H\"older regularity for the linearized porous medium equation in bounded domains

TL;DR

This paper establishes H"older regularity for weak solutions of a linearized porous medium equation in bounded domains, addressing singular and degenerate cases with boundary conditions.

Contribution

It provides the first systematic proof of H"older regularity for solutions of linearized porous medium equations with boundary conditions.

Findings

Weak solutions are H"older continuous in bounded domains.

Regularity results apply to both divergence and nondivergence forms.

Addresses singular and degenerate cases of the equation.

Abstract

In this paper, we systematically study weak solutions of a linear singular or degenerate parabolic equation in a mixed divergence form and nondivergence form, which arises from the linearized fast diffusion equation and the linearized porous medium equation with the homogeneous Dirichlet boundary condition. We prove the H\"older regularity of their weak solutions.

Equations

$$p>-1$$

$$\begin{array}{rl}a{\omega }^p{\partial }_{t}u-D_j(a_{ij}{D}_{i}u+d_{j}u)+b_i{D}_{i}u+{\omega }^{p}cu+c_{0}u& ={\omega }^{p}f+f_0-D_i{f}_i\text{mbox}in\Omega \times (-1,0],\\ u& =0\text{mbox}on\partial \Omega \times (-1,0],\end{array}$$

$$\forall (x,t)\in \Omega \times [-1,0],\lambda \le a(x,t)\le \Lambda ,\lambda |\xi {|}^2\le \sum _{i,j=1}^n{a}_{ij}(x,t){\xi }_i{\xi }_j\le \Lambda |\xi {|}^2\forall \xi \in {\mathbb{R}}^n,$$

$$\begin{array}{rl}{\partial }_t{v}^{p+1}& =\Delta v\text{mbox}in\Omega \times (0,\infty ),\\ v& =0\text{mbox}on\partial \Omega \times (0,\infty ).\end{array}$$

$$0<\underset{\Omega }{\inf }\frac{v(t,x)}{d(x)}\le \underset{\Omega }{\sup }\frac{v(t,x)}{d(x)}<\infty$$

$$Q_R^{+}(\bar{x},\bar{t})=B_R^{+}(\bar{x})\times [\bar{t}-R^2,\bar{t}],{\mathcal{Q}}_R^{+}(\bar{x},\bar{t})=B_R^{+}(\bar{x})\times [\bar{t}-R^{p+2},\bar{t}].$$

$$a{x}_n^p{\partial }_{t}u-D_j(a_{ij}{D}_{i}u+d_{j}u)+b_i{D}_{i}u+c{x}_n^{p}u+c_{0}u=x_n^{p}f+f_0-D_i{f}_i\text{mbox}in{Q}_1^{+}$$

$$u=0\text{mbox}on{\partial }^{\prime }{B}_1^{+}\times [-1,0],$$

$${\partial }^{\prime }{B}_R^{+}=B_R\cap \{x_n=0\}.$$

$${\partial }^{\prime \prime }{B}_R^{+}=\partial B_R^{+}\setminus {\partial }^{\prime }{B}_R^{+},$$

$${\partial }_{pa}{Q}_R^{+}\text{mbox}asthestandardparabolicboundaryof{Q}_R^{+}.$$

$$Q_{R,T}^{+}=B_R^{+}\times (-T,0].$$

$$W_2^{1,1}(Q_{R,T}^{+})$$

$$\Vert g{\Vert }_{W_2^{1,1}(Q_{R,T}^{+})}$$

$$V_2^{1,1}(Q_{R,T}^{+}):=\{g\in L^2(Q_{R,T}^{+}):{\partial }_{t}g\in L^2(Q_{R,T}^{+},x_n^p\mathrm{d}x\mathrm{d}t),D_{i}g\in L^2(Q_{R,T}^{+}),i=1,\cdots ,n\},$$

$$\Vert g{\Vert }_{V_2^{1,1}(Q_{R,T}^{+})}:=\Vert g{\Vert }_{L^2(Q_{R,T}^{+})}+\Vert {\partial }_{t}g{\Vert }_{L^2(Q_{R,T}^{+},x_n^p\mathrm{d}x\mathrm{d}t)}+\sum _{i=1}^n\Vert D_{i}g{\Vert }_{L^2(Q_{R,T}^{+})}$$

$$V_2(Q_{R,T}^{+})$$

$$\Vert u{\Vert }_{V_2(Q_{R,T}^{+})}$$

$$V_2^{1,0}(Q_{R,T}^{+})=C([-T,0];L^2(B_R^{+},x_n^p\mathrm{d}x))\cap L^2((-T,0];H^1(B_R^{+}))$$

$$W_2^{1,1}(Q_{R,T}^{+})\subset V_2^{1,1}(Q_{R,T}^{+})\subset V_2^{1,0}(Q_{R,T}^{+})\subset V_2(Q_{R,T}^{+}).$$

$$V_2^{1,1}(Q_{R,T}^{+})\subset W_2^{1,1}(Q_{R,T}^{+}),V_2^{1,1}(Q_{R,T}^{+})\subset V_2^{1,0}(Q_{R,T}^{+})\subset V_2(Q_{R,T}^{+}).$$

$${\mathring{W}}_2^{1,1}(Q_{R,T}^{+}),{\mathring{V}}_2^{1,1}(Q_{R,T}^{+}),{\mathring{V}}_2^{1,0}(Q_{R,T}^{+}),{\mathring{V}}_2(Q_{R,T}^{+})$$

$$W_2^{1,1}(Q_{R,T}^{+}),V_2^{1,1}(Q_{R,T}^{+}),V_2^{1,0}(Q_{R,T}^{+}),V_2(Q_{R,T}^{+})\text{mbox}vanishinga.e.on\partial B_R^{+}\times [-T,0]$$

$${\phi }_{\epsilon }(x,t):=e^{-\epsilon /x_n}\phi (x,t).$$

$${\partial }_t{\phi }_{\epsilon }=e^{-\epsilon /x_n}{\partial }_t\phi ,D_i{\phi }_{\epsilon }=e^{-\epsilon /x_n}{D}_i\phi ,i=1,\cdots ,n-1;$$

$$D_n{\phi }_{\epsilon }=e^{-\epsilon /x_n}{D}_n\phi +\frac{\epsilon e^{-\epsilon /x_n}}{x_n}\frac{\phi }{x_n}.$$

$${\int }_{Q_{R,T}^{+}}\frac{{\phi }^2}{x_n^2}\mathrm{d}x\mathrm{d}t\le C{\int }_{Q_{R,T}^{+}}|\nabla \phi {|}^2\mathrm{d}x\mathrm{d}t.$$

$$(u-k{)}^{+}:=\max (u-k,0)\in V_2^{1,0}(Q_{R,T}^{+}).$$

$$|(r_1-k{)}^{+}-(r_2-k{)}^{+}|\le |r_1-r_2|.$$

$$\Vert (u-k{)}^{+}(\cdot ,t+h)-(u-k{)}^{+}(\cdot ,t){\Vert }_{L^2(B_R^{+},x_n^p\mathrm{d}x)}\le \Vert u(\cdot ,t+h)-u(\cdot ,t){\Vert }_{L^2(B_R^{+},x_n^p\mathrm{d}x)}.$$

$$(u_j-k{)}^{+}\to (u-k{)}^{+}\text{mbox}in{V}_2^{1,0}(Q_{R,T}^{+})\text{mbox}asj\to \infty .$$

$$u_h(x,t)=\frac{1}{h}{\int }_{t-h}^{t}u(x,s)\mathrm{d}s$$

Full text

1 Introduction

Let $\Omega\subset\mathbb{R}^{n}$, $n\geq 1$, be a smooth bounded open set, and $\omega$ be a smooth function in $\overline{\Omega}$ comparable to the distance function $d(x):=\operatorname{dist}(x,\partial\Omega)$, that is, $0<\inf_{\Omega}\frac{\omega}{d}\leq\sup_{\Omega}\frac{\omega}{d}<\infty$. For example, $\omega$ can be taken as the positive normalized first eigenfunction of $-\Delta$ in $\Omega$ with Dirichlet zero boundary condition. Let

[TABLE]

be a fixed constant throughout the paper unless otherwise stated.

In this paper, we would like to study regularity of weak solutions to

[TABLE]

where all $a,a_{ij},d_{j},b_{i},c,c_{0},f,f_{0},f_{i}$ are functions of $(x,t)$, $D_{i}=\partial_{x_{i}}$, and the summation convention is used. Throughout this paper, we always assume the ellipticity condition, that is, $(a_{ij})$ is a matrix satisfying

[TABLE]

where $0<\lambda\leq\Lambda<\infty$.

The study of the equation (2) is motivated by the linearized equation of the fast diffusion equations (corresponding to $p>0$ in (4)) or slow diffusion equations (corresponding to $-1<p<0$ in (4), which are also called porous medium equations)

[TABLE]

From DiBenedetto-Kwong-Vespri [7], we know that the solution $v$ of (4) with $p>0$ satisfies the global Harnack inequality

[TABLE]

before its extinction time. See Bonforte-Figalli [2] for a survey. From Aronson-Peletier [1], we also know that the solution $v$ of (4) with $-1<p<0$ satisfies (5) as well after certain waiting time. Therefore, the linearized equation of (4), which plays an important role in proving optimal regularity of solutions to (4) in [19, 20, 21], falls into a form of the equation (2). In our earlier work [19], we have obtained many properties for equations like (2) with $p>0$, such as well-posedness, local boundedness and Schauder estimates. In this paper, we study the equation (2) in a more general and systematic way. The main goal of this paper is the Hölder regularity of its weak solutions to (2) up to the boundary $\{x_{n}=0\}$.

