Open Access

The boundary value condition of an evolutionary \(p(x)\)-Laplacian equation

Boundary Value Problems20152015:112

https://doi.org/10.1186/s13661-015-0377-6

Received: 29 January 2015

Accepted: 18 June 2015

Published: 2 July 2015

Abstract

Consider an evolutionary equation related to the \(p(x)\)-Laplacian: \({u_{t}}= \operatorname{div} ({\rho^{\alpha}}{ \vert {\nabla u} \vert ^{p(x) - 2}}\nabla u)+ {\frac{{\partial{b_{i}}(u,x,t)}}{{\partial {x_{i}}}}}\), \((x,t) \in{Q_{T}} = \Omega \times(0,T)\), which arises from electrorheological fluid mechanics. Since \(\rho(x) = \operatorname{dist} (x,\partial\Omega)\), the equation is degenerate on the boundary, one may expect that there is not flux across the boundary. The paper shows that the facts may be unexpected. The paper reviews Fichera-Oleinik theory, then uses the theory to discuss the boundary value condition related to the equation. If \(p^{-}>2\), the existence and the uniqueness of the solutions are researched. Finally, if \(b_{i}\equiv0\), the behavior of the solutions near the boundary is studied by the comparison theorem.

Keywords

evolutionary \(p(x)\)-Laplacian equation electrorheological fluid Fichera-Oleinik theory boundary degeneracy boundary value condition

MSC

35L65 35K85 35R35

1 Introduction

Consider the equation
$$ {u_{t}}= \operatorname{div} \bigl({\rho^{\alpha}} {\vert { \nabla u} \vert ^{p(x) - 2}}\nabla u\bigr)+ \sum _{i = 1}^{N} {\frac{{\partial {b_{i}}(u,x,t)}}{{\partial{x_{i}}}}},\quad (x,t) \in{Q_{T}} = \Omega \times (0,T), $$
(1.1)
where \(\Omega\subset R^{N}\) is a bounded domain with suitably smooth boundary Ω, \(\rho(x) = \operatorname{dist} (x,\partial\Omega)\) is the distance function from the boundary, \(p(x)\) is a measurable function.
If \(\alpha=0\), \(b_{i}(s,x,t)\equiv0\), (1.1) becomes the evolutionary \(p(x)\)-Laplacian equation
$$ u_{t} =\operatorname{div}\bigl(|\nabla u|^{p(x)-2}\nabla u \bigr), \quad (x,t) \in {Q_{T}} = \Omega \times(0,T), $$
(1.2)
which has been researched widely in recent years. The equation emerges in the so-called electrorheological fluid theory, in which \(p(x)\) is as a function of the external electromagnetic field (see [1, 2] and the references therein). Certainly, if \(p(x)\equiv p\) is a constant in (1.2), it is called the evolutionary p-Laplacian equation and emerges in the non-Newtonian fluid theory. It has been studied by very many papers, we only quote some basic references [38] here. By the way, the author also has researched (1.2) for a long time, cf. [919].
Throughout the paper we denote
$$ p^{+} = \operatorname{ess}\sup_{\overline {\Omega}}p(x), \qquad p^{-}= \operatorname{ess}\inf_{\overline {\Omega}}p(x). $$
To consider the posedness of the solutions to (1.1) and (1.2), a nature basic functional space is \(W^{1,p(x)}_{0}(\Omega)\). Let us introduce some basic definitions and properties of the function spaces with variable exponents; for more details, see [2023].
  1. 1.
    \(L^{p(x)}(\Omega)\) space,
    $$ L^{p(x)}(\Omega) =\biggl\{ u: u \mbox{ is a measurable real-valued function},\int_{\Omega}\bigl\vert u(x)\bigr\vert ^{p(x)} \, \mathrm{d}x < \infty\biggr\} $$
    is equipped with the following Luxemburg norm:
    $$|u|_{L^{p(x)}}(\Omega) = \inf\biggl\{ \lambda>0:\int_{\Omega} \biggl\vert \frac {u(x)}{\lambda}\biggr\vert ^{p(x)} \,\mathrm{d}x\leq1 \biggr\} . $$
    The space \((L^{p(x)}(\Omega),|\cdot|_{L^{p(x)}(\Omega)} )\) is a separable, uniformly convex Banach space.
     
  2. 2.
    \(W^{1,p(x)}(\Omega)\) space,
    $$W^{1,p(x)}(\Omega) =\bigl\{ u \in L^{p(x)}(\Omega): |\nabla u|\in L^{p(x)}(\Omega)\bigr\} $$
    is endowed with the following norm:
    $$ |u|_{W^{1,p(x)}}=|u|_{L^{p(x)}(\Omega)} + |\nabla u|_{L^{p(x)}(\Omega)},\quad \forall u\in W^{1,p(x)}(\Omega). $$
    (1.3)
    We use \(W_{0}^{1,p(x)}(\Omega)\) to denote the closure of \(C^{\infty }_{0}(\Omega)\) in \(W^{1,p(x)}\).
     
A very important property of the function spaces with variable exponents was found by Zhikov in [24]. He showed that
$$ W^{1,p(x)}_{0} (\Omega) \neq\bigl\{ v\in W^{1,p(x)}_{0} (\Omega): v|_{\partial\Omega} = 0\bigr\} =\mathring{W}^{1,p(x)}(\Omega). $$
Hence, the property of the space \(W^{1,p(x)}_{0}(\Omega)\) is different from that of the case when p is a constant. The following lemma gives some basic properties of \(W^{1,p(x)}(\Omega)\).

Lemma 1.1

  1. (i)

    The spaces \((L^{p(x)}(\Omega), | \cdot|_{L^{p(x)}(\Omega )} )\), \((W^{1,p(x)}(\Omega), |\cdot|_{W^{1,p(x)}(\Omega)} )\) and \(W^{1,p(x)}_{0}(\Omega)\) are reflexive Banach spaces.

     
  2. (ii)
    We have \(p(x)\)-Hölder’s inequality. Let \(q_{1}(x)\) and \(q_{2}(x)\) be real functions with \(\frac{1}{q_{1}(x)}+\frac{1}{q_{2}(x)} = 1\) and \(q_{1}(x) > 1\). Then the conjugate space of \(L^{q_{1}(x)}(\Omega)\) is \(L^{q_{2}(x)}(\Omega)\). For any \(u \in L^{q_{1}(x)}(\Omega)\) and \(v \in L^{q_{2}(x)}(\Omega)\), we have
    $$\biggl\vert \int_{\Omega}uv \,\mathrm{d}x\biggr\vert \leq2|u|_{L^{q_{1}(x)}(\Omega )}|v|_{L^{q_{2}(x)}(\Omega)}. $$
     
  3. (iii)

    If \(|u|_{L^{p(x)}(\Omega)} = 1\), then \(\int_{\Omega}|u|^{p(x)} \,\mathrm{d}x = 1\).

    If \(|u|_{L^{p(x)}(\Omega)} > 1\), then \(|u|^{p^{-}}_{L^{p(x)}}\leq\int_{\Omega}|u|^{p(x)} \,\mathrm{d}x\leq|u|^{p^{+}}_{L^{p(x)}}\).

    If \(|u|_{L^{p(x)}(\Omega)} < 1\), then \(|u|^{p^{+}}_{L^{p(x)}}\leq\int_{\Omega}|u|^{p(x)} \,\mathrm{d}x\leq|u|^{p^{-}}_{L^{p(x)}}\).

     
  4. (iv)
    If \(p_{1}(x)\leq p_{2}(x)\), then
    $$ L^{p_{1}(x)}(\Omega)\supset L^{p_{2}(x)}(\Omega). $$
     
  5. (v)
    If \(p_{1}(x)\leq p_{2}(x)\), then
    $$ W^{1,p_{2}(x)}(\Omega)\hookrightarrow W^{1,p_{1}(x)}(\Omega). $$
     
  6. (vi)
    We have the \(p(x)\)-Poincaré’s inequality. If \(p(x)\in C(\Omega )\), then there is a constant \(C>0\), such that
    $$ |u|_{L^{p(x)}}(\Omega) \leq C|\nabla u|_{L^{p(x)}(\Omega)},\quad \forall u\in W^{1,p(x)}_{0}(\Omega). $$
    (1.4)
    This implies that \(|\nabla u|_{L^{p(x)}}(\Omega)\) and \(|u|_{W^{1,p(x)}(\Omega)}\) are equivalent norms of \(W^{1,p(x)}_{0}\).
     
