An existence result for nonhomogeneous quasilinear parabolic equations beyond the duality pairing

TL;DR

This paper establishes the existence of very weak solutions for nonhomogeneous quasilinear parabolic equations using advanced weighted space estimates and compactness methods, extending beyond traditional duality approaches.

Contribution

It introduces a novel approach combining Calderón-Zygmund theory and Muckenhoupt weights to prove existence beyond duality pairing for these equations.

Findings

Proved existence of very weak solutions in weighted spaces.

Developed new a priori estimates using Calderón-Zygmund machinery.

Extended solution existence results beyond classical duality methods.

Abstract

In this paper, we prove existence of \emph{very weak solutions} to nonhomogeneous quasilinear parabolic equations beyond the duality pairing. The main ingredients are a priori esitmates in suitable weighted spaces combined with the compactness argument developed in \cite{bulicek2018well}. In order to obtain the a priori estimates, we make use of the full Calder\'on-Zygmund machinery developed in the past few years and combine it with some sharp bounds for the subclass of Muckenhoupt weights considered in this paper.

Equations

$$\left\{\begin{array}[]{ll}u_{t}-\operatorname{div}\mathcal{A}(x,t,\nabla u+\vec{h})=-\operatorname{div}|\vec{f}|^{p-2}\vec{f}&\qquad\text{in}\qquad\Omega\times(0,T),\\ u=0&\qquad\text{on}\qquad\partial_{p}(\Omega\times(0,T)),\end{array}\right.$$

$$\frac{2n}{n+2}<p<\infty .$$

$$\begin{array}[]{c}\langle\mathcal{A}(x,t,\zeta-\mathcal{A}(x,t,\eta),\zeta-\eta\rangle\geq\Lambda_{0}\left(|\zeta|^{2}+|\eta|^{2}\right)^{\frac{p-2}{2}}|\zeta-\eta|^{2},\\ |\mathcal{A}(x,t,\zeta)-\mathcal{A}(x,t,\eta)|\leq\Lambda_{1}|\zeta-\eta|\left(|\zeta|^{2}+|\eta|^{2}\right)^{\frac{p-2}{2}}.\end{array}$$

$$|\mathcal{A}(x,t,\zeta )|\le {\Lambda }_1|\zeta {|}^{p-1}.$$

$$B_r(0)\cap \{y_n>\gamma r\}\subset B_r(0)\cap \Omega \subset B_r(0)\cap \{y_n>-\gamma r\}.$$

$$|{\Omega }^c\cap B_r(x)|\ge m_e|B_r(x)|.$$

$$\Theta (\mathcal{A},Q_{\rho ,s}(\mathfrak{z}))(x,t):=\underset{\zeta \in {\mathbb{R}}^n\setminus \{0\}}{\sup }\frac{\left|\mathcal{A}(x,t,\zeta )-{\overline{\mathcal{A}}}_{B_{\rho }(\mathfrak{x})}(t,\zeta )\right|}{|\zeta {|}^{p-1}},$$

$${\overline{\mathcal{A}}}_{B_{\rho }(\mathfrak{x})}(t,\zeta ):={\text{fint}}_{B_{\rho }(\mathfrak{x})}\mathcal{A}(x,t,\zeta )dx.$$

$$[\mathcal{A}]_{\tau,S_{0}}:=\sup_{\begin{subarray}{c}0<\rho\leq S_{0}\\ 0<s\leq S_{0}^{2}\end{subarray}}\sup_{\begin{subarray}{c}\mathfrak{z}\in\mathbb{R}^{n+1}\\ \mathfrak{z}:=(\mathfrak{x},\mathfrak{t})\end{subarray}}\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-7.18916pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-6.15741pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-4.34457pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-3.59457pt}}\!\iint_{Q_{\rho,s}(\mathfrak{z})}\left|\Theta(\mathcal{A},B_{\rho}(\mathfrak{x}))(z)\right|^{\tau}\ dz\leq\gamma^{\tau}.$$

$$h_t=\mathrm{div}|\vec{H}{|}^{p-2}\vec{H}\text{in}{L}^{\frac{p-\beta }{p-1}}(0,T;W^{-1,\frac{p-\beta }{p-1}}(\Omega )).$$

$$\nabla h_k\to \nabla h=\vec{h}\text{ in }{L}^{p-\beta }({\Omega }_T).$$

$$\displaystyle\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-7.18916pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-6.15741pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-4.34457pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-3.59457pt}}\!\iint_{Q_{\rho,s}(\mathfrak{z})}|\Theta(\mathcal{A},Q_{\rho,s}(\mathfrak{z}))(z)|^{\mathfrak{t}}\ dz\leq C(\gamma,\Lambda_{1}),$$

$$\mathcal{M}(|f|)(x,t):=\sup_{\tilde{Q}\ni(x,t)}\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-7.18916pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-6.15741pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-4.34457pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-3.59457pt}}\!\iint_{\tilde{Q}}|f(y,s)|\ dy\ ds,$$

$$|\{z\in {\mathbb{R}}^{n+1}:\mathcal{M}(|f|)(z)>\alpha \}|\le \frac{5^{n+2}}{\alpha }\Vert f{\Vert }_{L^1({\mathbb{R}}^{n+1})},$$

$$\Vert \mathcal{M}(|f|){\Vert }_{L^{\vartheta }({\mathbb{R}}^{n+1})}\le C_{(n,\vartheta )}\Vert f{\Vert }_{L^{\vartheta }({\mathbb{R}}^{n+1})}.$$

$$\sup_{\mathfrak{z}\in\mathbb{R}^{n+1}}\sup_{\begin{subarray}{c}0<\rho<\infty,\\ 0<s<\infty\end{subarray}}\left(\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-7.18916pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-6.15741pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-4.34457pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-3.59457pt}}\!\iint_{Q_{\rho,s}(\mathfrak{z})}\omega(z)\ dz\right)\left(\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-7.18916pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-6.15741pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-4.34457pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-3.59457pt}}\!\iint_{Q_{\rho,s}(\mathfrak{z})}\omega^{\frac{-1}{q-1}}(z)\ dz\right)^{q-1}=:[\omega]_{q}<\infty.$$

$$\sup_{z\in\mathbb{R}^{n+1}}\sup_{\begin{subarray}{c}0<\rho<\infty,\\ 0<s<\infty\end{subarray}}\left(\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-7.18916pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-6.15741pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-4.34457pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-3.59457pt}}\!\iint_{Q_{\rho,s}(\mathfrak{z})}\omega(z)\ dz\right)\left|\left|\omega^{-1}\right|\right|_{L^{\infty}(Q_{\rho,s}(\mathfrak{z}))}=:[\omega]_{1}<\infty.$$

$${\left(\frac{1}{|Q|}{\iint }_{Q}f(x,t)dxdt\right)}^q\le \frac{c}{w(Q)}{\iint }_Q|f(x,t){|}^{q}w(x,t)dxdt,$$

$$\frac{1}{c}{\left(\frac{|E|}{|Q|}\right)}^q\le \frac{\omega (E)}{\omega (Q)}\le c{\left(\frac{|E|}{|Q|}\right)}^{\tau },$$

$${\left(\frac{1}{|Q|}{\iint }_Q\omega (z{)}^{1+\delta }dz\right)}^{\frac{1}{1+\delta }}\le C\frac{1}{|Q|}{\iint }_Q\omega (z)dz.$$

$$\omega (E)\le {\tau }_0{\left(\frac{|E|}{|Q|}\right)}^{{\tau }_1}\omega (Q).$$

$${\iint }_{\mathcal{O}}|\mathcal{M}(f){|}^q\omega dz\le C{\iint }_{\mathcal{O}}|f{|}^q\omega dz,$$

$$\Vert \varphi {\Vert }_{L^{\vartheta }(0,T;W^{1,\vartheta }(\Omega )}:={\left({\int }_0^T\Vert \varphi (\cdot ,t){\Vert }_{W^{1,\vartheta }(\Omega )}^{\vartheta }dt\right)}^{\frac{1}{\vartheta }}<\infty .$$

$${\int }_0^T\left({\int }_{\Omega }|\varphi (x,t){|}^{\vartheta }\omega (x,t)dx\right)dt<\infty .$$

$${\iint }_{{\Omega }_T}{g}^{\beta }w(z)dz=\beta {\int }_0^{\infty }{\lambda }^{\beta -1}w(\{z\in {\Omega }_T:g(z)>\lambda \})d\lambda =(\beta -\alpha ){\int }_0^{\infty }{\lambda }^{\beta -\alpha -1}\left({\iint }_{\{z\in {\Omega }_T:g(z)>\lambda \}}{g}^{\alpha }w(z)dz\right)d\lambda .$$

$$u_{h}(\cdot,t):=\left\{\begin{array}[]{ll}\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-13.5mu-$}$}}\kern-5.5225pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-13.5mu-$}$}}\kern-4.49075pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-13.5mu-$}$}}\kern-3.1779pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-13.5mu-$}$}}\kern-2.76125pt}}\!\int_{t}^{t+h}u(\cdot,\tau)\ d\tau&t\in(0,T-h),\\ 0&\text{else}.\end{array}\right.$$

$${\int }_{\Omega \times \{t\}}\frac{d[u{]}_h}{dt}\varphi +⟨[\mathcal{A}(x,t,\nabla u){]}_h,\nabla \varphi ⟩dx=0\text{for almost every}0<t<T-h.$$

$$X\subset B\subset Y\text{ with compact embedding }X\to B.$$

$$\left|\{x\in \tilde{\Omega }:f(x)=0\}\right|\ge m_e>0,$$

$${\int }_{\tilde{\Omega }}{\left|\frac{f}{\mathrm{diam}(\tilde{\Omega })}\right|}^{\vartheta }dx\le C_{(n,\vartheta ,m_e)}{\int }_{\tilde{\Omega }}|\nabla f{|}^{\vartheta }dx.$$

Full text

1 Introduction

In this paper, we are interested in obtaining existence of very weak solution to equations of the form

[TABLE]

where $\displaystyle\mathcal{A}(x,t,\zeta)$ is modelled after the well known $\displaystyle p$-Laplace operator, $\displaystyle\Omega$ is a bounded domain with possible nonsmooth boundary and $\displaystyle\vec{h}$ is a vector field that satisfies (iii) Assumptions on $\displaystyle\vec{h}$:.