After the De Giorgi-Nash-Moser theory on the Hölder regularity for uniformly elliptic and uniformly parabolic equations, there have been many investigations on regularity for degenerate or singular elliptic and parabolic equations. By the work of Fabes-Kenig-Serapioni [13], we still have Hölder regularity for elliptic equations whose coefficients are of $A_{2}$ weight. See also earlier work of Kruzkov [23], Murthy-Stampacchia [24], Trudinger [27, 28], as well as recent work Sire-Terracini-Vita [25, 26] and Wang-Wang-Yin-Zhou [29], on degenerate elliptic equations. However, Chiarenza-Serapioni [3] provided several counterexamples showing that the aforementioned elliptic results do not carry over directly to the parabolic case. Nevertheless, Hölder regularity and Harnack inequality for degenerate or singular parabolic equations with various conditions and structures have been obtained in, e.g., Chiarenza-Serapioni [4, 5] and Gutiérrez-Wheeden [16, 17], with either the same weight or different weights of singular/degenerate coefficients of $u_{t}$ and $D^{2}u$. Recently, in a series of papers [8, 9, 10, 11], Dong-Phan obtained results on the wellposedness and regularity estimates in weighted Sobolev spaces for parabolic equations with singular-degenerate coefficients, where the weights of singular/degenerate coefficients of $u_{t}$ and $D^{2}u$ appeared in a balanced way. Such Sobolev regularity was obtained later in Dong-Phan-Tran [12] for equations similar to our equation (2) for $-2<p<0$. Note that although our results on the boundedness of the weak solutions hold for $p>-2$ as well, our Hölder regularity results require the assumption (1) that $p>-1$, and thus, $x_{n}^{p}$ is locally integrable. The assumption (1) is used in Proposition 2.11, and also in the beginning of Section 5.1 when defining the measure $\mu_{p}$, that is the natural choice to measure the improvement of the oscillation of the solution. We need the measure $\mu_{p}$ to be locally finite in this step. Hölder estimates and Schauder estimates for $p=-1$ with a special structure that the coefficients in the drift terms are positive have been studied in Daskalopoulos-Hamilton [6], Koch [22] and Feehan-Pop [14]. The literature on regularity theory for degenerate elliptic and parabolic equations is vast, and one can refer to the above papers for more references.

Under the condition (3), the equation (2) is uniformly parabolic (in a mixed divergence and nondivergence form) when $x$ stays away from the boundary $\partial\Omega$. Therefore, to obtain global estimates for (2), we need to establish estimates near $\partial\Omega$, that is in $(B_{r}(x_{0})\cap\Omega)\times(-1,0]$, where $x_{0}\in\partial\Omega,r>0$ and $B_{r}(x_{0})$ is the open ball in $\mathbb{R}^{n}$ centered at $x_{0}$ with radius $r$. By the standard flattening the boundary techniques for studying boundary estimates, we only need to consider the equation in the half ball case.

Now we suppose $\Omega$ is a half ball. For $\bar{x}=(\bar{x}^{\prime},0)$, denote $B_{R}^{+}(\bar{x})=B_{R}(\bar{x})\cap\{(x^{\prime},x_{n}):x_{n}>0\}$,

[TABLE]

For brevity, we drop $(\bar{x})$ and $(\bar{x},\bar{t})$ in the above notations if $\bar{x}=0$ or $(\bar{x},\bar{t})=(0,0)$.

Consider the equation

[TABLE]

with partial Dirichlet condition

[TABLE]

where

[TABLE]

We also denote

[TABLE]

and

[TABLE]

We establish Hölder regularity estimates for solutions of (6) and (7) up to the boundary $\{x_{n}=0\}$, that is, in $\overline{B}_{1/2}^{+}\times[-1/2,0]$. If it additionally satisfies $u(\cdot,-1)=0$, then we also establish Hölder regularity up to the initial time, that is, in $\overline{B}_{1/2}^{+}\times[-1,0]$.

Our results are scattered in the following four sections.

• •

In Section 2, we introduce a corresponding weighted Sobolev space. We prove a weighted parabolic Sobolev inequality in Theorem 2.9 and Theorem 2.10, and a De Giorgi type isoperimetric inequality in Theorem 2.12.

• •

In Section 3, we introduce the definition of weak solutions in Definition 3.1, and establish the wellposedness in Theorem 3.7.

• •

in Section 4, we prove the local-in-time boundedness up to $\{x_{n}=0\}$ of weak solutions in Theorems 4.3, and space-time global boundedness in Theorem 4.5,

• •

In Section 5, we prove local-in-time Hölder estimates up to $\{x_{n}=0\}$ of weak solutions in Theorems 5.11, and space-time global Hölder estimates in Theorem 5.15. In the end of the paper, we show the well-posedness of the Cauchy-Dirichlet problem (2).

Our proof of the boundedness and Hölder estimates of weak solutions uses the De Giorgi iteration. The local-in-time boundedness and Hölder estimates for (6) with $-1<p<1$ and $a\equiv 1$ but without lower order terms follow from Gutiérrez-Wheeden [16, 17].

2 Sobolev spaces and inequalities

2.1 Some weighted Sobolev spaces

In this section, we will introduce several Sobolev spaces that will be needed to define and study weak solutions of (6). Denote

[TABLE]

Let

[TABLE]

be the standard Sobolev space with the standard Sobolev norm.

Let $p>-1$. Let

[TABLE]

be a weighted Sobolev space, with the weight $x_{n}^{p}$ only applied on $\partial_{t}g$. Let

[TABLE]

and

[TABLE]

be a subspace of $V_{2}(Q^{+}_{R,T})$ endowed with the norm (11).

Then all of $W^{1,1}_{2}(Q^{+}_{R,T})$, $V^{1,1}_{2}(Q^{+}_{R,T})$, $V_{2}^{1,0}(Q^{+}_{R,T})$ and $V_{2}(Q^{+}_{R,T})$ are Banach spaces. If $p\geq 0$, then

[TABLE]

If $-1<p<0$, then

[TABLE]

In fact, $V_{2}^{1,0}(Q^{+}_{R,T})$ is the closure of $V^{1,1}_{2}(Q^{+}_{R,T})$ under the norm $\|\cdot\|_{V_{2}(Q^{+}_{R,T})}$.

We also denote

[TABLE]

as the set of functions in

[TABLE]

in the trace sense, respectively.

Lemma 2.1**.**

For $p>0$, $\mathring{W}^{1,1}_{2}(Q^{+}_{R,T})$ is dense in $\mathring{V}^{1,1}_{2}(Q^{+}_{R,T})$.

Proof.

For $\varphi\in\mathring{V}^{1,1}_{2}(Q^{+}_{R,T})$ and $\varepsilon>0$, let

[TABLE]

Then

[TABLE]

and

[TABLE]

Hence, $\varphi_{\varepsilon}\in\mathring{W}^{1,1}_{2}(Q_{1}^{+})$. By Hardy’s inequality, we have

[TABLE]

Therefore, it follows from Lebesgue’s dominated convergence theorem that $\|\varphi_{\varepsilon}-\varphi\|_{V^{1,1}_{2}(Q_{1}^{+})}\to 0$ as $\varepsilon\to 0^{+}$. ∎

This density fact will be used for the existence of weak solutions to (6) (see Theorem 3.7).

Lemma 2.2**.**

Let $u\in V_{2}^{1,0}(Q^{+}_{R,T})$. Then for every $k\in\mathbb{R}$,

[TABLE]

Proof.

It is clear that $(u-k)^{+}\in V_{2}(Q^{+}_{R,T})$. For two real numbers $r_{1}$ and $r_{2}$, we have the pointwise estimate

[TABLE]

Hence

[TABLE]

Since $u\in C([-T,0];L^{2}(B_{R}^{+},x_{n}^{p}\mathrm{d}x))$, then $(u-k)^{+}\in C([-T,0];L^{2}(B_{R}^{+},x_{n}^{p}\mathrm{d}x))$ as well. ∎

Lemma 2.3**.**

Suppose $\{u_{j}\}\subset V_{2}^{1,0}(Q^{+}_{R,T})$ converges to $u$ in $V_{2}^{1,0}(Q^{+}_{R,T})$. Then for every $k\in\mathbb{R}$,

[TABLE]

Proof.

It follows from (13). ∎

Denote

[TABLE]

as the Steklov average of $u$.

Lemma 2.4**.**

Let $u\in V_{2}^{1,0}(Q^{+}_{R,T})$, and $\delta\in(0,T)$. Then for every $h\in(0,\delta)$, $u_{h}\in V^{1,1}_{2}(Q^{+}_{R,T-\delta})$, and

[TABLE]

Proof.

It is straightforward to verify that $u_{h}\in V^{1,1}_{2}(Q^{+}_{R,T-\delta})$. Also, by the Minkowski inequality, we have

[TABLE]

where $u\in V_{2}^{1,0}(Q^{+}_{R,T})$ is used in the last inequality. Similarly,

[TABLE]

where we used the continuity of Lebesgue integrals with respect to translations in the last inequality. ∎

2.2 Sobolev inequalities

Next, we will prove a Sobolev inequality for functions in $\mathring{V}_{2}(Q^{+}_{R,T})$ (in fact, in a slightly larger space). To accommodate the partial boundary condition (7), we define the following space:

[TABLE]

Then we have the well-known Hardy inequality.

Lemma 2.5** (Hardy’s inequality).**

For every $u\in H^{1}_{0,L}(B_{R}^{+})$, there holds

[TABLE]

Consequently, we have

Lemma 2.6**.**

Let $p>0$. For every $u\in H^{1}_{0,L}(B_{R}^{+})$ and every $\varepsilon>0$, there holds

[TABLE]

Proof.

We have

[TABLE]

where we used Hölder’s inequality, Young’s inequality and Lemma 2.5. ∎

By the usual Sobolev inequality, Hardy’s inequality, Hölder’s inequality, and a scaling argument, we have the following Sobolev inequality for functions in $H^{1}_{0,L}(B_{R}^{+})$.

Lemma 2.7** (Sobolev’s inequality).**

There exists $C>0$ depending only on $n$ such that for every $u\in H^{1}_{0,L}(B_{R}^{+})$, there holds

[TABLE]

Combining Hardy’s inequality and Sobolev’s inequality, we have the following Hardy-Sobolev inequality for functions in $H^{1}_{0,L}(B_{R}^{+})$.

Lemma 2.8** (Hardy-Sobolev inequality).**

Let $s\in(0,2)$. Then

[TABLE]

when $n\geq 3$, and for $s\leq r<\infty$,

[TABLE]

when $n=1,2$.

Proof.

By scaling, we only need to prove for $R=1$. If $n\geq 3$, using the Hölder inequality, Hardy inequality and Sobolev inequality, we have

[TABLE]

If $n=1,2$, we have

[TABLE]

Therefore, we complete the proof. ∎

The next theorem is a mild generalization of Lemma 2.2 in [19].

Theorem 2.9**.**

Let $p\geq 0$. For every $u\in L^{\infty}((-T,0];L^{2}(B_{R}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-T,0];H^{1}_{0,L}(B_{R}^{+}))$ (in particular, $u\in\mathring{V}_{2}(Q^{+}_{R,T})$), we have

[TABLE]

where $\chi=\frac{n+p+2}{n+p}$ and $C$ depends only on $n$ and $p$ if $n\geq 3$; while $\chi=\frac{p+2}{p+1}$ and $C=C(p)R^{\frac{p+2-n}{p+2}}$ with the constant $C(p)$ depending only on $p$ if $n=1,2$.

