}&\le \frac{C 2^{bj}}{R_{n+1}^{\frac{p^2}{N}}}\left\| u_+\right\| _{L^\infty \left( Q_{n+1}^-\right) }^{m-p\kappa }\left( \frac{1}{(1-\sigma )^{\frac{(N+p)^2}{N}}}+\frac{1}{(1-\sigma )^{\frac{p(N+p)}{N}}}\right) \nonumber \\&\quad \times \left( \frac{1}{{\tilde{k}}^{m-p}}+\frac{1}{{\tilde{k}}^{m-2}}\right) ^{1+\frac{p}{N}}Y_j^{1+\frac{p}{N}}\nonumber \\&\le \frac{C 2^{bj+dn}}{R_{n+1}^{\frac{p^2}{N}}}M_{n+1}^{m-p\kappa } \left( \frac{1}{{\tilde{k}}^{m-p}}+\frac{1}{{\tilde{k}}^{m-2}}\right) ^{1+\frac{p}{N}}Y_j^{1+\frac{p}{N}} \end{aligned}$$

(4.17)

with $b:=(1+p/N)(N+p+m)$ and $d=(N+p)^2/N$. For any $j\in {\mathbb {N}}$, define $W_j=Y_j/R_n^p$. Then we get

$$\begin{aligned} W_{j+1}\le C2^{bj+dn} M_{n+1}^{m-p\kappa } \left( \frac{1}{{\tilde{k}}^{m-p}}+\frac{1}{{\tilde{k}}^{m-2}}\right) ^{1+\frac{p}{N}}W_j^{1+\frac{p}{N}}. \end{aligned}$$

According to Lemma 4.5, we can see that

$$\begin{aligned} \lim _{j\rightarrow \infty }Y_j=0 \end{aligned}$$

if

$$\begin{aligned} W_{0} \le C 2^{-\frac{dnN}{p}-\frac{bN^2}{p^2}} M_{n+1}^{-\frac{N(m-p \kappa )}{p}}\left( \frac{1}{{\tilde{k}}^{m-p}}+\frac{1}{{\tilde{k}}^{m-2}}\right) ^{-\frac{p+N}{p}}. \end{aligned}$$

(4.18)

To ensure the above inequality, we need choose ${\tilde{k}}$ properly large. Indeed,

$$\begin{aligned} W_0=\frac{Y_0}{R_n^p}&=R_n^{-p}\int _{t_0-R_{n+1}^p}^{t_0}{\int \!\!\!\!-}_{B_{R_{n+1}}}u_+^m\,\mathrm{d}x\mathrm{d}t\\&\le 2^p{\int \!\!\!\!-}_{t_0-R_{n+1}^p}^{t_0}{\int \!\!\!\!-}_{B_{R_{n+1}}}u_+^m\,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

Namely,

$$\begin{aligned} 2^p{\int \!\!\!\!-}_{t_0-R_{n+1}^p}^{t_0}{\int \!\!\!\!-}_{B_{R_{n+1}}}u_+^m\,\mathrm{d}x\mathrm{d}t\le C2^{-\frac{dnN}{p}-\frac{bN^2}{p^2}} M_{n+1}^{-\frac{N(m-p\kappa )}{p}}\left( \frac{1}{{\tilde{k}}^{m-p}}+\frac{1}{{\tilde{k}}^{m-2}}\right) ^{-\frac{p+N}{p}}, \end{aligned}$$

which implies that

$$\begin{aligned} C 2^{\frac{dnN}{N+p}}M_{n+1}^{\frac{N(m-p\kappa )}{N+p}}\left( {\int \!\!\!\!-}_{t_0-R_{n+1}^p}^{t_0}{\int \!\!\!\!-}_{B_{R_{n+1}}}u_+^m\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{p}{N+p}}\le \left( \frac{1}{{\tilde{k}}^{m-p}}+\frac{1}{{\tilde{k}}^{m-2}}\right) ^{-1}. \end{aligned}$$

Therefore, we take

$$\begin{aligned} {\tilde{k}}&=C 2^{\frac{dnN}{(N+p)(m-p)}}M_{n+1}^{\frac{N(m-p\kappa )}{(N+p)(m-p)}}\left( {\int \!\!\!\!-}_{t_0-R_{n+1}^p}^{t_0}{\int \!\!\!\!-}_{B_{R_{n+1}}}u_+^m\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{p}{(N+p)(m-p)}}\\&\quad +C 2^{\frac{dnN}{(N+p)(m-2)}}M_{n+1}^{\frac{N(m-p\kappa )}{(N+p)(m-2)}}\left( {\int \!\!\!\!-}_{t_0-R_{n+1}^p}^{t_0}{\int \!\!\!\!-}_{B_{R_{n+1}}}u_+^m\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{p}{(N+p)(m-2)}}\\&\quad +\frac{\mathrm {Tail}_\infty \left( u_+;x_0,R_n,t_0-R_{n+1}^p, t_0\right) }{2}, \end{aligned}$$