Weak solutions to (1.1) are in the space $\displaystyle u\in L^{2}(0,T;L^{2}(\Omega))\cap L^{p}(0,T;W_{0}^{1,p}(\Omega))$ which allows one to use $\displaystyle u$ as a test function. But from the definition of weak solution, we see that the expression (see Definition 2.15) makes sense if we only assume $\displaystyle u\in L^{2}(0,T;L^{2}(\Omega))\cap L^{s}(0,T;W_{0}^{1,s}(\Omega))$ for some $\displaystyle s>\max\{p-1,1\}$. But under this milder notion of solution called very weak solution, we lose the ability to use $\displaystyle u$ as a test function. Subsequently, the technique of parabolic Lipschitz truncation was developed in the seminal paper of [24] where the authors could study certain properties of very weak solutions in the range of $\displaystyle(p-\beta,p)$ for all $\displaystyle\beta\in(0,\beta_{0})$ for sufficiently small $\displaystyle\beta_{0}$ depending only on the data.

One of the main questions was to obtain existence of very weak solutions in the range $\displaystyle(p-1,p)$. Though this problem is very formidable, nevertheless, there has been much progress achieved in the past few decades in understanding the question in the range $\displaystyle(p-\beta_{0},p)$ for all $\displaystyle\beta\in(0,\beta_{0})$. Here $\displaystyle\beta_{0}$ is some small universal constant depending only on the data. In this section, all the discussion will be restricted to the range $\displaystyle(p-\beta_{0},p)$ with the choice of $\displaystyle\beta_{0}$ appropriately chosen in the respective papers pertaining to the discussion.

The quasilinear elliptic analogue of the question considered in this paper was recently solved in [13] under suitable assumptions on the nonlinearity $\displaystyle\mathcal{A}(\cdot,\cdot)$ and the domain $\displaystyle\Omega$. In order to prove existence, they made use of a compactness argument developed in [12] along with a priori estimates in suitable weighted spaces. The weighted estimates in the linear case were first proved in [22] and using different techniques, a slightly weaker version was proved in [5] and a slightly more weaker version was proved in [12]. The weighted a priori estimates in the quasilinear case considered in [13] were obtained using the ideas that first originated in [27]. It should be noted that in [6], the required weighted a priori estimates in the elliptic case were also obtained, albeit with a stronger assumption on the nonlinear structure and a weaker assumption on the domain than those considered in [13]. This is the approach we follow to obtain the a priori weighted estimates in this paper. The precursor to all these elliptic estimates is the seminal paper of [25] which developed the method of Lipschitz truncation in the elliptic setting partially based on the ideas from [1].

In the parabolic case, the problem was recently addressed in [11] where the parabolic analogue of the compactness argument from [13] for the linear case was developed. The weighted/unweighted estimates were first proved in [22] for the linear case and a weaker version was reproved in [11]. While the compactness arguments from [11] were robust enough to work even in the quasilinear situation with relatively simple modifications, the a priori estimates developed in [22, 11] were specifically tailored to the linear case and hence could not be extended to handle quasilinear equations.

In this paper, we overcome this difficulty by obtaining the required a priori estimates using the ideas from [4] and combining them with several important observations regarding the weight class considered here. We then apply the modified compactness argument based on [11] to prove the required existence result (see Theorem 4.3). As discussed before, we consider the problem in the range $\displaystyle(p-\beta_{0},p)$ (see Definition 4.2 for the definition of $\displaystyle\beta_{0}$) and along the way, impose some restrictions on the nonlinearity $\displaystyle\mathcal{A}(\cdot,\cdot)$ and domain $\displaystyle\Omega$. It should be noted that our restriction on the domain $\displaystyle\Omega$ is less stringent whereas the restriction on the nonlinearity $\displaystyle\mathcal{A}(\cdot,\cdot)$ is more than that assumed in the elliptic analogues from [13]. One of the main tools used to obtain both the a priori estimates and the compactness argument is based on the parabolic Lipschitz truncation developed in the seminal paper of [24].

The plan of the paper is as follows: In Section 2, we collect some preliminary lemmas and assumptions and also describe the function spaces and Muckenhoupt weights, in Section 3, we collect few more important lemmas which will be useful in proving the compactness arguments, in Section 4, we describe the main theorems that will be proved, in Section 5, we shall prove the main weighted a priori estimates that will be needed to prove the existence of very weak solutions, in Section 6, we shall adapt the compactness arguments from [11] to our setting and finally in Section 7, we shall apply the results from Sections 5 and 6 to prove the existence of very weak solutions to Eq. 1.1.

2 Preliminaries

The following restriction on the exponent $\displaystyle p$ will always be enforced:

[TABLE]

Remark 2.1**.**

The restriction in (2.1) is necessary when dealing with parabolic problems because, we invariably have to deal with the $\displaystyle L^{2}$-norm of the solution which comes from the time-derivative. On the other hand, the following Sobolev embedding $\displaystyle W^{1,p}\hookrightarrow L^{2}$ is true provided (2.1) holds.

2.1 Assumptions on the Nonlinear structure

We shall now collect the assumptions on the nonlinear structure in (1.1). We assume that $\displaystyle\mathcal{A}(x,t,\nabla u)$ is a Carathéodory function, i.e., we have $\displaystyle(x,t)\mapsto\mathcal{A}(x,t,\zeta)$ is measurable for every $\displaystyle\zeta\in\mathbb{R}^{n}$ and $\displaystyle\zeta\mapsto\mathcal{A}(x,t,\zeta)$ is continuous for almost every $\displaystyle(x,t)\in\Omega\times(0,T)$. We also assume $\displaystyle\mathcal{A}(x,t,0)=0$ and $\displaystyle\mathcal{A}(x,t,\zeta)$ is differentiable in $\displaystyle\zeta$ away from the origin, i.e., $\displaystyle D_{\zeta}\mathcal{A}(x,t,\zeta)$ exists for a.e. $\displaystyle(x,t)\in\mathbb{R}^{n+1}$ and $\displaystyle\zeta\neq 0$.

We further assume that for a.e. $\displaystyle(x,t)\in\Omega\times(0,T)$ and for any $\displaystyle\eta,\zeta\in\mathbb{R}^{n}$, there exists two given positive constants $\displaystyle\Lambda_{0},\Lambda_{1}$ such that the following bounds are satisfied by the nonlinear structures:

[TABLE]

Note that from the assumption $\displaystyle\mathcal{A}(x,t,0)=0$, we get for a.e. $\displaystyle(x,t)\in\mathbb{R}^{n+1}$, there holds

[TABLE]

2.2 Structure of $\displaystyle\Omega$

The domain that we consider may be non-smooth but should satisfy some regularity condition. This condition would essentially say that at each boundary point and every scale, we require the boundary of the domain to be between two hyperplanes separated by a distance proportional to the scale.

Definition 2.2**.**

Given any $\displaystyle\gamma\in(0,1/8]$ and $\displaystyle S_{0}>0$, we say that $\displaystyle\Omega$ is $\displaystyle(\gamma,S_{0})$-Reifenberg flat domain if for every $\displaystyle x_{0}\in\partial\Omega$ and every $\displaystyle r\in(0,S_{0}]$, there exists a system of coordinates $\displaystyle\{y_{1},y_{2},\ldots,y_{n}\}$ (possibly depending on $\displaystyle x_{0}$ and $\displaystyle r$) such that in this coordinate system, $\displaystyle x_{0}=0$ and

[TABLE]

The class of Reifenberg flat domains is standard in obtaining Calderón-Zygmund type estimates, in the elliptic case, see [7, 16, 19] and references therein whereas for the parabolic case, see [10, 14, 15, 28] and references therein.

From the definition of $\displaystyle(\gamma,S_{0})$-Reifenberg flat domains, it is easy to see that the following property holds:

Lemma 2.3**.**

Suppose that $\displaystyle\Omega$ is a $\displaystyle(\gamma,S_{0})$-Reifenberg flat domain, then there exists an $\displaystyle m_{e}=m_{e}(\gamma,S_{0},n)\in(0,1)$ such that for every $\displaystyle x\in{\Omega}$ and every $\displaystyle r>0$, there holds

[TABLE]

2.3 Smallness Assumption

In order to prove the main results, we need to assume a smallness condition satisfied by $\displaystyle(\mathcal{A},\Omega)$.

Definition 2.4**.**

We say $\displaystyle(\mathcal{A},\Omega)$ is $\displaystyle(\gamma,S_{0})$-vanishing if the following assumptions hold:

(i) Assumption on $\displaystyle\mathcal{A}$:

For any parabolic cylinder $\displaystyle Q_{\rho,s}(\mathfrak{z})$ centered at $\displaystyle\mathfrak{z}:=(\mathfrak{x},\mathfrak{t})\in\mathbb{R}^{n+1}$, let us define the following:

[TABLE]

where we have used the notation

[TABLE]

Then $\displaystyle\mathcal{A}$ is said to be $\displaystyle(\gamma,S_{0})$-vanishing if for some $\displaystyle\tau\in[1,\infty)$, there holds

[TABLE]

Here we have used the notation $\displaystyle z:=(x,t)\subset\mathbb{R}^{n+1}$ and $\displaystyle dz:=dx\ dt$.

(ii) Assumption on $\displaystyle\partial\Omega$:

We require that $\displaystyle\Omega$ is a $\displaystyle(\gamma,S_{0})$-Reifenberg flat in the sense of Definition 2.2.

