Second-Order Regularity for Degenerate Parabolic Quasi-Linear Equations in the Heisenberg Group

Abstract

In the Heisenberg group ${\mathbb{H}}^n$, we obtain the local second-order $H{W}_{\mathrm{loc}}^{2,2}$-regularity for the weak solution u to a class of degenerate parabolic quasi-linear equations ${\partial }_{t}u={\displaystyle \sum _{i=1}^{2n}}{X}_i{A}_i(Xu)$ modeled on the parabolic p-Laplacian equation. Specifically, when $2\le p\le 4$, we demonstrate the integrability of ${({\partial }_{t}u)}^2$, namely, ${\partial }_{t}u\in L_{\mathrm{loc}}^2$; when $2\le p<3$, we demonstrate the $H{W}_{\mathrm{loc}}^{2,2}$-regularity of u, namely, $XXu\in L_{\mathrm{loc}}^2$. For the $H{W}_{\mathrm{loc}}^{2,2}$-regularity, when $p\ge 2$, the range of p is optimal compared to the Euclidean case.

1. Introduction

The study of partial differential equations with the p-Laplacian operator involved has a history of more than 50 years. It has always been a hot and difficult topic to analyze and study the regularity of solutions to these equations. The most classic one is the p-Laplacian equation, also known as the p-harmonic equation. In Euclidean space, Ural’ceva [1], Uhlenbeck [2], Evans [3], and Lewis [4] established, when $1<p<\infty$, the $C^{0,1}$- and $C^{1,\alpha }$-regularities of solutions to the p-Laplacian equation. Manfredi-Weitsman [5] adopted the Cordes condition [6,7,8] to prove, when $1<p<3+{\displaystyle \frac{2}{n-2}}$, the local second-order Sobolev $W^{2,2}$-regularity; also see [9] for a new proof. In recent years, there has been a trend that the regularity theories and achievements established in the Euclidean case have been expanded to sub-Riemannian manifolds. During this process, a large number of scholars made significant contributions and made breakthrough progress. In the Heisenberg group ${\mathbb{H}}^n$, Domokos-Manfredi [10,11], Manfredi-Mingione [12], Migione et al. [13], Ricciotti [14], and Zhong-Mukherjee [15,16] built up, when $1<p<\infty$, the $C^{0,1}$- and $C^{1,\alpha }$-regularities of solutions to the p-Laplacian equation; Domokos [17] and Liu et al. [18] proved, when $1<p\le 4$ with $n=1$ and $1<p<3+{\displaystyle \frac{1}{n-1}}$ with $n>1$, the local second-order Sobolev $W^{2,2}$-regularity. In the group SU(3), Domokos-Manfredi [19,20] established, when $1<p<\infty$, the $C^{0,1}$-regularity and, when $2\le p<\infty$, the $C^{1,\alpha }$-regularity for the p-Laplacian equation; Yu [21] built up, when $1<p<{\displaystyle \frac{7}{2}}$, the local second-order Sobolev $W^{2,2}$-regularity. Citti-Mukherjee [22] extended, via the Riemannian approximation technique [20,23,24], the method created by Zhong-Mukherjee [15,16] to the Hörmander vector fields in step two and successfully established the $C^{0,1}$- and $C^{1,\alpha }$-regularities of solutions to the p-Laplacian equation.

Given any open domain $\Omega$ in the Heisenberg group ${\mathbb{H}}^n$ and a positive number T, we consider a class of degenerate parabolic quasi-linear equations

$${\partial }_{t}u=\sum _{i=1}^{2n}{X}_i{A}_i(Xu)\mathrm{in}\Omega \times (0,T),$$

(1)

modeled on the parabolic p-Laplacian equation

$${\partial }_{t}u=\sum _{i=1}^{2n}{X}_i{(|Xu|}^{p-2}{X}_{i}u).$$

(2)

Here, $X=(X_1,X_2,\cdots ,X_{2n})$ is the horizontal gradient and, when $2\le p<\infty$, for any $\xi ,\eta \in {\mathbb{R}}^{2n}$, functions ${\left\{A_i\right\}}_{1\le i\le 2n}$ satisfy the condition

$$\left\{\begin{array}{c}{\lambda }^{\prime }{|\xi |}^{p-2}{|\eta |}^2\le {\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}(\xi ){\eta }_i{\eta }_j\le {\Lambda }^{\prime }{|\xi |}^{p-2}{|\eta |}^2,\hfill \\ |A_i(\xi )|\le {\Lambda }^{\prime }{|\xi |}^{p-1},\hfill \end{array}\right.$$

(3)

where $A_{i,{\xi }_j}(\xi ):={\partial }_{{\xi }_j}{A}_i(\xi )$ and $0<{\lambda }^{\prime }\le {\Lambda }^{\prime }<\infty$. If the equation

$${\int }_0^T{\int }_{\Omega }{\partial }_{t}u\varphi \mathrm{d}x\mathrm{d}t=-{\int }_0^T{\int }_{\Omega }\sum _{i=1}^{2n}{A}_i(Xu)X_i\varphi \mathrm{d}x\mathrm{d}t$$

(4)

holds for any test-function $\varphi \in C_0^{\infty }(\Omega \times (0,T))$, the function $u\in L^p((0,T),H{W}_{\mathrm{loc}}^{1,p}(\Omega ))$ is called as a weak solution to Equation (1). Here, $H{W}_{\mathrm{loc}}^{1,p}(\Omega )$ is a local horizontal Sobolev space, defined in Section 2. Let $A(\xi )={|\xi |}^{p-2}\xi$; then, the parabolic p-Laplacian Equation (2) is obtained from Equation (1). The research on parabolic p-Laplacian equations can be traced back to the work of DiBenedetto-Friedman [25]. One of the main achievements in their work is the establishment of the $C^{1,\alpha }$-regularity of weak solutions to parabolic p-Laplacian equations in Euclidean space; almost at the same time, Wiegner [26] also obtained the same result. In Euclidean space, DiBenedetto conducted further research on parabolic p-Laplacian equations and obtained many detailed results, which were compiled into a book [27]; Lindqvist [28] and Attouchi-Ruosteenoja [29], when $1<p<3$, demonstrated the $W^{2,2}$-regularity, where the range of p is optimal. There have always been difficulties in studying regularities for parabolic p-Laplacian equations in sub-Riemannian manifolds. In the Heisenberg group ${\mathbb{H}}^n$, letting u be a weak solution to the non-degenerate parabolic p-Laplacian equation

$${\partial }_{t}u=\sum _{i=1}^{2n}{X}_i\left(\right(1+{|Xu|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{X}_{i}u),$$