which makes (4.18) hold true. Here the constant C only depends on N, p, s, m and $\Lambda$. Under this choice, it follows from Lemma 4.5 that

$$\begin{aligned}&M_n=\underset{Q_{R_n}^-}{{\text {ess}}\,\sup } u_+\nonumber \\&\quad \le C 2^{\frac{dnN}{(N+p)(m-p)}}M_{n+1}^{\frac{N(m-p\kappa )}{(N+p)(m-p)}}\left( {\int \!\!\!\!-}_{t_0-R_{n+1}^p}^{t_0}{\int \!\!\!\!-}_{B_{R_{n+1}}}u_+^m\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{p}{(N+p)(m-p)}}\nonumber \\&\qquad +C 2^{\frac{dnN}{(N+p)(m-2)}}M_{n+1}^{\frac{N(m-p\kappa )}{(N+p)(m-2)}}\left( {\int \!\!\!\!-}_{t_0-R_{n+1}^p}^{t_0}{\int \!\!\!\!-}_{B_{R_{n+1}}}u_+^m\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{p}{(N+p)(m-2)}}\nonumber \\&\qquad +\frac{\mathrm {Tail}_\infty \left( u_+;x_0,R_n,t_0-R_{n+1}^p, t_0\right) }{2}. \end{aligned}$$

(4.19)

Obverse $m>\max \left\{ 2,\frac{N(2-p)}{p}\right\}$ and $\kappa =1+2/N$, which indicates that

$$\begin{aligned} 0<\frac{N(m-p\kappa )}{(N+p)(m-p)},\frac{N(m-p\kappa )}{(N+p)(m-2)}<1. \end{aligned}$$

Now we apply the Young’s inequality with $\varepsilon$ to (4.19), arriving at

$$\begin{aligned} M_{n}&\le \varepsilon M_{n+1}+C 2^{\frac{dnN}{(N+p)(m-p-\beta )}}\varepsilon ^{-\frac{\beta }{m-p-\beta }} \left( {\int \!\!\!\!-}_{t_0-R^p}^{t_0}{\int \!\!\!\!-}_{B_R}u_+^m\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{p}{(N+p)(m-p-\beta )}}\\&\quad +C 2^{\frac{dnN}{(N+p)(m-2-\beta )}}\varepsilon ^{-\frac{\beta }{m-2-\beta }}\left( {\int \!\!\!\!-}_{t_0-R^p}^{t_0}{\int \!\!\!\!-}_{B_R}u_+^m\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{p}{(N+p)(m-2-\beta )}}\\&\quad +\frac{\mathrm {Tail}_\infty \left( u_+;x_0,R/2,t_0-R^p,t_0\right) }{2} \end{aligned}$$

with $\beta =(m-p\kappa )N/(p+N)$, where we used the fact $R/2\le R_n<R$. Via the induction argument, we can derive

$$\begin{aligned} M_0&\le \varepsilon ^{n+1}M_{n+1}\\&\quad +C\varepsilon ^{-\frac{\beta }{m-p-\beta }}\left( {\int \!\!\!\!-}_{t_0-R^p}^{t_0}{\int \!\!\!\!-}_{B_R}u_+^m\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{p}{(N+p)(m-p-\beta )}}\sum _{i=0}^n \left( 2^{\frac{dN}{(N+p)(m-p-\beta )}}\varepsilon \right) ^i\\&\quad +C\varepsilon ^{-\frac{\beta }{m-2-\beta }}\left( {\int \!\!\!\!-}_{t_0-R^p}^{t_0}{\int \!\!\!\!-}_{B_R}u_+^m\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{p}{(N+p)(m-2-\beta )}}\sum _{i=0}^n \left( 2^{\frac{dN}{(N+p)(m-2-\beta )}}\varepsilon \right) ^i\\&\quad +\frac{\mathrm {Tail}_\infty \left( u_+;x_0,R/2,t_0-R^p,t_0\right) }{2}\sum _{i=0}^n\varepsilon ^i, \quad n=0,1,2,\ldots . \end{aligned}$$

It is easy to see that the sum on the right-hand side could be revised by a convergent series, provided that we take

$$\begin{aligned} \varepsilon =2^{-\left[ {\frac{dN}{(N+p)(m-2-\beta )}}+1\right] }. \end{aligned}$$

Finally, letting $n\rightarrow \infty$, we deduce that

$$\begin{aligned} \underset{Q_{R/2}^-}{{\text {ess}}\,\sup } u&\le \mathrm {Tail}_\infty \left( u_+;x_0,R/2,t_0-R^p, t_0\right) \nonumber \\&\quad +C\left( {\int \!\!\!\!-}_{Q_R^-} u_+^m\,\mathrm{d}x \mathrm{d}t\right) ^{\frac{p}{(N+p)(m-2-\beta )}} \vee \left( {\int \!\!\!\!-}_{Q_R^-} u_+^m\,\mathrm{d}x \mathrm{d}t\right) ^{\frac{p}{(N+p)(m-p-\beta )}}, \end{aligned}$$

(4.20)

where $C>0$ depends only on N, p, s, m and $\Lambda$. $\square$

5 Local Hölder Continuity

In this section, we aim at establishing the Hölder continuity of weak solutions to (1.1) in the case that $p>2$, based on the local boundedness results. Before verifying this conclusion, we introduce some notations.