(iii) Assumptions on $\displaystyle\vec{h}$:

In what follows, we shall assume that for any $\displaystyle\beta\in(0,\beta_{0})$ (where $\displaystyle\beta_{0}$ is from Definition 4.2), the vector field $\displaystyle\vec{h}$ satisfies the following hypothesis:

1. (a)

There exists a function $\displaystyle h\in L^{p-\beta}(0,T;W^{1,p-\beta}(\Omega))$ such that $\displaystyle\vec{h}=\nabla h$. 2. (b)

For the function $\displaystyle h$ from Item a, it satisfies $\displaystyle h_{t}\in L^{\frac{p-\beta}{p-1}}(0,T;W^{-1,\frac{p-\beta}{p-1}}(\Omega))$. Furthermore, there exists a vector field $\displaystyle\vec{H}\in L^{p-\beta}(\Omega_{T})$ such that

[TABLE] 3. (c)

For the function $\displaystyle h$ from Item a, there eixsts an approximating sequence $\displaystyle\{h_{k}\}_{k\geq 1}\subset C^{\infty}(\Omega_{T})$ such that

[TABLE]

Remark 2.5**.**

From (2.2), we see that $\displaystyle|\Theta(\mathcal{A},Q_{\rho,s}(\mathfrak{z}))(x,t)|\leq 2\Lambda_{1}$, thus combining this with the assumption (2.3), we see from standard interpolation inequality that for any $\displaystyle 1\leq\mathfrak{t}<\infty$, there holds

[TABLE]

with $\displaystyle C(\gamma,\Lambda_{1})\rightarrow 0$ whenever $\displaystyle\gamma\rightarrow 0$.

2.4 Some results about Maximal functions

For any $\displaystyle f\in L^{1}(\mathbb{R}^{n+1})$, let us now define the strong maximal function in $\displaystyle\mathbb{R}^{n+1}$ as follows:

[TABLE]

where the supremum is taken over all parabolic cylinders $\displaystyle\tilde{Q}_{a,b}$ with $\displaystyle a,b\in\mathbb{R}^{+}$ such that $\displaystyle(x,t)\in\tilde{Q}_{a,b}$. An application of the Hardy-Littlewood maximal theorem in $\displaystyle x-$ and $\displaystyle t-$ directions shows that the Hardy-Littlewood maximal theorem still holds for this type of maximal function (see [26, Lemma 7.9] for details):

Lemma 2.6**.**

If $\displaystyle f\in L^{1}(\mathbb{R}^{n+1})$, then for any $\displaystyle\alpha>0$, there holds

[TABLE]

and if $\displaystyle f\in L^{\vartheta}(\mathbb{R}^{n+1})$ for some $\displaystyle 1<\vartheta\leq\infty$, then there holds

[TABLE]

2.5 Muckenhoupt weights

In this subsection, let us collect all the properties of the weights that will be considered in the paper. See [23, Chapter 9] for the details concerning this subsection.

Definition 2.7** (Strong Muckenhoupt Weight).**

A non negative, locally integrable function $\displaystyle\omega$ is a strong weight in $\displaystyle A_{q}(\mathbb{R}^{n+1})$ for some $\displaystyle 1<q<\infty$ if

[TABLE]

In the case $\displaystyle q=1$, we define the strong $\displaystyle A_{1}(\mathbb{R}^{n+1})$ weight to be the class of non negative, locally integrable function $\displaystyle\omega\in A_{1}(\mathbb{R}^{n+1})$ satisfying

[TABLE]

The quantity $\displaystyle[w]_{q}$ for $\displaystyle 1\leq q<\infty$ will be called as the $\displaystyle A_{q}$ constant of the weight $\displaystyle\omega$.

We will need the following important characterization of Muckenhoupt weights:

Lemma 2.8**.**

A parabolic weight $\displaystyle w\in A_{q}$ for $\displaystyle 1<q<\infty$ if and only if

[TABLE]

holds for all non-negative, locally integrable functions $\displaystyle f$ and all cylinders $\displaystyle Q=Q_{\rho,s}$.

As a direct consequence of Lemma 2.8, the following Lemma holds:

Lemma 2.9**.**

Let $\displaystyle\omega\in A_{q}(\mathbb{R}^{n+1})$ for some $\displaystyle 1<q<\infty$, then there exists positive constants $\displaystyle c=c(n,q,[\omega]_{q})$ and $\displaystyle\tau=\tau(n,q,[\omega]_{q})\in(0,1)$ such that

[TABLE]

for all $\displaystyle E\subset Q$ and all parabolic cylinders $\displaystyle Q_{\rho,s}$.

It is well known that the class of Muckenhoupt weights satisfy a reverse Hölder inequality (see for example [23, Theorem 9.2.2] for the details) given by

Lemma 2.10**.**

Let $\displaystyle\omega\in A_{q}$ for some $\displaystyle 1\leq q<\infty$, then there exists constants $\displaystyle C=C(n,q,[w]_{A_{q}})$ and $\displaystyle\delta=\delta(n,q,[w]_{A_{q}})$ such that for every cube, there holds

[TABLE]

As a corollary, the following self-improvement property holds.

Lemma 2.11**.**

Let $\displaystyle 1<q<\infty$ and suppose $\displaystyle\omega\in A_{q}$ be a given weight, then there exists an $\displaystyle\varepsilon_{0}=\varepsilon_{0}(n,q,[\omega]_{q})>0$ such that $\displaystyle\omega\in A_{q-\varepsilon_{0}}$ with the estimate $\displaystyle[\omega]_{q-\varepsilon_{0}}\leq C[\omega]_{q}$ where $\displaystyle C=C{(q,n,[\omega]_{q})}$.

We will now define the $\displaystyle A_{\infty}$ class as follows:

Definition 2.12**.**

A weight $\displaystyle\omega\in A_{\infty}$ if and only if there are constants $\displaystyle\tau_{0},\tau_{1}>0$ such that for every parabolic cylinder $\displaystyle Q=Q_{\rho,s}\subset\mathbb{R}^{n+1}$ and every measurable $\displaystyle E\subset Q$, there holds

[TABLE]

Moreover, if $\displaystyle\omega$ is an $\displaystyle A_{q}$ weight with $\displaystyle[\omega]_{q}\leq\overline{\omega}$, then the constants $\displaystyle\tau_{0}$ and $\displaystyle\tau_{1}$ can be chosen such that $\displaystyle\max\{\tau_{0},1/\tau_{1}\}\leq c(\overline{\omega},n)$.

From the general theory of Muckenhoupt weights, we see that $\displaystyle A_{\infty}=\bigcup_{1\leq q<\infty}A_{q}$.

We now have the following important bounds for the Hardy Littlewood maximal function on weighted spaces (for example, see [23, Chapter 9] for more on this).

Theorem 2.13**.**

Let $\displaystyle 1<q<\infty$ and suppose that $\displaystyle\omega\in A_{q}(\mathbb{R}^{n+1})$ and let $\displaystyle\mathcal{O}$ be a bounded open subset of $\displaystyle\mathbb{R}^{n+1}$. Then for any $\displaystyle f\in L^{q}_{\omega}(\mathcal{O})$, there holds

[TABLE]

where $\displaystyle C=C(n,q,[\omega]_{q})$.

2.6 Function Spaces

Let $\displaystyle 1\leq\vartheta<\infty$, then $\displaystyle W_{0}^{1,\vartheta}(\Omega)$ denotes the standard Sobolev space which is the completion of $\displaystyle C_{c}^{\infty}(\Omega)$ under the $\displaystyle\|\cdot\|_{W^{1,\vartheta}}$ norm.

The parabolic space $\displaystyle L^{\vartheta}(0,T;W^{1,\vartheta}(\Omega))$ is the collection of measurable functions $\displaystyle\phi(x,t)$ such that for almost every $\displaystyle t\in(0,T)$, the function $\displaystyle x\mapsto\phi(x,t)$ belongs to $\displaystyle W^{1,\vartheta}(\Omega)$ with the following norm being finite:

[TABLE]

Analogously, the parabolic space $\displaystyle L^{\vartheta}(0,T;W_{0}^{1,\vartheta}(\Omega))$ is the collection of measurable functions $\displaystyle\phi(x,t)$ such that for almost every $\displaystyle t\in(0,T)$, the function $\displaystyle x\mapsto\phi(x,t)$ belongs to $\displaystyle W_{0}^{1,\vartheta}(\Omega)$.

Given a weight $\displaystyle\omega\in A_{\vartheta}$, the weighted Lebesgue space $\displaystyle L^{\vartheta}(0,T;L^{\vartheta}_{\omega}(\Omega))$ is the set of all measurable functions $\displaystyle\phi:\Omega_{T}\mapsto\mathbb{R}$ satisfying

[TABLE]

Let us recall the following important characterization of Lebesgue spaces:

Lemma 2.14**.**

Let $\displaystyle\Omega$ be a bounded domain in $\displaystyle\mathbb{R}^{n}$ and let $\displaystyle w\in L^{1}(\Omega_{T})$ be any non-negative function, then for all $\displaystyle\beta>\alpha>1$ and any non-negative measurable function $\displaystyle g(x,t):\Omega_{T}\mapsto\mathbb{R}$, there holds

[TABLE]

2.7 Very weak solution

There is a well known difficulty in defining the notion of solution for (1.1) due to a lack of time derivative of $\displaystyle u$. To overcome this, one can either use Steklov average or convolution in time. In this paper, we shall use the former approach (see also [20, Page 20, Equation (2.5)] for further details).

