Lipschitz and Second-Order Regularities for Non-Homogeneous Degenerate Nonlinear Parabolic Equations in the Heisenberg Group

Abstract

In the Heisenberg group ${\mathbb{H}}^n$, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form ${\partial }_{t}u-{\displaystyle \sum _{i=1}^{2n}}{X}_i{A}_i\left(Xu\right)=K(x,t,u,Xu)$, where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For $2\le p\le 4$, we establish the local Lipschitz regularity ($u\in C_{\mathrm{loc}}^{0,1}$), with the horizontal gradient satisfying $Xu\in L_{\mathrm{loc}}^{\infty }$; (ii) For $2\le p<3$, we establish the local second-order horizontal Sobolev regularity ($u\in H{W}_{\mathrm{loc}}^{2,2}$), with the second-order horizontal derivative satisfying $XXu\in L_{\mathrm{loc}}^2$. These results solve an open problem proposed by Capogna et al.

1. Introduction

The regularity theory for solutions to partial differential equations (PDEs) constitutes a cornerstone in modern analysis, with particular significance in the study of nonlinear operators such as the p-Laplacian. Over half a century of sustained investigation has established the p-Laplacian equation (p-harmonic equation), ${\Delta }_{p}u=0$, as a paradigmatic model for understanding nonlinear potential theory, while its regularity properties remain an active frontier in geometric analysis. The seminal works of Ural’ceva [1], Uhlenbeck [2], Evans [3], and Lewis [4] laid the analytical bedrock by proving optimal Lipschitz ($C^{0,1}$) and Hölder ($C^{1,\alpha }$) regularities for p-harmonic functions (weak solutions to p-harmonic equations) across the entire spectrum $1<p<\infty$. Subsequent refinements by Manfredi–Weitsman [5] through Cordes condition techniques [6,7,8] revealed second-order Sobolev regularity ($u\in W_{\mathrm{loc}}^{2,2}$) for $1<p<3+{\displaystyle \frac{2}{n-2}}$, later rederived via innovative methods by Dong et al. [9]. Recent decades have witnessed a paradigm shift toward non-Euclidean settings. In the Heisenberg group ${\mathbb{H}}^n$, the $C^{0,1}$- and $C^{1,\alpha }$-regularity theories were systematically developed in papers [10,11,12,13,14,15] for $2\le p<4$ (also see the book [16]). Zhong–Mukherjee’s breakthrough extension to $1<p<\infty$ [17,18] catalyzed further progress, enabling Domokos [19] and Liu et al. [20] to establish the second-order $W_{\mathrm{loc}}^{2,2}$-regularity with the following range: $1<p\le 4$ (for $n=1$) and $1<p<3+{\displaystyle \frac{1}{n-1}}$ (for $n>1$). The SU(3) group framework saw Domokos–Manfredi [21,22] achieve comprehensive regularity: the $C^{0,1}$-regularity for $1<p<\infty$ and the $C^{1,\alpha }$-regularity for $2\le p<\infty$. Yu’s work [23] pushed the second-order regularity to the range $1<p<{\displaystyle \frac{7}{2}}$, while Citti–Mukherjee [24] synthesized the Riemannian approximation technology [22,25,26] with Zhong–Mukherjee’s methods [17,18] to unify regularity theories for p-Laplacian equations with Hörmander vector fields of step two. Islam et al. [27] established analytical solutions to the modified Zakharov–Kuznetsov equation. Alam–Tunç [28] derived solitary wave solutions to the (2+1)-dimensional KD and KP equations with spatio-temporal dispersion. This evolutionary trajectory underscores the profound interplay between geometric structures and nonlinear analysis, continually reshaping our understanding of degenerate PDEs.

This article aims to solve the open problem (1) in paper [26]. We investigate the regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form

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

(1)

where $\mathsf{\Omega }\subset {\mathbb{H}}^n$ is a domain in the n-th Heisenberg group and $T>0$. This system generalizes the non-homogeneous parabolic p-Laplacian equation

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

serving as a canonical model for studying nonlinear diffusion processes in sub-Riemannian geometries. For exponents $2\le p<\infty$, the operators ${\left\{A_i\right\}}_{i=1}^{2n}$ and source term K satisfy the following uniform ellipticity/growth conditions:

$$\left\{\begin{array}{c}{\lambda (\delta +|\xi |}^2{)}^{{\displaystyle \frac{p-2}{2}}}{\left|\eta \right|}^2\le {\sum }_{i,j=1}^{2n}{\partial }_{{\xi }_j}{A}_i\left(\xi \right){\eta }_i{\eta }_j\le {\mathsf{\Lambda }(\delta +|\xi |}^2{)}^{{\displaystyle \frac{p-2}{2}}}{\left|\eta \right|}^2,\hfill \\ \left|A\left(\xi \right)\right|+\left|K(x,t,z,\xi )\right|+|D_x{K(x,t,z,\xi )|\le \mathsf{\Lambda }(\delta +\left|\xi \right|}^2{)}^{{\displaystyle \frac{p-1}{2}}},\hfill \\ |D_{\xi }K(x,t,z,\xi )|+|D_z{K(x,t,z,\xi )|\le \mathsf{\Lambda }(\delta +\left|\xi \right|}^2{)}^{{\displaystyle \frac{p-2}{2}}},\hfill \end{array}\right.$$

(2)

where $\delta \in [0,1]$, $0<\lambda \le \mathsf{\Lambda }<\infty$, and derivatives are interpreted in the distributional sense. A function $u\in L^p((0,T),H{W}_{\mathrm{loc}}^{1,p}\left(\mathsf{\Omega }\right))$ is a weak solution if it satisfies

$${\int }_0^T{\int }_{\mathsf{\Omega }}\left({\partial }_{t}u·\varphi +⟨A\left(Xu\right),X\varphi ⟩\right)\mathrm{d}x\mathrm{d}t={\int }_0^T{\int }_{\mathsf{\Omega }}K(·)\varphi \mathrm{d}x\mathrm{d}t$$

(3)

for all test functions $\varphi \in C_0^{\infty }(\mathsf{\Omega }\times (0,T))$, where $H{W}_{\mathrm{loc}}^{1,p}\left(\mathsf{\Omega }\right)$ denotes the local horizontal Sobolev space (see Section 2).

The study of parabolic p-Laplacian systems (obtained by setting $A\left(\xi \right)={\left|\xi \right|}^{p-2}\xi$ and $K\equiv 0$ in (1)) originated with DiBenedetto–Friedman’s pioneering $C^{1,\alpha }$-regularity theory in Euclidean spaces [29], paralleled by Wiegner’s contemporaneous work [30]. Subsequent breakthroughs by Lindqvist [31] and Attouchi–Ruosteenoja [32] established optimal $W^{2,2}$-regularity for $1<p<3$, later systematized in DiBenedetto’s monograph [33]. In sub-Riemannian settings, Capogna et al. [25] demonstrated striking smoothness ($u\in C_{\mathrm{loc}}^{\infty }$) for non-degenerate cases ($2\le p<\infty$) in ${\mathbb{H}}^n$ via the equation ${\partial }_{t}u={\sum }_{i=1}^{2n}{X}_i\left({(1+|Xu|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{X}_{i}u\right)$. For degenerate regimes, their recent work [26] with Zhong’s gradient estimates [17] achieved the $C^{0,1}$-regularity for $2\le p\le 4$ through innovative Riemannian approximation schemes. This main regularity result is also demonstrated in their work [34]. Based on Capogna et al.’s work [26,34], Yu et al. [35] demonstrated the second-order $H{W}_{\mathrm{loc}}^{2,2}$-regularity for $2\le p<3$. The method they established provided important insights for the proof of Theorem 2.