Let $({\overline{x}}_0,{\overline{t}}_0)\in Q_T$ and $r\in (0,R]$ for some $R\in (0,1)$. Let also $\alpha \in (0,\frac{p}{p-1})$ and $\sigma \in (0,\sigma _0)$ with $\sigma _0^\frac{p}{p-1}\le \frac{1}{4}$ be two constants to be determined later. Set

$$\begin{aligned} r_j:=\frac{\sigma ^jr}{2}, \quad \omega (r_0)=\omega (r/2):=M, \quad \omega (r_j):=\left( \frac{r_j}{r_0}\right) ^\alpha \omega (r_0), \quad j=0,1,2,3 \ldots \end{aligned}$$

(5.1)

and

$$\begin{aligned} M:=C\left[ \mathrm {Tail}_\infty \left( u;{\overline{x}}_0,r/2,{\overline{t}}_0-r^p, {\overline{t}}_0+r^p\right) +\left( {\int \!\!\!\!-}_{Q_r}|u|^p\,\mathrm{d}x\mathrm{d}t\right) ^{\frac{1}{2}} \vee 1\right] \end{aligned}$$

(5.2)

with C depending on $N,p,s,\Lambda$. Define

$$\begin{aligned} d_j:= {\left\{ \begin{array}{ll}{\left[ \varepsilon \sigma ^{(j-1)\alpha } M\right] ^{2-p}} &{} \text{ if } j \ge 1, \\ 1 &{} \text{ if } j=0,\end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \varepsilon =\sigma ^{\frac{p}{p-1}-\alpha }. \end{aligned}$$

Thus, it is easy to obtain

$$\begin{aligned} \frac{1}{d_{j+1}}=\left[ \varepsilon \omega (r_j)\right] ^{p-2} \text{ for } \text{ all } j \ge 0 . \end{aligned}$$

(5.3)

Denote

$$\begin{aligned} B_j:=B_{r_j}\left( {\overline{x}}_0\right) \text{ and } t_j:=d_j r_j^p \end{aligned}$$

and

$$\begin{aligned} Q_j:=Q_{r_j,t_j}\left( {\overline{x}}_0, {\overline{t}}_0\right) =B_j \times \left( {\overline{t}}_0-t_j, {\overline{t}}_0+t_j\right) . \end{aligned}$$

Hence, for $j\ge 1$, we have

$$\begin{aligned} 4\left( \sigma ^{\frac{p}{p-1}-\alpha }\right) ^{2-p} r_1^p \le r_0^p \quad \text{ and } \quad 4 \sigma ^{\alpha (2-p)} r_{j+1}^{p} \le r_j^p. \end{aligned}$$

The above inequalities combine with the definitions of $d_j$ and $t_j$ gives that

$$\begin{aligned} 4 t_{j+1} \le t_j \text{ for } \text{ all } j \ge 0. \end{aligned}$$

(5.4)

Now we are going to deduce an oscillation reduction on weak solutions.

Lemma 5.1

Let $p>2$ and u be a local weak solution to (1.1). Set $\left( {\overline{x}}_0,{\overline{t}}_0\right) \in Q_T$, $r\in (0,R]$ for some $R\in (0,1)$ and $Q_R\equiv B({\overline{x}}_0) \times \left( {\overline{t}}_0-R^p, {\overline{t}}_0+R^p\right)$ such that ${\overline{Q}}_R \subseteq Q_T$. Then

$$\begin{aligned} \underset{Q_j}{{\text {ess osc}}} u\le \omega (r_j) \text{ for } \text{ all } j=0,1,2, \ldots . \end{aligned}$$

(5.5)

Proof

We prove this Lemma by the induction argument. It follows from Theorem 1.2 and the definition of $\omega (r_0)$ that (5.5) holds true for $j=0$. We may assume that (5.5) is valid for each $i\in \{0,\ldots ,j\}$ with some $j\ge 0$. Then we devote to proving it holds for $i=j+1$.

