Harnack’s estimate for a mixed local–nonlocal doubly nonlinear parabolic equation

Abstract

We establish Harnack’s estimates for positive weak solutions to a mixed local and nonlocal doubly nonlinear parabolic equation, of the type

$$\begin{aligned} \partial _t\left( |u|^{p-2}u\right) -{\textrm{div}}\,\textbf{A}\left( x,t,u,Du\right) +\mathcal {L}u(x,t)=0, \end{aligned}$$

where the vector field $\textbf{A}$ satisfies the p-ellipticity and growth conditions and the integro-differential operator $\mathcal {L}$ whose model is the fractional p-Laplacian. All results presented in this paper are provided by using sharp tools in the doubly nonlinear theory together with quantitative estimates.

Appendices

Appendix A: Proof of Lemma 3.1

Proof of Lemma 3.1

Although the argument used here is almost same as Proposition 3.4 and Lemma 3.8 (also [36, Proposition 3.1]), for the sake of completeness we shall nevertheless give the full proof. For this, we divide it into four different steps.

Step 1: Beginning.   Let $t_1 \in (t_0-\varrho ^p,t_0)$ be an arbitrary number. Given $\delta >0$ small, take the cutoff function $\psi _\delta (t)$ with respect to time, defined by

$$\begin{aligned} \psi _\delta (t):={\left\{ \begin{array}{ll} 0,\quad &{}\,\,\,t \in (t_0-\varrho ^p, t_1), \\ \frac{1}{\delta }(t-t_1), \quad &{}\,\,\,t \in [t_1, t_1+\delta ),\\ 1,\quad &{}\,\,\,t \in [t_1+\delta , t_0). \end{array}\right. } \end{aligned}$$

In the weak formulation (2.10), we switch the test function $\varphi$ to $\varphi ^p\psi _\delta (t)u^{-\varepsilon }$ with nonnegative $\varphi \in C^\infty _0(Q^-_\varrho (z_0))$. Then the evolutional term is divided into two terms as follows:

$$\begin{aligned} \iint _{\Omega _T}\partial _t[|u|^{p-2}u]_h\varphi ^p\psi _\delta (t)u^{-\varepsilon }\,{\textrm{d}}x{\textrm{d}}t&=\iint _{Q^-_\varrho (z_0)}\varphi ^p\psi _\delta (t)\partial _t[u^{p-1}]_h\left( u^{-\varepsilon }-[u^{p-1}]_h^{-\frac{\varepsilon }{p-1}}\right) \,{\textrm{d}}x{\textrm{d}}t\\&\quad +\iint _{Q^-_\varrho (z_0)}\varphi ^p\psi _\delta (t)\partial _t[u^{p-1}]_h[u^{p-1}]_h^{-\frac{\varepsilon }{p-1}}\,{\textrm{d}}x{\textrm{d}}t\\&=:\textbf{I}_1+\textbf{I}_2, \end{aligned}$$

where the definition of $\textbf{I}_1$ and $\textbf{I}_2$ are obvious form the context. Abbreviating

$$\begin{aligned} f_h:=[u^{p-1}]_h^{\frac{1}{p-1}} \quad \iff \quad [u^{p-1}]_h=f_h^{p-1}, \end{aligned}$$

and Lemma 2.8-(ii) and (2.5) with $\alpha =1+\frac{p-1}{\varepsilon }$ in Lemma 2.3 imply that

$$\begin{aligned} \textbf{I}_1&=\iint _{Q^-_\varrho (z_0)}\varphi ^p\psi _\delta (t)\frac{u^{p-1}-[u^{p-1}]_h}{h}\cdot \frac{[u^{p-1}]_h^{\frac{\varepsilon }{p-1}}-u^\varepsilon }{[u^{p-1}]_h^{\frac{\varepsilon }{p-1}}u^\varepsilon }\,{\textrm{d}}x{\textrm{d}}t\\&=\iint _{Q^-_\varrho (z_0)}\varphi ^p\psi _\delta (t)\frac{u^{p-1}-f_h^{p-1}}{h}\cdot \frac{f_h^\varepsilon -u^\varepsilon }{f_h^\varepsilon u^\varepsilon }\,{\textrm{d}}x{\textrm{d}}t\leqq 0. \end{aligned}$$

