On the continuity of solutions to doubly singular parabolic equations

TL;DR

This paper proves local continuity of bounded weak solutions to a class of doubly singular parabolic equations with a time derivative singularity, especially as the parameter p approaches 2, ensuring stability of solutions.

Contribution

It establishes the local continuity of solutions for a range of p values close to 2, extending understanding of regularity in doubly singular parabolic equations.

Findings

Solutions are locally continuous for 2−ε₀ ≤ p < 2.

Continuity is stable as p approaches 2.

Results apply to equations with singularities in the time derivative.

Abstract

This paper considers a certain doubly singular parabolic equations with one singularity occurs in the time derivative, whose model is \begin{equation*} \partial_t\beta(u)-\operatorname{div}|Du|^{p-2}Du\ni0,\qquad \text{in}\quad \Omega\times(0,T)\end{equation*} where $\Omega\subset\mathbb{R}^N$ and $N\geq3$. We show that the bounded weak solutions are locally continuous in the range $$2-\epsilon_0\leq p<2,$$ provided $\epsilon_0>0$ is small enough, and the continuity is stable as $p\to2$.

Equations

$${\partial }_t\beta (u)-\mathrm{div}|Du{|}^{p-2}Du\ni 0,\text{in}\Omega \times (0,T)$$

$$2-{\epsilon }_0\le p<2,$$

$${\partial }_t\beta (u)-\mathrm{div}A(x,t,u,Du)\ni 0\text{in}{\Omega }_T.$$

$$\beta (r)=\{\begin{array}{ll}r,& r>0,\\ [-\bar{\nu },0],& r=0,\\ r-\bar{\nu },& r<0,\end{array}$$

$$\{\begin{array}{l}A(x,t,u,\zeta )\cdot \zeta \ge C_0|\zeta {|}^p,\\ |A(x,t,u,\zeta )|\le C_1|\zeta {|}^{p-1},\end{array}$$

$$[(\bar{x},\bar{t})+Q(d{R}^p,d_{\ast }R)],d={\left(\frac{\omega }{2}\right)}^{(1-p)(2-p)}\text{and}{d}_{\ast }={\left(\frac{\omega }{2^{n_{\ast }+1}}\right)}^{p-2}$$

$$[(\bar{x},\bar{t})+Q(d{R}^p,d_{\ast }R)],d={\left(\frac{\omega }{2^{m_1}}\right)}^{(1-p)(2-p)}\text{and}{d}_{\ast }={\left(\frac{\omega }{2^{m_2}}\right)}^{p-2}\text{where}{m}_1=\frac{p{m}_2}{p-1}$$

$$K_r=\{x=(x_1,x_2,\cdots ,x_N)\in {\mathbb{R}}^N:|x_i|\le r,i=1,2,\cdots ,N\}.$$

$$\zeta (x)=\prod _{i=1}^N{\zeta }_i(x_i)$$

$${\zeta }_i(x_i)=\{\begin{array}{ll}0,& \text{if}|x_i|>r^{\prime },\\ \frac{x_i+r^{\prime }}{r^{\prime }-r},& \text{if}-r^{\prime }<x_i<-r,\\ 1,& \text{if}|x_i|\le r,\\ \frac{r^{\prime }-x_i}{r^{\prime }-r},& \text{if}r<x_i<r^{\prime }.\end{array}$$

$$\begin{array}{rl}(v-k{)}_{+}& =\max \{v-k,0\}\\ (v-k{)}_{-}& =\max \{k-v,0\}.\end{array}$$

$$\Vert v{\Vert }_{L^q(0,T;B)}={\left({\int }_0^T\Vert v(t){\Vert }_B^{q}dt\right)}^{\frac{1}{q}}.$$

$$v\in L_{\mathrm{loc}}^p(0,T;W_0^{1,p}(\Omega ))\text{and}{\partial }_{t}v\in L_{\mathrm{loc}}^2({\Omega }_T).$$

$$u\in C_{\mathrm{loc}}(0,T;L_{\mathrm{loc}}^2(\Omega ))\cap L_{\mathrm{loc}}^p(0,T;W_{\mathrm{loc}}^{1,p}(\Omega ))$$

$$\int_{\Omega}w(\cdot,t)\varphi(\cdot,t)dx\bigg{|}_{t=t_{0}}^{t_{1}}+\int_{t_{0}}^{t_{1}}\int_{\Omega}\left[-w\partial_{t}\varphi+A(x,t,u,Du)\cdot D\varphi\right]dxdt=0$$

$$\underset{{\Omega }_T}{\mathrm{esssup}}|u|\le 1,$$

$${\partial }_{t}u\in L_{\mathrm{loc}}^2({\Omega }_T).$$

$${\left({\int }_{\Omega }\phi |v{|}^{p}dx\right)}^{\frac{1}{p}}\le \gamma \frac{(\mathrm{diam}\Omega {)}^N}{|[v=0]\cap [\phi =1]{|}^{\frac{N-1}{N}}}{\left({\int }_{\Omega }\phi |Dv{|}^{p}dx\right)}^{\frac{1}{p}}.$$

$${\iint }_{{\Omega }_T}|v{|}^{p\frac{N+2}{N}}dxdt\le \gamma \left({\iint }_{{\Omega }_T}|Dv{|}^{p}dxdt\right){\left(\underset{0<t<T}{\mathrm{esssup}}{\int }_{\Omega }|v{|}^{2}dx\right)}^{\frac{p}{N}}.$$

$${\Vert v(\cdot ,t)\Vert }_{L^{\frac{Np}{N-p}}(\Omega )}\le \frac{(N-1)p}{N-p}\Vert Dv(\cdot ,t){\Vert }_{L^p(\Omega )}\le 2(N-1)\Vert Dv(\cdot ,t){\Vert }_{L^p(\Omega )},$$

$$\begin{array}{rl}{\iint }_{{\Omega }_T}|v{|}^{p\frac{N+2}{N}}dxdt& \le {\int }_0^T{\Vert v(\cdot ,t)\Vert }_{L^{\frac{Np}{N-p}}(\Omega )}^p\Vert v(\cdot ,t){\Vert }_{L^2(\Omega )}^{\frac{2p}{N}}dt\\ & \le 4(N-1{)}^2\left({\iint }_{{\Omega }_T}|Dv{|}^{p}dxdt\right){\left(\underset{0<t<T}{\mathrm{esssup}}{\int }_{\Omega }|v{|}^{2}dx\right)}^{\frac{p}{N}},\end{array}$$

$$v(x,t)=\{\begin{array}{ll}\bar{\nu },& \text{on}[u<0],\\ -w(x,t),& \text{on}[u=0],\end{array}$$

$$\begin{split}-\int_{\Omega}v(\cdot,t)\chi_{[u\leq 0]}&\varphi(\cdot,t)dx\bigg{|}_{t=t_{0}}^{t_{1}}+\int_{t_{0}}^{t_{1}}\int_{\Omega}v\chi_{[u\leq 0]}\partial_{t}\varphi dxdt\\ &+\int_{t_{0}}^{t_{1}}\int_{\Omega}\left[\varphi\partial_{t}u+A(x,t,u,Du)\cdot D\varphi\right]dxdt=0\end{split}$$

$$U(\Omega,t_{0},t_{1},\varphi)=\int_{\Omega}v(\cdot,t)\chi_{[u\leq 0]}\varphi(\cdot,t)dx\bigg{|}_{t=t_{0}}^{t_{1}}-\int_{t_{0}}^{t_{1}}\int_{\Omega}v\chi_{[u\leq 0]}\partial_{t}\varphi dxdt.$$

$$0\le \zeta \le 1,|D\zeta |<\infty \text{and}\zeta =0\text{ on}\partial [\bar{x}+K].$$

$$\begin{array}{rl}\underset{t_0<t<t_1}{\mathrm{esssup}}& {\int }_{[\bar{x}+K]\times \{t\}}(u-k{)}_{\pm }^2{\zeta }^{p}dx+{\int }_{t_0}^{t_1}{\int }_{[\bar{x}+K]}|D(u-k{)}_{\pm }\zeta {|}^{p}dxdt\\ \le & {\int }_{[\bar{x}+K]\times \{t_0\}}(u-k{)}_{\pm }^2{\zeta }^{p}dx+{\gamma }_1{\int }_{t_0}^{t_1}{\int }_{[\bar{x}+K]}(u-k{)}_{\pm }^p|D\zeta {|}^{p}dxdt\\ & +{\gamma }_1{\int }_{t_0}^{t_1}{\int }_{[\bar{x}+K]}(u-k{)}_{\pm }^2|{\partial }_t\zeta |dxdt+U([\bar{x}+K],t_0,t_1,\pm (u-k{)}_{\pm }{\zeta }^p).\end{array}$$

$${\psi }^{\pm }(u)=\max \left\{\ln \frac{H_k^{\pm }}{H_k^{\pm }-(u-k{)}_{\pm }+c};0\right\},0<c<H_k^{\pm }$$

$$H_k^{\pm }\ge \underset{[\bar{x}+K]\times [t_0,t_1]}{\mathrm{esssup}}(u-k{)}_{\pm }.$$

$$\begin{array}{rl}\underset{t_0<t<t_1}{\mathrm{esssup}}& {\int }_{[\bar{x}+K]\times \{t\}}{\left({\psi }^{\pm }\right)}^2{\zeta }^{p}dx\le {\int }_{[\bar{x}+K]\times \{t_0\}}{\left({\psi }^{\pm }\right)}^2{\zeta }^{p}dx\\ & +{\gamma }_2{\int }_{t_0}^{t_1}{\int }_{[\bar{x}+K]}{\psi }^{\pm }|{\left({\psi }^{\pm }\right)}^{\prime }{|}^{2-p}|D\zeta {|}^{p}dxdt+U([\bar{x}+K],t_0,t_1,2{\psi }^{\pm }{\left({\psi }^{\pm }\right)}^{\prime }{\zeta }^p),\end{array}$$

$${\mu }_{-}=\underset{\overline{Q}}{\mathrm{essinf}}u,{\mu }_{+}=\underset{\overline{Q}}{\mathrm{esssup}}u\text{and}\underset{\overline{Q}}{\mathrm{essosc}}u={\mu }_{+}-{\mu }_{-}.$$

Full text

[TABLE]

where $\Omega\subset\mathbb{R}^{N}$ and $N\geq 3$. We show that the bounded weak solutions are locally continuous in the range

[TABLE]

provided $\varepsilon_{0}>0$ is small enough, and the continuity is stable as $p\to 2$.

1. Introduction

The aim of this paper is to establish a continuity result for solutions to singular parabolic equations related to the two phase Stefan problems of the type

[TABLE]

Here, the set $\Omega_{T}$ denotes the cylinder $\Omega\times(0,T)$ over an open bounded domain $\Omega\subset\mathbb{R}^{N}$ with dimension $N\geq 3$, and the function $\beta(\cdot)$ is defined by

[TABLE]

where $\bar{\nu}>0$ is a given constant. The vector field $A(x,t,u,\zeta)$ is measurable in $\Omega_{T}\times\mathbb{R}\times\mathbb{R}^{N}$ and satisfy the structure conditions:

[TABLE]

where $C_{0}$ and $C_{1}$ are given positive constants. No attempt has been made here to discuss the equations with lower order terms. A motivation for this study comes from phase transitions for the material obeying a Fourier’s law (see [7, 8, 9]). The continuity of weak solutions for the case $A(x,t,u,Du)=Du$ was first proved by Caffarelli and Evans [2]. This result was extended by DiBenedetto [4] to the quasilinear equations with general structure. The proof is based on the De Giorgi’s technique and a construction of logarithmic functions to prove the expansion of positivity in time direction. Furthermore, Urbano [10] established the continuity result for the degenerate case $p\geq 2$ by using the De Giorgi’s technique together with the intrinsic scaling method. This idea goes back at least as far as [4].