Denote

$$\begin{aligned} 2B_{j+1}:=B_{2r_{j+1}}({\overline{x}}_0), \quad 2Q_{j+1}:=B_{2r_{j+1}}({\overline{x}}_0)\times ({\overline{t}}_0-2t_{j+1},{\overline{t}}_0+2t_{j+1}). \end{aligned}$$

It is obvious that either

$$\begin{aligned} \frac{\left| 2 Q_{j+1} \cap \left\{ u \ge \underset{Q_j}{{\text {ess}}\,\inf } u+\omega (r_j) / 2\right\} \right| }{\left| 2 Q_{j+1}\right| } \ge \frac{1}{2} \end{aligned}$$

(5.6)

or

$$\begin{aligned} \frac{\left| 2 Q_{j+1} \cap \left\{ u \le \underset{Q_j}{{\text {ess}}\,\inf } u+\omega (r_j) / 2\right\} \right| }{\left| 2 Q_{j+1}\right| } \ge \frac{1}{2} \end{aligned}$$

(5.7)

must hold. In the case of (5.6), we set $u_j:=u-\underset{Q_j}{{\text {ess}}\,\inf } u$. In the case of (5.7), we set $u_j:=\omega (r_j)-\left( u-\underset{Q_j}{{\text {ess}}\,\inf } u\right)$. In all cases, we have

$$\begin{aligned} \frac{\left| 2 Q_{j+1} \cap \left\{ u_j \ge \omega (r_j)/2\right\} \right| }{\left| 2 Q_{j+1}\right| } \ge \frac{1}{2} \end{aligned}$$

(5.8)

and

$$\begin{aligned} 0\le \underset{Q_i}{{\text {ess}}\,\sup } u_j \le 2 \omega (r_i) \quad \text {for } i=0, \ldots , j. \end{aligned}$$

(5.9)

Now we provide an important estimate to be used later,

$$\begin{aligned}&\left[ \mathrm {Tail}_\infty \left( u_j;{\overline{x}}_0,r_j,{\overline{t}}_0-t_j, {\overline{t}}_0+t_j\right) \right] ^{p-1}\nonumber \\&\quad \le C \sigma ^{-\alpha (p-1)}\left[ \omega (r_j)\right] ^{p-1} \text{ for } j=0,1,2 \ldots , \end{aligned}$$

(5.10)

where C only depends on N, p, s, the difference of $p/(p-1)$ and $\alpha$. Indeed, it is easy to see the claim is true when $j=0$. For $j\ge 1$, we have

$$\begin{aligned}&\left[ \mathrm {Tail}_\infty \left( u_j; {\overline{x}}_0, r_j, {\overline{t}}_0-t_j, {\overline{t}}_0+t_j\right) \right] ^{p-1}\nonumber \\&\quad =r_j^p\underset{t \in ({\overline{t}}_0-t_j, {\overline{t}}_0+t_j)}{{\text {ess}}\,\sup } \sum _{i=1}^j \int _{B_{i-1} \backslash B_i} \frac{\left| u_j(x, t)\right| ^{p-1}}{\left| x-{\overline{x}}_0\right| ^{N+sp}}\,\mathrm{d}x\nonumber \\&\qquad +r_j^p \underset{t \in ({\overline{t}}_0-t_j, {\overline{t}}_0+t_j)}{{\text {ess}}\,\sup } \int _{{\mathbb {R}}^N \backslash B_0} \frac{\left| u_j(x, t)\right| ^{p-1}}{\left| x-{\overline{x}}_0\right| ^{N+sp}}\,\mathrm{d}x\nonumber \\&\quad \le r_j^p \sum _{i=1}^j\left( \underset{Q_{i-1}}{{\text {ess}}\,\sup } u_j\right) ^{p-1} \int _{{\mathbb {R}}^N \backslash B_i} \frac{1}{\left| x-{\overline{x}}_0\right| ^{N+sp}}\,\mathrm{d}x \nonumber \\&\qquad +r_j^p \underset{t \in ({\overline{t}}_0-t_j, {\overline{t}}_0+t_j)}{{\text {ess}}\,\sup } \int _{{\mathbb {R}}^N \backslash B_0} \frac{\left| u_j(x, t)\right| ^{p-1}}{\left| x-{\overline{x}}_0\right| ^{N+sp}}\,\mathrm{d}x\nonumber \\&\quad \le C \sum _{i=1}^{j}\left( \frac{r_j}{r_i}\right) ^p[\omega (r_{i-1})]^{p-1}, \end{aligned}$$

where in the last line we used (5.9) and the definition of $u_j$. Since $\sigma \le 1/4$ and $\alpha <p/(p-1)$, we estimate the right-hand side as follows:

$$\begin{aligned}&\sum _{i=1}^{j}\left( \frac{r_j}{r_i}\right) ^p[\omega (r_{i-1})]^{p-1}\\&\quad =\left[ \omega (r_0)\right] ^{p-1}\left( \frac{r_j}{r_0}\right) ^{\alpha (p-1)} \sum _{i=1}^j\left( \frac{r_{i-1}}{r_i}\right) ^{\alpha (p-1)}\left( \frac{r_j}{r_i}\right) ^{p-\alpha (p-1)}\\&\quad =\left[ \omega (r_j)\right] ^{p-1} \sigma ^{-\alpha (p-1)} \sum _{i=0}^{j-1} \sigma ^{i\left( p-\alpha (p-1)\right) }\\&\quad \le \left[ \omega (r_j)\right] ^{p-1} \frac{\sigma ^{-\alpha (p-1)}}{1-\sigma ^{p-\alpha (p-1)}} \\&\quad \le \frac{4^{p-\alpha (p-1)}}{\left( p-\alpha (p-1)\right) \log 4} \sigma ^{-\alpha (p-1)}\left[ \omega (r_j)\right] ^{p-1}. \end{aligned}$$

