Hölder regularity for porous medium systems

Abstract

We establish Hölder continuity for locally bounded weak solutions to certain parabolic systems of porous medium type, i.e.

$$\begin{aligned} \partial _t \mathbf{u}-\mathrm{div}(m|\mathbf{u}|^{m-1}D\mathbf{u})=0,\quad m>0. \end{aligned}$$

As a consequence of our local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given to ensure local boundedness of local weak solutions.

1 Introduction and main results

Let E be an open set in ${\mathbb {R}}^N$ and $E_T:=E\times (0,T]$ for some $T>0$. For a positive m, we are interested in the following parabolic system

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t u_i-{\text {div}}(m|\mathbf{u}|^{m-1}Du_i)=0\quad \text { weakly in }E_T,\\&i=1,\cdots , n,\quad \mathbf{u}\equiv (u_1,\cdots ,u_{n}). \end{aligned}\right. \end{aligned}$$

(I)

In the scalar case, i.e. $n=1$, it is commonly referred to as the porous medium equation. It presents degeneracy in the set $[|\mathbf{u}|=0]$ when $m>1$ and singularity when $0<m<1$.

As is well known, everywhere regularity is in general not expected from systems of elliptic or parabolic type (cf. [8]). Nevertheless, the special quasi-diagonal structure of (I) enables us to achieve the Hölder regularity of locally bounded solutions.

Let u be a solution to the system (I). Loosely speaking, our main result reads

$$\begin{aligned} \mathbf{u}\in L^{\infty }_{{\text {loc}}}(E_T)\implies \mathbf{u}\in C^{\alpha }_{{\text {loc}}}(E_T) \end{aligned}$$

for some $\alpha \in (0,1)$ depending only on n, N and the exponent m; see Theorem 1.1 for details. Based on our explicit local estimates, we are able to obtain a Liouville type result, which asserts bounded global solutions must be constant; see Corollary 1.1. In addition, we explore sufficient conditions that ensure local boundedness of solutions in Sect. 6.

The norm of u refers to the standard Euclidean norm, i.e. $|\mathbf{u}|=(u_1^2+u_2^2+\cdots +u_{n}^2)^\frac{1}{2}$. However, it is worth pointing out that our Hölder estimate for solutions to the system (I) still holds for any norm of u. This point will be clear from the proof. Thus, in particular, we could use $|\mathbf{u}|_1=|u_1|+|u_2|+\cdots +|u_{n}|$ in the system (I). Moreover, our results also hold for more general systems modeled on (I), which permit us to insert bounded and measurable diffusion coefficients in the system (I) (cf. (1.1) – (1.2)).

Our motivation of studying the system (I) with measurable coefficients as in (1.1) – (1.2) mainly stems from the $C^{1,\alpha }$ regularity for the parabolic p-Laplace equation, i.e.

$$\begin{aligned} u_t-{\text {div}}\big (|Du|^{p-2}Du\big )=0. \end{aligned}$$

The spatial gradient Du will satisfy a system with structures similar to (I). In fact, we may differentiate the above equation in $x_s$ formally to obtain

$$\begin{aligned} u_{tx_s}-\left( |Du|^{p-2}u_{x_jx_s}+(p-2)|Du|^{p-2}\frac{u_{x_i}u_{x_j}}{|Du|^2}u_{x_ix_s}\right) _{x_j}=0. \end{aligned}$$

Here the summation convention is evoked. Introduce the vector valued function $\mathbf{v}:=Du$. In terms of $\mathbf{v}$ the above differentiated equation can be written as

$$\begin{aligned} v_{s,t}-\left( a_{ij}|\mathbf{v}|^{p-2}v_{s,x_i}\right) _{x_j}=0,\quad s=1,\cdots ,N, \end{aligned}$$

where the measurable coefficients are defined by

$$\begin{aligned} a_{ij}=\delta _{ij}+(p-2)\frac{v_i v_j}{|\mathbf{v}|^2}, \end{aligned}$$

which in turn fulfills the condition

$$\begin{aligned} \min \{p-1,1\}|\xi |^2\le a_{ij}\xi _i\xi _j\le \max \{p-1,1\}|\xi |^2\quad \text { for any }\xi \in {\mathbb {R}}^N. \end{aligned}$$

In particular, when $N=1$, v reduces to a scalar function v that satisfies the one dimensional porous medium equation

$$\begin{aligned} v_t-(p-1)(|v|^{p-2}v_x)_x=0. \end{aligned}$$

Therefore Du satisfies a system with the same structure as (I) for $m=p-1$; see the general system (1.1) – (1.2). Particularly, our main result implies the following well-known $C^{1,\alpha }$ regularity of a weak solution u to the parabolic p-Laplace equation (cf. [2,3,4, 7, 17]):

$$\begin{aligned} Du\in L^{\infty }_{{\text {loc}}}\implies Du\in C^{\alpha }_{{\text {loc}}}. \end{aligned}$$

Our approach to the Hölder regularity for (I) is inspired by [5] where the $C^{1,\alpha }$ regularity for the elliptic p-Laplace equation has been investigated. The proof in [5] reflects a general idea due to De Giorgi: if the set where the equation is degenerate is confined within a small portion of a ball, then it is actually non-degenerate (cf. Sect. 3); if conversely the degenerate set occupies a seizable portion, then the solution can be compared with the radius of the ball (cf. Sect. 4).

The main new input of our argument lies in the introduction of proper intrinsically scaled cylinders, instead of balls, where the homogeneity of the parabolic systems is restored. We refer to [4, 6] for an account of the theory for intrinsic scaling. It would also be helpful to refer to the recent work [14] on the Hölder regularity for porous medium type equations. Some arguments we use here are closely related to this work. This kind of idea has been used in [4, 7] to treat the $C^{1,\alpha }$ regularity for parabolic systems of p-Laplacian type, via a combination of Campanato’s approach and De Giorgi’s iteration. Nevertheless, the quasi-diagonal structures of the system (I) allow us to implement the idea of intrinsic scaling more or less like we deal with scalar equations. Once proper energy estimates are derived from the system, we do not refer back to it any more. Thus we rely solely on De Giorgi’s techniques.

Finally, we mention that there is a “cousin” of the system (I), namely

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t u_i-\Delta (|\mathbf{u}|^{m-1}u_i)=0\quad \text { weakly in }E_T,\\&i=1,\cdots , n,\quad \mathbf{u}\equiv (u_1,\cdots ,u_{n}). \end{aligned}\right. \end{aligned}$$

(II)

The systems (I) and (II) are formally identical only in the scalar case, i.e. $n=1$. The system (II) has been derived in [19] from Bean’s critical-state model in the superconductivity theory. With the norm $|\mathbf{u}|_1$, it has been established in [11] modeling a competitive adsorption process among different species, via a multi-component isotherm of Freundlich type.

A study of Hölder regularity is carried out in [11] for non-negative solutions to (II) with the norm $|\mathbf{u}|_1$. The non-negativity of $\mathbf{u}$ turns out to be essential, since the argument relies on the fact that $|\mathbf{u}|_1$ satisfies the scalar porous medium equation. See [11, 19] for discussions on Dirichlet problems and [9, 16] on Cauchy problems.

Similar interest in the Hölder regularity of solutions to various parabolic systems also appears in [13, 15, 18]. Certain degeneracy is considered in [13] for systems with different structures from ours, whereas systems in [15, 18] are non-degenerate as the arguments are based on the regularity criterion in [8].

1.1 Main results

We postpone the formal definition of weak solutions to Sect. 1.3 and refer to Sect. 1.2 for the use of notations. Now we proceed to state in a precise manner our main results concerning the interior Hölder continuity of weak solutions to the system (I).

Theorem 1.1

Let $\mathbf{u}$ be a bounded, local, weak solution to the system (I) in $E_T$. Then $\mathbf{u}$ is locally Hölder continuous in $E_T$. More precisely, setting $M:=\Vert \mathbf{u}\Vert _{\infty ,E_T}$, there exist constants $\gamma >1$ and $\alpha \in (0,1)$ that can be determined a priori only in terms of m, n and N, such that for every compact set ${\mathcal {K}}\subset E_T$,

$$\begin{aligned} \begin{aligned}&\big |\mathbf{u}(x_1,t_1)-\mathbf{u}(x_2,t_2)\big | \le \gamma M \left( \frac{|x_1-x_2|+M^\frac{m-1}{2}|t_1-t_2|^{\frac{1}{2}}}{{\text {dist}}_1({\mathcal {K}};\Gamma )}\right) ^{\alpha },\quad m>1,\\&\big |\mathbf{u}(x_1,t_1)-\mathbf{u}(x_2,t_2)\big | \le \gamma M \left( \frac{M^\frac{1-m}{2}|x_1-x_2|+|t_1-t_2|^{\frac{1}{2}}}{{\text {dist}}_2({\mathcal {K}};\Gamma )}\right) ^{\alpha },\quad 0<m<1, \end{aligned} \end{aligned}$$

for every pair of points $(x_1,t_1), (x_2,t_2)\in {\mathcal {K}}$. Here ${\text {dist}}_1$ and ${\text {dist}}_2$ are intrinsic distances from ${\mathcal {K}}$ to the parabolic boundary $\Gamma$ of $E_T$ defined in Sect. 1.2.

Remark 1.1

We have stated Theorem 1.1 for globally bounded weak solutions. However the proof has a local thrust. As a matter of fact, for any $M>0$ verifying

for some cylinders $(x_o,t_o)+Q_\varrho (\theta )$ and $(x_o,t_o)+{\mathcal {Q}}_\varrho (\vartheta )$ included in $E_T$, we will show the following oscillation decay for $0<r<\varrho$:

See Sect. 1.2 for the meaning of these cylinders. The conclusion of Theorem 1.1 can be derived from this oscillation estimate via a standard covering argument.

Remark 1.2

Theorem 1.1 continues to hold for systems with more general structure modeled on (I); see (1.1) – (1.2). As a result, the constants $\alpha$ and $\gamma$ also depend on the structural constants $C_o$ and $C_1$ in (1.2).

The oscillation decay in Remark 1.1, while local in nature, has a global implication. Indeed, let $\mathbf{u}$ be a bounded, local weak solution to the system (I) in the semi-infinite strip ${\mathcal {S}}_T:={\mathbb {R}}^N\times (-\infty ,T)$ for some $T\in {\mathbb {R}}$. Then when $m>1$, we have for $0<r<\varrho$ that

for any cylinder $(x_o,t_o)+Q_r(\theta )$ compactly included in ${\mathcal {S}}_T$. Now fixing r and letting $\varrho \rightarrow \infty$, we immediately arrive at a Liouville-type result. The case $0<m<1$ can be analyzed similarly. Thus we arrive at the following Liouville type result.

Corollary 1.1

A bounded, local weak solution to the system (I) in ${\mathcal {S}}_T$ must be a constant.

Remark 1.3

Unlike elliptic equations, one-sided boundedness of solutions in ${\mathcal {S}}_T$ is generally not sufficient to imply solutions to parabolic equations are constants. This is evident from the non-negative solution $u(x,t)=e^{x+t}$ to the one dimensional heat equation. See [6, p. 105] for more on Liouville type results of parabolic equations.

Remark 1.4

The global boundedness condition in Corollary 1.1 can be easily relaxed to allow $|\mathbf{u}|$ to grow slower than $(|x|+|t|^{\frac{1}{2}})^\alpha$ as $|x|\rightarrow \infty$ and $t\rightarrow -\infty$.