From integration by parts, it follows that

$$\begin{aligned} \textbf{I}_2&=\frac{p-1}{p-1-\varepsilon }\iint _{Q^-_\varrho (z_0)}\varphi ^p\psi _\delta (t)\partial _t[u^{p-1}]_h^{\frac{p-1-\varepsilon }{p-1}}\,{\textrm{d}}x{\textrm{d}}t\\&=-\frac{p-1}{p-1-\varepsilon }\iint _{Q^-_\varrho (z_0)}\left( p\varphi ^{p-1}\varphi _t\psi _\delta +\varphi ^p\psi \delta ^\prime \right) [u^{p-1}]_h^{\frac{p-1-\varepsilon }{p-1}}\,{\textrm{d}}x{\textrm{d}}t. \end{aligned}$$

The preceding estimates above and the fact that $[u^{p-1}]_h^{\frac{1}{p-1}} \longrightarrow u$ in $L^p(\Omega _T)$ by Lemma 2.8-(iii) give that

(A.1)

The spatial term can be dealt with comparatively easily. Indeed, by the use of convergence $[|Du|^{p-2}Du]_h \longrightarrow |Du|^{p-2}Du$ in $L^{\frac{p}{p-1}}(\Omega _T)$ as $h\searrow 0$ via Lemma 2.8-(ii) and Young’s inequality with the exponents $\left( \frac{p}{p-1},p\right)$ yields that, for any $\theta >0$,

$$\begin{aligned}&\lim _{h\searrow 0}\iint _{\Omega _T}[|Du|^{p-2}Du]_h\cdot D\left( \varphi ^p\psi _\delta (t)u^{-\varepsilon }\right) \,{\textrm{d}}x{\textrm{d}}t\\&\quad =\iint _{\Omega _T}|Du|^{p-2}Du\cdot D\left( \varphi ^p\psi _\delta (t)u^{-\varepsilon }\right) \,{\textrm{d}}x{\textrm{d}}t\\&\quad =\iint _{\Omega _T}|Du|^{p-2}Du\cdot \left( p\varphi ^{p-1}D\varphi \,u^{-\varepsilon }+\varphi ^pDu^{-\varepsilon }\right) \,{\textrm{d}}x{\textrm{d}}t\\&\quad \leqq p\iint _{Q^-_\varrho (z_0)}\Big (|Du|u^{-\frac{\varepsilon +1}{p}}\varphi \psi _\delta ^{\frac{1}{p}}\Big )^{p-1}\Big (|D\varphi |u^{-\varepsilon +(\varepsilon +1)\frac{p-1}{p}}\psi _\delta ^{\frac{1}{p}}\Big )\,{\textrm{d}}x{\textrm{d}}t\\&\quad \quad -\varepsilon \iint _{\Omega _T}|Du|^pu^{-\varepsilon -1}\varphi ^p\psi _\delta \,{\textrm{d}}x{\textrm{d}}t\\&\quad \leqq \big [(p-1)\theta -\varepsilon \big ]\iint _{\Omega _T}|Du|^pu^{-\varepsilon -1}\varphi ^p\psi _\delta \,{\textrm{d}}x{\textrm{d}}t\\&\qquad +c(\theta )\iint _{Q^-_\varrho (z_0)}|D\varphi |^pu^{-\varepsilon p+(\varepsilon +1)(p-1)}\varphi ^p\psi _\delta \,{\textrm{d}}x{\textrm{d}}t, \end{aligned}$$

which, by taking $\theta =\frac{\varepsilon }{2(p-1)}$, yields in particular that

$$\begin{aligned}&\lim _{h\searrow 0}\iint _{\Omega _T}[|Du|^{p-2}Du]_h\cdot D\left( \varphi ^p\psi _\delta (t)u^{-\varepsilon }\right) \,{\textrm{d}}x{\textrm{d}}t\nonumber \\&\quad \leqq -\frac{\varepsilon }{2}\iint _{\Omega _T}|Du|^pu^{-\varepsilon -1}\varphi ^p\psi _\delta \,{\textrm{d}}x+c(\varepsilon )\iint _{Q^-_\varrho (z_0)}|D\varphi |^pu^{p-1-\varepsilon }\psi _\delta \,{\textrm{d}}x{\textrm{d}}t, \end{aligned}$$

(A.2)

where we have computed that $-\varepsilon p+(\varepsilon +1)(p-1)=p-1-\varepsilon$.

