Continuity of solutions to singular parabolic equations with coefficients from Kato-type classes

Abstract

We prove local boundedness and continuity of solutions to divergence type quasi-linear singular parabolic equations with measurable coefficients and lower order terms from nonlinear Kato classes.

1 Introduction and main results

In this paper we are concerned with divergence type quasi-linear singular parabolic equation with measurable coefficients and lower order terms. This class of equations has numerous applications and has been attracting attention for several decades (see, e.g. the monographs [7, 16, 28], survey [8] and reference therein).

Let $\varOmega$ be a domain in ${\mathbb {R}}^n,\ n\ge 2$, and for any $T>0$ let $\varOmega _T$ denote the cylindrical domain $\varOmega \times (0,T)$. We consider quasi-linear parabolic differential equation of the form

$$\begin{aligned} u_t-{{\mathrm{div}}}{\mathbb {A}}(x,t,u,\nabla u)=b(x,t,u,\nabla u),\quad (x,t)\in \varOmega _T. \end{aligned}$$

(1.1)

Throughout the paper we suppose that the functions ${\mathbb {A}}:\varOmega \times {\mathbb {R}}^+\times {\mathbb {R}}\times {\mathbb {R}} ^n\rightarrow {\mathbb {R}}^n$ and $b:\varOmega \times {\mathbb {R}}^+\times {\mathbb {R}}\times {\mathbb {R}} ^n\rightarrow {\mathbb {R}}^n$ are such that ${\mathbb {A}}(\cdot ,\cdot ,u,\xi ),b(\cdot ,\cdot ,u,\xi )$ are Lebesgue measurable for all $u\in {\mathbb {R}},\ \xi \in {\mathbb {R}}^n$, and ${\mathbb {A}}(x,t,\cdot ,\cdot ),b(x,t,\cdot ,\cdot )$ are continuous for almost all $(x,t)\in \varOmega _T$. We also assume that the following structure conditions are satisfied:

$$\begin{aligned}&{\mathbb {A}}(x,t,u,\xi )\, \xi \ge \mu _1|\xi |^p-g_1(x)|u|^p-f_1(x),\nonumber \\&|{\mathbb {A}}(x,t,u,\xi )|\le \mu _2|\xi |^{p-1}+g_2(x)|u|^{p-1}+f_2(x),\nonumber \\&|b(x,t,u,\xi )|\le h(x)|\xi |^{p-1}+g_3(x)|u|^{p-1}+f_3(x), \end{aligned}$$

(1.2)

where $\frac{2n}{n+1}<p<2,\mu _1,\mu _2$ are positive constants and $h(x),g_i(x),f_i(x),\ i=1,2,3$ are nonnegative functions, satisfying conditions which will be specified bellow.

The aim of this paper is to establish basic qualitative properties such as local boundedness of weak solutions and their continuity under minimal possible restrictions on the coefficients in structure conditions (1.2). These properties are indispensable in the qualitative theory of second-order elliptic and parabolic equations. For Eq. (1.1) with $g_i(x),f_i(x),\ i=1,2,3$ constants the local boundedness and Hölder continuity of solutions was known since mid-1980s (see [7] for the results, references and historical notes), and a resent breakthrough has been made in [9, 10], where the Harnack inequality has been proved. Before stating precisely our results we make several remarks related to lower order terms of (1.1) and refer the reader for an extensive survey of the regularity issues to [7–10].

Local boundedness and Hölder continuity of weak solutions to homogeneous linear divergence type second-order elliptic and parabolic equations with measurable coefficients without lower order terms is known since the famous results by De Giorgi [6] and Nash [20], and the Harnack inequality since Moser’s celebrated paper [18]. However, in presence of lower order term in the equation weak solutions may have singularities and/or internal zeroes, and the Harnack inequality in general may not be valid, as one can easily realize looking at the equation $-\varDelta u+\frac{c}{|x|^2}u=0$. It was Serrin [21] who generalized Moser’s result to the case of quasi-linear equations with lower order terms with conditions expressed in terms of $L^q$-spaces. Using probabilistic techniques Aizenman and Simon in their famous paper [1] proved the Harnack inequality and continuity of weak solutions to the equation $-\varDelta u+Vu=0$ under the local Kato class condition on the potential V. Moreover, they showed that the Kato-type condition on the potential V is necessary for the validity of the Harnack inequality. Soon after that Chiarenza et al. [5] developed a real variables technique to prove the Harnack inequality for a linear equation of divergence type with measurable coefficients and the potential from the Kato class, thus extending Aizenman, Simon’s result. Kurata [14] extended the method of Chiarenza, Fabes and Garofalo and proved the same to the equation $\sum _{i,j=1}^{n}\frac{\partial }{\partial x_i}(a_{ij}\frac{\partial u}{\partial x_j})+\sum _{i=1}^nb_i\frac{\partial u}{\partial x_i}+Vu=0$, with $|b|^2, V$ from the Kato class. Both papers [5] and [14] make a heavy use of Green’s functions which makes this approach inapplicable to quasi-linear equations. To treat the quasi-linear case of p-Laplacian with a lower order term Biroli [2] introduced the notion of the nonlinear Kato class and gave the Harnack inequality for positive solutions to $-\varDelta _pu+Vu^{p-1}=0$. This was extended in [25] to the general case of quasi-linear elliptic equations with lower order terms.

For second-order linear parabolic equations with measurable coefficients (without lower order terms) Hölder continuity of solutions was first proved by Nash [20]. Moser [19] proved the validity of the Harnack inequality which was extended to the case of quasi-linear equations with $p=2$ in the structure conditions and structure coefficients from $L^q$-classes in [26]. The continuity of weak solutions and the Harnack inequality for second-order linear elliptic equations with lower order coefficients from Kato classes was proved by Zhang [29, 30].

The parabolic theory for degenerate quasi-linear equations differs substantially from the “linear” case $p=2$ which can be already realized looking at the Barenblatt solution to the parabolic p-Laplace equation. Di Benedetto developed an innovative intrinsic scaling method (see [7] and the references to the original papers there; see also a nice exposition in [27] where some recent advances are included) and proved the Hölder continuity of weak solutions to (1.1) for $p\ne 2$ for the case $f_1,f_2,f_3,h$ from $L^q$-classes, and the Harnack inequality for the parabolic p-Laplace equations. For the case of measurable coefficients in the main part of (1.1) the Harnack inequality was proved in the recent breakthrough paper [9]. The Harnack inequality and continuity of solutions to the porous medium equations and to the degenerate ($p > 2$) parabolic equations with singular lower order terms was proved in [3, 4, 17]. It is natural to conjecture that the Harnack inequality holds for the singular ($p < 2$) parabolic p-Laplace equation perturbed by lower order terms with coefficients form Kato classes. The difficulty is that seemingly neither De Giorgy nor Moser iteration techniques work in this situation.

In this paper following the strategy of [10] but using a different iteration, namely the Kilpeläinen–Malý technique [13] properly adapted to the parabolic equations [17, 23, 24], we establish the local boundedness and continuity for solutions of (1.1).

In what follows we use the notion of the Wolf potential of a function g(x), which is defined by

$$\begin{aligned} W_{\alpha ,\beta }^g(x;\rho ):=\int _0^\rho \bigg \{\frac{1}{r^{n-\alpha \beta }} \int _{B_r(x)}|g(z)|\,\hbox {d}z\bigg \}^{\frac{1}{\alpha -1}}\frac{\hbox {d}r}{r},\quad \alpha >1,\quad n>\alpha \beta . \end{aligned}$$

(1.3)

The corresponding nonlinear Kato-type classes $K_{\alpha ,\beta }$ are defined by

$$\begin{aligned} K_{\alpha ,\beta }:=\left\{ g\in L^1(\varOmega ):\lim _{\rho \rightarrow 0}\sup _{x\in \varOmega }W_{\alpha ,\beta }^g(x;\rho )=0\right\} . \end{aligned}$$

(1.4)

As one can easily see, for $p=2$, the nonlinear Kato class $K_p:=K_{p,1}$ reduces to the standard definition of the Kato class with respect to the Laplacian [1, 22].

For the functions in the right-hand sides of (1.2) we assume that

$$\begin{aligned} g_1,f_1,g_2^{\frac{p}{p-1}},\quad f_2^{\frac{p}{p-1}}\in K_{p+1,\frac{p}{p+1}},\quad h^p, g_3, f_3\in K_p. \end{aligned}$$

(1.5)

Before formulating the main results let us remind the reader of the definition of a weak solution to Eq. (1.1). We say that the function $u\in V_\mathrm{loc}(\varOmega _T):=C_\mathrm{loc}(0,T;L_\mathrm{loc}^2(\varOmega ))\cap L_\mathrm{loc}^p(0,T;W_\mathrm{loc}^{1,p}(\varOmega ))$ is a local weak solution of equation (1.1) if for every sub-interval $[t_1,t_2]\subset (0,T]$ the following integral identity is valid

$$\begin{aligned} \int _\varOmega u\varphi \,\hbox {d}x\bigg |_{t_1}^{t_2}+\int _{t_1}^{t_2}\int _\varOmega \{ -u\,\varphi _t+{\mathbb {A}}(x,t,u,\nabla u)\nabla \varphi -b(x,t,u,\nabla u)\varphi \}\,\hbox {d}x\,\hbox {d}t=0 \end{aligned}$$

(1.6)

for any function $\varphi \in W_\mathrm{loc}^{1,2}(0,T:L^2(\varOmega ))\cap L_\mathrm{loc}^p(0,T;\overset{\circ }{W}{}_\mathrm{loc}^{1,p}(\varOmega ))$.

Further on we assume without loss of generality that $\frac{\partial u}{\partial t}\in L^2(\varOmega _T)$ since otherwise we can pass to the Steklov averages (see, e.g. [7]).

Remark 1

The parameters $\{n,p,\mu _1,\mu _2\}$ are the data and we say that generic constant $\gamma =\gamma (n,p,\mu _1,\mu _2)$ depends upon the data if it can be quantitatively determined a priori only in terms of the indicated parameters.

In what follows we use the following quantities

$$\begin{aligned}&W_f(\rho ):=\sup _{x\in \varOmega }W_{p+1,\frac{p}{p+1}}^{f_1+f_2^{\frac{p}{p-1}}}(x;\rho )+\sup _{x\in \varOmega }W_{p,1}^{f_3}(x;\rho ),\\&W_g(\rho ):=\sup _{x\in \varOmega }W_{p+1,\frac{p}{p+1}}^{g_1+g_2^{\frac{p}{p-1}}}(x;\rho )+\sup _{x\in \varOmega }W_{p,1}^{g_3}(x;\rho ),\\&W_h(\rho ):=\sup _{x\in \varOmega }W_{p,1}^{h^p}(x;\rho ). \end{aligned}$$

The first main result of this paper is the local boundedness of solutions.

Let $x\in \varOmega ,\ 0<s<T$, for any $\rho ,\tau >0$ we define $B_\rho (x)=\{y:|x-y|<\rho \},\ Q_{\rho ,\tau }(x,s):=Q_{\rho ,\tau }^-(x,s)\cup Q_{\rho ,\tau }^+(x,s)$, where $Q_{\rho ,\tau }^-(x,s):= B_\rho (x)\times (s-\tau ,s),\ Q_{\rho ,\tau }^+(x,s):=B_\rho (x)\times (s,s+\tau )$.

Theorem 1.1

Let the conditions (1.2), (1.5) be fulfilled and u be a local weak solution to Eq. (1.1). Then there exists $\nu _1\in (0,1)$ depending only on the data such that the inequality

$$\begin{aligned} W_h(32\rho )+W_g(32\rho )\le \nu _1 \end{aligned}$$

(1.7)

implies that either

$$\begin{aligned} \Big (\frac{t-s}{\rho ^p}\Big )^{\frac{1}{2-p}}\le W_f(32\rho ), \end{aligned}$$

(1.8)

or

$$\begin{aligned}&\mathop {{{\mathrm{ess\,sup}}}}\limits _{Q_{\frac{\rho }{2},t-s}^-(y,t)}|u|\le \gamma (t-s)^{-\frac{n}{\varkappa }}\bigg (\mathop {{{\mathrm{ess\,sup}}}}\limits _{2s-t<\tau <t}\int _{B_\rho (y)}|u(x,\tau )|\,\hbox {d}x\bigg )^{\frac{p}{\varkappa }}+\gamma \Big (\frac{t-s}{\rho ^p}\Big )^{\frac{1}{2-p}},\nonumber \\&\varkappa =p+n(p-2)>0, \end{aligned}$$

(1.9)

for all cylinders $Q_{\rho ,2(t-s)}^-(y,t)\subset \varOmega _T$.

Having established the local boundedness we proceed with the continuity.

Theorem 1.2

Let conditions (1.2), (1.5) be fulfilled and u be a bounded local weak solution to Eq. (1.1). Then u is continuous, that is $u\in C(\varOmega _T)$.

For a fixed cylinder $Q_{2\rho ,(2\rho )^p\theta }^-(y,s)\subset \varOmega _T$ denote by $\mu _{\pm }$ and $\omega$, nonnegative numbers such that

$$\begin{aligned} \mu _+\ge \mathop {{{\mathrm{ess\,sup}}}}\limits _{Q_{2\rho ,(2\rho )^p\theta }^-(y,s)}u,\quad \mu _-\le \mathop {{{\mathrm{ess\,inf}}}}\limits _{Q_{2\rho ,(2\rho )^p\theta }^-(y,s)}u,\quad \omega \ge \mu _+-\mu _-. \end{aligned}$$

The next is a De Giorgi-type lemma (cf. [10]), and its formulation is almost the same as in [10]. However, due to the different structure conditions the De Giorgi-type iteration cannot be used, so we adapt the Kilpeläinen–Malý iteration [13], combined with ideas from [17, 23, 24], where the Kilpeläinen–Malý technique was adapted to parabolic equations.