2. Main Theorems and Proof Methods

Consider u as a weak solution to Equation (1). We focus on the local Lipschitz regularity ($u\in C_{\mathrm{loc}}^{0,1}$). As a consequence, when $2\le p\le 4$, we demonstrate that $Xu\in L_{\mathrm{loc}}^{\infty }$.

Theorem 1.

Let $u\in L^p((0,T),H{W}_{\mathrm{loc}}^{1,p}\left(\mathsf{\Omega }\right))$ be a weak solution to the degenerate parabolic system (1) in ${\mathbb{H}}^n\times (0,T)$, where ${\left\{A_i\right\}}_{1\le i\le 2n}$ and K satisfy structural conditions (2) with $2\le p\le 4$. Then, u possesses the intrinsic local Lipschitz continuity ($u\in C_{\mathrm{loc}}^{0,1}$) with the following quantitative gradient estimate: there exists a geometric constant $C=C(n,p,\lambda ,\mathsf{\Lambda },r_0)>0$ such that for any parabolic cylinder $Q_{\mu ,2r}:=B(x_0,2r)\times (t_0-\mu {\left(2r\right)}^2,t_0)\subset \mathsf{\Omega }\times (0,T)$,

$$\underset{Q_{\mu ,r}}{sup}\left|Xu\right|\le C{\mu }^{{\displaystyle \frac{1}{2}}}max\left({\mathcal{E}}_p{(u,\mu ,2r)}^{1/2},{\mu }^{{\displaystyle \frac{p}{2(2-p)}}}\right),$$

(4)

where the normalized energy functional is defined by

$${\mathcal{E}}_p(u,\mu ,r):={\displaystyle \frac{1}{\mu r^{2n+4}}}{\int \int }_{Q_{\mu ,r}}{(1+|Xu|}^2{)}^{{\displaystyle \frac{p}{2}}}\mathrm{d}x\mathrm{d}t.$$

Drawing on Theorem 1, when $2\le p<3$, we demonstrate the local second-order $H{W}_{\mathrm{loc}}^{2,2}$-regularity for the weak solution u to Equation (1), specifically $XXu\in L_{\mathrm{loc}}^2$.

Theorem 2.

Let $u\in L^p((0,T),H{W}_{\mathrm{loc}}^{1,p}\left(\mathsf{\Omega }\right))$ be a weak solution to the degenerate parabolic system (1) in ${\mathbb{H}}^n\times (0,T)$, where ${\left\{A_i\right\}}_{1\le i\le 2n}$ and K satisfy structural conditions (2). If $2\le p<3$, then u possesses the local second-order horizontal Sobolev regularity ($u\in H{W}_{\mathrm{loc}}^{2,2}$) with the following second-order horizontal derivative estimate: there exists a geometric constant $C=C(n,p,\lambda ,\mathsf{\Lambda })>0$ such that for any non-negative test function $\eta \in C^1([0,T],C_0^{\infty }\left(\mathsf{\Omega }\right))$,

$${\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|XXu\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le C\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}\left|\mathrm{spt}\left(\eta \right)\right|{\omega }^{{\displaystyle \frac{4-p}{2}}}+C\mathcal{N}\left(\eta \right)\left|\mathrm{spt}\left(\eta \right)\right|\omega ,$$

(5)

where $\omega :={sup}_{\mathrm{spt}\left(\eta \right)}{(1+|Xu|}^2)$ and $\mathcal{N}\left(\eta \right):={\parallel \eta \parallel }_{L^{\infty }}^2+{\parallel X\eta \parallel }_{L^{\infty }}^2+{\parallel \eta R\eta \parallel }_{L^{\infty }}$.

Proof Architecture and Methodological Innovations

Theorem 1 (Section 4): The presence of non-homogeneous term $K(·)$ necessitates novel analytical machinery. The principal technical challenge resides in controlling integral terms involving K, $X_{l}K$, and $RK$ (see Section 3). Therefore, in condition (2), we add the following conditions for K:

$$\left\{\begin{array}{c}\left|K(x,t,z,\xi )\right|+|D_x{K(x,t,z,\xi )|\le \mathsf{\Lambda }(\delta +\left|\xi \right|}^2{)}^{{\displaystyle \frac{p-1}{2}}},\hfill \\ |D_{\xi }K(x,t,z,\xi )|+|D_z{K(x,t,z,\xi )|\le \mathsf{\Lambda }(\delta +\left|\xi \right|}^2{)}^{{\displaystyle \frac{p-2}{2}}}.\hfill \end{array}\right.$$

The proof employs Moser’s iteration scheme within the sub-Riemannian framework, hinging on the derivation of critical Caccioppoli-type estimates for horizontal gradients. Firstly, we construct two pivotal Caccioppoli-type inequalities for $XRu$ and $XXu$ (Lemmas 1 and 2), establishing control over second-order horizontal derivatives when $2\le p\le 4$. Secondly, through Lemma 1, we refine the integral bounds in (12), controlling each gradient integral term and establishing the crucial Caccioppoli-type inequality (14) when $2\le p\le 4$. Finally, synthesis of these estimates via Moser’s iteration yields the optimal $C_{\mathrm{loc}}^{0,1}$-regularity.

Theorem 2 (Section 5): This proof is based on Theorem 1 and some lemmas generated during the process of proving Theorem 1. The demonstration centers on strategic test-function selection and nonlinear potential analysis. Firstly, for $l=1,\dots ,2n$, we utilize the test function $\varphi =X_l[{\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u]$ in Equation (1), which allows us to decouple horizontal and vertical derivatives. Secondly, we apply the structural condition (2) with Young’s inequality to bound the $XXu$ energy by the vertical derivative, ${\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|XXu\right|}^2\mathrm{d}x\mathrm{d}t\lesssim C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t$. Finally, through the Caccioppoli-type inequality (18) and Hölder’s inequality, we establish the final control on ${\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t$ in (20).

3. Preliminaries

3.1. Heisenberg Group

The n-dimensional Heisenberg group ${\mathbb{H}}^n$ is canonically identified with the Euclidean space ${\mathbb{R}}^{2n+1}$ equipped with a stratified Lie group structure, possessing homogeneous (or Hausdorff) dimension $2n+2$. Its algebraic structure is characterized by the polynomial group law:

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

encoding noncommutative geometry through the symplectic term in the central coordinate.

The Lie algebra is generated by left-invariant vector fields forming the horizontal distribution:

$$\begin{array}{cc}\hfill X_i& :={\partial }_{x_i}-{\displaystyle \frac{x_{n+i}}{2}}{\partial }_{x_{2n+1}},1\le i\le n,\hfill \\ \hfill X_{n+i}& :={\partial }_{x_{n+i}}+{\displaystyle \frac{x_i}{2}}{\partial }_{x_{2n+1}},1\le i\le n,\hfill \end{array}$$

which satisfy Hörmander’s hypoellipticity condition of step two. The vertical direction is spanned by the commutator

$$R:={\partial }_{x_{2n+1}}=[X_i,X_{n+i}],1\le i\le n,$$

endowing ${\mathbb{H}}^n$ with its characteristic Carnot–Carathéodory geometry. Here, the Lie algebra decomposition is that ${\mathfrak{h}}^n=V_1\oplus V_2$ with $V_1=\mathrm{span}\{X_1,\dots ,X_{2n}\}$ and $V_2=\mathrm{span}\left\{R\right\}$; the dilation symmetry ${\delta }_{\lambda }\left(x\right)=(\lambda x_1,\dots ,\lambda x_{2n},{\lambda }^2{x}_{2n+1})$ preserves group structure; and the Carnot–Carathéodory distance is induced by horizontal curves.