Step 2: Fractional term.   Firstly, we will show that

$$\begin{aligned}&\int _0^T\iint _{{\mathbb {R}}^n\times {\mathbb {R}}^n}\left[ U(x,y,t)K(x,y,t)\right] _h\left( \phi (x,t)-\phi (y,t)\right) \,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\nonumber \\&\quad \longrightarrow \int _0^T\iint _{{\mathbb {R}}^n\times {\mathbb {R}}^n}U(x,y,t)K(x,y,t)\left( \phi (x,t)-\phi (y,t)\right) \,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\end{aligned}$$

(A.3)

in the limit $h\searrow 0$, where we abbreviated $\phi \equiv \varphi ^p\psi _\delta (t)u^{-\varepsilon }$ for short. In order to prove (A.3) we consider the quantity

$$\begin{aligned} (\textbf{II})_h&:=\left| \int _0^T\iint _{D_\Omega }\Big (\left[ U(x,y,t)K(x,y,t)\right] _h-U(x,y,t)K(x,y,t)\Big )\left( \phi (x,t)-\phi (y,t)\right) \,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\right| \\&\leqq \int _0^T\iint _{\Omega \times \Omega }\big |\cdots \big |\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t+2\int _0^T\iint _{\Omega \times \Omega ^c}\big |\cdots \big |\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&=:(\textbf{II}_1)_h+2(\textbf{II}_2)_h, \end{aligned}$$

with denoting $D_\Omega :=({\mathbb {R}}^n \times {\mathbb {R}}^n) \setminus (\Omega ^c \times \Omega ^c)$, where the definition of $(\textbf{II}_1)_h$ and $(\textbf{II}_2)_h$ are clear from the context. We now claim the following: As $h \searrow 0$,

$$\begin{aligned} \Big [U(x,y,t)K(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\Big ]_h \longrightarrow U(x,y,t)K(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\quad \text {in}\quad L^{\frac{p}{p-1}}(\Omega ^2_T), \end{aligned}$$

(A.4)

where $\Omega ^2_T:=\Omega \times \Omega \times (0,T)$. Indeed, since

$$\begin{aligned} \Big \Vert U(x,y,t)K(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\Big \Vert _{L^{\frac{p}{p-1}}(\Omega ^2_T)}^{\frac{p}{p-1}}<\infty \end{aligned}$$

Lemma 2.8-(i) with $E=\Omega ^2$ implies (A.4). Therefore, using (A.4), Hölder’s inequality and Lemma 2.6, we have

$$\begin{aligned} (\textbf{II}_1)_h&=\int _0^T\iint _{\Omega \times \Omega }\Bigg |\left[ U(x,y,t)K(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right] _h\\&\quad -U(x,y,t)K(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\Bigg | \\&\quad \times \frac{\left| \phi (x,t)-\phi (y,t)\right| }{|x-y|^{\frac{1}{p}(n+sp)}}\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&\leqq c\left( \int _0^T\iint _{\Omega \times \Omega }\Bigg |\left[ UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right] _h\right. \\&\left. \quad -UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\Bigg |^{\frac{p}{p-1}}\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\right) ^{\frac{p-1}{p}} \\&\quad \times \left( \int _{\Omega _T}|D\phi (x)|^p\,{\textrm{d}}x{\textrm{d}}t\right) ^{\frac{1}{p}} \\&\longrightarrow 0 \end{aligned}$$

as $h\searrow 0$, where we used the shorthand notation $UK(x,y,t)=U(x,y,t)K(x,y,t)$.