Theorem 1.3

Let the conditions (1.2), (1.5) be fulfilled and u be a bounded local weak solution to Eq. (1.1). Fix $\xi ,a\in (0,1)$, there exist numbers $\nu _1\in (0,1)$ depending only on the data and $B\ge 1,\nu \in (0,1)$ depending on $\theta ,\xi ,\omega ,a, {{\mathrm{ess\,sup}}}_{\varOmega _T}|u|$ and the data such that if

$$\begin{aligned} W_h(32\rho )\le \nu _1 \end{aligned}$$

(1.10)

and

$$\begin{aligned} |\{(x,t)\in Q_{2\rho ,(2\rho )^p\theta }^-(y,s):u(x,t)\le \mu _-+\xi \omega \}| \le \nu |Q_{2\rho ,(2\rho )^p\theta }^-(y,s)|, \end{aligned}$$

(1.11)

then either

$$\begin{aligned} \xi \omega \le B(W_f(32\rho )+W_g(32\rho )), \end{aligned}$$

(1.12)

or

$$\begin{aligned} u(x,t)\ge \mu _-+a\xi \omega \quad {\mathrm{for\, almost\, all\, (a.a.) }}\quad (x,t)\in Q_{\rho ,\rho ^p\theta }^-(y,s). \end{aligned}$$

(1.13)

Likewise if (1.10) holds and

$$\begin{aligned} |\{(x,t)\in Q_{2\rho ,(2\rho )^p\theta }^-(y,s):u(x,t)\le \mu _+-\xi \omega \}| \ge \nu |Q_{2\rho ,(2\rho )^p\theta }^-(y,s)|, \end{aligned}$$

(1.14)

then either (1.12) holds true, or

$$\begin{aligned} u(x,t)\le \mu _+-a\xi \omega \quad {\mathrm{for\, a.a. }}\quad (x,t)\in Q_{\rho ,\rho ^p\theta }^-(y,s). \end{aligned}$$

(1.15)

Moreover, if $g_1=g_2=g_3=0$, the constants $\nu ,B$ can be chosen independent from ${{\mathrm{ess\,sup}}}_{\varOmega _T}|u|$.

Next is a De Giorgi-type lemma involving “initial data”.

Theorem 1.4

Let the conditions (1.2), (1.5) be fulfilled and u be a bounded local weak solution to Eq. (1.1). Fix $\xi ,a\in (0,1)$, there exist numbers $\nu _1\in (0,1)$ depending only on the data and $B\ge 1,\nu \in (0,1)$ depending on $\theta ,\xi ,\omega ,a, {{\mathrm{ess\,sup}}}_{\varOmega _T}|u|$ and the data such that if (1.10) holds true and if

$$\begin{aligned} u(x,s)\ge \mu _-+\xi \omega \quad {\mathrm{for\, a.a. }}\quad x\in B_{2\rho }(y), \end{aligned}$$

(1.16)

and

$$\begin{aligned} |\{(x,t)\in Q_{2\rho ,(2\rho )^p\theta }^+(y,s):u(x,t)\le \mu _+-\xi \omega \}| \le \nu |Q_{2\rho ,(2\rho )^p\theta }^+(y,s)|, \end{aligned}$$

(1.17)

then either (1.12) holds true, or

$$\begin{aligned} u(x,t)\ge \mu _-+a\xi \omega \quad {\mathrm{for\, a.a. }}\quad (x,t)\in Q_{\rho ,(2\rho )^p\theta }^+(y,s). \end{aligned}$$

(1.18)

Likewise if (1.10) holds and if

$$\begin{aligned} u(x,s)\le \mu _+-\xi \omega \quad {\mathrm{for\, a.a. }}\quad x\in B_{2\rho }(y), \end{aligned}$$

(1.19)

and

$$\begin{aligned} |\{(x,t)\in Q_{2\rho ,(2\rho )^p\theta }^+(y,s):u(x,t)\ge \mu _+-\xi \omega \}| \le \nu |Q_{2\rho ,(2\rho )^p\theta }^+(y,s)|, \end{aligned}$$

(1.20)

then either (1.12) holds true, or

$$\begin{aligned} u(x,t)\le \mu _+-a\xi \omega \quad {\mathrm{for\, a.a. }}\;(x,t)\in Q_{\rho ,(2\rho )^p\theta }^+(y,s). \end{aligned}$$

(1.21)

If $g_1=g_2=g_3=0$, the constants $\nu ,B$ can be chosen independent from ${{\mathrm{ess\,sup}}}_{\varOmega _T}|u|$.

The following theorem is an expansion of positivity result, analogous in formulation as well as in the proof [10].

Theorem 1.5

Let the conditions (1.2), (1.5) be fulfilled and u be a bounded local weak solution to Eq. (1.1). Assume that for some $(y,s)\in \varOmega _T$ and some $\rho >0$

$$\begin{aligned} |\{x\in B_\rho (y):u(x,s)\le \mu _-+N\}|\le (1-\alpha ) |B_\rho (y)| \end{aligned}$$

(1.22)

for some $N>0$ and some $\alpha \in (0,1)$. Then there exist positive constants $\nu _1\in (0,1)$ depending only on the data and $B\ge 1,\ \sigma ,\varepsilon ,b\in (0,1)$ depending on the data and $\alpha$, such that if (1.10) holds true, then either

$$\begin{aligned} N\le B(W_f(32\rho )+W_g(32\rho )), \end{aligned}$$

(1.23)

or

$$\begin{aligned} u(x,t)\ge \mu _-+\sigma N\quad {\mathrm{for\, a.a. }}\;\; x\in B_{2\rho }(y), \end{aligned}$$

(1.24)

for all $s+b(1-\varepsilon )N^{2-p}\rho ^p\le t\le s+bN^{2-p}\rho ^p$.

If on the other hand

$$\begin{aligned} |\{x\in B_\rho (y):u(x,s)\ge \mu _+-N\}|\le (1-\alpha ) |B_\rho (y)|, \end{aligned}$$

(1.25)

and if (1.10) holds, then either (1.23) holds true, or

$$\begin{aligned} u(x,t)\le \mu _+-\sigma N\quad {\mathrm{for\, a.a. }}\;\;x\in B_{2\rho }(y), \end{aligned}$$

(1.26)

for all $s+b(1-\varepsilon )N^{2-p}\rho ^p\le t\le s+bN^{2-p}\rho ^p$.

The rest of the paper contains the proof of the above theorems.

2 Auxiliary material and integral estimates

2.1 Auxiliary properties and local energy estimates

The following lemmas will be used in the sequel. The first one is the well-known De Giorgi–Poincare lemma (see [15]).

Lemma 2.1

Let $u\in W^{1,1}(B_\rho (y))$ for some $\rho >0$ and $y\in {\mathbb {R}}^n$. Let k and l be real numbers such that $k<l$. Then exists a constant $\gamma >0$ depending only on n, such that

$$\begin{aligned} (l-k)|A_{k,\rho }||B_\rho (y){\setminus } A_{l,\rho }|\le \gamma \rho ^{n+1}\int _{A_{l,\rho }{\setminus } A_{k,\rho }}|\nabla u|\,\mathrm{d}x, \end{aligned}$$

(2.1)

where $A_{k,\rho }=\{x\in B_\rho (y):u(x)<k\}$.

The next lemma is an interpolation lemma.

Lemma 2.2

Let $\{y_j\},\ j=0,1,2,\ldots$ be a sequence of bounded positive numbers satisfying the recursive inequalities

$$\begin{aligned} y_j\le Aa^jy_{j+1}^\sigma , \end{aligned}$$

where $A,a>1$ and $\sigma \in (0,1)$ are given constants. Then there exists a constant $\gamma >0$ depending only on $a,\sigma$ such that

$$\begin{aligned} y_0\le \gamma A^{\frac{1}{1-\sigma }}. \end{aligned}$$

In what follows we will frequently use the following lemma.

Lemma 2.3

Let $\varphi \in W_0^{1,p}(B_\rho (y)),\ 0\le f\in L_\mathrm{loc}^1$. Then there exists $\gamma >0$ such that

$$\begin{aligned} \int _{B_\rho (y)}f|\varphi |^p\mathrm{d}x\le \gamma \sup _{x\in B_{2\rho }(y)}W_{p,1}^f(x;2\rho )^{p-1}\int _{B_\rho (y)} |\nabla \varphi |^p\mathrm{d}x. \end{aligned}$$

Proof

Let v be the weak solution to

$$\begin{aligned} - \varDelta _pv=f\quad \text {in }B_\rho (y),\ v\in \overset{\circ }{W}{}^{1,p}( B_\rho (y))=0. \end{aligned}$$

Then by [13] $\sup _{x\in B_\rho (y)}v(x)\le \gamma \sup _{x\in B_{2\rho (y)}}W_{p,1}^f(x,2\rho )$. Multiplying the equation $-\varDelta _pv=f$ by $\frac{|\varphi |^p}{(v+\varepsilon )^{p-1}},\ \varepsilon \rightarrow 0$, integrating by parts and letting $\varepsilon \rightarrow 0$ we obtain

$$\begin{aligned} \int _{B_\rho (y)}\frac{|\nabla v|^p|\varphi |^p}{v^p}\hbox {d}x+ \int _{B_\rho (y)}\frac{f|\varphi |^p}{v^{p-1}}\hbox {d}x\le \gamma \int _{B_\rho (y)}\frac{|\nabla v|^{p-1}|\varphi |^{p-1}}{v^{p-1}}|\nabla \varphi |\,\hbox {d}x. \end{aligned}$$

Using the Young’s inequality we get

$$\begin{aligned} \int _{B_\rho (y)}\frac{f|\varphi |^p}{v^{p-1}}\hbox {d}x\le \gamma \int _{B_\rho (y)}|\nabla \varphi |^p \hbox {d}x. \end{aligned}$$

Hence the requited inequality follows. $\square$

Lemma 2.4

Let u be a solution to Eq. (1.1) in $\varOmega _T$. Then there exists $\gamma >0$ depending only on the data, such that for every cylinder $Q^+_{\rho ,\rho ^p\theta }(y,s)\subset \varOmega _T$, and any $k\in {\mathbb {R}}$ and any smooth $\xi (x,t)$ which is equal to zero for $(x,t)\in \partial B_\rho (y)\times (s,s+\rho ^p\theta )$ one has

$$\begin{aligned}&\mathop {{{\mathrm{ess\,sup}}}}\limits _{s<t<s+\rho ^{p\theta }} \int _{B_\rho (y)}(u-k)_{\pm }^2 \xi ^p(x,t)\,\mathrm{d}x+ \iint _{Q^+_{\rho ,\rho ^p\theta }(y,s)}|\nabla (u-k)_\pm |^p\xi ^p\mathrm{d}x\,\mathrm{d}\tau \nonumber \\&\quad \le \int _{B_\rho (y)}(u-k)_{\pm }^2 \xi ^p(x,s)\,\mathrm{d}x+\gamma \iint _{Q^+_{\rho ,\rho ^p\theta }(y,s)}\left( (u-k)_\pm ^2|\xi _\tau |+(u-k)_\pm ^p|\nabla \xi |^p\right) \,\mathrm{d}x\,\mathrm{d}\tau \nonumber \\&\qquad +\, \gamma \iint _{Q^+_{\rho ,\rho ^p\theta }(y,s)}\left( |u|^p\left( g_1+g_2^{\frac{p}{p-1}}\right) \chi ((u-k)_\pm \!>\!0)\!+\!(u-k)_\pm |u|^{p-1}g_3\right) \xi ^p\,\mathrm{d}x\,\mathrm{d}\tau \nonumber \\&\qquad +\, \gamma \iint _{Q^+_{\rho ,\rho ^p\theta }(y,s)} (u-k)_\pm ^ph^p\xi ^p\mathrm{d}x\,\mathrm{d}\tau \nonumber \\&\qquad +\,\gamma \iint _{Q^+_{\rho ,\rho ^p\theta }(y,s)}\left( \left( f_1+f_2^{\frac{p}{p-1}}\right) \chi ((u-k)_\pm >0)+(u-k)_\pm f_3\right) \xi ^p\mathrm{d}x\,\mathrm{d}\tau . \end{aligned}$$

(2.2)

Proof

Test (1.6) by $\varphi =(u-k)_\pm \xi ^p$ and use conditions (1.2), the Hölder and Young inequalities. $\square$

2.2 Integral estimates of solutions

Fix a positive number $a\ge 1$ depending only on the data, which will be specified later. Set $v=u_++a W_f(32\rho )$ and

$$\begin{aligned} G(v)=\left\{ \begin{array}{ll} v&{}\quad \text {for }\quad v\ge 1\\ v^2&{}\quad \text {for }\quad 0<v\le 1. \end{array}\right. \end{aligned}$$

Lemma 2.5

Let the conditions of Theorem 1.1 be fulfilled. Then there exists a constant $\gamma >0$ depending only on the data, such that for any $l,\delta >0,\ 0<\lambda <\min (1,\frac{2-p}{p-1}),\ k>p$ and any cylinder $Q^-_{r,\theta }({\bar{x}},{\bar{t}})\subset Q^-_{\rho ,2(t-s)}(y,t)$ and any smooth $\xi (x,t)=\xi ^{(1)}(x)\xi ^{(2)}(t)$, where $\xi ^{(1)}(x)\in C_0^\infty (B_r({\bar{x}}))$ and $\xi ^{(2)}(t)$ is equal to zero for $t\le {\bar{t}}-\theta$ one has

$$\begin{aligned}&\sup _{{\bar{t}}-\theta <t<{\bar{t}}}\int _{L(t)}vG\Big (\frac{v-l}{\delta }\Big )\xi ^k\mathrm{d}x +\delta ^{-2}\iint _L\bigg (\int _l^v\Big (1+\frac{s-l}{\delta }\Big )^{-1-\lambda }\mathrm{d}s\nonumber \\&\qquad +\, v\Big (1+\frac{v-l}{\delta }\Big )^{-1-\lambda }\bigg )|\nabla v|^p\xi ^k\mathrm{d}x\,\mathrm{d}t \nonumber \\&\quad \le \gamma \iint _Lv\frac{v-l}{\delta }|\xi _t|\xi ^{k-p}\mathrm{d}x\,\mathrm{d}t+ \gamma \delta ^{p-2}\iint _Lv\frac{v-l}{\delta }|\nabla \xi |^p\xi ^{k-p}\mathrm{d}x\,\mathrm{d}t\nonumber \\&\qquad +\,\gamma \delta ^{-2}\iint _Lv^{p+1}\Big (1+\frac{v-l}{\delta }\Big )^{-1-\lambda }h_1\xi ^k\mathrm{d}x\,\mathrm{d}t \nonumber \\&\qquad +\,\gamma \delta ^{-2}\iint _Lv^p\int _l^v\Big (1+\frac{s-l}{\delta }\Big )^{-1-\lambda }\mathrm{d}sh_2 \, \xi ^k\mathrm{d}x\,\mathrm{d}t, \end{aligned}$$

(2.3)

where $L=Q^-_{r,\theta }({\bar{x}},{\bar{t}})\cap \{v>l\}$, $L(t)=L\cap \{\tau =t\}$, $h_1(x)=g_1(x)+g_2^{\frac{p}{p-1}}(x)+a^{-p}W_f^{-p}(32\rho )(f_1(x)+f_2^{\frac{p}{p-1}}(x))$, $h_2(x)=g_1(x)+g_3(x)+h^p(x)+a^{-p}W_f^{-p}(32\rho ) f_1(x)+a^{1-p}W_f^{1-p}(32\rho )f_3(x)$.