The treatment of the singular case $p<2$ is much more difficult. This problem was studied by Henriques and Urbano [6]. They introduced the intrinsic parabolic cylinders of the form

[TABLE]

and use this kind of cylinders to perform alternative arguments. However, we point out that there is a gap in the proof of Lemma 1 ([6, page 930, line 8]). The integral involving the term $|\nabla(\bar{\theta}-\bar{k}_{n})_{-}\bar{\xi}_{n}|^{p}$ must be multiplied by a factor $c_{0}^{-p}$ due to the chain rule. If we multiply such factor, the constant $\nu_{0}$ in [6, page 927] will depend upon $2^{-n_{*}}$ which is so small that we cannot determine the number $n_{*}$ in terms of $\nu_{0}$ (see [6, Lemma 4, page 938, line -1]). Of course, this is not meant to diminish the importance of Henriques and Urbano’s pioneering work, but rather to point out that it does not resolve the problem and a correct proof is also needed.

In this work, we try to find an idea to overcome the difficulty mentioned above. We shall use the following intrinsic parabolic cylinders

[TABLE]

for the alternative arguments. Since the length of the time interval is too large, we cannot simply use the idea from [6, Lemma 4]) to determine the quantity $m_{2}$ and the analysis of the second alternative is much more delicate. In order to obtain a result of expansion positivity in such large time interval, we shall apply an iterative argument. However, our approach only works for the range $2-\varepsilon_{0}\leq p<2,$ where $\varepsilon_{0}$ is a small positive constant depending only upon the data. We have to impose this artificial condition, because the integrals involving the quantity $[\psi_{\pm}^{\prime}]^{2-p}$, where $\psi_{\pm}$ is a logarithmic function, are too large in the singular range. Moreover, we can also show that the continuity is stable as $p\to 2$. As for the proof of the full range $1<p<2$, we leave this problem open to the further study.

An outline of this paper is as follows. We provide some preliminary material and state the main result in §2, while §3 provides an exposition of Caccioppoli estimates. In §4, we introduce the intrinsic parabolic cylinders for the alternative arguments and get started the proof of the main result. §5 is devoted to the analysis of the first alternative. We obtain a decay estimate for the oscillation of weak solution in a smaller cylinder. Subsequently, §6 is intended to prove a similar estimate in the second alternative. Finally, in §7, we finish the proof of the main result by a recursive argument.

2. Preliminary material and main result

We follow the notation of [5]. Throughout the paper, $\Omega$ denote a bounded open set in $\mathbb{R}^{N}$ with $N\geq 3$ and $\Omega_{T}=\Omega\times(0,T)$ is the associated space-time cylinder. We let $Du=(D_{x_{1}}u,D_{x_{2}}u,\cdots,D_{x_{N}}u)$ stand for the differentiation with respect to the space variables and let $\partial_{t}u$ denote the time derivative. The derivatives are teken in the weak sense. Points in $\mathbb{R}^{N+1}$ will be denoted by $(x,t)$, where $x\in\mathbb{R}^{N}$ and $t\in\mathbb{R}$. We shall use cubes of the form

[TABLE]

Concerning parabolic cylinders, we use the cylinders of the form $Q(r_{1},r_{2})=K_{r_{2}}\times(-r_{1},0)$. Let $\partial K_{r_{2}}$ be the boundary of $K_{r_{2}}$ and $\partial_{P}Q(r_{1},r_{2})=[\partial K_{r_{2}}\times(-r_{1},0)]\cup[K_{r_{2}}\times\{0\}]$ denotes the parabolic boundary of $Q(r_{1},r_{2})$. Let $r^{\prime}>r>0$, we construct the piecewise smooth function, with $\zeta\equiv 1$ in $K_{r}$, $\operatorname{supp}\zeta\subset K_{r^{\prime}}$ and $|D\zeta|\leq(r^{\prime}-r)^{-1}$, by

[TABLE]

where $x=(x_{1},x_{2},\cdots,x_{N})$ and

[TABLE]

We set $W^{1,p}(\Omega)$ for the Sobolev space of weakly differentiable functions $v:\Omega\to\mathbb{R}$ with $|v|,|Dv|\in L^{p}(\Omega)$. Similarly, a function $v\in W_{\operatorname*{loc}}^{1,p}(\Omega)$ if $v\in W^{1,p}(K)$ for all compact set $K\subset\Omega$. We denote by $W_{0}^{1,p}(\Omega)$ the closure of $C_{0}^{\infty}(\Omega)$ with respect to the norm in $W^{1,p}(\Omega)$. For any $k\in\mathbb{R}$ and a function $v\in W^{1,p}(\Omega)$, the truncations are defined by

[TABLE]

Let $B$ be a Banach space and $1\leq q<\infty$. We write $L^{q}(0,T,B)$ for the space of $L^{q}$ integrable functions from $[0,T]$ into $B$, which is a Banach space with the norm

[TABLE]

Moreover, we write $L_{\operatorname*{loc}}^{q}(0,T,B)$ for space of functions $v\in L^{q}(t_{1},t_{2},B)$ for all intervals $[t_{1},t_{2}]\subset(0,T]$. Let us denote by $C_{\operatorname*{loc}}(0,T;B)$ the space of locally continuous functions from $[0,T]$ into $B$. Finally, we denote by $W_{p}(\Omega_{T})$ the set of functions

[TABLE]

We are now ready to give the definition of the weak solution to the doubly singular parabolic equations (1.1)-(1.3):

Definition 2.1**.**

A function $u:\Omega_{T}\to\mathbb{R}$ is said to be a local weak solution of (1.1)-(1.3), if

[TABLE]

and there exists a function $w\subset\beta(u)$, with the inclusion being intended in the sense of the graphs, such that the identity

[TABLE]

holds for all testing functions $\varphi\in W_{p}(\Omega_{T})$ and all intervals $[t_{0},t_{1}]\subset(0,T]$.

Throughout the paper, we assume that the weak solution is bounded and its time derivative is locally square integrable. To be more precise, we assume that

[TABLE]

and the time derivative $\partial_{t}u$ exists and satisfies

[TABLE]

The statement that a constant $C$ depends only upon the data, means that it can be determined a priori only in terms of $\{\bar{\nu},\ N,\ C_{0},\ C_{1}\}$. Since we are concerned about the behaviour of constants as $p\to 2$, exponent $p$ should be excluded from the concept of the data. We can now state our main result:

Theorem 2.2**.**

There exists a constant $\varepsilon_{0}>0$, that can be determined a priori only in terms of the data, such that the following holds: Let $u$ be a weak solution to the singular parabolic equation (1.1)-(1.3) with $p\in[2-\varepsilon_{0},2)$. Suppose that the assumptions (2.4) and (2.5) are in force. Then the weak solution $u$ is locally continuous and the continuity is stable as $p\to 2$.

Finally, we collect here two results regarding Sobolev inequalities that will be of use later.

Lemma 2.3**.**

[5, Chapter I, Proposition 2.1]** Let $\Omega\subset\mathbb{R}^{N}$ be a bounded convex set and let $\varphi\in C(\Omega)$ satisfy $0\leq\varphi\leq 1$ and the sets $[\varphi\geq k]$ are convex for all $k\in(0,1)$. Let $v\in W^{1,p}(\Omega)$ with $1<p\leq 2$ and assume that $|[v=0]\cap[\varphi=1]|>0$. Then there exists a constant $\gamma$ depending only upon $N$, such that

[TABLE]

Lemma 2.4**.**

[5, Chapter I, Proposition 3.1]** Let $\Omega\subset\mathbb{R}^{N}$ be a bounded set and $N\geq 3$. Assume that $v\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{p}(0,T;W_{0}^{1,p}(\Omega))$ and $1<p\leq 2$. Then there exists a constant $\gamma$ depending only upon $N$ such that

[TABLE]

Proof.

We only need to check that the constant $\gamma$ is independent of $p$. For any fixed $t\in(0,T)$, we apply [7, Page 64 (2.13)] to obtain

[TABLE]

since $N\geq 3$ and $1<p\leq 2$. Based on the argument of DiBenedetto [5, page 8], we obtain

[TABLE]

where the constant on the right-hand side is independent of $p$. ∎

3. Local energy estimates

In this section we state two energy estimates for the cutoff functions $(u-k)_{\pm}$ where $k\in\mathbb{R}$ is a fixed number. From the assumption (2.5), the time derivative exists in the weak sense and square integrable. This enables us to proceed similarly to the discussions in [4, page 133]. To this end, we introduce the auxiliary function

[TABLE]

and the weak form (2.3) can be rewritten as

[TABLE]

where $\varphi\in W_{p}(\Omega_{T})$ and $[t_{0},t_{1}]\subset(0,T]$. The appearance of the first two terms in (3.1) is due to the singularity in time derivative and this can cause an extra difficulty in the proof. From now on, we denote the first two terms in (3.1) by the quantity

[TABLE]

Let $\bar{x}\in\Omega$ and $K$ be a cube centered at origin such that $[\bar{x}+K]\subset\Omega$. We denote by $\zeta$ a piecewise smooth function in $[\bar{x}+K]\times[t_{0},t_{1}]$ such that

[TABLE]

By substituting $\varphi=\pm(u-k)_{\pm}\zeta^{p}$ into (3.1), we obtain the following proposition, whose proof we omit (see [5, Chapter II, Proposition 3.1]).

Proposition 3.1**.**

Let $u$ be a weak solution of (1.1)-(1.3) in $\Omega_{T}$. There exists a constant $\gamma_{1}$ that can be determined a priori only in terms of the data such that

[TABLE]

We now turn to consider the local logarithmic estimates for weak solutions. To this end, we introduce the logarithmic function

[TABLE]

where $H_{k}^{\pm}$ is a constant chosen such that

[TABLE]

For simplicity of notation, we write $\psi^{\pm}$ instead of $\psi^{\pm}(u)$. We let $(\psi^{\pm})^{\prime}$ stand for the derivative of $\psi^{\pm}(u)$ with respect to $u$. If we plug $\varphi=2\psi^{\pm}\left(\psi^{\pm}\right)^{\prime}\zeta^{p}$ into (3.1), we obtain the following proposition.

Proposition 3.2**.**

Let $u$ be a weak solution of (1.1)-(1.3) in $\Omega_{T}$. There exists a constant $\gamma_{2}$ that can be determined a priori only in terms of the data such that

[TABLE]

where $\zeta=\zeta(x)$ is independent of $t\in[t_{0},t_{1}]$ and satisfies (3.2).

This is a standard result which can be found in [5, Chapter II, Proposition 3.2] and no proof will be given here. Finally, we remark that our approach does not follow the idea in [6] and [10] which concerns the approximate solutions to the equations with regularization of maximal monotone graph. Instead, we follow the approach in [4], which is more convenient to deal with the second alternative.

4. The intrinsic geometry

The continuity of $u$ will be a consequence of the following assertion. For any point $(x_{0},t_{0})\in\Omega_{T}$ there exists a family of nested and shrinking cylinders $[(x_{0},t_{0})+Q_{n}]$ such that the oscillation of $u$ in $[(x_{0},t_{0})+Q_{n}]$ tends to zero as $n\to\infty$. To begin the proof, we introduce a certain intrinsic parabolic cylinder which plays an important role in alternative arguments.