Thus, we have proved (5.10) with the constant C depending on N, p, s, the difference of $p/(p-1)$ and $\alpha$. Next, we define

$$\begin{aligned} v:=\min \left\{ \left[ \log \left( \frac{\omega (r_j)/2+d}{u_j+d}\right) \right] _+, k\right\} \quad \text{ for } k>0 . \end{aligned}$$

(5.11)

It follows from Corollary 3.3 with $a\equiv \omega (r_j)/2$ and $b\equiv \exp (k)$ that

$$\begin{aligned}&\int _{{\overline{t}}_0-2 t_{j+1}}^{{\overline{t}}_0+2 t_{j+1}} {\int \!\!\!\!-}_{2 B_{j+1}}\left| v(x,t)-(v)_{2B_{j+1}}(t)\right| ^p\,\mathrm{d}x\mathrm{d}t\\&\quad \le C t_{j+1} d^{1-p}\left( \frac{r_{j+1}}{r_j}\right) ^p\left[ \mathrm {Tail }_\infty \left( u_j;{\overline{x}}_0,r_j,{\overline{t}}_0-4 t_{j+1}, {\overline{t}}_0+4 t_{j+1}\right) \right] ^{p-1} \\&\qquad +C t_{j+1}+C d^{2-p} r_{j+1}^p+C t_{j+1} r_{j+1}^{p-sp}\\&\quad \le C t_{j+1} d^{1-p}\left[ \varepsilon \omega (r_j)\right] ^{p-1}+C t_{j+1}+C d^{2-p} r_{j+1}^p, \end{aligned}$$

where in the last line we used the fact $4 t_{j+1} \le t_j$ and (5.10). Choosing $d=\varepsilon \omega (r_j)$ in (5.11), we get

$$\begin{aligned} d^{2-p}=d_{j+1}. \end{aligned}$$

Since $\alpha <p/(p-1)$, we can verify

$$\begin{aligned} d^{1-p}\le r_{j+1}^{-p}. \end{aligned}$$

Thus, we arrive at

$$\begin{aligned} \int _{{\overline{t}}_0-2 t_{j+1}}^{{\overline{t}}_0+2 t_{j+1}} {\int \!\!\!\!-}_{2 B_{j+1}}\left| v(x,t)-(v)_{2B_{j+1}}(t)\right| ^p\,\mathrm{d}x\mathrm{d}t\le Ct_{j+1}, \end{aligned}$$

where C depends on $N,p,s,\Lambda$, the difference of $p/(p-1)$ and $\alpha$. Following the calculation in pp. 37–38 in [12], there holds that

$$\begin{aligned} \frac{\left| 2 Q_{j+1} \cap \left\{ u_j \le 2 \varepsilon \omega (r_j)\right\} \right| }{\left| 2 Q_{j+1}\right| } \le \frac{{\overline{C}}}{\log \left( \frac{1}{\sigma }\right) }, \end{aligned}$$

(5.12)

where ${\overline{C}}>0$ depends on $N,p,s,\Lambda$, the difference of $p/(p-1)$ and $\alpha$.

In what follows, we will proceed by a suitable iteration to infer the desired oscillation decay over the domain $Q_{j+1}$. For any $i=$ $0,1,2, \ldots$, we define

$$\begin{aligned}&\varrho _i=r_{j+1}+2^{-i} r_{j+1}, \quad {\tilde{\varrho }}_i:=\frac{\varrho _i+\varrho _{i+1}}{2}, \\&\theta _i:=t_{j+1}+2^{-i} t_{j+1}, \quad {\tilde{\theta }}_i:=\frac{\theta _i+\theta _{i+1}}{2}, \\&Q^i:=B^i \times \Gamma _i:=B_{\varrho _i}({\overline{x}}_0) \times ({\overline{t}}_0-\theta _{i}, {\overline{t}}_0+\theta _{i}), \\&{\tilde{Q}}^i:={\tilde{B}}^i \times {\tilde{\Gamma }}_i:=B_{{\tilde{\Gamma }}_i}({\overline{x}}_0) \times ({\overline{t}}_0-{\tilde{\theta }}_i, {\overline{t}}_0+{\tilde{\theta }}_i). \end{aligned}$$