Proof

First note that

$$\begin{aligned} \bigg (\int _l^v\Big (1+\frac{s-l}{\delta }\Big )^{-1-\lambda }\mathrm{d}s\bigg )_+\asymp \frac{v-l}{1+\frac{v-l}{\delta }}, \end{aligned}$$

(2.4)

and

$$\begin{aligned}&\int _l^vw\,\mathrm{d}w\int _l^w\Big (1+\frac{s-l}{\delta }\Big )^{-1-\lambda }\mathrm{d}s\ge \gamma v(v-l)\int _l^{\frac{v+l}{2}}\Big (1+\frac{s-l}{\delta }\Big )^{-1-\lambda }\mathrm{d}s\nonumber \\&\quad =\gamma \delta v(v-l)\int _0^{\frac{v-l}{2\delta }}(1+s)^{-1-\lambda }\mathrm{d}s\ge \gamma \delta ^2vG\Big ( \frac{v-l}{\delta }\Big ). \end{aligned}$$

(2.5)

Test (1.6) by $\varphi =v\varPhi (v)\xi ^k,\ \varPhi (v)=(\int _l^v(1+\frac{s-l}{\delta })^{-1-\lambda }\mathrm{d}s)_+$, using (1.2) and the Young inequality we have for any $t\in ({\bar{t}}-\theta ,{\bar{t}})$

$$\begin{aligned}&\int _{L(t)}\int _l^vw\varPhi (w)\,\mathrm{d}w\xi ^k\hbox {d}x+ \iint _L\Big (\varPhi (v)+v\Big (1+\frac{v-l}{\delta }\Big )^{-1-\lambda }\Big ) |\nabla v|^p\xi ^k\hbox {d}x\,\hbox {d}t\\&\quad \le \gamma \iint _L\int _l^vw\varPhi (w)\,\mathrm{d}w|\xi _t|\xi ^{k-1}\hbox {d}x\,\hbox {d}t\\&\qquad +\gamma \iint _L \varPhi ^p(v)\Big (1+\frac{v-l}{\delta }\Big )^{(1+\lambda )(p-1)} |\nabla \xi |^p\xi ^{k-p}\hbox {d}x\,\hbox {d}t\\&\qquad +\,\gamma \iint _L v\Big (1+\frac{v-l}{\delta }\Big )^{-1-\lambda } \left( v^p(g_1+g_2^{\frac{p}{p-1}})+f_1+f_2^{\frac{p}{p-1}}\right) \xi ^k\hbox {d}x\,\hbox {d}t \\&\qquad +\,\gamma \iint _L v^p\varPhi (v)(g_1+g_3+h^p)\xi ^k\hbox {d}x\,\hbox {d}t+ \gamma \iint _L\varPhi (v)\left( f_1+vf_3\right) \xi ^k\hbox {d}x\,\hbox {d}t. \end{aligned}$$

From this, using (2.4), (2.5) due to the choice of $\lambda$ we obtain the required (2.3) $\square$

Set

$$\begin{aligned} \psi (v)=\frac{1}{\delta }\bigg (\int _l^vs^{\frac{1}{p}}\Big (1+\frac{s-l}{\delta }\Big )^{-\frac{1+\lambda }{p}}\mathrm{d}s\bigg )_+. \end{aligned}$$

Lemma 2.6

Let the conditions of Lemma 2.5 be fulfilled and $0<\lambda <2-p$. Then there exists $\nu _1\in (0,1)$ depending only on the data such that the inequality

$$\begin{aligned} W_h(32\rho )+W_g(32\rho )+a^{-1}\le \nu _1 \end{aligned}$$

(2.6)

implies that

$$\begin{aligned}&\sup _{{\bar{t}}-\theta <t<{\bar{t}}}\int _{L(t)}vG\Big (\frac{v-l}{\delta }\Big )\xi ^k\mathrm{d}x +\delta ^{p-2}\iint _L|\nabla \psi |^p\xi ^k\mathrm{d}x\,\mathrm{d}t \le \gamma \iint _Lv\frac{v-l}{\delta }|\xi _t|\xi ^{k-p}\mathrm{d}x\,\mathrm{d}t \nonumber \\&\quad +\,\gamma \delta ^{p-2}\iint _L v\frac{v-l}{\delta }|\nabla \xi |^p\xi ^{k-p}\mathrm{d}x\,\mathrm{d}t+ \gamma l^{p+1}\delta ^{-2}\theta \int _{B_r({\bar{x}})}h_1\mathrm{d}x\!+\!\gamma l^p\delta ^{-1}\theta \int _{B_r({\bar{x}})}h_2\mathrm{d}x.\nonumber \\ \end{aligned}$$

(2.7)

Proof

Recall that $\varPhi (v)\asymp \frac{v-l}{1+\frac{v-l}{\delta }}$ and apply (2.6) and Lemma 2.3. Then since

$$\begin{aligned}&W_{p,1}^{h_1}(x;2\rho )\le \gamma W_{p+1,\frac{p}{p+1}}^{g_1+g_2^{\frac{p}{p-1}}}(x;2\rho )+\gamma a^{-\frac{p}{p-1}},\\&W_{p,1}^{h_2}(x;2\rho )\le \gamma W_{p+1,\frac{p}{p+1}}^{g_1}(x;2\rho )+\gamma W_{p,1}^{g_3+h^p}(x;2\rho )+\gamma (a^{-\frac{p}{p-1}}+a^{-1}), \end{aligned}$$

we have

$$\begin{aligned}&\gamma \iint _L(v-l)^{p+1} \Big (1+\frac{v-l}{\delta }\Big )^{-1-\lambda }h_1\xi ^k\hbox {d}x\,\hbox {d}t\nonumber \\&\quad \le \gamma \nu _1^{p-1}\iint _L(v-l)\Big (1+\frac{v-l}{\delta }\Big )^{-1-\lambda }|\nabla v|^p\xi ^k\hbox {d}x\,\hbox {d}t \nonumber \\&\qquad +\, \gamma \nu _1^{p-1}\iint _L(v-l)^{p+1}\Big (1+\frac{v-l}{\delta }\Big )^ {-1-\lambda }|\nabla \xi |^p \xi ^{k-p}\hbox {d}x\,\hbox {d}t\nonumber \\&\quad \le \gamma \nu _1^{p-1}\iint _Lv\Big (1+\frac{v-l}{\delta }\Big )^{-1-\lambda } |\nabla v|^p\xi ^k\hbox {d}x\,\hbox {d}t\nonumber \\&\qquad +\,\gamma \nu _1^{p-1}\delta ^p\iint _L v\frac{v-l}{\delta }|\nabla \xi |^p\xi ^{k-p}\hbox {d}x\,\hbox {d}t. \end{aligned}$$

(2.8)

Similarly

$$\begin{aligned}&\gamma \iint _L(v-l)^{p}\varPhi (v)h_2\xi ^k\hbox {d}x\,\hbox {d}t\le \gamma \iint _L\frac{(v-l)^{p+1}}{1+\frac{v-l}{\delta }}h_2\xi ^k\hbox {d}x\,\hbox {d}t \nonumber \\&\qquad \le \gamma \nu _1^{p-1}\iint _L\frac{v-l}{1+\frac{v-l}{\delta }}|\nabla v|^p\xi ^k\hbox {d}x\,\hbox {d}t+ \gamma \nu _1^{p-1}\iint _L\frac{(v-l)^{p+1}}{1+\frac{v-l}{\delta }} |\nabla \xi |^p\xi ^{k-p}\hbox {d}x\,\hbox {d}t \nonumber \\&\qquad \le \gamma \nu _1^{p-1}\iint _L\varPhi (v)|\nabla v|^p\xi ^k\hbox {d}x\,\hbox {d}t+ \gamma \nu _1\delta ^p\iint _Lv\frac{v-l}{\delta }|\nabla \xi |^p\xi ^{k-p}\hbox {d}x\,\hbox {d}t. \end{aligned}$$

(2.9)

From this, choosing $\nu _1$ sufficiently small, due to Lemma 2.5 we obtain the required (2.7). $\square$

3 $L_\mathrm{loc}^1-L_\mathrm{loc}^\infty$ estimate: Proof of Theorem 1.1

In what follows we suppose that

$$\begin{aligned} \Big (\frac{t-s}{\rho ^p}\Big )^{\frac{1}{2-p}}\ge W_f(32\rho ). \end{aligned}$$

(3.1)

For $\sigma \in (0,1)$ and $i=0,1,2,\ldots$ set $\rho ^{(i)}:=\frac{\rho }{2}(2-\sigma ^i)$, $t^{(i)}:=(t-s)(2-\sigma ^{ip})$. Fix a point $({\bar{x}},{\bar{t}})\in Q_{\rho ^{(i)},t^{(i)}}^-(y,t)$, and let $\rho _0:=\rho _0^{(i)}=\dfrac{\rho \sigma ^i(1-\sigma )}{2}$, $t_0:=t_0^{(i)}=(t-s)\sigma ^{ip}(1-\sigma ^p)$,

$$\begin{aligned} \delta _0:=\max \bigg \{\bigg (\frac{1}{\eta }\frac{1}{t_0^{1+\frac{n}{p}}}\iint _{Q^-_{\rho _0,t_0}({\bar{x}},{\bar{t}})}v^2\hbox {d}x\,\hbox {d}t\bigg )^{\frac{p}{p+\varkappa }}, \Big (\frac{t_0}{\rho _0^p}\Big )^{\frac{1}{2-p}}\bigg \}, \end{aligned}$$

(3.2)

where v is defined in Sect. 2, and choose a number $m\ge 0$ so that

$$\begin{aligned} \delta _0^{2-p}\rho _0^p=2^{mp}t_0. \end{aligned}$$

(3.3)

For $\alpha =\frac{\varkappa }{3-p},\ l>0$ and $j=0,1,2,\ldots$ set $r_j:=\frac{\rho _0}{2^{j+m}},\ t_j:=\frac{t_0}{2^{\alpha j}},\ \delta _j(l):=l-l_j$, $\theta _j(l):=\min (t_j,r_j^p\delta _j^{2-p}(l)),\ B_j:=B_{r_j}({\bar{x}})$, $Q_j(l):=Q^-_{r_j,\theta _j(l)}({\bar{x}},{\bar{t}}),\ L_j(l):=Q_j(l)\cap \{u>l_j\}$, $L_j(l,t):=L_j\cap \{\tau =t\},\ \xi _j(x,t)=\xi _j^{(1)}(x)\xi _j^{(2)}(t)$, where $\xi _j^{(1)}(x)\in C_0^\infty (B_j),\ \xi _j^{(1)}(x)=1$ in $B_{j+1}$, $\xi _j^{(2)}(t)=1$ for $t\ge {\bar{t}}-2^{-\alpha }\theta _j(l)$, $\xi _j^{(2)}(t)=0$ for $t\le {\bar{t}}-\theta _j(l)$, $0\le \xi _j(x,t)\le 1$ and $|\nabla \xi _j|\le \gamma r_j^{-1}$, $|\frac{\partial \xi _j}{\partial t}|\le \gamma \theta _j^{-1}(l)$, and set also

$$\begin{aligned} A_j(l):=\frac{\delta _j^{(2-p)\frac{n}{p}}(l)l^{-1}}{\theta _j^{1+\frac{n}{p}}(l)} \iint _{L_j(l)}v\frac{v-l_j}{\delta _j(l)}\xi _j^{k-p}\hbox {d}x\,\hbox {d}t. \end{aligned}$$

(3.4)

The sequences of positive numbers $\{l_j\}_{j\in {\mathbb {N}}}$ and $\{\delta _j\}_{j\in {\mathbb {N}}}$ are defined inductively as follows. Fix a positive number $\eta \in (0,1)$ depending only on the data, which will be specified later. Put $l_0=0$ and $l_1=\delta _0$, where $\delta _0$ is defined in (3.2). The following inequality is clear

$$\begin{aligned} A_0(l_1)\le \eta . \end{aligned}$$

Suppose we have chosen $l_1,\ldots ,l_j$ and $\delta _i=\delta _i(l_{i+1})=l_{i+1}-l_i,\ i=0,1,\ldots ,j-1$ such that $l_i+\frac{1}{2}\delta _{i-1}\le l_{i+1}\le l_i+2^{\frac{p-\alpha }{2-p}}\delta _{i-1},\ i=1,\ldots ,j-1$,

$$\begin{aligned} A_i(l_{i+1})\le \eta ,\quad i=0,1,\ldots ,j-1. \end{aligned}$$

(3.5)

Let us show how to choose $l_{j+1}$ and $\delta _j$. First we show that

$$\begin{aligned} A_j({\tilde{l}}_j)\le \eta ,\quad {\tilde{l}}_j=l_j+2^{\frac{p-\alpha }{2-p}}\delta _{j-1}. \end{aligned}$$

(3.6)

Further we show that $\theta _j(\tilde{l}_j)\le 2^{-\alpha }\theta _{j-1}(l_j)$ and so $Q_j(\tilde{l}_j)\subset Q_{j-1}(l_j)$ and $\{\xi _j\ne 0\}\subset \{\xi _{j-1}=1\}$.