However, if the exponent \(p(x)\) is required to satisfy a logarithmic Hölder continuity condition
$$ \bigl\vert p(x)-p(y)\bigr\vert \leq\omega\bigl(\vert x-y\vert \bigr), $$
(1.5)
\(\forall x,y\in Q_{T}\), \(|x-y|<\frac{1}{2}\) with
$$\varlimsup_{s \to{0^{+} }} \omega(s)\ln \biggl(\frac{1}{s}\biggr) = C < \infty, $$
then (see [25])
$$ W_{0}^{1,p(x)}(\Omega)= \mathring{W} ^{1,p(x)}( \Omega). $$
(1.6)
By (1.5) and (1.6), Antontsev-Shmarev [26] established the existence and uniqueness results of (1.2). Since then, using the logarithmic Hölder continuity condition, there were many papers in studying the solvability and the regularity of the equation related to (1.2); for examples, see [27, 28] etc. When \(p^{-}> 2\), Peng [29] had studied the existence of the solutions of the equation
$$ \operatorname{div}\bigl(|\nabla u|^{p(x)-2}\nabla u\bigr)+f(x,u)=0,\quad x \in \Omega, $$
(1.7)
without the condition (1.3). By adopting a time difference method, Lian et al. [30] generalized the method of [29] to study
$$ u_{t} =\operatorname{div}\bigl(|\nabla u|^{p(x)-2}\nabla u \bigr)+f(x,t,u),\quad (x,t) \in{Q_{T}} = \Omega \times(0,T), $$
(1.8)
provided that f satisfied some restrictions.
In our paper, we want to consider the initial boundary value problem of (1.1). By the paper of Yin and Wang [31], which studied the diffusion equation
$$ {u_{t}}= \operatorname{div} \bigl({\rho^{\alpha}} {\vert { \nabla u} \vert ^{p - 2}}\nabla u\bigr), \quad (x,t) \in{Q_{T}} = \Omega \times(0,T), $$
(1.9)
we know that the initial value condition
$$ u|_{t=0} = u_{0}(x),\quad x\in\Omega, $$
(1.10)
is always required. But due to the degeneracy of the diffusion \(\rho ^{\alpha}\) on the boundary, whether we can require the usual boundary value condition
$$ u(x,t) = 0, \quad (x,t) \in\partial\Omega\times (0,T), $$
(1.11)
is uncertain. From the point of physics, if we regard (1.9) as a heat transfer equation, since the diffusion coefficient vanishes on the boundary, it seems that there is not heat flux across the boundary. However, Yin and Wang [31] proved that, if \(\alpha\geq p-1\), the existence and uniqueness of solutions can be obtained without any boundary value condition. In other words, the solution of the equation is completely controlled by the initial value condition. Thus, whether there is heat flux across the boundary is unknown actually.
The first aim of our paper is to probe how to give a suitable boundary value condition of (1.1). We first review Fichera-Oleinik theory, and then we use it to discuss the suitable boundary value condition related to (1.1). The main point is that, to assure the posedness of the solutions to (1.1), instead of the whole boundary value condition (1.11), we can require only a partial boundary value condition,
$$ u(x,t) = 0, \quad (x,t) \in\Sigma _{p} \times (0,T), $$
(1.12)
where \(\Sigma _{p}\) is a subset of Ω. In some cases, \(\Sigma _{p}\) can be expressed clearly, whereas in some other cases, it is difficult to write out its explicit formulas.
We assume the following.
  1. (A)
    We call a bounded domain Ω has the integral non-singularity, if there are constants \(\alpha>0\), \(p^{-}>2\), such that
    $$ \int_{\Omega}\rho^{- \frac{{2\alpha}}{{p^{-} - 2}}}\,\mathrm{d}x \leqslant c. $$
    (1.13)
     
  2. (B)
    For any \(i \in\{ 1,2, \ldots, N\} \), \(b_{i}(s,x,t)\) is a \(C^{1}\) function on \(\mathbb{R}\times\overline{\Omega}\times[0,T]\), and there are constants β, c such that
    $$ \bigl\vert {{b_{i}}(s,x,t)} \bigr\vert \leqslant c{\vert s \vert ^{1 + \beta}}, \qquad \bigl\vert {{{b}_{is}}(s,x,t)} \bigr\vert \leqslant c{\vert s \vert ^{\beta}}, \qquad \bigl\vert {b_{ix_{i}}(s,x,t)} \bigr\vert \leqslant c, $$
    (1.14)
    where \(b_{is}=\frac{\partial b_{i}}{\partial s}\), \(b_{ix_{i}}=\frac {\partial b_{i}}{\partial x_{i}}\) as usual.
     

The main results in our paper are the following theorems.

Theorem 1.2

If \(p^{-}>2\) and \(\alpha<\frac{p^{-}-2}{2}\), the bounded domain Ω is with the integral non-singularity (A), \(b_{i}(s,x,t)\) and its partial derivatives satisfy the condition (B), and \(u_{0}\) satisfies
$$ {u_{0}} \in{L^{\infty}}(\Omega), \qquad {\rho^{\alpha}} \vert \nabla {u_{0}} \vert ^{p^{+}} \in{L^{1}}( \Omega), $$
(1.15)
then (1.1) with initial boundary values (1.10)-(1.12) has a solution. In particular, when \(\Sigma_{p}=\partial \Omega\), then the solution is unique.

Remark 1.3

The explicit formula of \(\Sigma_{p}\) of (1.12) used in the theorem is listed in Section 4.

Theorem 1.4

If \(p^{-}>2\) and \(\alpha>1\), let u be a viscous solution of (1.1), then there are constants \(c_{1}\), \(c_{2}\) such that
$$ c_{1}\rho(x)\leq u(x,t)\leq c_{2} \rho(x), $$
(1.16)
when x is near the boundary.

2 The stage of formal operation

Consider the second order equation with the form
$$ L(u) = {a^{rs}}(x){u_{{x_{r}}{x_{s}}}} + {b^{r}}(x){u_{{x_{r}}}} + c(x)u = f(x). $$
(2.1)
If for any real vector \(\xi = ({\xi_{1}},{\xi_{2}}, \ldots,{\xi_{m}})\) and any point \(x\in\Omega\),
$$ {a^{rs}} {\xi_{r}} {\xi_{s}} \geqslant0 $$
(2.2)
is true, then it is called the second order equation with nonnegative characteristic form in Ω. Obviously, it entails an elliptic equation, a parabolic equation, a first-order equation (the case \({a^{rs}}{\xi_{r}}{\xi_{s}} \equiv0\)), ultra parabolic equation, Brown motion equation, Tricomi equation on the half-plane and so on.
Consider the first boundary value problem of (2.1) in Ω, Fichera [32] first made thorough research in this problem. In what follows, we use the notations in [33, 34], especially, the pairs of the indices imply summation. Suppose on \(\overline {\Omega} = \Omega \cup\Sigma\), all the points x and all \(\xi\in R^{n}\) satisfy the condition (2.2), Ω is appropriately smooth, \({a^{rs}} \in{C^{2}}(\Omega)\), \({b^{r}} \in{C^{1}}(\Omega)\), \(c \in{C^{0}}(\Omega)\). Let \(\{n_{s}\}\) be the unit inner normal vector of \(\partial\widetilde{\Omega}\) and denote that
$$\Sigma^{0}=\bigl\{ x\in\Sigma: a^{rs}n_{r}n_{s}=0 \bigr\} . $$
In \(\Sigma^{0}\), let us consider the Fichera function
$$ b(x) \equiv\bigl({b^{r}} - a_{{x_{s}}}^{rs} \bigr){n_{r}}. $$
(2.3)
We denote
$$\begin{aligned}& \Sigma_{1}=\bigl\{ x\in\Sigma^{0}: \bigl(b_{r}-a^{rs}_{x_{s}} \bigr)n_{r}>0\bigr\} , \\& \Sigma_{2}=\bigl\{ x\in\Sigma^{0}: \bigl(b_{r}-a^{rs}_{x_{s}} \bigr)n_{r}< 0\bigr\} \end{aligned}$$
and
$$\Sigma_{0}=\bigl\{ x\in\Sigma^{0}: \bigl(b_{r}-a^{rs}_{x_{s}} \bigr)n_{r}=0\bigr\} , $$
\(\Sigma\setminus\Sigma^{0} \) is denoted as \(\Sigma_{3}\).
The first boundary value problem of (2.1) is quoted as follows: in \(\overline {\Omega} = \Omega \cup\Sigma\), to find a function u such that
$$\begin{aligned}& L(u) = f(x),\quad x\in\Omega, \end{aligned}$$
(2.4)
$$\begin{aligned}& u=g,\quad x\in\Sigma_{2} \cup\Sigma_{3}, \end{aligned}$$
(2.5)
where f is a given function, and g is a given function on \(\Sigma _{2} \cup\Sigma_{3}\). Clearly, if (2.1) is an elliptic equation, then (2.4)-(2.5) is the usual Dirichlet problem. For the cylindrical region, (2.4)-(2.5) consists of the mixed problem, also known as parabolic equations with the initial boundary values.
Now, if we consider (1.1) in our paper,
$$ {u_{t}} = \operatorname{div} \bigl(\rho^{\alpha} \vert \nabla u \vert ^{p(x) - 2}\nabla u\bigr) + \frac{\partial{b_{i}}(u,x,t)}{\partial{x_{i}}},\quad (x,t) \in {Q_{T}}, $$
(2.6)
then we can rewrite it as
$$ u_{t}=a^{ij}(x,t)\frac{\partial^{2} u}{\partial x_{i}\partial x_{j}}+{\beta_{i}}(x,t) \frac{{\partial u}}{{\partial{x_{i}}}}+\gamma (u,x,t), $$
(2.7)
where
$$\begin{aligned}& a^{ij}(x,t)=\rho^{\alpha} \vert {\nabla u} \vert ^{p(x) - 2}\bigl[\delta_{ij}+\bigl(p(x)-2\bigr)|\nabla u|^{-2}u_{x_{i}}u_{x_{j}}\bigr], \\& \beta_{i}=\alpha\rho^{\alpha-1}|\nabla u|^{p(x)-2} \rho_{x_{i}}+\rho ^{\alpha} \vert {\nabla u} \vert ^{p(x) - 2}\log|\nabla u|p_{x_{i}}+b_{iu}(u,x,t), \\& \gamma(u,x,t)=b_{ix_{i}}(u,x,t). \end{aligned}$$

If comparing (2.7) with (2.5), on the lateral boundary, when \(t=0\), the initial value condition (1.10) is required, and as we know when \(t=T\), no boundary value is necessary. The more interesting phases appear on the bottom boundary. Generally, only a portion of the bottom boundary can be required as the boundary value. Let us explain what happens as follows.

