Regularity of weak solutions to a certain class of parabolic system

TL;DR

This paper proves that weak solutions to a certain class of second order parabolic systems are locally Hölder continuous outside a measure-zero singular set, and establishes fractional differentiability and the Hausdorff dimension of the singular set.

Contribution

It introduces an $A$-caloric approximation method to establish regularity of weak solutions under minimal assumptions of continuous coefficients.

Findings

Weak solutions are locally Hölder continuous outside a zero measure singular set.

The regularity points form an open set with full measure.

The singular set has finite Hausdorff dimension.

Abstract

We study the regularity of weak solutions to a certain class of second order parabolic system under the only assumption of continuous coefficients. By using the $A-$caloric approximation argument, we claim that the weak solution $u$ to such system is locally H\"{o}lder continuous with any exponent $\alpha\in(0,1)$ outside a singular set with zero parabolic measure. In particular, we prove that the regularity point in $Q_T$ is an open set with full measure, and we obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. Finally, we deduce the fractional time and fractional space differentiability of $D u$, and at this stage, we obtain the Hausdorff dimension of singular set of $u$.

Equations

$${\partial }_{t}u-\mathrm{div}a(z,u,Du)=b(z,u,Du),$$

$${\partial }_{t}u-\mathrm{div}a(z,u,Du)=0.$$

$${\partial }_{t}u-\mathrm{div}(|Du{|}^{p-2}Du)=0,$$

$$\sup_{\mbox{\tiny$\begin{array}[]{c}x,y\in\Omega\\ |x-y|<w(y)\end{array}$}}\left|\ln\frac{w(x)}{w(y)}\right|<+\infty,$$

$$\theta =\frac{\frac{1}{p_0}-\frac{1}{q}+\frac{k}{n}}{\frac{1}{p_0}-\frac{1}{p_1}+\frac{r}{n}},$$

$$\Vert v{\Vert }_{W_{\alpha -(r-k)}^{k,q}(\Omega )}\le c(\Vert {\partial }^{r}v{\Vert }_{L_s^{p_1}(\Omega )}^{\theta }\Vert v{\Vert }_{L_{{\alpha }_0-r}^{p_0}(\Omega )}^{1-\theta }+\Vert v{\Vert }_{L_{{\alpha }_0-r}^{p_0}(\Omega )}),$$

$$\Vert v{\Vert }_{W_s^{s_1,s_2}(\Omega )}:=$$

$$\Vert v{\Vert }_{L_s^{k_1}(\Omega )}:=$$

$$|Q_T\setminus Q_0|=0andu\in C^{0;\alpha ,\alpha /2}(Q_0;{\mathbb{R}}^N)$$

$${\Sigma }_1:=$$

$${\Sigma }_2:=$$

$$\text{vbox}-{\int }_{Q_{\frac{\rho }{2}}(z_0)}w\cdot {\phi }_t-({\partial }_{F}a(z_0,l_{\rho }(x_0),D{l}_{\rho })Dw,D\phi )dz=0$$

$$\Phi (z_0,l,\rho ):=\text{vbox}-{\int }_{Q_{\rho }(z_0)}{\left|\frac{u-l}{\rho }\right|}^2+{\left|\frac{u-l}{\rho }\right|}^{p}dz.$$

$$\left|\frac{1}{|Q_{\rho }^{(\lambda )}(z_0)|}{\int }_{Q_{\rho }^{(\lambda )}(z_0)}udz\right|\approx \lambda ,$$

$$\omega (d(z,z_0{)}^2+|u-u_0{|}^2)\lesssim d(z,z_0)+1_K,$$

$$\text{dim}\mathcal{H}({\Sigma }_1)\le n+2-2\gamma ,\text{dim}\mathcal{H}({\Sigma }_2)\le n+2-2\gamma ,$$

$$B_{\rho }(x_0):=\{x\in {\mathbb{R}}^n:|x-x_0|<\rho \}$$

$$Q_{\rho }(x_0):=B_{\rho }(x_0)\times (t_0-{\rho }^2,t_0)\equiv B_{\rho }(x_0)\times {\Lambda }_{\rho }(t_0)$$

$$(f{)}_{x_0,\rho }=\text{vbox}-{\int }_{B_{\rho }(x_0)}fdx=\frac{1}{|B_{\rho }(x_0)|}{\int }_{B_{\rho }(x_0)}fdx,$$

$$(f{)}_{z_0,\rho }=\text{vbox}-{\int }_{Q_{\rho }(z_0)}fdz=\frac{1}{|Q_{\rho }(z_0)|}{\int }_{Q_{\rho }(z_0)}fdz.$$

$$Q_{\rho }^{(\lambda )}(z_0)=B_{\rho }(x_0)\times {\Lambda }_{\rho }^{(\lambda )}(t_0)$$

$${\Lambda }_{\rho }^{(\lambda )}(t_0):=(t_0-{\lambda }^{2-p}{\rho }^2,t_0).$$

$$d(z,z_0)=:max\left\{|x-x_0|,\sqrt{|t-t_0|}\right\}\approx \sqrt{|x-x_0{|}^2+|t-t_0|}.$$

$$u\in C^{k;\alpha ,\alpha /2}(Q_T;{\mathbb{R}}^N):=\left\{v\in C^k(Q_T;{\mathbb{R}}^N):\underset{z,z_0\in Q_T;z\ne z_0}{\sup }\frac{|v(z)-v(z_0)|}{d(z,z_0{)}^{\alpha }}<\infty \right\}.$$

$${\mathcal{H}}_s^{\delta }(E):=\inf \left\{\sum _{i=1}^{\infty }{r}_i^s:E\subset \bigcup _{i=1}^{\infty }{Q}_{r_i}(x_i,t_i),r_i\le \delta \right\},{\mathcal{H}}_s(E):=\underset{\delta >0}{\sup }{\mathcal{H}}_s^{\delta }(E).$$

$$\text{dim}\mathcal{H}(E):=\inf \left\{s>0:{\mathcal{H}}_s(E)=0\right\}=\sup \left\{s>0:{\mathcal{H}}_s(E)=\infty \right\}.$$

$${\tau }^{h}v(x,t):=v(x,t+h)-v(x,t),$$

$${\tau }_{1,2}^{h}a(z,u,Du):=a((x,t+h),u(x,t+h),Du)-a(z,u,Du)$$

$${\tau }_3^{h}a(z,u,Du):=a(z,u,Du(x,t+h))-a(z,u,Du)$$

$${\tau }_{h}v(x,t)\equiv {\tau }_{h,i}v(x,t):=v(x+h{e}_i,t)-v(x,t),$$

$${\tau }_h^{1,2}a(z,u,Du)\equiv {\tau }_{h,i}^{1,2}a(z,u,Du):=a((x+h{e}_i,t),u(x+h{e}_i,t),Du)-a(z,u,Du)$$

$${\tau }_h^{3}a(z,u,Du)\equiv {\tau }_{h,i}^{3}a(z,u,Du):=a(z,u,Du(x+h{e}_i,t))-a(z,u,Du)$$

$${△}_{h}v(x,t)\equiv {△}_{h,i}v(x,t):=\frac{v(x+h{e}_i,t)-v(x,t)}{h}.$$

$${\int }_{-T}^0{\int }_{\Omega }{\int }_{\Omega }\frac{|u(x,t)-u(y,t){|}^2}{|x-y{|}^{n+2\alpha }}dxdydt+{\int }_{\Omega }{\int }_{-T}^0{\int }_{-T}^0\frac{|u(x,t)-u(x,s){|}^2}{|t-s{|}^{1+2\theta }}dtdsdx=:[u{]}_{\alpha ,\theta ;Q_T}<\infty .$$

Full text

**Keywords ** Parabolic system ; regularity ; Hölder continuity ; weak solution ; Hausdorff dimension.

Corresponding author:Jianfeng Zhou, jianfengzhou@pku.edu.cn

This work was supported by the National Natural Science Foundation of China (No. 11271305, 11531010).

1. Introduction

Let $\Omega\subset\mathbb{R}^{n}$ ($n\geq 2$) be a bounded domain, the aim of this work is to give a study of regularity properties of weak solution to the following inhomogeneous parabolic system

[TABLE]

with $z=(x,t)\in\Omega\times(-T,0)\equiv Q_{T}$, and $T>0$. $a(\cdot):Q_{T}\times\mathbb{R}^{N}\times\mathbb{R}^{Nn}\rightarrow\mathbb{R}^{Nn}$, $u:Q_{T}\rightarrow\mathbb{R}^{N}$, $b:Q_{T}\times\mathbb{R}^{N}\times\mathbb{R}^{Nn}\rightarrow\mathbb{R}^{N}$, $N\in\mathbb{Z}^{+}$, $N\geq 1$. In general, the solution of parabolic systems (1.1) can not be expected to be regular everywhere on the domain, even the homogeneous case

[TABLE]

It is worth to note that everywhere regularity can be obtained only with special structure on $a(z,u,Du)$ such as the evolutionary $p-$Laplacian system

[TABLE]

for the regularity problem was settled by the fundamental contributions of Dibenedetto’s and Friedman [19, 20, 21], otherwise it fails in general see [47, 48, 49] for example.