Capogna et al. [23] demonstrated that $u\in C_{\mathrm{loc}}^{\infty }$ for $2\le p<\infty$; for the degenerate parabolic p-Laplacian equation, Capogna et al. [24] proved, via the Riemannian approximation technique and the idea from Zhong [15], when $2\le p\le 4$, the $C^{0,1}$-regularity. Unfortunately, the method established by Capogna et al. [24] is ineffective for $p\in (1,2)\cup (4,\infty )$; therefore, the $C^{0,1}$-regularity for $p\in (1,2)\cup (4,\infty )$ is still unknown.

In this paper, we focus on the local second-order $H{W}_{\mathrm{loc}}^{2,2}$-regularity for the weak solution u to (1) in the Heisenberg group ${\mathbb{H}}^n$. As a consequence, based on the work of Capogna et al. [24], we prove that ${\partial }_{t}u\in L_{\mathrm{loc}}^2$ for $2\le p\le 4$ and prove that $XXu\in L_{\mathrm{loc}}^2$ for $2\le p<3$. See Theorems 1 and 2 below for details. Here, the Hessian matrix $XXu$ is defined in Section 2.

Theorem 1. 

Assume that ${\left\{A_i\right\}}_{1\le i\le 2n}$ satisfy Condition (3) and that u is a weak solution to Equation (1) in $\Omega \times (0,T)$. When $2\le p\le 4$, we have ${\partial }_{t}u\in L_{\mathrm{loc}}^2(\Omega \times (0,T))$ and, moreover, there exists a constant $C=C(n,p,\lambda ,\Lambda )>0$ such that

$${\int }_{t_1}^{t_2}{\int }_{\Omega }{({\partial }_{t}u)}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le {C(\parallel X\eta \parallel }_{L^{\infty }}^2+{\parallel \eta R\eta \parallel }_{L^{\infty }})|\mathrm{spt}(\eta )|{\omega }^{p-1}+C\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}|\mathrm{spt}(\eta )|{\omega }^{{\displaystyle \frac{p}{2}}}$$

(5)

holds for any non-negative $\eta \in C^1([0,T],C_0^{\infty }(\Omega ))$, where $\omega ={sup}_{\mathrm{spt}(\eta )}{(\delta +|Xu|}^2)$ with $\mathrm{spt}(\eta )=\left\{\right(x,t)\in \Omega \times [0,T]:\eta (x,t)\ne 0\}$ and $R\eta ={\partial }_{x_{2n+1}}\eta$.

Theorem 2. 

Assume that ${\left\{A_i\right\}}_{1\le i\le 2n}$ satisfy Condition (3). When $2\le p<3$, the weak solution u to Equation (1) in $\Omega \times (0,T)$ have the uniform $H{W}_{\mathrm{loc}}^{2,2}$-regularity and, moreover, there exists a constant $C=C(n,p,\lambda ,\Lambda )>0$ such that

$$\begin{array}{cc}\hfill {\int }_{t_1}^{t_2}{\int }_{\Omega }{|XXu|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le & C\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}|\mathrm{spt}(\eta )|{\omega }^{{\displaystyle \frac{4-p}{2}}}\hfill \\ \hfill & +C(\parallel X\eta {\parallel }_{L^{\infty }}^2+\parallel \eta R\eta {\parallel }_{L^{\infty }})|\mathrm{spt}(\eta )|\omega ,\hfill \end{array}$$

(6)

holds for any non-negative $\eta \in C^1([0,T],C_0^{\infty }(\Omega ))$, where $\omega ={sup}_{\mathrm{spt}(\eta )}{(\delta +|Xu|}^2)$ with $\mathrm{spt}(\eta )=\left\{\right(x,t)\in \Omega \times [0,T]:\eta (x,t)\ne 0\}$ and $R\eta ={\partial }_{x_{2n+1}}\eta$.

When $p\ge 2$, the range of p shown in Theorem 2 is optimal compared to the Euclidean case; see [9] for details. Since our method is based on the work of Capogna et al. [24], we are currently unable to handle the case $p\in (1,2)$. Let u be a weak solution to Equation (1). The proofs of Theorems 1 and 2 rely on the study of the solution of the following regularized equation

$${\partial }_{t}v=\sum _{i=1}^{2n}{X}_i{A}_i^{\delta }(Xv)\mathrm{in}U\times (0,T);v=u\mathrm{on}U\times \{t=0\}\cup \partial U\times (0,T),$$

(7)

where, when $2\le p<\infty$, for any $\xi ,\eta \in {\mathbb{R}}^{2n}$ and $0<\delta <1$, functions ${\left\{A_i^{\delta }\right\}}_{1\le i\le 2n}$ satisfy the condition

$$\left\{\begin{array}{c}{\lambda (\delta +|\xi |}^2{)}^{{\displaystyle \frac{p-2}{2}}}{|\eta |}^2\le {\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}^{\delta }(\xi ){\eta }_i{\eta }_j\le {\Lambda (\delta +|\xi |}^2{)}^{{\displaystyle \frac{p-2}{2}}}{|\eta |}^2,\hfill \\ |A_i^{\delta }{(\xi )|\le \Lambda (\delta +|\xi |}^2{)}^{{\displaystyle \frac{p-1}{2}}},\hfill \end{array}\right.$$

(8)

where $A_{i,{\xi }_j}^{\delta }(\xi ):={\partial }_{{\xi }_j}{A}_i^{\delta }(\xi )$ and $0<\lambda \le \Lambda <\infty$.