Let us first define Steklov average as follows: let $\displaystyle h\in(0,T)$ be any positive number, then we define

[TABLE]

Definition 2.15** (Very weak solution).**

Let $\displaystyle\beta\in(0,1)$ and $\displaystyle h\in(0,T)$ be given and suppose $\displaystyle p-\beta>1$. We then say $\displaystyle u\in L^{2}(0,T;L^{2}(\Omega))\cap L^{p-\beta}(0,T;W_{0}^{1,p-\beta}(\Omega))$ is a very weak solution of (1.1) if for any $\displaystyle\phi\in W_{0}^{1,\frac{p-\beta}{1-\beta}}(\Omega)\cap L^{\infty}(\Omega)$, the following holds:

[TABLE]

2.8 Notation

3 Some well known Lemmas

Let us now recall a well known compactness lemma proved in [29].

Theorem 3.1**.**

Let $\displaystyle X,B$ and $\displaystyle Y$ are Banach spaces such that

[TABLE]

Let $\displaystyle 1\leq p<\infty$ be fixed, $\displaystyle F$ be a bounded subset in $\displaystyle L^{p}(0,T;X)$ and $\displaystyle\partial_{t}F=\{\partial_{t}f\mid f\in F\}$ be bounded in $\displaystyle L^{1}(0,T;Y)$. Then $\displaystyle F$ is relatively compact in $\displaystyle L^{p}(0,T;B)$.

Let us recall an important version of Chacon’s biting lemma proved in [8].

Lemma 3.2**.**

Let $\displaystyle\mathcal{O}\subset\mathbb{R}^{n+1}$ be a bounded domain and $\displaystyle\{f^{k}\}_{k\geq 1}$ be a bounded sequence in $\displaystyle L^{1}(\mathcal{O})$. Then there exists a non-decreasing sequence of measurable subsets $\displaystyle F_{j}\subset\mathcal{O}$ with $\displaystyle\left\lvert\mathcal{O}\setminus F_{j}\right\rvert\to 0$ as $\displaystyle j\to\infty$ such that $\displaystyle\{f^{k}\}_{k\geq 1}$ is pre-compact in $\displaystyle L^{1}(F_{j})$ for each $\displaystyle j\geq 1$.

Before we conclude this subsection, let us now recall the well known Poincaré’s inequality (see [3, Corollary 8.2.7] for the proof):

Theorem 3.3**.**

Let $\displaystyle 1\leq\vartheta<\infty$, and $\displaystyle f\in W^{1,\vartheta}(\tilde{\Omega})$ for some bounded domain $\displaystyle\tilde{\Omega}$ and suppose that the following measure density condition holds:

[TABLE]

then there holds

[TABLE]

4 Main Theorems

Let us first state the main weighted estimate that will be obtained in this paper:

Theorem 4.1**.**

Let $\displaystyle S_{0}\in(0,\infty)$ be given, then there exist $\displaystyle\gamma_{0}=\gamma_{0}(n,p,\Lambda_{0},\Lambda_{1},\Omega)$ and $\displaystyle\beta_{2}=\beta_{2}(n,p,\Lambda_{0},\Lambda_{1},\Omega)$ such that for any $\displaystyle\gamma\in(0,\gamma_{0}]$, if $\displaystyle(\mathcal{A},\Omega)$ is $\displaystyle(\gamma,S_{0})$-vanishing, then the following holds for any $\displaystyle\beta\in(0,\beta_{2}]$: Let $\displaystyle\phi\in L^{1}_{\operatorname{loc}}(\Omega_{T})$ be a given function such that \displaystyle\mathcal{M}(|\phi|\chi_{\mathchoice{\raisebox{-3.0pt}{\displaystyle\displaystyle\Omega_{T}}}{\raisebox{-3.0pt}{\displaystyle\textstyle\Omega_{T}}}{\raisebox{-3.0pt}{\displaystyle\scriptstyle\Omega_{T}}}{\raisebox{-3.0pt}{\displaystyle\scriptscriptstyle\Omega_{T}}}})(z)<\infty almost everywhere and denote

[TABLE]

Furthermore, let $\displaystyle\vec{f}\in L^{p-\beta}(\Omega_{T})\cap L^{p}_{\omega}(\Omega_{T})$ and $\displaystyle\vec{h}\in L^{p-\beta}(\Omega_{T})\cap L^{p}_{\omega}(\Omega_{T})$ be any two given vector fields such that (iii) Assumptions on $\displaystyle\vec{h}$: holds and let $\displaystyle u\in L^{p-\beta}(0,T;W_{0}^{1,p-\beta}(\Omega))$ with $\displaystyle|\nabla u|\in L^{p}_{\omega}(\Omega_{T})$ be a distributional solution of Eq. 1.1 in the sense of Eq. 2.5, then the following a priori estimate is satisfied

[TABLE]

where $\displaystyle d=d(n,p,\beta)\geq 1$ is a universal constant.

Before we state the existence result, let us first collect the restrictions on the exponent $\displaystyle\beta_{0}$:

Definition 4.2** (Definition of $\displaystyle\beta_{0}$).**

The following restriction on the exponent $\displaystyle\beta_{0}=\beta_{0}(n,p,\Lambda_{0},\Lambda_{1})$ shall be imposed.

We need $\displaystyle\beta_{0}\leq\beta_{1}$ so that Theorem 5.1 holds. 2. 2.

We need $\displaystyle\beta_{0}\leq\beta_{2}$ so that Theorem 4.1 holds. 3. 3.

We need $\displaystyle\beta_{0}\leq\beta_{4}$ so that 7.1 holds.

We are now ready to state the main theorem which is the existence of very weak solutions for Eq. 1.1.

Theorem 4.3**.**

There exists constant $\displaystyle\beta_{0}=\beta_{0}(n,p,\Lambda_{0},\Lambda_{1})$ (as quantified in Definition 4.2) such that for all $\displaystyle\beta\in(0,\beta_{0}]$ the following holds: Let $\displaystyle S_{0}>0$ be given, then there exists constants $\displaystyle\gamma_{0}=\gamma_{0}(n,p,\Lambda_{0},\Lambda_{1},\beta,\Omega)$ such that if $\displaystyle(\mathcal{A},\Omega)$ is $\displaystyle(\gamma,S_{0})$-vanishing for some $\displaystyle\gamma\in(0,\gamma_{0})$, then for any vector fields $\displaystyle\vec{f}\in L^{p-\beta}(\Omega_{T})$ and $\displaystyle\vec{h}\in L^{p-\beta}(\Omega_{T})$ satisfying (iii) Assumptions on $\displaystyle\vec{h}$:, there exists a very weak solution $\displaystyle u\in L^{p-\beta}(0,T;W_{0}^{1,p-\beta}(\Omega))$ solving Eq. 1.1 in the sense of Eq. 2.5.

5 A priori estimates

In this section, we shall obtain the main a priori estimates that will be needed to prove the existence results. The first is an unweighted estimate below the natural exponent for very weak solutions of Eq. 1.1. The proof of this result follows essentially as in [4, Theorem 6.1] and the theorem reads as follows:

Theorem 5.1**.**

Let the nonlinearity $\displaystyle\mathcal{A}$ satisfy Eq. 2.2 and $\displaystyle\Omega$ be a bounded domain that satisfies Lemma 2.3. Then there exists $\displaystyle\beta_{1}=\beta_{1}(n,p,\Lambda_{0},\Lambda_{1},m_{e})\in(0,1)$ such that for all $\displaystyle\beta\in(0,\beta_{1}]$ and any two given vector fields $\displaystyle\vec{f}\in L^{p-\beta}(\Omega_{T})$ and $\displaystyle\vec{h}\in L^{p-\beta}(\Omega_{T})$ satisfying (iii) Assumptions on $\displaystyle\vec{h}$:, the following holds: For any very weak solution $\displaystyle u\in L^{p-\beta}(0,T;W_{0}^{1,p-\beta}(\Omega))$ solving Eq. 1.1 in the sense of Definition 2.15, there holds

[TABLE]

To obtain the existence of very weak solutions to Eq. 1.1, we need to obtain suitable weighted esitmates which we will obtain as follows. Let us first recall several important properties of the special class of Muckenhoupt weights that we will consider. The first is a description of a subset of $\displaystyle A_{1}$ weights (see for example [23, Theorem 9.2.7] or [30, Pages 229-230] for the details).

Lemma 5.2**.**

Let $\displaystyle\phi\in L^{1}_{\operatorname{loc}}(\Omega_{T})$ be any function such that $\displaystyle\mathcal{M}(\phi)(z)<\infty$ almost everywhere and $\displaystyle\alpha\in(0,1)$ be a given exponent, then the following two conclusions hold: Firstly, the function $\displaystyle\mathcal{M}(\phi)^{\alpha}(z)\in A_{1}(\mathbb{R}^{n+1})$ and secondly, the $\displaystyle A_{1}$ norm of this weight is given by

[TABLE]

The second lemma that we will need is a way to construct $\displaystyle A_{p}$ weights from $\displaystyle A_{1}$ weights (see for example [23, Exercise 9.1.2] for the details).

Lemma 5.3**.**

Given any weight $\displaystyle\omega\in A_{1}$ and any $\displaystyle 1<p<\infty$, we have $\displaystyle\omega^{1-p}\in A_{p}$ with

[TABLE]

We shall make the following important observations that will enable us to obtain the desired weighted estimates in this section.

Remark 5.4**.**

From Lemma 5.2, Lemma 5.3 and Lemma 2.10, we have the following three observations:

Observation 1:

Given any function $\displaystyle\phi\in L^{1}_{\operatorname{loc}}(\mathbb{R}^{n+1})$, the function given by $\displaystyle\mathcal{M}(\phi)^{\alpha(1-p)}(z)$ for some $\displaystyle\alpha\in(0,1/2]$ and $\displaystyle 1<p<\infty$ is an $\displaystyle A_{p}$ weight.