However, one can expect partial regularity results, this is regularity away from a singular set that is in some sense small. The partial regularity for general parabolic (1.2) was a longstanding open problem until it was solved by Duzaar and Mingione [28], Duzaar, Mingione and Steffen [29], C. Scheven [43] and also Duzaar et al. [25, 8, 9], their proofs are based on the $A-$caloric approximation method to the parabolic setting. Subsequently, Scheven [43] derived an analogous result for the subquadratic case of (1.2). Moreover, Baroni [3] have showed the continuity of the gradient $Du$ while only assuming the Dini continuity of $a(\cdot,\cdot,Du)$. Under the assumption of continuous coefficients, Bögelein-Duzzar-Mingione [11] proved a partial Hölder continuity results for (1.2) with polynomial growth. When considering the boundary regularity of the parabolic system, the same authors [8, 9] have showed that almost every parabolic boundary point is a Hölder continuity point for $Du$. There have been many research articles on the regularity of weak solution to parabolic system, e.g., [1, 12, 31, 36, 42, 50] and the reference therein.

The above result for parabolic problems are analogous of results of elliptic case (cf. [40]), the application of the so called harmonic approximation to prove regularity theorems goes back to Simon [44, 46] and Duzaar et al. [26, 27]. Related results for problems with continuous coefficients, Campanato [17] (see also [16]) derived the Hölder continuity of the solutions of some nonlinear elliptic system in $\mathbb{R}$. In higher dimensions cases, Foss-Mingione [33] proved the partial Hölder continuity for solutions to elliptic system. The proof relies upon an iteration scheme of a decay estimate for a new type of excess functional measuring the oscillations in the solution and its gradient. Afterwards, Beck [4] showed the boundary regularity of elliptic system with Dirichlet condition. When considering the Dini continuous coefficients, Duzzar-Gastel [24] presented a general low-order partial regularity theory. In particular, for the system with variable exponent $p(x)$, Habermann [35] (see also [2]) derived the partial Hölder continuity for weak solution to a nonlinear problem with continuous growth exponent. For more details, one can also refer [5, 7, 22, 32, 34, 37, 52] and the reference therein.

Turning to the technically more challenging case of (1.1), as far as we are aware, there has been no previous work addressing partial regularity of weak solution $u$ to (1.1) with continuous coefficients available in the literature yet (cf. [11] for the homogeneous case (1.2)). Thus, in present paper, we aim to fill a gap in the partial regularity theory of quasi-linear parabolic system (1.1). This turns out to be a challenging task, since the nonhomogeneous term $b(z,u,Du)$ will lead to several new difficulties:

1. (1)

When establish the Poincaré inequality in Section 4, we are not able to obtain (4.13) directly, since we can not use the zero-boundary condition on $\partial B_{\rho}$ for any $B_{\rho}\subset\Omega$. In order to avoid this flaw, some iteration argument will be introduced; 2. (2)

For prove the Caccioppoli’s inequality (3.1), the key point is that, bound $b(z,u,Du)$ in terms of $Du-Dl$ or $u-l$ ($l$ be an affine function defined in later). However, one can not use the inequality $l(z)\leq l(z_{0})+Dl\leq M$ directly for a.e. $z\in Q_{\rho}$, $\rho\leq 1$ and $M\geq 1$ be a constant. Otherwise, the constant after (5.16) depends on $M$ with $M=H\lambda$ (see (5.7) and (Aj)). As a consequence, all constants in Lemma 5.3 depend on $\lambda$ so that the estimates could blow up during the iteration process. At this stage, we shall use a weighted Sobolev interpolation inequality (cf. [30, 13, 6]): for suitable function $w(\cdot):\Omega\longrightarrow\mathbb{R}_{+}$ satisfies

[TABLE]

and for any function $v$, $k\in\mathbb{N}_{0}$, $r\in\mathbb{N}$, $s\in\mathbb{R}$, $p_{0}\in[1,+\infty)$, $p_{1},q\in(1,+\infty)$, if $\partial^{r}v\in L^{p_{1}}_{s}(\Omega)$, $v\in L^{p_{0}}_{\alpha_{0}-r}(\Omega)$, $k<r$, $p_{0}\leq p_{1}\leq q$, $\frac{1}{q}>\frac{1}{p_{1}}-\frac{r-k}{n}$, $\alpha_{0}=s-n(\frac{1}{p_{0}}-\frac{1}{p_{1}})$, $\alpha=s-n(\frac{1}{q}-\frac{1}{p_{1}})$ and

[TABLE]

then, there holds

[TABLE]

where $c$ depends on $p_{0}$, $p_{1}$, $n$, $q$, $r$, $k$. Here, we have defined

[TABLE]

with $s_{1}\in\mathbb{N}$, $k_{1},s_{2}\geq 1$ and $s\in\mathbb{R}$.

The main result of the present paper is stated as following

Theorem 1.1**.**

Let $p\geq 2$ and $u\in C^{0}(-T,0;L^{2}(\Omega;\mathbb{R}^{N}))\cap L^{p}(-T,0;W^{1,p}(\Omega;\mathbb{R}^{N}))$ be a weak solution of the parabolic systems (1.1) in $Q_{T}$ under the assumptions (2.1)-(2.5). Then, there exists an open subset $Q_{0}\subset Q_{T}$ such that

[TABLE]

for every $\alpha\in(0,1)$. Moreover, we have the singular set satisfies $Q_{T}\setminus Q_{0}\subset\Sigma_{1}\cup\Sigma_{2}$, where

[TABLE]

The main technique we have used in the proof of Theorem 1.1 is the $A-$caloric approximation lemma. Here, $A$ is a bilinear form on $\mathbb{R}^{Nn}\times\mathbb{R}^{Nn}$ with constant coefficients. If $A$ satisfies certain growth and ellipticity conditions, then the weak solution $h$ to (5.6) is $A-$caloric and have nice decay properties. In order to look for such ‘good’ function, we shall use the $A-$caloric approximation lemma (cf. Lemma 2.5), from which, we can transfer the property of $A-$caloric to some ‘bad’ function (target function). When applying the $A-$caloric approximation lemma, we need to pay attention to three necessary conditions: i) the target function is bounded from above on the scale of $L^{2}$-norm and $L^{p}$-norm; ii) the target function satisfies a linearized system; iii) the target function satisfies the smallness condition in the sense of distribution. To satisfy these three conditions, we will establish the Caccioppoli inequality and linearize the system (1.1) in Sec. 3 and Sec. 5, respectively. On the other hand, with the help of linearization lemma (cf. Lemma 5.1), we would like to show $w:=u-l_{\rho}$ approximately solves

[TABLE]

for all $\varphi\in C_{0}^{\infty}(Q_{\rho}(z_{0});\mathbb{R}^{N})$. Here, $l_{\rho}:B_{\rho}\longrightarrow\mathbb{R}^{N}$ be the unique time independent affine map minimizing l\mapsto\mathchoice{{\vbox{\hbox{\textstyle- }}\kern-7.83337pt}}{{\vbox{\hbox{\scriptstyle- }}\kern-5.90005pt}}{{\vbox{\hbox{\scriptscriptstyle- }}\kern-4.75003pt}}{{\vbox{\hbox{\scriptscriptstyle- }}\kern-4.25003pt}}\!\int_{Q_{\rho}(z_{0})}|u-l|^{2}dz. At this stage, by the $A-$ caloric approximation lemma, then we can establish smallness of the first order excess functional

[TABLE]

From which, we are able to measure the oscillation in $u$ with respect to an affine mapping. Moreover, in order to provide a bilinear form that satisfies the growth and ellipticity bounds needed to apply the $A-$caloric approximation lemma, we may need the integral estimate on intrinsic cylinders, that is, parabolic cylinders stretched according to the size of the solution $u$ itself. The rough asymptotic is given by

[TABLE]

with $Q_{\rho}^{(\lambda)}:B_{\rho}(x_{0})\times(t_{0}-\lambda^{2-p}\rho^{2},t_{0})$.

According to Theorem 1.1, we immediately deduce that

[TABLE]

where $K:=\Sigma_{1}\cup\Sigma_{2}$ and $1_{K}=1$ for $x\in K$, otherwise, $1_{K}=0$. Then we have the following result.

Theorem 1.2**.**

Let the all assumptions in Theorem 1.1 be verified and $p=2$. Then, for any $\theta\in(0,\frac{1}{3})$, $\alpha\in(0,\frac{1}{2}]$, there holds $Du\in W^{\alpha,\theta;2}_{loc}(Q_{T})$. Furthermore, the singular set $\Sigma_{1}$ and $\Sigma_{2}$ in Theorem 1.1 satisfying

[TABLE]

where $\gamma\leq\alpha,$ $\frac{\gamma}{2}\leq\theta$.