By
$$\begin{aligned} a_{x_{j}}^{ij} =&\alpha{\rho^{\alpha- 1}} { \rho_{{x_{j}}}} {\vert {\nabla u} \vert ^{p(x) - 4}}\bigl[{ \delta_{ij}} {\vert {\nabla u} \vert ^{2}} + \bigl(p(x) - 2\bigr){u_{{x_{i}}}} {u_{{x_{j}}}}\bigr] \\ &{}+\rho^{\alpha} \vert {\nabla u} \vert ^{p(x) - 4} \bigl[p_{x_{j}}|\nabla u|^{2}\log|\nabla u|+\bigl(p(x)-2 \bigr)u_{x_{k}}u_{x_{k}x_{j}}+p_{x_{j}}u_{x_{j}}u_{x_{i}} \bigr] \\ &{}+\bigl(p(x)-2\bigr)\rho^{\alpha} \vert {\nabla u} \vert ^{p(x) - 6}u_{x_{j}}u_{x_{i}}\bigl[p_{x_{j}}\vert { \nabla u} \vert ^{2}\log|\nabla u|+\bigl(p(x)-4\bigr)u_{x_{k}}u_{x_{k}x_{j}} \bigr] \\ &{}+\bigl(p(x)-2\bigr)\rho^{\alpha} \vert {\nabla u} \vert ^{p(x)- 4}(u_{x_{i}x_{j}}u_{x_{j}}+u_{x_{i}}u_{x_{j}x_{j}}), \end{aligned}$$
if we notice that near the boundary \(\Sigma=\partial\Omega\), \(\rho ^{\alpha}(x)=o(\rho^{\alpha-1})\), then
$$\begin{aligned} \bigl({\beta^{i}} - \alpha_{{x_{j}}}^{ij} \bigr){n_{i}} =& \alpha{\rho^{\alpha- 1}} {\vert {\nabla u} \vert ^{p(x)- 4}}\bigl[{\rho_{{x_{i}}}} {\vert {\nabla u} \vert ^{2}} - {\rho_{{x_{j}}}}\bigl({\delta_{ij}} {\vert { \nabla u} \vert ^{2}} + \bigl(p(x)- 2\bigr){u_{{x_{i}}}} {u_{{x_{j}}}}\bigr)\bigr]{n_{i}} \\ &{}+ b_{iu}(u,x,t){n_{i}}+o \bigl(\rho^{\alpha-1}\bigr) \\ =&-\bigl(p(x)-2\bigr)\alpha{\rho^{\alpha- 1}}\bigl({\vert {\nabla u} \vert ^{p(x)- 4}} {u_{{x_{i}}}} {u_{{x_{j}}}} \bigr){n_{i}} {n_{j}}+ b_{iu}(u){n_{i}}(x)+o \bigl(\rho ^{\alpha-1}\bigr). \end{aligned}$$
(2.8)
Because the determinant of \(U_{x}=(u_{x_{i}}u_{x_{j}})_{N\times N}\),
$$| u_{x_{i}}u_{x_{j}}|_{N\times N}=\prod _{i=1}^{N}u_{x_{i}}| u_{x_{j}}|_{N\times N}=0, \quad x\in\overline{\Omega}, $$
and ith order principal minor determinants are all equal to 0, except that \(i=1\). Then according to the characteristic value theory, due to the symmetry of \(N\times N\) matrix \(U_{x}\), there exists an orthogonal matrix P such that \((u_{x_{i}}u_{x_{j}})_{N\times N}=PA_{N\times N}P^{-1}\), where A is a diagonal matrix which is just the characteristic matrix of \(U_{x}\). Let \(\lambda_{i}\geq0\) be the characteristic values of \(U_{x}\). By a direct calculation, we get
$$ \lambda_{1}=|\nabla u|^{2}, \quad \lambda_{i}=0, i=2,3,\ldots, N. $$
(2.9)
We have
$$\begin{aligned} \bigl({\beta^{i}} - \alpha_{{x_{j}}}^{ij} \bigr){n_{i}} =&- \bigl(p(x)- 2\bigr)\alpha{\rho ^{\alpha- 1}} \vert {\nabla u} \vert ^{p(x)- 4}\vec{n}PA_{N\times N}( \vec{n}P)^{T}+ b_{iu}(u,x,t){n_{i}}+o\bigl( \rho^{\alpha-1}\bigr) \\ =&- \bigl(p(x) - 2\bigr)\alpha{\rho^{\alpha - 1}}\vert {\nabla u} \vert ^{p(x)- 2}m_{1}^{2} \\ &{}+ b_{iu}(u,x,t){n_{i}}+o \bigl(\rho^{\alpha-1}\bigr) , \quad x\in \partial\Omega, \end{aligned}$$
(2.10)
where \(\vec{m}=\vec{n}P\).

Then it can be divided into the following cases.

1. \(\alpha>1\), then \(\rho^{\alpha}|_{\partial\Omega} =\rho ^{\alpha-1}|_{\partial\Omega} =0\).

1.1. \(b_{i}\equiv0\). Then \(\Sigma_{p}=\emptyset\), no boundary value is required.

1.2. \(b_{i}\) is not identical to 0,
$$ \Sigma_{p} = \bigl\{ x \in\partial\Omega: {b_{iu}}(0,x,t){n_{i}} < 0\bigr\} . $$
(2.11)
It shows that (2.6) still needs the partial boundary condition when \(\alpha>1\), this is different from the case of \(b_{i}\equiv0\), in which no boundary is required even when \(\alpha>1\). For example, considering the case of a one-dimensional space variable, and \(p=2\), \(x\in(0,1)\), \(b(u,x,t)=b(u)\), (2.6) becomes
$$ {u_{t}} = {\bigl({\rho^{\alpha}} {u_{x}} \bigr)_{x}} + b'(u){u_{x}}, $$
(2.12)
\(\Sigma _{p}\) in (2.11) means that (2.12) needs to give the boundary condition at \(x=0\) when \(b'(0) < 0\) and needs to give the boundary condition at \(x=1\) when \(b'(0) > 0\).

In this case, certainly, when \({b_{iu}}(0,x,t){n_{i}}(x) \geqslant0\) is true for all \(x\in\partial\Omega\), \(\Sigma _{p} \) is an empty set, then (2.6) does not require any boundary condition now.

2. \(\alpha=1\), then
$$I=\bigl({\beta^{i}} - \alpha_{{x_{j}}}^{ij} \bigr){n_{i}} =- \bigl(p(x) - 2\bigr)\vert {\nabla u} \vert ^{p(x) - 2}m_{1}^{2}+ b_{iu}(0,x,t){n_{i}}+o( \rho). $$
2.1. \(b_{i}\equiv0\),
$$I=\bigl({\beta^{i}} - \alpha_{{x_{j}}}^{ij} \bigr){n_{i}} =-\bigl(p(x)- 2\bigr)\vert {\nabla u} \vert ^{p(x)-2}m_{1}^{2}+o(\rho). $$
If \(N=1\), then
$$\Sigma_{p}=\bigl\{ x\in\partial\Omega: p(x)>2\bigr\} . $$
If \(N\geq2\), then
$$\Sigma_{p}=\bigl\{ x\in\partial\Omega: p(x)>2, m_{1}(x) \neq0\bigr\} . $$
2.2. \(b_{i}\) is not identical to 0,
$$I=\bigl({\beta^{i}} - \alpha_{{x_{j}}}^{ij} \bigr){n_{i}} =-\bigl(p(x)- 2\bigr)\vert {\nabla u} \vert ^{p(x) - 2}m_{1}^{2}+ b_{iu}(0,x,t){n_{i}}+o( \rho). $$
If \(N=1\), when for all \(x\in\partial\Omega\),
$$p(x)>2,\qquad b_{iu}(0,x,t)n_{i}(x)\leq0, $$
then \(\Sigma_{p}=\partial\Omega\). Generally, it is only a subset of Ω and it is difficult to write out the explicit formula.
If \(N\geq2\), when for all \(x\in\partial\Omega\),
$$p(x)>2, \qquad b_{iu}(0,x,t)n_{i}(x)\leq0,\qquad m_{1}(x)\neq0, $$
then \(\Sigma_{p}=\partial\Omega\).
If \(N\geq2\), when for all \(x\in\partial\Omega\),
$$b_{iu}(0,x,t)n_{i}(x)\geq0,\qquad m_{1}(x)=0, $$
then \(\Sigma_{p}=\emptyset\). Generally, it is only a subset of Ω and it is difficult to write out the explicit formula.

3. \(\alpha<1\).

3.1. \(b_{i}\equiv0\).

When \(N=1\), (2.10) becomes
$$I=- \alpha\bigl(p(x)-2\bigr) \rho^{\alpha-1}\bigl\vert u'(x) \bigr\vert ^{p(x)- 2}m_{1}^{2}+O\bigl( \rho^{\alpha}\bigr). $$
If \(p(x)\equiv p\), when \(p>2\), \(I<0\), then \(\Sigma_{p}=\partial\Omega\). When \(p\leq2\), \(I\geq0\), then \(\Sigma_{p}=\emptyset\).
If \(p(x)\) is just a continuous function, then
$$\Sigma_{p}=\bigl\{ x\in\partial\Omega: p(x)>2\bigr\} . $$
When \(N\geq2\), if \(p(x)\equiv p>2\), then
$$\Sigma_{p}=\bigl\{ x\in\partial\Omega: m_{1}(x)\neq0\bigr\} . $$
If \(p(x)\equiv p\leq2\), then \(\Sigma_{p}=\emptyset\).
If \(p(x)\) is just a function, then
$$\Sigma_{p}=\bigl\{ x\in\partial\Omega: p(x)>2, m_{1}(x) \neq0\bigr\} . $$

3.2. \(b_{i}\) is not identical to 0.