Proof.

We prove the case $n\geq 3$ first. Let $s\in(0,2)$ be such that $\frac{s(n-2)}{2-s}=p$. By (14) and the Hölder inequality, we have

[TABLE]

Integrating the above inequality in $t$, we have

[TABLE]

where we have used the Young inequality in the last inequality.

If $n=1,2$, using (15) and the Hölder inequality, we have

[TABLE]

Integrating the above inequality in $t$, we have

[TABLE]

where we have used the Young inequality in the last inequality. ∎

For $-2<p<0$, then we have another parabolic Sobolev inequality.

Theorem 2.10**.**

For every $u\in L^{\infty}((-T,0];L^{2}(B_{R}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-T,0];H^{1}_{0,L}(B_{R}^{+}))$ (in particular, $u\in\mathring{V}_{2}(Q^{+}_{R,T})$), where $-2<p<0$, we have

[TABLE]

where $\chi=\frac{n+2p+2}{n+p}$ and $C$ depends only on $n$ and $p$ if $n\geq 3$; while $\chi=\frac{3}{2}$ and $C=C(p)R^{\frac{p+4-n}{3}}$ with the constant $C(p)$ depending only on $p$ if $n=1,2$.

Proof.

We prove the case $n\geq 3$ first. By (14) and the Hölder inequality, we have

[TABLE]

Integrating the above inequality in $t$, we have

[TABLE]

where we have used the Young inequality in the last inequality.

If $n=1,2$, using (15) and the Hölder inequality, we have

[TABLE]

Integrating the above inequality in $t$, we have

[TABLE]

where we have used the Young inequality in the last inequality. Note that

[TABLE]

if $-2<p<0$ and $n=1,2$. ∎

Uing the idea of Fabes-Kenig-Serapioni [13], we have the following Poincaré inequality.

Proposition 2.11**.**

Let $n\geq 1$, $p>-1$ and $r>0$. Then there exists a constant $C>0$ depending only on $n$ and $p$ such that

[TABLE]

for all $u\in H^{1}(B_{r})$, where

[TABLE]

Proof.

By scaling, we only need to prove it for $r=1$. By a density argument, we only need to show it for Lipschitz continuous (in $\overline{B}_{1}$) functions.

Using the triangle inequality and Lemma 1.4 of Fabes-Kenig-Serapioni [13], we have for all $x\in B_{1}$ that

[TABLE]

where

[TABLE]

Then

[TABLE]

Since

[TABLE]

where we used $p>-1$ in the last inequality, we have

[TABLE]

Then the conclusion follows from the fact that

[TABLE]

∎

The last inequality is a De Giorgi type isoperimetric inequality.

Theorem 2.12**.**

Let $p>-1$, $k<\ell,r>0$ and $u\in H^{1}(B_{r})$. For every $0<\varepsilon<\min\left(\frac{1}{2},\frac{1}{2(p+1)}\right),$ there exists a positive constant $C$ depending only on $n,p$ and $\varepsilon$ such that

[TABLE]

Proof.

Let

[TABLE]

Then by Proposition 2.11,

[TABLE]

Using Hölder’s inequality, we have

[TABLE]

where we used $0<\varepsilon<\min\left(\frac{1}{2},\frac{1}{2(p+1)}\right),$ so that we can use Hölder’s inequality and $|x_{n}|^{-\frac{2p\varepsilon}{p-2\varepsilon}}$ is integrable.

On the other hand, we have

[TABLE]

Hence, the conclusion follows. ∎

3 Weak solutions

3.1 Definitions

Regarding the coefficients of the equation (6), besides (1), we assume that

• •

there exist $0<\lambda\leq\Lambda<\infty$ such that

[TABLE]

• •

[TABLE]

for some $q>\frac{\chi}{\chi-1}$;

• •

[TABLE]

where $\chi>1$ is the constant in Theorem 2.9 or Theorem 2.10 depending on the value of $p$.

Definition 3.1**.**

We say $u$ is a weak solution of (6) with the partial boundary condition (7) if $u\in C((-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ and satisfies

[TABLE]

for every $\varphi\in\mathring{V}^{1,1}_{2}(Q^{+}_{1})$ satisfying $\varphi(\cdot,-1)\equiv 0$ in $B_{1}^{+}$ (in the trace sense).

Using Theorem 2.9 and Theorem 2.10, one can verify that under the assumptions (1), (16), (17) and (18), each integral in (19) is finite.

Definition 3.2**.**

We say that $u$ is a weak solution of (6) with the partial boundary condition (7) and the initial condition $u(\cdot,-1)\equiv 0$, if $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$, $u(\cdot,-1)\equiv 0$, and satisfies (19) for all $\varphi\in\mathring{V}^{1,1}_{2}(Q^{+}_{1})$.

Definition 3.3**.**

We say that $u$ is a weak solution of (6) with the full boundary condition $u\equiv 0$ on $\partial_{pa}Q_{1}^{+}$, if $u\in\mathring{V}_{2}^{1,0}(Q^{+}_{1})$, $u(\cdot,-1)\equiv 0$, and satisfies (19) for all $\varphi\in\mathring{V}^{1,1}_{2}(Q^{+}_{1})$.

Definition 3.4**.**

Let $g\in V^{1,1}_{2}(Q_{1}^{+})$. We say that $u$ is a weak solution of (6) with the inhomogeneous boundary condition $u\equiv g$ on $\partial_{pa}Q_{1}^{+}$, if $u\in V_{2}^{1,0}(Q^{+}_{1})$, $u=g$ on $\partial_{pa}Q_{1}^{+}$, and $v:=u-g$ is a weak solution of

[TABLE]

with homogeneous boundary condition $v\equiv 0$ on $\partial_{pa}Q_{1}^{+}$.

3.2 Energy estimates, uniqueness and existence

We start with energy estimates.

Lemma 3.5**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7), where the coefficients of the equation satisfy (1), (16), (17) and (18). Let $k\geq\sup_{\partial_{pa}Q_{1}^{+}}|u|$ and $\varphi=(u-k)^{+}$. Then there exists $C>0$ depending only on $n,p,\lambda,\Lambda$ such that

[TABLE]

Proof.

If $u\in V^{1,1}_{2}(Q^{+}_{1})$ (cf. (9)), then $\varphi\in\mathring{V}^{1,1}_{2}(Q^{+}_{1})$ and $\varphi(\cdot,-1)\equiv 0$ in $B_{1}^{+}$. Then (20) follows from (19), by using (16) and Hölder’s inequality.

In the following, we will show that we do not need to assume $u\in V^{1,1}_{2}(Q^{+}_{1})$, and that $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ would be sufficient.

Denote

[TABLE]

as the Steklov average of $u$. Then for every $v\in V_{2}^{1,1}(B^{+}_{1}\times(-1+h,0))$ such that $v=0$ on $\partial(B^{+}_{1}\times(-1+h,0))$, by taking $v_{-h}$ as the test function in (19), we have

[TABLE]

By changing the order of the integration, we have

[TABLE]

and

[TABLE]

Furthermore,

[TABLE]

The proof of (24) is as follows. By Theorem 2.9 and Theorem 2.10, $u\in L^{2\chi}(Q_{1}^{+},x_{n}^{p}\mathrm{d}x\mathrm{d}t)$. For every $\varepsilon>0$, there exists $\phi\in C^{1}(\overline{Q}_{1}^{+})$ such that

[TABLE]

Using $\phi\in C^{1}(\overline{Q}_{1}^{+})$ and the dominated convergence theorem,

[TABLE]

Then

[TABLE]

Since $\varepsilon$ is arbitrary, the conclusion (24) follows.

For $0<\delta<1/4$, $3\delta-1<\tau<0$, define

[TABLE]

Take $v=\xi_{\delta}(t)(u_{h}-k)^{+}$. Combining (23) and (24), and using $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))$, Lemma 2.4, Theorem 2.9 and Theorem 2.10, we have

[TABLE]

Also, by the proof of Lemma 2.4, we have

[TABLE]

Therefore, (20) follows from (21), (22), (25), and the Cauchy-Schwarz inequality. ∎

We have the following uniqueness of weak solutions.

Theorem 3.6**.**

Suppose $u$ is a weak solution of (6) with the full boundary condition $u\equiv 0$ on $\partial_{pa}Q_{1}^{+}$, where the coefficients of the equation satisfy (1), (16), (17) and (18). Then there exists $C>0$ depending only on $n$, $\lambda,\Lambda$ and $p$ such that

[TABLE]

Consequently, there exists at most one weak solution of (6) with the full boundary condition $u\equiv 0$ on $\partial_{pa}Q_{1}^{+}$.

Proof.

By letting $k=0$ in Lemma 3.5, we have

[TABLE]

Since $q>\frac{\chi}{\chi-1}$, it follows from Theorem 2.9, Theorem 2.10 and Young’s inequality that

[TABLE]

Plugging these to (27) and using Lemma 2.6, we obtain

[TABLE]

where $F_{0}$ is defined in (18). In particular,

[TABLE]

By Gronwall’s inequality, we have

[TABLE]

Plugging this back to (28), the estimate (26) follows. Therefore, the uniqueness holds. ∎

Theorem 3.7**.**

Suppose $a$ is continuous in $\overline{Q}_{1}^{+}$, and the conditions (1), (16), (17) and (18) hold. Then there exists a unique weak solution of (6) with the full boundary condition $u\equiv 0$ on $\partial_{pa}Q_{1}^{+}$.

Proof.

For two real numbers $r_{1}$ and $r_{2}$, we denote $r_{1}\vee r_{2}=\max(r_{1},r_{2})$. We first consider the case with an additionally assume that $\partial_{t}a,c\in L^{q}(Q_{1}^{+},x_{n}^{p}\vee 1\,\mathrm{d}x\mathrm{d}t)$ and $f\in L^{\frac{2\chi}{\chi-1}}(Q_{1}^{+},x_{n}^{p}\vee 1\,\mathrm{d}x\mathrm{d}t)$, where $\chi$ is the constant in Theorem 2.9 or Theorem 2.10. An approximation argument in the end would remove this assumption.