This geometric scaffolding underpins the analysis of degenerate PDEs in sub-Riemannian settings, where the horizontal gradient $Xu=(X_{1}u,\dots ,X_{2n}u)$ and vertical derivatives interact non-trivially. Via the definition of vector fields $X_1,\dots ,X_{2n},R$, for any $i=1,\dots ,2n$, we have

$$[X_i,X_{i\pm n}]=R,[X_i,X_j]=0\mathrm{when}j\ne i\pm n,[X_i,R]=0.$$

(6)

3.2. Horizontal Sobolev Space

Let $\mathsf{\Omega }\subset {\mathbb{H}}^n$ be an open connected domain. For any differentiable function $\psi \in C^1\left(\mathsf{\Omega }\right)$, we define its horizontal gradient as the vector field

$$X\psi :=(X_1\psi ,\dots ,X_{2n}\psi )\in {\mathbb{R}}^{2n},$$

and for $\psi \in C^2\left(\mathsf{\Omega }\right)$, the second-order horizontal derivative as the matrix-valued distribution

$$XX\psi :={\left(X_i{X}_j\psi \right)}_{1\le i,j\le 2n}\in {\mathbb{R}}^{2n\times 2n}.$$

The associated seminorms are given by

$$\left|X\psi \right|:={\left(\sum _{i=1}^{2n}{\left|X_i\psi \right|}^2\right)}^{1/2},\left|XX\psi \right|:={\left(\sum _{i,j=1}^{2n}{\left|X_i{X}_j\psi \right|}^2\right)}^{1/2}.$$

The first-order horizontal Sobolev space $H{W}^{1,p}\left(\mathsf{\Omega }\right)$ for $1\le p<\infty$ is defined as

$$H{W}^{1,p}\left(\mathsf{\Omega }\right):=\left\{\psi \in L^p\left(\mathsf{\Omega }\right)|X\psi \in L^p(\mathsf{\Omega },{\mathbb{R}}^{2n})\right\},$$

equipped with the Banach norm

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

The local Sobolev space $H{W}_{\mathrm{loc}}^{1,p}\left(\mathsf{\Omega }\right)$ consists of functions satisfying $\psi \in H{W}^{1,p}\left(U\right)$ for every precompact subdomain $U⋐\mathsf{\Omega }$, forming the natural setting for local regularity analysis.

3.3. Notation

The following symbols are required for the subsequent proofs in this article. The notation $C^{k,\alpha }\left(\mathsf{\Omega }\right)$ represents the Hölder space, where k is a non-negative integer and $\alpha \in [0,1]$. The notation $C_0^{\infty }\left(\mathsf{\Omega }\right)$ represents the space of smooth functions with compact support in the domain $\mathsf{\Omega }$. The notation ${\parallel v\parallel }_{L^p\left(\mathsf{\Omega }\right)}$ represents the $L^p$-norm of the function v in the domain $\mathsf{\Omega }$, namely ${\parallel v\parallel }_{L^p\left(\mathsf{\Omega }\right)}:=({\int }_{\mathsf{\Omega }}{\left|v\right|}^p{\mathrm{d}x)}^{1/p}$. The notation ${\parallel v\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)}$ represents the $L^{\infty }$-norm of the function v in the domain $\mathsf{\Omega }$, namely ${\parallel v\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)}:={max}_{\mathsf{\Omega }}\left|v\right|$.

4. Crucial Caccioppli-Type Estimate

Using $XXu$ and a few Caccioppli-type inequalities, we develop the crucial Caccioppli-type estimate for $Xu$. The proofs of the ensuing lemmas depend on the following two lemmas. To simplify writing, we denote $A_{i,{\xi }_j}\left(\xi \right):={\partial }_{{\xi }_j}{A}_i\left(\xi \right)$, $K_{x_i}(x,t,z,\xi ):={\partial }_{x_i}K(x,t,z,\xi )$, $K_{{\xi }_i}(x,t,z,\xi ):={\partial }_{{\xi }_i}K(x,t,z,\xi )$, $K_z(x,t,z,\xi ):={\partial }_{z}K(x,t,z,\xi )$ and

$$\mathcal{N}\left(\eta \right):={\parallel \eta \parallel }_{L^{\infty }}^2+{\parallel X\eta \parallel }_{L^{\infty }}^2+{\parallel \eta R\eta \parallel }_{L^{\infty }}.$$

(7)

A Caccioppli-type inequality for $Ru$ involving $XRu$ is given by the following lemma.

Lemma 1.

Let u be a weak solution to (1) in ${\mathbb{H}}^n\times (0,T)$, where ${\left\{A_i\right\}}_{1\le i\le 2n}$ and K satisfy structural conditions (2). Then, there exists a geometric constant $C=C(n,\lambda ,\mathsf{\Lambda })>0$ such that for any non-negative test function $\eta \in C^1([0,T],C_0^{\infty }\left(\mathsf{\Omega }\right))$ and any $\beta \ge 0$,

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{\left|XRu\right|}^2{\eta }^{4+\beta }{\left|Ru\right|}^{\beta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {\displaystyle \frac{C}{\beta +1}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^{\beta +2}{\eta }^{3+\beta }\left|{\partial }_t\eta \right|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2}{2}}}({\eta }^2+{\left|X\eta \right|}^2){\eta }^{2+\beta }{\left|Ru\right|}^{\beta +2}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p}{2}}}{\eta }^{4+\beta }{\left|Ru\right|}^{\beta }\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(8)

Proof. 

Applying $\varphi =R\left[{\eta }^2\right|Ru{|}^{\beta }Ru]$ to test Equation (1), then by integration by parts, we obtain

$$\begin{array}{cc}\hfill L:=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}R\left[{\partial }_{t}u\right]{\eta }^2{\left|Ru\right|}^{\beta }Ru\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}R\left[X_i\left(A_i\left(Xu\right)\right)\right]{\eta }^2{\left|Ru\right|}^{\beta }Ru\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}R\left[K(x,t,u,Xu)\right]{\eta }^2{\left|Ru\right|}^{\beta }Ru\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{X}_i\left(A_{i,{\xi }_j}\left(Xu\right)X_{j}Ru\right){\eta }^2{\left|Ru\right|}^{\beta }Ru\mathrm{d}x\mathrm{d}t(\mathrm{Lie}\mathrm{bracket}[R,X_i]=[X_i,R])\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}RK(x,t,u,Xu){\eta }^2{\left|Ru\right|}^{\beta }Ru\mathrm{d}x\mathrm{d}t=:S_1+S_2.\hfill \end{array}$$

(9)