When \(N=1\), (2.10) becomes
$$I=- \alpha\bigl(p(x)-2\bigr) \rho^{\alpha- 1}\bigl\vert u'(x) \bigr\vert ^{p(x) - 2}m_{1}^{2}+b_{u}(0,x,t)n+O \bigl(\rho^{\alpha}\bigr). $$
If \(p(x)\equiv p\), when \(p>2\), \(I<0\), then \(\Sigma_{p}=\partial\Omega \). When \(p=2\),
$$\Sigma_{p}=\bigl\{ x\in\partial\Omega: b(u) (0,x,t)n< 0\bigr\} . $$
When \(p<2\), \(I>0\), then \(\Sigma_{p}=\emptyset\).
If \(p(x)\) is just a function, then
$$\Sigma_{p}=\bigl\{ x\in\partial\Omega: p(x)>2\bigr\} \cup\bigl\{ x\in \partial \Omega: p(x)=2, b_{iu}(0,x,t)n_{i}< 0\bigr\} . $$
Let \(N\geq2\), if \(p(x)\equiv p\), when \(p>2\), then
$$\Sigma_{p}=\bigl\{ x\in\partial\Omega: m_{1}(x)\neq0\bigr\} , $$
when \(p=2\),
$$\Sigma_{p}=\bigl\{ x\in\partial\Omega: b_{iu}(0,x,t)n_{i}(x)< 0 \bigr\} , $$
when \(p<2\), when for all \(x\in\partial\Omega\), \(m_{1}(x)\neq0\), then \(I>0\), \(\Sigma_{p}=\emptyset\). In general, it is just a subset of Ω, and it is difficult to write out the explicit formula.
If \(p(x)\) is just a continuous function, then
$$\begin{aligned} \Sigma_{p} =&\bigl\{ x\in\partial\Omega: p(x)>2\bigr\} \cup\bigl\{ x\in \partial \Omega: p(x)=2, b_{iu}(0,x,t)n_{i}< 0\bigr\} \\ &{}\cup\bigl\{ x\in\partial\Omega :p(x)< 2, m_{1}(x) \neq0,b_{iu}(0,x,t)n_{i}< 0\bigr\} . \end{aligned}$$

In other words, the boundary value condition of (1.1) is so complicated; it may depend on whether \(\alpha>1,=1, \mbox{or}<1\), whether \(N=1\), or \(N\geq2\), whether \(p(x)>2\) or not, whether \(b_{i}\equiv0\) or not. In Sections 3 and 4, we only consider the existence and the uniqueness of the solutions when \(p^{-}>2\). In last section, we only consider the behavior of the solutions near the boundary when \(\alpha \geq1\) and \(b_{i}\equiv0\).

Certainly, as we already know that a degenerate parabolic equation generally only has a weak solution, the above linearization is only formal. We only give some ideas of how to give the partial boundary value condition to assure the posedness of the weak solutions.

3 The existence of the solution related to the initial value

Let
$$ u_{0}\in L^{\infty}(\Omega), \qquad {\rho^{\alpha}} { \vert {\nabla {u_{0}}} \vert ^{p_{+}}} \in{L^{1}}( \Omega). $$
(3.1)

Definition 3.1

A function \(u(x,t)\) is said to be a solution of (1.1) with the initial value condition (1.10), if the initial condition is satisfied, in the sense of a trace, and u satisfies
$$ u \in L^{\infty}(Q_{T}),\qquad {\rho^{\alpha}} {\vert { \nabla u} \vert ^{p(x)}} \in{L^{1}}({Q_{T}}),\qquad u_{t}\in L^{2}(Q_{T}), $$
(3.2)
and for any function \(\varphi \in C_{0}^{\infty}({Q_{T}})\), the following integral equivalence holds:
$$ \iint_{{Q_{T}}} \bigl(-u\varphi_{t} + \rho^{\alpha} \vert \nabla u \vert ^{p(x)- 2}\nabla u \cdot\nabla\varphi+b_{i}(u,x,t) \varphi _{x_{i}}\bigr)\,\mathrm{d}x\,\mathrm{d}t = 0. $$
(3.3)
We consider the following regularized problem:
$$\begin{aligned}& u_{\varepsilon t} - \operatorname{div} \bigl(\rho^{\alpha}_{\varepsilon}\bigl(\vert \nabla{u_{\varepsilon}} \vert ^{2}+\varepsilon \bigr)^{\frac{p(x) - 2}{2}}\nabla{u_{\varepsilon}}\bigr) -{\frac{{\partial{b_{i}}(u,x,t)}}{{\partial{x_{i}}}}}= 0, \quad (x,t)\in {Q_{T}}, \end{aligned}$$
(3.4)
$$\begin{aligned}& {u_{\varepsilon}}(x,t) = 0, \quad (x,t) \in\partial\Omega \times (0,T), \end{aligned}$$
(3.5)
$$\begin{aligned}& {u_{\varepsilon}}(x,0) = {u_{\varepsilon,0}}(x),\quad x\in\Omega , \end{aligned}$$
(3.6)
where \({\rho_{\varepsilon}} = \rho\ast\delta_{\varepsilon }+\varepsilon\), \(\varepsilon > 0\), \(\delta_{\varepsilon}\) is the usual mollifier. For all \(\varepsilon>0\), selecting \({u_{\varepsilon,0}}\) such that \({\Vert {{u_{\varepsilon,0}}} \Vert _{{L^{\infty}}(\Omega)}}\) and \(\Vert \rho_{\varepsilon}^{\alpha} \vert \nabla u_{\varepsilon,0} \vert ^{p^{+}} \Vert _{L^{1}(\Omega)}\) are uniformly bounded, and \({u_{\varepsilon,0}}\) converges to \(u_{0}\) in \(W_{\operatorname {\mathrm{loc}}}^{1, p^{+}}(\Omega)\). For any \({u_{\varepsilon,0}} \in {C^{\infty}_{0} }(\Omega)\), \(\rho_{\varepsilon}^{\alpha}{ \vert {\nabla{u_{\varepsilon,0}}} \vert ^{p^{+}}}\in {L^{1}}(\Omega)\), it is well known that the above problem has a unique classical solution [35]. Hence for any \(\varphi \in C_{0}^{\infty}({Q_{T}})\), \(u_{\varepsilon}\) satisfies the following integral equivalence:
$$ \iint_{{Q_{T}}} \bigl(u_{\varepsilon t}\varphi+ {\rho_{\varepsilon}^{\alpha}} \vert \nabla u_{\varepsilon} \vert ^{p(x) - 2} \nabla u_{\varepsilon} \cdot\nabla\varphi+b_{i}(u_{\varepsilon },x,t)\varphi_{x_{i}} \bigr)\,\mathrm{d}x\,\mathrm{d}t = 0. $$
(3.7)

Lemma 3.2

If \(p^{-}>2\), Ω is a suitably smooth bounded domain, the assumptions (A) and (B) are true, then the solution \({u_{\varepsilon}}\) of the initial boundary value problem (3.4)-(3.6) is weakly star convergent to u and strongly convergent to \(u \in L_{\mathrm{loc}}^{2}(Q_{T})\), and its limit function u satisfies (3.2) and is the solution of (1.1) with the initial value condition (1.10).