Let $u_{\delta }\in L^p((0,T),H{W}_{\mathrm{loc}}^{1,p}(U))$ be a weak solution to Equation (7). When $2\le p<\infty$, the existence, uniqueness, and $C^{\infty }$-regularity of $u_{\delta }$ have been obtained in [23,24] and the references therein. When $2\le p<\infty$ and ${\left\{A_i^{\delta }\right\}}_{1\le i\le 2n}$ satisfy Condition (8), it is proven in [23,24] that $A_i^{\delta }\to A_i$ and $u_{\delta }\to u$ as $\delta \to 0$. Hence, to obtain Theorems 1 and 2, it then suffices to prove that ${\partial }_t{u}_{\delta }\in L_{\mathrm{loc}}^2$ and $XX{u}_{\delta }\in L_{\mathrm{loc}}^2$ uniformly in $\delta ,ϵ\in (0,1)$. Finally, setting $\delta \to 0$, from the subsequent theorems, we can conclude Theorems 1 and 2 in a standard way, as in [23,24].

Theorem 3. 

Assume that ${\left\{A_i^{\delta }\right\}}_{1\le i\le 2n}$ satisfy Condition (8) and that $u_{\delta }$ is a weak solution to Equation (7) in $U\times (0,T)$. When $2\le p\le 4$, we have ${\partial }_t{u}_{\delta }\in L_{\mathrm{loc}}^2(U\times (0,T))$ and, moreover, there exists a constant $C=C(n,p,\lambda ,\Lambda )>0$ such that

$${\int }_{t_1}^{t_2}{\int }_U{({\partial }_t{u}_{\delta })}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le {C(\parallel X\eta \parallel }_{L^{\infty }}^2+{\parallel \eta R\eta \parallel }_{L^{\infty }})|\mathrm{spt}(\eta )|{\omega }_{\delta }^{p-1}+C\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}|\mathrm{spt}(\eta )|{\omega }_{\delta }^{{\displaystyle \frac{p}{2}}}$$

(9)

holds for any non-negative $\eta \in C^1([0,T],C_0^{\infty }(U))$, where ${\omega }_{\delta }={sup}_{\mathrm{spt}(\eta )}(\delta +|X{u}_{\delta }{|}^2)$.

Theorem 4. 

Assume that ${\left\{A_i^{\delta }\right\}}_{1\le i\le 2n}$ satisfy Condition (8). When $2\le p<3$, the weak solution ${\left\{u_{\delta }\right\}}_{\delta \in (0,1)}$ to Equation (7) in $U\times (0,T)$ has the uniform $H{W}_{\mathrm{loc}}^{2,2}$-regularity in $\delta \in (0,1)$ and, moreover, there exists a constant $C=C(n,p,\lambda ,\Lambda )>0$ such that

$$\begin{array}{cc}\hfill {\int }_{t_1}^{t_2}{\int }_U{|XX{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le & C\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}|\mathrm{spt}(\eta )|{\omega }_{\delta }^{{\displaystyle \frac{4-p}{2}}}\hfill \\ \hfill & +{C(\parallel X\eta \parallel }_{L^{\infty }}^2+{\parallel \eta R\eta \parallel }_{L^{\infty }})|\mathrm{spt}(\eta )|{\omega }_{\delta }\hfill \end{array}$$

(10)

holds for any non-negative $\eta \in C^1([0,T],C_0^{\infty }(U))$, where ${\omega }_{\delta }={sup}_{\mathrm{spt}(\eta )}(\delta +|X{u}_{\delta }{|}^2)$.

Theorems 3 and 4 are demonstrated in Section 4. To obtain Theorem 3, from the divergence structure in (7) and Condition (8), we easily obtain the upper bound ${\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{p-2}{|XX{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t$ for the integral term of ${({\partial }_t{u}_{\delta })}^2$ uniformly in $\delta \in (0,1)$—see (16). Then, by a Caccioppoli-type inequality established by Capogna et al. [24] (see Lemma 2), we conclude ${\partial }_t{u}_{\delta }\in L_{\mathrm{loc}}^2$ uniformly in $\delta \in (0,1)$. Refer to Section 4 for particulars of the proof.

The key to proving Theorem 4 lies in selecting an appropriate test-function. From the divergence structure in (7) and Lie bracket, we easily establish a parabolic Equation (18) whose weak solution is $X_l{u}_{\delta }$ (see Lemma 4). To obtain Theorem 4, we choose $\varphi ={\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_l{u}_{\delta }$ to test (18). Then, via Condition (8) and Young’s inequality, we ultimately obtained that the integral term of ${(XX{u}_{\delta })}^2$ is bounded by ${\int }_{t_1}^{t_2}{\int }_U{|R{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t$—see (35). Refer to Section 4 for particulars of the proof. Finally, by a Caccioppoli-type inequality established by Capogna et al. [24] (see Lemma 1), we bound the integral term of ${(R{u}_{\delta })}^2$ uniformly in $\delta \in (0,1)$ (see Lemma 3). Refer to Section 3 for the particulars of the proof.

2. Preliminaries

In quantum mechanics, the Heisenberg group is used to describe the motion and interactions of particles, especially when dealing with the evolution of quantum systems, and Heisenberg groups provide an effective mathematical tool.