Without loss of generality, we assume that $(x_{0},t_{0})=(0,0)$. Let $R<1$ be a fixed number such that $Q(R,R^{\frac{1}{p}})\subset\Omega_{T}$ and we set $\overline{Q}=Q(R,R^{\frac{1}{p}})$ as a reference parabolic cylinder. We write

[TABLE]

Here and subsequently, $\omega$ stands for a fixed number and satisfies $\omega>\operatorname*{ess\,osc}_{\overline{Q}}u$. For $(\bar{x},\bar{t})\in\overline{Q}$, we introduce the intrinsic parabolic cylinders of the form $(\bar{x},\bar{t})+Q(dR^{p},d_{*}R)$ where

[TABLE]

The quantities $m_{1}$ and $m_{2}$ depend upon $\omega$ and will be determined in §6.2. Moreover, the relation $m_{1}=pm_{2}/(m-1)$ implies

[TABLE]

Let $L_{1}>1$ and $L_{2}>1$ denote the fixed constants which will be determined in §5. We first assume that $Q(c_{1}R^{p},c_{2}R)\subset\overline{Q}$ where $c_{1}=L_{1}d$ and $c_{2}=L_{2}d_{*}$. In the case $Q(c_{1}R^{p},c_{2}R)\nsubseteq\overline{Q}$, we shall derive a decay estimate for $\omega$ in terms of $R$ and the discussion is postponed until §6.2.1. Define $K_{\omega}=K_{(L_{2}-1)d_{*}R}$ and $I_{\omega}=[-(L_{1}-1)dR^{p},0]$. It is easy to check that $K_{\frac{1}{2}c_{2}R}\subset K_{\omega}$.

The main ingredient of the proof is to establish an estimate for the essential oscillation of $u$ in a smaller cylinder $Q\left(d\left(\frac{R}{2}\right)^{p},d_{*}\left(\frac{R}{2}\right)\right)$,

[TABLE]

where $\sigma(\omega)<1$. We now assume that $\mu_{+}-\mu_{-}\geq\omega/2$, otherwise (4.3) follows immediately. Moreover, for technical reasons, we have to consider the two cases $\mu_{+}\geq|\mu_{-}|$ and $\mu_{+}<|\mu_{-}|$ separately.

Motivated by the work of Henriques and Urbano [6], we consider the four complementary cases described below. In the case $\mu_{+}\geq|\mu_{-}|$. For a constant $\nu_{0}\in(0,1)$, that will be determined in §5, we have two possible alternatives.

The first alternative. There exists $\bar{t}\in I(\omega)$ such that for all $\bar{x}\in K_{\omega}$,

[TABLE]

The second alternative. For any $\bar{t}\in I(\omega)$ there exists $\bar{x}\in K_{\omega}$ such that

[TABLE]

In the case $\mu_{+}<|\mu_{-}|$. For a constant $\nu_{0}\in(0,1)$, that can be determined by $\omega$, we introduce the following two alternatives.

The third alternative. There exists $\bar{t}\in I(\omega)$ such that for all $\bar{x}\in K_{\omega}$,

[TABLE]

The fourth alternative. For any $\bar{t}\in I(\omega)$ there exists $\bar{x}\in K_{\omega}$ such that

[TABLE]

For simplicity, we concentrate in the next two sections only the case $\mu_{+}\geq|\mu_{-}|$, since the treatment of the case $\mu_{+}<|\mu_{-}|$ is similar. The proof of (4.3) in the case $\mu_{+}<|\mu_{-}|$ is left to the reader.

5. The first alternative

The aim in this section is to establish the estimate (4.3) for the first alternative. To start with, we establish the following DeGiorgi type lemma and determine the constant $\nu_{0}$ in terms of data and $\omega$.

Lemma 5.1**.**

There exists a constant $\nu_{0}\in(0,1)$, depending only upon data and $\omega$, such that if (4.4) holds for some $(\bar{x},\bar{t})\in K_{\omega}\times I_{\omega}$ then

[TABLE]

Proof.

Without loss of generality we may assume $(\bar{x},\bar{t})=(0,0)$. Define two decreasing sequences of numbers

[TABLE]

and construct the family of nested and shrinking cylinders $Q_{n}=Q(dR_{n}^{p},d_{*}R_{n})$. We construct smooth cutoff functions $0\leq\zeta_{n}\leq 1$, such that $\zeta_{n}\equiv 1$ in $Q_{n+1}$, $\zeta_{n}\equiv 0$ on $\partial_{P}Q_{n}$, $|D\zeta_{n}|\leq 2^{n+2}/(d_{*}R)$ and $0<\partial_{t}\zeta_{n}\leq 2^{2+np}/(dR^{p})$. Write the energy estimate (3.3) over the cylinder $Q_{n}$ for the functions $(u-k_{n})_{-}$. We first observe from [4, page 145] that

[TABLE]

Since $(u-k_{n})_{-}\leq\frac{\omega}{4}<1$ and $\partial_{t}\zeta_{n}\geq 0$, we obtain

[TABLE]

where $\bar{\gamma}=(1+2\bar{\nu})\gamma_{1}$. Moreover, the relation (4.2) implies

[TABLE]

since $\omega<1$. Then we conclude that

[TABLE]

Applying parabolic Sobolev’s inequality (2.7), we deduce

[TABLE]

At this point, we set

[TABLE]

The left-hand side of (5.2) is estimated below by

[TABLE]

Combining (5.3) with (5.2) and noting that $|Q_{n}|=dd_{*}^{N}R_{n}^{N+p}$, we obtain

[TABLE]

where we used (4.2) in the last step. Next, we set

[TABLE]

We observe that if $Y_{0}\leq\nu_{0}$, then

[TABLE]

Using a lemma on fast geometric convergence of sequences (cf. [5, Chapter I, Lemma 4.1]), we conclude that $Y_{n}\to 0$ as $n\to\infty$. This completes the proof of Lemma 5.1. ∎

From the definition of the first alternative, we can extend the estimate (5.1) to a larger domain by using a covering argument. To be more precise, we obtain the following corollary whose proof we omit.

Corollary 5.2**.**

Let $\nu_{0}$ be a constant chosen according to (5.4) and assume that (4.4) holds for some $\bar{t}\in I_{\omega}$ and for all $\bar{x}\in K_{\omega}$. Then there holds

[TABLE]

Next, we set $t_{*}=\bar{t}-d(R/2)^{p}$ and establish the following result regarding the expansion of the positivity in time direction.

Lemma 5.3**.**

Let $\nu_{0}$ be a constant chosen according to (5.4) and assume that (4.4) holds for some $\bar{t}\in I_{\omega}$ and for all $\bar{x}\in K_{\omega}$. For any fixed $\nu_{1}\in(0,1)$, the constant $L_{2}$ can be chosen in dependence on $N$, $C_{0}$, $C_{1}$, $\nu_{1}$ and $L_{1}$ such that, with $c_{2}=L_{2}d_{*}$, there holds

[TABLE]

for all $t\in(t_{*},0)$.

Proof.

We first recall that

[TABLE]

and

[TABLE]

where

[TABLE]

Next we let $k=\mu_{-}+\frac{\omega}{8}$ and choose $H_{k}^{-}=\frac{\omega}{8}$ which is admissible since $(u-k)_{-}\leq\frac{\omega}{8}$. We observe from (5.5) that $\psi^{-}(x,t_{*})=0$ for all $x\in K_{c_{2}\frac{R}{2}}$. The following proof will be divided into two steps.

Step 1: Let $\zeta\equiv\zeta(x)$ be a piecewise smooth function. We establish the estimate:

[TABLE]

In the case $k\leq 0$, we have $u<k-c<0$ and this yields

[TABLE]

which is our claim. We now turn to the case $k>0$. If $\mu_{-}\geq c$, then there holds $u\geq\mu_{-}\geq c>0$ and we obtain again the identity $U(K_{c_{2}\frac{R}{2}},t_{*},0,2\psi^{-}\left(\psi^{-}\right)^{\prime}\zeta^{p})=0$. It now remains to consider the case $\mu_{-}<c$. Since $\zeta\equiv\zeta(x)$, then $\partial_{t}\zeta\equiv 0$. We write

[TABLE]

with the obvious meaning of $S_{1}$ and $S_{2}$. We begin with the estimate for $S_{1}$. Noting that $-\left(\psi^{-}\right)^{\prime}=(u-\mu_{-}+c)^{-1}\leq c^{-1}$, we obtain

[TABLE]

To estimate $S_{2}$, we note that $\partial u/\partial t\equiv 0$ on $[u=0]$ and there holds

[TABLE]

At this stage, we introduce an auxiliary function $\Phi_{c}(u)$, defined by

[TABLE]

We observe that $0\leq\Phi_{c}(u)\leq-\psi^{-}\left(\psi^{-}\right)^{\prime}$, $\Phi_{c}(u)$ is Lipschitz with respect to $u$ and

[TABLE]

Integrating by parts, we deduce

[TABLE]

Combining (5.10) and (5.11), we conclude that

[TABLE]

since $\psi^{-}(x,t_{*})=0$ for all $x\in K_{c_{2}\frac{R}{2}}$. This proves the claim (5.9).

Step 2: Proof of (5.3). Since $\psi^{-}(x,t_{*})=0$ for all $x\in K_{c_{2}\frac{R}{2}}$, we find that

[TABLE]

Plugging this into (3.4) and taking into account (5.9), we obtain

[TABLE]

We take cutoff function $\zeta=\zeta(x)$ which satisfies $0\leq\zeta\leq 1$ in $K_{c_{2}\frac{R}{2}}$, $\zeta\equiv 1$ in $K_{c_{2}\frac{R}{4}}$ and $|D\zeta|\leq 4(c_{2}R)^{-1}$. Now we take $c=2^{-6}\omega$ and deduce from (5.7) and (5.8) the estimate

[TABLE]

Keeping in mind $|t_{*}|\leq L_{1}dR^{p}$, we deduce from (4.2) the estimate

[TABLE]

where $C=2^{N+12}\gamma_{2}>1$. At this point, we choose

[TABLE]

For such a choice of $L_{2}$, the above estimate yields

[TABLE]

The left-hand side of (5.12) is estimated below by integrating over the smaller set

[TABLE]

On such a set, $\zeta\equiv 1$ and

[TABLE]

This gives

[TABLE]

for all $t\in(t_{*},0)$. Combining (5.13) with (5.12), we conclude that for any $t\in(t_{*},0)$ there holds

[TABLE]

which proves the lemma. ∎

With the help of the preceding two lemmas we can now prove the following proposition which is the main result in this section.

Proposition 5.4**.**

Suppose that $Q(c_{1}R^{p},c_{2}R)\subset\overline{Q}\subset\Omega_{T}$. Let $\nu_{0}$ be a constant chosen according to (5.4) and assume that (4.4) holds for some $\bar{t}\in I_{\omega}$ and for all $\bar{x}\in K_{\omega}$. Then there exists a constant $\nu_{1}\in(0,1)$ depending only upon the data such that the following holds. For the constant $L_{2}>0$ which is determined a priori only in terms of $\{N,C_{0},C_{1},L_{1}\}$ such that

[TABLE]

and $c_{2}=L_{2}d_{*}$, there holds

[TABLE]

Proof.