1.2 Notations

Suppose $\mathbf{u}$ is a bounded solution to (I) in $E_T$. Let $\Gamma :=\partial E_T-{\overline{E}}\times \{T\}$ be the parabolic boundary of $E_T$, and for a compact set ${\mathcal {K}}\subset E_T$ introduce the intrinsic, parabolic m-distance from ${\mathcal {K}}$ to $\Gamma$ by

$$\begin{aligned} \begin{aligned} {\text {dist}}_1({\mathcal {K}};\,\Gamma )&:=\inf _{\begin{array}{c} (x,t)\in {\mathcal {K}}\\ (y,s)\in \Gamma \end{array}} \left\{ |x-y|+\Vert \mathbf{u}\Vert ^{\frac{m-1}{2}}_{\infty }|t-s|^{\frac{1}{2}}\right\} ,\quad m>1,\\ {\text {dist}}_2({\mathcal {K}};\,\Gamma )&:=\inf _{\begin{array}{c} (x,t)\in {\mathcal {K}}\\ (y,s)\in \Gamma \end{array}} \left\{ \Vert \mathbf{u}\Vert ^{\frac{1-m}{2}}_{\infty }|x-y|+|t-s|^{\frac{1}{2}}\right\} ,\quad 0<m<1. \end{aligned} \end{aligned}$$

For $\varrho >0$ let $K_\varrho (x_o)$ be the cube with center at $x_o\in {\mathbb {R}}^{N}$ and edge $2\varrho$. We define backward cylinders scaled by a positive parameter $\theta$ by

$$\begin{aligned} (x_o,t_o)+Q_\varrho (\theta ):= K_{\varrho }(x_o)\times (t_o-\theta \varrho ^2,t_o]. \end{aligned}$$

We also need another type of cylinders

$$\begin{aligned} (x_o,t_o)+{\mathcal {Q}}_{\varrho }(\vartheta ):=K_{\vartheta \varrho } (x_o)\times (t_o-\varrho ^2,t_o], \end{aligned}$$

for some positive parameter $\vartheta$. In Sect. 6, we will use the following type of cylinders

$$\begin{aligned} (x_o,t_o)+Q_{\varrho ,\tau }:=K_{\varrho }(x_o)\times (t_o-\tau ,t_o]. \end{aligned}$$

When $(x_o,t_o)=(0,0)$ or $\theta =\vartheta =1$, we will omit them from the notation.

In what follows, we will use $\gamma$ as a generic positive constant in our estimates, which may change from line to line. Nevertheless, such $\gamma$ can be determined a priori in terms of given data.

1.3 Notion of solution

It is always very important to pinpoint what we mean by a weak solution, as different notions may lead to different implications. In the following we may consider a general system modeled on the system (I):

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t u_i-{\text {div}}\mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})=0\quad \text { weakly in }E_T,\\&i=1,\cdots ,n,\quad \mathbf{u}\equiv (u_1,\cdots ,u_{n}), \end{aligned}\right. \end{aligned}$$

(1.1)

where the functions

$$\begin{aligned} \mathbf{A}^{(i)}:=\left( A_1^{(i)},\cdots ,A_N^{(i)}\right) :\, E_T\times {\mathbb {R}}^{N}\times {\mathbb {R}}^{nN}\rightarrow {\mathbb {R}}^N \end{aligned}$$

are Carathéodory, i.e. $A_j^{(i)}(x,t,\mathbf{u}, \mathbf{z})$ is measurable in (x, t) for all $(\mathbf{u}, \mathbf{z})\in {\mathbb {R}}^{N}\times {\mathbb {R}}^{nN}$ and continuous in $(\mathbf{u}, \mathbf{z})$ for a.e. $(x,t)\in E_T$, and satisfy the structure conditions

$$\begin{aligned} \left\{ \begin{aligned}&\mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\cdot Du_i\ge C_o |\mathbf{u}|^{m-1}|Du_i|^2,\\&|\mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})|\le C_1 |\mathbf{u}|^{m-1}|Du_i|, \end{aligned}\right. \end{aligned}$$

(1.2)

for all $i=1,\cdots ,n$ and for some positive m, $C_o$ and $C_1$. Together with n and N, they will be referred to as the data in the sequel.

A vector valued function $\mathbf{u}$ satisfying

$$\begin{aligned} \left\{ \begin{array}{ll} u_i\in C\big (0,T;L^2_{{\text {loc}}}(E)\big ),\, |\mathbf{u}|^{m-1}u_i,\,u_i\in L^2_{{\text {loc}}}\big (0,T; W^{1,2}_{{\text {loc}}}(E)\big ),\,&{}m>1,\\ u_i\in C\big (0,T;L^{m+1}_{{\text {loc}}}(E)\big ),\, |\mathbf{u}|^{m-1}u_i,\,|u_i|^{m-1}u_i\in L^2_{{\text {loc}}}\big (0,T; W^{1,2}_{{\text {loc}}}(E)\big ),\,&{}0<m<1, \end{array} \right. \end{aligned}$$

is a local, weak sub(super)-solution to (1.1) with the structure conditions (1.2), if for every compact set $K\subset E$ and every sub-interval $[t_1,t_2]\subset (0,T]$,

$$\begin{aligned} \int _K u_i\phi \,\mathrm {d}x\bigg |_{t_1}^{t_2}+\iint _{K\times (t_1,t_2)} \big [-u_i\phi _t+\mathbf{A}^{(i)} (x,t,\mathbf{u},D\mathbf{u})\cdot D\phi \big ]\mathrm {d}x\mathrm {d}t\le (\ge )0 \end{aligned}$$

(1.3)

for all non-negative test functions

$$\begin{aligned} \phi \in W^{1,2}_{{\text {loc}}}\big (0,T;L^2(K)\big )\cap L^2_{{\text {loc}}}\big (0,T;W_o^{1,2}(K) \big ). \end{aligned}$$

Remark 1.5

The notion of solution may generate discrepancies, as it is not entirely clear even in the scalar case. An interesting study on the equivalence of different notions of solution for the prototype porous medium equation is carried out in [1].

In what follows, special care needs to be taken for the time variable of a solution since we will use test functions that involve the solution itself. Lack of regularity in the time variable notwithstanding, they become admissible provided we employ a proper mollification in the time variable.

To this end, we introduce for any $v\in L^1(E_T)$,

$$\begin{aligned} \llbracket v \rrbracket _h(x,t) := \tfrac{1}{h} \int _0^t \mathrm e^{\frac{s-t}{h}} v(x,s) \, \mathrm {d}s,\quad \llbracket v \rrbracket _{{\bar{h}}}(x,t) := \tfrac{1}{h} \int _t^T \mathrm e^{\frac{t-s}{h}} v(x,s) \, \mathrm {d}s. \end{aligned}$$

Properties of this mollification can for instance be found in [10, Lemma 2.2]. In particular, we need the following identities:

$$\begin{aligned} \partial _t \llbracket v \rrbracket _h = \tfrac{1}{h} \big (v-\llbracket v \rrbracket _h\big ) ,\quad \partial _t \llbracket v \rrbracket _{{\bar{h}}} = \tfrac{1}{h} \big (\llbracket v \rrbracket _{{\bar{h}}}- v\big ). \end{aligned}$$

(1.4)

From the weak form (1.3) of the differential inequality for sub(super)-solutions we may deduce the mollified version:

$$\begin{aligned}&\iint _{E_T} \Big [\partial _t \llbracket u_i\rrbracket _h \phi + \llbracket {\mathbf {A}}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\rrbracket _h\cdot D\phi \Big ]\,\mathrm {d}x\mathrm {d}t\nonumber \\&\quad \le (\ge ) \int _E u_i(x,0)\cdot \tfrac{1}{h}\int _0^T \mathrm e^{-\frac{s}{h}}\phi (x,s)\,\mathrm {d}s\,\mathrm {d}x, \end{aligned}$$

(1.5)

for any non-negative test function $\phi \in L^p(0,T;W^{1,p}_0(E))$ with compact support in $E_T$. Indeed, we may take the test function $\llbracket \phi \rrbracket _{{\bar{h}}}$ in the weak formulation (1.3) of sub(super)-solutions to (I). Since $\phi$ is compactly supported in $E_T$, we have after an application of Fubini’s theorem

$$\begin{aligned}&\iint _{E_T} \Big [-u_i\partial _t\llbracket \phi \rrbracket _{{\bar{h}}} + \llbracket {\mathbf {A}}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\rrbracket _h\cdot D\phi \Big ]\,\mathrm {d}x\mathrm {d}t\nonumber \\&\quad \le (\ge ) \int _E u_i(x,0)\cdot \tfrac{1}{h}\int _0^T \mathrm e^{-\frac{s}{h}}\phi (x,s)\,\mathrm {d}s\,\mathrm {d}x. \end{aligned}$$

On the other hand, we calculate the term involving the time derivative using (1.4)

$$\begin{aligned} \iint _{E_T} u_i\partial _t\llbracket \phi \rrbracket _{{\bar{h}}}\,\mathrm {d}x\mathrm {d}t&= \tfrac{1}{h}\iint _{E_T} u_i(\llbracket \phi \rrbracket _{{\bar{h}}}-\phi )\,\mathrm {d}x\mathrm {d}t\\&=\tfrac{1}{h}\iint _{E_T} (\llbracket u_i\rrbracket _{h}-u_i)\phi \,\mathrm {d}x\mathrm {d}t=-\iint _{E_T}\partial _t \llbracket u_i\rrbracket _{h}\phi \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

This yields the desired claim in (1.5).

2 Energy estimates

This section is devoted to local energy estimates for the general system (1.1) – (1.2). Due to the degenerate or singular nature of the system, we need to consider the case $0<m<1$ (Proposition 2.1) and the case $m>1$ (Proposition 2.2) separately.

Proposition 2.1

Let $0<m<1$ and $\mathbf{u}$ be a local weak sub(super)-solution to (1.1) – (1.2) in $E_T$. There exists a constant $\gamma >0$, such that for all cylinders $Q_{R,S}:=K_R (x_o)\times (t_o-S,t_o]\subset E_T$, every $k\in {\mathbb {R}}$, every $i\in \{1,\cdots ,n\}$, and every non-negative, piecewise smooth cutoff function $\zeta$ in $Q_{R,S}$ vanishing on $\partial K_R(x_o)$,

where we have defined

$$\begin{aligned} {\mathfrak {g}}_\pm (u_i,k):=\pm \int _{k}^{u_i} \big (\varphi (s)-\varphi (k)\big )_\pm \,\mathrm {d}s,\quad \varphi (s):=|s|^{m-1}s. \end{aligned}$$

Proof

We prove the statement for sub-solutions only, as the other case is similar. For fixed $t_o-S<t_1<t_2 <t_o$ and $\varepsilon >0$ small enough we define the cutoff function in time by

$$\begin{aligned} \psi _\varepsilon (t) := \left\{ \begin{array}{cl} 0, &{} \text { for } t_o - S \le t \le t_1-\varepsilon , \\ 1+ \frac{t-t_1}{\varepsilon }, &{} \text { for } t_1-\varepsilon< t \le t_1, \\ 1, &{} \text { for } t_1< t \le t_2,\\ 1-\frac{t-t_2}{\varepsilon }, &{} \text { for } t_2< t \le t_2 +\varepsilon , \\ 0, &{} \text { for } t_2+\varepsilon < t \le t_o. \end{array} \right. \end{aligned}$$

Now, we choose in (1.5) the test function

$$\begin{aligned} Q_{R,S}\ni (x,t)\mapsto \phi (x,t) = \zeta ^2(x,t)\psi _\varepsilon (t) \big (\varphi (u_i(x,t))-\varphi (k)\big )_+. \end{aligned}$$

It is an admissible test function thanks to the notion of solution and the choice of cutoff functions. In the following we omit the reference to the vertex $(x_o,t_o)$ in the notation.