Indeed, $\delta _j(\tilde{l}_j)=2^{\frac{p-\alpha }{2-p}}\delta _{j-1}$, thus $r_j^p\delta _j^{2-p}(\tilde{l}_j)=2^{-\alpha }r_{j-1}^p\delta _{j-1}^{2-p}$, therefore $\theta _j({\tilde{l}}_j)\le 2^{-\alpha }\theta _{j-1}(l_j)$ and

$$\begin{aligned} A_j(\tilde{l}_j)\le & {} 2^{-\frac{p-\alpha }{2-p}\frac{\varkappa }{p}} \delta _{j-1}^{-\frac{\varkappa }{p}} \theta _j^{-1-\frac{n}{p}}({\tilde{l}}_j) l_j^{-1} \iint _{L_j({\tilde{l}}_j)}v(v-l_j)\xi _{j-1}^{k-p}\hbox {d}x\,\hbox {d}t\\\le & {} 2^{-\frac{p-\alpha }{2-p}\frac{\varkappa }{p}+\alpha (1+\frac{n}{p})} \delta _{j-1}^{(2-p)\frac{n}{p}} \theta _{j-1}^{-1-\frac{n}{p}}( l_j) l_{j-1}^{-1} \iint _{L_{j-1}( l_j)}v\frac{v-l_{j-1}}{\delta _{j-i}}\xi _{j-1}^{k-p}\hbox {d}x\,\hbox {d}t\\= & {} A_{j-1}(l_j)\le \eta . \end{aligned}$$

Thus inequality (3.6) is proved. If $A_j(l_j+\frac{1}{2}\delta _{j-1})\le \eta$ we set $l_{j+1}=l_j+\frac{1}{2}\delta _{j-1}$. Note that $A_j(l)$ is continuous as a function of l. So if $A_j(l_j+\frac{1}{2}\delta _{j-1})>\eta$, the equation $A_j(l)=\eta$ has roots. Denote $l_{j+1}$ the largest root $A_j(l_{j+1})=\eta$ and in both cases we set $\delta _j=\delta _j(l_{j+1})=l_{j+1}-l_j$. Note that our choices guarantee that $\delta _j\le 2^{\frac{p-\alpha }{2-p}j}\delta _0=(\frac{t_j}{r_j^p})^{\frac{1}{2-p}}$ and

$$\begin{aligned} A_j(l_{j+1})\le \eta . \end{aligned}$$

(3.7)

Further we set $\theta _j=\theta _j(l_{j+1})$, $Q_j=Q_j(l_{j+1})$, $L_j=L_j(l_{j+1})$ and $L_j(t)=L_j(l_{j+1},t)$.

The following lemma is a key in the Kilpëlainen–Malý technique [13].

Lemma 3.1

For all $j\ge 2$ there exists $\gamma >0$ depending only on the data, such that

$$\begin{aligned} \delta _j\le \frac{1}{2}\delta _{j-1}+\gamma l_j\bigg (r_j^{p-n}\int _{B_j}h_1\mathrm{d}x\bigg )^{\frac{1}{p}}+\gamma l_j\bigg (r_j^{p-n}\int _{B_j}h_2\mathrm{d}x\bigg )^{\frac{1}{p-1}}. \end{aligned}$$

(3.8)

Proof

Fix $j\ge 2$ and without loss assume that

$$\begin{aligned} \delta _j>\frac{1}{2}\delta _{j-1}, \end{aligned}$$

(3.9)

since otherwise (3.8) is evident. This inequality guarantees that $A_j(l_{j+1})=\eta$. Let us estimate the term in the right-hand side of (3.4) with $l=l_{j+1}$. For this we decompose $L_j$ as $L_j=L'_l\cup L''_j$, $L'_j=\{(x,t)\in L_j:\frac{v(x,t)-l_j}{\delta _j}\le \varepsilon \}$, $L''_j=L_j{\setminus } L'_j$, where $\varepsilon >0$ depending on the data is small enough to be determined later. Observe that our choices guarantee that $\theta _j=\delta _j^{2-p}r_j^p\le t_j$ and for $(x,t)\in L_j$ one has

$$\begin{aligned} \frac{u(x,t)-l_{j-1}}{\delta _{j-1}}=1+\frac{u(x,t)-l_j}{\delta _{j-1}}\ge 1. \end{aligned}$$

Since $\xi _{j-1}=1$ on $Q_j$ we obtain

$$\begin{aligned}&\delta _j^{p-2}l_{j+1}^{-1}r_j^{-n-p} \iint _{L'_j}v\frac{v-l_j}{\delta _j}\xi _j^{k-p}\hbox {d}x\,\hbox {d}t \le \varepsilon \delta _j^{p-2} l_j^{-1}r_j^{-p-n}\iint _{L'_j}v\xi _{j-1}^{k-p}\hbox {d}x\,\hbox {d}t \nonumber \\&\quad \le \varepsilon \gamma \delta _{j-1}^{p-2} l_j^{-1}r_{j-1}^{-p-n}\iint _{L_{j-1}}v\frac{v-l_{j-1}}{\delta _{j-1}} \xi _{j-1}^{k-p}\hbox {d}x\,\hbox {d}t \le \varepsilon \gamma A_{j-1}(l_j)\le \varepsilon \gamma \eta . \end{aligned}$$

(3.10)

Let

$$\begin{aligned} \psi _j=\frac{1}{\delta _j}\bigg (\int _{l_j}^vs^{\frac{1}{p}}\Big (1+\frac{s-l_j}{\delta _j}\Big )^{-\frac{1+\lambda }{p}}\mathrm{d}s\Bigg )_+. \end{aligned}$$

Using the evident inequality $\gamma ^{-1}(\varepsilon )\psi _j^{p}\le v(\frac{v-l_j}{\delta _j})^{p-1-\lambda }\le \gamma (\varepsilon ) \psi _j^{p}$ for $(x,t)\in L_j''$, the Sobolev inequality and Lemma 2.6 with $l=l_{j+1},\ \delta =\delta _j,\ \theta =\theta _j$ and $0<\lambda <\min (2-p,\frac{\varkappa }{n})$, we obtain

$$\begin{aligned}&\delta _j^{p-2}l_{j+1}^{-1}r_j^{-n-p} \iint _{L_j''}v\frac{v-l_j}{\delta _j}\xi _j^{k-p}\hbox {d}x\,\hbox {d}t \le \gamma (\varepsilon ) \delta _j^{p-2}l_j^{-1}r_j^{-n-p} \iint _{L_j''}\Big (\frac{v-l_j}{\delta _j}\Big )^{2-p+\lambda }\psi _j^p \xi _j^{k-p}\hbox {d}x\,\hbox {d}t \nonumber \\&\quad \le \gamma (\varepsilon )\delta _j^{p-2}l_j^{-1}r_j^{-n-p} \bigg (\sup _{0<t<T}\int _{L_j''(t)}\Big (\frac{v-l_j}{\delta _j}\Big )^ {(2-p+\lambda ) \frac{n}{p}}\xi _j^{\frac{(k-p)n}{n+p}}\hbox {d}x\bigg )^{\frac{p}{n}} \iint _{L_j}|\nabla (\psi _j\xi _j^{\frac{(k-p)n}{(n+1)p}})|^p\hbox {d}x\,\hbox {d}t \nonumber \\&\quad \le \gamma (\varepsilon )\delta _j^{p-2}l_j^{-1}r_j^{-n-p} \bigg (\sup _{0<t<T}\int _{L_j''(t)}\frac{v}{l_j}\Big (\frac{v-l_j}{\delta _j}\Big )\xi _j^{\frac{(k-p)n}{n+p}} \hbox {d}x\bigg )^{\frac{p}{n}}\nonumber \\&\qquad \times \iint _{L_j}\Big (|\nabla \psi _j|^p+v\frac{v-l_j}{\delta _j}\xi _j^ {-p}|\nabla \xi _j|^p \Big )\xi _j^{\frac{(k-p)n}{n+p}}\hbox {d}x\,\hbox {d}t \nonumber \\&\quad \le \gamma (\varepsilon )\left\{ \frac{\delta _j^{p-2}l_j^{-1}}{r_j^{n+p}} \iint _{L_j} v\frac{v-l_j}{\delta _j}\xi _j^{\frac{(k-p)n}{n+p}-p}\hbox {d}x\,\hbox {d}t\right. \nonumber \\&\qquad \qquad \qquad \left. +\, l_j^p\delta _j^{-p}r_j^{p-n}\int _{B_j}h_1\hbox {d}x + l_j^{p-1}\delta _j^{1-p}r_j^{p-n} \int _{B_j}h_2\hbox {d}x\right\} ^{1+\frac{p}{n}}. \end{aligned}$$

(3.11)

Choosing k such that $\frac{(k-p)n}{n+p}-p\ge 1$, and using (3.9), we obtain

$$\begin{aligned} \frac{\delta _j^{p-2}l_j^{-1}}{r_j^{n+p}}\iint _{L_j}v\frac{v-l_j}{\delta _j}\xi _j\hbox {d}x\,\hbox {d}t\le \gamma A_{j-1}(l_j)\le \gamma \eta . \end{aligned}$$

(3.12)

Combining (3.10)–(3.12) we get

$$\begin{aligned} \eta \le \varepsilon \gamma \eta +\gamma (\varepsilon )\bigg \{\eta +l_j ^p\delta _j^{-p} r_j^{p-n}\int _{B_j}h_1\hbox {d}x+l_j^{p-1}\delta _j^{1-p}r_j^{p-n} \int _{B_j}h_2\hbox {d}x\bigg \}^{1+\frac{p}{n}}. \end{aligned}$$

(3.13)

Choose $\varepsilon$ such that $\varepsilon \gamma =\frac{1}{4}$, and $\eta$ such that $\gamma (\varepsilon )\eta ^{\frac{p}{n}}=\frac{1}{4}$. Hence (3.13) yields (3.8), which completes the proof of the lemma. $\square$

In order to complete the proof of Theorem 1.1 we sum up (3.8) with respect to j from 2 to $J-1$,

$$\begin{aligned} l_J\le \gamma (\delta _1+l_1)+\gamma l_JW_{p+1,\frac{p}{p+1}}^{h_1}(2r_0)+\gamma l_JW_{p,1}^{h_2}(2r_0). \end{aligned}$$

(3.14)

Inequality (1.7) implies that $\gamma W_{p+1,\frac{p}{p+1}}^{h_1}(2r_0)+\gamma W_{p,1}^{h_2}(2r_0)\le \frac{1}{2}$, then by (3.14) we obtain

$$\begin{aligned} l_J\le \gamma (\delta _1+l_1)\le \gamma \delta _0, \end{aligned}$$

(3.15)

where $\delta _0$ is defined in (3.2).

Hence the sequence $\{l_j\}_{j\in {\mathbb {N}}}$ is convergent, and $\delta _j\rightarrow 0\ (j\rightarrow \infty )$, and we can pass to the limit $J\rightarrow \infty$ in (3.15). Let $l=\lim _{j\rightarrow \infty }l_j$, from (3.7) we conclude that

$$\begin{aligned} \frac{1}{|Q_j|}\iint _{Q_j}v(v-l)_+\hbox {d}x\,\hbox {d}t\le \gamma l\delta _j\rightarrow 0\quad (j\rightarrow \infty ). \end{aligned}$$

(3.16)

Choosing $({\bar{x}},{\bar{t}})$ as a Lebesgue point of the function $v(v-l)_+$ we conclude that $v({\bar{x}},{\bar{t}})\le l$ and hence $v({\bar{x}},{\bar{t}})$ is estimated from above by $\delta _0$. Applicability of the Lebesgue differentiation theorem follows from [12, Chapter II,Section 3]. Taking essential supremum over $Q_{\rho ^{(i)},t^{(i)}}^-(y,t)$ we get for any $i=0,1,2,\ldots$

$$\begin{aligned} M_i:= & {} \mathop {{{\mathrm{ess\,sup}}}}\limits _{Q_{\rho ^{(i)},t^{(i)}}^-(y,t)}v \le \gamma \sigma ^{-i\gamma }\bigg ( (t-s)^{-1-\frac{n}{p}}\iint _{Q_{\rho ^{(i+1)},t^{(i+1)}}^-(y,t)}v^2\hbox {d}x\,\hbox {d}t\bigg )^{\frac{p}{p+\varkappa }}\nonumber \\&+\,\gamma \sigma ^{-\gamma }\Big (\frac{t-s}{\rho ^p}\Big )^{\frac{1}{2-p}}. \end{aligned}$$

(3.17)

If for some $i_0\ge 0$

$$\begin{aligned} \sigma ^{-i_0\gamma }\bigg ( (t-s)^{-1-\frac{n}{p}}\iint _{Q_{\rho ^{(i_0+1)},t^{(i_0+1)}}^-(y,t)}v^2\hbox {d}x\,\hbox {d}t\bigg )^{\frac{p}{p+\varkappa }}\le \Big (\frac{t-s}{\rho ^p}\Big )^{\frac{1}{2-p}}, \end{aligned}$$

then

$$\begin{aligned} M_0\le M_{i_0}\le \gamma \sigma ^{-\gamma } \Big (\frac{t-s}{\rho ^p}\Big )^{\frac{1}{2-p}}. \end{aligned}$$

otherwise inequality (3.17) implies that

$$\begin{aligned} M_i\le \gamma \sigma ^{-i\gamma }M_{i+1}^{\frac{p}{p+\varkappa }} \bigg ((t-s)^{-\frac{n}{p}}\sup _{2s-t<\tau <t} \int _{B_\rho (y)}v\,\hbox {d}x\bigg )^{\frac{p}{p+\varkappa }}. \end{aligned}$$

From this, by Lemma 2.2, and taking into account (3.1), we conclude that

$$\begin{aligned} M_0\le \gamma \bigg ((t-s)^{-\frac{n}{p}}\sup _{2s-t<\tau <t} \int _{B_\rho (y)}u_+\,\hbox {d}x\bigg )^{\frac{p}{\varkappa }}+\gamma \Big (\frac{t-s}{\rho ^p}\Big )^{\frac{1}{2-p}}, \end{aligned}$$

this completes the proof of Theorem 1.1.