Let $\gamma$ and $\gamma_{1}$ be the constants determined by Lemma 2.4 and 3.3, respectively, depending only upon the data $\{N,C_{0},C_{1}\}$. We now choose

[TABLE]

and choose $L_{2}$ satisfying (5.14). By Lemma 5.3, we conclude from (5.6) that

[TABLE]

Next, we define two decreasing sequences of numbers

[TABLE]

We set $Q_{n}=Q(-t_{*},c_{2}R_{n})$ and choose piecewise smooth cutoff functions $\zeta_{n}(x)$ defined in $K_{c_{2}R_{n}}$ and satisfying $0\leq\zeta_{n}\leq 1$ in $K_{c_{2}R_{n}}$, $\zeta_{n}\equiv 1$ in $K_{c_{2}R_{n+1}}$ and $|D\zeta_{n}|\leq 2^{n+4}(c_{2}R)^{-1}$. We recall that from (5.1) there holds $(u-k_{n})_{-}(\cdot,t_{*})=0$ in $K_{c_{2}R_{n}}$. Write the energy estimate (3.3) for $(u-k_{n})_{-}$ over the cylinder $Q_{n}$, we obtain

[TABLE]

since $\partial_{t}\zeta_{n}\equiv 0$. Next, we claim that

[TABLE]

To prove (5.18), we first consider the case $k_{n}<0$. In this case, either $(u-k_{n})_{-}=0$ or $u\leq k_{n}<0$. We obtain

[TABLE]

In the case $k_{n}\geq 0$, we note that $(u-k_{n})_{-}\chi_{[u\leq 0]}=k_{n}\chi_{[u\leq 0]}+u_{-}$ and $\partial_{t}(u-k_{n})_{-}\chi_{[u\leq 0]}=\partial_{t}u_{-}$. Taking into account $(u-k_{n})_{-}(\cdot,t_{*})=0$ in $K_{c_{2}R_{n}}$, we get

[TABLE]

which proves the claim. At this stage, we arrive at

[TABLE]

Set

[TABLE]

With this notation, the estimate (5.17) reads

[TABLE]

Keeping in mind $(u-k_{n})_{-}\leq 2^{-6}\omega$ and $|D\zeta_{n}|\leq 2^{n+4}(c_{2}R_{n})^{-1}$, we deduce

[TABLE]

Applying the parabolic Sobolev’s inequality (2.7), we obtain

[TABLE]

On the other hand, we estimate below

[TABLE]

Combining the estimates above we infer that

[TABLE]

Taking into account $|Q_{n}|=L_{2}^{N}d_{*}^{N}R_{n}^{N}t_{*}$, $8^{-1}R\leq R_{n}\leq 4^{-1}R$, $t_{*}\leq L_{1}dR^{p}$ and the relation (4.2), we obtain

[TABLE]

since

[TABLE]

Recalling from (5.16) and (5.19) that

[TABLE]

By the lemma on fast geometric convergence of sequences (cf. [5, Chapter I, Lemma 4.1]), we conclude that $Y_{n}\to 0$ as $n\to\infty$, which proves the proposition. ∎

From now on, we choose $L_{1}=2$. For such a choice of $L_{1}$, we determine $L_{2}$ by

[TABLE]

where $\nu_{1}$ is the constant chosen according to (5.16). By these choices, taking (5.15) into account, we obtain the desired estimate (4.3) under the assumption of first alternative.

6. The second alternative

In this section we will establish the estimate (4.3) for the second alternative. We start with the following lemma whose proof we omit.

Lemma 6.1**.**

Let $\nu_{0}$ be a constant chosen according to (5.4) and assume that (4.5) holds for some fixed $\bar{t}\in I_{\omega}$ and $\bar{x}\in K_{\omega}$. There exists a time level $t_{0}\in[\bar{t}-dR^{p},\bar{t}-\frac{1}{2}\nu_{0}dR^{p}]$ such that

[TABLE]

The estimate (6.1) is the starting point for the analysis of the second alternative and we can now ignore the inequality (4.5). Next, since $\mu_{+}\geq|\mu_{-}|$ and $\mu_{+}-\mu_{-}>\frac{\omega}{2}$, then $\mu_{+}\geq 0$ and $4\mu_{+}>\omega$. For $k\geq\mu_{+}-\frac{\omega}{4}$, we observe that if $(u-k)_{+}>0$ then $u>k>0$. So we can always discard the term involving $U$ in the energy estimates. Let $m_{2}\geq 2$ be the quantity which will be determined in §6.2. We give the following lemma regarding the expansion of positivity in time direction.

Lemma 6.2**.**

There exists $\tilde{n}>1$ depending only upon the data and $\nu_{0}$, and independent of $m_{2}$ such that

[TABLE]

for all $t\in(t_{0},\tilde{t})$ where

[TABLE]

Proof.

Without loss of generality, we assume that $\bar{x}=0$. From (6.3), we observe that

[TABLE]

This enable us to use the logarithmic estimate (3.4) over the cylinder $K_{d_{*}R}\times(t_{0},\tilde{t}]$. Set $k=\mu_{+}-\frac{\omega}{2^{m_{2}}}$ and $c=\frac{\omega}{2^{m_{2}+\tilde{n}}}$ where $\tilde{n}>1$ is to be determined. We consider the logarithmic function

[TABLE]

Then we have $\psi^{+}\leq\tilde{n}\ln 2$ and

[TABLE]

Choose a piecewise smooth cutoff function $\zeta(x)$, defined in $K_{d_{*}R}$, and satisfying $0\leq\zeta\leq 1$ in $K_{d_{*}R}$, $\zeta\equiv 1$ in $K_{(1-\sigma)d_{*}R}$ and $|D\zeta|\leq(\sigma d_{*}R)^{-1}$, where $\sigma\in(0,1)$ is to be determined. Since $k>\mu_{+}-\frac{\omega}{4}$, then $U(K_{d_{*}R},t_{0},\tilde{t},2\psi^{+}\left(\psi^{+}\right)^{\prime}\zeta^{p})=0$. Then we deduce from (3.4) the logarithmic estimate

[TABLE]

Keeping in mind the definition of $\tilde{t}$ and (6.1), we obtain

[TABLE]

At this stage, we proceed similarly as in [6, Page 938-939]. To estimate below the integral on the left-hand side, we consider a smaller set defined by

[TABLE]

On such a set, $\zeta\equiv 1$ and

[TABLE]

It follows that

[TABLE]

for all $t\in(t_{0},\tilde{t})$. To prove the lemma choose

[TABLE]

With this choice we see that $\tilde{n}$ is independent of $m_{2}$. ∎

6.1. Expansion of positivity in space variable

In this subsection we will establish an expansion of positivity result for the weak solutions in a larger cylinder. We have to work with an equation in dimensionless form and our proof is in the spirit of [1, 3, 5, 6]. To start with, we introduce the change of variables

[TABLE]

This transformation maps $[\bar{x}+K_{d_{*}R}]\times(t_{0},\tilde{t})\to K_{\frac{1}{2L_{2}}}\times(-4^{p},0)$ and $[\bar{x}+K_{8c_{2}R}]\times(t_{0},\tilde{t})\to K_{4}\times(-4^{p},0)$. Next, we set the new functions

[TABLE]

With these notations, the estimate (6.2) can be rewritten as

[TABLE]

for all $t^{\prime}\in(-4^{p},0]$. On the other hand, we set $\tilde{w}(x^{\prime},t^{\prime})=w(x,t)$. The function $v(x,t)$ in (3.1) can be written in the new variable as

[TABLE]

For any $\varphi\in W_{p}(Q_{4})$ and $[t_{1},t_{2}]\subset(-4^{p},0]$, we rewrite the weak form (3.1) in terms of the new variables and new functions as follows

[TABLE]

where

[TABLE]

From (1.3), we obtain the structure conditions for $\bar{A}$ as follows

[TABLE]

where

[TABLE]

According to (6.3), the above identity reads

[TABLE]

which is independent of $m_{2}$. By abuse of the notation, we write $(x,t)$ instead of the new variables. The next lemma is an analogue of [5, Chapter II, Lemma 1.1].

Lemma 6.3**.**

Let $k\in(-1,0)$. Then there holds

[TABLE]

for all $\varphi\in W_{p}(Q_{4})$ and $\varphi\geq 0$.

Proof.

We proceed similarly as in [5, page 18-19]. In (6.7) choose the testing function

[TABLE]

In view of $\omega<4\mu_{+}$, we find that the first two terms in (6.7) vanish and there holds

[TABLE]

Consequently, the inequality (6.10) follows by passing to the limit $\varepsilon\downarrow 0$. ∎

With the previous result at hand, we can now give the following lemma which concerns the expansion of positivity in space variable. Since the constants in the proof should be uniformly bounded in $p$ when $p\to 2$, we employ the idea from Alkhutov and Zhikov [1]. Moreover, this approach enables us to obtain explicit estimates for the parameters which will play a crucial role in the next subsection.

Lemma 6.4**.**

Suppose that $\frac{3}{2}\leq p<2$. For any $\nu\in(0,1)$, there exists $\tilde{m}>1$ depending only upon the data, $\nu$ and $\nu_{0}$, such that

[TABLE]

for all $t\in[-2^{p},0]$.

Proof.

Before proceeding to the proof, we introduce the auxiliary functions

[TABLE]

and

[TABLE]

where $V=(-k)(-\delta)^{\frac{p}{p-1}}$ and $k$, $\delta\in(-\frac{1}{8},0)$ are to be determined. Recalling the definition of $\bar{u}$, we have $\bar{u}\leq 0$ and $(\bar{u}-k)_{+}\leq-k$. Moreover, we obtain the inequality

[TABLE]

for any $0\leq\tau\leq-k$. Taking into account

[TABLE]

and $1-(1-z)^{2-p}\leq-(2-p)\ln(1-z)$, we conclude that

[TABLE]

Let $\zeta=\zeta_{1}(x)\zeta_{2}(t)$ be a piecewise smooth cutoff function defined in $Q_{4}$. Suppose that $0\leq\zeta\leq 1$ in $Q_{4}$, $\zeta\equiv 1$ in $Q_{2}$, $\zeta\equiv 0$ on $\partial_{P}Q_{4}$, $|D\zeta_{1}|\leq 1$, $0\leq\partial_{t}\zeta_{2}\leq 1$ and the sets $\{x\in K_{4}:\zeta_{1}(x)>-k\}$ are convex for all $k\in(-\frac{1}{8},0)$. Testing the weak formulation (6.10) with the function

[TABLE]

and taking (6.8) into account, we have

[TABLE]

for any $[t_{1},t_{2}]\subset(-4^{p},0]$. In view of $p\geq\frac{3}{2}$, $\Lambda(\omega)\leq 1$, and therefore, Young’s inequality and (6.12) allow us to conclude that

[TABLE]

where

[TABLE]

Next, we observe that

[TABLE]

for any $t\in(-4^{p},0]$. Applying Lemma 2.3 on each of the time slice and keeping in mind (6.5), we deduce

[TABLE]

Multiplying both sides of the above inequality by $\zeta_{2}(t)^{p}$ and integrating over $[t_{1},t_{2}]$, we obtain

[TABLE]

where

[TABLE]

Recalling that $u\in C_{\operatorname*{loc}}(0,T;L_{\operatorname*{loc}}^{2}(\Omega))$, we obtain $\phi_{k}(\bar{u}),\ \psi_{k}(\bar{u})^{p}\in C_{\operatorname*{loc}}(0,T;L_{\operatorname*{loc}}^{1}(\Omega))$ and for any $t\in(-4^{p},0]$ there holds

[TABLE]

where

[TABLE]

At this stage, we introduce the quantities

[TABLE]

To prove this lemma, it suffices to determine constants $\delta$ and $i_{*}$ depending only upon the data, $\nu$ and $\nu_{0}$, such that $Y_{i_{*}}\leq\nu$.