The n-th Heisenberg group ${\mathbb{H}}^n$ is identified with the Euclidean space ${\mathbb{R}}^{2n+1}$, and its homogeneous dimension is $2n+2$. For any two points $x=(x_1,\cdots ,x_{2n},x_{2n+1})$, $y=(y_1,\cdots ,y_{2n},y_{2n+1})\in {\mathbb{H}}^n$, we endow the group multiplication on the Heisenberg group ${\mathbb{H}}^n$ with

$$x\circ y:=\left(x_1+y_1,\cdots ,x_{2n}+y_{2n},x_{2n+1}+y_{2n+1}+{\displaystyle \frac{1}{2}}\sum _{i=1}^n(x_i{y}_{n+i}-x_{n+i}{y}_i)\right).$$

For all $1\le i\le n$, we denote by $X_i={\partial }_{x_i}-{\displaystyle \frac{x_{n+i}}{2}}{\partial }_{x_{2n+1}}$ and $X_{n+i}={\partial }_{x_{n+i}}+{\displaystyle \frac{x_i}{2}}{\partial }_{x_{2n+1}}$ the left invariant vector fields, which form a canonical basis of the Lie algebra of the Heisenberg group ${\mathbb{H}}^n$, and by $R={\partial }_{x_{2n+1}}$ the only non-trivial commutator. We usually refer to the vector fields $X_1,\cdots ,X_{2n}$ as the horizontal vector fields and R as the vertical vector field.

Given any open and connected domain $\Omega \subset {\mathbb{H}}^n$, we denote by $X\psi =(X_1\psi ,\cdots ,X_{2n}\psi )$ the horizontal gradient of a function $\psi \in C^1(\Omega )$ and by $XX\psi ={(X_i{X}_j\psi )}_{2n\times 2n}$ the second-order horizontal derivative of a function $\psi \in C^2(\Omega )$. Here, the lengths of $X\psi$ and $XX\psi$ are written as $|X\psi |={\left({\sum }_{i=1}^{2n}{|X_i\psi |}^2\right)}^{1/2}$ and $|XX\psi |={\left({\sum }_{i,j=1}^{2n}{|X_i{X}_j\psi |}^2\right)}^{1/2}$.

For $1\le p<\infty$, we denote by $H{W}^{1,p}(\Omega )$ the first-order p-th integrable horizontal Sobolev space, which is the set composed of all functions $\psi \in L^p(\Omega )$ that fulfill $X\psi \in L^p(\Omega ,{\mathbb{R}}^{2n})$. We make $H{W}^{1,p}(\Omega )$ become a Banach space by equipping it with the norm

$${\parallel \psi \parallel }_{H{W}^{1,p}(\Omega )}={\parallel \psi \parallel }_{L^p(\Omega )}+{\parallel X\psi \parallel }_{L^p(\Omega ,{\mathbb{R}}^{2n})}.$$

The first-order p-th integrable local horizontal Sobolev space $H{W}_{\mathrm{loc}}^{1,p}(\Omega )$ is the set composed of all functions $\psi :\Omega \to \mathbb{R}$ that fulfill $\psi \in H{W}^{1,p}(U)$ for every domain $U⋐\Omega$.

Throughout the remainder of this section, we review several uniform estimates of weak solutions $u_{\delta }$ in $\delta \in (0,1)$ established by Capogna et al. [24]. Letting $\beta =0$ in [24], Lemma 3.3, we derive the subsequent lemma.

Lemma 1. 

Assume that $u_{\delta }$ is a weak solution to Equation (7). When $1<p<\infty$, there exists a constant $C=C(n,\lambda ,\Lambda )>0$ such that

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{|XR{u}_{\delta }|}^2{\eta }^4\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{\int }_{t_1}^{t_2}{\int }_U|R{u}_{\delta }{|}^2{\eta }^2|\eta {\partial }_t\eta |\mathrm{d}x\mathrm{d}t+C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{|X\eta |}^2{\eta }^2{|R{u}_{\delta }|}^2\mathrm{d}x\mathrm{d}t\hfill \end{array}$$

(11)

holds for any non-negative $\eta \in C^1([0,T],C_0^{\infty }(U))$.

The subsequent lemma is [24], Proposition 4.4.

Lemma 2. 

Assume that $u_{\delta }$ is a weak solution to Equation (7). When $2\le p\le 4$, there exists a constant $C=C(n,p,\lambda ,\Lambda )>0$ such that

$$\begin{array}{cc}\hfill & \underset{t_1<t<t_2}{sup}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{\beta +2}{2}}}{\eta }^2\mathrm{d}x+{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}{|XX{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{(p+\beta )}^7{(\parallel X\eta \parallel }_{L^{\infty }}^2+{\parallel \eta R\eta \parallel }_{L^{\infty }})\int {\int }_{\mathrm{spt}(\eta )}(\delta +|X{u}_{\delta }{{|}^2)}^{{\displaystyle \frac{p+\beta }{2}}}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{(p+\beta )}^7\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}{|\mathrm{spt}(\eta )|}^{{\displaystyle \frac{p-2}{p+\beta }}}{\left(\int {\int }_{\mathrm{spt}(\eta )}(\delta +|X{u}_{\delta }{{|}^2)}^{{\displaystyle \frac{p+\beta }{2}}}\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{\beta +2}{p+\beta }}}\hfill \end{array}$$

(12)

holds for any $\beta \ge 0$ and non-negative $\eta \in C^1([0,T],C_0^{\infty }(U))$.

Capogna et al. [24] also arrived at the uniform gradient estimate and convergence that are shown below.

Theorem 5. 