Observation 2:

The $\displaystyle A_{p}$ constant of the weight given by $\displaystyle\mathcal{M}(\phi)^{\alpha(1-p)}$ is independent of the function $\displaystyle\phi$ and $\displaystyle\alpha\in[0,1/2]$ and depends only $\displaystyle n$ and $\displaystyle p$. In particular,

[TABLE]

**Observation 3: **

As a consequence of Eq. 5.1 and Lemma 2.10, we see that the weight given by $\displaystyle\mathcal{M}(\phi)^{\alpha(1-p)}$ is in $\displaystyle A_{p-\delta_{0}}$ for some universal constant $\displaystyle\delta_{0}=\delta_{0}(n,p)$. In particular, the self improvement property is independent of $\displaystyle\phi$ and $\displaystyle\alpha$.

5.1 Covering arguments needed for the proof of Theorem 4.1

The proof of this theorem crucially uses some of the a priori estimates proved in [4, 18]. The a priori estimates below the natural exponent are proved in [4] and the covering argument we follow is obtained in [18] based on the techniques developed in the [2].

Let $\displaystyle\varepsilon\in(0,1)$ be given (which will eventually be chosen) and for $\displaystyle\beta\in(0,\beta_{2})$ (note that $\displaystyle\beta_{2}$ will eventually be fixed to depend on data), consider the weight given by

[TABLE]

For a given $\displaystyle\varepsilon\in(0,1)$, we can find a $\displaystyle\delta_{\varepsilon}=\delta_{\varepsilon}(n,p,\Lambda_{0},\Lambda_{1},\varepsilon)\in(0,1)$ such that [4, Theorem 5.6 and Theorem 5.7] holds. Here we take the largest possible choice of $\displaystyle\delta_{\varepsilon}$.

Claim 5.5**.**

There exists $\displaystyle\tilde{\beta}_{1}:=\frac{\delta_{\varepsilon}}{2(p-\delta_{\varepsilon})}$ such that for all $\displaystyle\beta\in(0,\tilde{\beta}_{1})$, we have

[TABLE]

Proof of 5.5.

We have the following sequence of estimates:

[TABLE]

To obtain (a), from the choice of $\displaystyle\tilde{\beta}_{1}$, we see that $\displaystyle\frac{\beta(p-\delta_{\varepsilon})}{\delta_{\varepsilon}}\leq\frac{1}{2}$ and $\displaystyle\frac{p}{p-\delta_{\varepsilon}}-1=\frac{\delta_{\varepsilon}}{p-\delta_{\varepsilon}}\leq 1$. ∎

Claim 5.6**.**

For any $\displaystyle\beta\in(0,\tilde{\beta}_{2})$, with $\displaystyle\tilde{\beta}_{2}:=\frac{p-1}{2}$, we have

[TABLE]

Proof of 5.6.

We have the following sequence of estimates:

[TABLE]

To obtain (a), we made use of the fact that $\displaystyle 2\beta\leq p-1$ from the hypothesis. ∎

From 5.5 and Lemma 2.11, we see that there exists constants $\displaystyle\tilde{\delta}=\tilde{\delta}(n,p,\delta_{\varepsilon})$ and $\displaystyle C=C(n,p,\delta_{\varepsilon})$ such that

[TABLE]

It is important to note that

[TABLE]

Claim 5.7**.**

If we further restrict $\displaystyle\beta\in(0,\tilde{\beta}_{3})$ where $\displaystyle\tilde{\beta}_{3}:=\frac{\delta_{\varepsilon}-\tilde{\delta}(p-\delta_{\varepsilon})}{2(p-\delta_{\varepsilon})}$, then we have the following ameliorated bound

[TABLE]

Proof of 5.7.

We have the following sequence of estimates:

[TABLE]

To obtain (a), we used the hypothesis which implies $\displaystyle\frac{\beta}{\frac{p}{p-\delta_{\varepsilon}}-\tilde{\delta}-1}\leq\frac{1}{2}$ along with the bound $\displaystyle\frac{\delta_{\varepsilon}}{p-\delta_{\varepsilon}}-\tilde{\delta}\leq 2$. ∎

Let us now take $\displaystyle\beta\leq\min\{\tilde{\beta}_{1},\tilde{\beta}_{2},\tilde{\beta}_{3}\}$ where $\displaystyle\tilde{\beta}_{1}$ is such that 5.5 holds, $\displaystyle\tilde{\beta}_{2}$ is such that 5.6 holds and $\displaystyle\tilde{\beta}_{3}$ is such that 5.7 holds. We now define

[TABLE]

where $\displaystyle d$ is defined to be

[TABLE]

and for any $\displaystyle\lambda>0$, denote

[TABLE]

We have the following important lemma proved in [4, Lemma 7.2]:

Lemma 5.8**.**

Let $\displaystyle\gamma\in(0,1)$ be any constant and $\displaystyle S_{0}$ be from Definition 2.4, then for any $\displaystyle\lambda\geq c_{e}\alpha$ where

[TABLE]

there exists a family of disjoint cylinders $\displaystyle\{K_{r_{i}}^{\lambda}(z_{i})\}_{i\in\mathbb{N}}$ with $\displaystyle z_{i}\in E_{\lambda}$ and $\displaystyle r_{i}\in(0,S_{0})$ such that

[TABLE]

Using the previous lemma, we have the following important weighted estimate:

Lemma 5.9**.**

There exists a constant $\displaystyle c^{*}=c^{*}(n,p)$ such that

[TABLE]

Proof.

Applying Hölder’s inequality, we have

[TABLE]

From 5.7 and Definition 2.7, we see that

[TABLE]

Analogously, we can estimate \displaystyle\mathchoice{{\vbox{\hbox{\displaystyle\textstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}}}\kern-7.18916pt}}{{\vbox{\hbox{\displaystyle\scriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}}}\kern-6.15741pt}}{{\vbox{\hbox{\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}}}\kern-4.34457pt}}{{\vbox{\hbox{\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}}}\kern-3.59457pt}}\!\iint_{K_{r_{i}}^{\lambda}(z_{i})}\left(\frac{\lvert\vec{h}\rvert+\lvert\vec{f}\rvert}{\gamma}\right)^{p-\delta_{\varepsilon}}\ dz. Thus combining Eq. 5.8 and Eq. 5.9 with Eq. 5.7, we get

[TABLE]

We take

[TABLE]

then from a simple calculation, we see that

[TABLE]

An analogous estimate also holds for the second term on the right hand side of Eq. 5.10. Thus combining Eq. 5.10 with Eq. 5.12, the proof of the lemma follows. ∎

We now have the following important lemma proved in [17, Lemma 4.3].

Lemma 5.10**.**

For any $\displaystyle\lambda\geq c_{e}\alpha$ as in Lemma 5.8, there exists a constant $\displaystyle N=N(\Lambda_{0},\Lambda_{1},n,p)>1$ such that for any $\displaystyle\varepsilon\in(0,1)$, there exists a small $\displaystyle\gamma=\gamma(\Lambda_{0},\Lambda_{1},\varepsilon,n,p)$ such that if $\displaystyle(\mathcal{A},\Omega)$ is $\displaystyle(\gamma,S_{0})$-vanishing in the sense of Definition 2.4, then there holds

[TABLE]

We now have the following important weighted estimates on the level sets:

Lemma 5.11**.**

Let the notation from Eq. 5.6 be in force and $\displaystyle c^{*}$ be as obtained in Lemma 5.9. Furthermore let $\displaystyle N$ be given from Lemma 5.10, then for any $\displaystyle\lambda\geq c_{e}\alpha$ as in Lemma 5.8 and for any $\displaystyle\varepsilon\in(0,1)$, there holds

[TABLE]

Here $\displaystyle\tau_{1}=\tau_{1}(n,p)$ is the exponent from Definition 2.12 applied with 5.6 under consideration.

Proof.

We observe that

[TABLE]

Combining the above estimate with Lemma 5.8 gives the proof of the lemma. ∎

5.2 Proof of Theorem 4.1

We are now ready to combine all the estimates to prove the main theorem:

Let $\displaystyle c_{e}$ be as given in Lemma 5.8 and $\displaystyle\alpha$ be from Eq. 5.5, then from Lemma 2.14, we get

[TABLE]

Estimate for $\displaystyle I$:

This term is estimated as follows:

[TABLE]

Note that the constant in the above estimate is independent of $\displaystyle\delta_{\varepsilon}$ since we have $\displaystyle p-1\leq p-\delta_{\varepsilon}\leq p$.

Estimate for $\displaystyle II$:

We estimate this term as follows:

[TABLE]

We now choose $\displaystyle\varepsilon=\varepsilon(n,p,\Lambda_{0},\Lambda_{1})$ small such that $\displaystyle C(n,p,\Lambda_{0},\Lambda_{1})\varepsilon^{(p-1)\tau_{1}}=\frac{1}{2}$ where $\displaystyle C(n,p,\Lambda_{0},\Lambda_{1})$ is the constant appearing in the above inequality.

Once we choose $\displaystyle\varepsilon=\varepsilon(n,p,\Lambda_{0},\Lambda_{1})$ based on Eq. 5.13, this fixes the choice of $\displaystyle\delta_{\varepsilon}=\delta_{\varepsilon}(n,p,\Lambda_{0},\Lambda_{1})$ which in turn fixes $\displaystyle\tilde{\delta}$. Now we take

[TABLE]

where $\displaystyle\tilde{\beta}_{1}$ is from 5.5, $\displaystyle\tilde{\beta}_{2}$ is from 5.6, $\displaystyle\tilde{\beta}_{3}$ is from 5.7 and $\displaystyle\beta_{1}$ is from Theorem 5.1. This completes the proof of the theorem.