For all $\varepsilon\in(0,1)$, let $a^{\varepsilon}\in C^{2}(\overline{Q}_{1}^{+})$ be such that $a^{\varepsilon}\to a$ uniformly on $Q_{1}^{+}$, and $\partial_{t}a^{\varepsilon}\to\partial_{t}a$ in $L^{q}(Q_{1}^{+},x_{n}^{p}\vee 1\,\mathrm{d}x\mathrm{d}t)$. Then there exists a unique energy weak solution $u_{\varepsilon}\in C([-1,0];L^{2}(B_{1}^{+}))\cap L^{2}((-1,0];H_{0}^{1}(B_{1}^{+}))$ to the uniformly parabolic equation

[TABLE]

with $u_{\varepsilon}\equiv 0$ on $\partial_{pa}Q_{1}^{+}$. That is,

[TABLE]

for every $\varphi\in\mathring{W}^{1,1}_{2}(Q^{+}_{1})$ satisfying $\varphi(\cdot,-1)\equiv 0$ in $B_{1}^{+}$ (in the trace sense). By the same proof of (26), we have

[TABLE]

where

[TABLE]

Hence, if $p\geq 0$, then

[TABLE]

If $-1<p<0$, then by (31) and the proof of Theorem 2.10, we have

[TABLE]

Therefore, by Theorem 2.9 and Theorem 2.10, for all $p>-1$, there exist $u\in L^{2\chi}(Q_{1}^{+})\cap L^{2}((-1,0];H_{0}^{1}(B_{1}^{+}))$ and a subsequence $\{u_{\varepsilon_{j}}\}$, such that $u_{\varepsilon_{j}}\rightharpoonup u$ weakly in $L^{2\chi}(Q_{1}^{+})$ and $Du_{\varepsilon_{j}}\rightharpoonup Du$ weakly in $L^{2}(Q_{1}^{+})$. Let $\varphi\in C^{\infty}(Q^{+}_{1})$ be such that $\varphi\equiv 0$ near the parabolic boundary $\partial_{pa}Q^{+}_{1}$ and let

[TABLE]

By (30), (31), (32), Theorem 2.9, and the absolute continuity of Lebesgue integrals (applying to the right hand side of (30)), we know that $h_{j}$ is uniformly bounded and equicontinuous on $[-1,0]$. By the Ascoli-Arzela Theorem, there is a subsequence of $\{h_{j}\}$, which is still denoted by $\{h_{j}\}$, such that $h_{j}$ uniformly converges to a function $h\in C([-1,0])$. On the other hand, since $u_{\varepsilon_{j}}\rightharpoonup u$ weakly in $L^{2\chi}(Q_{1}^{+})$, we have that for every interval $I\subset[-1,0]$,

[TABLE]

Hence,

[TABLE]

Therefore, if one considers such a $\varphi$ independent of the time variable, then we know from (31) that $u\in L^{\infty}([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))$, and it is straightforward to verify by sending $\varepsilon_{j}\to 0$ in (30) that $u$ satisfies (19) for every $\varphi\in C^{\infty}(Q^{+}_{1})$ being such that $\varphi\equiv 0$ near the parabolic boundary $\partial_{pa}Q^{+}_{1}$.

When $p\geq 0$, then by a standard density argument, it is straightforward to verify that $u$ satisfies (19) for every $\varphi\in\mathring{W}^{1,1}_{2}(Q^{+}_{1})$ satisfying $\varphi(\cdot,-1)\equiv 0$ in $B_{1}^{+}$ (in the trace sense). By Lemma 2.1, this $u$ satisfies (19) for every $\varphi\in\mathring{V}^{1,1}_{2}(Q^{+}_{1})$ satisfying $\varphi(\cdot,-1)\equiv 0$ in $B_{1}^{+}$.

When $-1<p<0$, we also use approximation arguments. Let $\varphi\in\mathring{V}^{1,1}_{2}(Q^{+}_{1})$ satisfy $\varphi(\cdot,-1)\equiv 0$ in $B_{1}^{+}$. Using Minkowski’s integral inequality, for every $\delta>0$, there exists $\mu>0$ such that

[TABLE]

where $\chi$ is the one in Theorem 2.10. Let $\eta_{\mu}$ be a smooth cut-off function such that $\eta\equiv 1$ on $[\mu,+\infty)$ and $\eta\equiv 0$ on $[0,\mu/2]$. Let $\varphi_{1}(x,t)=\eta(x_{n})\varphi(x,t)$ and $\varphi_{2}(x,t)=(1-\eta(x_{n}))\varphi(x,t)$. Using the fact that $V^{1,1}_{2}(Q^{+}_{1})\subset W^{1,1}_{2}(Q^{+}_{1})$ when $-1<p<0$, we have (30). Similar to the above, by using the weak convergence of $u_{\varepsilon}$, it is straightforward to verify that

[TABLE]

and

[TABLE]

By using Theorem 2.10, Hölder’s inequality, (31) and (32), we can verify that

[TABLE]

Then by sending $\varepsilon\to 0$ and then $\delta\to 0$ in (30), it follows that (19) holds for every $\varphi\in\mathring{V}^{1,1}_{2}(Q^{+}_{1})$ satisfying $\varphi(\cdot,-1)\equiv 0$ in $B_{1}^{+}$.

Next, we want to verify that $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))$. Note that we have

[TABLE]

where

[TABLE]

Hence, we know that $g_{j}\in L^{2}(Q_{1}^{+}),j=1,\cdots,n$, and $g_{0}\in L^{\frac{2\chi}{2\chi-1}}(Q_{1}^{+})$. Moreover, we clearly have

[TABLE]

Denote

[TABLE]

Then we know that $I$ is of measure zero. We can redefine $u(x,s)$ such that (33) and (34) for $s\in[-1,0]\setminus I$. Indeed, because of $\eqref{eq:continuityV2-2}$, for every $s_{0}\in[-1,0]\setminus I$, there exists $\{s_{k}\}\subset I$ such that $s_{k}\to s_{0}$ and $u(\cdot,s_{k})\rightharpoonup v(\cdot)$ in $L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x)$. We redefine $u(\cdot,s_{0})=v(\cdot)$. Then (33) and (34) hold for $s_{0}$, and moreover, by (33), this $v(\cdot)$ is independent on the choice of the sequence $\{s_{k}\}$. Thus, we can assume that (33) and (34) hold for all $s\in[-1,0]$.

Let $Q_{1,s,h}^{+}=B_{1}^{+}\times(s,s+h)$ when $s,s+h\in(-1,0)$ (here, we assume $h>0$, and the argument for the case $h<0$ can be modified correspondingly). From (33), we obtain

[TABLE]

By choosing $\varphi$ as a function in $C^{\infty}_{c}(B_{1}^{+})$, and since $a$ is continuous in $\overline{Q}_{1}^{+}$, we have

[TABLE]

Hence,

[TABLE]

By a density argument, (36) holds for all $\varphi\in L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x)$.

Choose $\varepsilon>0$ small such that $s\pm\varepsilon,s+h\pm\varepsilon\in(-1,0)$. Let

[TABLE]

and

[TABLE]

Then $\varphi_{\varepsilon}\in\mathring{V}^{1,1}_{2}(Q^{+}_{1})$ and (35) holds for $\varphi_{\varepsilon}$. Note that

[TABLE]

Using the continuity of $a$ and (36), we have

[TABLE]

Setting $\varphi=\varphi_{\varepsilon}$ in (35) and then letting $\varepsilon\to 0$, we have by using (36) that

[TABLE]

Hence,

[TABLE]

Since $a\in C(\overline{Q}_{1}^{+})$, we have

[TABLE]

and thus,

[TABLE]

It follows from (36) and (38) that

[TABLE]

Since $a\geq\lambda>0$, we obtain

[TABLE]

Hence, $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))$, and thus, $u\in\mathring{V}_{2}^{1,0}(Q^{+}_{1})$.

Now let us use another approximation to remove the assume that $\partial_{t}a,c\in L^{q}(Q_{1}^{+},x_{n}^{p}\vee 1\,\mathrm{d}x\mathrm{d}t)$ and $f\in L^{\frac{2\chi}{\chi-1}}(Q_{1}^{+},x_{n}^{p}\vee 1\,\mathrm{d}x\mathrm{d}t)$. Suppose (17) and (18) hold. Let $a^{\varepsilon}\in C^{2}(\overline{Q}_{1}^{+})$ be such that $a^{\varepsilon}\to a$ uniformly on $Q_{1}^{+}$ and $\partial_{t}a^{\varepsilon}\to\partial_{t}a$ in $L^{q}(Q_{1}^{+},x_{n}^{p}\,\mathrm{d}x\mathrm{d}t)$, $c^{\varepsilon}\in C^{2}(\overline{Q}_{1}^{+})$ be such that $c^{\varepsilon}\to c$ in $L^{q}(Q_{1}^{+},x_{n}^{p}\,\mathrm{d}x\mathrm{d}t)$, and $f^{\varepsilon}\in C^{2}(\overline{Q}_{1}^{+})$ be such that $f^{\varepsilon}\to f$ in $L^{\frac{2\chi}{\chi-1}}(Q_{1}^{+},x_{n}^{p}\,\mathrm{d}x\mathrm{d}t)$. Then as proved in the above, there exists a weak solution $u_{\varepsilon}\in\mathring{V}_{2}^{1,0}(Q^{+}_{1})$ to the parabolic equation

[TABLE]

with $u_{\varepsilon}\equiv 0$ on $\partial_{pa}Q_{1}^{+}$. Then by the energy estimate in Theorem 3.6 and the same argument as above, one can show that $u_{\varepsilon}$ will converge to a weak solution of (6) with the full boundary condition $u\equiv 0$ on $\partial_{pa}Q_{1}^{+}$.

Finally, the uniqueness follows from Theorem 3.6. ∎

3.3 $W^{1,1}_{2}$ regularity

Next, we want to study the $W^{1,1}_{2}$ regularity of weak solutions to the equation (6) with slightly stronger assumptions on the coefficients. Consider the following equation

[TABLE]

where $x_{n}^{p/2}\wedge 1=\min(x_{n}^{p/2},1)$. For the coefficients, besides (16), we suppose that

[TABLE]

for some $q>\frac{\chi}{\chi-1}$. We also suppose that $-\mathrm{div}(A\nabla)+c_{0}$ is coercive, where $A=(a_{ij})$, i.e., there exists a constant $\bar{\lambda}>0$ such that

[TABLE]

Note that (41) implies that there exists $\tilde{\lambda}>0$ depending only on $\bar{\lambda},\lambda,\Lambda$ and $n$ such that

[TABLE]

Theorem 3.8**.**

Suppose $a$ is continuous in $\overline{Q}_{1}^{+}$, $A$ is symmetric, and the conditions (1), (16), (40) and (41) hold. Suppose that $f\in L^{2}(Q_{1}^{+},x^{p}\mathrm{d}x\mathrm{d}t)$. Let $u$ be the weak solution of (39) with the full boundary condition $u\equiv 0$ on $\partial_{pa}Q_{1}^{+}$. Then

[TABLE]

where $C>0$ depends only on $\lambda,\bar{\lambda},\Lambda,n$ and $p$.