For the integral in (1.5) containing the time derivative we compute

$$\begin{aligned}&\iint _{E_T} \partial _t \llbracket u_i\rrbracket _h\phi \,\mathrm {d}x\mathrm {d}t= \iint _{Q_{R,S}} \zeta ^2 \psi _\varepsilon \partial _t \llbracket u_i\rrbracket _h \big (\varphi (\llbracket u_i\rrbracket _h)-\varphi (k)\big )_+ \mathrm {d}x\mathrm {d}t\\&\qquad + \iint _{Q_{R,S}} \zeta ^2 \psi _\varepsilon \partial _t\llbracket u_i\rrbracket _h\Big [ \big (\varphi (u_i(x,t))-\varphi (k)\big )_+-\big (\varphi (\llbracket u_i\rrbracket _h)-\varphi (k)\big )_+\Big ]\, \mathrm {d}x\mathrm {d}t\\&\quad \ge \iint _{Q_{R,S}} \zeta ^2 \psi _\varepsilon \partial _t {\mathfrak {g}}_+ (\llbracket u_i\rrbracket _h,k) \,\mathrm {d}x\mathrm {d}t\\&\quad = - \iint _{Q_{R,S}} \big (\zeta ^2 \psi _\varepsilon '+ \psi _\varepsilon \partial _t\zeta ^2\big ) {\mathfrak {g}}_+ (\llbracket u_i\rrbracket _h,k)\, \mathrm {d}x\mathrm {d}t. \end{aligned}$$

Here we have used in the second line the identity (1.4)$_1$ and the fact that the map $s\mapsto (\varphi (s)-\varphi (k))_+$ is a monotone increasing function, which implies that the term in the second line is non-negative. Since $\llbracket u_i \rrbracket _h\rightarrow u_i$ in $L^2(\Omega _T)$ (cf. [10, Lemma 2.2]), we can pass to the limit as $h\rightarrow 0$ in the integral on the right-hand side. We therefore get

$$\begin{aligned} \liminf _{h\rightarrow 0} \iint _{Q_{R,S}} \partial _t \llbracket u_i\rrbracket _h \phi \,\mathrm {d}x\mathrm {d}t&\ge - \iint _{Q_{R,S}} \big (\zeta ^2 \psi _\varepsilon ' + \psi _\varepsilon \partial _t\zeta ^2\big ) {\mathfrak {g}}_+ (u_i,k) \,\mathrm {d}x\mathrm {d}t\\&=: -\big [{\mathbf {I}}_{\varepsilon } +\mathbf {II}_{\varepsilon }\big ], \end{aligned}$$

with the obvious meaning of ${\mathbf {I}}_{\varepsilon }$ and $\mathbf {II}_{\varepsilon }$. We now pass to the limit $\varepsilon \rightarrow 0$. For the term ${\mathbf {I}}_{\varepsilon }$ we obtain for any $t_o-S<t_1<t_2<t_o$ that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}{\mathbf {I}}_\varepsilon = \int _{K_R} \zeta ^2(x,t_1) {\mathfrak {g}}_+(u_i(x,t_1),k)\,\mathrm {d}x- \int _{K_R} \zeta ^2(x,t_2) {\mathfrak {g}}_+(u_i(x,t_2),k)\,\mathrm {d}x, \end{aligned}$$

whereas for $\mathbf {II}_{\varepsilon }$ we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\mathbf {II}_{\varepsilon } = \iint _{K_R\times (t_1,t_2)} \partial _t\zeta ^2{\mathfrak {g}}_+(u_i,k)\,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

Next, we observe that the term on the right-hand side of (1.5) disappears as $h\rightarrow 0$, since by construction $\phi (\cdot ,0)\equiv 0$ on E, i.e. we have

$$\begin{aligned} \lim _{h\rightarrow 0} \int _E u_i(x,0)\cdot \tfrac{1}{h}\int _0^T \mathrm {e}^{-\frac{s}{h}}\phi (x,s)\,\mathrm {d}s\,\mathrm {d}x= \int _E u_i(x,0)\phi (x,0)\,\mathrm {d}x=0. \end{aligned}$$

It remains to consider the diffusion term. After passing to the limit as $h\rightarrow 0$, we use the structure conditions (1.2) for the vector-field ${\mathbf {A}}^{(i)}$, and subsequently Young’s inequality to estimate

$$\begin{aligned}&\lim _{h\rightarrow 0} \iint _{E_T} \llbracket {\mathbf {A}}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\rrbracket _h\cdot D\phi \, \mathrm {d}x\mathrm {d}t\\&\quad = \iint _{Q_{R,S}} \psi _\varepsilon {\mathbf {A}}^{(i)} (x,t,\mathbf{u},D\mathbf{u}) \cdot \big [\zeta ^2 D\big (\varphi (u_i)-\varphi (k)\big )_+ \\&\qquad + 2\zeta \big (\varphi (u_i)-\varphi (k)\big )_+D\zeta \big ]\,\mathrm {d}x\mathrm {d}t\\&\quad \ge C_o \iint _{Q_{R,S}} \psi _\varepsilon |\mathbf{u}|^{2(m-1)}|D(u_i-k)_+|^2\zeta ^2 \,\mathrm {d}x\mathrm {d}t\\&\qquad -2C_1\iint _{Q_{R,S}} \zeta |D\zeta | \psi _\varepsilon |\mathbf{u}|^{(m-1)}\big (\varphi (u_i)-\varphi (k)\big )_+|D(u_i-k)_+|\, \mathrm {d}x\mathrm {d}t\\&\quad \ge \frac{C_o}{2} \iint _{Q_{R,S}} \psi _\varepsilon |\mathbf{u}|^{2(m-1)}|D(u_i-k)_+|^2\zeta ^2\,\mathrm {d}x\mathrm {d}t\\&\qquad -\gamma \iint _{Q_{R,S}}\psi _\varepsilon \big (\varphi (u_i)-\varphi (k)\big )^2_+|D\zeta |^2 \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

Combining the preceding estimates and letting $\varepsilon \rightarrow 0$ we arrive at

$$\begin{aligned}&\int _{K_R\times \{t_2\}} \zeta ^2 {\mathfrak {g}}_+(u_i,k)\,\mathrm {d}x+ \frac{C_o}{2} \iint _{K_R\times (t_1,t_2)}|\mathbf{u}|^{2(m-1)}|D(u_i-k)_+|^2\zeta ^2\,\mathrm {d}x\mathrm {d}t\\&\quad \le \gamma \iint _{K_R\times (t_1,t_2)}\Big [ \big (\varphi (u_i)-\varphi (k)\big )^2_+|D\zeta |^2 + \partial _t\zeta ^2{\mathfrak {g}}_+(u_i,k)\Big ]\,\mathrm {d}x\mathrm {d}t\\&\qquad +\int _{K_R\times \{t_1\}} \zeta ^2 {\mathfrak {g}}_+(u_i,k)\,\mathrm {d}x, \end{aligned}$$

whenever $t_o-S<t_1<t_2<t_o$. Remembering $0<m<1$, the terms in the bracket are estimated by

$$\begin{aligned} \big (\varphi (u_i)-\varphi (k)\big )_+\le (u_i-k)^m_+, \end{aligned}$$

and

$$\begin{aligned} {\mathfrak {g}}_+(u_i,k)=\int _k^{u_i} \big (\varphi (s)-\varphi (k)\big )_+\,\mathrm {d}s\le \int _k^{u_i} (s-k)^m_+\,\mathrm {d}s=\tfrac{1}{1+m}(u_i-k)^{1+m}_+. \end{aligned}$$

Recalling $u_i\in C\big ( 0,T;L^{m+1}_{\mathrm{loc}}(E)\big )$, then one easily obtains the desired estimate. $\square$

Proposition 2.2

Let $m>1$ and $\mathbf{u}$ be a local weak sub(super)-solution to (1.1) – (1.2) in $E_T$. There exists a constant $\gamma >0$, such that for all cylinders $Q_{R,S}:=K_R (x_o)\times (t_o-S,t_o]\subset E_T$, every $k\in {\mathbb {R}}$, every $i\in \{1,\cdots ,n\}$, and every non-negative, piecewise smooth cutoff function $\zeta$ in $Q_{R,S}$ vanishing on $\partial K_{R}(x_o)$,

Proof

In the weak formulation of the sub(super)-solution to the i-th equation of the system (1.1) – (1.2) we take the test function $\pm (u_i-k)_{\pm }\psi _\varepsilon \zeta ^2$ over the cylinder $K_{\varrho }(x_o)\times (t_o-S,t_o]$, where $\psi _\varepsilon$ is defined in the proof of Proposition 2.1. Then a time mollification as in Proposition 2.1 will justify the use of the test function. We refrain from further elaboration and refer the reader to [6, Chapter 3, Proposition 6.1] for analogous calculations. $\square$

Remark 2.1

The energy estimate in Proposition 2.2 does not hold when $0<m<1$ in general, since the integral containing $|\mathbf{u}|^{m-1}$ might diverge. Nevertheless if we restrict ourselves to $k>0$ for sub-solutions with positive truncations, or $k<0$ for super-solutions with negative truncations, then the above estimate still holds for locally bounded solutions, when $0<m<1$. Indeed, one observes that the notion of solution and $\mathbf{u}\in L^{\infty }_{{\text {loc}}}(E_T)$ implies that $D\mathbf{u}\in L^2_{{\text {loc}}}(E_T)$, when $0<m<1$.

3 De Giorgi-type Lemmas

Suppose $(x_o,t_o)+Q_{\varrho }(\theta )\subset E_T$, and for a locally bounded function $\mathbf{u}$ in $E_T$, we introduce a quantity M satisfying

(3.1)

First of all, we have the following De Giorgi type lemma for the general system (1.1) – (1.2).

Lemma 3.1

Let $\mathbf{u}$ be a locally bounded, local weak sub(super)-solution to (1.1) – (1.2) in $E_T$. There exists a positive constant $\nu \in (0,1)$ depending only on M, $\theta$ and the data, such that if for some $i\in \{1,\cdots ,n\}$,

$$\begin{aligned} |[\mp u_i\le M]\cap [(x_o,t_o)+Q_{\varrho }(\theta )]|\le \nu |Q_{\varrho }(\theta )|, \end{aligned}$$

then

$$\begin{aligned} \mp u_i\ge \tfrac{1}{2}M\quad \text { a.e. in }(x_o,t_o)+Q_{\frac{\varrho }{2}}(\theta ). \end{aligned}$$

Moreover,

$$\begin{aligned} \nu =\frac{\gamma M^{1-m} / \theta }{\big (1+M^{1-m} / \theta \big )^\frac{N+2}{2}} \end{aligned}$$

for some $\gamma >0$ depending only on the data.

Remark 3.1

If we set $\theta =M^{1-m}$ in Lemma 3.1, then $\nu$ becomes a positive constant depending only on the data. Moreover, Lemma 3.1 can be stated for cylinders of the type ${\mathcal {Q}}_{\varrho }(\vartheta )$, with the choice $\vartheta =M^{\frac{m-1}{2}}$.