Then we take the cut-off function $\psi _i \in C_0^\infty ({\tilde{B}}^i)$ and $\eta _i \in C_0^\infty ({\tilde{\Gamma }}_i)$ such that

$$\begin{aligned} 0 \le \psi _i \le 1,\quad \left| \nabla \psi _i\right| \le C 2^i r_{j+1}^{-1} \text{ in } {\tilde{B}}^i, \quad \psi _i \equiv 1 \text{ in } B^{i+1} \end{aligned}$$

and

$$\begin{aligned} 0 \le \eta _i \le 1,\quad \left| \partial _t \eta _i\right| \le C 2^i t_{j+1}^{-1} \text{ in } {\tilde{\Gamma }}_i, \quad \eta _i \equiv 1 \text{ in } \Gamma _{i+1}. \end{aligned}$$

Define

$$\begin{aligned} k_i:=(1+2^{-i}) \varepsilon \omega (r_j), \quad v_i:=(k_i-u_j)_+. \end{aligned}$$

Taking $\ell =\theta _i-\theta _{i+1}, \tau _1={\overline{t}}_0-\theta _{i+1}$ and $\tau _2={\overline{t}}_0+\theta _{i+1}$ in Lemma 3.1, we get

$$\begin{aligned}&\int _{\Gamma _{i+1}} {\int \!\!\!\!-}_{B^{i}}|\nabla v_i(x,t)|^p\psi ^p_i(x)\,\mathrm{d}x\mathrm{d}t+\underset{t\in \Gamma _{i+1}}{{\text {ess}}\,\sup }{\int \!\!\!\!-}_{B^i} v_i^2(x,t) \psi _i^p(x)\,\mathrm{d}x\nonumber \\&\qquad +\int _{\Gamma _{i+1}} \int _{B^i} {\int \!\!\!\!-}_{B^i} \frac{\left| v_i(x,t) \psi _i(x)-v_i(y,t) \psi _i(y)\right| ^p}{|x-y|^{N+sp}}\,\mathrm{d}x\mathrm{d}y\mathrm{d}t\\&\quad \le C \int _{\Gamma _i} {\int \!\!\!\!-}_{B^i} |\nabla \psi _i(x)|^p v_i^p(x,t)\eta _i^2(t)\,\mathrm{d}x\mathrm{d}t\\&\qquad +C \int _{\Gamma _i} \int _{B^i} {\int \!\!\!\!-}_{B^i} \max \left\{ v_i(x,t), v_i(y,t)\right\} ^p\left| \psi _i(x)-\psi _i(y)\right| ^p \eta _i^2(t)\,\mathrm{d}\mu \mathrm{d}t \\&\qquad +C\underset{{\mathop {x \in \mathrm {supp} \psi _i}\limits ^{t\in \Gamma _i}}}{\mathrm {ess}\,\sup }\int _{{\mathbb {R}}^N \backslash B^i} \frac{v_i^{p-1}(y,t)}{|x-y|^{N+sp}}\,\mathrm{d}y \int _{\Gamma _i} {\int \!\!\!\!-}_{B^i} v_i(x, t) \psi _i^p(x) \eta _i^2(t)\,\mathrm{d}x\mathrm{d}t \\&\qquad +C \int _{\Gamma _i} {\int \!\!\!\!-}_{B^i} v_i^2(x,t) \psi _i^p(x) \eta _i(t)\left| \partial _t \eta _i(t)\right| \,\mathrm{d}x\mathrm{d}t\\&\quad =:I_1+I_2+I_3+I_4. \end{aligned}$$

We estimate $I_1$ as

$$\begin{aligned} I_1&\le C k_i^p2^{pi}r_{j+1}^{-p}\int _{{\Gamma }_i}{\int \!\!\!\!-}_{B_i} \chi _{\left\{ u_j \le k_i\right\} }(x,t)\,\mathrm{d}x\mathrm{d}t\\&\le C 2^{pi}r_{j+1}^{-p}[\varepsilon \omega (r_j)]^p\int _{{\Gamma }_i}{\int \!\!\!\!-}_{B_i} \chi _{\left\{ u_j \le k_i\right\} }(x,t)\,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

where we used the properties of $\psi _i$. As the computations in pp. 39–40 in [12], we derive

$$\begin{aligned} I_2&\le C 2^{pi} r_{j+1}^{-p}[\varepsilon \omega \left( r_j\right) ]^p \int _{{\Gamma }_i}{\int \!\!\!\!-}_{B_i} \chi _{\left\{ u_j \le k_i\right\} }(x,t)\,\mathrm{d}x\mathrm{d}t, \\ I_3&\le C 2^{(N+sp)i}r_{j+1}^{-p}[\varepsilon \omega \left( r_j\right) ]^p \int _{{\Gamma }_i}{\int \!\!\!\!-}_{B_i} \chi _{\left\{ u_j \le k_i\right\} }(x,t)\,\mathrm{d}x\mathrm{d}t \end{aligned}$$

and

$$\begin{aligned} I_4\le C 2^{spi}r_{j+1}^{-p}[\varepsilon \omega \left( r_j\right) ]^p \int _{{\Gamma }_i}{\int \!\!\!\!-}_{B_i} \chi _{\left\{ u_j \le k_i\right\} }(x,t)\,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