4 A De Giorgi-type lemmas: Proof of Theorems 1.3 and 1.4

In this section we prove a De Giorgi-type lemmas [10]. Here we assume the structure conditions

$$\begin{aligned}&{\mathbb {A}}(x,t,u,\xi )\, \xi \ge \mu _1|\xi |^p-F_1(x),\nonumber \\&|{\mathbb {A}}(x,t,u,\xi )|\le \mu _2|\xi |^{p-2}+F_2(x),\nonumber \\&|b(x,t,u,\xi )|\le h(x)|\xi |^{p-1}+F_3(x), \end{aligned}$$

(4.1)

where $F_i (x) = f_i (x) + g_i (x) ,$ $i = 1, 2,3,$ these assumptions follow from (1.2) due to the boundedness of u.

We provide the proof of (1.13), while the proof of (1.15) is completely similar.

Lemma 4.1

Let u be a weak solution to Eq. (1.1). Set $v=u-\mu _-$, then for any $l,\delta >0$, $0<\lambda <\min (1,\frac{2-p}{p-1}),\ k\ge p$ and any cylinder $Q_{r,\tau }^-({\bar{x}},{\bar{t}})\subset Q_{2\rho ,(2\rho )^p\theta }^-(y,s)$ and any smooth function $\xi (x,t)=\xi ^{(1)}(x)\xi ^{(2)}(t)$, where $\xi ^{(1)}(x)\in C_0^\infty (B_r({\bar{x}}))$ and $\xi ^{(2)}(t)$ is equal to zero for $t\le {\bar{t}}-\tau$ one has

$$\begin{aligned}&\sup _{{\bar{t}}-\tau <t<{\bar{t}}}\int _{L(t)}(v+l)^{1-2p}G\Big (\frac{l-v}{\delta }\Big )\xi ^k\mathrm{d}x \nonumber \\&\qquad +\, \delta ^{-2}\iint _L\bigg \{\int _v^l\Big (1+\frac{l-s}{\delta }\Big )^{-1-\lambda }\mathrm{d}s +(v+l)\Big (1+\frac{l-v}{\delta }\Big )^{-1-\lambda }\bigg \} (v+l)^{-2p}|\nabla v|^p\xi ^k\mathrm{d}x\,\mathrm{d}t \nonumber \\&\quad \le \gamma \iint _L(l+v)^{1-2p}\frac{l-v}{\delta }(|\xi _t|+\delta ^{p-2}|\nabla \xi |^p)\xi ^{k-p}\mathrm{d}x\,\mathrm{d}t \nonumber \\&\qquad +\, \gamma \delta ^{-2}\iint _L(v+l)^{-2p}\int _v^l\Big (1+\frac{l-s}{\delta }\Big )^ {-1-\lambda }\mathrm{d}s (F_1+(v+l) F_3)\xi ^k\mathrm{d}x\,\mathrm{d}t \nonumber \\&\qquad +\, \gamma \delta ^{-2} \iint _L(v+l)^{1-2p}(F_1 + F_2^{\frac{p}{p-1}})\xi ^k\mathrm{d}x\,\mathrm{d}t\nonumber \\&\qquad + \gamma \delta ^{-2}\iint _L(l+v)^{-p}\int _v^l\Big (1+\frac{l-s}{\delta }\Big )^ {-1-\lambda }\mathrm{d}s\, h^p\xi ^k\mathrm{d}x\,\mathrm{d}t. \end{aligned}$$

(4.2)

Proof

The proof is similar to that of Lemma 2.5 with the choice of the test function $\varphi =(v+l)^{1-2p}(\int _v^l(1+\frac{l-s}{\delta })^{-1-\lambda }\mathrm{d}s) _+\xi ^k$. $\square$

Set

$$\begin{aligned} \psi (v)=\frac{1}{\delta }\bigg (\int _v^l\Big (1+\frac{l-s}{\delta }\Big )^{-\frac{1+\lambda }{p}} \mathrm{d}s\bigg )_+. \end{aligned}$$

Lemma 4.2

Let the conditions of Lemma 4.1 be fulfilled and $0<\lambda <2-p$. Then there exists $\nu _1\in (0,1)$ depending only on the data such that the inequality

$$\begin{aligned} W_h(32\rho )\le \nu _1 \end{aligned}$$

(4.3)

implies that

$$\begin{aligned}&\sup _{{\bar{t}}-\tau <t<{\bar{t}}}\int _{L(t)} G\Big (\frac{l-v}{\delta }\Big )\xi ^k\mathrm{d}x +\delta ^{p-2}\iint _L |\nabla \psi |^p\xi ^k\mathrm{d}x\,\mathrm{d}t\nonumber \\&\quad \le \gamma \iint _L\frac{l-v}{\delta }(|\xi _t|+\delta ^{p-2}|\nabla \xi |^p)\xi ^{k-p}\mathrm{d}x\,\mathrm{d}t \nonumber \\&\qquad +\, \gamma l^{-1}\delta ^{-1}\tau \int _{B_r({\bar{x}})}F_1\mathrm{d}x + \gamma \delta ^{-1}\tau \int _{B_r({\bar{x}})}F_3\mathrm{d}x +\gamma \delta ^{-1}\tau \int _{B_r({\bar{x}})}\left( F_1+F_2^{\frac{p}{p-1}}\right) \,\mathrm{d}x\nonumber \\&\qquad +\, \gamma l^{p-1}\delta ^{-1}\tau \int _{B_r({\bar{x}})}h^p\mathrm{d}x . \end{aligned}$$

(4.4)

Proof

Note that $(\int _v^l(1+\frac{l-s}{\delta })^{-1-\lambda }\mathrm{d}s)_+\asymp \frac{l-v}{1+\frac{l-s}{\delta }}$, and apply Lemma 2.3 and (4.3), we have

$$\begin{aligned}&\iint _L\Big (\frac{1}{v+l}-\frac{1}{2l}\Big )^p \frac{l-v}{1+\frac{l-v}{\delta }}h^p\xi ^k\hbox {d}x\,\hbox {d}t \le \gamma \nu _1l^{-p}\iint _L(v+l)^{-p}\frac{l-v}{1+\frac{l-v}{\delta }} |\nabla v|^p\xi ^k\hbox {d}x\,\hbox {d}t \\&\qquad +\, \gamma \nu _1\iint _L(v+l)^{-2p}\frac{l-v}{1+\frac{l-v}{\delta }}|\nabla v|^p\xi ^k\hbox {d}x\,\hbox {d}t\\&\qquad +\, \gamma \nu _1l^{-p}\iint _L \frac{(l-v)^{p+1}}{(v+l)^p(1+\frac{l-v}{\delta })}|\nabla \xi |^p\xi ^{k-p}\hbox {d}x\,\hbox {d}t\\&\quad \le \gamma \nu _1\iint _L(v+l)^{-2p}\int _v^l\Big (1+\frac{s-l}{\delta }\Big )^{-1-\lambda }\mathrm{d}s|\nabla v|^p\xi ^k\hbox {d}x\,\hbox {d}t \\&\qquad +\, \gamma \nu _1\delta ^p\iint _L(v+l)^{1-2p}\frac{v-l}{\delta }|\nabla \xi |^p\xi ^{k-p}\hbox {d}x\,\hbox {d}t. \end{aligned}$$

Therefore, multiplying (4.2) by $l^{2p-1}$ and using the evident inequality $l\le v(x,t)+l\le 2l$ for $(x,t)\in L$, and choosing $\nu _1$ sufficiently small, we get from (4.2) the required (4.4). $\square$

Further on we assume that

$$\begin{aligned} \xi \omega \ge B \left( W_f(32\rho ) +W_g(32\rho ) \right) . \end{aligned}$$

(4.5)

Fix a point $({\bar{x}},{\bar{t}})\in Q_{\rho ,\rho ^{p}\theta }^-(y,s)$ and for $\alpha =\frac{\varkappa }{3-p},\ l>0$ and $j=0,1,2,\ldots$ set $r_j:=\frac{\rho }{2^j}$, $\delta _j(l):=l_j-l$, $\tau _j(l):=r_j^p\delta _j^{2-p}(l)$, $B_j:=B_{r_j}({\bar{x}})$, $Q_j:=Q_{r_j,\tau _j(l)}^-({\bar{x}},{\bar{t}})$, $L_j(l)=Q_j\cap \{v<l_j\}$, $L_j(l,t)=L_j(l)\cap \{\tau =t\}$, $\xi _j(x,t)=\xi _j^{(1)}(x)\xi _j^{(2)}(t)$, where $\xi _j^{(1)}(x)\in C_0^\infty (B_j),\ \xi _j^{(1)}(x)=1$ in $B_{j+1}$, $\xi _j^{(2)}(t)=1$ for $t\ge {\bar{t}}-2^{-\alpha }\tau _j(l)$, $\xi _j^{(2)}(t)=0$ for $t\le {\bar{t}}-\tau _j(l)$ and $|\nabla \xi _j|\le \gamma r_j^{-1}$, $|\frac{\partial \xi _j}{\partial t}|\le \gamma \tau _j^{-1}(l)$ and set also

$$\begin{aligned} A_j(l):=\frac{\delta _j^{p-2}(l)}{r_j^{n+p}} \iint _{L_j(l)}\frac{l_j-v}{\delta _j(l)}\xi _j^{k-p}\hbox {d}x\,\hbox {d}t. \end{aligned}$$

(4.6)

Define also the sequences $\{\alpha _j\},\ \{\beta _j\}$ by

$$\begin{aligned}&\alpha _j:=\int _0^{r_j}\bigg (r^{p-n}\int _{B_r({\bar{x}})}(F_1+F_2^{\frac{p}{p-1}})\,\hbox {d}x\bigg )^{\frac{1}{p}}+\int _0^{r_j}\bigg (r^{p-n}\int _{B_r({\bar{x}})} F_3\,\hbox {d}x\bigg )^{\frac{1}{p-1}}\frac{\hbox {d}r}{r},\\&\beta _j:=c\int _0^{r_j}\bigg (r^{p-n}\int _{B_r({\bar{x}})}h^p\hbox {d}x\bigg ) ^{\frac{1}{p-1}}\frac{\hbox {d}r}{r},\quad j=-1,0,1,\ldots \end{aligned}$$

where $c>1$ is fixed number, depending only on the known parameters, which will be defined later.

We start wit the choice of the sequences $l_j,\delta _j$. Put $l_0=\xi \omega ,\ \delta _0=\min \left( \frac{(1-a)\xi \omega }{4(1+2^{\frac{2-\alpha }{2-p}})}, \theta ^{\frac{1}{2-p}}\right)$, $l_1=l_0-\delta _0$. From (1.9) it follows that

$$\begin{aligned} A_0(l_1)\le & {} \gamma \xi \omega \theta ^{-1-\frac{1}{2-p}}\rho ^{-n-p} (1+((1-a)\xi \omega )^{p-2}\theta )^{\frac{3-p}{2-p}} |\{ Q_{2\rho ,(2\rho )^p\theta }^-(y,s):u\le \mu _-+\xi \omega \}|\\\le & {} \gamma \nu \xi \omega \theta ^{-\frac{1}{2-p}}(1+((1-a)\xi \omega )^{p-2}\theta ) ^{\frac{3-p}{2-p}}. \end{aligned}$$

Fix a number $\eta \in (0,1)$ depending on the known data, which will be specified later, and choose $\nu$ from the condition

$$\begin{aligned} \gamma \nu \xi \omega \theta ^{-\frac{1}{2-p}} (1+((1-a)\xi \omega )^{p-2}\theta )^{\frac{3-p}{2-p}}\le \eta , \end{aligned}$$

then we obtain that $A_0(l_1)\le \eta$. If $c\nu _1\le \frac{1}{8}$, then obviously we have

$$\begin{aligned} \frac{l_0}{2}+\frac{B}{4}\alpha _0+l_0\beta _0\le l_1\le l_0-\frac{1}{4}(\alpha _{-1}-\alpha _0)-l_0(\beta _{-1}-\beta _0). \end{aligned}$$

Lemma 4.3

Suppose we have chosen $l_1,\ldots ,l_j$ and $\delta _i=\delta _i(l_{i+1})=l_i-l_{i+1},\ i=0,1,\ldots ,j-1$ such that

$$\begin{aligned}&\max \Big (\frac{l_i}{2}+\frac{B}{4}\alpha _i+l_i\beta _i,l_i-2^{\frac{p-\alpha }{2-p}}\delta _{i-1}\Big )\nonumber \\&\quad \le l_{i+1}\le l_i-\min \Big (\frac{1}{4}(\alpha _{i-1}-\alpha _1)+l_i(\beta _{i-1}-\beta _i), \frac{1}{2}\delta _{i-1}\Big ),\nonumber \\&\qquad i=1,\ldots ,j-1, \end{aligned}$$

(4.7)

$$\begin{aligned}&l_{i+1}\ge \frac{B}{2}\alpha _i+2l_i\beta _i,\quad i=1,\ldots ,j-1,\nonumber \\&\qquad A_i(l_{i+1})\le \eta ,\quad i=1,\ldots ,j-1, \end{aligned}$$