We next prove that $(\textbf{II}_1)_h \longrightarrow 0$. Take a ball $B_R\equiv B_R(0)$ satisfying $B_R \Supset \Omega$. By $\phi (y,t)=0$ for any $(y,t) \in (B_R\setminus \Omega )\times (0,T)$ and Hölder’s inequality, we obtain

$$\begin{aligned} (\textbf{II}_2)_h(R)&:=\int _0^T\iint _{\Omega \times (B_R\setminus \Omega )}\Big |\left[ UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right] _h-UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\Big | \nonumber \\&\quad \times \frac{\left| \phi (x,t)\right| }{|x-y|^{\frac{1}{p}(n+sp)}}\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\nonumber \\&\leqq \left( \int _0^T\iint _{\Omega \times (B_R\setminus \Omega )} \frac{\left| \phi (x,t)\right| ^p}{|x-y|^{n+sp}}\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\right) ^{\frac{1}{p}} \nonumber \\&\quad \times \left( \int _0^T\iint _{\Omega \times (B_R\setminus \Omega )}\Bigg |\left[ UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right] _h\nonumber \right. \\&\left. \quad -UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\Bigg |^{\frac{p}{p-1}}\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\right) ^{\frac{p-1}{p}}\nonumber \\&\leqq (\textbf{III})^{\frac{1}{p}}(\textbf{IV})^{\frac{p-1}{p}}, \end{aligned}$$

(A.5)

where

$$\begin{aligned} \textbf{III}:=\int _0^T\iint _{\Omega \times (B_R\setminus \Omega )} \frac{\left| \phi (x,t)\right| ^p}{|x-y|^{n+sp}}\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\end{aligned}$$

and

$$\begin{aligned} \textbf{IV}:= & {} \int _0^T\iint _{\Omega \times (B_R\setminus \Omega )}\Big |\left[ UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right] _h\\{} & {} -UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\Big |^{\frac{p}{p-1}}\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t. \end{aligned}$$

Since

$$\begin{aligned} |x-y| \geqq d_R:=\textrm{dist}\left( \,{\textrm{supp}}\,\phi (\cdot ,t),\, B_R \setminus \Omega \right) \qquad \forall (x,y) \in \,{\textrm{supp}}\,\phi (\cdot ,t) \times (B_R\setminus \Omega ) \end{aligned}$$

and $\,{\textrm{supp}}\,\phi =Q_\varrho ^-(z_0)$, we estimate

$$\begin{aligned} \textbf{III}&=\iint _{Q_\varrho ^-(z_0)}\left( \int _{B_R\setminus \Omega }\frac{\,{\textrm{d}}y}{|x-y|^{n+sp}} \right) |\phi (x,t)|\,{\textrm{d}}x{\textrm{d}}t\nonumber \\&\leqq m^{-\varepsilon }\Vert \varphi \Vert _{L^\infty (Q_\varrho ^-(z_0))}|Q_\varrho ^-(z_0)|\left( \int _{\mathbb {R}^n \setminus B_{d_R}(x)}\frac{\,{\textrm{d}}y}{|x-y|^{n+sp}} \right) \nonumber \\&=m^{-\varepsilon }\Vert \varphi \Vert _{L^\infty (Q_\varrho ^-(z_0))}|Q_\varrho ^-(z_0)|\cdot \frac{c(n)}{sp}d_R^{-sp}. \end{aligned}$$

(A.6)

Next, we estimate the integrand of $\textbf{IV}$:

$$\begin{aligned}&\Bigg |\left[ UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right] _h-UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\Bigg |^{\frac{p}{p-1}} \\&\quad \leqq c(p)\left( \left| \left[ UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right] _h\right| ^{\frac{p}{p-1}}+\left| UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right| ^{\frac{p}{p-1}}\right) . \end{aligned}$$

By the use of Hölder’s inequality and the assumption (2.3), we infer that

$$\begin{aligned} \left| \left[ UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right] _h\right|&=\left| \frac{1}{h}\int _0^te^{\frac{s-t}{h}}UK(x,y,s)|x-y|^{\frac{1}{p}(n+sp)}\,{\textrm{d}}s\right| \\&\leqq \Lambda \cdot \frac{1}{h}\int _0^te^{\frac{s-t}{h}}\frac{|U(x,y,s)|}{|x-y|^{\frac{p-1}{p}(n+sp)}}\,{\textrm{d}}s\\&\leqq \Lambda \left( \frac{1}{h}\int _0^te^{\frac{s-t}{h}}\,{\textrm{d}}s\right) ^{\frac{1}{p}}\left( \frac{1}{h}\int _0^te^{\frac{s-t}{h}}\frac{|U(x,y,s)|^{\frac{p}{p-1}}}{|x-y|^{n+sp}}\,{\textrm{d}}s\right) ^{\frac{p-1}{p}} \\&=\Lambda (1-e^{-\frac{t}{h}})^{\frac{1}{p}}\left[ \frac{|U(x,y,t)|^{\frac{p}{p-1}}}{|x-y|^{n+sp}}\right] _h^{\frac{p-1}{p}}, \end{aligned}$$