6 Modified compactness theory from [11]

Let us first recall a Whitney type decomposition Lemma proved in [21, Lemma 3.1] or [9, Chapter 3]:

Lemma 6.1**.**

Let $\displaystyle\mathbb{E}$ be any closed set and $\displaystyle\lambda\in(0,\infty)$ be a fixed constant. Define $\displaystyle\kappa:=\lambda^{2-p}$, then there exists an $\displaystyle\kappa$-parabolic Whitney covering $\displaystyle\{Q_{i}(z_{i})\}$ of $\displaystyle\mathbb{E}^{c}$ in the following sense:

(W1)

$\displaystyle Q_{j}(z_{j})=B_{j}(x_{j})\times I_{j}(t_{j})$* where $\displaystyle B_{j}(x_{j})=B_{r_{j}}(x_{j})$ and $\displaystyle I_{j}(t_{j})=(t_{j}-\kappa r_{j}^{2},t_{j}+\kappa r_{j}^{2})$.*

(W2)

$\displaystyle d_{\lambda}(z_{j},\mathbb{E})=16r_{j}$.

(W3)

$\displaystyle\bigcup_{j}\frac{1}{2}Q_{j}(z_{j})=\mathbb{E}^{c}$.

(W4)

for all $\displaystyle j\in\mathbb{N}$, we have $\displaystyle 8Q_{j}\subset\mathbb{E}^{c}$ and $\displaystyle 16Q_{j}\cap\mathbb{E}\neq\emptyset$.

(W5)

if $\displaystyle Q_{j}\cap Q_{k}\neq\emptyset$, then $\displaystyle\frac{1}{2}r_{k}\leq r_{j}\leq 2r_{k}$.

(W6)

$\displaystyle\frac{1}{4}Q_{j}\cap\frac{1}{4}Q_{k}=\emptyset$* for all $\displaystyle j\neq k$.*

(W7)

\displaystyle\sum_{j}\chi_{\mathchoice{\raisebox{-3.0pt}{\displaystyle\displaystyle 4Q_{j}}}{\raisebox{-3.0pt}{\displaystyle\textstyle 4Q_{j}}}{\raisebox{-3.0pt}{\displaystyle\scriptstyle 4Q_{j}}}{\raisebox{-3.0pt}{\displaystyle\scriptscriptstyle 4Q_{j}}}}(z)\leq c(n)* for all $\displaystyle z\in\mathbb{E}^{c}$.*

For a fixed $\displaystyle k\in\mathbb{N}$, let us define $\displaystyle A_{k}:=\left\{j\in\mathbb{N}:\frac{3}{4}Q_{k}\cap\frac{3}{4}Q_{j}\neq\emptyset\right\}$, then we have

(W8)

For any $\displaystyle i\in\mathbb{N}$, we have $\displaystyle\#A_{i}\leq c(n)$.

(W9)

Let $\displaystyle i\in\mathbb{N}$ be given and let $\displaystyle j\in A_{i}$, then $\displaystyle\max\{|Q_{j}|,|Q_{i}|\}\leq C(n)|Q_{j}\cap Q_{i}|.$

(W10)

Let $\displaystyle i\in\mathbb{N}$ be given and let $\displaystyle j\in A_{i}$, then $\displaystyle\max\{|Q_{j}|,|Q_{i}|\}\leq\left|\frac{3}{4}Q_{j}\cap\frac{3}{4}Q_{i}\right|.$

(W11)

Let $\displaystyle i\in\mathbb{N}$ be given, then for any $\displaystyle j\in A_{i}$, we have $\displaystyle\frac{3}{4}Q_{j}\subset 4Q_{i}$.

Subordinate to the above Whitney covering, we have an associated partition of unity which we recall in the following lemma.

Lemma 6.2**.**

Associated to the covering given in Lemma 6.1, there exists functions $\displaystyle\{\Psi_{j}\}_{j\in\mathbb{N}}\in C_{c}^{\infty}\left(\frac{3}{4}Q_{j}\right)$ such that the following holds:

(W12)

\displaystyle\chi_{\mathchoice{\raisebox{-3.0pt}{\displaystyle\displaystyle\frac{1}{2}Q_{j}}}{\raisebox{-3.0pt}{\displaystyle\textstyle\frac{1}{2}Q_{j}}}{\raisebox{-3.0pt}{\displaystyle\scriptstyle\frac{1}{2}Q_{j}}}{\raisebox{-3.0pt}{\displaystyle\scriptscriptstyle\frac{1}{2}Q_{j}}}}\leq\Psi_{j}\leq\chi_{\mathchoice{\raisebox{-3.0pt}{\displaystyle\displaystyle\frac{3}{4}Q_{j}}}{\raisebox{-3.0pt}{\displaystyle\textstyle\frac{3}{4}Q_{j}}}{\raisebox{-3.0pt}{\displaystyle\scriptstyle\frac{3}{4}Q_{j}}}{\raisebox{-3.0pt}{\displaystyle\scriptscriptstyle\frac{3}{4}Q_{j}}}}.

(W13)

$\displaystyle\|\Psi_{j}\|_{\infty}+r_{j}\|\nabla\Psi_{j}\|_{\infty}+r_{j}^{2}\|\nabla^{2}\Psi_{j}\|_{\infty}+\lambda^{2-p}r_{j}^{2}\|\partial_{t}\Psi_{j}\|_{\infty}\leq C(n)$.

(W14)

Let $\displaystyle i\in\mathbb{N}$ be given, then $\displaystyle\sum_{j\in A_{i}}\Psi_{j}(z)=1$ for all $\displaystyle z\in\frac{3}{4}Q_{i}$.

In this section, let us take any exponent $\displaystyle q$ such that

[TABLE]

and let us denote

[TABLE]

We consider following problem: Let $\displaystyle\vec{F},\vec{H}\in L^{p}(\Omega_{T},\mathbb{R}^{n})$ be given and suppose that $\displaystyle w\in C(0,T;L^{2}(\Omega))\cap L^{p}(0,T;W_{0}^{1,p}(\Omega))$ be the unique weak solution of

[TABLE]

Let us now define the following function:

[TABLE]

where $\displaystyle\mathcal{M}$ is the Hardy Littlewood maximal function defined in (2.4).

For a fixed $\displaystyle\lambda>0$, let us define the good set by

[TABLE]

Since $\displaystyle E_{\lambda}^{c}$ is open, from Lemma 6.1 and Lemma 6.2, we define following extension:

[TABLE]

where

[TABLE]

Lemma 6.3**.**

We have the following estimates for the function constructed in Eq. 6.5:

1. (i)

For any $\displaystyle z\in E_{\lambda}^{c}$, there holds

[TABLE] 2. (ii)

For a given $\displaystyle i\in\mathbb{N}$ and any $\displaystyle j\in A_{i}$, there holds

[TABLE] 3. (iii)

Given any \displaystyle z\in E_{\mathchoice{\raisebox{-1.0pt}{\displaystyle\displaystyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\textstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptscriptstyle\lambda}}}^{c}, we have $\displaystyle z\in\frac{3}{4}Q_{i}$ for some $\displaystyle i\in\mathbb{N}$. Then there holds

[TABLE] 4. (iv)

For any $\displaystyle\vartheta\geq 1$, we have the following bound:

[TABLE]

Proof.

The proof of Item i follows from [4, Lemma 4.9], the proof of Item ii follows from [4, Lemma 4.10], the proof of Item iii follows from [4, Lemma 4.11] and finally the proof of Item iv follows from [4, Lemma 4.14]. ∎

We now prove an important pointwise estimate which follows the same idea as in [11, Lemma 3.1.1]:

Lemma 6.4**.**

Let $\displaystyle i\in\mathbb{N}$ and suppose $\displaystyle\frac{3}{4}Q_{i}\subset\Omega\times(0,\infty)$, then for any $\displaystyle j\in A_{i}$ and a.e. $\displaystyle(x,t)=z\in\frac{3}{4}Q_{i}$, we have

[TABLE]

Proof.

Without loss of generality we only need to prove

[TABLE]

since otherwise, we can apply triangle inequality and Eq. 6.6 to get

[TABLE]

We shall now split the proof into three cases, depending on where the cylinder $\displaystyle\frac{3}{4}Q_{i}$ lies.

Case $\displaystyle\frac{3}{4}Q_{i}\subset\Omega\times(0,T)$:

In this case, let us fix any $\displaystyle z_{0}=(x_{0},t_{0})\in\frac{3}{4}Q_{i}$, from which we get

[TABLE]

Let us now estimate each of the terms as follows:

Estimate for $\displaystyle I$:

We estimate this term as follows, without loss of generality, we shall assume $\displaystyle Q_{i}$ is centered at $\displaystyle(0,0)$.

[TABLE]

To obtain (a), we made use of (W13) along with the bound $\displaystyle|\tilde{x}-x_{0}|\apprle r_{i}$, to obtain (b), we enlarged the cylinder noting that $\displaystyle s\in[0,1]$ and finally to obtain (c), we made use of (W4).

Estimate for $\displaystyle II$:

Note that we have assumed without loss of generality that $\displaystyle Q_{i}$ is centered at $\displaystyle(0,0)$. Since $\displaystyle s\in[0,1]$, we see that $\displaystyle\frac{\tilde{x}+sx_{0}-x_{0}}{s}\leq\frac{3}{4}$ whenever $\displaystyle x_{0}\in B_{\frac{3}{4}r_{i}}(0)$ and $\displaystyle\tilde{x}\in B_{\frac{3}{4}r_{i}}(0)$. Thus we can make use of Eq. 6.2 to estimate $\displaystyle II$ as follows:

[TABLE]

To obtain (a), we made use of the weak formulation of Eq. 6.2, to obtain (b), we made use of (W13) and (W1), to obtain (c), we made use of Eq. 2.2 and finally to obtain (d), we made use of (W4).

From (W12), we note that $\displaystyle\frac{1}{\lVert\Psi_{i}\rVert_{L^{1}(\frac{3}{4}Q_{i})}}\approx\frac{1}{|Q_{i}|}$ and hence combining Eq. 6.9 and Eq. 6.10 into Eq. 6.8 proves the desired estimate.