Since $\psi _i\equiv 1$ in $B^{i+1}$, we can deduce that

$$\begin{aligned}&\int _{\Gamma _{i+1}} {\int \!\!\!\!-}_{B^{i+1}}|\nabla v_i(x,t)|^p\,\mathrm{d}x\mathrm{d}t+\int _{\Gamma _{i+1}} \int _{B^{i+1}} {\int \!\!\!\!-}_{B^{i+1}} \frac{\left| v_i(x,t)-v_i(y,t)\right| ^p}{|x-y|^{N+sp}}\,\mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\qquad +\underset{t \in \Gamma _{i+1}}{{\text {ess}}\,\sup }{\int \!\!\!\!-}_{B^{i+1}} v_i^2(x,t)\,\mathrm{d}x\nonumber \\&\quad \le C 2^{(N+p) i}r_{j+1}^{-p}[\varepsilon \omega (r_j)]^p \int _{\Gamma _i} {\int \!\!\!\!-}_{B^i} \chi _{\left\{ u_j \le k_i\right\} }(x,t)\,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

(5.13)

From (5.3), we get

$$\begin{aligned} \underset{t \in \Gamma _{i+1}}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B^{i+1}} v_i^p(x,t)\,\mathrm{d}x&\le k_i^{p-2} \underset{t \in \Gamma _{i+1}}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B^{i+1}} v_i^2(x,t)\,\mathrm{d}x\nonumber \\&\le C d_{j+1}^{-1} \underset{t \in \Gamma _{i+1}}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B^{i+1}} v_i^2(x,t)\,\mathrm{d}x. \end{aligned}$$

(5.14)

Combining (5.13) with (5.14) gives that

$$\begin{aligned}&r_{j+1}^p{\int \!\!\!\!-}_{\Gamma _{i+1}} {\int \!\!\!\!-}_{B^{i+1}} |\nabla v_i(x,t)|^p\,\mathrm{d}x\mathrm{d}t+\underset{t \in \Gamma _{i+1}}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B^{i+1}} v_i^p(x,t)\,\mathrm{d}x\nonumber \\&\quad \le d_{j+1}^{-1}\int _{\Gamma _{i+1}} {\int \!\!\!\!-}_{B^{i+1}} |\nabla v_i(x,t)|^p\,\mathrm{d}x\mathrm{d}t+d_{j+1}^{-1}\underset{t \in \Gamma _{i+1}}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B^{i+1}} v_i^2(x,t)\,\mathrm{d}x\nonumber \\&\quad \le C 2^{(N+p)i} r_{j+1}^{-p} d_{j+1}^{-1}[\varepsilon \omega (r_j)]^p \int _{\Gamma _i} {\int \!\!\!\!-}_{B^i} \chi _{\left\{ u_j \le k_i\right\} }(x,t)\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad \le C2^{(N+p)i}[\varepsilon \omega (r_j)]^p A_i, \end{aligned}$$

(5.15)

where $A_i$ is denoted by

$$\begin{aligned} A_i:=\frac{\left| Q_i \cap \left\{ u_j \le k_i\right\} \right| }{\left| Q_i\right| }. \end{aligned}$$

In view of Lemma 2.4, we can see

$$\begin{aligned} \int _{\Gamma _{i+1}} {\int \!\!\!\!-}_{B^{i+1}} v_i^{p(1+\frac{p}{N})}(x,t)\,\mathrm{d}x\mathrm{d}t&\le Cr_{j+1}^p \int _{\Gamma _{i+1}} {\int \!\!\!\!-}_{B^{i+1}} |\nabla v_i(x,t)|^p\,\mathrm{d}x\mathrm{d}t\nonumber \\&\quad \times \left( \underset{t \in \Gamma _{i+1}}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B^{i+1}} v_i^p(x,t)\,\mathrm{d}x\right) ^{\frac{p}{N}}. \end{aligned}$$

(5.16)

Merging (5.15) and (5.16) leads to

$$\begin{aligned}&A_{i+1}\left( k_i-k_{i+1}\right) ^{p(1+\frac{p}{N})} \\&\quad \le {\int \!\!\!\!-}_{\Gamma _{i+1}} {\int \!\!\!\!-}_{B^{i+1} \cap \left\{ u_j\le k_{i+1}\right\} } v_i^{p(1+\frac{p}{N})}(x,t)\,\mathrm{d}x \mathrm{d}t\\&\quad \le Cr_{j+1}^p {\int \!\!\!\!-}_{\Gamma _{i+1}} {\int \!\!\!\!-}_{B^{i+1}} |\nabla v_i(x,t)|^p\,\mathrm{d}x\mathrm{d}t\left( \underset{t \in \Gamma _{i+1}}{{\text {ess}}\,\sup } {\int \!\!\!\!-}_{B^{i+1}} v_i^p(x,t)\,\mathrm{d}x\right) ^{\frac{p}{N}}\\&\quad \le C\left[ 2^{(N+p)i}(\varepsilon \omega (r_j))^p A_i\right] ^{1+\frac{p}{N}}. \end{aligned}$$