To bound L, we combine the Lie bracket operation and integration by parts. Specifically, we have

$$\begin{array}{cc}\hfill L=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\partial }_{t}Ru{\eta }^2{\left|Ru\right|}^{\beta }Ru\mathrm{d}x\mathrm{d}t(\mathrm{Lie}\mathrm{bracket}[R,{\partial }_t]=[{\partial }_t,R])\hfill \\ \hfill =& {\displaystyle \frac{1}{\beta +2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\partial }_t{\left(\right|Ru|}^{\beta +2}){\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -{\displaystyle \frac{2}{\beta +2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^{\beta +2}\eta {\partial }_t\eta \mathrm{d}x\mathrm{d}t(\mathrm{by}\mathrm{integration}\mathrm{by}\mathrm{parts}),\hfill \end{array}$$

which yields

$$\left|L\right|\le {\displaystyle \frac{2}{\beta +2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^{\beta +2}\eta \left|{\partial }_t\eta \right|\mathrm{d}x\mathrm{d}t.$$

For $S_1$, by integration by parts, we have

$$\begin{array}{cc}\hfill S_1=& -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_{j}Ru2\eta X_i\eta {\left|Ru\right|}^{\beta }Ru\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -(\beta +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_{j}Ru{\eta }^2{\left|Ru\right|}^{\beta }{X}_{i}Ru\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =:& -S_{11}-S_{12}.\hfill \end{array}$$

(10)

For $S_2$, we have

$$\begin{array}{cc}\hfill S_2=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}{K}_{x_i}(x,t,u,Xu)R{x}_i{\eta }^2{\left|Ru\right|}^{\beta }Ru\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}{K}_{{\xi }_i}(x,t,u,Xu)R{X}_{i}u{\eta }^2{\left|Ru\right|}^{\beta }Ru\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}{K}_z(x,t,u,Xu)Ru{\eta }^2{\left|Ru\right|}^{\beta }Ru=:S_{21}+S_{22}+S_{23}.\hfill \end{array}$$

(11)

Combining (9), (10), and (11), we obtain

$$S_{12}=-L-S_{11}+S_{21}+S_{22}+S_{23}.$$

We now estimate each term in the above equation independently using the condition inequality. When we estimate $S_{12}$ using condition (2), we obtain

$$S_{12}\ge (\beta +1)\lambda {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{\left|XRu\right|}^2{\eta }^2{\left|Ru\right|}^{\beta }\mathrm{d}x\mathrm{d}t.$$

When we estimate $S_{11}$ using condition (2), we obtain

$$|S_{11}|\le 2\mathsf{\Lambda }{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{\left|XRu\right|\eta \left|X\eta \right|\left|Ru\right|}^{\beta +1}\mathrm{d}x\mathrm{d}t.$$

When we estimate $S_{12}$ using condition (2), we obtain

$$|S_{21}|\le C\mathsf{\Lambda }{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-1}{2}}}{\eta }^2{\left|Ru\right|}^{\beta +1}\mathrm{d}x\mathrm{d}t.$$

To estimate $S_{22}$, by $R{X}_i=X_{i}R$ and condition (2), we obtain

$$|S_{22}|\le \mathsf{\Lambda }{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{\eta }^2{\left|Ru\right|}^{\beta +1}\left|XRu\right|\mathrm{d}x\mathrm{d}t.$$

When we estimate $S_{23}$ using condition (2), we obtain

$$|S_{23}|\le \mathsf{\Lambda }{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{\eta }^2{\left|Ru\right|}^{\beta +2}\mathrm{d}x\mathrm{d}t.$$

Combining above estimates, then using Young’s inequality, we obtain

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2}{2}}}{\left|XRu\right|}^2{\eta }^2{\left|Ru\right|}^{\beta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {\displaystyle \frac{C}{{(\beta +1)}^2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^{\beta +2}\eta \left|{\partial }_t\eta \right|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2}{2}}}\left[{\displaystyle \frac{1}{{(\beta +1)}^2}}({\eta }^2+{\left|X\eta \right|}^2)+{\displaystyle \frac{1}{\beta +1}}{\eta }^2\right]{\left|Ru\right|}^{\beta +2}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p}{2}}}{\eta }^2{\left|Ru\right|}^{\beta }\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Setting $\eta \to {\eta }^{2+\beta /2}$ in the above inequality, we obtain (8). □

A Caccioppli-type inequality for $XXu$ is given by the following lemma.

Lemma 2.

Let u be a weak solution to (1) in ${\mathbb{H}}^n\times (0,T)$, where ${\left\{A_i\right\}}_{1\le i\le 2n}$ and K satisfy structural conditions (2). Then, there exists a geometric constant $C=C(n,p,\lambda ,\mathsf{\Lambda })>0$ such that for any non-negative test function $\eta \in C^1([0,T],C_0^{\infty }\left(\mathsf{\Omega }\right))$ and any $\beta \ge 0$,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\beta +2}}{\displaystyle \underset{t_1<t<t_2}{sup}}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta +2}{2}}}{\eta }^2\mathrm{d}x+{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}{\left|XXu\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{(\beta +1)}^2\mathcal{N}\left(\eta \right)\int {\int }_{\mathrm{spt}\left(\eta \right)}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p+\beta }{2}}}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{(\beta +1)}^4{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}{\left|Ru\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+{\displaystyle \frac{C}{\beta +2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta +2}{2}}}\left|{\partial }_t\eta \right|\eta \mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(12)

Proof. 

For $l=1,\dots ,2n$, applying $\varphi =X_l[{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_{l}u]$ to test Equation (1), then by integration by parts, we obtain

$$\begin{array}{cc}\hfill L^l:=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{X}_l[{\partial }_{t}u]{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}{X}_l\left[X_i{A}_i\left(Xu\right)\right]{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}K(x,t,u,Xu)X_l[{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_{l}u]\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}{X}_i{X}_l{A}_i\left(Xu\right){\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t(\mathrm{Lie}\mathrm{bracket}{X}_l{X}_i=[X_l,X_i]+X_i{X}_l)\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}[X_l,X_i]A_i\left(Xu\right){\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}K(x,t,u,Xu)X_l[{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_{l}u]\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_l{X}_{j}u2\eta X_i{\eta (\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t(\mathrm{by}\mathrm{integration}\mathrm{by}\mathrm{parts})\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_l{X}_{j}u{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_i{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\displaystyle \frac{\beta }{2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_l{X}_{j}u{\eta }^2{X}_l{u(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta -2}{2}}}{X}_i{\left(\right|Xu|}^2)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}{A}_i\left(Xu\right)2\eta [X_l,X_i]{\eta (\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}{A}_i\left(Xu\right){\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}[X_l,X_i]X_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -\beta {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}{A}_i\left(Xu\right){\eta }^2{X}_l{u(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta -2}{2}}}{\displaystyle \sum _{k=1}^{2n}}{X}_{k}u[X_l,X_i]X_{k}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}K(x,t,u,Xu)X_l[{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_{l}u]\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & =:-S_1^l-S_2^l-S_3^l-S_4^l-S_5^l-S_6^l-S_7^l.\hfill \end{array}$$

For $S_2^l$, we use $X_l{X}_j=X_j{X}_l+[X_l,X_j]$ to obtain

$$\begin{array}{cc}\hfill S_2^l=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_j{X}_{l}u{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_i{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)[X_l,X_j]u{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{X}_i{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =:& S_{21}^l+S_{22}^l.\hfill \end{array}$$

For $S_3^l$, we use $X_l{X}_j=X_j{X}_l+[X_l,X_j]$ to obtain

$$\begin{array}{cc}\hfill S_3^l=& {\displaystyle \frac{\beta }{4}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_j\left({\left(X_{l}u\right)}^2\right){\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta -2}{2}}}{X}_i{\left(\right|Xu|}^2)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\displaystyle \frac{\beta }{2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)[X_l,X_j]u_{\eta }^2{X}_l{u(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta -2}{2}}}{X}_i{\left(\right|Xu|}^2)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =:& S_{31}^l+S_{32}^l.\hfill \end{array}$$