Case $\displaystyle\frac{3}{4}Q_{i}$ crosses the lateral boundary:

In this case, we see that $\displaystyle\phi_{h}^{i}=0$, thus we have

[TABLE]

The terms $\displaystyle I$ and $\displaystyle II$ are estimated exactly as Eq. 6.9 and Eq. 6.10 respectively. In order to estimate $\displaystyle III$, we can apply Theorem 3.3 since $\displaystyle w_{h}=0$ outside the lateral boundaries. Thus we get

[TABLE]

Case $\displaystyle\frac{3}{4}Q_{i}$ crosses the initial boundary:

In this case, again we can proceed as in Eq. 6.11 to get

[TABLE]

and pick up $\displaystyle III$ as the error term which needs to be estimated. Since we are at the initial boundary, we cannot directly apply Theorem 3.3 to bound $\displaystyle III$ and instead we proceed analogously to [4, Estimate (4.15) and (4.16)] to again get the same bound as Eq. 6.12.

Combining all the estimates completes the proof of the lemma. ∎

We now have the Lipschitz regularity of the function constructed in Eq. 6.5, the proof of which can be found in [4, Lemma 4.16].

Lemma 6.5**.**

The function $\displaystyle\varphi_{\lambda,h}$ constructed in Eq. 6.5 is $\displaystyle C^{0,1}(\mathbb{R}^{n}\times[0,T])$ with respect to the parabolic metric given by

[TABLE]

In particular, the following bound holds for any $\displaystyle z_{1},z_{2}\in\mathbb{R}^{n+1}$:

[TABLE]

where $\displaystyle C$ depends on $\displaystyle n,p,q,\Lambda_{0},\Lambda_{1},\lambda,T$ and $\displaystyle\lVert w\rVert_{L^{1}(\Omega_{T})}$.

We now prove a weighted estimate which follows similarly to [11, Theorem 3.4].

Lemma 6.6**.**

Suppose that $\displaystyle\omega\in A_{\frac{p}{q}}(\Omega_{T})$ where $\displaystyle q$ as defined in Eq. 6.1 and $\displaystyle\lvert\nabla w\rvert^{q},\lvert\vec{F}\rvert^{q},\lvert\vec{H}\rvert^{q}\in L^{\frac{p}{q}}_{\omega}(\Omega_{T})$ hold. Then the function $\displaystyle g(z)$ constructed in (6.3) satisfies

[TABLE]

and

[TABLE]

where $\displaystyle C=C\left(n,p,q,\Lambda_{0},\Lambda_{1},[\omega]_{\frac{p}{q}}\right)$.

Proof.

From Theorem 2.13, we get

[TABLE]

which proves the first assertion.

To obtain the second assertion, we proceed as follows:

[TABLE]

∎

Lemma 6.7**.**

Suppose that $\displaystyle\omega\in A_{\frac{p}{q}}(\Omega_{T})$ where $\displaystyle q$ as defined in Eq. 6.1 and $\displaystyle\lvert\nabla w\rvert^{q},\lvert\vec{F}\rvert^{q},\lvert\vec{H}\rvert^{q}\in L^{\frac{p}{q}}_{\omega}(\Omega_{T})$ hold. Then

[TABLE]

Proof.

From Eq. 6.5, we see that $\displaystyle\{\varphi_{\lambda,h}\neq w_{h}\}\subseteq E_{\lambda}^{c}$. Thus we have the following sequence of estimates:

[TABLE]

This completes the proof of the lemma. ∎

We now prove a second weighted estimate which follows similarly to [11, Theorem 3.1].

Lemma 6.8**.**

Suppose that $\displaystyle\omega\in A_{\frac{p}{q}}(\Omega_{T})$ where $\displaystyle q$ as defined in Eq. 6.1 and $\displaystyle\lvert\nabla w\rvert^{q},\lvert\vec{F}\rvert^{q},\lvert\vec{H}\rvert^{q}\in L^{\frac{p}{q}}_{\omega}(\Omega_{T})$ hold. Then

[TABLE]

Proof.

From Eq. 6.5, we see that if \displaystyle z\in E_{\mathchoice{\raisebox{-1.0pt}{\displaystyle\displaystyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\textstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptscriptstyle\lambda}}}, then $\displaystyle\varphi_{\lambda,h}(z)=w_{h}(z)$ and hence the term on the left hand side of Eq. 6.13 is zero. Thus, we only have to integrate the term on the left hand side of Eq. 6.13 over \displaystyle E_{\mathchoice{\raisebox{-1.0pt}{\displaystyle\displaystyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\textstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptscriptstyle\lambda}}}^{c}.

Let \displaystyle z\in E_{\mathchoice{\raisebox{-1.0pt}{\displaystyle\displaystyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\textstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptscriptstyle\lambda}}}^{c}, then there exists an $\displaystyle i\in\mathbb{N}$ such that $\displaystyle z\in\frac{3}{4}Q_{i}$. Thus we get

[TABLE]

Making use of the previous pointwise bound, we get

[TABLE]

∎

6.1 Proof of the compactness theory.

We now prove the quasilinear analogue of [11, Theorem 3.1].

Theorem 6.9**.**

Let $\displaystyle 1<p<\infty$ and $\displaystyle\max\{1,p-1\}\leq q<p$ be given. Let $\displaystyle\omega\in A_{\frac{p}{q}}$ be an Muckenhoupt weight and let $\displaystyle\{\vec{F}^{k}\}_{k\geq 1}$, $\displaystyle\{\vec{H}^{k}\}_{k\geq 1}$ be any two sequences in $\displaystyle L^{q}(\Omega_{T})\cap L^{p}_{\omega}(\Omega_{T})$. Corresponding to these vector fields, let $\displaystyle w^{k}\in L^{q}(0,T;W^{1,q}_{0}(\Omega))\cap L^{p}_{\omega}(0,T;W^{1,p}_{\omega}(\Omega))$ be a very weak solution of

[TABLE]

Furthermore, assume that there exists constant $\displaystyle\mathbf{C}_{1}$ such that

[TABLE]

and

[TABLE]

Then for any fixed $\displaystyle\lambda>1$, there exists a sequence $\displaystyle\{\varphi_{\lambda,h}^{k}\}_{k\geq 1}\subset L^{q}(0,T;W^{1,q}_{0}(\Omega))$ satisfying following:

[TABLE]

Proof.

For each $\displaystyle k\geq 1$, let

[TABLE]

For each $\displaystyle k\geq 1$ and fixed $\displaystyle\lambda>1$, we shall find a suitable $\displaystyle\lambda_{k}$ and by an abuse of notation, we will denote $\displaystyle\varphi^{k}_{\lambda,h}=\varphi^{k}_{\lambda_{k},h}$. Using Lemma 6.6 along with Eq. 6.15, we have

[TABLE]

Let $\displaystyle m_{0}\in\mathbb{N}$ be a number to be eventually chosen, then for each $\displaystyle k\geq 1$, there exists an $\displaystyle 0\leq m_{k}\leq m_{0}$ such that

[TABLE]

Let us now define

[TABLE]

then we have

[TABLE]

∎

Let us now make the choice

[TABLE]

then we have

[TABLE]

which combined with Eq. 6.23 gives

[TABLE]

We are now ready to prove each of the conclusions of the theorem:

Proof of Eq. 6.16:

This follows from Lemma 6.5 since the Lipschitz extension is given by $\displaystyle\varphi_{\lambda_{k},h}^{k}$. It is important to note that the bounds are independent of $\displaystyle k\in\mathbb{N}$ because of Eq. 6.14 and Eq. 6.15.

Proof of Eq. 6.17:

This follows from Lemma 6.6 and making use of Eq. 6.14.

Proof of Eq. 6.18:

This follows from Lemma 6.8 along with the bound from Eq. 6.25.

Proof of Eq. 6.19:

Since \displaystyle\{z\in\Omega_{T}\mid\varphi_{\lambda,h}^{k}(z)\neq w_{h}^{k}(z)\}\subseteq\Omega_{T}\setminus E_{\mathchoice{\raisebox{-1.0pt}{\displaystyle\displaystyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\textstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptscriptstyle\lambda}}}^{k}, we can directly use Eq. 6.25 and Eq. 6.24 to prove this estimate.

Proof of Eq. 6.20:

Since \displaystyle\{z\in\Omega_{T}\mid\varphi_{\lambda,h}^{k}(z)\neq w_{h}^{k}(z)\}\subseteq\Omega_{T}\setminus E_{\mathchoice{\raisebox{-1.0pt}{\displaystyle\displaystyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\textstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptstyle\lambda}}{\raisebox{-1.0pt}{\displaystyle\scriptscriptstyle\lambda}}}^{k}, we can apply the weak type $\displaystyle 1-1$ estimate from Lemma 2.6 to get

[TABLE]

7 Proof of Theorem 4.3

In this section, let us take $\displaystyle\beta=\beta_{0}$ where $\displaystyle\beta_{0}$ is from Definition 4.2. Subsequently, we shall apply the results from Section 6 with

[TABLE]

Let us define

[TABLE]

Let us apply Theorem 4.1 with the above choice of $\displaystyle\phi$. Thus, we have

[TABLE]

From the properties of the Hardy-Littlewood Maximal function, we also have

[TABLE]

This implies that the assumption $\displaystyle\vec{f}\in L^{p-\beta}(\Omega_{T})$ gives $\displaystyle\vec{f}\in L^{p}_{\omega}(\Omega_{T})$ where we have set

[TABLE]

Claim 7.1**.**

For $\displaystyle\beta_{4}:=\min\left\{\frac{1}{p-1},p-1\right\}$, the following holds for all $\displaystyle\delta\in(0,\beta_{4})$:

[TABLE]

Proof.

Let us first consider the case $\displaystyle p\geq 2$. From Remark 5.4, we see that

[TABLE]

provided there exists $\displaystyle\alpha\in(0,1)$ such that the following holds

[TABLE]

In particular, we would need $\displaystyle\alpha=\delta(p-1)<1$, and this is possible if we choose $\displaystyle\delta<\frac{1}{p-1}$.