that is,

$$\begin{aligned} \left| \left[ UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right] _h\right| ^{\frac{p}{p-1}}\leqq \Lambda ^{\frac{p}{p-1}} (1-e^{-\frac{t}{h}})^{\frac{1}{p-1}}\left[ \frac{|U(x,y,t)|^{\frac{p}{p-1}}}{|x-y|^{n+sp}}\right] _h. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \left| UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right| ^{\frac{p}{p-1}}\leqq \Lambda ^{\frac{p}{p-1}}\frac{|U(x,y,t)|^{\frac{p}{p-1}}}{|x-y|^{n+sp}}. \end{aligned}$$

Collecting the preceding estimates above, the integrand of $\textbf{IV}$ is estimated as

$$\begin{aligned}&\Bigg |\left[ UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\right] _h-UK(x,y,t)|x-y|^{\frac{1}{p}(n+sp)}\Bigg |^{\frac{p}{p-1}} \\&\quad \leqq c\Lambda ^{\frac{p}{p-1}} \left( \left[ \frac{|U(x,y,t)|^{\frac{p}{p-1}}}{|x-y|^{n+sp}}\right] _h+\frac{|U(x,y,t)|^{\frac{p}{p-1}}}{|x-y|^{n+sp}}\right) \\&\quad =c\Lambda ^{\frac{p}{p-1}} \left( \left[ \frac{\big |u(x,t)-u(y,t)\big |^{p}}{|x-y|^{n+sp}}\right] _h+\frac{\big |u(x,t)-u(y,t)\big |^{p}}{|x-y|^{n+sp}}\right) \\&\quad =:c\Lambda ^{\frac{p}{p-1}}W_h(x,y,t). \end{aligned}$$

Since $W_h \in L^1(\Omega \times (B_R \setminus \Omega )\times (0,T))$ holds true, Lemma 2.8-(i) applied with $E=\Omega \times (B_R \setminus \Omega )$ yields that

$$\begin{aligned} W_h \longrightarrow W:=2\frac{\big |u(x,t)-u(y,t)\big |^{p}}{|x-y|^{n+sp}}\quad \text {in}\,\,\,L^1\left( \Omega \times (B_R\setminus \Omega )\times (0,T)\right) \end{aligned}$$

as $h\searrow 0$. Therefore using the dominated convergence theorem, we conclude that

$$\begin{aligned} \lim \limits _{h\searrow 0}\textbf{IV}=0. \end{aligned}$$

(A.7)

Merging (A.6) with (A.7) in (A.5) and, subsequently, sending $R \nearrow \infty$, we finally arrive at the resulting convergence (A.3).

Step 3: Taking the limit as $h\searrow 0$ and $\delta \searrow 0$.   Since by $u \geqq m>0$ in ${\mathbb {R}}^n \times (t_0-\varrho ^p,t_0)$, we have

$$\begin{aligned} \lim _{h\searrow 0}\int _\Omega |u|^{p-2}u(0)\left( \frac{1}{h}\int _0^Te^{-\frac{s}{h}}\phi (x,s)\,{\textrm{d}}s\right) \,{\textrm{d}}x=0 \end{aligned}$$

with $\phi =\varphi ^p\psi _\delta (t)u^{-\varepsilon }$. Therefore, combining this with the observations (A.1)–(A.3) and passing to the limit as $h\searrow 0$ in the weak formulation (2.10) with the testing function $\varphi ^p\psi _\delta (t)u^{-\varepsilon }$, we gain