For $\delta \in [0,1)$, assume that ${\left\{A_i^{\delta }\right\}}_{1\le i\le 2n}$ satisfy Condition (8) and that $u_{\delta }\in L^p((0,T),H{W}_{\mathrm{loc}}^{1,p}(U))$ is a weak solution to Equation (7) in $U\times (0,T)$, where we denote $u_0=u$. If $2\le p\le 4$, then $X{u}_{\delta }\in L_{\mathrm{loc}}^{\infty }(U\times (0,T))$ uniformly in $\delta \in [0,1)$ and there exists a constant $C=C(n,p,\lambda ,\Lambda )>0$ such that

$$\underset{Q_{\mu ,r}}{sup}|X{u}_{\delta }|\le C{\mu }^{{\displaystyle \frac{1}{2}}}max\left({\left({\displaystyle \frac{1}{\mu r^{2n+4}}}\int {\int }_{Q_{\mu ,2r}}(\delta +|X{u}_{\delta }{{|}^2)}^{{\displaystyle \frac{p}{2}}}\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{1}{2}}},{\mu }^{{\displaystyle \frac{p}{2(2-p)}}}\right)$$

(13)

holds for any $Q_{\mu ,2r}\subset U\times (0,T)$, where $Q_{\mu ,r}:=B(x_0,r)\times (t_0-\mu r^2,t_0)$. Moreover, $u_{\delta }\to u$ in $C^0((0,T),\overline{U})$.

3. Integrability of ${({\mathbf{Ru}}_{\mathbf{\delta }})}^{\mathbf{2}}$

In this section, based on Lemma 1 and Theorem 5, we built up the integrability of ${(R{u}_{\delta })}^2$ and organize it into the subsequent lemma, which is essential to the proof of Theorem 4.

Lemma 3. 

Assume that $u_{\delta }$ is a weak solution to Equation (7). Then, when $2\le p\le 4$, for all non-negative $\eta \in C^1([0,T],C_0^{\infty }(U))$, we have

$${\int }_{t_1}^{t_2}{\int }_U|R{u}_{\delta }{|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le C\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}|\mathrm{spt}(\eta )|{\omega }_{\delta }^{{\displaystyle \frac{4-p}{2}}}+{C\parallel X\eta \parallel }_{L^{\infty }}^2|\mathrm{spt}(\eta )|{\omega }_{\delta },$$

(14)

where $C=C(n,\lambda ,\Lambda )>0$ and ${\omega }_{\delta }={sup}_{\mathrm{spt}(\eta )}(\delta +|X{u}_{\delta }{|}^2)$.

Proof. 

Noting that $R=[X_1,X_{n+1}]$, we have

$${\int }_{t_1}^{t_2}{\int }_U{|R{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t={\int }_{t_1}^{t_2}{\int }_U{X}_1{X}_{n+1}{u}_{\delta }R{u}_{\delta }{\eta }^2\mathrm{d}x\mathrm{d}t-{\int }_{t_1}^{t_2}{\int }_U{X}_{n+1}{X}_1{u}_{\delta }R{u}_{\delta }{\eta }^2\mathrm{d}x\mathrm{d}t.$$

Adopting integration by parts to estimate ${\int }_{t_1}^{t_2}{\int }_U{X}_1{X}_{n+1}{u}_{\delta }R{u}_{\delta }{\eta }^2\mathrm{d}x\mathrm{d}t$, we have

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_U{X}_1{X}_{n+1}{u}_{\delta }R{u}_{\delta }{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & =-{\int }_{t_1}^{t_2}{\int }_U{X}_{n+1}{u}_{\delta }{X}_{1}R{u}_{\delta }{\eta }^2\mathrm{d}x\mathrm{d}t-{\int }_{t_1}^{t_2}{\int }_U{X}_{n+1}{u}_{\delta }R{u}_{\delta }2\eta X_1\eta \mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {\int }_{t_1}^{t_2}{\int }_U|X{u}_{\delta }||XR{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t+2{\int }_{t_1}^{t_2}{\int }_U|X{u}_{\delta }||R{u}_{\delta }|\eta |X\eta |\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Similarly, we estimate $-{\int }_{t_1}^{t_2}{\int }_U{X}_{n+1}{X}_1{u}_{\delta }R{u}_{\delta }{\eta }^2\mathrm{d}x\mathrm{d}t$. Thus,

$${\int }_{t_1}^{t_2}{\int }_U|R{u}_{\delta }{|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le 2{\int }_{t_1}^{t_2}{\int }_U|X{u}_{\delta }||XR{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t+4{\int }_{t_1}^{t_2}{\int }_U|X{u}_{\delta }||R{u}_{\delta }|\eta |X\eta |\mathrm{d}x\mathrm{d}t.$$

Adopting Young’s inequality to the final integral term in the above inequality, we have

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_U{|R{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le 4{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{4-p}{4}}+{\displaystyle \frac{p-2}{4}}}|XR{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t+64{\int }_{t_1}^{t_2}{\int }_{\mathrm{spt}(\eta )}|X{u}_{\delta }{|}^2{|X\eta |}^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Adopting Young’s inequality again to the above inequality, for any $\tau >0$, we have

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_U{|R{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {\displaystyle \frac{16}{\tau }}{\int }_{t_1}^{t_2}{\int }_{\mathrm{spt}(\eta )}(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{4-p}{2}}}\mathrm{d}x\mathrm{d}t+\tau {\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{|XR{u}_{\delta }|}^2{\eta }^4\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+64{\int }_{t_1}^{t_2}{\int }_{\mathrm{spt}(\eta )}|X{u}_{\delta }{|}^2{|X\eta |}^2\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

from which, by Lemma 1, when $2\le p\le 4$, we have

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_U{|R{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {\displaystyle \frac{16}{\tau }}{\int }_{t_1}^{t_2}{\int }_{\mathrm{spt}(\eta )}(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{4-p}{2}}}\mathrm{d}x\mathrm{d}t+64{\int }_{t_1}^{t_2}{\int }_{\mathrm{spt}(\eta )}|X{u}_{\delta }{|}^2{|X\eta |}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+\tau C{\int }_{t_1}^{t_2}{\int }_U|R{u}_{\delta }{|}^2{\eta }^2|\eta {\partial }_t\eta |\mathrm{d}x\mathrm{d}t+\tau C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{|X\eta |}^2{\eta }^2{|R{u}_{\delta }|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {\displaystyle \frac{16}{\tau }}|\mathrm{spt}(\eta )|{\omega }_{\delta }^{{\displaystyle \frac{4-p}{2}}}+{64\parallel X\eta \parallel }_{L^{\infty }}^2|\mathrm{spt}(\eta )|{\omega }_{\delta }\hfill \\ \hfill & \phantom{\le }+\tau C(\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}+{\parallel X\eta \parallel }_{L^{\infty }}^2{\omega }_{\delta }^{{\displaystyle \frac{p-2}{2}}}){\int }_{t_1}^{t_2}{\int }_U{|R{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