Proof.

We first assume that $f\in L^{2}(Q_{1}^{+},x^{p}\vee 1\mathrm{d}x\mathrm{d}t)$ and $\partial_{t}a\in L^{q}(Q_{1}^{+},x_{n}^{p}\vee 1\,\mathrm{d}x\mathrm{d}t)$.

For $\varepsilon>0$, let $a^{\varepsilon},a_{ij}^{\varepsilon},d_{j}^{\varepsilon},c_{0}^{\varepsilon}\in C^{\infty}(\mathbb{R}^{n})$ be such that $a^{\varepsilon}\to a$, $a_{ij}^{\varepsilon}\to a_{ij}$, $d_{j}^{\varepsilon}\to d_{j}$, $c_{0}^{\varepsilon}\to c_{0}$ uniformly on $\overline{Q}_{1}^{+}$, $\partial_{t}a^{\varepsilon}\to\partial_{t}a$ in $L^{q}(Q_{1}^{+},x_{n}^{p}\vee 1\,\mathrm{d}x\mathrm{d}t)$, and

[TABLE]

Let $b_{i}^{\varepsilon},c^{\varepsilon}\in C^{\infty}(\mathbb{R}^{n})$ be such that $b_{i}^{\varepsilon}\to b_{i}$, $c^{\varepsilon}\to c$ in $L^{q}(\overline{Q}_{1}^{+})$ for some $q>\frac{\chi}{\chi-1}$, and

[TABLE]

Let $f_{\varepsilon}\in C_{c}^{\infty}(Q_{1}^{+})$ be such that $f_{\varepsilon}\to f$ in $L^{2}(Q_{1}^{+},x^{p}\vee 1\,\mathrm{d}x\mathrm{d}t)$ as $\varepsilon\to 0$.

Let $u_{\varepsilon}\in C([-1,0];L^{2}(B_{1}^{+}))\cap L^{2}((-1,0];H_{0}^{1}(B_{1}^{+}))$ be the unique weak solution of

[TABLE]

with $u_{\varepsilon}\equiv 0$ on $\partial_{pa}Q_{1}^{+}$. By the Schauder regularity theory, we know that $D_{x}u_{\varepsilon},\partial_{t}u_{\varepsilon}\in C(\overline{Q}_{1}^{+})$.

For small $h>0$, denote

[TABLE]

for all $-1\leq t\leq-h$, and denote the left hand side of (44) as $I(x,t)$. Then we have for all $-1<t<-h$,

[TABLE]

Using the symmetry of $A$, we have

[TABLE]

where we used that $D_{x}u\in C^{0}(\overline{B}_{1}^{+}\times[-1,0])$ and $u(x,-1)\equiv 0$. Here, we used $u_{\varepsilon}^{h}(x,t)$ instead of $\partial_{t}u_{\varepsilon}$ to avoid involving $D_{x}\partial_{t}u_{\varepsilon}$ in the calculation. Also,

[TABLE]

Using similar arguments, by sending $h\to 0$ in (45), and using (16), (40), (41) (or (42)) and Hölder’s inequality, we have

[TABLE]

Then it follows from (31) that

[TABLE]

Therefore, $\int_{(B_{1}^{+}\cap\{x_{n}>\delta\})\times(-1,0]}|\partial_{t}u_{\varepsilon}|^{2}\leq C(\delta)$ for every $\delta>0$. This implies the existence of weak derivative $\partial_{t}u$, and that $\partial_{t}u_{\varepsilon}$ weakly converges to $\partial_{t}u$ in $L^{2}((B_{1}^{+}\cap\{x_{n}>\delta\})\times(-1,0])$ for every $\delta$. Since

[TABLE]

we have from (46) by sending $\varepsilon\to 0$ that

[TABLE]

Then, (43) follows by sending $\delta\to 0$ and using the monotone convergence theorem.

Now let us use another approximation to remove the assume that $f\in L^{2}(Q_{1}^{+},x^{p}\vee 1\mathrm{d}x\mathrm{d}t)$ and $\partial_{t}a\in L^{q}(Q_{1}^{+},x_{n}^{p}\vee 1\,\mathrm{d}x\mathrm{d}t)$. Let $a^{\varepsilon}\in C^{2}(\overline{Q}_{1}^{+})$ be such that $a^{\varepsilon}\to a$ uniformly on $Q_{1}^{+}$ and $\partial_{t}a^{\varepsilon}\to\partial_{t}a$ in $L^{q}(Q_{1}^{+},x_{n}^{p}\,\mathrm{d}x\mathrm{d}t)$, and $f_{\varepsilon}\in C^{2}_{c}(\overline{Q}_{1}^{+})$ be such that $f_{\varepsilon}\to f$ in $L^{2}(Q_{1}^{+},x_{n}^{p}\,\mathrm{d}x\mathrm{d}t)$. Then there exists a weak solution $u_{\varepsilon}\in\mathring{V}_{2}^{1,0}(Q^{+}_{1})$ to the parabolic equation

[TABLE]

with $u_{\varepsilon}\equiv 0$ on $\partial_{pa}Q_{1}^{+}$. By the argument of Theorem 3.7, $u_{\varepsilon}$ will converge to a weak solution of (39) with the full boundary condition $u\equiv 0$ on $\partial_{pa}Q_{1}^{+}$. By the same argument as above, one can show that

[TABLE]

Then the conclusion follows by sending $\varepsilon\to 0$. ∎

4 Boundedness of weak solutions

4.1 A maximum principle

Suppose

[TABLE]

where $q$ is the one in (17).

Theorem 4.1**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7), where the coefficients of the equation satisfy (1), (16), (17) and (47). Suppose that all $d_{j}=0$ and $c\geq 0$. Then

[TABLE]

where $|Q_{1}^{+}|$ denotes the Lebesgue measure of $Q_{1}^{+}$, $|Q_{1}^{+}|_{p}$ is defined in (48) in the below, and $C>0$ depends only on $\lambda,\Lambda,n,p$ and $q$.

Proof.

It follows from Lemma 3.5 that (20) holds. Let $\varphi$ be the one there. Using (17), Theorem 2.9, Theorem 2.10 and Hölder’s inequality, one obtains for $p\geq 0$ that

[TABLE]

and for $-1<p<0$ that

[TABLE]

Similarly, we have

[TABLE]

where

[TABLE]

By choosing $\varepsilon>0$ small, and using Theorem 2.9 and Theorem 2.10, we have for $p\geq 0$ that

[TABLE]

and for $-1<p<0$ that

[TABLE]

When $s+1$ is sufficiently small, we have for $p\geq 0$ that

[TABLE]

and for $-1<p<0$ that

[TABLE]

Hence, we have

[TABLE]

For $h>k$, we have

[TABLE]

Hence, if we denote

[TABLE]

then

[TABLE]

where $\beta=(1-\frac{1}{q})\chi>1$ by the assumption of $q$. Define

[TABLE]

Then

[TABLE]

Similar to (55) and (56), we can choose $C>0$ such that for $d\geq CF_{1}|B_{1}^{+}\times(-1,s)|^{\frac{1}{2}(1-\frac{1}{q}-\frac{1}{\chi})}$, we have

[TABLE]

That is,

[TABLE]

Keeping iterating for $s$ with a uniform step size, we obtain that

[TABLE]

Applying the same result to the equation of $-u$, the conclusion follows. ∎

4.2 A local maximum principle

The following is the Caccioppoli inequality of weak solutions to (6) and (7), which is the starting point of the De Giorgi iteration.

Theorem 4.2**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7), where the coefficients of the equation satisfy (1), (16), (17) and (47). Let $x_{0}\in\partial^{\prime}B_{1}^{+}$, $Q_{\rho,\tau}^{+}=B^{+}_{\rho}(x_{0})\times(t_{0},t_{0}+\tau]\subset Q_{1}^{+}$, $k\geq 0$, $\varepsilon\in(0,1]$, and $\xi\in V_{2}^{1,1}(Q_{\rho,\tau}^{+})$ such that $\xi=0$ on $\partial^{\prime\prime}B_{\rho}(x_{0})\times(t_{0},t_{0}+\tau]$ and $0\leq\xi\leq 1$. Then

[TABLE]

where $|\cdot|_{p}$ is defined in (48), $C>0$ depends only on $\lambda,\Lambda,n,p$ and $q$, $\kappa>0$ depends only on $q$ and $\chi$, and $F_{1}$ is given in (47).

Proof.

Similar to the proof of Lemma 3.5, we can assume $u\in V^{1,1}_{2}(Q^{+}_{1})$, since otherwise we can use its Steklov average to remove this assumption.

Taking $\varphi=\xi^{2}(u-k)^{+}$ in (19), and using (16) and Hölder’s inequality, one obtains that

[TABLE]

Using (17), Theorem 2.9, Theorem 2.10 and Hölder’s inequality one obtains

[TABLE]

and

[TABLE]

By choosing $\varepsilon>0$ small, and using Theorem 2.9 and Theorem 2.10, the conclusion follows from (49). ∎

Now we can prove the local-in-time boundedness of weak solutions up to $\{x_{n}=0\}$.

Theorem 4.3**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7), where the coefficients of the equation satisfy (1), (16), and

[TABLE]

for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Denote $\mathcal{Q}_{R}^{+}=B_{R}^{+}(x_{0})\times(t_{0}-R^{p+2},t_{0}]\subset Q_{1}^{+}$, where $x_{0}\in\partial^{\prime}B_{1}^{+}$. Then we have, for any $\gamma>0$,

[TABLE]

where $C>0$ depends only on $\lambda,\Lambda,n,p$, $q$ and $\gamma$.

Proof.

Let $\theta\in(0,1)$. We will consider $R=1$ first, and then scale it back. We would like to first show that

[TABLE]

where $\beta=1-\frac{1}{q}-\frac{1}{\chi}$. We only need to prove (52) for $\theta\in(1/2,1)$.