Fix $i\in\mathbb{N}$, we choose $k=-|\delta|^{i}$ and $V=|\delta|^{i+\frac{p}{p-1}}$. For any $\varepsilon>0$, there exists $t_{0}\in(-4^{p},0]$ such that

[TABLE]

Let

[TABLE]

In the case $t_{0}\in\mathcal{C}^{+}$, we choose $\varepsilon\in(0,\nu/2]$. In (6.15) take $t=t_{0}$, $k=-|\delta|^{i}$ and there holds

[TABLE]

To estimate below the integral on the left-hand side, take into account the domain of integration $K_{4}\cap[\bar{u}>-|\delta|^{i+1}]$. On such a set, $(\bar{u}+|\delta|^{i})_{+}>|\delta|^{i}-|\delta|^{i+1}$. Taking into account $V\leq|\delta|^{i+1}$, we deduce

[TABLE]

Since $|\delta|<\frac{1}{8}$, then there holds

[TABLE]

We use these estimates to conclude that

[TABLE]

and therefore

[TABLE]

Then $Y_{i+1}\leq\nu$, provided

[TABLE]

To this end, we choose

[TABLE]

We now turn our attention to the case $t_{0}\notin\mathcal{C}^{+}$. Denote by $t_{*}$ the least upper bound of the set

[TABLE]

In the case $t_{0}=t_{*}$, there exists a sequence $t_{k}^{\prime}\in\mathcal{C}_{t_{0}}^{+}$ such that $t_{k}^{\prime}\to t_{0}$ as $k\to\infty$, and

[TABLE]

Passing to the limit $t_{k}^{\prime}\to t_{0}$ and taking into account that $\psi_{-|\delta|^{i}}(\bar{u})^{p}\in C_{\operatorname*{loc}}(0,T;L_{\operatorname*{loc}}^{1}(\Omega))$, we deduce

[TABLE]

We may now repeat the same arguments as in the previous proof and obtain $Y_{i+1}\leq\nu$, provided $\delta$ satisfies (6.16).

Finally, we come to the case $t_{0}>t_{*}$. Let $j$ be an integer that can be determined a priori only in terms of the data, $\nu$ and $\nu_{0}$. Let us initially assume $Y_{i}>\nu$ for all $i=1,2,\cdots,j$. Next, we claim that

[TABLE]

Fix $i\in\{1,2,\cdots,j\}$. Since $t_{0}>t_{*}$, then there holds

[TABLE]

for any $t\in(t_{*},t_{0})$. It follows that

[TABLE]

Recalling the definition of $t_{*}$, we have

[TABLE]

with the obvious meaning of $C_{\delta}$. Consider the set $K_{4}\cap[(\bar{u}+|\delta|^{i})_{+}>\tau|\delta|^{i}]$ where $\tau\in[0,1]$. On this set

[TABLE]

since $V\leq|\delta|^{i+1}$. Combining this estimate with (6.19), we obtain

[TABLE]

At this stage, we set

[TABLE]

Since $Y_{i}>\nu$, then we have

[TABLE]

with the obvious meaning of $\sigma$. For a technical reason, we introduce a constant $\sigma^{\prime}>\sigma$ defined by

[TABLE]

At this point, we claim that

[TABLE]

For the value of $|\delta|$ given by (6.16), we have

[TABLE]

for all $\frac{3}{2}\leq p<2$. Then there holds

[TABLE]

From (6.16), we observe that $|\delta|<(2+2\sqrt{2})^{-1}$ and this implies $|\delta|^{-1}(1+|\delta|)<(2|\delta|)^{-2}.$ Combining this inequality with (6.21), we obtain

[TABLE]

By (6.16), this choice of $\delta$ yields $|\delta|<[4(3+2\sqrt{2})]^{-1}$ and there holds $(1-\sqrt{2|\delta|})(1+2|\delta|)\leq 1-\sqrt{|\delta|}$. Then we have

[TABLE]

This implies the claimed estimate (6.20). We now proceed to estimate the right-hand side of (6.18). Using a change of variable $\tau^{\prime}=|\delta|^{-i}\tau$, we obtain

[TABLE]

and this yields

[TABLE]

Taking into account that $Y_{i}>\nu$, we obtain

[TABLE]

where

[TABLE]

Moreover, we rewrite the above estimate as

[TABLE]

where the function $f(\delta)$ satisfies

[TABLE]

To estimate below the integral on the left-hand side of (6.18), take into account the domain of integration $K_{4}\cap[\bar{u}>-|\delta|^{i+1}]$. Recalling that $V=|\delta|^{i+\frac{p}{p-1}}$, we deduce

[TABLE]

for any $\tau<|\delta|^{i}$. Then we find that

[TABLE]

Applying a change of variable $\tau^{\prime}=|\delta|^{-i}\tau$, we obtain

[TABLE]

Combining this estimate with (6.22), we conclude that

[TABLE]

Finally, we need to show that $f(\delta)>\delta^{2}$. The strategy of proof is exactly the same as in [1, Page 378-379] and we include the proof here for the sake of completeness. Since $\sigma^{\prime}>\sigma$ and

[TABLE]

It follows that

[TABLE]

Then we arrive at

[TABLE]

Taking into account that $\sigma^{\prime}(1+2|\delta|)<1-\sqrt{|\delta|}$, we obtain

[TABLE]

By (6.16), this choice of $\delta$ yields $|\delta|<e^{-64\ln 2}$ and therefore

[TABLE]

From this estimate, we find that

[TABLE]

since $2^{2-p}-1\geq(2-p)\ln 2$. Taking into account that $\ln(1+|\delta|^{-1})\leq|\delta|^{-1}$, we obtain

[TABLE]

From (6.24) and (6.25), we obtain

[TABLE]

Therefore, we conclude from (6.23) that $Y_{i+1}\leq(1-|\delta|)Y_{i}$ and the claim (6.17) follows.

From (6.17), by iteration

[TABLE]

Having fixed $\nu\in(0,1)$, one can choose

[TABLE]

where $[\cdot]$ denotes the integer portion of the number. For such a choice, $Y_{j+1}\leq\nu$ and hence

[TABLE]

Moreover, by (6.16), there exists $\gamma_{3}>2^{10}$, depending only upon the data $\{N,C_{0},C_{1}\}$, such that

[TABLE]

We choose

[TABLE]

Taking into account that

[TABLE]

the choice (6.26) guarantees that $2^{-\tilde{m}}\leq|\delta|^{j+1}$. For such a choice of $\tilde{m}$, we obtain the desired estimate (6.11). Finally, if $Y_{i_{0}}\leq\nu$ for some $i_{0}\in\{1,2,\cdots,j\}$, then the estimate (6.11) holds as well, for the same choice of $\tilde{m}$ as in (6.26). This concludes the proof of the lemma. ∎

From (6.9), (6.14) and (6.26), we remark that $\tilde{m}$ can be chosen independent of $m_{2}$. Transforming back to the original function $u$ and original variables $(x,t)$, we obtain an estimate for the measure of the level sets

[TABLE]

for all $\tilde{t}-2^{-p}(\tilde{t}-t_{0})<t<\tilde{t}$. With the help of this estimate we can now prove a DeGiorgi-type lemma.

Lemma 6.5**.**

Let $\tilde{n}>1$ be the constant chosen according to (6.4). Then there exist a constant $\tilde{m}>1$ depending only upon the data and $\nu_{0}$, and a time level $t_{\omega}\in(t_{0},\tilde{t})$ such that

[TABLE]

Proof.

Without loss of generality, we may assume $(\bar{x},\tilde{t})=(0,0)$. For $\nu\in(0,1)$ to be determined later we take $\tilde{m}=\tilde{m}(\nu)>1$ according to (6.26). For $n=0,1,2,\cdots$, set

[TABLE]

where the time level $t_{*}$ is taken as

[TABLE]

From (6.4), (6.9), (6.14) and (6.26), we observe that

[TABLE]

for any $\nu\in(0,1)$. So, we conclude from (6.3) that $t_{\omega}\in(t_{0},\tilde{t})$. Take piecewise smooth cutoff functions $\zeta_{n}$ in $Q_{n}$, such that $0\leq\zeta_{n}\leq 1$, $\zeta_{n}\equiv 1$ in $Q_{n+1}$, $\zeta_{n}\equiv 0$ on $\partial_{P}Q_{n}$, $|D\zeta_{n}|\leq 2^{n}/(c_{2}R)$ and $0<\partial_{t}\zeta_{n}\leq 2^{p+1}2^{pn}/(\tilde{t}-t_{\omega})$. Write down the energy estimates (3.3) for the truncated functions $(u-k_{n})_{+}\zeta_{n}^{p}$ over the cylinders $Q_{n}$. Taking into account that

[TABLE]

we deduce

[TABLE]

At this point, we set

[TABLE]

Applying parabolic Sobolev’s inequality (2.7), we obtain

[TABLE]

for a constant $\gamma_{5}$ depending only upon the data. The integral on the left-hand side is estimated below by

[TABLE]

Combining the estimates above and keeping in mind

[TABLE]

we infer that

[TABLE]

for a constant $\gamma_{6}$ depending only upon the data. At this point, we set

[TABLE]

Choose $\nu=\nu_{2}$ in Lemma 6.4, and hence $\tilde{m}=\tilde{m}(\nu_{2})$, from this and (6.26). Moreover, we conclude from (6.27) that $Y_{0}\leq\nu_{2}.$ By the lemma on fast geometric convergence of sequences, we infer that $Y_{n}\to 0$, as $n\to\infty$, which proves the lemma. ∎

6.2. Expansion of positivity in time variable

The aim of this subsection is to establish a DeGiorgi-type result similar to that of Lemma 6.5. To start with, we determine the constant $m_{2}$ in terms of the data and $\omega$. Let $\tilde{n}$ and $\nu_{2}$ be the constants determined by (6.4) and (6.29), respectively. In (6.26), take $\nu=\nu_{2}$ and choose $\tilde{m}=\tilde{m}(\nu_{2})$. At this point, we choose $m_{2}>2$ be such that

[TABLE]

and hence $m_{1}=m_{2}p/(p-1)$. This determines the precise formulation of the intrinsic parabolic cylinders. Next, we consider some geometric properties of these cylinders.

From now on, we assume that $p>\frac{1+\sqrt{5}}{2}$ and set $\kappa=\frac{1}{2}(p-1-\frac{1}{p})>0$. Moreover, we will need the following assumption:

[TABLE]

In the case $2^{-m_{2}}\omega\geq R$. Keeping in mind $L_{1}=2$, we conclude from (6.31) that $Q(8c_{1}R^{p},8c_{2}R)\subset\overline{Q}$. While in the case $2^{-m_{2}}\omega\leq R$ we conclude from (5.4), (6.4), (6.9), (6.14), (6.26), (6.29) and (6.30) that

[TABLE]

for some constants $\gamma_{7}$ and $\gamma_{8}$ depending only upon the data. From this inequality, we obtain a decay estimate for $\omega$ as follows

[TABLE]

We now turn our attention to the case $Q(8c_{1}R^{p},8c_{2}R)\subset\overline{Q}$. In this case we have already established the estimate (6.28). Before proceeding further, let us remark that this estimate implies

[TABLE]

The next lemma deals with the expansion of positivity in time, starting from $\tilde{t}$.