Proof

By the maximum principle, there is a constant c, only dependent on \({\Vert {{u_{0}}} \Vert _{{L^{\infty}}(\Omega)}}\) but independent of ε, such that
$$ {\Vert {{u_{\varepsilon}}} \Vert _{{L^{\infty}}({Q_{T}})}} \leqslant c. $$
(3.8)
Multiplying (3.4) by \(u_{\varepsilon}\) and integrating over \(Q_{T}\), we get
$$\begin{aligned}& \frac{1}{2}\int_{\Omega}{u_{\varepsilon}^{2}{ \,\mathrm{d}}x} + \iint _{{Q_{T}}} {\rho_{\varepsilon}^{\alpha}\bigl(|\nabla u_{\varepsilon}|^{2}+\varepsilon\bigr)^{\frac{p(x)-2}{2}}| \nabla u_{\varepsilon}|^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t} \\& \quad = \iint_{{Q_{T}}} {{u_{\varepsilon}} {\frac{{\partial {b_{i}}({u_{\varepsilon}}x,t)}}{{\partial{x_{i}}}}} {\,\mathrm{d}}x{\, \mathrm{d}}t}+\frac{1}{2}\int_{\Omega}{u_{\varepsilon,0}^{2}{ \,\mathrm{d}}x}. \end{aligned}$$
By the fact
$$\begin{aligned}& \iint_{{Q_{T}}} {{u_{\varepsilon}} {\frac{{\partial{b_{i}}({u_{\varepsilon}}x,t)}}{{\partial{x_{i}}}}} {\,\mathrm{d}}x{ \,\mathrm{d}}t} \\& \quad = - {\iint _{{Q_{T}}} {\frac{{\partial{u_{\varepsilon}}}}{{\partial {x_{i}}}}{b_{i}}({u_{\varepsilon}},x,t){ \,\mathrm{d}}x{\,\mathrm{d}}t}} \\& \quad = -\iint_{{Q_{T}}}{\frac{\partial}{{\partial{x_{i}}}}\int_{0}^{{u_{\varepsilon}}} {{b_{i}}(s,x,t){\,\mathrm{d}}s{\,\mathrm{d}}x} } +\iint_{{Q_{T}}} \int_{0}^{{u_{\varepsilon}}} b_{ix_{i}}(s,x,t){\, \mathrm{d}}s{\,\mathrm{d}}x \\& \quad = \iint_{{Q_{T}}}\int_{0}^{{u_{\varepsilon}}} b_{ix_{i}}(s,x,t){\,\mathrm{d}}s{\,\mathrm{d}}x , \end{aligned}$$
(3.9)
and by (B),
$$\frac{1}{2}\int_{\Omega}{u_{\varepsilon}^{2}{ \,\mathrm{d}}x} +\iint _{{Q_{T}}} {\rho_{\varepsilon}^{\alpha}\bigl(|\nabla u_{\varepsilon}|^{2}+\varepsilon\bigr)^{\frac{p(x)-2}{2}}| \nabla u_{\varepsilon}|^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t} \leqslant c. $$
Let \(\Omega_{\lambda}=\{x\in\Omega: \operatorname{dist}(x,\partial\Omega )>\lambda\}\). Since \(p^{+}\geq p^{-}>2\), we have
$$ \int_{0}^{T}\!\!\int_{\Omega_{\lambda}}| \nabla u_{\varepsilon }|^{2}\,\mathrm{d}x\,\mathrm{d}t\leq c \biggl( \int_{0}^{T}\!\!\int_{\Omega_{\lambda}}|\nabla u_{\varepsilon}|^{p^{-}}\,\mathrm{d}x\,\mathrm{d}t \biggr)^{\frac{1}{p^{-}}} \leq c(\lambda ). $$
(3.10)
Multiplying (3.4) by \(u_{\varepsilon t}\), integrating over \(Q_{T}\),
$$\begin{aligned}& \iint_{{Q_{T}}} {{{({u_{\varepsilon t}})}^{2}} {\,\mathrm{d}}x{ \,\mathrm{d}}t} \\& \quad = \iint_{{Q_{T}}} {\operatorname{div} \bigl( \rho_{\varepsilon}^{\alpha}{{ \vert {\nabla{u_{\varepsilon}}} \vert }^{p(x) - 2}}\nabla {u_{\varepsilon}}\bigr) \cdot{u_{\varepsilon t}} {\, \mathrm{d}}x{\,\mathrm{d}}t} +\iint_{{Q_{T}}}u_{\varepsilon t} \frac{\partial b_{i}(u_{\varepsilon},x,t)}{\partial x_{i}}\,\mathrm{d}x\,\mathrm{d}t . \end{aligned}$$
(3.11)
We notice that
$$\bigl(|\nabla u_{\varepsilon}|^{2}+\varepsilon\bigr)^{\frac{p(x)-2}{2}} \nabla {u_{\varepsilon}} \cdot\nabla{u_{\varepsilon t}} = \frac{1}{2} \frac {{\,\mathrm{d}}}{{{\,\mathrm{d}}t}}\int_{0}^{{{\vert {\nabla u_{\varepsilon }(x,t)} \vert }^{2}}+\varepsilon} {{s^{\frac{{p(x) - 2}}{2}}} {\,\mathrm{d}}s} . $$
Thus,
$$\begin{aligned}& \iint_{{Q_{T}}} {\operatorname{div}} \bigl(\rho_{\varepsilon}^{\alpha}\bigl(|\nabla u_{\varepsilon}|^{2}+\varepsilon\bigr)^{\frac{p(x)-2}{2}} \nabla {u_{\varepsilon}}\bigr) \cdot{u_{\varepsilon t}} {\,\mathrm{d}}x{\, \mathrm{d}}t \\& \quad = - \iint_{{Q_{T}}} \rho_{\varepsilon}^{\alpha}\bigl(| \nabla u_{\varepsilon }|^{2}+\varepsilon\bigr)^{\frac{p(x)-2}{2}} \nabla{u_{\varepsilon}}\nabla u_{\varepsilon t}{\,\mathrm{d}}x{\,\mathrm{d}}t \\& \quad = -\frac{1}{2}\iint_{{Q_{T}}} \rho_{\varepsilon}^{\alpha}\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{0}^{\vert {\nabla u_{\varepsilon}(x,t)} \vert ^{2}+\varepsilon} s^{\frac{p(x) - 2}{2}}\, \mathrm{d}s \,\mathrm{d}x{\,\mathrm{d}}t. \end{aligned}$$
(3.12)
By condition (B),
$$\begin{aligned}& \iint_{Q_{T}} u_{\varepsilon t} \frac{\partial{b_{i}}(u_{\varepsilon},x,t)}{ \partial{x_{i}}} \,\mathrm{d}x\, \mathrm{d}t \\& \quad \leqslant \iint_{Q_{T}} \bigl\vert b_{iu}(u_{\varepsilon},x,t) \bigr\vert \vert {u_{\varepsilon}}_{x_{i}} \vert \vert u_{\varepsilon t} \vert \,\mathrm{d}x\,\mathrm{d}t + \iint_{Q_{T}} \bigl\vert b_{ix_{i}}(u_{\varepsilon},x,t)\bigr\vert \vert u_{\varepsilon t} \vert \,\mathrm{d}x\,\mathrm{d}t \\& \quad \leqslant \frac{1}{4}\iint_{{Q_{T}}} {{{({u_{\varepsilon t}})}^{2}} {\,\mathrm{d}}x{\,\mathrm{d}}t} + c\iint_{{Q_{T}}} {{{\vert {{u_{\varepsilon}}} \vert }^{2\beta}} {{\vert {\nabla{u_{\varepsilon}}} \vert }^{2}} {\,\mathrm{d}}x{\,\mathrm{d}}t} \\& \qquad {}+\frac{1}{4}\iint_{{Q_{T}}} {{{({u_{\varepsilon t}})}^{2}} {\,\mathrm{d}}x{\,\mathrm{d}}t} + c. \end{aligned}$$
(3.13)
Here, we have used the fact that \(|u_{\varepsilon}|\) is bounded, \(b_{i}(s,x,t)\in C^{1}(\mathbb{R}\times\overline{\Omega}\times[0,T])\).
By Hölder’s inequality and \(\alpha\leq\frac{p^{-}-2}{2}\),
$$\begin{aligned}& \iint_{{Q_{T}}} {{{\vert {{u_{\varepsilon}}} \vert }^{2\beta }} {{\vert {\nabla{u_{\varepsilon}}} \vert }^{2}} {\,\mathrm{d}}x{\, \mathrm{d}}t} \\& \quad \leqslant c\iint_{{Q_{T}}} {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{2}} {\,\mathrm{d}}x{\,\mathrm{d}}t} = c\iint_{{Q_{T}}} {{\rho _{\varepsilon}}^{ - \frac{{2\alpha}}{p^{-}}} \cdot{\rho_{\varepsilon}}^{\frac{{2\alpha}}{p^{-}}}{{\vert {\nabla{u_{\varepsilon}}} \vert }^{2}} { \,\mathrm{d}}x{\,\mathrm{d}}t} \\& \quad \leq c \biggl(\iint_{{Q_{T}}} {\rho_{\varepsilon}^{ - \frac{{2\alpha }}{{p^{-} - 2}}}{\, \mathrm{d}}x{\,\mathrm{d}}t} \biggr)^{\frac{{p^{-} - 2}}{p^{-}}} \cdot \biggl( \iint_{{Q_{T}}} \rho_{\varepsilon}^{a}\vert { \nabla{u_{\varepsilon}}} \vert ^{p^{-}}{\,\mathrm{d}}x{\,\mathrm{d}}t \biggr)^{\frac{2}{p^{-}}} \leq c. \end{aligned}$$
(3.14)
Combining (3.11)-(3.14), we have
$$\iint_{Q_{T}} (u_{\varepsilon t})^{2}{\,\mathrm{d}}x{\, \mathrm{d}}t + \iint_{Q_{T}} \rho_{\varepsilon}^{\alpha}\frac{\mathrm{d}}{\mathrm{d}t}\int_{0}^{{{\vert {\nabla{u_{\varepsilon}}(x,t)} \vert }^{2}}} s^{\frac{p - 2}{2}}\,\mathrm{d}s \, \mathrm{d}x\,\mathrm{d}t \leqslant c, $$
by the inequality, we have
$$ \iint_{{Q_{T}}} {{{({u_{\varepsilon t}})}^{2}} {\,\mathrm{d}}x{ \,\mathrm{d}}t} \leqslant c + c\int_{\Omega}{ \rho_{\varepsilon}^{\alpha}{{ \vert {\nabla{u_{\varepsilon,0}}} \vert }^{p(x)}}\,\mathrm{d}x} \leqslant c. $$
(3.15)
By (3.10), (3.15), we know that
$$ \int_{0}^{T}\!\!\int_{\Omega_{\lambda}}| \nabla u_{\varepsilon }|^{2}\,\mathrm{d}x\,\mathrm{d}t\leq c,\qquad \int _{0}^{T}\!\!\int_{\Omega}|u_{\varepsilon t}|^{2} \,\mathrm{d}x\,\mathrm{d}t\leq c. $$
(3.16)
By (3.10), (3.15), and (3.16), we know that there exists a subsequence (still denoted \(u_{\varepsilon}\)) of \(u_{\varepsilon}\), which is weakly star convergent to u, and strongly convergent to \(u \in L_{\mathrm{loc}}^{2}(Q_{T})\), and it satisfies (3.2). In particular, \(u_{\varepsilon}\rightarrow u\) a.e. in \(Q_{T}\), and there exists an n-dimensional vector function \(\vec{\zeta} = ({\zeta_{1}}, \ldots,{\zeta_{n}})\),
$$\vert {\vec{\zeta}} \vert \in L^{\frac{p(x)}{p(x) - 1}}({Q_{T}}), $$
such that
$$\begin{aligned}& {u_{\varepsilon}} \rightharpoonup^{*} u\quad \mbox{in } {L^{\infty}(Q_{T})}, \qquad u_{\varepsilon}\rightarrow u \quad \mbox{in }L^{2}_{\mathrm{loc}}(Q_{T}), \\& {\nabla u_{\varepsilon}} \rightharpoonup\nabla u \quad \mbox{in } L_{\mathrm{loc}}^{p(x)}(Q_{T}), \\& \rho_{\varepsilon}^{\alpha}{ \vert {\nabla{u_{\varepsilon}}} \vert ^{p(x) - 2}}\nabla{u_{\varepsilon}} \rightharpoonup\vec{ \zeta}\quad \mbox{in } {L^{\frac{p(x)}{p(x)-1}}}({Q_{T}}). \end{aligned}$$
So u satisfies (1.10) in the sense of a trace. In order to prove that u satisfies equivalence (3.1), we notice that, for any function \(\varphi \in C_{0}^{\infty}({Q_{T}})\),
$$ \iint_{{Q_{T}}} {\bigl(-u_{\varepsilon}\varphi_{t} + \rho_{\varepsilon}^{\alpha}{\bigl(|\nabla u_{\varepsilon}|^{2}+ \varepsilon \bigr)^{\frac{p(x)-2}{2}}} \nabla{u_{\varepsilon}}\cdot\nabla \varphi+b_{i}(u_{\varepsilon},x,t)\varphi_{x_{i}} \bigr){\, \mathrm{d}}x{\,\mathrm{d}}t} = 0. $$
(3.17)
By \(u_{\varepsilon}\rightarrow u\) a.e. in \(Q_{T}\), then \(b_{i}(u_{\varepsilon},x,t)\rightarrow b_{i}(u,x,t)\), and so
$$ \iint_{{Q_{T}}} {\biggl(\frac{{\partial u}}{{\partial t}}\varphi + \vec{\varsigma}\cdot \nabla\varphi+ b_{i}(u,x,t)\varphi_{x_{i}} \biggr){\, \mathrm{d}}x{\,\mathrm{d}}t} = 0. $$
(3.18)
Now, it is not difficult to prove that (cf. [36, 37])
$$ \iint_{{Q_{T}}} {{\rho^{\alpha}} {{\vert {\nabla u} \vert }^{p(x) - 2}}\nabla u \cdot\nabla\varphi{\,\mathrm{d}}x{\,\mathrm{d}}t} = \iint _{{Q_{T}}} {\vec{\zeta} \cdot\nabla\varphi{\,\mathrm{d}}x{\, \mathrm{d}}t}, $$
(3.19)
for any function \(\varphi \in C_{0}^{\infty}({Q_{T}})\), then u satisfies (3.1) and it is the solution of (1.1) with the initial value (1.10). Thus, we have proved Lemma 3.2. □