Combining the above equations, we obtain

$$L^l+S_{21}^l+S_{31}^l=-S_1^l-S_{22}^l-S_{32}^l-S_4^l-S_5^l-S_6^l-S_7^l.$$

(13)

We now estimate each term in (13) independently using the condition inequality. Noting

$$\begin{array}{cc}\hfill L^l=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\partial }_t{X}_{l}u{X}_{l}u{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}\mathrm{d}x\mathrm{d}t(\mathrm{Lie}\mathrm{bracket}[X_l,{\partial }_t]=[{\partial }_t,X_l])\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}{\partial }_t\left({\left(X_{l}u\right)}^2\right){\eta }^2\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

by condition (2), we bound the first term in the left-hand side of (12) as below:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\beta +2}}\left({\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta +2}{2}}}{\eta }^2\mathrm{d}x\right){|}_{t_1}^{t_2}=2{\displaystyle \sum _{l=1}^{2n}}{L}^l+{\displaystyle \frac{2}{\beta +2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta +2}{2}}}\eta {\partial }_t\eta \mathrm{d}x\mathrm{d}t;\hfill \\ \hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}{\left|XXu\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le {\displaystyle \frac{1}{\lambda }}{\displaystyle \sum _{l=1}^{2n}}{S}_{21}^l;\hfill \\ \hfill & 0\le {\displaystyle \frac{\lambda \beta }{4}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-4+\beta }{2}}}{\left|X\right(\left|Xu\right|}^2{\left)\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le {\displaystyle \sum _{l=1}^{2n}}{S}_{31}^l.\hfill \end{array}$$

When we estimate $S_1^l$ using condition (2), we obtain

$$|S_1^l|\le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-1+\beta }{2}}}\left|XXu\right|\eta \left|X\eta \right|\mathrm{d}x\mathrm{d}t.$$

Applying (6) and condition (2) to estimate $S_{22}^l$ and $S_{32}^l$, we obtain

$$|S_{22}^l|+|S_{32}^l|\le C(\beta +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}\left|Ru\right|{\eta }^2\left|XXu\right|\mathrm{d}x\mathrm{d}t.$$

Applying (6) and condition (2) to estimate $S_4^l$, we obtain

$$|S_4^l|\le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p+\beta }{2}}}\eta \left|R\eta \right|\mathrm{d}x\mathrm{d}t.$$

We estimate $S_5^l$ below. Applying (6) to $S_5^l$, we obtain

$$\begin{array}{cc}\hfill & S_5^l={\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{A}_{l+n}\left(Xu\right){\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}R{X}_{l}u\mathrm{d}x\mathrm{d}t\mathrm{when}l=1,\dots ,n;\hfill \\ \hfill & S_5^l=-{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{A}_{l-n}\left(Xu\right){\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}R{X}_{l}u\mathrm{d}x\mathrm{d}t\mathrm{when}l=n+1,\dots ,2n;\hfill \\ \hfill & S_5^l=0\mathrm{when}l=2n+1.\hfill \end{array}$$

From this, when $l=1,\dots ,n$, by $R{X}_l=X_{l}R$ and then by integration by parts, we obtain

$$\begin{array}{cc}\hfill S_5^l=& -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{j=1}^{2n}}{A}_{l+n,{\xi }_j}\left(Xu\right)X_l{X}_{j}u{\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}Ru\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\displaystyle \frac{\beta }{2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{A}_{l+n}\left(Xu\right){\eta }^2{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta -2}{2}}}{\displaystyle \sum _{k=1}^{2n}}{X}_{k}u{X}_l{X}_{k}uRu\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -2{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{A}_{l+n}\left(Xu\right)\eta X_l{\eta (\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta }{2}}}Ru\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

which, together with condition (2), yields

$$\begin{array}{cc}\hfill |S_5^l|\le & C(\beta +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}\left|XXu\right|{\eta }^2\left|Ru\right|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-1+\beta }{2}}}\eta \left|X\eta \right|\left|Ru\right|\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

When $l=n+1,\dots ,2n$, we can estimate $S_5^l$ in a same way.

For $S_6^l$, we employ the same method used for estimating $S_5^l$ and derive the following bound,

$$\begin{array}{cc}\hfill |S_6^l|\le & C\beta (\beta +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}\left|XXu\right|{\eta }^2\left|Ru\right|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C\beta {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-1+\beta }{2}}}\eta \left|X\eta \right|\left|Ru\right|\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

For $S_7^l$, by condition (2), we obtain

$$\begin{array}{cc}\hfill |S_7^l|\le & C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p+\beta }{2}}}\eta \left|X\eta \right|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C(\beta +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-1+\beta }{2}}}\left|XXu\right|{\eta }^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

We derive (12) by using Young’s inequality after combining all estimates into (13). □

We derive the crucial Caccioppoli-type estimate for $Xu$ involving $XXu$ based on Lemmas 1 and 2.

Lemma 3.

Let u be a weak solution to (1) in ${\mathbb{H}}^n\times (0,T)$, where ${\left\{A_i\right\}}_{1\le i\le 2n}$ and K satisfy structural conditions (2) with $2\le p\le 4$. Then, there exists a geometric constant $C=C(n,p,\lambda ,\mathsf{\Lambda })>0$ such that for any non-negative test function $\eta \in C^1([0,T],C_0^{\infty }\left(\mathsf{\Omega }\right))$ and any $\beta \ge 0$,

$$\begin{array}{cc}\hfill & {\displaystyle \underset{t_1<t<t_2}{sup}}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{\beta +2}{2}}}{\eta }^2\mathrm{d}x+{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}{\left|XXu\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{(p+\beta )}^7\mathcal{N}\left(\eta \right)\int {\int }_{\mathrm{spt}\left(\eta \right)}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p+\beta }{2}}}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{(p+\beta )}^7\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}{\left|\mathrm{spt}\left(\eta \right)\right|}^{{\displaystyle \frac{p-2}{p+\beta }}}{\left(\int {\int }_{\mathrm{spt}\left(\eta \right)}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p+\beta }{2}}}\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{\beta +2}{p+\beta }}}.\hfill \end{array}$$

(14)

Proof. 