(15)

where ${\omega }_{\delta }={sup}_{\mathrm{spt}(\eta )}(\delta +|X{u}_{\delta }{|}^2)$. Letting ${\tau }^{-1}=2C(\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}+{\parallel X\eta \parallel }_{L^{\infty }}^2{\omega }_{\delta }^{{\displaystyle \frac{p-2}{2}}})$ in (15), we deduce (14). □

4. Proofs of Theorems 3 and 4

Firstly, we demonstrate Theorem 3.

Proof of Theorem 3. 

From (7), we have

$${({\partial }_t{u}_{\delta })}^2={\partial }_t{u}_{\delta }\left[\sum _{i=1}^{2n}{X}_i(A_i^{\delta }(X{u}_{\delta }))\right],$$

which, together with Condition (8), yields

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_U{({\partial }_t{u}_{\delta })}^2{\eta }^2\mathrm{d}x\mathrm{d}t={\int }_{t_1}^{t_2}{\int }_U\sum _{i,j=1}^{2n}{\partial }_t{u}_{\delta }{A}_{i,{\xi }_j}^{\delta }(X{u}_{\delta })X_i{X}_j{u}_{\delta }{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{\int }_{t_1}^{t_2}{\int }_U|{\partial }_t{u}_{\delta }|(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2}{2}}}|XX{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Adopting Young’s inequality to the above inequality, we obtain

$${\int }_{t_1}^{t_2}{\int }_U{({\partial }_t{u}_{\delta })}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{p-2}{|XX{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t.$$

(16)

Using Lemma 2 with $\beta =p-2$ to bound the right side of (16), we have

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{p-2}{|XX{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {C(\parallel X\eta \parallel }_{L^{\infty }}^2+{\parallel \eta R\eta \parallel }_{L^{\infty }})\int {\int }_{\mathrm{spt}(\eta )}(\delta +|X{u}_{\delta }{{|}^2)}^{p-1}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}{|\mathrm{spt}(\eta )|}^{{\displaystyle \frac{p-2}{2p-2}}}{\left(\int {\int }_{\mathrm{spt}(\eta )}(\delta +|X{u}_{\delta }{{|}^2)}^{p-1}\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{p}{2p-2}}}.\hfill \end{array}$$

(17)

Combining (16) and (17), we conclude (9). □

Here is a prerequisite for demonstrating Theorem 4.

Lemma 4. 

Assume that $u_{\delta }$ is a weak solution to Equation (7). Then, $v_l^{\delta }=X_l{u}_{\delta }$, with $l=1,\cdots ,2n$, is a weak solution to the equation

$${\partial }_t{v}_l^{\delta }=\sum _{i,j=1}^{2n}{X}_i(A_{i,{\xi }_j}^{\delta }(X{u}_{\delta })X_l{X}_j{u}_{\delta })+\sum _{i=1}^{2n}[X_l,X_i]A_i^{\delta }(X{u}_{\delta }).$$

(18)

Proof. 

For $l=1,\cdots ,2n$, noting that ${\partial }_t{X}_l=X_l{\partial }_t$, from (7) and Lie bracket, we obtain

$$\begin{array}{cc}\hfill {\partial }_t{v}_l^{\delta }& =X_l{\partial }_t{u}_{\delta }=\sum _{i=1}^{2n}{X}_l(X_i{A}_i^{\delta }(X{u}_{\delta }))\hfill \\ \hfill & =\sum _{i=1}^{2n}{X}_i(X_l{A}_i^{\delta }(X{u}_{\delta }))+\sum _{i=1}^{2n}[X_l,X_i]A_i^{\delta }(X{u}_{\delta })\hfill \\ \hfill & =\sum _{i=1}^{2n}{X}_i(A_{i,{\xi }_j}^{\delta }(X{u}_{\delta })X_l{X}_j{u}_{\delta })+\sum _{i,j=1}^{2n}[X_l,X_i]A_i^{\delta }(X{u}_{\delta }).\hfill \end{array}$$

□

Here, we show the proof of Theorem 4.

Proof of Theorem 4. 

For every $\beta \in (0,1)$, applying $\varphi ={\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_l{u}_{\delta }$ to Test (18), then, by integration by parts, we obtain

$$\begin{array}{cc}\hfill L^l:=& {\int }_{t_1}^{t_2}{\int }_U{\partial }_t(X_l{u}_{\delta }){\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_l{u}_{\delta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -{\int }_{t_1}^{t_2}{\int }_U\sum _{i,j=1}^{2n}{A}_{i,{\xi }_j}^{\delta }(X{u}_{\delta })X_l{X}_j{u}_{\delta }{X}_i({\eta }^2(\delta +|X{u}_{\delta }{|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_l{u}_{\delta })\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_U\sum _{i=1}^{2n}[X_l,X_i]A_i^{\delta }(X{u}_{\delta }){\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_l{u}_{\delta }\mathrm{d}x\mathrm{d}t=:-I_1^l+I_2^l.\hfill \end{array}$$

(19)

For $L^l$, by integration by parts, we have

$$\begin{array}{cc}\hfill \sum _{i=1}^{2n}{L}^l=& {\displaystyle \frac{1}{2-\beta }}{\int }_{t_1}^{t_2}{\int }_U{\partial }_t\left(\right(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{2-\beta }{2}}}){\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -{\displaystyle \frac{2}{2-\beta }}{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{{|}^2)}^{{\displaystyle \frac{2-\beta }{2}}}\eta {\partial }_t\eta \mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

thus,

$$|\sum _{i=1}^{2n}{L}^l|\le {\displaystyle \frac{C}{2-\beta }}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{2-\beta }{2}}}|\eta {\partial }_t\eta |\mathrm{d}x\mathrm{d}t.$$