Lemma 6.6**.**

For any $\nu\in(0,1)$, there exist a constant $\gamma_{9}$ depending only upon the data, and a time level $t^{(0)}=\tilde{t}+\gamma_{9}\nu\omega^{(1-p)(2-p)}R^{p}$ such that

[TABLE]

for any $t\in(\tilde{t},t^{(0)})$.

Proof.

Set $k=\mu_{+}-\frac{\omega}{2^{m_{2}p-4}}$ and $c=\frac{\omega}{2^{m_{2}p-1}}$. We consider the logarithmic function

[TABLE]

Then we have $\psi^{+}\leq 8\ln 2$ and

[TABLE]

Choose a piecewise smooth cutoff function $\zeta(x)$, defined in $K_{c_{2}R}$, and satisfying $0\leq\zeta\leq 1$ in $K_{c_{2}R}$, $\zeta\equiv 1$ in $K_{\frac{1}{2}c_{2}R}$ and $|D\zeta|\leq 2(c_{2}R)^{-1}$. Since $k>\mu_{+}-\frac{\omega}{4}$, it is easy to check that $U(K_{c_{2}R},\tilde{t},t^{(0)},2\psi^{+}\left(\psi^{+}\right)^{\prime}\zeta^{p})=0$. Then we obtain from (3.4) the logarithmic estimate

[TABLE]

From (6.33), we observe that $\int_{K_{c_{2}R}\times\{\tilde{t}\}}\left(\psi^{+}\right)^{2}\zeta^{p}dx=0$ and there holds

[TABLE]

provided

[TABLE]

To estimate below the integral on the left-hand side, we consider a smaller set defined by

[TABLE]

On such a set, $\zeta\equiv 1$ and $\psi^{+}\geq 2\ln 2>1$. Combining this inequality with (6.35), we obtain (6.34), which proves the lemma. ∎

With the help of this lemma we can now prove the following DeGiorgi-type result.

Lemma 6.7**.**

There exists a constant $\nu_{3}\in(0,1)$ depending only upon the data such that

[TABLE]

where $t^{(0)}=\tilde{t}+\gamma_{9}\nu_{3}\omega^{(1-p)(2-p)}R^{p}$.

Proof.

Let $t^{(0)}=\tilde{t}+\gamma_{9}\nu\omega^{(1-p)(2-p)}R^{p}$ where $\nu\in(0,1)$ is to be determined. Consider two decreasing sequences of real numbers

[TABLE]

We set $Q_{n}=K_{R_{n}}\times(\tilde{t},t^{(0)}]$. Take piecewise smooth cutoff functions $\zeta_{n}(x)$ in $K_{R_{n}}$, such that $0\leq\zeta_{n}\leq 1$ in $K_{R_{n}}$, $\zeta_{n}\equiv 1$ in $K_{R_{n+1}}$ and $|D\zeta_{n}|\leq 4^{n+2}(c_{2}R)^{-1}$. Write down the energy estimates (3.3) for the truncated functions $(u-k_{n})_{+}$ over the cylinders $Q_{n}$. Taking into account that

[TABLE]

we obtain

[TABLE]

Set

[TABLE]

Applying the parabolic Sobolev’s inequality (2.7), we get

[TABLE]

for a constant $\gamma_{10}$ depending only upon the data. The integral on the left-hand side is estimated below by

[TABLE]

Combining (6.37) and (6.38), we have

[TABLE]

Taking into account that

[TABLE]

and $\nu<1$, we obtain

[TABLE]

for a constant $\gamma_{11}$ depending only upon the data. At this point, we set

[TABLE]

We now choose $\nu=\nu_{3}$. By Lemma 6.6, $Y_{0}\leq\nu_{3}$. Applying the lemma on fast geometric convergence of sequences, we deduce $Y_{n}\to 0$ as $n\to\infty$, which proves the lemma. ∎

6.3. Iterative arguments: time propagation of positivity from $t^{(0)}$ to $t^{(1)}$

In this subsection we first remark that the time level $t^{(0)}$ could be lower than $\bar{t}$. So the estimate (6.36) is insufficient for the proof. We have to use an iterative argument to obtain the estimate similar to (6.36) in the cylinder with a larger time interval.

The starting point is a space propagation of positivity similar to Lemma 6.4. Starting from (6.36), we set $R^{(0)}=\frac{1}{4}c_{2}R$ and introduce the change of variables

[TABLE]

This transformation maps $K_{R^{(0)}}\times(\tilde{t},t^{(0)}]\to K_{\frac{1}{20}}\times(-1,0]$ and $K_{20R^{(0)}}\times(\tilde{t},t^{(0)}]\to K_{1}\times(-1,0]$. Moreover, we set the new functions

[TABLE]

With these notations, the estimate (6.36) implies

[TABLE]

for all $t^{\prime}\in(-1,0]$. On the other hand, we set $\tilde{w}(x^{\prime},t^{\prime})=w(x,t)$ and $v(x,t)$ in (3.1) can be written in the new variable as

[TABLE]

We rewrite the weak form (3.1) in terms of the new variables and new functions

[TABLE]

for any $\varphi\in W_{p}(Q_{1})$ and $[t_{1},t_{2}]\subset(-1,0]$. We observe that the vector field $\bar{A}$ satisfies the structure condition

[TABLE]

where

[TABLE]

Recalling that $t^{(0)}=\tilde{t}+\gamma_{9}\nu_{3}\omega^{(1-p)(2-p)}R^{p}$ and $R^{(0)}=\frac{1}{4}c_{2}R$, we deduce

[TABLE]

The constant $\Lambda_{1}$ depends only upon the data. To simplify notation, we continue to write $(x,t)$ for the new variables. In the same fashion as in the proof of Lemma 6.3, we conclude that the truncated functions $(\bar{u}-k)_{+}$ are subsolutions to parabolic equations. In a precise way we have

[TABLE]

for all $k\in(-1,0)$, $t_{1},t_{2}\in(-1,0)$ and all nonnegative $\varphi\in W_{p}(Q_{1})$.

Lemma 6.8**.**

Suppose that $\frac{3}{2}\leq p<2$. For any $\nu\in(0,1)$, there exists $\bar{m}>1$ depending only upon the data and $\nu$, such that

[TABLE]

for all $t\in\left(-\left(\frac{3}{4}\right)^{p},0\right]$.

Proof.

We proceed similarly as in the proof of Lemma 6.4. To this end, we introduce the auxiliary functions

[TABLE]

and

[TABLE]

where $k,\ \delta\in(-\frac{1}{8},0)$ and $V=(-k)(-\delta)^{\frac{p}{p-1}}$. Take a piecewise smooth, cutoff function $\zeta=\zeta_{1}(x)\zeta_{2}(t)$ in $Q_{1}$, such that $0\leq\zeta\leq 1$ in $Q_{1}$, $\zeta\equiv 1$ in $Q_{\frac{3}{4}}$, $\zeta\equiv 0$ on $\partial_{P}Q_{1}$, $|D\zeta_{1}|\leq 4$, $0\leq\partial_{t}\zeta_{2}\leq 4^{p}$ and the sets $\{x\in K_{1}:\zeta_{1}(x)>-k\}$ are convex for all $k\in(-\frac{1}{8},0)$. In the weak formulation (6.44) take the test function

[TABLE]

This gives

[TABLE]

for any $[t_{1},t_{2}]\in(-1,0]$. Next, we observe from (6.40) that

[TABLE]

for any $t\in(-1,0]$. Applying Sobolev’s inequality (2.6) slicewise, we obtain

[TABLE]

With the same argument as in the proof of Lemma 6.4, we derive the estimate

[TABLE]

where the constant $\Lambda_{2}$ depends only upon the data. At this stage, we follow the proof of Lemma 6.4 to conclude that there exists a constant $\gamma^{\prime}$ depending only on the data, such that the following holds. If we choose

[TABLE]

then the estimate (6.45) follows. Moreover, we observe that the constant $\bar{m}$ depends only upon the data. ∎

Lemma 6.9**.**

There exist a constant $\bar{m}>1$ depending only upon the data and $p$, and a time level $t_{\omega}^{\prime}\in\left(-\left(\frac{3}{4}\right)^{p},0\right)$ such that

[TABLE]

and the constant $\bar{m}$ is stable as $p\to 2$.

Proof.

For $\nu\in(0,1)$ to be determined later we take $\bar{m}>1$ according to (6.46). For $n=0,1,2,\cdots$, set

[TABLE]

where $t_{\omega}^{\prime}=-2^{(\bar{m}+1)(p-2)}\left(\frac{3}{4}\right)^{p}$. Take piecewise smooth cutoff functions $\zeta_{n}$ in $Q_{n}$, such that $0\leq\zeta_{n}\leq 1$, $\zeta_{n}\equiv 1$ in $Q_{n+1}$, $\zeta_{n}\equiv 0$ on $\partial_{P}Q_{n}$, $|D\zeta_{n}|\leq 2^{n+3}$ and $0<\partial_{t}\zeta_{n}\leq 2^{p+p(n+1)}(-t_{\omega}^{\prime})^{-1}$.

In the weak formulation (6.41) take the test function $\varphi=(\bar{u}-k_{n})_{+}\zeta_{n}^{p}$ . Observe that the first two terms vanish. By a standard argument, we derive a Caccioppoli’s estimate for $(\bar{u}-k_{n})_{+}$ over $Q_{n}$ as follows

[TABLE]

Applying parabolic Sobolev’s inequality (2.7), we obtain

[TABLE]

for a constant $C_{3}$ depending only upon the data. At this point, we set

[TABLE]

The left-hand side of (6.48) is estimated below by

[TABLE]

Combining this estimate with (6.48), we obtain

[TABLE]

for a constant $C_{4}$ depending only upon the data. At this point, we set

[TABLE]

Choose $\nu=\nu_{4}$, and hence $\bar{m}=\bar{m}(\nu_{4})$, from this and (6.46). From Lemma 6.8, $Y_{0}\leq\nu_{4}$. By the lemma on fast geometric convergence of sequences, we infer that $Y_{n}\to 0$, as $n\to\infty$. This proves the lemma. ∎

Transforming back to the original function $u$ and original variables $(x,t)$, we obtain the following DeGiorgi-type result.

Lemma 6.10**.**

Let $\bar{m}>1$ be the constant as in Lemma 6.9. Then there exists a time level $t_{\omega}^{(0)}\in(\tilde{t},t^{(0)})$ such that

[TABLE]

where $t_{\omega}^{(0)}=t^{(0)}+(t^{(0)}-\tilde{t})t_{\omega}^{\prime}.$

In order to obtain an expansion of positivity result at a higher time level $t^{(1)}$, we need the following lemma that is similar to Lemma 6.6.

Lemma 6.11**.**

For any $\nu\in(0,1)$, there exist a constant $\delta_{1}$ depending only upon the data and $p$ , and a time level

[TABLE]

such that

[TABLE]

for any $t\in(t^{(0)},t^{(1)})$, and the constant $\delta_{1}$ is stable as $p\to 2$.

Proof.