The rest of paper is organised as follows. First of all, in Sec. 2 we state some assumption of the structure function $a(\cdot)$ and the inhomogeneity term $b(\cdot)$. Moreover, we present some notation, definition of weak solution to (1.1), and some useful lemma which will be used in our proof. Next, we provide some preliminary material in Sec. 3 and Sec. 4, which will be quite useful in the proof of main result. The first step of our proof is to establish a Caccioppoli’s type inequality. Subsequently, we establish a Poincaré type inequality in Section 4, which is useful to show the boundness of $|Dl|$, with $l$ be an affine function. In Sec. 5, we first provide a linearization strategy for context, we show a decay estimate of $\Phi_{\lambda_{j}}(\vartheta^{j}\rho)$, and then obtain a Campanato type estimate. This, combined with a standard argument implies the Theorem 1.1. Finally, in Sec. 6, we derive the fractional time and space differentiability of $Du$, from which, we estimate the Hausdorff dimension of singular set of weak solution $u$ to (1.1).

2. preliminaries

2.1. Notation

Let $x_{0}\in\mathbb{R}^{n}$, $t_{0}\in\mathbb{R}$, $z_{0}=(x_{0},t_{0})$, we denote

[TABLE]

as an open ball in $\mathbb{R}^{n}$, and let

[TABLE]

as a cylinder in $\mathbb{R}^{n+2}$. Let $B_{\rho}(x_{0})$, $Q_{\rho}(z_{0})\subset Q_{T}$, and $f(x,t)$ integrable on $B_{\rho}(x_{0})$ and $Q_{\rho}(z_{0})$, then the average integral of $f$ over $B_{\rho}(x_{0})$ and $Q_{\rho}(z_{0})$ are defined by

[TABLE]

and

[TABLE]

In what follows, we shall repeatly use the scaled parabolic cylinders of the form

[TABLE]

with radius $\rho>0$, scaling factor $\lambda>0$, and

[TABLE]

In particular, when $\lambda=1,$ then $Q_{\rho}^{(1)}(z_{0})\equiv Q_{\rho}(z_{0})$, and $\Lambda_{\rho}^{(1)}(t_{0})\equiv\Lambda_{\rho}(t_{0})$. Furthermore, The parabolic metric is defined as usual by

[TABLE]

Based on the parabolic metric, the space $C^{k;\alpha_{1},\alpha_{2}}(Q_{T})$ are those of functions $u\in C^{k}(Q_{T})$ which are $\alpha_{1}$-Hölder continuous in the space variables $\alpha_{2}$-Hölder continuous in the time variables. More precisely, we call $u\in C^{k;\alpha,\alpha/2}(\Omega_{T};\mathbb{R}^{N})$ ($k\geq 0$ be an integer), if

[TABLE]

We said $u\in C_{loc}^{k;\alpha,\alpha/2}(Q_{T};\mathbb{R}^{N})$ if and only if for all $A\subset Q_{T}$, there holds $u\in C^{k;\alpha,\alpha/2}(A;\mathbb{R}^{N})$. Finally, we note that in the whole paper, we use the notation $(\cdot,\cdot)$ denote the inner product.

For $s\in[0,n+2]$ and $E\subset\mathbb{R}^{n+1}$, we define the (parabolic) Hausdorff measure:

[TABLE]

From above, then the Hausdorff dimension is usually defined by

[TABLE]

Moreover, in this paper we use $D$ or $\nabla$ denotes the ‘gradient’, and we will use the following natation:

[TABLE]

Here $e_{i}=(0,\cdots 0,1_{i-th},0,\cdots,0)$, $i=1,\cdots,n$. Finally, let us recall the definition of parabolic fractional Sobolev space (refer to [38] for details). We say $u\in L^{2}(Q_{T})$ belongs to the fractional Sobolev space $W^{\alpha,\theta;2}(Q_{T})$, $\alpha,\theta\in(0,1)$, if

[TABLE]

2.2. Assumption on the structure function $a(\cdot)$ and $b(\cdot)$

In the following, we impose the condition on the structure function $a(z,u,F)$ and $b(z,u,F)$ for $p\geq 2$.

• •

The growth condition

[TABLE]

with $(z,u,F)\in Q_{T}\times{\mathbb{R}^{N}}\times\mathbb{R}^{Nn},L\geq 1$ be a constant.

• •

The ellipticity condition

[TABLE]

for all $(z,u,F)\in Q_{T}\times\mathbb{R}^{N}\times\mathbb{R}^{Nn}$, $\widetilde{F}\in\mathbb{R}^{Nn}$, $0<\nu\leq 1\leq L$ be a constant.

Moreover, we also need the following two continuity conditions:

• •

Continuity of lower order term

[TABLE]

• •

Continuity of higher order term

[TABLE]

for all $z,z_{0}\in Q_{T}$, $u,u_{0}\in\mathbb{R}^{N}$ and $F,F_{0}\in\mathbb{R}^{Nn}$. Here, $\omega,\ \mu:[0,\infty)\rightarrow[0,1]$ are two bounded, concave, and non-decreasing functions satisfy

[TABLE]

The term $b(z,u,F)$ satisfies the following growth conditions:

• •

Controllable growth condition

[TABLE]

for all $(z,u,F)\in Q_{T}\times\mathbb{R}^{N}\times\mathbb{R}^{Nn}$ with $q_{1}\in[0,\frac{(n+2)(p-1)}{n}]$, where the upper bound of $q_{1}$ depends on the Ladyzhenskaya inequality.

2.3. Definition of weak solution

Let $p\geq 2$, we call $u\in C^{0}(-T,0;L^{2}(\Omega;\mathbb{R}^{N}))\cap L^{p}(-T,0;W^{1,p}(\Omega;\mathbb{R}^{N}))$ is a weak solution to (1.1), if and only if the following identity

[TABLE]

holds for all $\varphi\in C_{0}^{\infty}(Q_{T};\mathbb{R}^{N})$.

From [19] (see also [38]) we recall the definition of the Steklov averages that allow us to restate (2.6) in an equivalent way. Let $v\in L^{1}(Q_{T})$ and $0<h<T$, the Steklov averages $v_{h}$ and $v_{\bar{h}}$ are defined by

[TABLE]

and

[TABLE]

respectively, for all $t\in(-T,0)$. We note that if $v\in L^{r}(-T,0;L^{q}(\Omega))$ with $r,q\geq 1$, then $v_{h}\longrightarrow v$ in $L^{r}(-T+\varepsilon,0;L^{q}(\Omega))$ as $h\longrightarrow 0$, for every $t\in(-T+\varepsilon,0)$ and $\varepsilon\in(0,T)$, and the same result holds for $v_{\bar{h}}$.

In virtue of the convergence properties of the Steklov averages, then we have a equivalent definition of weak solution to (1.1):

Definition 2.1**.**

(A equivalent definition of weak solution).  Let $2\leq p<\infty$ and $u_{0}\in L^{2}(\Omega)$. Then $u\in L^{\infty}(-T,0;L^{2}(\Omega))\cap L^{p}(-T,0;W^{1,p}(\Omega))$ is called a weak solution to (1.1) if

[TABLE]

holds for all $\varphi\in C_{0}^{\infty}(Q_{T};\mathbb{R}^{N})$.

Employing ($H_{1}$) and ($H_{2}$), we have the following result.

Lemma 2.1**.**

Let $2\leq p<\infty$, then there exists a constant $c=c(L,n,p)>0$ such that for any $F_{1},F_{2}\in\mathbb{R}^{Nn}$, it holds that

[TABLE]

Next, the following lemma as an auxiliary tool will be heavily used (cf. [14]).

Lemma 2.2**.**

Let $A,B\in\mathbb{R}^{k}$, $k\geq 1$ and $\sigma>-1$, then there exists a constant $c=c(\sigma)$, such that

[TABLE]

As a consequence, from Lemma 2.2 and (2.2), it follows that the monotonicity of $a(z,u,\cdot)$:

[TABLE]

where $c=c(n,p,\nu)$.

In the next proposition we recall the parabolic version of the well known relation between Nikolski spaces and Fractional Sobolev spaces (cf. [45]).

Proposition 2.1**.**

Let $u\in L^{2}(Q_{T};\mathbb{R}^{N})$, suppose

[TABLE]

where $Q^{\prime}:=\Omega^{\prime}\times(-T+\delta,-\delta)$ and $\Omega^{\prime}\subset\subset\Omega$ for every $h\in\mathbb{R}$, such that $|h|\leq\min\left\{\delta,1\right\}$ with $\delta\in(0,\frac{T}{8})$.Then there exists a constant $c^{\prime}=c^{\prime}(\theta,\gamma,\delta,\|u\|_{L^{2}(Q_{T})})>0$ such that

[TABLE]

for all $\gamma\in(0,\theta)$. Furthermore, suppose that

[TABLE]

for every $|h|\leq\min\{dist(\Omega^{\prime},\partial\Omega),1\}$, $s\in\{1,\cdots,n\}$, with $\{e_{s}\}_{s=1}^{n}$ is the standard basis of $\mathbb{R}^{n}$. Then, for every $\tilde{\Omega}\subset\subset\Omega^{\prime}$ there exists a constants $c^{\prime\prime}=c^{\prime\prime}(\delta,\theta,\gamma,c_{2},dist(\Omega^{\prime},\partial\Omega),dist(\partial\Omega^{\prime},\tilde{\Omega}),\|u\|_{L^{2}(Q_{T})})$ such that

[TABLE]

for all $\gamma\in(0,\theta)$.