(20)

For $I_1^l$, we have

$$\begin{array}{cc}\hfill I_1^l=& {\int }_{t_1}^{t_2}{\int }_U\sum _{i,j=1}^{2n}{A}_{i,{\xi }_j}^{\delta }(X{u}_{\delta })X_l{X}_j{u}_{\delta }2\eta X_i\eta (\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_l{u}_{\delta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\displaystyle \frac{\beta }{2}}{\int }_{t_1}^{t_2}{\int }_U\sum _{i,j=1}^{2n}{A}_{i,{\xi }_j}^{\delta }(X{u}_{\delta })X_l{X}_j{u}_{\delta }{\eta }^2(\delta +|X{u}_{\delta }{|}^2{)}^{-{\displaystyle \frac{\beta }{2}}-1}{X}_i(|X{u}_{\delta }{|}^2)X_l{u}_{\delta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_U\sum _{i,j=1}^{2n}{A}_{i,{\xi }_j}^{\delta }(X{u}_{\delta })X_l{X}_j{u}_{\delta }{\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_i{X}_l{u}_{\delta }\mathrm{d}x\mathrm{d}t:=I_{11}^l-I_{12}^l+I_{13}^l.\hfill \end{array}$$

(21)

For $I_{11}^l$, by Condition (8), we have

$$|I_{11}^l|\le C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-1-\beta }{2}}}|XX{u}_{\delta }|\eta |X\eta |\mathrm{d}x\mathrm{d}t.$$

(22)

For $I_{12}^l$, by $X_l{X}_j=[X_l,X_j]+X_j{X}_l$, we have

$$\begin{array}{cc}\hfill I_{12}^l=& {\displaystyle \frac{\beta }{2}}{\int }_{t_1}^{t_2}{\int }_U\sum _{i,j=1}^{2n}{A}_{i,{\xi }_j}^{\delta }(X{u}_{\delta })[X_l,X_j]u_{\delta }{\eta }^2(\delta +|X{u}_{\delta }{|}^2{)}^{-{\displaystyle \frac{\beta }{2}}-1}{X}_i(|X{u}_{\delta }{|}^2)X_l{u}_{\delta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\displaystyle \frac{\beta }{2}}{\int }_{t_1}^{t_2}{\int }_U\sum _{i,j=1}^{2n}{A}_{i,{\xi }_j}^{\delta }(X{u}_{\delta })X_j{X}_l{u}_{\delta }{\eta }^2(\delta +|X{u}_{\delta }{|}^2{)}^{-{\displaystyle \frac{\beta }{2}}-1}{X}_i(|X{u}_{\delta }{|}^2)X_l{u}_{\delta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& :I_{121}^l+I_{122}^l.\hfill \end{array}$$

(23)

For $I_{121}^l$, by Condition (8), we have

$$|I_{121}^l|\le C\beta {\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}|R{u}_{\delta }||XX{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t.$$

(24)

For $I_{122}^l$, by Condition (8), we have

$$\begin{array}{cc}\hfill \sum _{l=1}^{2n}{I}_{122}^l=& {\displaystyle \frac{\beta }{4}}{\int }_{t_1}^{t_2}{\int }_U\sum _{i,j=1}^{2n}{A}_{i,{\xi }_j}^{\delta }(X{u}_{\delta })X_j(|X{u}_{\delta }{|}^2)X_i(|X{u}_{\delta }{|}^2){\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}-1}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill \ge & {\displaystyle \frac{\lambda \beta }{4}}{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-4-\beta }{2}}}|X(|X{u}_{\delta }{|}^2{)|}^2{\eta }^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(25)

For $I_{13}^l$, by $X_l{X}_j=[X_l,X_j]+X_j{X}_l$, we have

$$\begin{array}{cc}\hfill I_{13}^l=& {\int }_{t_1}^{t_2}{\int }_U\sum _{i,j=1}^{2n}{A}_{i,{\xi }_j}^{\delta }(X{u}_{\delta })[X_l,X_j]u_{\delta }{\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_i{X}_l{u}_{\delta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_U\sum _{i,j=1}^{2n}{A}_{i,{\xi }_j}^{\delta }(X{u}_{\delta })X_j{X}_l{u}_{\delta }{\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}-1}{X}_i{X}_l{u}_{\delta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& :I_{131}^l+I_{132}^l.\hfill \end{array}$$

(26)

For $I_{131}^l$, by Condition (8), we have

$$|I_{121}^l|\le C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}|R{u}_{\delta }||XX{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t.$$

(27)

For $I_{132}^l$, by Condition (8), we have

$$\sum _{l=1}^{2n}{I}_{132}^l\ge \lambda {\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}{|XX{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t.$$

(28)

For $I_2^l$, when $l=1,\cdots ,n$, by integration by parts, and then by $[R{X}_k=X_{k}R]$, we have

$$\begin{array}{cc}\hfill I_2^l=& {\int }_{t_1}^{t_2}{\int }_U\sum _{i=1}^{2n}R{A}_i^{\delta }(X{u}_{\delta }){\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_l{u}_{\delta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& {\int }_{t_1}^{t_2}{\int }_U\sum _{i=1}^{2n}{A}_i^{\delta }(X{u}_{\delta })2\eta R\eta (\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_l{u}_{\delta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +\beta {\int }_{t_1}^{t_2}{\int }_U\sum _{i,k=1}^{2n}{A}_i^{\delta }(X{u}_{\delta }){\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}-1}{X}_k{u}_{\delta }{X}_{k}R{u}_{\delta }{X}_l{u}_{\delta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_U\sum _{i=1}^{2n}{A}_i^{\delta }(X{u}_{\delta }){\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}R{u}_{\delta }\mathrm{d}x\mathrm{d}t=:I_{21}^l+I_{22}^l+I_{23}^l.\hfill \end{array}$$