(4.8)

then

$$\begin{aligned} A_j({\bar{l}})\le \eta ,\quad {\bar{l}}=\max \Big (\frac{l_j}{2}+\frac{B}{4}\alpha _j+l_j\beta _j, l_j-2^{\frac{p-\alpha }{2-p}}\delta _{j-1}\Big ). \end{aligned}$$

(4.9)

Proof

Let us decompose $L_j({\bar{l}}_j)$ as $L_j({\bar{l}})=L'_j({\bar{l}})\cup L''_j({\bar{l}})$, $L'_j({\bar{l}})=\{\frac{l_j-v}{\delta _j({\bar{l}})}<\varepsilon \}$, $L_j''({\bar{l}})=L_j({\bar{l}} ){\setminus } L_j'({\bar{l}})$, where $\varepsilon >0$ depending on the data is small enough to be determined later. Note that

$$\begin{aligned}&\frac{l_j}{2}-\frac{B}{4}\alpha _j-l_j\beta _j\ge \frac{l_{j-1}}{4}+\frac{B}{8}\alpha _{j-1}-\frac{B}{4}\alpha _j+\frac{1}{2}l_{j-1}\beta _{j-1}-l_j\beta _j\\&\quad =\frac{\delta _{j-1}}{4}+\frac{l_j}{4}+\frac{B}{8}\alpha _{j-1}-\frac{B}{4}\alpha _j+\frac{1}{2}l_{j-1}\beta _{j-1} -l_i\beta _j \\&\quad \ge \frac{\delta _{j-1}}{4}+\frac{B}{4}(\alpha _{j-1}-\alpha _j)+l_j(\beta _{j-1}-\beta _j), \end{aligned}$$

therefore we have

$$\begin{aligned} \min \Big (\frac{\delta _{j-1}}{4}+\frac{B}{4}(\alpha _{j-1}-\alpha _j)+l_j(\beta _{j-1}-\beta _j),2^{\frac{p-\alpha }{2-p}} \delta _{j-1}\Big ) \le \delta _j({\bar{l}})\le 2^{\frac{p-\alpha }{2-p}}\delta _{j-1}. \end{aligned}$$

(4.10)

By (4.10) $r_j^p\delta _j^{2-p}({\bar{l}})\le 2^{-\alpha }r_{j-1}^p\delta _{j-1}^{2-p}$ and hence $\xi _{j-1}(x,t)\equiv 1$ for $(x,t)\in Q_j({\bar{l}})$, so if $\delta _j({\bar{l}})=2^{\frac{p-\alpha }{2-p}}\delta _j$, then by (4.8)

$$\begin{aligned} A_j({\bar{l}})\le & {} 2^{-\frac{p-\alpha }{2-p}(3-p)}\delta _{j-1}^{p-3}r_j^{-n-p}\iint _{L_j({\bar{l}})}(l_j-v)\xi _{j-1}^{k-p}\hbox {d}x\,\hbox {d}t \nonumber \\\le & {} 2^{-\frac{p-\alpha }{2-p}(3-p)+n+p}\delta _{j-1}^{p-2}r_{j-1} ^{-n-p}\iint _{L_{j-1}( l_j)}\frac{l_{j-1}-v}{\delta _{j-1}}\xi _{j-1} ^{k-p}\hbox {d}x\,\hbox {d}t=A_{j-1}(l_j)\le \eta .\nonumber \\ \end{aligned}$$

(4.11)

If $\delta _j({\bar{l}})\ge \frac{\delta _{j-1}}{4}+\frac{B}{4}(\alpha _{j-1}-\alpha _j)+l_j(\beta _{j-1}-\beta _j)$, then

$$\begin{aligned}&\delta _{j}^{p-2}({\bar{l}})r_j^{-n-p}\iint _{L_{j}'({\bar{l}})} \frac{l_{j}-v}{\delta _{j}({\bar{l}})} \xi _{j}^{k-p}\hbox {d}x\,\hbox {d}t\le \gamma \varepsilon \delta _{j-1}^{p-2}r_{j-1}^{-n-p}\iint _{L_{j}({\bar{l}})}\xi _{j-1} ^{k-p}\hbox {d}x\,\hbox {d}t\nonumber \\&\quad \le \gamma \varepsilon \delta _{j-1}^{p-2}r_{j-1}^{-n-p}\iint _{L_{j-1}( l_j)}\frac{l_{j-1}-v}{\delta _{j-1}}\xi _{j-1}^{k-p}\hbox {d}x\, \hbox {d}t\le \gamma \varepsilon A_{j-1}(l_j)\le \gamma \varepsilon \eta . \end{aligned}$$

(4.12)

Define

$$\begin{aligned} \psi _j=\frac{1}{\delta _j({\bar{l}})}\bigg (\int _v^{l_j}\Big ( 1+\frac{l_j-s}{\delta _j({\bar{l}})}\Big )^{-\frac{1+\lambda }{p}}\mathrm{d}s\bigg )_+, \end{aligned}$$

using the evident inequalities $\gamma ^{-1}(\varepsilon )\psi _j^{\rho (\lambda )}\le \frac{l_j-v}{\delta _j({\bar{l}})}\le \gamma (\varepsilon )\psi _j^{\rho (\lambda )}$ for $(x,t)\in L_j''$, $\rho (\lambda )=\frac{p}{p-1-\lambda }$, the Sobolev inequality, and Lemma 4.2 with $l=l_j,\ \delta =\delta _j({\bar{l}}),\ \tau =\tau _j({\bar{l}})$, $0<\lambda <\min \{1,\frac{2-p}{p-1},\frac{\varkappa }{n}\}$, and k such that $\frac{(k-p)n}{n+p}-p\ge 1$, similar to (3.11), we obtain

$$\begin{aligned}&\frac{\delta _j^{p-2}({\bar{l}})}{r_j^{n+p}}\iint _{L_j''({\bar{l}})} \frac{l_j-v}{\delta _j({\bar{l}})}\xi _j^{k-p}\hbox {d}x\,\hbox {d}t\nonumber \\&\quad \le \gamma (\varepsilon )\bigg \{ \frac{\delta _j^{p-2}({\bar{l}})}{r_j^ {n+p}}\iint _{L_j({\bar{l}})}\frac{l_j-v}{\delta _j({\bar{l}})}\,\hbox {d}x\,\hbox {d}t + l_j^{-1}\delta _j^{1-p}({\bar{l}})r_j^{p-n}\int _{B_j}F_1\hbox {d}x \nonumber \\&\qquad \qquad \qquad +\, \delta _j^{-p}({\bar{l}})r_j^{p-n}\int _{B_j}(F_1 +F_2^{\frac{p}{p-1}})\,\hbox {d}x +\delta _j^{1-p}({\bar{l}})r_j^{p-n}\int _{B_j}\!\! F_3\hbox {d}x \nonumber \\&\qquad \qquad \qquad +\delta _j^{1-p}({\bar{l}})l_j^{p-1}r_j^{p-n}\int _{B_j} h^p\hbox {d}x \bigg \}^{1+\frac{p}{n}}. \end{aligned}$$

(4.13)

Using the inequality $\delta _j({\bar{l}})\ge \frac{\delta _{j-1}}{4}$, similar to (4.12). we obtain

$$\begin{aligned} \frac{\delta _j^{p-2}({\bar{l}})}{r_j^{n+p}}\iint _{L_j({\bar{l}})} \frac{l_j-v}{\delta _j({\bar{l}})}\,\hbox {d}x\,\hbox {d}t\le \gamma \eta . \end{aligned}$$

Furthermore, (4.10) implies that $\delta _j({\bar{l}})\ge l_j(\beta _{j-1}-\beta _j)+\frac{B}{4}(\alpha _{j-1}-\alpha _j)$, therefore we have

$$\begin{aligned}&l_j^{-1}\delta _j^{1-p}({\bar{l}})r_j^{p-n}\int _{B_j}F_1\hbox {d}x+\delta _j ^{-p}({\bar{l}})r_j^{p-n}\int _{B_j}(F_1+F_2^{\frac{p}{p-1}})\,\hbox {d}x\nonumber \\&\quad +\, \delta _j^{1-p}({\bar{l}})r_j^{p-n}\int _{B_j}F_3\hbox {d}x+\delta _j ^{1-p}({\bar{l}})l_j^{p-1}r_j^{p-n}\int _{B_j}h^p\hbox {d}x\le \gamma (B^{-p}+B^{1-p}+c^{1-p}).\nonumber \\ \end{aligned}$$

(4.14)

Combining estimates (4.12)–(4.14) we have

$$\begin{aligned} A_j({\bar{l}})\le \gamma \varepsilon \eta +\gamma (\varepsilon ) \{\eta +B^{-p}+B^{1-p}+c^{1-p}\}^{1+\frac{p}{n}}. \end{aligned}$$

(4.15)

First choose $\varepsilon$ from the condition $\gamma \varepsilon \le \frac{1}{4}$, next fix $\eta$ by $\gamma (\varepsilon )\eta ^{\frac{p}{n}}=\frac{1}{4}$ and choosing B and c large enough so that $B^{-p}+B^{1-p}+c^{1-p}\le \eta$, we conclude from (4.15) that $A_j({\bar{l}})\le \eta$, which completes the proof of Lemma 4.3. $\square$

Note that $A_j(l)$ is continuous as a function of l. So if $A_j(l_j-\min (\frac{1}{4}(\alpha _{j-1}-\alpha _j)+l_j(\beta _{j-1}-\beta _j), \frac{1}{2}\delta _{j-1}))>\eta$ the equation $A_j(l)=\eta$ has roots. Denote $l_{j+1}$ the largest root $A_j(l_{j+1})=\eta$. If $A_j(l_j-\min (\frac{1}{4}(\alpha _{j-1}-\alpha _j)+l_j(\beta _{j-1}-\beta _j), \frac{1}{2}\delta _{j-1}))\le \eta$, we set $l_{j+1}=l_j-\min (\frac{1}{4}(\alpha _{j-1}-\alpha _j)+l_j(\beta _{j-1}-\beta _j), \frac{1}{2}\delta _{j-1})$ and in both cases we set $\delta _j=\delta _j(l_{j+1})=l_j-l_{j+1}$. Note that our choice guarantee that $\delta _j\le 2^{\frac{p-\alpha }{2-p}}\delta _{j-1}$ and $A_j(l_{j+1})\le \eta$ for $j=1,2,\ldots$. Similar to Lemma 3.1 we prove the following lemma.

Lemma 4.4

For all $j\ge 2$ there exists $\gamma >0$ depending only on the data, such that

$$\begin{aligned} \delta _j\le & {} \frac{1}{2}\delta _{j-1}+\gamma l_j\bigg (r_j^{p-n}\int _{B_j}h^p\mathrm{d}x\bigg )^{\frac{1}{p-1}}+ \gamma \bigg (r_j^{p-n} \int _{B_j}(F_1+F_2^{\frac{p}{p-1}})\,\mathrm{d}x\bigg )^{\frac{1}{p}}\nonumber \\&+\, \gamma \bigg (r_j^{p-n}\int _{B_j}F_3\mathrm{d}x\bigg )^{\frac{1}{p-1}}. \end{aligned}$$

(4.16)

Summing up inequality (4.16) with respect to $j=2,\ldots ,J-1$, we obtain

$$\begin{aligned} l_2-l_J\le \delta _1+\gamma l_0W_h(32\rho )+\gamma W_f(32\rho ) +\gamma W_g(32\rho ), \end{aligned}$$

or the same

$$\begin{aligned} l_0\le & {} l_J+2\delta _0+\delta _1+\gamma l_0W_h(32\rho )+\gamma W_f(32\rho ) +\gamma W_g(32\rho ) \nonumber \\\le & {} l_J+2(1+2^{\frac{2-\alpha }{2-p}})\delta _0 +\gamma l_0W_h(32\rho )+\gamma W_f(32\rho ) +\gamma W_g(32\rho ). \end{aligned}$$

(4.17)

Let $l=\lim _{j\rightarrow \infty }l_j$, passing to the limit in (4.17) as $J\rightarrow \infty$ and choosing $({\bar{x}}, {\bar{t}})$ as a Lebesgue point of the function $(l-v)_+$, using the definition of $\delta _0$, we conclude that

$$\begin{aligned} \frac{1+a}{2}\xi \omega \le v({\bar{x}},{\bar{t}})+\gamma \xi \omega W_h(32\rho )+\gamma W_f(32\rho ) +\gamma W_g(32\rho ). \end{aligned}$$

Fix $\nu _1\in (0,1)$ and B large enough so that $\nu _1\gamma +B^{-1}\gamma \le \frac{1-a}{2}$, we obtain

$$\begin{aligned} u({\bar{x}},{\bar{t}})-\mu _-=v({\bar{x}},{\bar{t}})\ge a\xi \omega . \end{aligned}$$

(4.18)

Since $({\bar{x}},{\bar{t}})$ is an arbitrary point in $Q^-_{\rho ,\rho ^p\theta }(y,s)$, from (4.18) the required (1.13) follows, which proves Theorem 1.3.

The proof of Theorem 1.4 is similar to that of Theorem 1.3. Moreover, by taking $l\le \xi \omega$ and a cutoff function $\xi =\xi ^{(1)}(x)$ independent of t, the integral involving $\xi _t$ in the right-hand side of (4.4) vanishes. We may now repeat the same arguments as in previous proof for $(l_j-v)_+$ and $A_j(l)$ over the cylinders $Q_j:=B_j\times (s,s+(2\rho )^p\theta )$, $A_j(l):=\frac{\delta _j^{p-2}(l)}{r_j^{n+p}}\iint _{L_j(l)} \frac{l_j-v}{\delta _j(l)}\xi _j^{k-p}\hbox {d}x\,\hbox {d}t$, $\delta _j(l)=l_j-l,\ L_j(l)=Q_j\cap \{v<l_j\}$.

5 The expansion of positivity: Proof of Theorem 1.5

In the proof we closely follow [10].