In the case $\displaystyle 1<p<2$, we proceed analogously to require would need $\displaystyle\alpha=\frac{\delta}{p-1}<1$ which again holds provided $\displaystyle\delta<p-1$. ∎

A simple application of Hölder’s inequality gives

[TABLE]

Since we are given $\displaystyle\vec{f}\in L^{p-\beta}(\Omega_{T})$ and $\displaystyle\vec{h}\in L^{p-\beta}(\Omega_{T})$, we shall consider the following approximation sequence for any $\displaystyle k\in(0,\infty)$:

1. (i)

Define $\displaystyle\vec{f}_{k}:=T_{k}(\vec{f})$ where $\displaystyle T_{k}(s)=s$ if $\displaystyle|s|\leq k$ and $\displaystyle T_{k}(s)=k$ if $\displaystyle|s|\geq k$. Note that we have used the notation $\displaystyle T_{k}(\vec{f})$ to denote $\displaystyle T_{k}$ acting on each component of $\displaystyle\vec{f}$ separately. 2. (ii)

Define $\displaystyle\vec{h}_{k}=\nabla h_{k}$ where $\displaystyle h_{k}\in C^{\infty}(\Omega_{T})$ is from Item c of Definition 2.4.

From Eq. 7.2 and the above construction, we see that for every $\displaystyle k\geq 1$, we have $\displaystyle\vec{f}_{k}\in L^{\infty}(\Omega_{T})$, $\displaystyle\vec{h}_{k}\in C^{\infty}(\Omega_{T})$ and

[TABLE]

As a consequence, for each $\displaystyle k\in\mathbb{N}$, there exists a unique weak solution $\displaystyle u^{k}\in C(0,T;L^{2}(\Omega))\cap L^{p}(0,T;W_{0}^{1,p}(\Omega))$ of

[TABLE]

We have the following observations:

• •

From Theorem 5.1 applied to Eq. 7.5, we see that

[TABLE]

• •

From Theorem 4.1 applied to Eq. 7.5 with weight defined as in Eq. 7.3, we have

[TABLE]

• •

Furthermore, applying the above two observations to Eq. 7.5, we see that

[TABLE]

As a consequence of the above observations, we have the following convergences (upto relabelling a suitable subsequence):

[TABLE]

To get the above convergence results, we made use of Theorem 3.1 along with the restriction Eq. 2.1 which ensures that the Sobolev exponent $\displaystyle p^{*}:=\frac{np}{n-p}>2$.

From Eq. 7.7, we see that the function $\displaystyle u\in L^{p-\beta}(0,T;W_{0}^{1,p-\beta}(\Omega))$ is a distributional solution of

[TABLE]

where the operator $\displaystyle\bar{\mathcal{A}}$ is a formal limit as obtained in Eq. 7.7.

Denote $\displaystyle v^{k}:=u^{k}-u$, then $\displaystyle v^{k}$ is a weak solution of

[TABLE]

From Eq. 7.7, we see that all the hypothesis of Theorem 6.9 are satisfied for the weak solutions of Eq. 7.8. Hence for each fixed $\displaystyle\lambda>1$, there exists a a family of functions $\displaystyle\{v_{\lambda,h}^{k}\}_{k\geq 1}$ satisfying the conclusions of Theorem 6.9.

Claim 7.2**.**

Suppose 7.3 holds, then the following holds:

[TABLE]

Proof.

From Eq. 7.7, Eq. 7.8 and Theorem 6.9, we see that $\displaystyle\{\langle\mathcal{A}^{k},\nabla u^{k}-\nabla u\rangle\omega\}_{k\geq 1}$, $\displaystyle\{\lvert\nabla u^{k}-\nabla u\rvert^{p}\omega\}_{k\geq 1}$, $\displaystyle\{|\nabla v^{k}|\omega\}_{k\geq 1}$ and $\displaystyle\{|\nabla v_{\lambda,h}^{k}|\omega\}_{k\geq 1}$ are bounded sequences in $\displaystyle L^{1}(\Omega_{T})$. As a consequence, we can apply Lemma 3.2 to get a sequence of measurable sets $\displaystyle\{F_{j}\}_{\{j\geq 1\}}$ such that $\displaystyle F_{j}\subset\Omega_{T}$, $\displaystyle\lvert\Omega_{T}\setminus F_{j}\rvert\to 0$ as $\displaystyle j\to\infty$ and having the property that $\displaystyle\{\langle\mathcal{A}^{k},\nabla u^{k}-\nabla u\rangle\omega\}_{k\geq 1}$ and $\displaystyle\{\lvert\nabla u^{k}-\nabla u\rvert^{p}\omega\}_{k\geq 1}$ are precompact in $\displaystyle L^{1}(F_{j})$ for all $\displaystyle j\geq 1$.

Therefore taking subsequence if necessary, for any $\displaystyle\varepsilon>0$ and $\displaystyle j\geq 1$, there exists a $\displaystyle\delta(\epsilon,j)>0$ such that for any $\displaystyle\mathcal{O}\subset F_{j}$, there holds

[TABLE]

From 7.3, we see that the following holds with $\displaystyle F=F_{j}$ for any $\displaystyle j\in\mathbb{N}$:

[TABLE]

Therefore for any $\displaystyle\Theta\in L^{p}_{\omega}(\Omega_{T})$, the following sequence of estimates hold:

[TABLE]

Let us take $\displaystyle\Theta=\nabla u+\varepsilon\left(\frac{\bar{\mathcal{A}}-\mathcal{A}(z,\nabla u+\vec{h})}{1+\lvert\bar{\mathcal{A}}-\mathcal{A}(z,\nabla u+\vec{h})\rvert}\right)$ where second term belong to $\displaystyle L^{\infty}(\Omega_{T})$, we get

[TABLE]

Since $\displaystyle\omega\geq 0$, we now let $\displaystyle j\to\infty$ to obtain we obtain (7.9). ∎

We are now left to prove the following claim:

Claim 7.3**.**

Let $\displaystyle j\in\mathbb{N}$ and $\displaystyle F_{j}$ be a measurable set obtained in 7.2. We then claim that the following limit holds:

[TABLE]

Proof.

From Eq. 6.5, Item iv and Eq. 7.7, we see that

[TABLE]

Therefore by Eq. 6.16 and Arzelà-Ascoli Theorem, taking a subsequence if necessary, we get

[TABLE]

Also, since for any $\displaystyle\eta\in C_{0}^{\infty}(\Omega_{T})$, there holds

[TABLE]

from which we have

[TABLE]

Making use of Eq. 7.11, for any $\displaystyle\eta\in C^{\infty}(\Omega_{T})$ with $\displaystyle\eta=0$ on $\displaystyle\partial_{p}(\Omega_{T})$, we have

[TABLE]

Making use of Eq. 7.4 along with Eq. 7.8, Eq. 7.11 and Eq. 7.12, we see that Eq. 7.13 becomes

[TABLE]

A simple calculation shows

[TABLE]

Using Eq. 7.7 along with Eq. 7.11, we see that the first two terms on the right hand side of Eq. 7.14 goes to zero as $\displaystyle k\rightarrow\infty$. Thus, Eq. 7.13 becomes

[TABLE]

From Eq. 7.7 and Eq. 6.16, we see that

[TABLE]

from which (after possibly taking a subsequence if necessary), for any $\displaystyle\eta\in L^{\left(\frac{p-\beta}{p-1}\right)^{\prime}}(\Omega_{T})$, we have

[TABLE]

Since our weight function from Eq. 7.3 $\displaystyle\omega\in L^{\infty}(\Omega_{T})$, we take \displaystyle\eta=\omega{\mathchoice{\raisebox{0.0pt}{\displaystyle\displaystyle\chi}}{\raisebox{0.0pt}{\displaystyle\textstyle\chi}}{\raisebox{0.0pt}{\displaystyle\scriptstyle\chi}}{\raisebox{0.0pt}{\displaystyle\scriptscriptstyle\chi}}}_{F_{j}} where $\displaystyle F_{j}$ is some measurable subset of $\displaystyle\Omega_{T}$. Now applying Hölder’s inequality and Eq. 6.18 to the expression on the right hand side of (7.15), we get

[TABLE]

And since \displaystyle{\mathchoice{\raisebox{0.0pt}{\displaystyle\displaystyle\chi}}{\raisebox{0.0pt}{\displaystyle\textstyle\chi}}{\raisebox{0.0pt}{\displaystyle\scriptstyle\chi}}{\raisebox{0.0pt}{\displaystyle\scriptscriptstyle\chi}}}_{F_{j}}\omega\bar{\mathcal{A}}\in L^{1}(\Omega_{T}), from (7.12), we see that \displaystyle[\bar{\mathcal{A}}]_{h}\nabla v_{\lambda,h}^{k}\omega{\mathchoice{\raisebox{0.0pt}{\displaystyle\displaystyle\chi}}{\raisebox{0.0pt}{\displaystyle\textstyle\chi}}{\raisebox{0.0pt}{\displaystyle\scriptstyle\chi}}{\raisebox{0.0pt}{\displaystyle\scriptscriptstyle\chi}}}_{F_{j}}\rightarrow 0. Thus we have

[TABLE]

Making use of Eq. 7.17 in Eq. 7.16, we have the following sequence of estimates:

[TABLE]

where to obtain (a), we made use of Eq. 7.7 and Eq. 6.17 along with Eq. 6.20. Here $\displaystyle o\left(\frac{1}{\lambda}\right)$ denotes a quantity that goes to zero as $\displaystyle\lambda\rightarrow\infty$ which holds due to Theorem 6.9 and the choice of measurable set $\displaystyle F_{j}$. We now let $\displaystyle\lambda\rightarrow\infty$ which proves the claim. ∎

This completes the proof of the existence result.