(29)

For $I_{21}^l$, by Condition (8), we have

$$|I_{21}^l|\le C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-\beta }{2}}}|\eta R\eta |\mathrm{d}x\mathrm{d}t.$$

(30)

For $I_{22}^l$, by integration by parts, we have

$$I_{22}^l=-\beta {\int }_{t_1}^{t_2}{\int }_U\sum _{i,k=1}^{2n}{X}_k[A_i^{\delta }(X{u}_{\delta }){\eta }^2(\delta +|X{u}_{\delta }{|}^2{)}^{-{\displaystyle \frac{\beta }{2}}-1}{X}_k{u}_{\delta }]R{u}_{\delta }{X}_l{u}_{\delta }\mathrm{d}x\mathrm{d}t,$$

which, together with Condition (8), yields

$$\begin{array}{cc}\hfill |I_{22}^l|\le & C{(\beta +1)}^2{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}|XX{u}_{\delta }||R{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C\beta {\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-1-\beta }{2}}}|R{u}_{\delta }|\eta |X\eta |\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(31)

For $I_{23}^l$, we use the same method to obtain the same estimate as $I_{22}^l$, namely,

$$\begin{array}{cc}\hfill |I_{23}^l|\le & C(\beta +1){\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}|XX{u}_{\delta }||R{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-1-\beta }{2}}}|R{u}_{\delta }|\eta |X\eta |\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

which, together with (29), (30), and (31), yields

$$\begin{array}{cc}\hfill |I_2^l|\le & C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-\beta }{2}}}|\eta R\eta |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C{(\beta +1)}^2{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}|XX{u}_{\delta }||R{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C(\beta +1){\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-1-\beta }{2}}}|R{u}_{\delta }|\eta |X\eta |\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(32)

We combine (19), (21), (23), and (26) to obtain

$$I_{132}^l-I_{122}^l=-L^l-I_{11}^l+I_{121}^l-I_{131}^l+I_2^l.$$

(33)

Combining (33), (20), (22), (24), (25), (27), (28), and (32), then, by $|X|X{u}_{\delta }||\le |XX{u}_{\delta }|$, we have

$$\begin{array}{cc}\hfill & (1-\beta ){\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}{|XX{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {\displaystyle \frac{C}{2-\beta }}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{2-\beta }{2}}}|\eta {\partial }_t\eta |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C(\beta +1){\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-1-\beta }{2}}}|XX{u}_{\delta }|\eta |X\eta |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{(\beta +1)}^2{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}|XX{u}_{\delta }||R{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-\beta }{2}}}|\eta R\eta |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C(\beta +1){\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{p-1-\beta }{2}}}|R{u}_{\delta }|\eta |X\eta |\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(34)

Here, $\beta <1$ ensures that the coefficient of the integral term in the left side of (34) is positive. When $2\le p<3$, setting $\beta =p-2$ in (34), we have

$$\begin{array}{cc}\hfill (3-p){\int }_{t_1}^{t_2}{\int }_U{|XX{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t& \le C{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{4-p}{2}}}|\eta {\partial }_t\eta |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{1}{2}}}|XX{u}_{\delta }|\eta |X\eta |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_U|XX{u}_{\delta }||R{u}_{\delta }|{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2)|\eta R\eta |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{1}{2}}}|R{u}_{\delta }|\eta |X\eta |\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Adopting Young’s inequality to the above inequality, we obtain

$$\begin{array}{cc}\hfill {\int }_{t_1}^{t_2}{\int }_U{|XX{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t& \le C{\int }_U(\delta +|X{u}_{\delta }{|}^2{)}^{{\displaystyle \frac{4-p}{2}}}|\eta {\partial }_t\eta |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_U(\delta +|X{u}_{\delta }{|}^2{)(|X\eta |}^2+|\eta R\eta |)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_U{|R{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(35)

Finally, adopting Lemma 3 to control the final integral term in the right side of (35), we deduce (10). □

5. Conclusions

In this article, we study the regularity of the weak solution u to a class of degenerate parabolic quasi-linear equations ${\partial }_{t}u={\displaystyle \sum _{i=1}^{2n}}{X}_i{A}_i(Xu)$ modeled on the parabolic p-Laplacian equation in the Heisenberg group ${\mathbb{H}}^n$. When $2\le p\le 4$, we built up, via a Caccioppoli-type inequality (Equation (12)) established by Capogna et al. [24], the integrability of ${({\partial }_{t}u)}^2$, namely, ${\partial }_{t}u\in L_{\mathrm{loc}}^2$. When $2\le p<3$, we construct an appropriate test-function $\varphi ={\eta }^2(\delta +|X{u}_{\delta }{{|}^2)}^{-{\displaystyle \frac{\beta }{2}}}{X}_l{u}_{\delta }$ to Test (18) and obtain that the integral term of ${(XX{u}_{\delta })}^2$ is bounded by ${\int }_{t_1}^{t_2}{\int }_U{|R{u}_{\delta }|}^2{\eta }^2\mathrm{d}x\mathrm{d}t$—see Inequality (35). Basing on (35), we demonstrate the $H{W}_{\mathrm{loc}}^{2,2}$-regularity of u, namely, $XXu\in L_{\mathrm{loc}}^2$. For the $H{W}_{\mathrm{loc}}^{2,2}$-regularity, when $p\ge 2$, the range of p is optimal compared to the Euclidean case.

Since our method is based on the work of Capogna et al. [24], we are currently unable to handle the case $p\in (1,2)$. Hence, our next effort will be challenging and focused on establishing the regularity for the range $p\in (1,2)$.

To sum up, these results presented in this article are original. Our findings, we think, will find extensive use in the study of regularities for equations with the p-Laplacian operator involved.