Lemma 5.1

Assume that for some $(y,s)\in \varOmega _T$ and some $\rho >0$

$$\begin{aligned} |\{x\in B_\rho (y):u(x,s)\le \mu _-+N\}|\le (1-\alpha )|B_\rho (y)| \end{aligned}$$

(5.1)

for some $N>0$ and some $0<\alpha <1$. There exist $\nu _1,\varepsilon _0,b\in (0,1),\ B\ge 1$ depending only on the data and $\alpha$, such that the inequality

$$\begin{aligned} W_h(32\rho )\le \nu _1 \end{aligned}$$

(5.2)

implies that either

$$\begin{aligned} N\le B \left( W_f(32\rho ) +\gamma W_g(32\rho ) \right) \end{aligned}$$

(5.3)

or

$$\begin{aligned} |\{x\in B_\rho (y):u(x,t)\le \mu _-+\varepsilon _0N\}|\le \Big (1- \frac{\alpha ^2}{4}\Big )|B_\rho (y)| \end{aligned}$$

(5.4)

for all $t\in (s,s+bN^{2-p}\rho ^p)$.

Proof

For $k>0$ and $t>s$ set $A_{k,\rho }(t)=\{x\in B_\rho (y):u(x,t)\le k+\mu _-\}.$ Write down the estimate (2.2) for the function $(N+\mu _-u)_+$ over the cylinder $Q_{\rho ,\rho ^p\theta }^+(y,s)$, where $\theta >0$ is to be chosen. The cutoff function $\xi$ is taken independent on t, nonnegative, and such that $\xi =1$ on $B_{\rho (1 -\sigma )}(y)$, $|\nabla \xi |\le \frac{\gamma }{\sigma \rho }$, where $\sigma \in (0,1)$ is to be chosen. Lemma 2.4 yields

$$\begin{aligned}&\int _{B_{\rho (1-\sigma )}(y)\times \{t\}} (N+\mu _-u)^2_+\hbox {d}x+ \iint _{Q_{\rho ,\rho ^p\theta }^+(y,s)}|\nabla (N+\mu _-u)_+|^p\xi ^p\hbox {d}x\,\hbox {d}\tau \nonumber \\&\quad \le \int _{B_{\rho }(y)\times \{s\}} (N+\mu _-u)^2_+\hbox {d}x+\gamma (\sigma \rho )^{-p} \iint _{Q_{\rho ,\rho ^p\theta }^+(y,s)} (N+\mu _-u)_+^p\hbox {d}x\,\hbox {d}\tau \nonumber \\&\qquad +\, \gamma \iint _{Q_{\rho ,\rho ^p\theta }^+(y,s)} (F_1+F_2^{\frac{p}{p-1}}+(N+\mu _-u)_+F_3)\,\hbox {d}x\,\hbox {d}\tau \nonumber \\&\qquad +\,\gamma \iint _{Q_{\rho ,\rho ^p\theta }^+(y,s)}(N+\mu _-u)_ +^ph^p\xi ^p\hbox {d}x\,\hbox {d}\tau . \end{aligned}$$

(5.5)

The last integral in the right-hand side of (5.5) we estimate using Lemma 2.3

$$\begin{aligned}&\gamma \iint _{Q_{\rho ,\rho ^p\theta }^+(y,s)} (N+\mu _-u)_+^ph^p\xi ^p\hbox {d}x\,\hbox {d}\tau \le \gamma \nu _1^{p-1}\iint _{Q_{\rho ,\rho ^p\theta } ^+(y,s)}|\nabla (N+\mu _-u)_+|^p \xi ^p\hbox {d}x\,\hbox {d}\tau \\&\qquad +\, \gamma \nu _1^{p-1}(\sigma \rho )^{-p} \iint _{Q_{\rho ,\rho ^p\theta }^+(y,s)}(N+\mu _-u)_+^p\hbox {d}x\,\hbox {d}\tau . \end{aligned}$$

Choosing $\nu _1$ sufficiently small, and using (5.1), (5.3) from (5.5) we conclude that

$$\begin{aligned}&\int _{B_{\rho (1-\sigma )}(y)\times \{t\}} (N+\mu _-u)^2_+\hbox {d}x\le N^2 \bigg \{1-\alpha +\gamma \theta \sigma ^{-p}N^{p-2}\nonumber \\&\qquad +\,\gamma \theta N^{-2}\rho ^{p-n}\int _{B_\rho (y)}(F_1+F_2^{\frac{p}{p-1}})\,\hbox {d}x +\gamma \theta N^{-1}\rho ^{p-n}\int _{B_\rho (y)}F_3\hbox {d}x\bigg \} |B_\rho (y)|\nonumber \\&\quad \le N^2\{1-\alpha +\gamma \theta \sigma ^{-p}N^{p-2}\}|B_\rho (y)|. \end{aligned}$$

(5.6)

The left-hand side of (5.6) is estimated below by

$$\begin{aligned} \int _{B_{\rho (1-\sigma )}(y)\times \{t\}} (N+\mu _-u)^2_+\hbox {d}x\ge N^2(1-\varepsilon _0)^2|A_{\varepsilon _0N,\rho (1-\sigma )}(t)|, \end{aligned}$$

where $\varepsilon _0 \in (0,1)$ is to be chosen. Next we have

$$\begin{aligned} |A_{\varepsilon _0N,\rho }(t)|\le & {} |A_{\varepsilon _0N,\rho (1-\sigma )}(t)|+\sigma n|B_\rho (y)|\\\le & {} \{\sigma n+(1-\varepsilon _0)^{-2}(1-\alpha +\gamma \theta \sigma ^ {-p}N^{p-2})\}|B_\rho (y)|. \end{aligned}$$

Choose $\sigma =\frac{\alpha ^2}{4n}$, $\varepsilon _0=1-\sqrt{\frac{1-\alpha }{1-\alpha ^2}}$, $b=\frac{\sigma ^p\gamma ^{-1}(1-\varepsilon _0)^2}{4}\alpha ^2$ and $\theta =bN^{2-p}$, then the last inequality implies the required (5.4). $\square$

Let the cylinder $Q^+_{16\rho ,bN^{2-p}\rho ^p}(y,s)$ be contained in $\varOmega _T$. In the same way as in [10] we consider the function

$$\begin{aligned} w(z,\tau )=e^{\frac{\tau }{2-p}}N^{-1}(u(y+z\rho ,s+bN^{2-p} \rho ^p(1-e^{-\tau }))-\mu _-). \end{aligned}$$

Inequality (5.4) translates into w as

$$\begin{aligned} |\{z\in B_8(0):w(z,\tau )\le \varepsilon _0 e^{\frac{\tau _0}{2-p}}\}| \le \Big (1-\frac{\alpha ^2}{4\cdot 8^n}\Big )|B_8(0)|, \end{aligned}$$

(5.7)

for all $\tau \in (\tau _0,+\infty )$, where $\tau _0>0$ to be chosen.

Since $w\ge 0$, formal differentiation, which can be justified in a standard way, gives

$$\begin{aligned} w_\tau =\frac{w}{2-p}+{{\mathrm{div}}}\tilde{{\mathbb {A}}}(z,\tau ,w,\nabla w)+{\tilde{b}}(z,\tau ,w,\nabla w), \end{aligned}$$

where $\tilde{{\mathbb {A}}}(z,\tau ,w,\nabla w),\ {\tilde{b}}(z,\tau ,w,\nabla w)$ satisfy the inequality

$$\begin{aligned}&\tilde{{\mathbb {A}}}(z,\tau ,w,\nabla w) \, \nabla w \ge b\mu _1|\nabla w|^p-{\tilde{F}}_1(x,\tau ),\nonumber \\&|\tilde{{\mathbb {A}}}(z,\tau ,w,\nabla w)|\le b\mu _2|\nabla w|^{p-1}+{\tilde{F}}_2(z,\tau ),\nonumber \\&|b(z,\tau ,w,\nabla w)|\le b{\tilde{h}}|\nabla w|^{p-1}+{\tilde{F}}_3(z,\tau ), \end{aligned}$$

(5.8)

where $\mu _1,\mu _2$ are the constants in the structure conditions (1.2), b is a number claimed by Lemma 5.1 and

$$\begin{aligned}&\tilde{F_1}(z,\tau )=bN^{-p}\rho ^pe^{\frac{p\tau }{2-p}}F_1(y+z\rho ), \quad \tilde{F}_{2}(z,\tau )=bN^{1-p}\rho ^{p-1}e^{\frac{p-1}{2-p}\tau }F_2(y+z\rho ), \\&{\tilde{F}}_3(z,\tau )=bN^{1-p}\rho ^pe^{\frac{p-1}{2-p}\tau }F_3(y+z\rho ), \quad {\tilde{h}}(z,\tau )=\rho h(y+z\rho ). \end{aligned}$$

Set $k_0=\varepsilon _0e^{\frac{\tau _0}{2-p}}$, and $k_s=\frac{k_0}{2^s},\ s=0,1,\ldots ,s_*$, where $s_*$ to be chosen later. Then (5.7) yields

$$\begin{aligned} |\{z\in B_8(0):w(z,\tau )\le k_s\}| \le \Big (1-\frac{\alpha ^2}{4\cdot 8^n}\Big )|B_8(0)| \end{aligned}$$

(5.9)

for all $\tau \in (\tau _0,+\infty )$ and for all $0\le s\le s_*$.

Let $Q_{\tau _0}=B_8(0)\times (\tau _0+k_0^{2-p},\tau _0+2k_0^{2-p})$ and $Q_{\tau _0}'=B_8(0)\times (\tau _0,\tau _0+2k_0^{2-p})$ and a nonnegative cutoff function in $Q'_{\tau _0}$, $\xi (z,\tau )=\xi _1(z),\xi _2(\tau )$, where $\xi _1\in C_0^\infty (B_{16}(0))$, $\xi _1=1$ in $B_8(0)$, $\xi _2=1$ for $\tau \ge \tau _0+k_0^{2-p}$, $\xi _2=0$ for $\tau \le \tau _0$ and $|\nabla \xi _1|\le \frac{\gamma }{8}$, $|\frac{\partial \xi _2}{\partial \tau }|\le \gamma k_0^{p-2}$.

Using Lemma 2.4 we have

$$\begin{aligned}&b\iint _{Q'_{\tau _0}}|\nabla (k_s-w)|_+^p\xi ^p\hbox {d}z\,\hbox {d}\tau \le \gamma \iint _{Q'_{\tau _0}}((k_s-w)_+^2|\xi _\tau |+b(k_s-w)_+^p|\nabla \xi |^p)\,\hbox {d}z\,\hbox {d}\tau \\&\qquad + \gamma \iint _{Q'_{\tau _0}}({\tilde{F}}_1+{\tilde{F}}_2^{\frac{p}{p-1}}+(k_s-w)_+{\tilde{F}}_3)\,\hbox {d}z\,\hbox {d}\tau +\gamma b\iint _{Q'_{\tau _0}}(k_s-w)_+^p{\tilde{h}}^p\xi ^p\hbox {d}z\,\hbox {d}\tau . \end{aligned}$$

Taking into account the expressions of ${\tilde{h}},{\tilde{f}}_1,{\tilde{f}}_2,{\tilde{f}}_3$ and $k_0$, we estimate

$$\begin{aligned}&\gamma \iint _{Q'_{\tau _0}}(k_s-w)_+^p{\tilde{h}}^p\xi ^p\hbox {d}z\,\hbox {d}\tau \le \gamma \nu _1^{p-1}\iint _{Q'_{\tau _0}}|\nabla (k_s-w)_+|^p \xi ^p \hbox {d}z\,\hbox {d}\tau \nonumber \\&\qquad +\, \gamma \nu _1^{p-1}\iint _{Q'_{\tau _0}}(k_s-w)_+^p|\nabla \xi |^p\hbox {d}z\,\hbox {d}\tau , \end{aligned}$$

(5.10)

and

$$\begin{aligned}&\iint _{Q'_{\tau _0}}({\tilde{F}}_1+{\tilde{F}}_2^{\frac{p}{p-1}}+(k_s-w)_+{\tilde{F}}_3)\,\hbox {d}z\,\hbox {d}\tau \nonumber \\&\quad \le \gamma b\varepsilon _0^{-p}k_0^2 e^{\frac{2p}{2-p}k_0^{2-p}}\bigg \{ N^{-p}\rho ^{p-n}\int _{B_{16\rho }(y)}(F_1+F_2^{\frac{p}{p-1}})\,\hbox {d}x +N^{1-p}\rho ^{p-n}\int _{B_{16\rho }(y)}F_3\hbox {d}x\bigg \}.\nonumber \\ \end{aligned}$$

(5.11)

From this, choosing $\nu _1$sufficiently small, we obtain

$$\begin{aligned}&\iint _{Q_{\tau _0}}|\nabla (k_s-w)_+|^p\hbox {d}x\,\hbox {d}\tau \le \gamma k_s^p|Q_{\tau _0}| \left\{ 1+\varepsilon _0^{-p}2^{s_*p}e^{\frac{2p}{2-p}k_0^{2-p}} N^{1-p} \right. \\&\quad \times \left. \left( W_f(32\rho ) + W_g(32\rho )\right) ^{p-1}(1+N^{-1}W_f(32\rho ) +N^{-1}W_g(32\rho ))\right\} . \end{aligned}$$

Suppose for the moment that $s_*$ and $k_0$ have been chosen, and set

$$\begin{aligned} \gamma _*=\gamma (s_*,\varepsilon _0,\tau _0)= \varepsilon _0^{-p}2^{s_*p}e^{\frac{2p}{2-p} k_0^{2-p}}. \end{aligned}$$

Therefore either $N\le \gamma _* \left( W_f(32\rho ) + W_g(32\rho ) \right)$ or the previous inequality yields

$$\begin{aligned} \iint _{Q_{\tau _0}}|\nabla (k_s-w)_+|^p\hbox {d}x\,\hbox {d}\tau \le \gamma k_s^p|Q_{\tau _0}| \end{aligned}$$

with constant $\gamma$ depending only on the data, ${{\mathrm{ess\,sup}}}_{\varOmega _T}|u|$ and b.