From Proposition 2.1, we can see that in order to prove the fractional differentiability of $Du$ in Theorem 1.2, it is only need to prove

[TABLE]

for all $\theta\in(0,\frac{2}{3})$, and

[TABLE]

for $\gamma\in(0,1].$

On the other hand, for estimate the Hausdorff dimension of singular set of $u$ defined in Theorem 1.1, we shall use the following arguments (cf. [23, 41]).

Lemma 2.3**.**

Let $\Omega\subset\mathbb{R}^{n}$ ($n\geq 2$), $v\in W^{\beta,\beta/2;2}_{loc}(Q_{T};\mathbb{R}^{N})$ with $\beta>0$ and $N\geq 1$. Let

[TABLE]

Then, there holds

[TABLE]

2.4. Minimizing affine function

Let $z_{0}\in\mathbb{R}^{n+2}$, $\rho$, $\lambda>0$. For a given function $u\in L^{2}(Q_{\rho}^{(\lambda)}(z_{0});\mathbb{R}^{N})$, we denote by $l_{z_{0},\rho}^{(\lambda)}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$ the unique affine function (in space) minimizing

[TABLE]

amongst all affine function $l(z)=l(x)$ independent of $t$. We note that such a unique minimizing affine function exists and takes the form

[TABLE]

where $\xi_{z_{0},\rho}^{(\lambda)}\in\mathbb{R}^{N}$, $A_{z_{0},\rho}^{(\lambda)}\in\mathbb{R}^{Nn}$. A straightforward calculation shows that

[TABLE]

for any $a(x)=\xi+A(x-x_{0})$ with $\xi\in\mathbb{R}^{N}$, $A\in\mathbb{R}^{Nn}$. This implies in particular that

[TABLE]

Furthermore, we need the following argument, which can be proven analogously to [51]. For any $\xi\in\mathbb{R}^{n}$ and $A\in\mathbb{R}^{Nn}$ there holds

[TABLE]

Finally, we introduce the following conclusion (cf. [10] Lemma 3.8), which provide a connection between the minimizing affine functions $l(z)$ and $l_{z_{0},\rho}^{(\lambda)}(z)$.

Lemma 2.4**.**

Let $p\geq 2$, $Q_{\rho}^{(\lambda)}(z_{0})\in\mathbb{R}^{n+2}$ with $z_{0}\in\mathbb{R}^{n+2}$ and $\rho,\lambda>0$ be a scaled parabolic cylinder and $u\in L^{p}(Q_{\rho}^{(\lambda)}(z_{0});\mathbb{R}^{N})$, and let $l:\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$ be an affine function independent of $t$. Then, we have

[TABLE]

2.5. $A-$caloric approximation

A strongly elliptic bilinear form $A$ on $\mathbb{R}^{Nn}$ means that

[TABLE]

for all $F,\widetilde{F}\in\mathbb{R}^{Nn}$ with ellipticity constant $\nu>0$ and upper bound $L>0$. We shall say that a function $h\in L^{p}(\Lambda_{\rho}(t_{0});W^{1,p}(B_{\rho}(x_{0});\mathbb{R}^{N}))$ is $A-caloric$ on $Q_{\rho}(z_{0})$ if it satisfies

[TABLE]

for all $\varphi\in C_{0}^{\infty}(Q_{\rho}(z_{0});\mathbb{R}^{N})$.

In order to obtain the decay estimate (5.11), we introduce the following $A-$caloric approximation lemma (cf. [29]).

Lemma 2.5**.**

There exists a positive function $\delta_{0}=\delta_{0}(n,p,\nu,L,\varepsilon)\in(0,1]$ with the following property, for each $\gamma\in(0,1]$, and each bilinear form $A$ in $\mathbb{R}^{Nn}$ with ellipticity constant $\nu$ and upper bound $L,\varepsilon$ is a positive number, whenever $u\in L^{p}(\Lambda_{\rho}(t_{0});W^{1,p}(B_{\rho}(x_{0});\mathbb{R}^{N}))$ satisfying

[TABLE]

is approximately $A-caloric$, in the sense that for each some $\delta\in(0,\delta_{0}]$ there holds

[TABLE]

for all $\varphi\in C_{0}^{\infty}(Q_{\rho}(z_{0});\mathbb{R}^{N})$. Then, there exists an $A-$caloric function $h$ such that

[TABLE]

and

[TABLE]

3. Caccioppoli type inequality

In this section, we propose to derive a Caccioppoli type inequality under the conditions (2.1)-(2.3), (2.5). Such result provide the smallness condition in the iteration process of decay estimate in Section 5.

Lemma 3.1**.**

(Caccioppoli type inequality) Let $u\in C^{0}(-T,0;L^{2}(\Omega;\mathbb{R}^{N}))\cap L^{p}(-T,0;W^{1,p}(\Omega;\mathbb{R}^{N}))$ be a weak solution to (1.1) under the assumption (2.1)-(2.3), (2.5), $Q_{\rho}^{(\lambda)}\subset Q_{T}$ is a scaled parabolic cylinder with reference point $z_{0}=(x_{0},t_{0})$ and $0<\rho\leq 1$ suitable small, scaling factor $\lambda\geq 1$ and affine function $l:\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$ such that $\lambda\leq 1+|Dl|$. Then there holds

[TABLE]

with $c=c(n,p,\nu,L)$.

Proof.

Let $0<\rho/2\leq r<\sigma\leq\rho<1$, let $\eta(x)\in C_{0}^{1}(B_{\sigma}(x_{0});\mathbb{R})$ be a cut-off function, $i.e$. $0\leq\eta\leq 1$, $\eta=1$ in $B_{r}(x_{0})$ and $|\nabla\eta|\leq\frac{c^{\prime}}{\sigma-r}$ with $c^{\prime}$ is a positive constant independent of $\sigma$ and $r$. Moreover, we choose $\xi\in C_{0}^{1}(\Lambda_{\sigma}^{(\lambda)}(t_{0}))$ be a cut-off function in time, such that, with $0<\delta_{1}<r$ being arbitrary

[TABLE]

For simplicity, in what follows, we will omit the reference point $z_{0}$, and denote $Q_{\rho}^{(\lambda)}(z_{0})$,$B_{\rho}^{(\lambda)}(x_{0}),\Lambda_{\rho}^{(\lambda)}(t_{0})$ as $Q_{\rho}^{(\lambda)},B_{\rho}^{(\lambda)},$ $\Lambda_{\rho}^{(\lambda)}$, respectively. Let $\varphi=\eta^{q_{0}+1}\xi^{2}(u-l)$ with

[TABLE]

as a test function in the weak formulation $(\ref{2.1})$, which implies that

[TABLE]

Observe that

[TABLE]

and

[TABLE]

Thus, inserting (3)-(3.4) into (3) and note that $l(z)=l(x)$, we arrive at

[TABLE]

Firstly, we focus our attention on estimating the term in the left side of (3). Appealing to (2.2) and Lemma 2.2, we infer that

[TABLE]

Now, we turn to estimate the terms $I-V$ in (3). For the term $I$, we first note that, from (2.1) there holds

[TABLE]

and hence

[TABLE]

where $\varepsilon\in(0,1)$ will be specified in later, and in the previous inequality we have taken into account that $|Du|\leq|Dl|+|Du-Dl|$.

Next, using (2.3), we deduce that

[TABLE]

For the term $IV$, note that $\lambda\leq 1+|Dl|$, we have

[TABLE]

Finally, we estimate the term $V$. From $(H_{6})$, we have

[TABLE]

By the Young’s inequality, it is clearly that

[TABLE]

with $\varepsilon_{1}\in(0,1)$ will be specified in later.

For the term $V_{2}$, first, we divide $B_{\sigma}$ into two parts: $B_{\sigma}:=B_{\sigma}^{1}\cup B_{\sigma}^{2}$ and

[TABLE]

Next, we take $k$, $q$, $p_{1}$, $p_{0}$, $\alpha$, $r$, $s$, $\alpha_{0}$ in (1.3) as [math], $\frac{(n+2)p}{n}$, $p$, $2$, $q_{0}+2$, $1$, $q_{0}+1$, $q_{0}+1-n(\frac{1}{2}-\frac{1}{p})$, respectively, and at this stage, we have $\theta=\frac{n}{n+2}$ in (1.3). Moreover, by the definition of $q_{0}$, we can see that

[TABLE]

Therefore, by weighted Sobolev interpolation inequality (1.3) and the Hölder’s inequality, we are in a position to obtain

[TABLE]

[TABLE]

where $\varepsilon_{2}\in(0,1)$ will be specified in later.