3.1 Proof of Lemma 3.1 when $m>1$

Assume $(x_o,t_o)=(0,0)$. We will present the proof for super-solutions only, sub-solutions being similar. In order to employ the energy estimate in Proposition 2.2, we set

$$\begin{aligned} k_n=\frac{M}{2}+\frac{M}{2^{n+1}},\quad \varrho _n=\varrho +\frac{\varrho }{2^n},\quad K_n=K_{\varrho _n},\quad Q_n=K_n\times (-\theta \varrho _n^2,0]. \end{aligned}$$

Introduce the test function $\zeta$ vanishing on the parabolic boundary of $Q_{n}$ and equal to identity in $Q_{n+1}$, such that

$$\begin{aligned} |D\zeta |\le \gamma \frac{2^n}{\varrho }\quad \text { and }\quad |\zeta _t|\le \gamma \frac{4^n}{\theta \varrho ^2}. \end{aligned}$$

In this setting, the energy estimate may be written as

where we have set $A_n:=[u_i<k_n]\cap Q_n$. To estimate the left-hand side from below we introduce

$$\begin{aligned} {\tilde{u}}_i:=\max \left\{ u_i,\,\tfrac{1}{4}M\right\} . \end{aligned}$$

Then for the time part we estimate

$$\begin{aligned} \int _{K_n}(u_i-k_n)^2_-\zeta ^2(x,t)\,\mathrm {d}x\ge \int _{K_n}({\tilde{u}}_i-k_n)^2_-\zeta ^2(x,t)\,\mathrm {d}x. \end{aligned}$$

For the space part, we first observe that

$$\begin{aligned}&\left( \tfrac{1}{4} M\right) ^{m-1}\iint _{Q_n}|D[({\tilde{u}}_i-k_n)_-\zeta ]|^2\,\mathrm {d}x\mathrm {d}t\\&\quad \le \iint _{Q_n}|{\tilde{u}}_i|^{m-1}|D[({\tilde{u}}_i-k_n)_-\zeta ]|^2\,\mathrm {d}x\mathrm {d}t\\&\quad =\iint _{Q_n\cap [{\tilde{u}}_i=u_i]}|{\tilde{u}}_i|^{m-1}|D[({\tilde{u}}_i-k_n)_-\zeta ]|^2\,\mathrm {d}x\mathrm {d}t\\&\qquad +\iint _{Q_n\cap [u_i<{\tilde{u}}_i]}|{\tilde{u}}_i|^{m-1}|D[({\tilde{u}}_i-k_n)_-\zeta ]|^2\,\mathrm {d}x\mathrm {d}t\\&\quad \le \iint _{Q_n}|\mathbf{u}|^{m-1}|D[(u_i-k_n)_-\zeta ]|^2\,\mathrm {d}x\mathrm {d}t+\gamma \frac{4^n}{\varrho ^2}M^{m+1}|A_n|. \end{aligned}$$

Here in the last line we have used the fact that $[u_i<k_n]$ and $[{\tilde{u}}_i<k_n]$ coincide.

Collecting all the estimates above gives us that

An application of Hölder’s inequality and Sobolev’s imbedding [4, Chapter I, Proposition 3.1] gives that

In terms of $Y_n=|A_n|/|Q_n|$, this can be rewritten as

$$\begin{aligned} Y_{n+1}\le \gamma 2^{4n}\bigg (\frac{\theta }{M^{1-m}}\bigg )^{\frac{2}{N+2}}\bigg ( 1+ \frac{M^{1-m}}{\theta }\bigg ) Y_n^{1+\frac{2}{N+2}}. \end{aligned}$$

Hence by [4, Chapter I, Lemma 4.1] $Y_n\rightarrow 0$, if we require that

$$\begin{aligned} Y_o\le \gamma \frac{M^{1-m}}{\theta } \bigg [1+\frac{M^{1-m}}{\theta }\bigg ]^{-\frac{N+2}{2}}=:\nu . \end{aligned}$$

The proof is concluded.

3.2 Proof of Lemma 3.1 when $0<m<1$

Assume $(x_o,t_o)=(0,0)$. In order to employ the energy estimate in Proposition 2.1, we notice first that, for ${\tilde{k}}<k$ we have

$$\begin{aligned} \int _{u_i}^k (\varphi (s)-\varphi (k))_-\,\mathrm {d}s\ge \int _{u_i}^{{\tilde{k}}} (\varphi (s)-\varphi (k))_-\,\mathrm {d}s\ge (\varphi (k)-\varphi ({\tilde{k}}))(u_i-{\tilde{k}})_-. \end{aligned}$$

As a result, the energy estimate reads

In order to use this energy estimate, we set for $n=0,1,\cdots ,$

$$\begin{aligned} \begin{array}{l} \displaystyle k_n=\frac{M}{2}+\frac{M}{2^{n+1}},\quad {\tilde{k}}_n=\frac{k_n+k_{n+1}}{2},\quad \varrho _n=\varrho +\frac{\varrho }{2^n},\quad {\tilde{\varrho }}_n=\frac{\varrho _n+\varrho _{n+1}}{2},\\ K_n=K_{\varrho _n}, \quad {\widetilde{K}}_n=K_{{\tilde{\varrho }}_n}, \quad Q_n=K_n\times (-\theta \varrho _n^2,0],\quad {\widetilde{Q}}_n={\widetilde{K}}_n\times (-\theta {\tilde{\varrho }}_n^2,0]. \end{array} \end{aligned}$$

Introduce the test function $\zeta$ vanishing on the parabolic boundary of $Q_{n}$ and equal to identity in ${\widetilde{Q}}_{n}$, such that

$$\begin{aligned} |D\zeta |\le \gamma \frac{2^n}{\varrho }\quad \text { and }\quad |\zeta _t|\le \gamma \frac{4^n}{\theta \varrho ^2}. \end{aligned}$$

In this setting, the energy estimate may be written as

where $A_n=[u_i<k_n]\cap Q_n$.

Noting that $k_n\le M$ and $\varphi '(s)=m|s|^{m-1}$, we may estimate the terms on the left-hand side of the energy estimate from below by

$$\begin{aligned} \begin{array}{l} \displaystyle \varphi (k_n)-\varphi ({\tilde{k}}_n) =\int ^{k_n}_{{\tilde{k}}_n}\varphi '(s)\,\mathrm {d}s\ge mM^{m-1}(k_n-{\tilde{k}}_n) =\frac{mM^m}{2^{n+3}},\\ \displaystyle \iint _{{\widetilde{Q}}_n}|\mathbf{u}|^{2(m-1)}|D(u_i-{\tilde{k}}_n)_-|^2\,\mathrm {d}x\mathrm {d}t\ge (\gamma _o M)^{2(m-1)}\iint _{{\widetilde{Q}}_n}|D(u_i-{\tilde{k}}_n)_-|^2\,\mathrm {d}x\mathrm {d}t. \end{array} \end{aligned}$$

In the last line we have used (3.1), and consequently $|\mathbf{u}|\le 2n^{\frac{1}{2}}M$, where n is the number of equations in (I). In order not to confuse it with the integral index n, we have used $\gamma _o$ to denote $2n^{\frac{1}{2}}$.

Collecting all the estimates above gives us that

Now setting $\phi$ to be a cutoff function which vanishes on the parabolic boundary of ${\widetilde{Q}}_n$ and equals identity in $Q_{n+1}$, an application of the Sobolev imbedding [4, Chapter I, Proposition 3.1] gives that

In terms of $Y_n=|A_n|/|Q_n|$, this can be rewritten as

$$\begin{aligned} Y_{n+1}\le \gamma b^n\bigg (\frac{\theta }{M^{1-m}}\bigg )^{\frac{2}{N+1}}\bigg ( 1+ \frac{M^{1-m}}{\theta }\bigg )^{\frac{N+2}{N+1}} Y_n^{1+\frac{2}{N+1}} \end{aligned}$$

where b and $\gamma$ depend only on the data. Hence by [4, Chapter I, Lemma 4.1], $Y_n\rightarrow 0$ if we require that

$$\begin{aligned} Y_o\le \gamma \frac{M^{1-m}}{\theta } \bigg [1+\frac{M^{1-m}}{\theta }\bigg ]^{-\frac{N+2}{2}}=:\nu . \end{aligned}$$

The proof is concluded.

4 Reduction of the supreme of solutions

Let M be defined in (3.1) and $\nu$ be the constant determined in Lemma 3.1 by taking $\theta =M^{1-m}$. Suppose for all $i\in \{1,\cdots ,n\}$, the following measure theoretical information holds

$$\begin{aligned} \left\{ \begin{array}{c} |[u_i\le M]\cap [(x_o,t_o)+Q_{\varrho }(\theta )]|>\nu |Q_{\varrho }(\theta )|,\\ |[u_i\ge -M]\cap [(x_o,t_o)+Q_{\varrho }(\theta )]|>\nu |Q_{\varrho }(\theta )|. \end{array}\right. \end{aligned}$$

(4.1)

We claim that if (4.1)$_1$ holds, then there exists $\eta \in (0,1)$ depending only on the data, such that

$$\begin{aligned} u_i\le 2M(1-\eta )\quad \text { a.e. in }(x_o,t_o)+Q_{\frac{\varrho }{2}}(\nu \theta ). \end{aligned}$$

Similarly, if (4.1)$_2$ holds, then there exists $\eta \in (0,1)$ depending only on the data, such that

$$\begin{aligned} u_i\ge -2M(1-\eta )\quad \text { a.e. in }(x_o,t_o)+Q_{\frac{\varrho }{2}}(\nu \theta ). \end{aligned}$$

Combining the two estimates, the supreme of $|\mathbf{u}|$ is reduced in a smaller cylinder, i.e.

(4.2)

Remark 4.1

The above statement can be made for cylinders of the type ${\mathcal {Q}}_{\varrho }(\vartheta )$ with $\vartheta =M^{\frac{m-1}{2}}$. Namely, omitting the reference to $(x_o,t_o)$, the following implications hold:

$$\begin{aligned} |[u_i\le M]\cap {\mathcal {Q}}_{\varrho }(\vartheta )|>\nu |{\mathcal {Q}}_{\varrho }(\vartheta )|&\implies u_i\le 2M(1-\eta )\quad \text { a.e. in }{\mathcal {Q}}_{\frac{\varrho }{2}}(\nu ^{-\frac{1}{2}}\vartheta ),\\ |[u_i\ge -M]\cap {\mathcal {Q}}_{\varrho }(\vartheta )|>\nu |{\mathcal {Q}}_{\varrho }(\vartheta )|&\implies u_i\ge -2M(1-\eta )\quad \text { a.e. in }{\mathcal {Q}}_{\frac{\varrho }{2}}(\nu ^{-\frac{1}{2}}\vartheta ). \end{aligned}$$

Let us treat (4.1)$_1$ for instance. For simplicity, we assume $(x_o,t_o)=(0,0)$. Recalling the definition of M in (3.1), we always have $|\mathbf{u}|\le 2n^{\frac{1}{2}}M$ in $Q_{\varrho }(\theta )$. We will work with $\mathbf{u}$ as a sub-solution within the smaller set

$$\begin{aligned}{}[M\le |\mathbf{u}|\le 2n^{\frac{1}{2}}M]\cap Q_{\varrho }(\theta ). \end{aligned}$$

(4.3)

This is realized by working with the energy estimate in Proposition 2.2 about $(u_i-k)_+$ for some $k\ge M$. As we have already mentioned in Remark 2.1, the energy estimate in Proposition 2.2 holds for all $m>0$ here, since the level $k>0$. This energy estimate is the basis for the following Lemmas 4.1 – 4.3.

To start with, we observe that there exists some $s\in [-\theta \varrho ^2, -\tfrac{1}{2}\nu \theta \varrho ^2]$, such that

$$\begin{aligned} \left| \left[ u_i(\cdot , s)\le M\right] \cap K_{\varrho }\right| >\tfrac{1}{2}\nu |K_{\varrho }|. \end{aligned}$$

(4.4)

Indeed, if the above inequality were not to hold for any s in the given interval, then we would have arrived at a contradiction to (4.1)$_1$:

$$\begin{aligned} \left| \left[ u_i\le M\right] \cap Q_{\varrho }(\theta )\right|&= \int _{-\theta \varrho ^2}^{-\frac{1}{2}\nu \theta \varrho ^2}\left| \left[ u_i(\cdot , s)\le M\right] \cap K_{\varrho }\right| \,\mathrm {d}s\\&\quad +\int ^{0}_{-\frac{1}{2}\nu \theta \varrho ^2}\left| \left[ u_i(\cdot , s)\le M\right] \cap K_{\varrho }\right| \,\mathrm {d}s\\&\le \nu |Q_{\varrho }(\theta )|. \end{aligned}$$

4.1 Propagation of measure information

Now we introduce a lemma concerning the time propagation of measure theoretical information in (4.4).