4 The existence and the uniqueness of solutions

Definition 4.1

Let \(\alpha< p^{-}-1\), \(p^{-}>2\). The function \(u(x,t)\) is said to be the weak solution of (1.1) with the initial value (1.10) and with the boundary value condition
$$ u|_{\Sigma_{p}\times(0,T)}=0, $$
(4.1)
if u satisfies Definition 3.1, and for any function
$$\phi(x,t) \in{C^{2}}({\overline {Q}_{T}}), \qquad {\operatorname{supp}} \phi(x,t) \subset \overline {\Omega} \times(0,T), $$
\(\phi=0\) near \(\Sigma_{p}'= \partial\Omega\setminus\Sigma_{p}\), and the following integral equivalence holds:
$$\begin{aligned}& \iint_{{Q_{T}}} {{u_{t}}\phi \,\mathrm{d}x\,\mathrm{d}t} + \iint_{{Q_{T}}} {{\rho^{\alpha}} {{\vert {\nabla u} \vert }^{p(x)- 2}}\nabla u} \cdot\nabla\phi \,\mathrm{d}x\,\mathrm{d}t \\& \qquad {}+ \iint_{{Q_{T}}} {{b_{i}}(u,x,t){\phi_{{x_{i}}}}\,\mathrm{d}x \,\mathrm{d}t} \\& \quad = - \int_{0}^{T} \!\!{\int_{\Sigma _{p}} {{b_{i}}(0,x,t){n_{i}}\phi} \, \mathrm{d}\sigma \, \mathrm{d}t}, \end{aligned}$$
(4.2)
where \(\Sigma_{p}\) is defined in Section 2 in detail. We quote it as follows.
  1. (i)
    If \(\alpha>1\), then for any given \(t\in(0,T)\),
    $$\Sigma_{p}=\bigl\{ x\in\partial\Omega: b_{iu}(0,x,t)n_{i}(x)< 0 \bigr\} . $$
     
  2. (ii)
    If \(\alpha\leq1\), then for any given \(t\in(0,T)\),
    $$\Sigma_{p}=\partial\Omega. $$
     
  3. (iii)

    If \(\alpha=1\), then for any given \(t\in(0,T)\), \(\Sigma _{p}\) is just a subset of Ω, and it is difficult to write out its explicit formula, except some special cases.

     
For examples, we have the following special cases.
Case 1.: 
When \(N=1\), for any given \(t\in(0,T)\), if for all \(x\in \partial\Omega\), we have
$$b_{iu}(0,x,t)n_{i}\leq0, $$
then
$$\Sigma_{p}=\partial\Omega. $$
Case 2.: 
When \(N\geq2\), for any given \(t\in(0,T)\), if for all \(x\in \partial\Omega\), we have
$$b_{iu}(0,x,t)n_{i}\leq0,\qquad m_{1}(x)\neq0, $$
then
$$\Sigma_{p}=\partial\Omega. $$

Theorem 4.2

Let \(\alpha< \frac{{p^{-} - 2}}{2}\), conditions (A) and (B) be true. Suppose
$${u_{0}} \in{L^{\infty}}(\Omega),\qquad {\rho^{\alpha}} { \vert {\nabla {u_{0}}} \vert ^{p^{+}}} \in{L^{1}}( \Omega), $$
then there is a solution of (1.1) with the initial value condition (1.10) and with the partial boundary value condition (4.1).

Proof

For all \(\varepsilon>0\), selecting \({u_{\varepsilon ,0}}\) such that \(\Vert u_{\varepsilon,0} \Vert _{{L^{\infty}}(\Omega)}\) and \(\Vert \rho_{\varepsilon}^{\alpha} \vert \nabla u_{\varepsilon,0} \vert ^{p^{+}} \Vert _{{L^{1}}(\Omega)}\) are uniformly bounded, and \({u_{\varepsilon,0}}\) converges to \(u_{0}\) in \(W_{\operatorname {\mathrm{loc}} }^{1, p^{+}}(\Omega)\). Let \(u_{\varepsilon}\) be the solution of the initial boundary value problem (3.4)-(3.6). By the condition \(\alpha < \frac{{p^{-} - 2}}{2}\), we have Lemma 3.2, and \(u_{\varepsilon}\) converges to u in \({L^{2}_{\mathrm{loc}}}({Q_{T}})\), and the limit function u is a weak solution of (1.1) with the initial condition (1.10). Now, just as in [31], we can prove that there is \(\gamma \in(1, p^{-}-\frac{\alpha}{\beta})\) such that
$$\iint_{{Q_{T}}} {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{\gamma}} {\,\mathrm{d}}x{\,\mathrm{d}}t}\leq c. $$

Here c is independent of ε. So \(\nabla{u_{\varepsilon}}\) is uniformly bounded in \({L^{\gamma}}({Q_{T}})\), and u has a trace on the boundary.

Let \(\phi(x,t) \in{C^{2}}({\overline {Q}_{T}})\), \({\operatorname{supp}}\phi(x,t) \subset \overline {\Omega} \times(0,T)\), \(\varphi=0\) near \(\Sigma _{p}'\). Equation (1.1) is multiplied by \(\varphi(x,t)\) on both sides, integrated over \(Q_{T}\), then
$$\begin{aligned}& \iint_{{Q_{T}}} {{u_{t}}\phi \,\mathrm{d}x\,\mathrm{d}t} + \iint_{{Q_{T}}} {{\rho^{\alpha}} {{\vert {\nabla u} \vert }^{p(x) - 2}}\nabla u} \cdot\nabla\phi \,\mathrm{d}x\,\mathrm{d}t \\& \qquad {}+ \iint_{{Q_{T}}} {{b_{i}}(u,x,t){\phi_{{x_{i}}}}\,\mathrm{d}x \,\mathrm{d}t} \\& \quad = - \int_{0}^{T} \!\!{\int _{\Sigma _{p} } {{b_{i}}(0,x,t){n_{i}}\phi} \, \mathrm{d}\sigma\, \mathrm{d}t} \end{aligned}$$
That means (4.2) is true. □

Theorem 4.3

Let conditions (A) and (B) be true and \(\alpha< p^{-}-1\),
$${u_{0}} \in{L^{\infty}}(\Omega), \qquad {\rho^{\alpha}} \vert {\nabla {u_{0}}} \vert ^{p^{+}} \in{L^{1}}(\Omega). $$
If \(\Sigma_{p}=\partial \Omega\), the solution of the problem (1.1)-(1.10)-(4.1) is unique.