To arrive at (14), we re-estimate the term ${\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}{\left|Ru\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t$ in (12) via Hölder’s inequality as below:

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}{\left|Ru\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {\left(\int {\int }_{\mathrm{spt}\left(\eta \right)}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p+\beta }{2}}}\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{p-2+\beta }{p+\beta }}}{\left({\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^{p+\beta }{\eta }^{p+\beta }\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{2}{p+\beta }}}=:U^{{\displaystyle \frac{p-2+\beta }{p+\beta }}}{V}^{{\displaystyle \frac{2}{p+\beta }}}.\hfill \end{array}$$

Noting

$$\begin{array}{cc}\hfill {\left|Ru\right|}^{p+\beta }& ={\left|Ru\right|}^{p-2+\beta }{\left(Ru\right)}^2\hfill \\ \hfill & ={\left|Ru\right|}^{p-2+\beta }Ru(X_1{X}_{n+1}u-X_{n+1}{X}_{1}u)(\mathrm{commutator}R=[X_1,X_{n+1}]),\hfill \end{array}$$

we rewrite V via integration by parts as below:

$$\begin{array}{cc}\hfill V=& -(p-2-\beta ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^{p-3+\beta }Ru{\eta }^{p+\beta }(X_1\left|Ru\right|X_{n+1}u-X_{n+1}\left|Ru\right|X_{1}u)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^{p-2+\beta }{\eta }^{p+\beta }(X_{1}Ru{X}_{n+1}u-X_{n+1}Ru{X}_{1}u)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -(p+\beta ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^{p-2+\beta }Ru{\eta }^{p-1+\beta }(X_1\eta X_{n+1}u-X_{n+1}\eta X_{1}u)\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

which implies

$$\begin{array}{cc}\hfill \left|V\right|\le & 2(p+\beta ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Xu\right|\left|Ru\right|}^{p-2+\beta }\left|XRu\right|{\eta }^{p+\beta }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +2(p+\beta ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Xu\right|\left|Ru\right|}^{p-1+\beta }\left|XRu\right|\left|X\eta \right|{\eta }^{p-1+\beta }\mathrm{d}x\mathrm{d}t=:I_1+I_2.\hfill \end{array}$$

(15)

For $I_1$, Hölder’s inequality with exponents $\left({\displaystyle \frac{2(p+\beta )}{2p-4+\beta }},{\displaystyle \frac{2(p+\beta )}{4-p}}\right)$ yields

$$I_1\le 2(p+\beta )W^{{\displaystyle \frac{1}{2}}}{U}^{{\displaystyle \frac{4-p}{2(p+\beta )}}}{V}^{{\displaystyle \frac{2p-4+\beta }{2(p+\beta )}}},$$

(16)

where

$$W:={\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Xu\right|}^{p-2}{\left|Ru\right|}^{\beta }{\left|XRu\right|}^2{\eta }^{4+\beta }\mathrm{d}x\mathrm{d}t.$$

For $I_2$, Hölder’s inequality yields

$$I_2\le 2(p+\beta ){\parallel X\eta \parallel }_{L^{\infty }}{U}^{{\displaystyle \frac{1}{p+\beta }}}{V}^{{\displaystyle \frac{p-1+\beta }{p+\beta }}}.$$

(17)

Invoking Lemma 1 to bound W, we obtain

$$W\le {C\parallel X\eta \parallel }_{L^{\infty }}^2{U}^{{\displaystyle \frac{p-2}{p+\beta }}}{V}^{{\displaystyle \frac{\beta +2}{p+\beta }}}+C\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}{\left|\mathrm{spt}\left(\eta \right)\right|}^{{\displaystyle \frac{p-2}{p+\beta }}}{V}^{{\displaystyle \frac{\beta +2}{p+\beta }}}+C{\parallel \eta \parallel }_{L^{\infty }}^4{U}^{{\displaystyle \frac{p}{p+\beta }}}{V}^{{\displaystyle \frac{\beta }{p+\beta }}}.$$

Substituting this into (16) yields refined estimates for $I_1$,

$$\begin{array}{cc}\hfill I_1\le & {C(p+\beta )\parallel X\eta \parallel }_{L^{\infty }}{U}^{{\displaystyle \frac{1}{p+\beta }}}{V}^{{\displaystyle \frac{p-1+\beta }{p+\beta }}}\hfill \\ \hfill & +C(p+\beta )\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}^{{\displaystyle \frac{1}{2}}}{\left|\mathrm{spt}\left(\eta \right)\right|}^{{\displaystyle \frac{p-2}{2(p+\beta )}}}{U}^{{\displaystyle \frac{4-p}{2(p+\beta )}}}{V}^{{\displaystyle \frac{p-1+\beta }{p+\beta }}}\hfill \\ \hfill & +{C(p+\beta )\parallel \eta \parallel }_{L^{\infty }}^2{U}^{{\displaystyle \frac{2}{p+\beta }}}{V}^{{\displaystyle \frac{p-2+\beta }{p+\beta }}}.\hfill \end{array}$$

Synthesizing the estimates for $I_1$ and $I_2$ through (15) and (17), we arrive at

$$\begin{array}{cc}\hfill V\le & {C(p+\beta )\parallel X\eta \parallel }_{L^{\infty }}{U}^{{\displaystyle \frac{1}{p+\beta }}}{V}^{{\displaystyle \frac{p-1+\beta }{p+\beta }}}\hfill \\ \hfill & +C(p+\beta )\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}^{{\displaystyle \frac{1}{2}}}{\left|\mathrm{spt}\left(\eta \right)\right|}^{{\displaystyle \frac{p-2}{2(p+\beta )}}}{U}^{{\displaystyle \frac{4-p}{2(p+\beta )}}}{V}^{{\displaystyle \frac{p-1+\beta }{p+\beta }}}\hfill \\ \hfill & +{C(p+\beta )\parallel \eta \parallel }_{L^{\infty }}^2{U}^{{\displaystyle \frac{2}{p+\beta }}}{V}^{{\displaystyle \frac{p-2+\beta }{p+\beta }}}.\hfill \end{array}$$

A strategic normalization via division by $V^{{\displaystyle \frac{p-2+\beta }{p+\beta }}}$, followed by application of Young’s inequality with exponents $\left({\displaystyle \frac{p+\beta }{2}},{\displaystyle \frac{p+\beta }{p+\beta -2}}\right)$, culminates in

$$V^{{\displaystyle \frac{2}{p+\beta }}}\le C{(p+\beta )}^2\mathcal{N}{\left(\eta \right)}^2{U}^{{\displaystyle \frac{2}{p+\beta }}}+C{(p+\beta )}^2\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}{\left|\mathrm{spt}\left(\eta \right)\right|}^{{\displaystyle \frac{p-2}{p+\beta }}}{U}^{{\displaystyle \frac{4-p}{p+\beta }}},$$

(18)

which implies

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2+\beta }{2}}}{\left|Ru\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{(p+\beta )}^2\mathcal{N}\left(\eta \right)U+C{(p+\beta )}^2\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}{\left|\mathrm{spt}\left(\eta \right)\right|}^{{\displaystyle \frac{p-2}{p+\beta }}}{U}^{{\displaystyle \frac{\beta +2}{p+\beta }}},\hfill \end{array}$$

which, together with (12), yields (14). □

Summary of This Section

In this section, we establish a crucial Caccioppoli-type estimate (14) for the horizontal gradient $Xu$ involving the second-order horizontal derivative $XXu$ and an estimate (18) for the vertical derivative $Ru$.

5. Proof of Theorem 1

The proof of Theorem 1. 

The proof of Theorem 1 is divided into two steps.

Step 1. To simplify writing, we denote $f:={(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p+\beta }{4}}}$ for all $\beta \ge 0$. Rewriting inequality (14) yields

$$\begin{array}{cc}\hfill & {\displaystyle \underset{t_1<t<t_2}{sup}}{\int }_{\mathsf{\Omega }}{f}^{{\displaystyle \frac{2(\beta +2)}{p+\beta }}}{\eta }^2\mathrm{d}x+{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Xf\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{(p+\beta )}^9\mathcal{N}\left(\eta \right)\int {\int }_{\mathrm{spt}\left(\eta \right)}{f}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{(p+\beta )}^9\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}{\left|\mathrm{spt}\left(\eta \right)\right|}^{{\displaystyle \frac{p-2}{p+\beta }}}{\left(\int {\int }_{\mathrm{spt}\left(\eta \right)}{f}^2\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{\beta +2}{p+\beta }}},\hfill \end{array}$$

(19)

where $\eta \in C^1([0,T],C_0^{\infty }\left(\mathsf{\Omega }\right))$ is any non-negative cut-off function with ${\parallel \eta \parallel }_{L^{\infty }\left(Q\right)}\le 1$, and $\mathcal{N}\left(\eta \right)$ is defined in (7) and is the same as below.