Now, we choose $\rho$ suitable small such that

[TABLE]

Therefore, we have

[TABLE]

This implies that

[TABLE]

As a consequence, from (3)–(3.13), it follows that

[TABLE]

Inserting (3)-(3) and (3) into (3), we conclude that

[TABLE]

Now, we choose $\varepsilon=\frac{\nu}{8c(p)}$ and $\varepsilon_{1}=2^{-(p+2)}\frac{\nu}{c(p)}$, $\varepsilon_{2}=\frac{\nu}{8c(n,p)c(p)}$, note that $2q_{0},\frac{pq_{0}}{p-1}\geq q_{0}+1$ and $\frac{p}{p-1}\in(1,2]$, moving the same terms into left side, and dividing by $(1+|Dl|)^{p}$ in both side, taking mean values and Jensen’s inequality

[TABLE]

letting $\delta_{1}\rightarrow 0$, then we have (3.1). ∎

4. Poincaré type inequality

In this section, we aim at establishing a Poincaré type inequality of weak solution to (1.1) under the assumptions (2.1), (2.3), (2.5). We note that such inequality plays a key role in the whole paper, that will be used in Sec. 5, from which, we are able to show that for every $z_{0}\in Q_{T}\setminus(\Sigma_{1}\cup\Sigma_{2})$ and suitable $0\leq\rho_{0}\leq 1$, the assumption of Lemma 5.3 is valid.

Lemma 4.1**.**

(Poincaré type inequality). Let $u\in L^{p}(-T,0;W^{1,p}(\Omega;\mathbb{R}^{N}))\cap C^{0}(-T,0;L^{2}(\Omega;\mathbb{R}^{N}))$ be a weak solution of (1.1) in $Q_{T}$ under the assumption (2.1), (2.3), (2.5), for any $Q_{\rho}{(z_{0})}\subset Q_{T}$ be a parabolic cylinder with reference $z_{0}=(x_{0},t_{0})$ and $0<\rho\leq 1$. Then, there holds

[TABLE]

for any $q\in[1,p]$, and

[TABLE]

where $c=c(n,N,p,L)$.

Proof.

For simplicity, we may also omit the reference point $z_{0}$ of $Q_{\rho}{(z_{0})}$, $B_{\rho}{(z_{0})}$, $\Lambda_{\rho}{(z_{0})}$, instead by $Q_{\rho}$, $B_{\rho}$, and $\Lambda_{\rho}$, respectively, if there is no danger of any confusion. Let $\eta\in C_{0}^{\infty}(B_{\rho})$ be a nonnegative weight function satisfying

[TABLE]

where $c_{\eta}=c_{\eta}(n)$, define

[TABLE]

as a weighted mean of $u(x,t)$ on $B_{\rho}$ for a.e. $t\in(-T,0)$. To begin with, we shall show the following argument for a.e. $t,\tau\in\Lambda_{\rho}$:

[TABLE]

and

[TABLE]

where $c=c(n,N,p,L)$.

Now, we concentrate our attention on the proof of (4.3)-(4), without loss of generality, we may assume $t>\tau$, let $\xi_{\theta}(s)\in C_{0}^{\infty}((\tau,t))$ be a cut-off function, defined by

[TABLE]

with $\theta\in(0,(t-\tau)/2)$. We now choose $\varphi_{\theta}:\mathbb{R}^{n+2}\rightarrow\mathbb{R}^{N}$ be a test function in the weak formulation (2.6) with $(\varphi_{\theta})_{i}=\eta\xi_{\theta}$ and $(\varphi_{\theta})_{j}=0$ for $j\neq i$ and $i,j\in\{1,\cdots,N\}$, which implies

[TABLE]

Taking into account the Steklov arguments and the definition of $(u)_{\eta}(t)$, we first deduce that

[TABLE]

as $\theta\rightarrow 0$.

Next, letting $\theta\rightarrow 0$ in the right side of (4.5), we arrive at

[TABLE]

In virtue of (2.5), (4.6), and note that $t,\tau\in\Lambda_{\rho}$, we infer that

[TABLE]

Now we focus our attention on estimating the term $W$. Employing interpolation inequality (G-N-S inequality), it holds that

[TABLE]

where in the last inequality we have taken into account that

[TABLE]

It is clearly that the term $W_{1}$ can be split as

[TABLE]

where $\Lambda_{\rho}=J_{1}\cup J_{2}$, and

[TABLE]

Thus, we are in a position to obtain

[TABLE]

and by iteratively estimating, we have

[TABLE]

Plugging (4)-(4) into (4), we conclude that

[TABLE]

Now, combining (4.13) and (4), and summing up over $i=1,\cdots,N$, then we have (4.3). Hence, it remains to prove (4).

Observing that

[TABLE]

Making use of (4.6), then we infer that

[TABLE]

Applying (2.1) and Lemma 2.2, for the term $K_{1}$, we have

[TABLE]

In addition, making use of (2.3) and Jensen’s inequality, the term $K_{2}$ and $K_{3}$ can be estimated as

[TABLE]

For the term $K_{4}$, in view of (4.6)-(4) and (4.13), we have

[TABLE]

Inserting (4)-(4.17) into (4), summing up over $i=1,\cdots,N$, whence (4).

Now, we turn to prove (4.1)-(4.1). First, appealing to (4.3), Poincaré’s inequality with weighed function, Hölder’s inequality, we infer that

[TABLE]

where $\tilde{z}=(x,\tau)$ and $c=c(n,N,p,L)$. Thus, we have (4.1).

Next, by (4), Poincaré and Hölder’s inequality, we obtain

[TABLE]

with $c=c(n,N,p,L)$, where in the second inequality, we have used the Poincaré’s inequality for a.e. $t\in\Lambda_{\rho}$ and the fact

[TABLE]

Taking into account the concavity of $\omega(\cdot)$ and (4.1) for $q=2$ implies

[TABLE]

Thus, Combining (4) and (4), we are in a position to obtain

[TABLE]

whence (4.1). ∎

5. Partial regularity of $u$

According to Lemma 3.1, now, we define some excess functionals. For reference point $z_{0}=(x_{0},t_{0})\in Q_{T}$, $u\in L^{p}(-T,0;W^{1,p}(\Omega;\mathbb{R}^{N}))$, affine function $l:\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$, and $l(z)=l(x)$, in what follows, we denote

[TABLE]

[TABLE]

and hybrid excess functional:

[TABLE]

5.1. Linearization

The following lemma is a prerequisite for applying the $A-$caloric approximation technique.

Lemma 5.1**.**

Let $u\in L^{p}(-T,0;W^{1,p}(\Omega;\mathbb{R}^{N}))\cap C^{0}(-T,0;L^{2}(\Omega;\mathbb{R}^{N}))$ is a weak solution to (1.1) in $Q_{T}$ under the assumption (2.1)-(2.5) and $Q_{\rho}^{(\lambda)}(z_{0})\subset Q_{T}$ is a parabolic cylinder with reference point $z_{0}=(x_{0},t_{0})$, $0<\rho\leq 1$ and scaling factor $\lambda\geq 1$. Let $l:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{N}$ be any affine function. Then, there holds

[TABLE]

for all $\varphi\in C_{0}^{\infty}(Q_{\rho/2}^{(\lambda)}(z_{0});\mathbb{R}^{N})$ with $c=c(n,L,p,\nu)$.

Proof.

Without loss of generality, we may assume $\sup_{Q_{\rho/2}^{(\lambda)}(z_{0})}|D\varphi|\leq 1$ and we also denote $Q^{\lambda}_{\rho}$, $B_{\rho}$, $\Lambda_{\rho}$ instead of $Q^{\lambda}_{\rho}(z_{0})$, $B_{\rho}(x_{0})$, $\Lambda_{\rho}(t_{0})$, respectively, if there is no danger of any confusion. Note that

[TABLE]

then, from weak formulation (2.6), we deduce that

[TABLE]

Now, we start to estimate $I_{1}$–$I_{4}$. For the term $I_{1}$, applying (2.4), the Hölder and Young’s inequality, we have

[TABLE]

with $c=c(n,p,L,\nu)$, where in the last inequality, we have taken into account the Caccioppoli’s type inequality (3.1), Jensen’s inequality for $\mu(\cdot)$ and $\mu^{s}(\cdot)\leq\mu(\cdot)$ for $s=2$ or $p$.

Likewise, applying (2.3), Young’s inequality, and note that $z\in Q_{\rho/2}^{(\lambda)}(z_{0})$, then $I_{2}$ and $I_{3}$ can be estimated as

[TABLE]

where $c=c(n,p,\nu,L)$.

Taking into account the fact $\sup_{Q_{\rho/2}^{(\lambda)}(z_{0})}|\varphi|\leq\rho\leq 1$, similar with (4), we infer that

[TABLE]

where $c=c(n,p,L,\nu)$ and $\theta_{2}\in(0,1)$ is same with $\theta_{1}$ in (4.9).