Lemma 4.1

Assume (4.4) holds. There exist $\delta$ and $\varepsilon$ in (0, 1), depending only on the data and $\nu$, such that

$$\begin{aligned} \left| \big [u_i(\cdot , s)\le 2M-\varepsilon M\big ]\cap K_{\varrho }(y)\right| \ge \tfrac{1}{4}\nu |K_\varrho (y)| \quad \text { for all }t\in (s,s+\delta M^{1-m}\varrho ^2]. \end{aligned}$$

Proof

Use the energy estimate in Proposition 2.2 for sub-solutions in the cylinder $Q=K_{\varrho }\times (0,\delta M^{1-m}\varrho ^2]$, with $k=M$ and choose a standard non-negative cutoff function $\zeta (x)$ that equals 1 on $K_{(1-\sigma )\varrho }$ and vanishes on $\partial K_{\varrho }$ satisfying $|D\zeta |\le (\sigma \varrho )^{-1}$; in such a case, we have for all $0<t<\delta M^{1-m}\varrho ^2$,

$$\begin{aligned} \int _{K_\varrho } (u_i-k)^2_+\zeta ^2(x,t)\,\mathrm {d}x&\le \int _{K_\varrho } (u_i-k)^2_+\zeta ^2(x,0)\,\mathrm {d}x\\&\quad +\gamma \iint _{Q}|\mathbf{u}|^{m-1}(u_i-k)^{2}_+|D\zeta |^2\,\mathrm {d}x\mathrm {d}t\\&\le \int _{K_\varrho } (u_i-k)^2_+\zeta ^2(x,0)\,\mathrm {d}x+\gamma |K_\varrho |\frac{\delta M^{2}}{\sigma ^2}\\&\le M^2\left[ (1-\tfrac{1}{2}\nu )+\gamma \frac{\delta }{\sigma ^2}\right] |K_\varrho |. \end{aligned}$$

In estimating the term containing $|\mathbf{u}|^{m-1}$ we have evoked (4.3).

The left-hand side of the energy estimate can be bounded from below by

$$\begin{aligned} \int _{K_\varrho } (u_i-k)^2_+\zeta ^2(x,t)\,\mathrm {d}x&\ge \int _{K_{(1-\sigma )\varrho }\cap [u_i\ge 2M-\varepsilon M]} (u_i-k)^2_+\zeta ^2(x,t)\,\mathrm {d}x\\&\ge (1-\varepsilon )^2 M^{2}|A_{\ell ,(1-\sigma )\varrho }(t)| \end{aligned}$$

where we have defined

$$\begin{aligned} A_{\ell ,(1-\sigma )\varrho }(t)=[u_i(\cdot ,t)\ge \ell ]\cap K_{(1-\sigma )\varrho }\quad \text { and }\quad \ell =2M-\varepsilon M. \end{aligned}$$

Notice that

$$\begin{aligned} |A_{\ell ,\varrho }(t)|&=|A_{\ell ,(1-\sigma )\varrho }(t)\cup (A_{\ell ,\varrho }(t)-A_{\ell ,(1-\sigma )\varrho }(t))|\\&\le |A_{\ell ,(1-\sigma )\varrho }(t)|+|K_\varrho - K_{(1-\sigma )\varrho }|\\&\le |A_{\ell ,(1-\sigma )\varrho }(t)|+N\sigma |K_\varrho |. \end{aligned}$$

Collecting all the above estimates yields that

$$\begin{aligned} |A_{\ell ,\varrho }(t)|&\le \frac{1}{(1-\varepsilon )^2}\left[ (1-\tfrac{1}{2}\nu )+\gamma \frac{\delta }{\sigma ^2}\right] |K_\varrho | +N\sigma |K_\varrho |. \end{aligned}$$

Finally, this allows us to choose the various parameters quantitatively. Indeed, we may assume $\varepsilon$ is so small that $(1-\varepsilon )^2\ge \frac{1}{2}$ and choose

$$\begin{aligned} \sigma =\frac{\nu }{16N},\quad \gamma \frac{\delta }{\sigma ^2}\le \frac{\nu }{32}. \end{aligned}$$

Then the above estimate is simplified as

$$\begin{aligned} |A_{\ell ,\varrho }(t)|\le \bigg [\frac{1-\tfrac{1}{2}\nu }{(1-\varepsilon )^2}+\frac{\nu }{8}\bigg ]|K_\varrho |. \end{aligned}$$

To conclude, we may choose $\varepsilon$ such that

$$\begin{aligned} \frac{1-\tfrac{1}{2}\nu }{(1-\varepsilon )^2}+\frac{\nu }{8}<1-\frac{\nu }{4},\quad \text { i.e. }\quad \varepsilon \approx \frac{\nu }{8}. \end{aligned}$$

This completes the proof. $\square$

4.2 A shrinking lemma

Based on the measure theoretical information obtained in Lemma 4.1 for each slice $K_{\varrho }\times \{t\}$, where $t\in \left( s,s+\delta \theta \varrho ^2\right]$, we may prove the following shrinking lemma.

Lemma 4.2

Assume (4.4) holds. There exists $\gamma >0$ depending only on the data, such that for any positive integer $j_*$, we have

$$\begin{aligned} \left| \left[ u_i\ge 2M-\frac{\varepsilon M}{2^{j_*}}\right] \cap Q\right| \le \frac{\gamma }{\nu \sqrt{\delta j_*}}|Q|,\quad \text { where }Q=K_{\varrho }\times \left( s,s+\delta \theta \varrho ^2\right] . \end{aligned}$$

Proof

We employ the energy estimate of Proposition 2.2 in a larger cylinder $Q'=K_{2\varrho }\times (s,s+\delta \theta \varrho ^2]$, with levels $k_j=2M\left( 1-\frac{\varepsilon }{2^{j+1}}\right)$ for $j\ge 0$ and a cutoff function $\zeta$ in $K_{2\varrho }$ that is equal to 1 in $K_{\varrho }$ and vanishes on $\partial K_{2\varrho }$, such that $|D\zeta |\le 2\varrho ^{-1}$. Then, we obtain

$$\begin{aligned} \iint _{Q}|\mathbf{u}|^{m-1}|D(u_i-k_j)_+|^2\,\mathrm {d}x\mathrm {d}t&\le \gamma \int _{K_{2\varrho }}(u_i-k_j)_+^2(x,s)\,\mathrm {d}x\\&\quad +\gamma \iint _{Q'}|\mathbf{u}|^{m-1}(u_i-k_j)_+^2|D\zeta |^2\,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

Recalling (3.1) and observing that on the set $[u_i>k_j]$, there holds

$$\begin{aligned} M<u_i\le |\mathbf{u}|\le 2n^{\frac{1}{2}}M. \end{aligned}$$

Hence the above energy estimate yields

$$\begin{aligned} \iint _{Q}|D(u_i-k_j)_+|^2\,\mathrm {d}x\mathrm {d}t\le \frac{\gamma }{\delta \varrho ^2}\left( \frac{\varepsilon M}{2^j}\right) ^{2}|Q|. \end{aligned}$$

Next, we apply [4, Chapter I, Lemma 2.2] to $u_i(\cdot ,t)$ for $t\in \left( s,s+\delta \theta \varrho ^2\right]$ over the cube $K_\varrho$, for levels $k_{j}<k_{j+1}$. Taking into account the measure theoretical information from Lemma 4.1:

$$\begin{aligned} \left| \left[ u_i(\cdot , t)\le 2M-\varepsilon M\right] \cap K_{\varrho }\right| \ge \tfrac{1}{4}\nu |K_\varrho | \quad \text { for all }t\in (s,s+\delta \theta \varrho ^2], \end{aligned}$$

together with an application of Hölder’s inequality, this gives

$$\begin{aligned}&\frac{\varepsilon M}{2^{j+1}}|[u_i(\cdot ,t)>k_{j+1}]\cap K_\varrho |\\&\quad \le \frac{\gamma \varrho ^{N+1}}{|[u_i(\cdot ,t)<k_j]\cap K_\varrho |}\int _{[k_j<u_i(\cdot ,t)<k_{j+1}]\cap K_{\varrho }}|Du_i|\,\mathrm {d}x\\&\quad \le \frac{\gamma \varrho }{\nu }\bigg (\int _{[k_j<u_i(\cdot ,t)<k_{j+1}]\cap K_{\varrho }}|Du_i|^2\,\mathrm {d}x\bigg )^{\frac{1}{2}}\\&\qquad \times |([u_i(\cdot ,t)>k_j]-[u_i(\cdot ,t)>k_{j+1}])\cap K_\varrho |^\frac{1}{2}. \end{aligned}$$

Set $A_j=[u_i>k_j]\cap Q$ and integrate the above estimate in $\mathrm {d}t$ over $(s,s+\delta \theta \varrho ^2]$; we obtain by using the energy estimate and the Fubini theorem that

$$\begin{aligned} \frac{\varepsilon M}{2^j}|A_{j+1}|&\le \frac{\gamma \varrho }{\nu }\bigg (\iint _{Q}|D(u_i-k_j)_+|^2\,\mathrm {d}x\mathrm {d}t\bigg )^\frac{1}{2}(|A_j|-|A_{j+1}|)^\frac{1}{2}\\&\le \frac{\gamma }{\nu \sqrt{\delta }}\frac{\varepsilon M}{2^j}|Q|^{\frac{1}{2}}(|A_j|-|A_{j+1}|)^\frac{1}{2}. \end{aligned}$$

Now square both sides of the above inequality to obtain

$$\begin{aligned} |A_{j+1}|^2\le \frac{\gamma }{\nu ^2\delta }|Q|(|A_j|-|A_{j+1}|). \end{aligned}$$

Add these inequalities from 0 to $j_*-1$ to obtain

$$\begin{aligned} j_* |A_{j_*}|^2\le \sum _{j=0}^{j_*-1}|A_{j+1}|^2\le \frac{\gamma }{\nu ^2 \delta }|Q|^2. \end{aligned}$$

From this we conclude

$$\begin{aligned}|A_{j_*}|\le \frac{\gamma }{\nu \sqrt{\delta j_*}}|Q|.\end{aligned}$$

This is exactly the desired conclusion. $\square$

4.3 A De Giorgi-type lemma near the supreme

Recall that $Q=K_{\varrho }\times \left( s,s+\delta \theta \varrho ^2\right]$. We need the following De Giorgi-type lemma in the set Q. The proof is analogous to the one for Lemma 3.1.