Proof

Let u and v be two weak solutions, \(u(x,0)=v(x,0)\). We have \({\rho^{\alpha}}|\nabla u|^{p(x)}, {\rho^{\alpha}}|\nabla v{|^{p(x)}} \in{L^{1}}(Q)\), and for all \(\varphi \in C_{0}^{\infty}({Q_{T}})\),
$$\begin{aligned} \iint_{{Q_{T}}} \varphi\frac{\partial(u - v)}{\partial t}\,\mathrm{d}x\,\mathrm{d}t =& - \iint_{{Q_{T}}} {\rho^{\alpha}} \bigl({{\vert {\nabla u} \vert }^{p(x) - 2}}\nabla u - {{\vert {\nabla v} \vert }^{p(x) - 2}}\nabla v \bigr) \cdot\nabla\varphi{\,\mathrm{d}}x{\,\mathrm{d}}t \\ &{}- \iint_{{Q_{T}}} {b_{i}}(u,x,t) \cdot{\varphi_{{x_{i}}}} \,\mathrm{d}x\,\mathrm{d}t . \end{aligned}$$
For any given positive integer n, let \(g_{n}(s)\) be an odd function. When \(s>0\) it is defined as
$${g_{n}}(s) = \left \{ { \textstyle\begin{array}{l@{\quad}l} {1,}&{s > \frac{1}{n},} \\ {{n^{2}}{s^{2}}{{e}^{1 - {n^{2}}{s^{2}}}},}&{s \leqslant\frac{1}{n}.} \end{array}\displaystyle } \right . $$
Choosing \({g_{n}}(u - v)\) as the test function, then
$$\begin{aligned}& \iint_{{Q_{T}}} {{g_{n}}(u - v)\frac{{\partial(u - v)}}{{\partial t}}{ \,\mathrm{d}}x{\,\mathrm{d}}t} \\& \quad {}+ \iint_{{Q_{T}}} {{\rho^{\alpha}}\bigl({{\vert {\nabla u} \vert }^{p(x) - 2}}\nabla u - {{\vert {\nabla v} \vert }^{p(x) - 2}}\nabla v\bigr) \cdot\nabla(u - v){g'_{n}} {\,\mathrm{d}}x{\,\mathrm{d}}t} \\& \quad {}+ {\iint_{{Q_{T}}} {\bigl({b_{i}}(u,x,t) - {b_{i}}(v,x,t)\bigr) \cdot{{(u - v)}_{{x_{i}}}} {g'_{n}} {\,\mathrm{d}}x{\,\mathrm{d}}t}} = 0. \end{aligned}$$
(4.3)
Since for any given \(s>0\), \(g_{n}(s)\) is a monotone increasing sequence of n, and clearly
$$\lim_{n\rightarrow\infty}g_{n}(s)=1,\quad s>0 $$
and
$$\lim_{n\rightarrow\infty}g_{n}(s)=\operatorname{sgn}(x),\quad s \in\mathbb{R}, $$
where \(\operatorname{sgn}(x)\) is the sign function. Thus, we have
$$ \lim_{n\rightarrow\infty}\int_{\Omega}{{g_{n}}(u - v)\frac {{\partial(u - v)}}{{\partial t}}{\,\mathrm{d}}x} = \frac{{\,\mathrm{d}}}{{{\,\mathrm{d}}t}}{\Vert {u - v} \Vert _{1}}. $$
(4.4)
At the same time, it is clear that
$$ \iint_{{Q_{T}}} {{\rho^{\alpha}}\bigl({{\vert {\nabla u} \vert }^{p(x) - 2}}\nabla u - {{\vert {\nabla v} \vert }^{p(x) - 2}}\nabla v \bigr) \cdot \nabla(u - v){g'_{n}} {\,\mathrm{d}}x{\, \mathrm{d}}t} \geqslant0. $$
(4.5)
Now, according to the definition of \({g_{n}}(s)\),
$$\bigl\vert {g_{n}'(s)} \bigr\vert \leqslant \frac{c}{s}, \qquad \vert s \vert \leqslant\frac{1}{n}. $$
We use the following facts:
$$\begin{aligned}& \biggl\vert \iint_{{Q_{T}} \cap\{ |u - v| < \frac{1}{n}\} } {\bigl[{b_{i}}(u,x,t) - {b_{i}}(v,x,t)\bigr]{g_{n}} {{(u - v)}_{{x_{i}}}}}\, \mathrm{d}x\,\mathrm{d}t\biggr\vert \\& \quad = \biggl\vert \iint_{Q \cap\{ |u - v| < \frac{1}{n}\} } {\bigl[{b_{i}}(u,x,t) - {b_{i}}(v,x,t)\bigr]g_{n}'(u - v){{(u - v)}_{{x_{i}}}}}\,\mathrm{d}x\,\mathrm{d}t\biggr\vert \\& \quad \leqslant c\iint_{Q \cap\{ |u - v| < \frac{1}{n}\} } \biggl\vert \frac{{b_{i}}(u,x,t) - {b_{i}}(v,x,t)}{u - v}\biggr\vert \bigl\vert {(u - v)_{{x_{i}}}}\bigr\vert \,\mathrm{d}x\, \mathrm{d}t \\& \quad = c\iint_{Q \cap\{ |u - v| < \frac{1}{n}\} } \biggl\vert {\rho^{ - \frac{\alpha}{p^{-}}}} \frac{{b_{i}}(u,x,t) - {b_{i}}(v,x,t)}{u - v}\biggr\vert \bigl\vert \rho^{\frac{\alpha}{p^{-}}}{(u - v)_{{x_{i}}}}\bigr\vert \,\mathrm{d}x\,\mathrm{d}t \\& \quad \leqslant c\biggl[\iint_{Q \cap\{ |u - v| < \frac{1}{n}\} } {\biggl(\biggl\vert { \rho^{ - \frac{\alpha}{p^{-}}}}\frac{{{b_{i}}(u,x,t) - {b_{i}}(v,x,t)}}{ u - v}\biggr\vert \biggr)^{\frac{p^{-}}{{p^{-} - 1}}}}\, \mathrm{d}x\,\mathrm{d}t\biggr]^{\frac{{p^{-} - 1}}{p^{-}}} \\& \qquad {}\cdot\biggl(\iint_{Q \cap\{ |u - v| < \frac{1}{n}\} } \bigl\vert {\rho^{\alpha}} \nabla(u - v) \bigr\vert ^{p^{-}}\,\mathrm{d}x\,\mathrm{d}t \biggr)^{\frac{1}{p^{-}}} . \end{aligned}$$
(4.6)
Since \(\alpha< p^{-}-1\),
$$\begin{aligned}& \iint_{Q \cap \{ {\vert {u - v} \vert < \frac{1}{n}} \}} {{{\biggl(\biggl\vert {\rho^{ - \frac{\alpha}{p^{-}}}} \frac{{{b_{i}}(u,x,t) - {b_{i}}(v,x,t)}}{{u - v}}\biggr\vert \biggr)}^{\frac{p}{{p^{-} - 1}}}}\,\mathrm{d}x\, \mathrm{d}t} \\& \quad \leqslant\iint_{Q} {{\rho^{ - \frac{\alpha}{{p^{-} - 1}}}}\bigl\vert b_{i}'(\xi,x,t )\bigr\vert ^{\frac{p}{{p^{-} - 1}}}\, \mathrm{d}x\,\mathrm{d}t} \leqslant c\iint_{Q} {{\rho^{ - \frac{\alpha}{{p^{-} - 1}}}} \,\mathrm{d}x\,\mathrm{d}t} \leqslant c, \end{aligned}$$
(4.7)
where \(b_{i}'(\xi,x,t)=\frac{\partial b_{i}(s,x,t)}{\partial s}| _{s=\xi}\), which is bounded by the assumption (B).
In (4.6), let \(n\rightarrow\infty\). If \(\{ x \in\Omega:|u - v| = 0\} \) is a set with 0 measure, then
$$\begin{aligned}& \lim_{n \to\infty} \iint_{Q \cap\{ |u - v| < \frac{1}{n}\} } {\bigl\vert { \rho^{\frac{\alpha}{{p^{-} - 1}}}} b_{i}'(\xi,x,t )\bigr\vert \, \mathrm{d}x\,\mathrm{d}t} \\& \quad = \iint_{Q \cap\{ |u - v| = 0\} } {\bigl\vert { \rho^{\frac{\alpha}{{p^{-} - 1}}}} b_{i}'(\xi,x,t)\bigr\vert \, \mathrm{d}x\,\mathrm{d}t} = 0. \end{aligned}$$
(4.8)
If the set \(\{ x \in\Omega:|u - v| = 0\}\) has a positive measure, then
$$\begin{aligned}& \lim_{n \to\infty} \iint_{Q \cap\{ |u - v| < \frac{1}{n}\} } {\rho^{\alpha}}\bigl\vert \nabla(u - v)\bigr\vert ^{p^{-}}\,\mathrm{d}x\,\mathrm{d}t \\& \quad = \iint_{Q \cap\{ |u - v| = 0\} } {\rho^{\alpha}}\bigl\vert \nabla(u - v)\bigr\vert ^{p^{-}}\,\mathrm{d}x\,\mathrm{d}t = 0. \end{aligned}$$
(4.9)
Therefore, in both cases, (4.6) tends to 0 as \(n\rightarrow\infty\).
Thus we have
$$ { \lim_{n \to\infty} } \iint_{{Q_{T}}} {\bigl({b_{i}}(u,x,t) - {b_{i}}(v,x,t)\bigr)g_{n}'(u - v){{(u - v)}_{{x_{i}}}}\,\mathrm{d}x\,\mathrm{d}t} = 0. $$
(4.10)
Now, let \(n\rightarrow\infty\) in (4.2). Then, by (4.3)-(4.10), we have
$$\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}{\Vert {u - v} \Vert _{1}} \leqslant0. $$
It implies that
$$\int_{\Omega}{\bigl\vert {u(x,t) - v(x,t)} \bigr\vert {\, \mathrm{d}}x} \leqslant\int_{\Omega}{ \vert {{u_{0}} - {v_{0}}} \vert {\,\mathrm{d}}x} = 0, \quad \forall t \in[0,T). $$
By the arbitrariness of t,
$$u(x,t) = v(x,t) \quad \mbox{a.e. }(x,t) \in{Q_{T}}. $$
Theorem 4.3 is proved. □

5 The behavior of solutions near the boundary

Without loss the generality, we also assume that the boundary Ω is of class \(C^{2}\). That is, there exists a number \(\rho_{0}\in(0,1)\) such that for all \(x_{0}\in\partial\Omega\) the portion of Ω within the ball \(B_{\rho_{0}}(x_{0})\) can be represented, in a local system of coordinates, as the graph of a \(C^{2}\) function \(\varphi^{(x_{0})}\) such that \(\varphi^{(x_{0})}(x_{0})=0\), and for \(x\in B_{\rho_{0}}(x_{0})\cap \Omega=\{x=(x_{1}, x_{2}, \ldots, x_{N-1}, x_{N}): x_{N}>0\}\), \(x\in B_{\rho_{0}}(x_{0})\cap\partial\Omega=\{x=(x_{1}, x_{2}, \ldots, x_{N-1}, x_{N}): x_{N}=0\}\). We call this local coordinate transform a planarization technique. In this section, we use some ideas of [38, 39].

Definition 5.1

If \(u_{0}(x)\) satisfies (3.1), u is the limit of the solutions \(\{u_{n}\}\) of the following equations:
$$\begin{aligned}& \frac{\partial u}{\partial t}=\operatorname{div}\biggl[\biggl(\rho^{\alpha}+ \frac{1}{n}\biggr) \biggl(|\nabla u|^{2}+\frac{1}{n} \biggr)^{\frac{p(x)-2}{2}}\nabla u\biggr],\quad (x,t)\in Q_{T}, \end{aligned}$$
(5.1)
$$\begin{aligned}& u(x,0)=u_{0,n}(x), \quad x\in\Omega, \end{aligned}$$
(5.2)
$$\begin{aligned}& u(x,t)=0,\quad (x,t)\in\partial\Omega\times(0,T), \end{aligned}$$
(5.3)
where \(u_{0,n}(x)\) is the smoothly mollified functions of \(u_{0}(x)\). Then we say u is a viscous solution of (1.1).