Plugging (5.1)-(5.1) into (5.1), then we have

[TABLE]

where $c=c(n,p,\nu,L)$. By scaling argument for general, then we have (5.1). ∎

5.2. Decay estimate

The aim of this section is to provide a decay estimate of $\Phi_{\lambda_{j}}(z_{0},\vartheta^{j}\rho,l_{j})$ with $\lambda_{j}$, $\vartheta$, $l_{j}$ will be specified in later, from which we can obtain a Campanato type estimate of weak solution $u$ to (1.1), then we deduce the regularity of $u$ by a standard argument of Campanato space. First, we introduce a standard estimate for weak solution to linear parabolic systems with constant coefficients (cf. [15] Lemma 5.1), which is necessary in the proof of decay estimate of $\|u-(u)_{z_{0},r}\|_{L^{2}(Q_{r}(z_{0}))}$.

Lemma 5.2**.**

Let $h\in L^{2}(\Lambda_{\rho}(t_{0});W^{1,2}(B_{\rho}(x_{0});\mathbb{R}^{N}))$ be a weak solution in $Q_{\rho}(z_{0})$ of the following linear parabolic system with constant coefficients

[TABLE]

for all $\varphi\in C_{0}^{\infty}(Q_{\rho}(z_{0});\mathbb{R}^{N})$, where the coefficients $A$ satisfy

[TABLE]

for any $F,\widetilde{F}\in\mathbb{R}^{Nn}$. Then, $h$ is smooth in $Q_{\rho}(z_{0})$ and for all $s\geq 1$, $\theta\in(0,1]$, there holds

[TABLE]

for a constant $c_{pa}=c_{pa}(n,N,L/\nu)\geq 1$.

The $A-$caloric approximation lemma (Lemma 2.5) allows one to translate these decay estimates on $h$ into a certain excess functional, e.g., $v$ in (5.15). This eventually allows one to derive the partial regularity of $u$. Based on Lemma 5.1-5.2, we have the following result.

Lemma 5.3**.**

(Decay estimate.) Given $\alpha\in(0,1)$ be a constant. Suppose $H\geq 1$ be a constant, and $\rho_{0}=\rho_{0}(n,N,p,\nu,L,H,\alpha,\omega(\cdot),\mu(\cdot))\in(0,1]$. Let $u\in C^{0}(-T,0;L^{2}(\Omega;\mathbb{R}^{N}))\cap L^{p}(-T,0;W^{1,p}(\Omega;\mathbb{R}^{N}))$ is a weak solution to (1.1) in $Q_{T}$ under the assumption (2.1)-(2.5) and $Q_{\rho}^{(\lambda)}(z_{0})\subset Q_{T}$ is a parabolic cylinder with reference point $z_{0}\in Q_{T}$, and $\rho\in(0,\rho_{0}]$. For the scaling factor $\lambda\geq 1$, if there exist constants $\vartheta\in(0,1)$, $\varepsilon_{0}=\varepsilon_{0}(n,N,p,\nu,L,H,\alpha,\mu(\cdot))\in(0,1)$ and $c_{1}=c_{1}(n,N,p,\nu,L,H)$, such that

[TABLE]

and the smallness condition

[TABLE]

holds, and for $\lambda=1$ on $Q_{\rho}(z_{0})\equiv Q_{\rho}^{(1)}(z_{0})$, there holds

[TABLE]

with $\varepsilon_{1}=c_{1}\vartheta^{2}\varepsilon_{0}\leq\frac{\varepsilon_{0}}{3}$. Then, there exist numbers $\{\lambda_{j}\}_{j=0}^{\infty}$ such that

[TABLE]

and for any $r\in(0,\rho]$, there holds

[TABLE]

where $c=c(n,N,p,\nu,L,H,\alpha)$.

Proof.

For the convenience of notation, we shall once again omit the reference point $z_{0}=(x_{0},t_{0})$ in the notation, and we denote $Q_{\rho}^{(\lambda)}$, $l_{j},l_{\rho}^{(\lambda)}$, $E_{\lambda}(\rho)$, $E_{\lambda_{j}}$, $\Phi_{\lambda_{j}}(\vartheta^{j}\rho)$, $\Psi_{\lambda_{j}}(\vartheta^{j}\rho)$ instead of $Q_{\rho}^{(\lambda)}(z_{0})$, $l_{z_{0},\vartheta^{j}\rho}^{(\lambda_{j})}(z_{0})$, $l_{z_{0},\rho}^{(\lambda)}(z_{0})$, $E_{\lambda}(z_{0},\rho,l_{\rho}^{(\lambda)})$, $E_{\lambda_{j}}(z_{0},\vartheta^{j}\rho,l_{j})$, $\Phi_{\lambda_{j}}(z_{0},\vartheta^{j}\rho,l_{j})$, $\Psi_{\lambda_{j}}(z_{0},\vartheta^{j}\rho,l_{j}(z_{0}))$, respectively.

Suppose (Aj) holds, then a direct consequence of (Aj) is

[TABLE]

In fact, by (Aj), we have

[TABLE]

Before proving (Aj), first, we propose to prove that from (5.7)–(5.8) there exist $\lambda^{\prime}\in[\frac{\lambda}{2},2H\lambda]$ such that

[TABLE]

and there exists a constant $c_{1}$ such that the decay estimate

[TABLE]

holds. To prove (5.13) and (5.14), we first define

[TABLE]

for all $(x,t)\in Q_{\rho}\equiv Q_{\rho}^{(1)}$ with $c_{2}\geq 1$ will be specified in later. In virtue of (3.1) we can see that

[TABLE]

where $c=c(n,p,\nu,L)$. From (5.8), we define

[TABLE]

Thus, by the aid of (5.16) and the definition of $E_{\lambda}(\rho)$, we deduce that

[TABLE]

Indeed, we only need to choose $c_{2}=c_{2}(n,p,\nu,L)\geq 1$ large enough, then the previous inequality is automatically satisfied. Next, we define the bilinear form

[TABLE]

for all $F,\widetilde{F}\in\mathbb{R}^{Nn}$. Taking into account (2.2), (2.3), and (5.7), there holds

[TABLE]

for all $F,\widetilde{F}\in\mathbb{R}^{Nn}$. Appealing to (5.1) and (5.7), then we have

[TABLE]

where in the last inequality, we have taken into account the fact $c_{2}\geq 1$ large enough.

Let $\varepsilon>0$ from Lemma 2.5, which will be specified in later and $\delta\equiv\delta(n,p,\nu,LH^{p-2},\varepsilon)$ are constants from the $A-$caloric approximation Lemma 2.5, here, we replace $L$ in Lemma 2.5 with $LH^{p-2}$. From (5.2), (5.2) and the definition of $v$, $A$, we can see that the all assumptions of Lemma 2.5 are satisfied, if we proved the smallness condition

[TABLE]

holds. Thus, applying Lemma 2.5, there exists a $A-$caloric function $h\in L^{2}(\Lambda_{\rho/4};W^{1,2}(B_{\rho/4};\mathbb{R}^{N}))$ on $Q_{\rho/4}$, such that

[TABLE]

and

[TABLE]

Taking into account Lemma 5.2 and (5.20), for $\theta\in(0,\frac{1}{4}]$, $s=2$ and $s=p$, there exists a constant $c_{pa}\equiv c_{pa}(n,N,\nu,LH^{p-2})=c_{pa}(n,N,\nu,L,p,H)$, such that for the $A-$caloric function $h$ satisfies

[TABLE]

Employing (5.21) and (5.2), for all $\theta\in(0,\frac{1}{4}]$, we further obtain

[TABLE]

Now, we choose $\varepsilon:=\theta^{n+2+2s}$ with $\theta\in(0,\frac{1}{4}]$ is a fixed parameter will be specified in later, at this stage, we have also determined the constant $\delta\equiv\delta(n,p,\nu,L,H,\varepsilon)=\delta(n,p,\nu,L,H,\theta)$ in (5.19).

From the definition of $v$ and (5.7), (5.2), by scaling back, for $s=2$ or $s=p$, there holds

[TABLE]

with $c=c(n,N,p,\nu,L,H)$.

As a consequence, from the minimizing property of $l_{\theta\rho}^{(\lambda)}$ in Lemma 2.4 and (5.2), it follows that for $s=2$ or $s=p$

[TABLE]

with $c_{3}=c_{3}(n,N,p,\nu,L,H)$.

Now, we concentrate our attention on the proof of (5.13) and (5.14). Define

[TABLE]

which implies $\vartheta\in(0,2^{-\frac{p+2}{2}}]$ due to $\theta\in(0,\frac{1}{4}]$. For some $\bar{\mu}\in(\frac{\lambda}{2},2H\lambda]$, then we have $Q_{\vartheta\rho}^{(\bar{\mu})}\subset Q_{\theta\rho}^{(\lambda)}$. Hence, by $(\ref{5.22})$, there holds

[TABLE]

where in the first inequality we have used (2.13) with $\xi\equiv l_{\rho}^{(\lambda)}(z_{0})$, $w=Dl_{\rho}^{(\lambda)}$ on $Q_{\vartheta\rho}^{(\bar{\mu})}$ and in the last inequality, we have taken into account smallness assumption of $E_{\lambda}(\rho)$, that is

[TABLE]

with $c_{4}=c_{4}(n,\vartheta,p,H)$.