Recall the Soberev inequality

$${\left({\int }_{\mathsf{\Omega }}{\left(f\eta \right)}^{{\displaystyle \frac{2n+2}{n}}}\mathrm{d}x\right)}^{{\displaystyle \frac{n}{n+1}}}\le {\int }_{\mathsf{\Omega }}{\left|X\left(f\eta \right)\right|}^2\mathrm{d}x.$$

By the power exponent of the first term in the left-hand side of (19) and the power exponent of the Soberev inequality, we calculate the power exponent $m:={\displaystyle \frac{2(\beta +2)}{p+\beta }}·{\displaystyle \frac{1}{n+1}}+{\displaystyle \frac{2n+2}{n}}·{\displaystyle \frac{n}{n+1}}=2+{\displaystyle \frac{2(\beta +2)}{(n+1)(p+\beta )}}$. Then, Hölder’s inequality implies

$${\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{f}^m{\eta }^{2m}\mathrm{d}x\mathrm{d}t\le {\int }_{t_1}^{t_2}{\left({\int }_{\mathsf{\Omega }}{f}^{{\displaystyle \frac{2(\beta +2)}{p+\beta }}}{\eta }^2\mathrm{d}x\right)}^{{\displaystyle \frac{1}{n+1}}}{\left({\int }_{\mathsf{\Omega }}{\left(f\eta \right)}^{{\displaystyle \frac{2n+2}{n}}}\mathrm{d}x\right)}^{{\displaystyle \frac{n}{n+1}}}\mathrm{d}t,$$

which, together with the Soberev inequality, yields

$${\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{f}^m{\eta }^{2m}\mathrm{d}x\mathrm{d}t\le C{\left(\underset{t_1<t<t_2}{sup}{\int }_{\mathsf{\Omega }}{f}^{{\displaystyle \frac{2(\beta +2)}{p+\beta }}}{\eta }^2\mathrm{d}x\right)}^{{\displaystyle \frac{1}{n+1}}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left[\right|Xf|}^2{\eta }^2+{\left|X\eta \right|}^2{f}^2]\mathrm{d}x\mathrm{d}t.$$

Integrating with (19), we derive the key estimate

$$\begin{array}{cc}\hfill {\left({\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{f}^m{\eta }^{2m}\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{n+1}{n+2}}}\le & C{(p+\beta )}^9\mathcal{N}\left(\eta \right)\int {\int }_{\mathrm{spt}\left(\eta \right)}{f}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C{(p+\beta )}^9\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}{\left|\mathrm{spt}\left(\eta \right)\right|}^{{\displaystyle \frac{p-2}{p+\beta }}}{\left(\int {\int }_{\mathrm{spt}\left(\eta \right)}{f}^2\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{\beta +2}{p+\beta }}}.\hfill \end{array}$$

Step 2. Define the parabolic cylinder $Q_{\mu ,r}:=B(x_0,r)\times (t_0-\mu r^2,t_0)\subset \mathsf{\Omega }\times (0,T)$ for all $\mu ,r>0$. For $i=0,1,2,\dots$, define sequences $r_i=(1+2^{-i})r$ and ${\beta }_i=2\left({\left({\displaystyle \frac{n+2}{n+1}}\right)}^i-1\right)$, satisfying the recurrence relation

$$p+{\beta }_{i+1}=(p+{\beta }_i)\left(1+{\displaystyle \frac{{\beta }_i+2}{(n+1)(p+{\beta }_i)}}\right).$$

Select cut-off functions ${\eta }_i\in C^{\infty }\left(Q_i\right)$ satisfying

$$\left\{\begin{array}{cc}{\eta }_i=1\hfill & \mathrm{in}{Q}_{i+1},\hfill \\ |X{\eta }_i|\le 2^{i+8}/r,|R{\eta }_i|\le 2^{2i+8}/r^2,\left|{\partial }_t{\eta }_i\right|\le 2^{2i+8}/\left(\mu r^2\right)\hfill & \mathrm{in}{Q}_i,\hfill \end{array}\right.$$

where $Q_i:=Q_{\mu ,r_i}\subset Q_{\mu ,2r_0}\subset \mathsf{\Omega }\times (0,T)$.

Letting $\eta ={\eta }_i$ and $\beta ={\beta }_i$ in the key estimate in Step 1, we have

$$\begin{array}{cc}\hfill & {\left(\int {\int }_{Q_{i+1}}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p+{\beta }_{i+1}}{2}}}\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{n+1}{n+2}}}\hfill \\ \hfill & \le C{4}^i{(p+{\beta }_i)}^9(r^{-2}+1)\left[{\left(\int {\int }_{Q_i}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p+{\beta }_i}{2}}}\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{p-2}{p+{\beta }_i}}}+{\mu }^{-1}{\left(\mu r^{2n+4}\right)}^{{\displaystyle \frac{p-2}{p+{\beta }_i}}}\right]\hfill \\ \hfill & \phantom{\le }\times {\left(\int {\int }_{Q_i}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p+{\beta }_i}{2}}}\mathrm{d}x\mathrm{d}t\right)}^{{\displaystyle \frac{{\beta }_i+2}{p+{\beta }_i}}}.\hfill \end{array}$$

Let ${\mathcal{E}}_i=max\left({\left(\int {\int }_{Q_i}{(\delta +|Xu|}^2{)}^{(p+{\beta }_i)/2}\mathrm{d}x\mathrm{d}t\right)}^{1/(p+{\beta }_i)},{\mu }^{1/(2-p)}\right)$. Then, the above inequality simplifies to

$${\mathcal{E}}_{i+1}^{{\displaystyle \frac{\left(n+1\right)(p+{\beta }_{i+1})}{n+2}}}\le C{\mu }^{{\displaystyle \frac{2}{n+2}}}{4}^i{(p+{\beta }_i)}^9{\mathcal{E}}_i^{p+{\beta }_i}.$$

Employing Moser’s iteration on the above inequality, we ultimately obtain

$$\underset{Q_{\mu ,r}}{sup}\left|Xu\right|\le \underset{i\to \infty }{lim\; sup}{\mathcal{E}}_i\le C{\mu }^{{\displaystyle \frac{1}{2}}}{\mathcal{E}}_0^{{\displaystyle \frac{p}{2}}},$$

which yields the desired inequality (4). □

6. Proof of Theorem 2

The proof of Theorem 2. 