Lemma 4.3

Let $\xi \in (0,1)$. There exists a positive constant $\nu _o$ depending only on the data and $\delta$, but independent of $\xi$, such that if

$$\begin{aligned} |[u_i\ge 2M-\xi M]\cap Q|\le \nu _o|Q|, \end{aligned}$$

then

$$\begin{aligned} u_i\le 2M-\tfrac{1}{2}\xi M\quad \text { a.e. in }K_{\frac{\varrho }{2}}\times \left( s+\tfrac{3}{4}\delta \theta \varrho ^2,s+\delta \theta \varrho ^2\right] . \end{aligned}$$

Proof

For simplicity, we may fix the top of the cylinder Q, i.e. $s+\delta \theta \varrho ^2$, at 0. For $n=0,1,\dots$ we set

$$\begin{aligned} \begin{array}{l} \displaystyle k_n =2M-\frac{\xi M}{2}-\frac{\xi M}{2^{n+1}},\quad \varrho _n=\frac{\varrho }{2}+\frac{\varrho }{2^{n+1}},\\ \displaystyle K_n=K_{\varrho _n},\quad Q_n=K_n\times \left( -\delta \theta \varrho _n^2,0\right] . \end{array} \end{aligned}$$

Write down the energy estimate in Proposition 2.2 over the cylinder $Q_n$. Taking also into account the set $[u_i>k_n]$, we are working in the set (4.3), i.e. there holds

$$\begin{aligned} M<u_i\le |\mathbf{u}|\le \gamma _o M. \end{aligned}$$

Here $\gamma _o=2n^{\frac{1}{2}}$ and n represents the number of equations in the system (1.1). We also use n as a generic integral index in the proof here.

This gives the energy estimate

where we have set $A_n:=[u_i>k_n]\cap Q_n$.

Applying Hölder’s inequality and the Sobolev embedding [4, Chapter I, Proposition 3.1], and recalling that $\zeta =1$ on $Q_{n+1}$, we get

for a constant $\gamma$ depending only upon N. Combine this with the energy estimate to get

$$\begin{aligned} \begin{aligned} |A_{n+1}|\le&\gamma \frac{2^{4n}}{\varrho ^2} M^{\frac{2(m-1)}{N+2}} \Big (1+\frac{M^{1-m}}{\delta \theta }\Big )|A_n|^{1+\frac{2}{N+2}}. \end{aligned} \end{aligned}$$

In terms of $Y_n=|A_n|/|Q_n|$ this can be rewritten as

$$\begin{aligned} Y_{n+1}\le \gamma 2^{4n} \Big (\frac{\delta \theta }{M^{1-m}}\Big )^{\frac{2}{N+2}}\Big (1+\frac{M^{1-m}}{\delta \theta }\Big ) \,Y_n^{1+\frac{2}{N+2}}. \end{aligned}$$

By [4, Chapter I, Lemma 4.1], $Y_n\rightarrow 0$ as $n\rightarrow \infty$, provided

$$\begin{aligned} \begin{aligned} Y_o=\frac{|A_o|}{|Q_o|}\le \frac{1}{\gamma }\frac{M^{1-m} / (\delta \theta )}{\big (1+M^{1-m} / (\delta \theta )\big )^\frac{N+2}{2}}\,=:\,\nu _o. \end{aligned} \end{aligned}$$

This finishes the proof. $\square$

4.4 Proof of (4.2)

Now we are in a position to prove (4.2). The measure theoretical information (4.1)$_1$ gives (4.4) at some s in $[-\theta \varrho ^2,-\tfrac{1}{2}\nu \theta \varrho ^2]$. Whereas (4.4) can be written in $K_{2\varrho }$, i.e.

$$\begin{aligned} \left| \left[ u_i(\cdot , s)\le M\right] \cap K_{2\varrho }\right| >\tfrac{1}{2} \tfrac{1}{4^{N}}\nu |K_{2\varrho }|. \end{aligned}$$

Next we may apply Lemma 4.1 with $\varrho$ and $\nu$ replaced by $2\varrho$ and $\tfrac{1}{4^N}\nu$ respectively. With $\delta$ and $\varepsilon$ fixed in terms of these parameters as in Lemma 4.1, we may fix $\nu _o$ in Lemma 4.3 depending on the data and also on $\delta$. Then we may choose $\xi =\varepsilon 2^{-j_*}$, whereas $j_*$ is chosen according to Lemma 4.2, such that

$$\begin{aligned}\frac{\gamma 4^N}{\nu \sqrt{\delta j_*}}\le \nu _o.\end{aligned}$$

Thus Lemma 4.3 yields that

$$\begin{aligned} u_i\le 2M-\frac{\varepsilon M}{2^{j_*+1}}\quad \text { a.e. in }K_{\varrho }\times \left( s+\tfrac{3}{4}\delta \theta (2\varrho )^2,s+\delta \theta (2\varrho )^2\right] . \end{aligned}$$

(4.5)

If $s+\delta \theta (2\varrho )^2\ge 0$, we are done. If not, we may run the machinery of Lemmas 4.1 – 4.3 again, keeping in mind that we are always in the set (4.3) where degeneracy or singularity is avoided. In fact, the quantitative information in (4.5) gives a similar condition as (4.4), with $\tfrac{1}{2}\nu =1$. Then reasoning as above using Lemmas 4.1 – 4.3 would yield ${{\bar{\delta }}}$ and ${{\bar{\xi }}}$ depending only on the data such that,

$$\begin{aligned} u_i\le 2M-{{\bar{\xi }}} \xi M\quad \text { a.e. in }K_{\varrho }\times \left( s+\tfrac{3}{4}\delta \theta (2\varrho )^2,s+(\delta +{{\bar{\delta }}})\theta (2\varrho )^2\right] . \end{aligned}$$

After L steps, we arrive at

$$\begin{aligned} u_i\le 2M-{{\bar{\xi }}}^L \xi M\quad \text { a.e. in }K_{\varrho }\times \left( s+\tfrac{3}{4}\delta \theta (2\varrho )^2,s+(\delta +L{{\bar{\delta }}})\theta (2\varrho )^2\right] . \end{aligned}$$

(4.6)

The quantitative information as in (4.6) will reach the top of $Q_{\varrho }(\theta )$ when L is so large that $4(\delta +L{{\bar{\delta }}})\ge 1$. Therefore, we may conclude with $\eta ={{\bar{\xi }}}^L \xi$.

5 Proof of Theorem 1.1

With all the ingredients prepared so far, we are ready to prove Theorem 1.1. The proof splits into two parts according to either $m>1$ or $0<m<1$. In the first part, we will work with cylinders of the type $Q_{\varrho }(\theta )$, whereas in the second part we use cylinders of the type ${\mathcal {Q}}_{\varrho }(\vartheta )$.

5.1 When $m>1$

We may assume that $(x_o,t_o)$ coincides with the origin and set

Constructing the cylinder $Q_{\varrho }(\theta )$ with $\theta =M^{1-m}$, we may assume

5.1.1 Reduction of oscillation near zero

Let us first suppose (4.1) holds for all $i=1,\cdots ,n$. Then according to (4.2), we have

If we introduce

$$\begin{aligned} Q_1=Q_{\varrho _1}(\theta _1)\quad \text { with }\varrho _1=c\varrho ,\quad c=\tfrac{1}{2}\nu ^{\frac{1}{2}}(1-\eta )^{\frac{m-1}{2}},\quad \theta _1=[M(1-\eta )]^{1-m}, \end{aligned}$$

then the above estimate yields that

Now we may proceed by induction. Suppose $\varrho _o=\varrho$, $M_o=M$, and up to $j=1,\cdots , \ell -1$ we have built

For all the indices $j=1,\cdots , \ell -1$, we alway assume that (4.1) holds for all $i=1,\cdots ,n$, i.e.,

$$\begin{aligned} \begin{aligned}&|[u_i\le M_j]\cap Q_j|>\nu |Q_j|\quad \text {and}\quad |[u_i\ge -M_j]\cap Q_j|>\nu |Q_j|. \end{aligned} \end{aligned}$$

In this way the previous argument can be repeated and we have for all $j=1,\cdots , \ell$,

(5.1)

Consequently, iterating the above recursive inequality we obtain for all $j=1,2,\cdots , \ell$,

(5.2)

5.1.2 Reduction of oscillation away from zero

Let us suppose $\ell$ is the first index such that (4.1) is violated, that is, for some $i\in \{1,\cdots , n\}$ there holds

$$\begin{aligned} \text {either}\quad |[u_i\le M_\ell ]\cap Q_\ell |\le \nu |Q_\ell |\quad \text {or}\quad |[u_i\ge -M_\ell ]\cap Q_\ell |\le \nu |Q_\ell |. \end{aligned}$$

According to Lemma 3.1, we must have

$$\begin{aligned} \text {either}\quad u_i\ge \tfrac{1}{2}M_\ell \quad \text {or}\quad u_i\le -\tfrac{1}{2}M_\ell \quad \text {a.e. in }Q_{\frac{1}{2}\varrho _\ell }(\theta _\ell ). \end{aligned}$$

In either case we end up with

$$\begin{aligned} |\mathbf{u}|\ge |u_i|\ge \tfrac{1}{2}M_\ell \quad \text {a.e. in }Q_{\frac{1}{2}\varrho _\ell }(\theta _\ell ). \end{aligned}$$

(5.3)

On the other hand, taking (5.1) for $j=\ell$ into consideration, we obtain

(5.4)

The lower bound (5.3) and the upper bound (5.4) of $\mathbf{u}$ permit us to realize the oscillation decay of $\mathbf{u}$ by an appeal to the classical parabolic theory in [12]. In fact, we may introduce new variables

$$\begin{aligned} {\tilde{x}}=\frac{x}{\varrho _\ell },\qquad {\tilde{t}}=\frac{t}{M_\ell ^{1-m}\varrho _\ell ^2}, \end{aligned}$$

and a new function

$$\begin{aligned} \mathbf{w}({\tilde{x}},{\tilde{t}}):=M_\ell ^{-1} \mathbf{u}\big (\varrho _\ell {\tilde{x}},M_\ell ^{1-m}\varrho _\ell ^2 {\tilde{t}}\big )\quad \text {for }({\tilde{x}},{\tilde{t}})\in Q_{\frac{1}{2}}. \end{aligned}$$

As a result, the function $\mathbf{w}$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{{\tilde{t}}} w_i-{\text {div}}_{{\tilde{x}}}\widetilde{\mathbf{A}}^{(i)}({\tilde{x}}, {\tilde{t}},\mathbf{w},D_{{\tilde{x}}}{} \mathbf{w})=0\quad \text { weakly in }Q_{\frac{1}{2}},\\&i=1,\cdots ,n, \end{aligned}\right. \end{aligned}$$

where the functions

$$\begin{aligned} \widetilde{\mathbf{A}}^{(i)}({\tilde{x}},{\tilde{t}},\mathbf{w},D_{{\tilde{x}}}{} \mathbf{w}):= M_\ell ^{-m}\varrho _\ell \mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u}) \end{aligned}$$

satisfy the structure conditions

$$\begin{aligned} \left\{ \begin{aligned}&\widetilde{\mathbf{A}}^{(i)}({\tilde{x}},{\tilde{t}},\mathbf{w},D_{{\tilde{x}}}{} \mathbf{w})\cdot D_{{\tilde{x}}}w_i\ge C_o |\mathbf{w}|^{m-1} |D_{{\tilde{x}}}w_i|^2,\\&|\widetilde{\mathbf{A}}^{(i)}({\tilde{x}},{\tilde{t}},\mathbf{w},D_{{\tilde{x}}}{} \mathbf{w})|\le C_1|\mathbf{w}|^{m-1} |D_{{\tilde{x}}}w_i|. \end{aligned}\right. \end{aligned}$$

Appealing to (5.3) – (5.4), we have $\tfrac{1}{2}\le |\mathbf{w}|\le 2n^{\frac{1}{2}}$ in $Q_\frac{1}{2}$. This becomes a non-degenerate, diagonal system about w. According to [12], there exist $\alpha _1\in (0,1)$ and $\gamma >1$ depending only on the data, such that for all $0<r<\tfrac{1}{2}$,

Rephrasing this oscillation decay in terms of $\mathbf{u}$, we have for all $0<r<\varrho _\ell$,

(5.5)

Combining (5.2) and (5.5), we arrive at the desired conclusion, i.e., for all $0<r<\varrho$ we have

Here we have used $m>1$ such that $\theta \le \theta _\ell$, and thus $Q_r(\theta )\subset Q_r(\theta _\ell )$.