Let

[TABLE]

where $k>0$ to be fixed later. For brevity, we denote $Q_{m}^{+}=Q^{+}_{\rho_{m}}=B_{\rho_{m}}^{+}(x_{0})\times(t_{0}-\rho_{m}^{p+2},t_{0}]$, and we take $\xi_{m}$ to be a cut-off function such that $\xi\in V_{2}^{1,1}(Q_{m}^{+})$, $\xi_{m}\equiv 1$ on $Q_{m+1}^{+}$, $\xi=0$ on $Q_{m}^{+}\setminus Q^{+}_{(\rho_{m}+\rho_{m+1})/2}$, and $|D\xi_{m}|^{2}+|\partial_{t}\xi_{m}|\leq C(n)(\rho_{m}-\rho_{m+1})^{-2}.$

Let

[TABLE]

Case 1: Suppose $p\geq 0$. By Theorem 4.2 and Theorem 2.9, we have

[TABLE]

where

[TABLE]

Take $k\geq F_{1}$. Then,

[TABLE]

Notice that

[TABLE]

Hence,

[TABLE]

Case 2: Suppose $-1<p<0$. By Theorem 4.2 and Theorem 2.10, we have

[TABLE]

where

[TABLE]

Take $k\geq F_{1}$. Then,

[TABLE]

Notice that

[TABLE]

Hence, (54) also holds.

Now let us start from (54) which holds for all $p>-1$. If we further take $k\geq\|u\|_{L^{2}(Q_{1}^{+})}+\|u\|_{L^{2}(Q_{1}^{+},\ x_{n}^{p}\,\mathrm{d}x\mathrm{d}t)}$, then

[TABLE]

Thus,

[TABLE]

If

[TABLE]

then one can show by induction that

[TABLE]

and thus,

[TABLE]

That is,

[TABLE]

Therefore, we only need to choose

[TABLE]

This proves (52).

Now we will use a scaling argument. For any $R\in(0,1]$, define

[TABLE]

Then

[TABLE]

Note that since $q>\max(\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$, then

[TABLE]

and

[TABLE]

Hence, it follows from (52) that

[TABLE]

where in the second inequality we used that $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2})$ and $R\leq 1$. Scaling the estimate of $\tilde{u}$ back to $u$, we then obtain for $p\geq 0$ that

[TABLE]

where $\alpha=\max(\frac{n+p+2}{\gamma},\frac{2}{\beta\gamma})$. By an iterative lemma, Lemma 1.1 in Giaquinta-Giusti [15], we have

[TABLE]

Applying this estimate to $\tilde{u}$ again, and scaling it back to $u$, we obtain the desired estimate.

Similarly, for $-1<p<0$ and $\gamma\in(0,2)$, we let $\tilde{\gamma}=\frac{(1+p)\gamma}{1-p}<\gamma$. Then we have

[TABLE]

where $\tilde{\alpha}=\max(\frac{n+2p+2}{\tilde{\gamma}},\frac{2}{\beta\tilde{\gamma}})$. By an iterative lemma, Lemma 1.1 in Giaquinta-Giusti [15], we have

[TABLE]

Since

[TABLE]

which follows from Hölder’s inequality, then

[TABLE]

Applying this estimate to $\tilde{u}$ again, and scaling it back to $u$, we obtain the desired estimate. ∎

If it additionally satisfies $u(\cdot,-1)=0$, then we can show the boundedness up to the initial time.

Theorem 4.4**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7) and the initial condition $u(\cdot,-1)=0$, where the coefficients of the equation satisfy (1), (16), (50) and (51) for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Denote $\widetilde{\mathcal{Q}}_{R}^{+}=B_{R}^{+}(x_{0})\times(-1,-1+R^{p+2}]\subset Q_{1}^{+}$, where $x_{0}\in\partial^{\prime}B_{1}^{+}$. Then we have, for any $\gamma>0$,

[TABLE]

where $C>0$ depends only on $\lambda,\Lambda,n,p$, $q$ and $\gamma$.

The proof of this theorem is almost identical to that of Theorem 4.3 (which is actually simpler since we do not need to cut off in the time variables). We omit the details.

Combining Theorems 4.3 and 4.4, we have

Theorem 4.5**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7) and the initial condition $u(\cdot,-1)=0$, where the coefficients of the equation satisfy (1), (16), (50) and (51) for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Then we have, for any $\gamma>0$,

[TABLE]

where $C>0$ depends only on $\lambda,\Lambda,n,p$, $q$ and $\gamma$.

Proof.

It follows from Theorems 4.3 and 4.4.∎

5 Hölder regularity

5.1 Improvement of oscillations centered at the boundary

Throughout this subsection, we assume all the assumptions in Theorem 4.3 and let $u$ be as in Theorem 4.3. Suppose

[TABLE]

Let

[TABLE]

and

[TABLE]

Recall that for $x_{0}\in\partial R^{n}_{+}$,

[TABLE]

We simply write it as $\mathcal{Q}_{R}$ and $\mathcal{Q}_{R}^{+}$ if $(x_{0},t_{0})=(0,0)$.

Lemma 5.1**.**

There exists $C>0$ depending only on $n$ and $p$ such that for every $\varepsilon\in(0,1)$, every $R\in(0,1]$, every $\delta>0$, every $\widetilde{A}\subset B_{R}^{+}\times[0,\delta R^{p+2}]$, if

[TABLE]

then

[TABLE]

Proof.

If $-1<p<0$, then

[TABLE]

If $p\geq 0$, then we have

[TABLE]

∎

We have the following De Giorgi lemmas.

Lemma 5.2**.**

Let $0<R\leq 1$ and

[TABLE]

Then there exists $0<\gamma_{0}<1$ depending only on $\lambda,\Lambda,n,p$ and $q$ such that for $0\leq k<\mu$, if

[TABLE]

and

[TABLE]

then

[TABLE]

Proof.

Let

[TABLE]

Let $\eta_{j}$ be a smooth cut-off function satisfying

[TABLE]

[TABLE]

Case 1: $p\geq 0$. Let us consider $n\geq 3$ first. Since $k_{j}>k\geq 0$, By Theorem 4.2 and Theorem 2.9, we have

[TABLE]

where $v=(u-k_{j})^{+}$. Let $A(k_{j},\rho)=\{(x,t)\in\mathcal{Q}_{\rho}^{+}:u>k_{j}\}$ for $0<\rho\leq R$. Then

[TABLE]

and

[TABLE]

It follows that

[TABLE]

where we used the assumption on $H$, and $|A(k_{j},r_{j})|\leq|\mathcal{Q}_{R}^{+}|\leq CR^{n+p+2}$. Hence

[TABLE]

where we used that $\chi=\frac{n+p+2}{n+p}$. Therefore, similarly to (55) and (56), there exists $\theta\in(0,1)$ such that if $\frac{|A(k_{0},r_{0})|}{|\mathcal{Q}_{R}^{+}|}\leq\theta$, then

[TABLE]

By Lemma 5.1, we only need to choose $\gamma_{0}=(\theta/C)^{p+1}$.

Now, let us consider $n=1,2$. By Theorem 4.2 and Theorem 2.9, (61) would become

[TABLE]

By using $\chi=\frac{p+2}{p+1}$, one will still obtain (62). Then the left proof is the same as above.

Case 2: $-1<p<0$. We still consider $n\geq 3$ first. By Theorem 4.2 and Theorem 2.10, we have

[TABLE]

where $v=(u-k_{j})^{+}$. Since

[TABLE]

and

[TABLE]

it follows that

[TABLE]

where we used the assumption on $H$, and $|A(k_{j},r_{j})|_{\nu_{p}}\leq|\mathcal{Q}_{R}^{+}|_{\nu_{p}}\leq CR^{n+2p+2}$. Hence

[TABLE]

where we used that $\chi=\frac{n+2p+2}{n+p}$. Hence, there exists $\theta\in(0,1)$ such that if $\frac{|A(k_{0},r_{0})|_{\nu_{p}}}{|\mathcal{Q}_{R}^{+}|_{\nu_{p}}}\leq\theta$, then

[TABLE]

If $n=1,2$, then by Theorem 4.2 and Theorem 2.10, (63) would become

[TABLE]

By using $\chi=\frac{3}{2}$, one will still obtain (64). Then the left proof is the same as above. ∎

Lemma 5.3**.**

Let $0<R\leq\frac{1}{2}$. Suppose

[TABLE]

Then there exists $C>1$ depending only on $\lambda,\Lambda,n,p$ and $q$ such that for every $\ell\in\mathbb{Z}^{+}$, there holds either

[TABLE]

or

[TABLE]

Proof.

We extend $u$ to be identically zero in $(B_{1}\setminus B_{1}^{+})\times[-1,0]$, which will still be denoted as $u$. Let

[TABLE]

and

[TABLE]

Since $k_{j}\geq 0$, we have $A(k_{j},R;t)=B_{R}^{+}\cap\{u(\cdot,t)>k_{j}\}$ and

[TABLE]

Then by Theorem 2.12, we have

[TABLE]

where we choose $\varepsilon=\min\left(\frac{1}{4},\frac{1}{4(p+1)}\right)$. Integrating in the time variable, and using Hölder’s inequality again, we have

[TABLE]

It follows from Theorem 4.2 (with $\xi$ independent of $t$) that

[TABLE]

If (65) fails for some $\ell$, then we have

[TABLE]

for $j\leq\ell$, where we used that $R^{\frac{p+2}{2}-\frac{n+2p+2}{2q}}\geq R^{1-\frac{n+p+2}{2q}}$ for $p<0$. Hence,

[TABLE]

or

[TABLE]

Taking a summation, we have

[TABLE]

The lemma follows. ∎

Now we can prove the Hölder continuity on the boundary.

Theorem 5.4**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7), where the coefficients of the equation satisfy (1), (16), (50) and (51) for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Let $\bar{x}\in\partial^{\prime}B_{1/2}$ and $\bar{t}\in(-1/4,0])$. Then there exist $\alpha>0$ and $C>0$, both of which depend only on $\lambda,\Lambda,n,p$ and $q$, such that

[TABLE]

for every $(x,t)\in B_{1/2}^{+}\times(-1/4,0]$.

Proof.

Without loss of generality, we assume $(\bar{x},\bar{t})=(0,0)$. For $R\in(0,1/2]$, denote

[TABLE]

Let $\gamma_{0}$ be the one in Lemma 5.2. We can choose $\ell$ sufficiently large so that

[TABLE]

where $C$ is the one in (66). Then it follows from Lemma 5.2 and Lemma 5.3 that either

[TABLE]

or

[TABLE]

Applying these estimates to $-u$, we have either

[TABLE]

or

[TABLE]

In any case, we will obtain

[TABLE]

By an iterative lemma, e.g. Lemma 3.4 in Han-Lin [18] (or Lemma B.2 in [19]), there exist $\alpha$ and $C$, both of which depend only on $\lambda,\Lambda,n,p$ and $q$, such that

[TABLE]

from which the conclusion follows. ∎

5.2 Interior Hölder estimates

When $x$ is away from the boundary $\partial^{\prime}B_{1}^{+}$, the equation (6) is uniformly parabolic. We observe that all the assumptions in Theorem 4.3 are stronger than those in the uniformly parabolic case (which corresponds to $p=0$ and $\chi=\frac{n+2}{n}$). Therefore, using the same proof for uniformly parabolic equations, with a small adaptation to the existence of the coefficient $a$ in front of $\partial_{t}u$, one can show the following interior Hölder estimate.