Let $k=\mu_{+}-\frac{\omega}{2^{m_{2}p+\bar{m}+1}}$ and $c=\frac{\omega}{2^{m_{2}p+\bar{m}+4}}$. We consider the logarithmic function

[TABLE]

Then we get $\psi^{+}\leq 8\ln 2$ and

[TABLE]

Choose a piecewise smooth cutoff function $\zeta(x)$, defined in $K_{10R^{(0)}}$, and satisfying $0\leq\zeta\leq 1$ in $K_{10R^{(0)}}$, $\zeta\equiv 1$ in $K_{6R^{(0)}}$ and $|D\zeta|\leq(4R^{(0)})^{-1}$. Such kind of cutoff functions can be chosen explicitly via the formulas (2.1)-(2.2). Since $k>\mu_{+}-\frac{\omega}{4}$, we check at once that $U(K_{10R^{(0)}},t^{(0)},t^{(1)},2\psi^{+}\left(\psi^{+}\right)^{\prime}\zeta^{p})=0$. Then we obtain from (3.4) the logarithmic estimate

[TABLE]

By Lemma 6.10, we see that $\int_{K_{10R^{(0)}}\times\{t^{(0)}\}}\left(\psi^{+}\right)^{2}\zeta^{p}dx=0$ and therefore

[TABLE]

At this stage, we choose the time level

[TABLE]

Combining the estimates above we infer that

[TABLE]

On the other hand, we introduce the smaller set

[TABLE]

In this set, $\zeta\equiv 1$ and $\psi^{+}\geq 2\ln 2>1$. Then we conclude that

[TABLE]

for all $t\in(t^{(0)},t^{(1)})$ and the lemma follows with the choice of $\delta_{1}$ in (6.53). ∎

With the help of Lemma 6.11, we establish a DeGiorgi-type lemma and determine the value of $\nu$ and the time level $t^{(1)}$ in (6.51). This result extends (6.36) to a larger time interval.

Lemma 6.12**.**

Let $\delta_{1}>0$ be the constant chosen according to (6.53). Then there exists a constant $\nu_{5}$ depending only upon the data and $p$ such that the following holds. For the choice

[TABLE]

there holds

[TABLE]

and the constant $\nu_{5}$ is stable as $p\to 2$.

Proof.

For $\nu\in(0,1)$ to be determined later we take $\bar{m}=\bar{m}(\nu)>1$ according to (6.46). For $n=0,1,2,\cdots$, set

[TABLE]

Let $\zeta_{n}=\zeta_{n}(x)$ be defined via (2.1)-(2.2) with $r$ and $r^{\prime}$ replaced by $R_{n+1}$ and $R_{n}$. Then the cutoff function $\zeta_{n}$ satisfies $0\leq\zeta_{n}\leq 1$ in $K_{R_{n}}$, $\zeta_{n}\equiv 1$ in $K_{R_{n+1}}$ and $|D\zeta_{n}|\leq 2^{n}(2R^{(0)})^{-1}$. Taking into account that $U(K_{R_{n}},t^{(0)},t^{(1)},(u-k_{n})_{+}\zeta_{n}^{p})=0$, we proceed similarly as in the proof of Proposition 6.7. Write down the energy estimates (3.3) for $(u-k_{n})_{+}$ over $Q_{n}$, we obtain

[TABLE]

Set $A_{n}=\iint_{Q_{n}}\chi_{[(u-k_{n})_{+}>0]}dxdt$. Applying the parabolic Sobolev’s inequality (2.7), we deduce

[TABLE]

The integral on the left-hand side is estimated below by

[TABLE]

Combining the estimates above we see that

[TABLE]

Set $Y_{n}=A_{n}/|Q_{n}|$ and we shall derive a recursive inequality. Keeping in mind that

[TABLE]

we deduce

[TABLE]

Moreover, this estimate implies

[TABLE]

Since $0<\nu<1$, then we conclude that

[TABLE]

where

[TABLE]

At this point, we set $\nu_{5}=\delta_{2}^{-\frac{N}{p}}b^{-\frac{N^{2}}{p^{2}}}$. Recalling the definition of $\delta_{1}$ from (6.53), we see that

[TABLE]

which is even independent of $\bar{m}$. Therefore, we conclude that if $Y_{0}\leq\delta_{2}^{-\frac{N}{p}}b^{-\frac{N^{2}}{p^{2}}}$ then $Y_{n}\to 0$ as $n\to\infty$. To this end, we choose $\nu=\nu_{5}$ and the lemma follows. ∎

6.4. Iterative arguments: time propagation of positivity from $t^{(1)}$ to $t^{(2)}$

Starting from (6.54), we will repeat the argument of §6.3 to obtain an estimate similar to (6.54) in a space-time cylinder containing a higher time level $t^{(2)}$. The argument is divided into four steps. To start with, we set $R^{(1)}=2R^{(0)}$.

Step 1: We introduce the change of variables

[TABLE]

This transformation maps $K_{R^{(1)}}\times(t^{(0)},t^{(1)})\to K_{\frac{1}{20}}\times(-1,0)$ and $K_{20R^{(1)}}\times(t^{(0)},t^{(1)})\to Q_{1}$. We introduce the new functions

[TABLE]

From (6.54), we see that \big{|}\{x^{\prime}\in K_{1}:\bar{u}(x^{\prime},t^{\prime})<-1\}\big{|}\geq 20^{-N} for all $t^{\prime}\in(-1,0)$. Set $\tilde{w}(x^{\prime},t^{\prime})=w(x,t)$ and $v(x,t)$ in (3.1) can be written in the new variable as

[TABLE]

We rewrite the weak form (3.1) in terms of the new variables and new functions

[TABLE]

for any $\varphi\in W_{p}(Q_{1})$ and $[t_{1},t_{2}]\subset(-1,0]$. It is easy to check that the vector field $\bar{A}$ satisfies the structure condition

[TABLE]

where

[TABLE]

According to (6.53)-(6.54), we deduce that

[TABLE]

which depends only upon the data and $p$, and is stable as $p\to 2$. For any $\nu\in(0,1)$, we proceed similarly as in the proof of Lemma 6.8 to conclude that there exists a constant $\gamma^{\prime\prime}$, depending only upon the data, such that the following holds. For the constant $m^{\prime}$ with the expression

[TABLE]

there holds

[TABLE]

Step 2: Based on the estimate (6.60), we are now in a position to obtain an upper bound for $u$ in a space-time cylinder with a larger cube in space. By the proof of Lemma 6.9 and Lemma 6.10, we conclude that there exists a time level $t_{\omega}^{(1)}\in(t^{(0)},t^{(1)})$, such that

[TABLE]

where

[TABLE]

and the constant $\nu_{4}$ is chosen according to (6.49).

Step 3: Let $\delta_{1}>0$ be the constant chosen according to (6.53). For any $\nu\in(0,1)$, we claim that there exist a constant $M_{1}>0$ depending only upon the data and $p$, and a time level

[TABLE]

such that

[TABLE]

for any $t\in(t^{(1)},t^{(2)}]$.

Proof of the claim.

Let $k=\mu_{+}-\frac{\omega}{2^{m_{2}p+\bar{m}+5+m^{\prime}+1}}$ and $c=\frac{\omega}{2^{m_{2}p+\bar{m}+5+m^{\prime}+4}}$. We consider the logarithmic function

[TABLE]

Then we have $\psi^{+}\leq 8\ln 2$ and

[TABLE]

Let $\zeta=\zeta(x)$ be defined via (2.1)-(2.2) with $r$ and $r^{\prime}$ replaced by $6R^{(1)}$ and $10R^{(1)}$. It follows that $0\leq\zeta\leq 1$ in $K_{10R^{(1)}}$, $\zeta\equiv 1$ in $K_{6R^{(1)}}$ and $|D\zeta|\leq(4R^{(1)})^{-1}$. From (6.61), we deduce $\int_{K_{10R^{(1)}}\times\{t^{(1)}\}}\left(\psi^{+}\right)^{2}\zeta^{p}dx=0$. Taking into account that $U(K_{10R^{(1)}},t^{(1)},t^{(2)},2\psi^{+}\left(\psi^{+}\right)^{\prime}\zeta^{p})=0$, we obtain from (3.4) the logarithmic estimate

[TABLE]

Recalling the definition of $\delta_{1}$ from (6.53), we deduce

[TABLE]

At this stage, we set

[TABLE]

Then we find that

[TABLE]

On the other hand, we introduce the smaller set

[TABLE]

In this set, $\zeta\equiv 1$ and $\psi^{+}\geq 2\ln 2>1$. Then we conclude that

[TABLE]

for all $t\in(t^{(1)},t^{(2)})$ and the estimate (6.63) follows. ∎

Step 4: Take $\nu_{5}=\delta_{2}^{-\frac{N}{p}}b^{-\frac{N^{2}}{p^{2}}}$ be the constant as in Lemma 6.12. We claim that for the choice

[TABLE]

there holds

[TABLE]

Proof of the claim.

Proceed similarly as in the proof of Lemma 6.12, we consider two decreasing sequences of real numbers

[TABLE]

and set $Q_{n}=K_{R_{n}}\times(t^{(1)},t^{(2)}]$. Let $\zeta_{n}=\zeta_{n}(x)$ be defined via (2.1)-(2.2) with $r$ and $r^{\prime}$ replaced by $R_{n+1}$ and $R_{n}$. Then the cutoff function $\zeta_{n}$ satisfies $0\leq\zeta_{n}\leq 1$ in $K_{R_{n}}$, $\zeta_{n}\equiv 1$ in $K_{R_{n+1}}$ and $|D\zeta_{n}|\leq 2^{n}(2R^{(1)})^{-1}$. Taking into account that $U(K_{R_{n}},t^{(1)},t^{(2)},(u-k_{n})_{+}\zeta_{n}^{p})=0$, we write the energy estimate (3.3) for $(u-k_{n})_{+}$ over the cylinder $Q_{n}$ and obtain

[TABLE]

where $A_{n}=\iint_{Q_{n}}\chi_{[(u-k_{n})_{+}>0]}dxdt$. Applying the parabolic Sobolev’s inequality (2.7), we have

[TABLE]

To estimate below for the left-hand side, we observe that

[TABLE]

and therefore

[TABLE]

Then we deduce that

[TABLE]

Taking into account that

[TABLE]

then there holds

[TABLE]

and

[TABLE]

Set $Y_{n}=A_{n}/|Q_{n}|$. Recalling from the definition of $M_{1}$ and noting that $0<\nu_{5}<1$, we deduce

[TABLE]

since

[TABLE]

Then we conclude that if $Y_{0}\leq\nu_{5}=\delta_{2}^{-\frac{N}{p}}b^{-\frac{N^{2}}{p^{2}}}$ then $Y_{n}\to 0$ as $n\to\infty$. To this end, we choose $\nu=\nu_{5}$ in (6.63) and the claim follows. ∎

6.5. Iterative arguments: time propagation of positivity from $t^{(k)}$ to $t^{(k+1)}$ ($k\geq 2$)

Starting from (6.66), we can repeat the argument of §6.4 at any time to obtain a sequence of time levels $\{t^{(k)}\}_{k=1}^{\infty}$. Here and subsequently, set

[TABLE]

We now claim that

[TABLE]

for any $k\geq 2$.

Proof of (6.68).

We first observe from (6.66) that (6.68) holds with $k=2$. Assuming (6.68) to hold for $k$, we will prove it for $k+1$. We proceed similarly as in §6.4 and divide the proof into four steps.