5.2 When $0<m<1$

We set up M just like in Sect. 5.1. Constructing the cylinder ${\mathcal {Q}}_{\varrho }(\vartheta )$ with $\vartheta :=M^{\frac{m-1}{2}}$, we may assume

5.2.1 Reduction of oscillation near zero

Let us first suppose the following holds for all $i=1,\cdots ,n$:

$$\begin{aligned} |[u_i\le M]\cap {\mathcal {Q}}_{\varrho }(\vartheta )|>\nu |{\mathcal {Q}}_{\varrho }(\vartheta )|\quad \text {and}\quad |[u_i\ge -M]\cap {\mathcal {Q}}_{\varrho }(\vartheta )|>\nu |{\mathcal {Q}}_{\varrho }(\vartheta )|. \end{aligned}$$

(5.6)

Then according to Remark 4.1, we have

If we introduce

$$\begin{aligned} {\mathcal {Q}}_1={\mathcal {Q}}_{\varrho _1}(\vartheta _1)\quad \text { with } \varrho _1=c\varrho ,\,c=\min \left\{ \tfrac{1}{2},\tfrac{1}{2}\nu ^{-\frac{1}{2}}(1-\eta )^{\frac{1-m}{2}}\right\} ,\quad \vartheta _1=[M(1-\eta )]^{\frac{m-1}{2}}, \end{aligned}$$

then the above estimate yields that

Now we may proceed by induction. Suppose $\varrho _o=\varrho$, $M_o=M$, and up to $j=1,\cdots , \ell -1$ we have built

For all the indices $j=1,\cdots , \ell -1$, we alway assume that for all $i=1,\cdots ,n$,

$$\begin{aligned} \begin{aligned}&|[u_i\le M_j]\cap {\mathcal {Q}}_j|>\nu |{\mathcal {Q}}_j|\quad \text {and}\quad |[u_i\ge -M_j]\cap {\mathcal {Q}}_j|>\nu |{\mathcal {Q}}_j|. \end{aligned} \end{aligned}$$

In this way the previous argument can be repeated and we have for all $j=1,\cdots , \ell$,

(5.7)

Consequently, iterating the above recursive inequality we obtain for all $j=1,\cdots , \ell$,

(5.8)

5.2.2 Reduction of oscillation away from zero

Let us suppose $\ell$ is the first index such that (5.6) is violated, that is, for some $i\in \{1,\cdots , n\}$ there holds

$$\begin{aligned} \text {either}\quad |[u_i\le M_\ell ]\cap {\mathcal {Q}}_\ell |\le \nu |{\mathcal {Q}}_\ell | \quad \text {or}\quad |[u_i\ge -M_\ell ]\cap {\mathcal {Q}}_\ell |\le \nu |{\mathcal {Q}}_\ell |. \end{aligned}$$

According to Lemma 3.1, we must have

$$\begin{aligned} \text {either}\quad u_i\ge \tfrac{1}{2}M_\ell \quad \text {or}\quad u_i\le -\tfrac{1}{2}M_\ell \quad \text {a.e. in }{\mathcal {Q}}_{\frac{1}{2}\varrho _\ell }(\vartheta _\ell ). \end{aligned}$$

In either case we end up with

$$\begin{aligned} |\mathbf{u}|\ge |u_i|\ge \tfrac{1}{2}M_\ell \quad \text {a.e. in }{\mathcal {Q}}_{\frac{1}{2}\varrho _\ell }(\theta _\ell ). \end{aligned}$$

(5.9)

On the other hand, taking (5.7) for $j=\ell$ into consideration, we obtain

(5.10)

Next, we may introduce new variables

$$\begin{aligned} {\tilde{x}}=\frac{x}{\varrho _\ell M_\ell ^{\frac{m-1}{2}}},\qquad {\tilde{t}}=\frac{t}{\varrho _\ell ^2}, \end{aligned}$$

and a new function

$$\begin{aligned} \mathbf{w}({\tilde{x}},{\tilde{t}}):=M_\ell ^{-1} \mathbf{u}\big (\varrho _\ell M_\ell ^{\frac{m-1}{2}}{\tilde{x}},\varrho _\ell ^2 {\tilde{t}}\big )\quad \text {for }({\tilde{x}},{\tilde{t}})\in {\mathcal {Q}}_{\frac{1}{2}}. \end{aligned}$$

As a result, the function $\mathbf{w}$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{{\tilde{t}}} w_i-{\text {div}}_{{\tilde{x}}}\widetilde{\mathbf{A}}^{(i)}({\tilde{x}}, {\tilde{t}},\mathbf{w},D_{{\tilde{x}}}{} \mathbf{w})=0\quad \text { weakly in }{\mathcal {Q}}_{\frac{1}{2}},\\&i=1,\cdots ,n, \end{aligned}\right. \end{aligned}$$

where the functions

$$\begin{aligned} \widetilde{\mathbf{A}}^{(i)}({\tilde{x}},{\tilde{t}},\mathbf{w},D_{{\tilde{x}}}{} \mathbf{w}):= M_\ell ^{-\frac{m+1}{2}}\varrho _\ell \mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u}) \end{aligned}$$

satisfy the structure conditions

$$\begin{aligned} \left\{ \begin{aligned}&\widetilde{\mathbf{A}}^{(i)}({\tilde{x}},{\tilde{t}},\mathbf{w},D_{{\tilde{x}}}{} \mathbf{w})\cdot Dw_i\ge C_o |\mathbf{w}|^{m-1} |D_{{\tilde{x}}}w_i|^2,\\&|\widetilde{\mathbf{A}}^{(i)}({\tilde{x}},{\tilde{t}},\mathbf{w},D_{{\tilde{x}}}{} \mathbf{w})|\le C_1|\mathbf{w}|^{m-1} |D_{{\tilde{x}}}w_i|. \end{aligned}\right. \end{aligned}$$

Appealing to (5.9) – (5.10), we have $\tfrac{1}{2}\le |\mathbf{w}|\le 2 n^{\frac{1}{2}}$ in ${\mathcal {Q}}_\frac{1}{2}$. This becomes a diagonal system about w. According to [12], there exist $\alpha _1\in (0,1)$ and $\gamma >1$ depending only on the data, such that for all $0<r<\tfrac{1}{2}$,

Rephrasing this oscillation decay in terms of $\mathbf{u}$, we have for all $0<r<\varrho _\ell$,

(5.11)

Combining (5.8) and (5.11), we arrive at the desired conclusion, i.e., for all $0<r<\varrho$ we have

Here we have used $0<m<1$ such that $\vartheta \le \vartheta _\ell$, and thus ${\mathcal {Q}}_r(\vartheta )\subset {\mathcal {Q}}_r(\vartheta _\ell )$.

6 Local boundedness of solutions

In this section, we propose some sufficient conditions that imply the local boundedness of solutions. In the following we still consider the general system (1.1). However, we need to impose more structure conditions, namely

$$\begin{aligned} \left\{ \begin{aligned}&\mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\cdot Du_i\ge C_o |\mathbf{u}|^{m-1}|Du_i|^2,\\&\sum _{i=1}^{n}{} \mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\cdot u_i D|\mathbf{u}|\ge C_o |\mathbf{u}|^{m}\big |D|\mathbf{u}|\big |^2,\\&|\mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})|\le C_1 |\mathbf{u}|^{m-1}|Du_i|, \end{aligned}\right. \end{aligned}$$

(6.1)

for all $i=1,\cdots ,n$ and for some positive m, $C_o$ and $C_1$. Note that (6.1)$_2$ is additional comparing with (1.2). Nevertheless they are all verified by the prototype system (1.1).

For $m>0$, a vector valued function $\mathbf{u}$ satisfying

$$\begin{aligned} \begin{aligned} u_i\in C\big (0,T;L^2_{{\text {loc}}}(E)\big ), \, |\mathbf{u}|^{m-1}u_i,\,u_i\in L^2_{{\text {loc}}}\big (0,T; W^{1,2}_{{\text {loc}}}(E)\big ), \end{aligned} \end{aligned}$$

is a local, weak sub(super)-solution to (1.1) with the structure conditions in (6.1), if for every compact set $K\subset E$ and every sub-interval $[t_1,t_2]\subset (0,T]$, the integral formulation (1.3) holds.

We have the following result concerning quantitative supreme estimates of local weak solutions, whose proof we use Moser’s iteration to present.

Theorem 6.1

Let $\mathbf{u}$ be a local weak solution to (1.1) in $E_T$ with conditions (6.1). There exists $\gamma >1$ depending only on the data, such that for any $\sigma \in (0,1)$ and $(x_o,t_o)+Q_{\varrho ,\tau }\subset E_T$, for $m>1$, we have

where ${\mathcal {A}}$ is defined in (6.6) and $\kappa :=\frac{N+2}{N}$; whereas for $0<m<1$, requiring that $\lambda :=N(m-1)+2r>0$ for some $r\ge 2$ and $\mathbf{u}\in L^r_{{\text {loc}}}(E_T)$, we have

where $\widetilde{{\mathcal {A}}}$ is defined in (6.7).

6.1 Energy estimates

We first present an energy estimate that will serve as the starting point of Moser’s iteration scheme.

Proposition 6.1

Let $\mathbf{u}$ be a local weak solution to (1.1) in $E_T$ with conditions in (6.1). Set $f(\cdot )$ to be a non-negative, non-decreasing, locally Lipschitz function in ${\mathbb {R}}_+$. There exists a constant $\gamma >0$, such that for all cylinders $Q_{R,S}:=K_R (x_o)\times (t_o-S,t_o]\subset E_T$, and every non-negative, piecewise smooth cutoff function $\zeta$ in $Q_{R,S}$ vanishing on $\partial K_{R}(x_o)$,

provided

$$\begin{aligned} |\mathbf{u}|^{m+1}f(|\mathbf{u}|)\quad \text {and}\quad \int _0^{|\mathbf{u}|} s f(s)\,\mathrm {d}s\in L^1_{{\text {loc}}}(E_T). \end{aligned}$$

Proof

Define the truncated version of f by $f_k:=\min \{f,\,k\}$ for some $k>0$. Let $\psi _\varepsilon$ be as in Proposition 2.1. We take $\phi :=\llbracket u_i\rrbracket _h f_k\big (|\llbracket \mathbf{u}\rrbracket _h|\big )\psi _\varepsilon \zeta ^2$ as a test function against the i-th equation of (I) in $Q:=K_{R}(x_o)\times (t_o-S,\tau ]$ for $\tau <t_o$. In the following we omit the reference to $x_o$ and sum over the index $i\in \{1,\cdots ,n\}$ tacitly. As in Proposition 2.1, the right-hand side of (1.5) vanishes as $h\rightarrow 0$. The time part is calculated as

$$\begin{aligned}&\iint _Q \partial _t \llbracket u_i\rrbracket _h \llbracket u_i\rrbracket _h f_k\big (|\llbracket \mathbf{u}\rrbracket _h|\big )\zeta ^2\,\mathrm {d}x\mathrm {d}t\\&\quad =\tfrac{1}{2}\iint _Q \partial _t{|\llbracket \mathbf{u}\rrbracket _h|^2}f_k(|\llbracket \mathbf{u}\rrbracket _h|)\zeta ^2\,\mathrm {d}x\mathrm {d}t=\iint _Q \partial _t\int _0^{|\llbracket \mathbf{u}\rrbracket _h|}s f_k(s)\mathrm {d}s\,\zeta ^2\,\mathrm {d}x\mathrm {d}t\\&\quad =\int _{K_{R}}\int _0^{|\llbracket \mathbf{u}\rrbracket _h|}s f_k(s)\mathrm {d}s\,\zeta ^2\,\mathrm {d}x\Big |_{t_o-S}^\tau -2\iint _Q \int _0^{|\llbracket \mathbf{u}\rrbracket _h|}s f_k(s)\mathrm {d}s\,\zeta \zeta _t\,\mathrm {d}x\mathrm {d}t\\&\quad \rightarrow \int _{K_{R}}\int _0^{| \mathbf{u}|}s f_k(s)\mathrm {d}s\,\zeta ^2\,\mathrm {d}x\Big |_{t_o-S}^\tau -2\iint _Q \int _0^{| \mathbf{u}|}s f_k(s)\mathrm {d}s\,\zeta \zeta _t\,\mathrm {d}x\mathrm {d}t\quad \text { as }h\rightarrow 0. \end{aligned}$$