We can readily get the recursive inequality

$$\begin{aligned} A_{i+1} \le {\tilde{C}} 2^{(N+p)(1+\frac{p}{N})i} A_i^{1+\frac{p}{N}}, \end{aligned}$$

where ${\tilde{C}}$ depends on $N,p,s,\Lambda$, the difference of $p/(p-1)$ and $\alpha$. Set

$$\begin{aligned} \nu ^*:={\tilde{C}}^{-\frac{N}{p}} 2^{\frac{-N(N+p)^2}{p^2}}. \end{aligned}$$

Then we take

$$\begin{aligned} \sigma =\min \left\{ \frac{1}{4},\sigma _0,\exp \left( -\frac{{\overline{C}}}{\nu ^*}\right) \right\} , \end{aligned}$$

where we need to note that $\sigma _0$ only depends on p. Utilizing the definition of $A_i$, we obtain

$$\begin{aligned} A_0=\frac{\left| 2 Q_{j+1} \cap \left\{ u_j \le 2 \varepsilon \omega (r_j)\right\} \right| }{\left| 2 Q_{j+1}\right| } \le \nu ^*. \end{aligned}$$

It follows from Lemma 4.5 that

$$\begin{aligned} \lim _{i\rightarrow \infty }A_i=0, \end{aligned}$$

which implies that

$$\begin{aligned} u_j(x,t) \ge \varepsilon \omega (r_j) \text{ in } Q_{j+1}. \end{aligned}$$

Thereby, recalling (5.1) and the definition of $u_j$, we have

$$\begin{aligned}&\underset{Q_{j+1}}{{\text {ess}}\,osc}\,u=\underset{Q_{j+1}}{{\text {ess}}\,\sup }u_j-\underset{Q_{j+1}}{{\text {ess}}\,\inf }u_j\le (1-\varepsilon ) \omega (r_j)\nonumber \\&\quad =(1-\varepsilon ) \sigma ^{-\alpha } \omega (r_{j+1}). \end{aligned}$$

(5.17)

Now, we pick $\alpha \in (0,p/(p-1))$ such that

$$\begin{aligned} \sigma ^\alpha \ge 1-\varepsilon =1-\sigma ^{\frac{p}{p-1}-\alpha }, \end{aligned}$$

which together with (5.17) ensures that

$$\begin{aligned} \underset{Q_{j+1}}{{\text {ess osc}}} u\le \omega (r_{j+1}). \end{aligned}$$

Now we finish the proof. $\square$

Proof of Theorem 1.4

Let $p>2$. Suppose that u is a local weak solution of (1.1). Let $(x_0,t_0) \in Q_T$, $R \in (0,1)$ and $Q_R \equiv B_R(x_0) \times (t_0-R^p,t_0+R^p)$ such that ${\overline{Q}}_R\subseteq Q_T$. Taking $r=R$ in Lemma 5.1 we can get

$$\begin{aligned} \underset{Q_j}{{\text {ess osc}}} u \le C\left( \frac{r_j}{R}\right) ^{\alpha } \omega \left( \frac{R}{2}\right) \text{ for } \text{ all } j \in {\mathbb {N}} \end{aligned}$$

(5.18)

with $\alpha <p/(p-1)$, $\sigma <1/4$, where $C\ge 1$ depends on $N,p,s,\Lambda ,$ and

$$\begin{aligned} \omega \left( \frac{R}{2}\right) =\mathrm {Tail}_\infty \left( u;x_0,R/2,t_0-R^p, t_0+R^p\right) +\left( {\int \!\!\!\!-}_{Q_R}|u|^p \,\mathrm{d}x\mathrm{d}t\right) ^{\frac{1}{2}} \vee 1. \end{aligned}$$

(5.19)

For every $\rho \in (0,R/2]$, we have $\rho \in \left( r_{j_0+1}, r_{j_0}\right]$ for $j_0 \in {\mathbb {N}}$. Choosing $d=[C\omega (R/2)]^{2-p}$, it follows that $Q_{\rho , d \rho ^p} \subseteq Q_{j_0}$. By applying (5.18), we get

$$\begin{aligned} \underset{Q_{\rho , d\rho ^p} }{{\text {ess osc}}} u \le \underset{Q_{j_0}}{{\text {ess osc}}} u \le C \sigma ^{-\alpha }\left( \frac{r_{j_0+1}}{R}\right) ^{\alpha } \omega \left( \frac{R}{2}\right) \le C \sigma ^{-\alpha }\left( \frac{\rho }{R}\right) ^{\alpha } \omega \left( \frac{R}{2}\right) . \end{aligned}$$

We can obtain the Hölder continuity from the above inequality along with (5.19). $\square$