For each parameter $\beta \in (0,1)$, we select the test function $\varphi =X_l[{\eta }^2{(\delta +|Xu|}^2{)}^{-\beta /2}{X}_{l}u]$ in Equation (1). Through integration by parts, we derive the following identity:

$$\begin{array}{cc}\hfill L^l:=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{X}_l\left[{\partial }_{t}u\right]{\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}{X}_l\left[X_i{A}_i\left(Xu\right)\right]{\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}K(x,t,u,Xu)X_l[{\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u]\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}{X}_i{X}_l{A}_i\left(Xu\right){\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t(\mathrm{Lie}\mathrm{bracket}{X}_l{X}_i=[X_l,X_i]+X_i{X}_l)\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}[X_l,X_i]A_i\left(Xu\right){\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}K(x,t,u,Xu)X_l[{\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u]\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_l{X}_{j}u{X}_i({\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u)\mathrm{d}x\mathrm{d}t(\mathrm{by}\mathrm{integration}\mathrm{by}\mathrm{parts})\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^{2n}}[X_l,X_i]A_i\left(Xu\right){\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}K(x,t,u,Xu)X_l[{\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u]\mathrm{d}x\mathrm{d}t=:-I_1^l+I_2^l-I_3^l.\hfill \end{array}$$

Considering the temporal integral term $L^l$, repeated integration by parts yields

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i=1}^{2n}}{L}^l=& {\displaystyle \sum _{i=1}^{2n}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\partial }_t{X}_{l}u{\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t(\mathrm{Lie}\mathrm{bracket}[X_l,{\partial }_t]=[{\partial }_t,X_l])\hfill \\ \hfill =& {\displaystyle \frac{1}{2-\beta }}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\partial }_t\left(\right(\delta +{\left|Xu\right|}^2{)}^{{\displaystyle \frac{2-\beta }{2}}}){\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -{\displaystyle \frac{2}{2-\beta }}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{2-\beta }{2}}}\eta {\partial }_t\eta \mathrm{d}x\mathrm{d}t(\mathrm{by}\mathrm{integration}\mathrm{by}\mathrm{parts}),\hfill \end{array}$$

which establishes the temporal estimate

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

The principal term $I_1^l$ decomposes into three components through differentiation,

$$\begin{array}{cc}\hfill I_1^l=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_l{X}_{j}u2\eta X_i{\eta (\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\displaystyle \frac{\beta }{2}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_l{X}_{j}u{\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}-1}{X}_i{\left(\right|Xu|}^2)X_{l}u\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,j=1}^{2n}}{A}_{i,{\xi }_j}\left(Xu\right)X_l{X}_{j}u{\eta }^2{(\delta +|Xu|}^2{)}^{-{\displaystyle \frac{\beta }{2}}}{X}_i{X}_{l}u\mathrm{d}x\mathrm{d}t=:I_{11}^l-I_{12}^l+I_{13}^l.\hfill \end{array}$$

Invoking the structural condition (2), we bound the first component

$$|I_{11}^l|\le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-1-\beta }{2}}}\left|XXu\right|\eta \left|X\eta \right|\mathrm{d}x\mathrm{d}t.$$

Applying the commutator relation $X_l{X}_j=[X_l,X_j]+X_j{X}_l$ to $I_{12}^l$ produces

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

The commutator term $I_{121}^l$ satisfies

$$|I_{121}^l|\le C\beta {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}\left|Ru\right|\left|XXu\right|{\eta }^2\mathrm{d}x\mathrm{d}t.$$

while the principal term $I_{122}^l$ provides the crucial positive contribution

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

Similar decomposition techniques applied to $I_{13}^l$ yield

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

with corresponding estimates

$$|I_{121}^l|\le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}\left|Ru\right|\left|XXu\right|{\eta }^2\mathrm{d}x\mathrm{d}t,$$

and

$${\displaystyle \sum _{l=1}^{2n}}{I}_{132}^l\ge \lambda {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}{\left|XXu\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t.$$

For the commutator term $I_2^l$ with $l=1,\dots ,n$, integration by parts and structural conditions yield

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

with component estimates

$$|I_{21}^l|\le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-\beta }{2}}}\left|\eta R\eta \right|\mathrm{d}x\mathrm{d}t,$$

and

$$\begin{array}{cc}\hfill |I_{22}^l|\le & C{(\beta +1)}^2{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}\left|XXu\right|\left|Ru\right|{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C\beta {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-1-\beta }{2}}}\left|Ru\right|\eta \left|X\eta \right|\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Combining these estimates with analogous bounds for $I_{23}^l$, we obtain

$$\begin{array}{cc}\hfill |I_2^l|\le & C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-\beta }{2}}}\left|\eta R\eta \right|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C{(\beta +1)}^2{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-2-\beta }{2}}}\left|XXu\right|\left|Ru\right|{\eta }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C(\beta +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-1-\beta }{2}}}\left|Ru\right|\eta \left|X\eta \right|\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

The remaining term $I_3^l$ satisfies

$$\begin{array}{cc}\hfill |I_3^l|\le & C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-\beta }{2}}}\eta \left|X\eta \right|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C(\beta +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{p-1-\beta }{2}}}\left|XXu\right|{\eta }^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Synthesizing the above equations, we derive the key inequality

$$\begin{array}{c}\hfill I_{132}^l-I_{122}^l=-L^l-I_{11}^l+I_{121}^l-I_{131}^l+I_2^l-I_3^l.\end{array}$$

Combining all estimates, together with $\left|X\right|Xu\left|\right|\le \left|XXu\right|$, and applying Young’s inequality with parameter $\beta =p-2$ for $2\le p<3$ yields

$$\begin{array}{cc}\hfill (3-p){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|XXu\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le & {\displaystyle \frac{C}{4-p}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{(\delta +|Xu|}^2{)}^{{\displaystyle \frac{4-p}{2}}}\left|\eta {\partial }_t\eta \right|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C\mathcal{N}\left(\eta \right)\int {\int }_{\mathrm{spt}\left(\eta \right)}{(\delta +|Xu|}^2)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

where $\mathcal{N}\left(\eta \right)$ is defined in (7) and is the same as below.

Finally, applying Hölder’s inequality and (18) with $\beta =4-p$ to the last term produces

$${\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\left|Ru\right|}^2{\eta }^2\mathrm{d}x\mathrm{d}t\le C\mathcal{N}\left(\eta \right)\left|\mathrm{spt}\left(\eta \right)\right|\omega +C\parallel \eta {\partial }_t{\eta \parallel }_{L^{\infty }}\left|\mathrm{spt}\left(\eta \right)\right|{\omega }^{{\displaystyle \frac{4-p}{2}}},$$

(20)

where $\omega ={sup}_{\mathrm{spt}\left(\eta \right)}{(\delta +|Xu|}^2)$. This completes the proof of the main estimate (5).

□

7. Conclusions

In the Heisenberg group ${\mathbb{H}}^n$, we establish the regularity theory for the weak solution u to the non-homogeneous degenerate nonlinear parabolic equation ${\partial }_{t}u-{\displaystyle \sum _{i=1}^{2n}}{X}_i{A}_i\left(Xu\right)=K(x,t,u,Xu)$ modeled on the non-homogeneous parabolic p-Laplacian equation. In conclusion, we establish the local $C^{0,1}$-regularity for $2\le p\le 4$ and the local second-order $H{W}_{\mathrm{loc}}^{2,2}$-regularity for $2\le p<3$. These results solve the open problem (1) in paper [26].

Notably, the condition $p\in [2,4]$ is essential in the proof of Lemma 3, as we rely on Hölder’s inequality to derive (16). This restriction currently prevents us from addressing the cases $p\in (1,2)$ and $p\in (1,2)$. Consequently, our future efforts will be focused on the challenging task of establishing regularities for the ranges $p\in (1,2)\cup (4,\infty )$.

We formulate two pivotal open problems to guide subsequent investigations:

(i)

Establish the corresponding regularity results, by improving the above methods and techniques, when the non-homogeneous term $K(·)$ does not satisfy condition (2).

(ii)

Extend the above regularity results to more general sub-Riemannian manifolds by improving the above methods and techniques.