Passing to the limit $\delta \searrow 0$ combined with Lebegue’s differential theorem and the dominated convergence theorem implies that

$$\begin{aligned}&\int _{B_\varrho (x_0)\times \{t_1\}}u^{p-1-\varepsilon }\varphi ^p\,{\textrm{d}}x+\int _{t_1}^{t_0}\int _{B_\varrho (x_0)}|Du|^pu^{-\varepsilon -1}\varphi ^p\,{\textrm{d}}x{\textrm{d}}t\nonumber \\&\quad \leqq \left( p+\tfrac{2p(p-1)}{\varepsilon (p-1-\varepsilon )}\right) \int _{t_1}^{t_0}\int _{B_\varrho (x_0)}\varphi ^{p-1}|\varphi _t|u^{p-1-\varepsilon }\,{\textrm{d}}x{\textrm{d}}t\nonumber \\&\quad \quad +c\left( \tfrac{p-1-\varepsilon }{p-1}+\tfrac{2}{\varepsilon }\right) \int _{t_1}^{t_0}\int _{B_\varrho (x_0)}|D\varphi |^pu^{p-1-\varepsilon }\,{\textrm{d}}x{\textrm{d}}t\nonumber \\&\quad \quad +c\left( \tfrac{p-1-\varepsilon }{p-1}+\tfrac{2}{\varepsilon }\right) \int _{t_1}^{t_0}\iint _{{\mathbb {R}}^n\times {\mathbb {R}}^n}UK(x,y,t)\left( \varphi ^p(x,t)u(x,t)^{-\varepsilon }\nonumber \right. \\&\left. \qquad -\varphi ^p(y,t)u(y,t)^{-\varepsilon }\right) \,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t. \end{aligned}$$

(A.8)

Since $\frac{p-1-\varepsilon }{p-1}<1<p$ and $\frac{p-1}{p-1-\varepsilon }>1$ the constant

$$\begin{aligned} C(\varepsilon ,p):=c\left( p+\frac{2p(p-1)}{\varepsilon (p-1-\varepsilon )}\right) \end{aligned}$$

bounds the all constants appearing on the right-hand side of (A.8), and we know that $C(\varepsilon ,p)$ blows up as $\varepsilon \searrow 0$ or $\varepsilon \nearrow p-1$. In the first term on the left-hand side of (A.8) we take the supremum over $t_1 \in (t_0-\varrho ^p, t_0)$, while in the others we let $t_1\searrow t_0-\varrho ^p$. This finally leads to

$$\begin{aligned}&\sup _{t_1 \in (t_0-\varrho ^p,t_0)}\int _{B_\varrho (x_0)\times \{t_1\}}u^{p-1-\varepsilon }\varphi ^p\,{\textrm{d}}x+\iint _{Q_\varrho ^-(z_0)}|Du|^pu^{-\varepsilon -1}\varphi ^p\,{\textrm{d}}x{\textrm{d}}t\nonumber \\&\quad \leqq C\iint _{Q_\varrho ^-(z_0)}\varphi ^{p-1}|\varphi _t|u^{p-1-\varepsilon }\,{\textrm{d}}x{\textrm{d}}t+C\iint _{Q_\varrho ^-(z_0)}|D\varphi |^pu^{p-1-\varepsilon }\,{\textrm{d}}x{\textrm{d}}t\nonumber \\&\quad +C\int _{t_0-\varrho ^p}^{t_0}\iint _{{\mathbb {R}}^n\times {\mathbb {R}}^n}U(x,y,t)K(x,y,t)\left( \varphi ^p(x,t)u(x,t)^{-\varepsilon }-\varphi ^p(y,t)u(y,t)^{-\varepsilon }\right) \,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t. \end{aligned}$$

(A.9)

Step 4: Conclusion.   In this final step, we are ready to conclude the whole proof, again by estimating the fractional term appearing on the right-hand side of (A.9). For this, we estimate separately as follows:

$$\begin{aligned} \textbf{V}&:=\int _{t_0-\varrho ^p}^{t_0}\iint _{{\mathbb {R}}^n\times {\mathbb {R}}^n}U(x,y,t)K(x,y,t)\left( \varphi ^p(x,t)u(x,t)^{-\varepsilon }-\varphi ^p(y,t)u(y,t)^{-\varepsilon }\right) \,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&=\int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times B_\varrho (x_0) }(\cdots )\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t+2\int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times ({\mathbb {R}}^n \setminus B_\varrho (x_0) )}(\cdots )\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&=:\textbf{V}_1+2\textbf{V}_2, \end{aligned}$$

with the obvious meaning of $\textbf{V}_1$ and $\textbf{V}_2$. Applying Lemma 2.4 with $a=u(y,t)$, $\tau _1=\varphi (y,t)$ and $b=u(y,t)$, $\tau _2=\varphi (x,t)$, the integrand of $\textbf{V}_1$ is estimated as