Theorem 5.5**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7), where the coefficients of the equation satisfy (1), (16), (50) and (51) for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Then there exist $\alpha>0$ and $C>0$, both of which depend only on $\lambda,\Lambda,n,p$ and $q$, such that for every $(x,t),(y,s)\in B_{1/4}(e_{n}/2)\times(-1/4,0]$, there holds

[TABLE]

where $e_{n}=(0,\cdots,0,1)$.

The proof Theorem 5.5 will be proved as follows. We only need to prove the Hölder continuity at the point $(e_{n}/2,0)$.

Similar to Theorem 4.2, we have the Caccipolli inequality around the point $(e_{n}/2,0)$. Let $Q_{\rho,\tau}=B_{\rho}(e_{n}/2)\times(t_{0},t_{0}+\tau]\subset Q_{1/2}(e_{n}/2,0)$, $k\in\mathbb{R}$, $\varepsilon\in(0,1]$, and $\xi\in V_{2}^{1,1}(Q_{\rho,\tau})$ such that $\xi=0$ on $\partial B_{\rho}(e_{n}/2)\times(t_{0},t_{0}+\tau]$ and $0\leq\xi\leq 1$. Then

[TABLE]

Lemma 5.6**.**

Let $0<R\leq 1/2$ and

[TABLE]

Then there exists $0<\gamma_{0}<1$ depending only on $\lambda,\Lambda,n,p$ and $q$ such that for $k<\mu$, if

[TABLE]

and

[TABLE]

then

[TABLE]

The proof of Lemma 5.6 is almost identical to that of Lemma 5.2, and thus, we omit it.

Lemma 5.7**.**

Let $0<\delta\leq 1$, $0<R\leq\frac{1}{4}$, $0<\sigma<1$ and

[TABLE]

Suppose that $k<\mu$ and

[TABLE]

Then there exists $C>1$ depending only on $\lambda,\Lambda,n,p$ and $q$ such that for every $\ell\in\mathbb{Z}^{+}$, there holds either

[TABLE]

or

[TABLE]

Proof.

The proof is very similar to that of Lemma 5.3 with the following two changes. The first is that $k_{j}$ should be defined as $k_{j}=\mu-\frac{H}{2^{j}}$ instead. The second is that the estimate (67) should be replaced by the assumption (73). The left proofs are identical so that we omit it. ∎

The next lemma was not needed in the proof of Theorem 5.4, and its proof is slightly different from the uniformly parabolic equations with $a\equiv 1$. Thus, we provide a proof.

Lemma 5.8**.**

Let $0<\sigma<1$. There exist $R_{0}\in(0,\frac{1}{4})$ and $s_{0}>1$ depending only on $\lambda,\Lambda,n,p,q$ and $\sigma$ such that the following holds. Let $R\in(0,R_{0}]$ and

[TABLE]

Suppose that $k<\mu$ and

[TABLE]

Then either

[TABLE]

or

[TABLE]

Proof.

Let $\eta$ be a cut-off function supported in $B_{R}(e_{n}/2)$ and $\eta=1$ in $B_{\beta R}(e_{n}/2)$, where $0<\beta<1$ will be fixed later. Let $0<\delta\leq 1$ and

[TABLE]

Let $k_{1}>1$. By (72), we have

[TABLE]

where $v=(u-k)^{+}$. Note that

[TABLE]

It follows that if (75) fails, then for all $t\in[t_{0},t_{0}+\delta R^{p+2}]$,

[TABLE]

where

[TABLE]

Hence,

[TABLE]

By choosing $\beta$ such that

[TABLE]

we have

[TABLE]

For every $t_{0}\leq\tau_{1}\leq\tau_{2}<t_{0}+R^{p+2}$, we have

[TABLE]

where we used (50) in the third inequality, and

[TABLE]

Then (5.2) becomes

[TABLE]

If we let

[TABLE]

then

[TABLE]

Since

[TABLE]

we fix an $a$ such that

[TABLE]

We choose $\delta$ slightly smaller if necessary to make $\delta^{-1}$ to be an integer. Let $N=\delta^{-1}$ and denote

[TABLE]

We will inductively prove that there exist $s_{1}<s_{2}<\cdots<s_{N}$ such that

[TABLE]

for all $t_{j-1}\leq t\leq t_{j}$, where all the $s_{j}$ depend only on $\lambda,\Lambda,n,p,q$ and $\sigma$, from which the conclusion of this lemma follow.

Let us consider $j=1$ first.

Since $2-\frac{n+2}{q}>0$, there exist $R_{0}$ small and $k_{0}$ large, depending on $\sigma$, such that for all $k_{1}\geq k_{0}$ and $R\leq R_{0}$, we have

[TABLE]

Then,

[TABLE]

for all $t\in[t_{0},t_{1}]$. Applying Lemma 5.7, for every $k_{2}>k_{1}$, we have

[TABLE]

Hence,

[TABLE]

Hence, we can choose $k_{2}$ large enough such hat

[TABLE]

Let $k_{1}=k_{0}$ and $s_{1}=k_{1}+k_{2}$. By replacing $H$ by $H2^{-k_{2}}$ in (5.2), it follows that

[TABLE]

This prove (78) for $j=1$. The proof for $j=2,3,\cdots,N$ is similar, and we omit it. ∎

Combining the above three lemmas, we will have the following improvement of oscillations.

Lemma 5.9**.**

Let $0<\sigma<1$. There exist $R_{0}\in(0,\frac{1}{2})$ and $s>1$ depending only on $\lambda,\Lambda,n,p,q$ and $\sigma$ such that the following holds. Let $R\in(0,R_{0}]$ and

[TABLE]

Suppose that $k<\mu$ and

[TABLE]

Then either

[TABLE]

or

[TABLE]

Proof.

Let $R_{0}$ and $s_{0}$ be those from Lemma 5.8. Suppose (79) fails for some $s>s_{0}$, which will be fixed in the end. Then it follows from Lemma 5.8 that

[TABLE]

Then using Lemma 5.7, we have

[TABLE]

Let $\gamma_{0}$ be the one in Lemma 5.6. We can choose $s$ sufficiently large so that

[TABLE]

Then it follows from Lemma 5.6 that

[TABLE]

∎

Remark 5.10**.**

From the above proof, for $\delta_{0}\leq\delta\leq\delta_{0}^{-1}$, if we consider the problem in $B_{2R}(e_{n}/2)\times[-\delta R^{p+2},0]$ instead of $B_{2R}(e_{n}/2)\times[-R^{p+2},0]$, then the conclusion in Lemma 5.9 still holds, where the constant $s$ would additionally depend on $\delta_{0}$.

Proof of Theorem 5.5.

We only need to prove the Hölder continuity at the point $(e_{n}/2,0)$. Let $R_{0}$ be the one in Lemma 5.9 with $\sigma=1/2$. For $R\in(0,R_{0}]$, denote

[TABLE]

Then one of the following two inequalities must hold:

[TABLE]

If (80) holds, then by Lemma 5.9, there exists $s>1$ such that either

[TABLE]

or

[TABLE]

If (81) holds, then by applying the above estimates to $-u$, one has either (82) or

[TABLE]

In any case, we obtain

[TABLE]

By an iterative lemma, e.g. Lemma 3.4 in Han-Lin [18] (or Lemma B.2 in [19]), there exist $\alpha$ and $C$, both of which depend only on $\lambda,\Lambda,n,p$ and $q$, such that

[TABLE]

from which the conclusion follows. ∎

5.3 Hölder estimates near the boundary

Together with the Hölder regularity at the boundary in Theorem 5.4 and the interior Hölder regularity in Theorem 5.5, one can obtain the Hölder regularity up to the boundary.

Theorem 5.11**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7), where the coefficients of the equation satisfy (1), (16), (50) and (51) for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Then for every $\gamma>0$, there exist $\theta>0$ and $C>0$, both of which depend only on $\lambda,\Lambda,n,p,\gamma$ and $q$, such that for every $(x,t),(y,s)\in B_{1/2}^{+}\times(-1/4,0]$, there holds

[TABLE]

Proof.

By normalization, we assume $\sup_{B_{3/4}\times[-3/4,0]}|u|+F_{1}=1$. For any $\bar{x}=(0,\bar{x}_{n})\in B_{1/2}^{+}$, we let $R:=\bar{x}_{n}>0$, and rescale the solution and the coefficients as in (57) with $x_{0}=0$. Then (58), (59) and (60) hold. By Theorem 5.5, there exist $C>1$ and $0<\beta<1$, both of which depend only on $\lambda,\Lambda,n,p$ and $q$, such that

[TABLE]

Consider $t\in(-1/2,0]$. If $|t|\leq R^{2p+4}$, then we have

[TABLE]

where we used (85) in the first inequality. If $|t|\geq R^{2p+4}$, then we have

[TABLE]

where we used Theorem 5.4 in the second inequality. This shows that $u$ is Hölder continuous in the time variable.

Consider $\tilde{x}=(\tilde{x}^{\prime},\tilde{x}_{n})\in B_{1/2}^{+}$ such that $\tilde{x}_{n}\leq\bar{x}_{n}$. If $\tilde{x}\in B_{R^{2}}(\bar{x})$, then we have

[TABLE]

where we used (85) in the first inequality. If $\tilde{x}\not\in B_{R^{2}}(\bar{x})$, then we have

[TABLE]

where we used Theorem 5.4 in the second inequality. This shows that $u$ is Hölder continuous in the spatial variables.

Together with Theorem 4.3, we finish the proof of this theorem. ∎

5.4 Hölder estimates up to the initial time

We can also show Hölder estimates up to the initial time.

Theorem 5.12**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7) and the initial condition $u(\cdot,-1)=0$, where the coefficients of the equation satisfy (1), (16), (50) and (51) for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Let $\bar{x}\in\partial^{\prime}B_{1/4}$. Then for every $\gamma>0$, there exist $\alpha>0$ and $C>0$, both of which depend only on $\lambda,\Lambda,n,p,\gamma$ and $q$, such that

[TABLE]

for every $(x,t)\in B_{1/4}^{+}\times[-1,-\frac{3}{4}]$.

Proof.