Applying (5.7), (5.2) we can see that

[TABLE]

and

[TABLE]

Define $f:[\frac{\lambda}{2},2H\lambda]\rightarrow\mathbb{R}$ by

[TABLE]

then $f$ is a continuous function. Appealing to (5.28)-(5.29), we have $f(\frac{\lambda}{2})\leq 0$ and $f(2H\lambda)\geq 0$. Thus, there exists $\lambda^{\prime}\in[\frac{\lambda}{2},2H\lambda]$ such that $f(\lambda^{\prime})=0$, that is $\lambda^{\prime}=1+Dl_{\vartheta\rho}^{(\lambda^{\prime})}$, whence (5.13).

Next, for $s=2$ or $s=p$, we once again using the minimizing property of $l$, (5.13), (5.25) and the definition of $\vartheta$, we deduce that

[TABLE]

with $c_{1}=c_{1}(n,N,p,\nu,L,H)$. Whence (5.14).

From now on, we have determined $\delta\equiv\delta(n,p,\nu,L,H,\theta=2^{\frac{p-2}{2}}\vartheta$)=$\delta(n,p,\vartheta,L,H,\nu)$ in (5.19), and hence from (5.19) and (5.27) we have also determined $\varepsilon_{0}=\varepsilon_{0}(n,p,\nu,L,$ $H,\vartheta,\mu(\cdot))$, $c_{1}=c_{1}(n,N,p,\nu,L,$ $H)$, which is close to fulfilled the conditions of the Lemma 5.3, it remains to determine the constant $\vartheta\in(0,1)$. For $\alpha\in(0,1)$, let

[TABLE]

and

[TABLE]

Joining (5.30) with (5.31), we can see that once $\vartheta$ is chosen, which is dependent on $n,N,p,\nu,L,\alpha,H$, then $\varepsilon_{0}$ from (5.8) is determined, that is $\varepsilon_{0}=\varepsilon_{0}(n,N,p,\nu,L,\alpha,H)$. Moreover, there holds $\varepsilon_{1}\leq\frac{\varepsilon_{0}}{3}$.

According to the conclusion above, now, we focus our attention on proving (Aj). We shall use the induction argument, first, consider the case (A0). Taking into account the assumption (5.9), let $\lambda_{0}=1$, we have (A0) holds. We now choose $\rho_{0}=\rho_{0}(n,N,p,\nu,L,H,\alpha,\omega(\cdot),\mu(\cdot))\in(0,1]$ suitable small such that

[TABLE]

and suppose that (Aj) holds for some $j\in\{0,1,2\cdots\}$, we proceed to prove (Aj+1) holds. By claimed as before, from (Aj) we have (Bj) holds, then using the assumptions (Aj) and (5.30) we deduce that

[TABLE]

Thus, in virtue of (5.32), we infer that

[TABLE]

Similarly, by $(\ref{5.30})$, we also have

[TABLE]

Taking into account (5.33), (5.34) and the induction assumption (Aj)3, we infer that

[TABLE]

Finally, by (5.35) and the induction assumption (Aj)2, we can replace $(\rho,\lambda)$ in (5.7), (5.8) by $(\vartheta^{j}\rho,\lambda_{j})$, then, from (5.13), (5.14) and (5.28), there exists a number $\lambda_{j+1}\in\left[\frac{\lambda_{j}}{2},2H\lambda_{j}\right]$ such that

[TABLE]

and

[TABLE]

In view of (5.36) and (5.37) we have (Aj+1)2, (Aj+1)3 hold. Furthermore, applying (Aj)1, we refer that (Aj+1)1 holds since $\lambda_{j+1}\leq 2H\lambda_{j}$. Thus, we have proved (Aj+1). For simplicity, here, we represent the recursive relationship as follows

[TABLE]

Based on (5.38), now, it remains to prove (5.11). First, from (Bj), we can see that for some $j\in\{1,2\cdots\}$

[TABLE]

where $\alpha_{n}$ denotes the volume of unit ball in $\mathbb{R}^{n}$ and in the previous inequality we have used the fact $\xi_{z_{0},\rho}^{(\lambda)}=(u)_{z_{0},\rho}^{(\lambda)}$ in (2.12).

Now, we define

[TABLE]

which implies the inclusion $Q_{\theta^{j}\rho}(z_{0})\subset Q_{\vartheta^{j}\rho}^{(\lambda_{j})}(z_{0})$, using (5.2), (5.30), and the minimizing property of $(u)_{z_{0},\vartheta^{j}\rho}$ we infer that

[TABLE]

For any $r\in(0,\rho]$, there exists $j\in\{0,1,2,\cdots\}$ such that $\theta^{j+1}\rho<r\leq\theta^{j}\rho$, making use of (5.2), we obtain

[TABLE]

with $c=c(n,N,p,\nu,L,H,\alpha)$, whence $(\ref{5.10})$. ∎

Remark 5.1**.**

Recalling Lemma 3.1 and (5.2), we can see that, in the whole paper, we only need $\rho\in(0,1)$ suitable small, and it does not tend to zero as $j\longrightarrow\infty$.

From Lemma 4.1 and Lemma 5.3, now, we are able to prove Theorem 1.1.

Proof of Theorem 1.1.

Let $z_{0}\in Q_{T}\backslash(\Sigma_{1}\cup\Sigma_{2})$, then by the definition of \Sigma_{1}\and $\Sigma_{2}$, there exist some constants $\varepsilon_{2}\in(0,1]$, $M_{0}\geq 1$ such that

[TABLE]

and

[TABLE]

Now, in virtue of (2.12), (5.41), (5.42) and (4.1), we are in a position to obtain

[TABLE]

where $c=c(n,N,p,L)$.

Since $\varepsilon_{2}\leq 1\leq M_{0}$, then the previous inequality implies that

[TABLE]

Furthermore, by the minimality of $l_{z_{0},\rho}$, (4.1) and (5.41)–(5.42), we can see that

[TABLE]

This implies that

[TABLE]

with $c=c(n,N,p,L)$.

Appealing to $(\ref{5.42})$-$(\ref{5.46})$, for suitable small $\varepsilon_{2}\in(0,1)$, we can deduce the existence of $H\geq 1$ and $0<\rho\leq\rho_{0}(H)$ such that $Q_{2\rho}(z_{0})\subset Q_{T}$, and at this stage, we further obtain

[TABLE]

and

[TABLE]

Note that the mappings

[TABLE]

are continuous. Thus, there exists $0<R\leq\frac{\rho}{2}$ such that

[TABLE]

and

[TABLE]

for all $z\in Q_{R}(z_{0})$. Hence, we infer for the suitable $\rho_{0}(H)\in(0,1]$, the smallness condition of Lemma 5.3 is uniformly hold for $z\in Q_{R}(z_{0})$. Observe that, for all $z\in Q_{R}(z_{0})$, there holds $Q_{\rho}(z)\subset Q_{2\rho}(z_{0})\subset Q_{T}$, then for all $r\in(0,\rho)$ and $z\in Q_{R}(z_{0})$ we obtain

[TABLE]

with $c=c(n,N,p,\nu,L,H,\alpha)$. By the Campanato space argument (cf. [18, 39]), we have $u\in C^{0;\alpha,\alpha/2}$ in a neighborhood of any point $z_{0}\in Q_{T}\backslash(\Sigma_{1}\cup\Sigma_{2})$, and we further obtain $\left|\Sigma_{1}\cup\Sigma_{2}\right|=0$, which means $|Q_{0}|=|Q_{T}|$. ∎

6. Estimate of singular set

In this section, with Theorem 1.1 in hand, now, we proceed to prove Theorem 1.2. Such result will be proved by combining the fractional time and fractional space differentiability of gradient of weak solution $u$ to (1.1).

6.1. Fractional time differentiability

In this subsection, we aim to proving the fractional time differentiability of $Du$ for $p=2$. First, we estimate the $L^{2}$-norm of $\tau^{h}u$.

Lemma 6.1**.**

Let $u\in L^{\infty}(-T,0;L^{2}(\Omega;\mathbb{R}^{N}))\cap L^{2}(-T,0;H^{1}(\Omega;\mathbb{R}^{N}))$ be a weak solution to (1.1). Let $(t_{0},t_{1})\subset\subset(-T,0)$ and $\eta\in C_{0}^{\infty}(\Omega)$ be a cut-off function with $supp(\eta)\subset\subset\Omega$. Then, whenever $0<|h|\leq\frac{1}{2}\min\{|t_{1}|,T-|t_{0}|,1\}$, the following estimate holds

[TABLE]

where $c=c(L)$.

Proof.