$$\begin{aligned}&U(x,y,t)K(x,y,t)\left( \varphi ^p(x,t)u(x,t)^{-\varepsilon }-\varphi ^p(y,t)u(y,t)^{-\varepsilon }\right) \\&\quad \leqq -\Lambda c(p) \zeta (\varepsilon )\frac{\Big |\varphi (x,t)u(x,t)^{\frac{\alpha }{p}}-\varphi (y,t)u(y,t)^{\frac{\alpha }{p}}\Big |^p}{|x-y|^{n+sp}} \\&\quad \quad +\Lambda \left( \zeta (\varepsilon )+1+\varepsilon ^{-(p-1)}\right) \big |\varphi (x,t)-\varphi (y,t)\big |^p\cdot \frac{u(x,t)^\alpha +u(y,t)^\alpha }{|x-y|^{n+sp}} \end{aligned}$$

and therefore

$$\begin{aligned} \textbf{V}_1&\leqq -\Lambda c(p) \zeta (\varepsilon )\int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times B_\varrho (x_0)}\frac{\Big |\varphi (x,t)u(x,t)^{\frac{\alpha }{p}}-\varphi (y,t)u(y,t)^{\frac{\alpha }{p}}\Big |^p}{|x-y|^{n+sp}} \,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&\quad +\Lambda \left( \zeta (\varepsilon )+1+\varepsilon ^{-(p-1)}\right) \\&\quad \int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times B_\varrho (x_0)}\frac{\big |\varphi (x,t)-\varphi (y,t)\big |^p\left( u(x,t)^\alpha +u(y,t)^\alpha \right) }{|x-y|^{n+sp}}\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t. \end{aligned}$$

On the other hand, using the positivity of u, the term $\textbf{V}_2$ is estimated as

$$\begin{aligned} \textbf{V}_2&\leqq \Lambda \int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times ({\mathbb {R}}^n \setminus B_\varrho (x_0))}\frac{U(x,y,t)}{|x-y|^{n+sp}}\varphi ^p(x,t)u(x,t)^{-\varepsilon }\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&=\Lambda \int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times ({\mathbb {R}}^n \setminus B_\varrho (x_0)) \cap \{u(x,t) \,\geqq \,u(y,t)\}}(\cdots )\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&\quad +\Lambda \underbrace{\int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times ({\mathbb {R}}^n \setminus B_\varrho (x_0)) \cap \{u(x,t) \,<\,u(y,t)\}}(\cdots )\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t}_{<0} \\&\leqq \Lambda \int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times ({\mathbb {R}}^n \setminus B_\varrho (x_0)) \cap \{u(x,t) \,\geqq \,u(y,t)\}}\frac{(u(x,t)-u(y,t))^{p-1}}{|x-y|^{n+sp}}\varphi ^p(x,t)u(x,t)^{-\varepsilon }\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&\leqq \Lambda \int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times ({\mathbb {R}}^n \setminus B_\varrho (x_0)) \cap \{u(x,t) \,\geqq \,u(y,t)\}}\frac{u(x,t)^{p-1-\varepsilon }}{|x-y|^{n+sp}}\varphi ^p(x,t)\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&\leqq \Lambda \left( \sup \limits _{x\in \,{\textrm{supp}}\,\varphi (\cdot ,t)}\int _{{\mathbb {R}}^n \setminus B_\varrho (x_0)}\frac{\,{\textrm{d}}y}{|x-y|^{n+sp}}\right) \iint _{Q_\varrho ^-(z_0)}u(x,t)^\alpha \varphi ^p(x,t)\,{\textrm{d}}x{\textrm{d}}t. \end{aligned}$$