Let $M=\|u\|_{L^{\infty}(B_{3/4}^{+}\times(-1,-1/4))}$,

[TABLE]

[TABLE]

For brevity, we denote

[TABLE]

Let $\eta_{j}(x)$ be a smooth cut-off function satisfying

[TABLE]

[TABLE]

Case 1: $p\geq 0$. Let us consider $n\geq 3$ first. By Theorem 4.2 and Theorem 2.9, we have

[TABLE]

where $v=(u-k_{j})^{+}$. Let $A(k,r_{j})=\{(x,t)\in\mathcal{Q}_{j,\delta}^{+}:u>k\}$. Then

[TABLE]

and

[TABLE]

If

[TABLE]

then

[TABLE]

where we used $|A(k_{j},r_{j})|\leq\delta|\mathcal{Q}_{j,\delta}^{+}|\leq C\delta R^{n+p+2}$. Hence

[TABLE]

where we used that $\chi=\frac{n+p+2}{n+p}$. Therefore, similarly to (55) and (56), there exists $\delta_{0}\in(0,1)$ such that if $\delta\leq\delta_{0}$, then

[TABLE]

Now, let us consider $n=1,2$. By Theorem 4.2 and Theorem 2.9, (61) would become

[TABLE]

By using $\chi=\frac{p+2}{p+1}$, one will still obtain (87) and (88). Then the left proof is the same as above.

Case 2: $-1<p<0$. Again, we consider $n\geq 3$ first. By Theorem 4.2 and Theorem 2.10, we have

[TABLE]

where $v=(u-k_{j})^{+}$. Then

[TABLE]

and

[TABLE]

If

[TABLE]

then

[TABLE]

where we used $|A(k_{j},r_{j})|_{\nu_{p}}\leq\delta|\mathcal{Q}_{j,\delta}^{+}|_{\nu_{p}}\leq C\delta R^{n+2p+2}$. Hence

[TABLE]

where we used that $\chi=\frac{n+2p+2}{n+p}$. Therefore, there exists $\delta_{0}\in(0,1)$ such that if $\delta\leq\delta_{0}$, then

[TABLE]

Now, let us consider $n=1,2$. By Theorem 4.2 and Theorem 2.9, (61) would become

[TABLE]

By using $\chi=\frac{3}{2}$, one will still obtain (90) and (91). Then the left proof is the same as above.

In each case, we have that if $0<\delta\leq\delta_{0}$, then

[TABLE]

Applying this estimate to $-u$, one have

[TABLE]

Meanwhile, it follows from Lemma 5.2 and Lemma 5.3 that there exists $\ell>0$ such that either

[TABLE]

or

[TABLE]

and either

[TABLE]

or

[TABLE]

In any case, we obtain

[TABLE]

By an iterative lemma, e.g. Lemma 3.4 in Han-Lin [18] (or Lemma B.2 in [19]), there exist $\alpha$ and $C$, both of which depend only on $\lambda,\Lambda,n,p$ and $q$, such that

[TABLE]

The conclusion follows from the above and Theorem 4.5. ∎

It has been pointed by the referee that Theorem 5.12 also follows from applying Theorem 5.4 to the solution that is extended to be zero for $t<-1$.

Similar to the justifications of Theorem 5.5 and Theorem 5.12, we also have

Theorem 5.13**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7) and the initial condition $u(\cdot,-1)=0$, where the coefficients of the equation satisfy (1), (16), (50) and (51) for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Then for every $\gamma>0$, there exist $\alpha>0$ and $C>0$, both of which depend only on $\lambda,\Lambda,n,p,\gamma$ and $q$, such that for every $(x,-1),(y,s)\in B_{1/4}(e_{n}/2)\times[-1,-\frac{3}{4}]$, there holds

[TABLE]

where $e_{n}=(0,\cdots,0,1)$.

Together with Theorem 5.5 and Theorem 5.13, using similar scaling arguments to those in the proof of Theorem 5.11, we have

Theorem 5.14**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7) and the initial condition $u(\cdot,-1)=0$, where the coefficients of the equation satisfy (1), (16), (50) and (51) for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Then for every $\gamma>0$, there exist $\alpha>0$ and $C>0$, both of which depend only on $\lambda,\Lambda,n,p,\gamma$ and $q$, such that for every $(x,t),(y,s)\in B_{1/4}(e_{n}/2)\times[-1,0]$, there holds

[TABLE]

where $e_{n}=(0,\cdots,0,1)$.

Proof.

The proof is in the same spirit as that of Theorem 5.11. We omit the details, and one can also refer to the proof of Theorem 5.15 in the below. ∎

Finally, we have the space-time global Hölder estimate:

Theorem 5.15**.**

Suppose $u\in C([-1,0];L^{2}(B_{1}^{+},x_{n}^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0,L}^{1}(B_{1}^{+}))$ is a weak solution of (6) with the partial boundary condition (7) and the initial condition $u(\cdot,-1)=0$, where the coefficients of the equation satisfy (1), (16), (50) and (51) for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Then for every $\gamma>0$, there exist $\alpha>0$ and $C>0$, both of which depend only on $\lambda,\Lambda,n,p,\gamma$ and $q$, such that for every $(x,t),(y,s)\in B_{1/2}^{+}\times[-1,0]$, there holds

[TABLE]

Proof.

By Theorem 4.5 and normalization, we assume $\sup_{B_{3/4}^{+}\times[-1,0]}|u|+F_{1}=1$. For any $\bar{x}=(0,\bar{x}_{n})\in B_{1/4}^{+}$ and $\bar{t}\in(-1,0]$, we let $R:=\max(\bar{x}_{n},(\bar{t}+1)^{\frac{1}{p+2}})>0$, and rescale the solution and the coefficients as in (57) with $x_{0}=0$. Then we have (58) in $\mathcal{Q}^{+}_{1/R}$. Also, (59) and (60) hold with $Q_{1}^{+}$ replaced by $\widetilde{Q}^{+}=B_{2}^{+}\times(-R^{-p-2},-R^{-p-2}+1])$.

Case 1: $R=\bar{x}_{n}$.

Consider $s\in(-1,\bar{t}]$. If $|\bar{t}-s|\leq R^{2p+4}$, then by Theorem 5.14, we have

[TABLE]

If $|\bar{t}-s|\geq R^{2p+4}$, then we have

[TABLE]

This shows that $u$ is Hölder continuous in the time variable.

Consider $\tilde{x}=(\tilde{x}^{\prime},\tilde{x}_{n})\in B_{1/2}^{+}$ such that $\tilde{x}_{n}\leq\bar{x}_{n}$. If $\tilde{x}\in B_{R^{2}}(\bar{x})$, then by Theorem 5.14, we have

[TABLE]

If $\tilde{x}\not\in B_{R^{2}}(\bar{x})$, then we have

[TABLE]

where we used Theorem 5.12 in the second inequality. This shows that $w$ is Hölder continuous in the spatial variables.

Case 2: $R=(\bar{t}+1)^{\frac{1}{p+2}}$.

Consider $s\in(-1,\bar{t}]$. If $|\bar{t}-s|\leq R^{2p+4}$, then by Theorem 5.11, we have

[TABLE]

If $|\bar{t}-s|\geq R^{2p+4}$, then by Theorem 5.12, we have

[TABLE]

This shows that $u$ is Hölder continuous in the time variable.

Consider $\tilde{x}=(\tilde{x}^{\prime},\tilde{x}_{n})\in B_{1/2}^{+}$ such that $\tilde{x}_{n}\leq\bar{x}_{n}$. If $\tilde{x}\in B_{R^{2}}(\bar{x})$, then by Theorem 5.5, we have

[TABLE]

If $\tilde{x}\not\in B_{R^{2}}(\bar{x})$, then we have

[TABLE]

where we used Theorem 5.12 in the second inequality. This shows that $u$ is Hölder continuous in the spatial variables.

Together with Theorem 4.5, we finish the proof of this theorem. ∎

5.5 The Cauchy-Dirichlet problem

In the end, let us go back to the Cauchy-Dirichlet problem in general domains mentioned at the beginning:

[TABLE]

where $\Omega\subset\mathbb{R}^{n}$, $n\geq 1$, is a smooth bounded open set, and $\omega$ is a smooth function in $\overline{\Omega}$ comparable to the distance function $d(x):=\operatorname{dist}(x,\partial\Omega)$, that is, $0<\inf_{\Omega}\frac{\omega}{d}\leq\sup_{\Omega}\frac{\omega}{d}<\infty$, and $p>-1$ is a constant.

Suppose there exist $0<\lambda\leq\Lambda<\infty$ such that

[TABLE]

and

[TABLE]

for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$, where $\chi>1$ is the constant in Theorem 2.9 or Theorem 2.10 depending on the value of $p$.

We say that $u$ is a weak solution of (92) if $u\in C((-1,0];L^{2}(\Omega,\omega^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0}^{1}(\Omega))$, $u(\cdot,-1)\equiv 0$, and satisfies

[TABLE]

for every $\varphi\in\{g\in L^{2}(\Omega\times(-1,0]):\partial_{t}g\in L^{2}(\Omega\times(-1,0],\omega^{p}\mathrm{d}x\mathrm{d}t),D_{i}g\in L^{2}(\Omega\times(-1,0]),i=1,\cdots,n,g=0\mbox{ on }\partial\Omega\times(-1,0]\}.$

Theorem 5.16**.**

Suppose $p>-1$, (93), (94) and (95) hold for some $q>\max(\frac{\chi}{\chi-1},\frac{n+p+2}{2},\frac{n+2p+2}{p+2})$. Then there exists a unique weak solution $u\in C((-1,0];L^{2}(\Omega,\omega^{p}\mathrm{d}x))\cap L^{2}((-1,0];H_{0}^{1}(\Omega))$ of (92). Furthermore, for every $\gamma>0$, there exist $\alpha>0$ and $C>0$, both of which depend only on $\lambda,\Lambda,n,\Omega,p,\gamma$ and $q$, such that for every $(x,t),(y,s)\in\Omega\times[-1,0]$, there holds

[TABLE]

Proof.

The Hölder estimate of the weak solution follows from Theorem 5.15, Theorem 5.14, the flattening boundary technique and a covering argument.

The uniqueness of the weak solution follows from a similar energy estimate to that in Theorem 3.6.

The existence of weak solutions follows by a similar argument to the proof of Theorem 3.7. Here, we do not need to assume $a$ to be continuous, since the approximating solutions in the proof of Theorem 3.7 under the assumption of this theorem will be uniformly Hölder continuous up to the boundary. The argument there will go through without the assumption of the continuity of $a$. We leave the details to the readers. ∎