First, we restrict ourselves to the case $h>0$, choosing $\eta^{2}\tau^{h}u$ as a test function in the Steklov averages formulation of (2.7), and integrating with respect to $t\in(t_{0},t_{1})$, we deduce

[TABLE]

Taking into account ($H_{1}$), ($H_{3}$) and Young’s inequality, there holds

[TABLE]

For the term $I_{2}$, from the Hölder’s inequality, it follows that

[TABLE]

Similarly, for $p=2$, applying (4)-(4.13), the term $I_{3}$ can be estimated as

[TABLE]

Finally, we note that the estimation in the other one being the same using $u_{\bar{h}}$ instead of $u_{h}$. Now, inserting (6.3)-(6.1) into (6.1), we obtain

[TABLE]

where $c=c(L)$. Thus, we have (6.1). ∎

From Lemma 6.1, we have a direct result:

Remark 6.1**.**

Let $u\in L^{\infty}(-T,0;L^{2}(\Omega;\mathbb{R}^{N}))\cap L^{2}(-T,0;H^{1}(\Omega;\mathbb{R}^{N}))$ be a weak solution to (1.1). Let $(t_{0},t_{1})\subset\subset(-T,0)$ and $\Omega^{\prime}\subset\subset\Omega$. Then, whenever $0<|h|\leq\frac{1}{2}\min\{|t_{1}|,T-|t_{0}|,1\}$, there holds

[TABLE]

where $c=c(L,dist(\Omega^{\prime},\partial\Omega))$.

Based on Lemma 6.1, we now propose to estimate the time derivative of $Du$, which will be as the starting point of an iteration process.

Lemma 6.2**.**

Let $u\in L^{\infty}(-T,0;L^{2}(\Omega;\mathbb{R}^{N}))\cap L^{2}(-T,0;H^{1}(\Omega;\mathbb{R}^{N}))$ be a weak solution to (1.1). Let $(t_{0},t_{1})\subset\subset(-T,0)$ and $\eta\in C_{0}^{\infty}(\Omega)$ be a cut-off function with $supp(\eta)\subset\subset\Omega$. Then, there exists a constant $c=c(L,\nu,dist(supp(\eta),\partial\Omega),t_{0},t_{1})$, such that whenever $0<|h|<\min\{|t_{1}|,T-|t_{0}|,1,dist(supp(\eta),\partial\Omega)\}$, there holds

[TABLE]

Proof.

We choose $\chi_{\varepsilon}:\mathbb{R}\longmapsto[0,1]$ be a continuous affine function satisfying

[TABLE]

which approximating the characteristic function of $(-\infty,l)$ with $l\in(-T,0)$. Let $\xi(t)$, $\eta(x)$ are cut-off functions in the time and space variables, respectively, such that

[TABLE]

We take $\phi(x,t)=\tau^{h}(\eta^{2}(x)\xi^{2}(t)\chi_{\varepsilon}(t)\tau^{-h}u_{\lambda}(x,t))$ as a test function in (2.7), and integrating in time respect to $(-T,0)$, yields that

[TABLE]

We first note that

[TABLE]

and

[TABLE]

This, combined with (6.9) implies that

[TABLE]

Recalling the definition of $\chi_{\varepsilon}$, we can find that

[TABLE]

Moreover, note that $H_{4}=0$. Then, by passing to the limit for $\lambda\longrightarrow 0$ in (6.1), we obtain

[TABLE]

Observe that

[TABLE]

and

[TABLE]

By the aid of (2.8), there holds

[TABLE]

where $c_{1}$ depends on $\nu$. Furthermore, applying ($H_{3}$), (1.4) and Young’s inequality, it holds that

[TABLE]

For the term $H_{51}$, making use of (6.6), we obtain

[TABLE]

with $c=c(L,dist(supp(\eta),\partial\Omega))$.

Concerning $H_{6}$, in virtue of (4.13) and (6.6), we have

[TABLE]

where $c=c(L,dist(supp(\eta),\partial\Omega))$.

Note that $H_{2}\leq 0$, combining (6.1)-(6.1) and (6.6), we finally obtain

[TABLE]

where $c=c(L,\nu,dist(supp(\eta),\partial\Omega))$.

Applying the property of $\xi(t)$ and $\eta(x)$, letting $\varepsilon\longrightarrow 0$ and $l\longrightarrow 0$, then we have (6.7). ∎

According to (6.7), we can re-estimate the term $I_{2}$ in (6.1):

[TABLE]

From (6.1) and note that $|h|<1$, then we can rewrite (6.6) as follows

[TABLE]

where $c_{2}=c_{2}(L,\nu,dist(supp(\eta),\partial\Omega))$.

Similarly, in the proof of Lemma 6.2, we can use (6.16) instead of (6.6), then we arrive at

[TABLE]

Like the estimation of (6.1)-(6.17), by iteration argument, we finally obtain

[TABLE]

with $a_{k}=\frac{1}{2}(1+\frac{a_{k-1}}{2})$ and $a_{0}=\frac{1}{2}$, $c_{k}=c_{k}(L,\nu,dist(supp(\eta),\partial\Omega))$. Letting $k\longrightarrow\infty$, we have $a_{k}\longrightarrow\frac{2}{3}$, applying the property of $\eta(x)$, then we obtain (2.9).

6.2. Fractional space differentiability

In this subsection, we propose to deduce the fractional space differentiability of the gradient of weak solution $u$ to (1.1). To begin with, we discuss the general case for $p\geq 2$.

Lemma 6.3**.**

Let $\Omega\subset\mathbb{R}^{n}$ $(n=2,3)$ be a bounded domain. Let $u\in L^{\infty}(-T,0;L^{2}(\Omega;\mathbb{R}^{N}))\cap L^{p}(-T,0;$ $W^{1,p}(\Omega;\mathbb{R}^{N}))$ $(p\geq 2)$ be a weak solution to (1.1). Then, we have

[TABLE]

and for any $Q_{2\rho}(z_{0})\subset Q_{T}$, there holds

[TABLE]

where $\theta\in(0,\frac{1}{2}]$ and $c=c(n,p,L,\nu)$.

Proof.

First, replace the test function $\varphi$ in (2.6) by $\tau_{-h}\varphi$ with $0<|h|<\min\{dist(supp(\varphi),\partial Q_{T}),1,\rho\}$, then we infer that

[TABLE]

By approximation, we choose $\varphi\equiv\phi\tau_{h}u$ in the previous equation with $\phi\in C_{0}^{\infty}(Q_{T})$. Thus, we are in a position to obtain that

[TABLE]

Now, let us choose $\phi(x,t)=\bar{\chi}(t)\chi(t)\psi^{2}(x)$ with $\chi\in W^{1,\infty}((-T,0))$, $\chi(-T)=0$, $\partial_{t}\chi\geq 0$ and $0\leq\chi\leq 1$, $\psi\in C_{0}^{\infty}(B_{\rho}(x_{0}))$, $0\leq\psi\leq 1$ and $\bar{\chi}:(-T,0)\longrightarrow\mathbb{R}$ is a Lipschitz continuous function, defined by

[TABLE]

with $-T<t_{0}<t_{0}+\delta<0$, $\delta>0$. Thus, letting $\delta\longrightarrow 0$, then (6.2) becomes

[TABLE]

where $Q_{t_{0}}:=B_{\rho}(x_{0})\times(-T,t_{0})$.

Observing that

[TABLE]

and

[TABLE]

Then, applying ($H_{2}$) and Lemma 2.2, we can find that

[TABLE]

Here we have used abbreviated notation

[TABLE]

Thus, we conclude that

[TABLE]

Moreover, from ($H_{3}$) and (1.4), it follows that

[TABLE]

Using ($H_{1}$), we further obtain

[TABLE]

Therefore, applying (6.2) and the Young’s inequality, we have

[TABLE]

where $c(\varepsilon)=c(p)L^{2}\varepsilon^{-1}$.

Furthermore, in virtue of ($H_{3}$), it holds that

[TABLE]

Inserting (6.23)-(6.27) into (6.2), and choosing $\varepsilon=\frac{\nu}{2c(p)}$, then we obtain

[TABLE]

where $c=c(n,L,p,\nu)$.

Let $Q_{\rho}(z_{0})\subset Q_{T}$ with $\rho$ suitable small such that $2\rho<\min\{dist(x_{0},\partial\Omega),1\}$. Now, we choose $\chi\in W^{1,\infty}(\mathbb{R})$ and $\psi\in C_{0}^{\infty}(B_{\rho}(x_{0}))$ satisfying

[TABLE]

and

[TABLE]

Note that, for any $\theta\in(0,\frac{1}{2}]$, the right-most term of (6.2) can be estimated as

[TABLE]

Employing (6.2), (6.2), we finally obtain

[TABLE]

By Lemma 2.1, the previous inequality implies that

[TABLE]

Dividing both side in (6.2) by $|h|^{2\theta}$. Then, we can see that

[TABLE]

Using the standard estimate for difference quotients, letting $|h|\longrightarrow 0$, then we have (6.3) and (6.19). ∎

Proof of Theorem 1.2.

Taking into account Proposition 2.1, (2.9) and Sec. 6.1 for $p=2$, we have proved the fractional time differentiability of $Du$. Hence, it is remains to prove the fractional space differentiability of $Du$. From (6.3) we can see that for $p=2$

[TABLE]

This, combined with (2.10) implies the fractional space differentiability of $Du$. Finally, applying Lemma 2.3 we obtain (1.5). Thus, we have completed the proof of Theorem 1.2. ∎