Joining the preceding estimates for $\textbf{V}_1$ and $\textbf{V}_2$ we get

$$\begin{aligned} \textbf{V}&\leqq -\Lambda c(p) \zeta (\varepsilon )\int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times B_\varrho (x_0)}\frac{\Big |\varphi (x,t)u(x,t)^{\frac{\alpha }{p}}-\varphi (y,t)u(y,t)^{\frac{\alpha }{p}}\Big |^p}{|x-y|^{n+sp}} \,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&\quad +\Lambda \left( \zeta (\varepsilon )+1+\varepsilon ^{-(p-1)}\right) \int _{t_0-\varrho ^p}^{t_0}\iint _{B_\varrho (x_0) \times B_\varrho (x_0)}\frac{\big |\varphi (x,t)-\varphi (y,t)\big |^p\left( u(x,t)^\alpha +u(y,t)^\alpha \right) }{|x-y|^{n+sp}}\,{\textrm{d}}x{\textrm{d}}y{\textrm{d}}t\\&\quad +2\Lambda \left( \sup \limits _{x\in \,{\textrm{supp}}\,\varphi (\cdot ,t)}\int _{{\mathbb {R}}^n \setminus B_\varrho (x_0)}\frac{\,{\textrm{d}}y}{|x-y|^{n+sp}}\right) \iint _{Q_\varrho ^-(z_0)}u(x,t)^\alpha \varphi ^p(x,t)\,{\textrm{d}}x{\textrm{d}}t. \end{aligned}$$

Inserting this estimate back to (A.9), we finally arrive at the desired estimate and therefore, the lemma is completely proved. $\square$

Appendix B: Proof of Lemmata 3.5 and 3.9

In this final appendix, we report the proof of Lemmata 3.5 and 3.9.

Proof of Lemma 3.5

Since by $u \geqq m >0$ in ${\mathbb {R}}^n \times (t_1,t_2)$ there holds that, for all $t \in (t_1,t_2)$

$$\begin{aligned} {[}u^{p-1}]_h= \frac{1}{h}\int _{0}^{t}e^{\frac{s-t}{h}}u(s)^{p-1}\,{\textrm{d}}s&\geqq m^{p-1}\left( \frac{1}{h}\int _0^te^{\frac{s-t}{h}}\,{\textrm{d}}s\right) \nonumber \\&=m^{p-1}(1-e^{-\frac{t}{h}}). \end{aligned}$$

(B.1)

This together with an elementary estimate $|\log s| \leqq \dfrac{|s-1|}{\min \{s,1\}}$ for $s>0$ implies that

$$\begin{aligned} \Big |\log [u^{p-1}]_h-\log u^{p-1}\Big |=\left| \log \left( \frac{[u^{p-1}]_h}{u^{p-1}}\right) \right|&\leqq \frac{\big |[u^{p-1}]_h-u^{p-1}\big |}{\min \{[u^{p-1}]_h,\,u^{p-1}\}} \\&\leqq \frac{1}{m^{2(p-1)}(1-e^{-\frac{t}{h}}) }\big |[u^{p-1}]_h-u^{p-1}\big | \end{aligned}$$

for every $(x,t) \in \Omega _{t_1,t_2}$. Therefore, Lemma 2.8-(ii) concludes that

$$\begin{aligned}{} & {} \Big \Vert \log [u^{p-1}]_h-\log u^{p-1}\Big \Vert _{L^{\frac{p}{p-1}}(\Omega _{t_1,t_2})} \\{} & {} \quad \leqq \frac{1}{m^{2(p-1)}(1-e^{-\frac{t_1}{h}})}\Big \Vert [u^{p-1}]_h-u^{p-1}\Big \Vert _{L^{\frac{p}{p-1}}(\Omega _{t_1,t_2})} \longrightarrow 0, \end{aligned}$$

as desired. $\square$

Proof of Lemma 3.9

Again, by (B.1)

$$\begin{aligned} \big |[u^{p-1}]_h^{-1}-u^{1-p}\big | \leqq \frac{1}{m^{2(p-1)}(1-e^{-\frac{t_1}{h}})}\big |[u^{p-1}]_h-u^{p-1}\big | \end{aligned}$$

holds whenever $(x,t) \in \Omega _{t_1,T}$. Hence

$$\begin{aligned} \Big \Vert [u^{p-1}]_h^{-1}-u^{1-p}\Big \Vert _{L^{\frac{p}{p-1}}(\Omega _{t_1,T})}&\leqq \frac{1}{m^{2(p-1)}(1-e^{-\frac{t_1}{h}})}\Big \Vert [u^{p-1}]_h-u^{p-1}\Big \Vert _{L^{\frac{p}{p-1}}(\Omega _{t_1,T})} \longrightarrow 0, \end{aligned}$$

finishing the proof. $\square$