Step 1: Let $(x^{\prime},t^{\prime})$ be the new variables defined by

[TABLE]

This transformation maps $K_{R^{(k)}}\times(t^{(k-1)},t^{(k)})\to K_{\frac{1}{20}}\times(-1,0)$ and $K_{20R^{(k)}}\times(t^{(k-1)},t^{(k)})\to Q_{1}$. We introduce the new functions

[TABLE]

By induction hypothesis, we see that \big{|}\{x^{\prime}\in K_{1}:\bar{u}(x^{\prime},t^{\prime})<-1\}\big{|}\geq 20^{-N} for all $t^{\prime}\in(-1,0)$. Set $\tilde{w}(x^{\prime},t^{\prime})=w(x,t)$ and $v(x,t)$ in (3.1) can be written in the new variable as

[TABLE]

We rewrite the weak form (3.1) in terms of the new variables and new functions

[TABLE]

for any $\varphi\in W_{p}(Q_{1})$ and $[t_{1},t_{2}]\subset(-1,0]$. We check at once that $\bar{A}$ satisfies the structure condition

[TABLE]

where

[TABLE]

Taking into account the definitions of $s_{k}$, $R^{(k)}$, $t^{(k)}$, $M_{1}$ and $\Lambda_{2}$, we infer that

[TABLE]

At this point, the proof now is exactly the same as in the step 1 in §6.4. For any $\nu\in(0,1)$, we choose $m^{\prime}$ as in (6.59) and conclude that

[TABLE]

Step 2: With the previous result at hand, we proceed similarly as in the step 1 in §6.4 to conclude that there exists a time level $t_{\omega}^{(k)}\in(t^{(k-1)},t^{(k)})$, such that

[TABLE]

where $m^{\prime}$ is the same constant as in (6.62).

Step 3: For any $\nu\in(0,1)$, we claim that there exists a time level

[TABLE]

such that

[TABLE]

for any $t\in(t^{(k)},t^{(k+1)}]$.

Proof of the claim.

Let $k^{\prime}=\mu_{+}-\frac{\omega}{2^{s_{k}+m^{\prime}+1}}$ and $c=\frac{\omega}{2^{s_{k}+m^{\prime}+4}}$. We consider the logarithmic function

[TABLE]

Taking into account the definition of $s_{k}$, we have $\psi^{+}\leq 8\ln 2$ and

[TABLE]

Let $\zeta=\zeta(x)$ be defined via (2.1)-(2.2) with $r$ and $r^{\prime}$ replaced by $6R^{(k)}$ and $10R^{(k)}$. It is easily seen that $0\leq\zeta\leq 1$ in $K_{10R^{(k)}}$, $\zeta\equiv 1$ in $K_{6R^{(k)}}$ and $|D\zeta|\leq(4R^{(k)})^{-1}$. It follows from (6.72) that $\int_{K_{10R^{(k)}}\times\{t^{(k)}\}}\left(\psi^{+}\right)^{2}\zeta^{p}dx=0$. Taking into account that $U(K_{10R^{(k)}},t^{(k)},t^{(k+1)},2\psi^{+}\left(\psi^{+}\right)^{\prime}\zeta^{p})=0$, we obtain from (3.4) the logarithmic estimate

[TABLE]

Keeping in mind the definitions of $\delta_{1}$ and $M_{1}$, we obtain

[TABLE]

On the other hand, we introduce the smaller set

[TABLE]

In this set, $\zeta\equiv 1$ and $\psi^{+}\geq 2\ln 2>1$. Then we conclude that

[TABLE]

for all $t\in(t^{(k)},t^{(k+1)})$ and the estimate (6.73) follows. ∎

Step 4: Proof of the estimate (6.68) for $k+1$. For $n=0,1,2,\cdots$, set

[TABLE]

Let $\zeta_{n}=\zeta_{n}(x)$ be defined via (2.1)-(2.2) with $r$ and $r^{\prime}$ replaced by $R_{n+1}$ and $R_{n}$. Then the cutoff function $\zeta_{n}$ satisfies $0\leq\zeta_{n}\leq 1$ in $K_{R_{n}}$, $\zeta_{n}\equiv 1$ in $K_{R_{n+1}}$ and $|D\zeta_{n}|\leq 2^{n}(2R^{(k)})^{-1}$. Taking into account that $U(K_{R_{n}},t^{(k)},t^{(k+1)},(u-k_{n})_{+}\zeta_{n}^{p})=0$, we write the energy estimate (3.3) for $(u-k_{n})_{+}$ over the cylinder $Q_{n}$ and obtain

[TABLE]

where $A_{n}=\iint_{Q_{n}}\chi_{[(u-k_{n})_{+}>0]}dxdt$. Applying the parabolic Sobolev’s inequality (2.7), we have

[TABLE]

To estimate below for the left-hand side, we observe that

[TABLE]

and therefore

[TABLE]

Then we deduce that

[TABLE]

Taking into account that

[TABLE]

then there holds

[TABLE]

and also

[TABLE]

Set $Y_{n}=A_{n}/|Q_{n}|$. Recalling from the definition of $M_{1}$ and noting that $\nu_{5}\in(0,1)$, we deduce from (6.67) that

[TABLE]

Then we conclude that if $Y_{0}\leq\nu_{5}=\delta_{2}^{-\frac{N}{p}}b^{-\frac{N^{2}}{p^{2}}}$ then $Y_{n}\to 0$ as $n\to\infty$. To this end, we choose $\nu=\nu_{5}$ in (6.73) and the estimate (6.68) follows for $k+1$. ∎

6.6. Analysis of the second alternative concluded

Now we are ready to proceed to obtain a DeGiorgi-type result similar to Proposition 5.4. We conclude from (6.36), (6.54) and (6.68) that for any $j\geq 0$ there holds

[TABLE]

Note that the time level $t^{(j)}$ should be taken so large that $\bar{t}\in(\tilde{t},t^{(j)}]$. For this purpose, we have to impose a condition that $M_{1}\geq 1$. Taking into account that $m^{\prime}$ is uniformly bounded in $p$, we set $m^{\prime\prime}=\sup_{\frac{3}{2}<p\leq 2}m^{\prime}$ and impose a condition for $p$ as follows

[TABLE]

In this case, we see that $t^{(j)}\to+\infty$ as $j\to+\infty$ and we can find an integer $j_{*}$ with the expression

[TABLE]

such that $\bar{t}\in(\tilde{t},t^{(j_{*})}]$.

In order to perform the iteration, we observe that any subcylinder should be contained in the reference cylinder $\overline{Q}=Q(R,R^{\frac{1}{p}})\subset\Omega_{T}$. To this end, we conclude that the estimate (6.74) holds for $j=j_{*}$ under the assumption that $2^{j_{*}}\leq 20^{-1}R^{-\kappa}$ and $2^{-m_{2}}\omega\geq R$, where $\kappa=\frac{1}{2}(p-1-\frac{1}{p})$. However, if $2^{j_{*}}>20^{-1}R^{-\kappa}$, then there exists a constant $C^{\prime}$ depending only upon the data and $p$ such that

[TABLE]

According to (6.30), there exist positive constants $\bar{C}$ and $\bar{\alpha}$ depending only upon the data and $p$ such that

[TABLE]

We observe that $\omega\downarrow 0$ as $R\downarrow 0$.

To proceed further, we now turn our attention to the case $K_{20R^{(j_{*})}}\subset K_{R^{\frac{1}{p}}}$. If there exists $j\in\mathbb{N}^{+}$ such that $\bar{t}=t^{(j)}$, then (6.74) is the desired estimate for the second alternative. If $\bar{t}\neq t^{(j)}$ for any $j\geq 0$, then there exists $\bar{j}\in\mathbb{N}^{+}$ such that $t^{(\bar{j})}<\bar{t}<t^{(\bar{j}+1)}$ and $\bar{j}+1<j_{*}$. The iteration scheme only need a slight modification. We perform the iterative arguments $\bar{j}$ times to obtain (6.74) for $j=\bar{j}$. Starting from this estimate, we repeat the arguments from step 1 to step 2 in §6.5 to obtain the estimate (6.73) with $k=\bar{j}$. Furthermore, we can repeat the the arguments in step 3 and step 4 in §6.5 with $t^{(k+1)}=\bar{t}$, since the proof still works with the condition $t^{(k+1)}=t^{(k)}+\delta_{1}\nu_{5}\omega^{(1-p)(2-p)}M_{1}^{k}R^{p}$ replaced by

[TABLE]

Then we conclude that the DeGiorgi-type estimate (6.74) holds in the cylinder $K_{\frac{1}{4}c_{2}R}\times(\tilde{t},\bar{t}]$, with $s_{j}$ replaced by $s_{\bar{j}+1}$.

In conclusion, we have proved the following proposition.

Proposition 6.13**.**

Suppose that the assumptions (6.31) and (6.75) are in force. Then there exists a constant $s$ depending only upon the data and $\omega$, such that, either

[TABLE]

for some constants $\kappa$, $\bar{C}$ and $\bar{\alpha}$ depending only upon the data and $p$. Moreover, $s\uparrow+\infty$ as $\omega\downarrow 0$.

7. Proof of the main result concluded

As we have already discussed in §4, the strategy of the proof is to study oscillation of the weak solution in a sequence of nested and shrinking cylinders with common vertex and prove that the essential oscillation converges to zero. We follow the notation used in §4 and assume that the common vertex coincides with $(0,0)$. To start with, we set $R_{1}=R$,

[TABLE]

From Proposition 5.4 and 6.13, we conclude that there exists $\sigma(\omega)$ such that either

[TABLE]

and $\sigma(\omega)$ satisfies $\sigma(\omega)<1$ and $\sigma(\omega)\uparrow 1$ as $\omega\downarrow 0$. Next, we set $\omega_{1}=\sigma(\omega)\omega$, $R_{2}=2^{-p}R_{1}^{p}$ and construct the reference parabolic cylinder $\overline{Q}_{2}=Q(R_{2},R_{2}^{\frac{1}{p}})$. We see that $\overline{Q}_{2}\subset Q_{1}\cap\overline{Q}_{1}$ and $\operatorname*{ess\,osc}_{\overline{Q}_{2}}u\leq\omega$. Moreover, we choose the cylinder $Q_{2}$ by

[TABLE]

At this point, we apply Proposition 5.4 and 6.13 again, with $Q_{1}$ and $\overline{Q}_{1}$ replaced by $Q_{2}$ and $\overline{Q}_{2}$, to obtain either

[TABLE]

[TABLE]

We continue this process to find two sequences $\{R_{n}\}_{n=1}^{\infty}$ and $\{\omega_{n}\}_{n=1}^{\infty}$ such that

[TABLE]

Then we have $R_{n}\downarrow 0$ as $n\to\infty$. From [9, Page 222], we see that $\omega_{n}\downarrow 0$ as $n\to\infty$. With these choices, we define $\overline{Q}_{n+1}=Q(R_{n+1},R_{n+1}^{\frac{1}{p}})$ and determine the cylinder $Q_{n+1}$ by

[TABLE]

where $n=2,3,\cdots.$ These cylinders are nested with common vertex $(0,0)$ and shrinking to this point as $n\to\infty$. Next, we remark that there is no any subsequence $\{n_{k}\}_{k=1}^{\infty}$ such that $n_{k}\to\infty$ and

[TABLE]

This is because if this estimate holds true for any $k=1,2,\cdots$, then $\omega=0$, which contradicts to the assumption $\omega>0$. Repeated application of Proposition 5.4 and 6.13, we conclude that there exist subsequences $\{n_{k}\}_{k=1}^{\infty}$ and $\{n_{k}^{\prime}\}_{k=1}^{\infty}$ such that either

[TABLE]

Therefore, we conclude that either $\operatorname*{ess\,osc}_{Q_{n_{k}}}u\downarrow 0$ or $\operatorname*{ess\,osc}_{\overline{Q}_{n_{k}^{\prime}}}u\downarrow 0$ as $k\to\infty$. This proves the theorem.

Acknowledgement

The author wishes to thank Eurica Henriques, Peter Lindqvist, Irina Markina and José Miguel Urbano for the valuable discussions.