Set $A_s(\tau )=\{B_8(0):w(\cdot ,\tau )<k_s\}$, and $A_s=\{Q_{\tau _0}:w<k_s\}$. By Lemma 2.1, (5.9) and (5.11) we have

$$\begin{aligned} |A_{s+1}|\le \gamma |Q_{\tau _0}|^{\frac{1}{p}}|A_s{\setminus } A_{s+1}|^{\frac{p-1}{p}}. \end{aligned}$$

Taking the $\frac{p}{p-1}$ power and summing up the last inequality with respect to $s=0,\ldots ,s_*-1$, we conclude that

$$\begin{aligned} s_*|A_{s_*}|^{\frac{p}{p-1}}\le \gamma |Q_{\tau _0}|^{\frac{p}{p-1}}. \end{aligned}$$

Choosing $s_*$ from the condition $\gamma s^{-\frac{p-1}{p}}_*=\nu$, we obtain

$$\begin{aligned} |\{(z,\tau )\in Q_{\tau _0}:w(z,\tau )\le k_{s_*}\}|\le \nu |Q_{\tau _0}|. \end{aligned}$$

Without loss of generality we assume that $2^{s_*(2-p)}$ is an integer, and subdivide cylinder $Q_{\tau _0}$ into $2^{s_*(2-p)}$ cylinders, each of length $k_{s_*}^{2-p}$, by setting

$$\begin{aligned} Q_i= & {} B_8(0)\times (\tau _0+k_0^{2-p}+ik_{s_*}^{2-p},\tau _0+k_0^{2-p}\\&+(i+1)k_{s_*}^{2-p}), \quad \text {for }\quad i=0,1,\ldots ,2^{s_*(2-p)}-1. \end{aligned}$$

For at least one of these, say $Q_{i_0}$ there must hold

$$\begin{aligned} |\{(z,\tau )\in Q_{i_0}:w(z,\tau )\le k_{s_*}\}|\le \nu |Q_{i_0}|. \end{aligned}$$

(5.12)

Apply Theorem 1.3 to w over $Q_{i_0}$ with $\mu _-=0,\ \xi \omega =k_{s_*},\ a=\frac{1}{2},\ \theta =k_{s_*}^{2-p}$. It gives

$$\begin{aligned} w(z,\tau _0+k_0^{2-p}+(i_0+1)k_{s_*}^{2-p})\ge \frac{k_{s_*}}{2},\quad \text {a.a. in } B_4(0), \end{aligned}$$

(5.13)

and hence, there exists a time level $\tau _1\in (\tau _0+k_0^{2-p},\tau _0+2k_0^{2-p})$, such that

$$\begin{aligned} w(z,\tau _1)\ge \sigma _0e^{\frac{\tau _0}{2-p}},\quad \sigma _0=\varepsilon _02^{-(s_*+1)}. \end{aligned}$$

(5.14)

In terms of the original coordinates (5.14) implies

$$\begin{aligned} u(x,t_1)\ge \mu _-+\sigma _0Ne^{-\frac{\tau _1-\tau _0}{2-p}}= \mu _-+N_0\quad \text {in } B_{4\rho }(y), \end{aligned}$$

(5.15)

$t_1=s+bN^{2-p}\rho ^p(1-e^{-\tau _1})$.

Apply Lemma 2.4 with $k=\mu _-+N_0,\ \theta =\nu _0N_0^{2-p}$, and $\xi :=\xi (x)\in C_0^\infty (B_{4\rho }(y))$, $\xi (x)=1$ in $B_{3\rho }(y)$, $|\nabla \xi |\le \gamma \rho ^{-1}$, where $\nu _0\in (0,1)$ to be chosen.

$$\begin{aligned}&\int _{B_{3\rho }(y)\times \{t\}} (N_0+\mu _-u)_+^2\hbox {d}x+\iint _{Q_{4\rho ,(4\rho )^p\theta }^+(y,s)} |\nabla (N_0+\mu _-u)_+|^p\xi ^p\hbox {d}x\,\hbox {d}t \nonumber \\&\quad \le \gamma \rho ^{-p} \iint _{Q_{4\rho ,(4\rho )^p\theta }^+(y,s)} (N_0+\mu _-u)_+^p\hbox {d}x\,\hbox {d}t\nonumber \\&\qquad +\,\gamma \iint _{Q_{4\rho ,(4\rho )^p\theta }^+(y,s)} (F_1+F_2^{\frac{p}{p-1}}+(N_0+\mu _-u)F_3)\,\hbox {d}x\,\hbox {d}t \nonumber \\&\qquad +\, \gamma \iint _{Q_{4\rho ,(4\rho )^p\theta }^+(y,s)} (N_0+\mu _-u)_+^ph^p\xi ^p\hbox {d}x\,\hbox {d}t. \end{aligned}$$

(5.16)

The last two terms in the right-hand side of (5.16) we estimate similar to (5.10), (5.11), hence (5.16) yields

$$\begin{aligned}&\int _{B_{3\rho }(y)\times \{t\}} (N_0+\mu _-u)_+^2\hbox {d}x \le \gamma \nu _0 N_0^2 \left( 1+N_0^{1-p} \left( W_f(32\rho ) + W_g(32\rho )\right) ^{p-1} \right. \\&\quad \times \left. \left( 1+N_0^{-1}\left( W_f(32\rho ) + W_g(32\rho )\right) \right) \right) . \end{aligned}$$

Therefore either $N\le \gamma (\sigma ,\tau _0,\tau )W_f(32\rho )$ or the previous inequality yields

$$\begin{aligned} |\{B_{3\rho }(y):u(\cdot ,t)\le \mu _-+\frac{3}{4}N_0\}| \le \gamma \nu _0|B_{3\rho }(y)|, \end{aligned}$$

(5.17)

for all $t\in (t_1,t_1+\nu _0N_0^{2-p}(4\rho )^p)$.

Using Theorem 1.4 over $Q_{3\rho ,(4\rho )^p\nu _0N_0^{2-p}}^+(y,t_1)$ with $\xi \omega =\frac{3}{4}N_0$, $a=\frac{2}{3},\ \nu =\gamma \nu _0$, we get

$$\begin{aligned} u(x,t)\ge \mu _-+\frac{N_0}{2}=\mu _-+\frac{N\sigma _0e^{ -\frac{\tau _1-\tau _0}{2}}}{2}\ge \mu _-+\frac{N\sigma _0}{2}e^{-\frac{2k_0^{2-p}}{2-p}}, \end{aligned}$$

(5.18)

in $B_{2\rho }(y)$ and for all times $t_1\le t\le t_1+\nu _0N_0^{2-p}(4\rho )^p$.

Now we define $\tau _0$ so that

$$\begin{aligned} t_1+\nu _0N_0^{2-p}(4\rho )^p=s+bN^{2-p}\rho ^p, \end{aligned}$$

which implies

$$\begin{aligned} e^{\tau _0}=\frac{b}{4^p\nu _0\sigma _0^{2-p}}. \end{aligned}$$

From the previous $t_1\le s+(1-\varepsilon )b N^{2-p}\rho ^p$, where $\varepsilon =e^{-\tau _0-2\varepsilon _0^{2-p}e^{\tau _0}}$. Choose B so large that $B\ge \max (\gamma _*,\gamma (\sigma _0,\tau _0,\tau _1)),$ we get the required (1.24). This proves Theorem 1.5.

6 Continuity of solutions: Proof of Theorem 1.2

Fix a point $(x_0,t_0)\in \varOmega _T$, let the cylinder $Q_{R,R^pM^{2-p}}^-(x_0,t_0)$ be contained in $\varOmega _T$, $M={{\mathrm{ess\,sup}}}_{\varOmega _T}|u|$ and R is so small, that $W_h(R)\le \nu _1$, where $\nu _1\in (0,1)$ is defined in Theorem 1.5.

Let $0<R_1<\frac{R}{32}$ and set

$$\begin{aligned} M_0=\mathop {{{\mathrm{ess\,sup}}}}\limits _{Q_{R,R^pM^{2-p}}^-(x_0,t_0)}u,\quad m_0=\mathop {{{\mathrm{ess\,inf}}}}\limits _{Q_{R,R^pM^{2-p}}^-(x_0,t_0)}u,\quad \omega _0=M_0-m_0. \end{aligned}$$

Lemma 6.1

There exist constants $B\ge 1$ and $b,\delta \in (0,1)$ that can be quantitatively determined only in terms of the data, such that, if $\omega _0\ge BW_f(32R_1)$, setting $\rho _0=R_1$ and for $j=0,1,2,\ldots$

$$\begin{aligned} \omega _{j+1}:=\delta \omega _j,\quad \theta _j:=b\omega _j^{2-p},\quad \rho _j:=2^{-j}R_1,\quad Q_j:=Q_{\rho _j,\rho _j^p\theta _j}^-(x_0,t_0), \end{aligned}$$

(6.1)

either

$$\begin{aligned} \mathop {{{\mathrm{ess\,osc}}}}\limits _{Q_j}u\le 2BW_f(R_1)\quad {\text {or}}\quad \mathop {{{\mathrm{ess\,osc}}}}\limits _{Q_j}u\le \omega _j. \end{aligned}$$

(6.2)

Proof

We assume that statement (6.2) holds for j and prove for $j+1$. Set $\mu _j^+={{\mathrm{ess\,sup}}}_{Q_j}u,\mu _j^-={{\mathrm{ess\,inf}}}_{Q_j}u$ and ${\bar{t}}_j=t_0-Ab\rho _j^p\omega _j^{2-p}$, $A=\frac{2^{2(p-1)}}{2^p-\delta ^{2-p}}$. At least one of the two inequalities

$$\begin{aligned}&|\{B_{\rho _j}(x_0):u(\cdot ,{\bar{t}}_j)\le \mu _j^-+\frac{\omega _j}{2}\} |\le \frac{1}{2}|B_{\rho _j}(x_0)|,\\&|\{B_{\rho _j}(x_0):u(\cdot ,{\bar{t}}_j)\ge \mu _j^+-\frac{\omega _j}{2}\} |\le \frac{1}{2}|B_{\rho _j}(x_0)| \end{aligned}$$

must hold. Assuming the first holds true, apply Theorem 1.5 with $\alpha =\frac{1}{2},\ N=\frac{\omega _j}{2}$, either

$$\begin{aligned} \frac{\omega _j}{2}\le B \left( W_f(32\rho _j) + W_g(32\rho _j) \right) \le B \left( W_f(32R_1) +W_g(32R_1) \right) \end{aligned}$$

(6.3)

or

$$\begin{aligned} u(x,{\bar{t}}_{j+1})\ge \mu _j^-+\frac{\sigma \omega _j}{2}\quad \text {for a.a. } \quad x\in B_{\rho _{j+1}}(x_0), \end{aligned}$$

and therefore

$$\begin{aligned} \mathop {{{\mathrm{ess\,inf}}}}\limits _{Q_{j+1}}u\ge \mu _j^-+\frac{\sigma \omega _j}{2}. \end{aligned}$$

(6.4)

Choose $\delta =1-\frac{\sigma }{2}$, then if (6.3) occurs, ${{\mathrm{ess\,osc}}}_{Q_{j+1}}u \le 2B \left( W_f(32R_1) +W_g(32R_1)\right) ,$ if (6.4) occurs, then ${{\mathrm{ess\,osc}}}_{Q_{j+1}}u\le \delta \omega _j=\omega _{j+1}.$ This proves Lemma 6.1. $\square$

From the construction of Lemma 6.1 it follows that

$$\begin{aligned} \mathop {{{\mathrm{ess\,osc}}}}\limits _{Q_{j}}u\le 2B \left( W_f(32R_1) +W_g(32R_1) \right) +\omega _j, \end{aligned}$$

by iteration

$$\begin{aligned} \mathop {{{\mathrm{ess\,osc}}}}\limits _{Q_{j}}u\le 2B \left( W_f(32R_1) +W_g(32R_1) \right) +\delta ^j\omega _0. \end{aligned}$$

(6.5)

Let now $0<r<R$ be fixed, set $R_1=r^\mu R^{1-\mu },\ \mu \in (0,1)$, there exists a nonnegative integer j such that $R_12^{-j-1}\le r\le R_12^{-j}$, this implies

$$\begin{aligned} \mathop {{{\mathrm{ess\,osc}}}}\limits _{Q_{j}}u\le \frac{1}{\delta }\Big (\frac{r}{R_1}\Big )^{\log _2\frac{1}{\delta }}\omega _0+2B \left( W_f(32R_1) +W_g(32R_1) \right) , \end{aligned}$$

that is

$$\begin{aligned} \mathop {{{\mathrm{ess\,osc}}}}\limits _{Q_{j}}u\le \frac{1}{\delta }\Big (\frac{r}{R}\Big )^\alpha \omega _0+2B \left( W_f(32r^\mu R^{1-\mu }) +W_g(32r^\mu R^{1-\mu }) \right) ,\quad \alpha =(1-\mu )\log _2\frac{1}{\delta }. \end{aligned}$$

To conclude the proof, we observe that since $\delta ^j\ge (\frac{r}{R_1})^{\log _2\frac{1}{\delta }}\ge (\frac{r}{R})^\alpha$, the cylinder $Q_{r,r^p\theta _0}(x_0,t_0),\ \theta _0=b(\frac{r}{R})^{\alpha (2-p)}\omega _0^{2-p}$ is included in $Q_j$, and therefore

$$\begin{aligned} \mathop {{{\mathrm{ess\,osc}}}}\limits _{Q_{r,r^p\theta _0}(x_0,t_0)}u\le \frac{1}{\delta }\Big (\frac{r}{R}\Big )^\alpha \omega _0+2B \left( W_f(32r^\mu R^{1-\mu }) + W_g(32r^\mu R^{1-\mu }) \right) . \end{aligned}$$

(6.6)

This completes the proof of the continuity of solution to Eq. (1.1).