The space part can be calculated by sending $h\rightarrow 0$:

$$\begin{aligned}&\iint _Q \llbracket \mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\rrbracket _h\cdot D\big [\llbracket u_i\rrbracket _h f_k\big (|\llbracket \mathbf{u}\rrbracket _h|\big )\zeta ^2\big ]\,\mathrm {d}x\mathrm {d}t\\&\quad \rightarrow \iint _Q \mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\cdot \big [Du_i f_k(|\mathbf{u}|)\zeta ^2\big ]\,\mathrm {d}x\mathrm {d}t\\&\qquad +\iint _Q\mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\cdot \big [u_i Df_k(|\mathbf{u}|)\zeta ^2\big ]\,\mathrm {d}x\mathrm {d}t\\&\qquad +2\iint _Q\mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\cdot \big [u_i f_k(|\mathbf{u}|)\zeta D\zeta \big ]\,\mathrm {d}x\mathrm {d}t\\&\quad =:I_1+I_2+I_3. \end{aligned}$$

Next we estimate the above three terms separately. First, according to (6.1)$_1$, we have

$$\begin{aligned} I_1\ge C_o\iint _Q |\mathbf{u}|^{m-1}|D\mathbf{u}|^2 f_k(|\mathbf{u}|)\zeta ^2\,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

With regard to $I_2$, we calculate using (6.1)$_2$:

$$\begin{aligned} I_2&=\iint _Q\mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\cdot Df_k(|\mathbf{u}|)\zeta ^2\,\mathrm {d}x\mathrm {d}t\\&\ge C_o\iint _Q |\mathbf{u}|^{m} \big |D|\mathbf{u}|\big |^2 f_k^{\prime }(|\mathbf{u}|)\zeta ^2\,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

For $I_3$, we estimate using (6.1)$_3$ and Young’s inequality:

$$\begin{aligned} I_3&=2\iint _Q\mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\cdot D\zeta u_i f_k(|\mathbf{u}|)\zeta \,\mathrm {d}x\mathrm {d}t\\&\ge -\tfrac{1}{2} C_o\iint _Q |\mathbf{u}|^{m-1}|D\mathbf{u}|^2 f_k(|\mathbf{u}|)\zeta ^2\,\mathrm {d}x\mathrm {d}t\\&\quad - \gamma \iint _Q|D\zeta |^2|\mathbf{u}|^{m+1}f_k(|\mathbf{u}|)\,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

Collecting all above estimates and sending $k\rightarrow \infty$, we may conclude. $\square$

Remark 6.1

It is not hard to see from the proof that the energy estimate in Proposition 6.1 still holds if we replace (6.1)$_{1,3}$ by the weaker conditions

$$\begin{aligned} \left\{ \begin{aligned}&\sum _{i=1}^{n}{} \mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})\cdot Du_i\ge C_o |\mathbf{u}|^{m-1}|D\mathbf{u}|^2,\\&|\mathbf{A}^{(i)}(x,t,\mathbf{u},D\mathbf{u})|\le C_1 |\mathbf{u}|^{m-1}|D\mathbf{u}|,\quad i=1,\cdots , n. \end{aligned}\right. \end{aligned}$$

6.2 Proof of Theorem 6.1 when $m>1$

For ease of notation, we set $v:=|\mathbf{u}|$. In the energy estimate of Proposition 6.1, we take $f(v)=v^\beta$ for $\beta \ge 0$, apply the inequality $|Dv|\le |D\mathbf{u}|$ to the second term on the left, and dump the third term. The test function $\zeta$ is chosen to vanish on the parabolic boundary of $Q_{R,S}$ and equal 1 on $Q_{\varrho ,\tau }$, such that $|D\zeta |\le (R-\varrho )^{-1}$ and $|\zeta _t|\le (S-\tau )^{-1}$. As a result, the energy estimate in Proposition 6.1 gives

(6.2)

In the last line we have applied Young’s inequality.

Written in terms of $w:=v^{\frac{m+1+\beta }{2}}$, the above estimate gives

This is the starting point of Moser’s iteration scheme. In order to use this energy estimate properly, we introduce for $\varrho ,\,\tau >0$, $\sigma \in (0,1)$ and $n=0,1,\cdots$,

$$\begin{aligned} \left\{ \begin{array}{c} \displaystyle \varrho _n=\sigma \varrho +\frac{(1-\sigma )\varrho }{2^n},\quad \tau _n=\sigma \tau +\frac{(1-\sigma )\tau }{2^n},\\ \displaystyle {\tilde{\varrho }}_n=\frac{\varrho _n+\varrho _{n+1}}{2},\quad {\tilde{\tau }}_n=\frac{\tau _n+\tau _{n+1}}{2},\\ \displaystyle K_{n}=K_{\varrho _n},\quad {\widetilde{K}}_{n}=K_{{\tilde{\varrho }}_n}, \quad Q_{n}=K_n\times (-\tau _n,0],\quad {\widetilde{Q}}_n={\widetilde{K}}_n\times (-{\tilde{\tau }}_n,0],\\ \displaystyle p_n=m+1+\beta _{n},\quad q_n=\frac{2(2+\beta _n)}{m+1+\beta _n},\quad \beta _{n+1}=\kappa \beta _n+\frac{4}{N},\quad \kappa =1+\frac{2}{N}. \end{array} \right. \end{aligned}$$

Note that the choice of sequences in the last line implies

$$\begin{aligned} p_{n+1}=(m+1+\beta _n)\frac{N+q_n}{N}. \end{aligned}$$

Now set $\zeta$ to be a standard cutoff function that vanishes on $\partial _{p}{\widetilde{Q}}_n$ and equals identity in $Q_{n+1}$, such that $|D\zeta |\le 2^n/\varrho$. We apply the Sobolev imbedding (cf. [4, Chapter I, Proposition 3.1]), together with the energy estimate in ${\widetilde{Q}}_n\subset Q_n$ and the choice $\beta _o=0$, to obtain

for some $b,\, \gamma >1$ depending only on the data. To simplify the above recurrence inequality, we take the power $p_{n+1}^{-1}$ on both sides, set

$$\begin{aligned} Y_n=\left[ \frac{1}{|Q_n|}\iint _{Q_n}(v^{p_n}+1)\,\mathrm {d}x\mathrm {d}t\right] ^{\frac{1}{p_n}}, \end{aligned}$$

(6.3)

and rewrite it as

$$\begin{aligned} Y_{n+1}\le B^{\frac{n}{p_{n+1}}} Y_n^{\frac{\kappa p_n}{p_{n+1}}}, \end{aligned}$$

(6.4)

where

$$\begin{aligned} B:=\frac{\gamma }{(1-\sigma )^{2\kappa }} \left[ \left( \frac{\tau }{\varrho ^2}\right) ^{\frac{2}{N+2}}+\left( \frac{\varrho ^2}{\tau }\right) ^{\frac{N}{N+2}}\right] ^{\kappa }. \end{aligned}$$

Iterating this inequality yields

$$\begin{aligned} Y_n\le B^{\frac{1}{p_{n+1}}[n+\kappa (n-1)+\kappa ^2(n-2)+\cdots +\kappa ^{n-1}]}Y_o^{\frac{p_o}{p_{n+1}}\kappa ^{n+1}} \end{aligned}$$

(6.5)

One calculates the limits

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{p_{n+1}}\sum _{j=0}^{n}\kappa ^j(n-j)=\frac{N^2}{8},\quad \lim _{n\rightarrow \infty }\frac{p_o}{p_{n+1}}\kappa ^{n+1}=\frac{m+1}{2}. \end{aligned}$$

As a result, sending $n\rightarrow \infty$ in (6.5) gives

where

$$\begin{aligned} {\mathcal {A}}:= \left[ \left( \frac{\varrho ^2}{\tau }\right) ^{\frac{N}{N+2}}+\left( \frac{\tau }{\varrho ^2}\right) ^{\frac{2}{N+2}}\right] ^{\frac{N^2\kappa }{8}}. \end{aligned}$$

(6.6)

6.3 Proof of Theorem 6.1 when $0<m<1$

The proof in this case is similar to the previous section. We indicate some key differences. First of all, the energy estimate departs from (6.2), which now becomes

Introduce the sequences $\varrho _n$, ${\tilde{\varrho }}_n$, $\tau _n$, ${\tilde{\tau }}_n$, $q_n$, $K_n$, ${\widetilde{K}}_n$, $Q_n$, ${\widetilde{Q}}_n$, and $\kappa$ as in Sect. 6.2. In this section, the sequences $p_n$ and $\beta _n$ are defined by

$$\begin{aligned} p_n=2+\beta _n,\quad \beta _{n+1}=\kappa \beta _n+\frac{4}{N}+m-1,\,\text { such that } p_{n+1}=(m+1+\beta _n)\frac{N+q_n}{N}. \end{aligned}$$

Next, an application of the Sobolev imbedding as in Sect. 6.2 would yield that

$$\begin{aligned} \iint _{Q_{n+1}}v^{p_{n+1}}\,\mathrm {d}x\mathrm {d}t\le \frac{\gamma b^{n}}{(1-\sigma )^{2\kappa }}\left( \frac{1}{\varrho ^2}+\frac{1}{\tau }\right) ^{\kappa }\left[ \iint _{Q_n}(v^{p_n}+1)\,\mathrm {d}x\mathrm {d}t\right] ^{\kappa }, \end{aligned}$$

for some $\gamma ,\,b>1$ depending only on the data.

The sequence $\{\beta _n\}$ is increasing and the first number $\beta _o$ is chosen to ensure $p_{n+1}>p_n$. A simple calculation indicates this amounts to

$$\begin{aligned} \lambda :=N(m-1)+2r>0\quad \text { where }r:=\beta _o+2. \end{aligned}$$

As in Sect. 6.2, one defines $Y_n$ as in (6.3) and obtains a recurrence inequality as in (6.4) and (6.5). Next, one calculates the limits

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{p_{n+1}}\sum _{j=0}^{n}\kappa ^j(n-j)=\frac{N^2}{2\lambda },\quad \lim _{n\rightarrow \infty }\frac{p_o}{p_{n+1}}\kappa ^{n+1}=\frac{2r}{\lambda }. \end{aligned}$$

Hence sending $n\rightarrow \infty$ in (6.5) gives

where

$$\begin{aligned} \widetilde{{\mathcal {A}}}:= \left[ \left( \frac{\varrho ^2}{\tau }\right) ^{\frac{N}{N+2}}+\left( \frac{\tau }{\varrho ^2}\right) ^{\frac{2}{N+2}}\right] ^{\frac{N^2\kappa }{2\lambda }}. \end{aligned}$$

(6.7)