We shall get estimates above and near Ω.

Theorem 5.2

Let u, \(0\leq u\leq M\), be a nonnegative bounded viscous solution of (1.1) in the sense of Definition  5.1. If \(\alpha \geq 1\), then for any given \(s\in(0,T)\), we have
$$ u(x,t)\leq kC\rho(x), \quad (x,t)\in\Omega\times(s,T), $$
(5.4)
where the constant C depending upon M, N, p, s, and the constant k is a constant independent of s, M.

Proof

Fix \((x_{0},t_{0})\in\partial\Omega\times(s,T)\). By the planarization technique, we may assume that \((x_{0},t_{0})\equiv(0,0)\) and in the vicinity of \((0,0)\), after flattening of Ω near \(x_{0}\), without loss of generality, let us assume that Ω coincides with the portion of hyperplane \(\{x_{N}=0\}\), and the inclusion \(\Omega\cap\{|x|<\rho_{0}\}\subset\{x_{N}>0\}\) is true. Let \(y=(0,\ldots,0,-1)\), and define the set
$$ \aleph_{k}=\biggl\{ (x,t):x_{N}>0,1< |x-y|< 1+ \frac{1}{k},-s_{n}\leq t\leq0\biggr\} . $$
(5.5)
We assume k is so large that \(\aleph_{k}\subset B_{\rho_{0}}^{+}\times(s,0]\). Consider the following problem:
$$\begin{aligned}& \frac{\partial v}{\partial t}=\operatorname{div}\biggl[\biggl(\rho^{\alpha}+ \frac{1}{n}\biggr) \biggl(|\nabla v|^{2}+\frac{1}{n} \biggr)^{\frac{p(x)-2}{2}}\nabla v\biggr],\quad (x,t)\in\aleph_{k}, \end{aligned}$$
(5.6)
$$\begin{aligned}& v(x,-s_{n})=u_{n}(x,-s_{n}),\quad x\in B_{k}^{+}, \end{aligned}$$
(5.7)
$$\begin{aligned}& v(x,t)=u_{n}(x,t)-\frac{1}{n},\quad (x,t)\in\partial B_{k}^{+}\times [-s_{n},0], \end{aligned}$$
(5.8)
where \(u_{n}\) is the solution of the problem (5.1)-(5.3), \(0< s_{n}< s< T\), \(s_{n}n\) is small enough, and
$$B^{+}_{k}=\biggl\{ x: x_{N}>0, 1< |x-y|< 1+ \frac{1}{k}\biggr\} \cap B_{\rho_{0}}^{+}. $$
By the comparison theorem ([40], p.119), we have
$$ v\leq u_{n}. $$
(5.9)
Let us construct a barrier for u in \(\aleph_{k}\). Consider the function
$$\eta_{k}(x,t)=e^{-k(|x-y|-1)}e^{t}, $$
and the barrier is given by
$$\Psi_{k}=CM\bigl(1-\eta_{k}(x,t)\bigr)+\gamma t,\quad (x,t)\in\aleph_{k}, $$
where the constants γ, C are to be chosen later so large that \(v\leq\Psi_{k}\) on the parabolic boundary of \(\aleph_{k}\). This holds true on the portion of such a boundary lying on the hyperplane \(x_{N}=0\),
$$\Psi_{k}=CM\bigl(1-\eta_{k}(x,t)\bigr)+\gamma t\geq CM \bigl(1-e^{-k(|x-y|-1)}\bigr)-\gamma s_{n}\geq-\frac{1}{n} $$
provided that \(\gamma\leq\frac{1}{ns_{n}}\). On the portion \(\{|x-y|=1+\frac{1}{k}\}\cap\{x_{N}\geq0\}\), we have
$$\Psi_{k}\geq CM\bigl(1-e^{-1+t}\bigr)-\gamma s_{n}\geq2M\geq v, $$
if \(C\geq3(1-e^{-1})^{-1}\). On the bottom of \(\aleph_{k}\) we have
$$\Psi_{k}\geq CM\bigl(1-e^{-s}\bigr)-\gamma s_{n}\geq2M\geq v, $$
provided that
$$C\geq\frac{2M+\gamma s}{M(1-e^{-s})}. $$
By direct calculation,
$$\begin{aligned}& \Psi_{k,x_{j}}=kCM\eta_{k}\frac{x_{j}-y_{j}}{\vert x-y\vert }, \\& \Psi_{k,x_{l}x_{j}}=-k^{2}CM\eta_{k}\frac {(x_{j}-y_{j})(x_{l}-y_{l})}{\vert x-y\vert ^{2}} +kCM\eta_{k}\frac{1}{\vert x-y\vert ^{2}}\biggl[\delta_{jl}\vert x-y\vert -\frac {(x_{j}-y_{j})(x_{l}-y_{l})}{\vert x-y\vert }\biggr], \\& \Psi_{k,x_{i}x_{i}}=-k^{2}CM\eta_{k}+kCM \eta_{k}\frac {1}{\vert x-y\vert ^{2}}\bigl(N\vert x-y\vert -\vert x-y \vert \bigr) \\& \hphantom{\Psi_{k,x_{i}x_{i}}}=kCM\eta_{k}\biggl(-k+\frac{N-1}{\vert x-y\vert }\biggr), \\& {\biggl[\biggl(\vert \nabla \Psi_{k}\vert ^{2}+ \frac{1}{n}\biggr)^{\frac{p(x)-2}{2}}\Psi _{k,x_{i}}\biggr]_{x_{i}}} \\& \quad =\frac{p(x)-2}{2}\biggl(\vert \nabla \Psi_{k}\vert ^{2}+\frac{1}{n}\biggr)^{\frac{p(x)-4}{2}}2\Psi_{k,x_{l}}\Psi _{k,x_{l}x_{i}}\Psi_{k,x_{i}} \\& \qquad {}+\biggl(\vert \nabla \Psi_{k}\vert ^{2}+ \frac{1}{n}\biggr)^{\frac{p(x)-2}{2}}\Psi _{k,x_{i}x_{i}}+\ln\biggl(\vert \nabla \Psi_{k}\vert ^{2}+\frac{1}{n} \biggr)p_{x_{i}}(x)\cdot\frac {x_{i}-y_{i}}{\vert x-y\vert }KCM\eta_{k} \\& \quad =\bigl(p(x)-2\bigr)\biggl[(kCM\eta_{k})^{2}+ \frac{1}{n}\biggr]^{\frac{p(x)-4}{2}}(kCM\eta _{k})^{2} \frac{(x_{j}-y_{j})(x_{l}-y_{l})}{\vert x-y\vert ^{2}} \\& \qquad {}\cdot\biggl\{ -k^{2}CM\eta_{k}\frac {(x_{j}-y_{j})(x_{l}-y_{l})}{\vert x-y\vert ^{2}}+ \frac{kCM\eta_{k}}{\vert x-y\vert ^{2}} \biggl[\delta_{jl}\vert x-y\vert - \frac{(x_{j}-y_{j})(x_{l}-y_{l})}{\vert x-y\vert }\biggr]\biggr\} \\& \qquad {}+\biggl[(kCM\eta_{k})^{2}+\frac{1}{n} \biggr]^{\frac{p(x)-2}{2}}kCM\eta _{k}\biggl(-k+\frac{N-1}{\vert x-y\vert }\biggr) \\& \qquad {}+\ln\biggl[(KCM\eta_{k})^{2}+\frac{1}{n} \biggr]p_{x_{i}}(x)\cdot\frac {x_{i}-y_{i}}{\vert x-y\vert }KCM\eta_{k} \\& \quad =G_{n}(x)kCM\eta_{k}\biggl[-k+\frac{1}{\vert x-y\vert ^{2}} \bigl(\vert x-y\vert -\vert x-y\vert \bigr)\biggr] \\& \qquad {}+\biggl[(kCM \eta_{k})^{2}+\frac{1}{n}\biggr]^{\frac{p(x)-2}{2}}kCM \eta _{k}\biggl(-k+\frac{N-1}{\vert x-y\vert }\biggr) \\& \qquad {}+\ln\biggl[(KCM\eta_{k})^{2}+\frac{1}{n} \biggr]p_{x_{i}}(x)\cdot\frac {x_{i}-y_{i}}{\vert x-y\vert }KCM\eta_{k} \\& \quad =-G_{n}(x)k^{2}CM\eta_{k} +\biggl[(kCM \eta_{k})^{2}+\frac{1}{n}\biggr]^{\frac{p(x)-2}{2}}kCM \eta _{k}\biggl(-k+\frac{N-1}{\vert x-y\vert }\biggr) \\& \qquad {}+\ln\biggl[(KCM\eta_{k})^{2}+\frac{1}{n} \biggr]p_{x_{i}}(x)\cdot\frac {x_{i}-y_{i}}{\vert x-y\vert }KCM\eta_{k}, \end{aligned}$$
where \(G_{n}(x)=(p(x)-2)[(kCM\eta_{k})^{2}+\frac{1}{n}]^{\frac {p(x)-4}{2}}(kCM\eta_{k})^{2}\),
$$\begin{aligned}& \Psi_{k,t}-\operatorname{div}\biggl[\biggl(\rho^{\alpha}+ \frac{1}{n}\biggr) \biggl(|\nabla\Psi _{k}|^{2}+ \frac{1}{n}\biggr)^{\frac{p(x)-2}{2}}\nabla\Psi_{k}\biggr] \\& \quad =-CMe^{-k(|x-y|-1)}e^{t}+\gamma-\alpha \rho^{\alpha-1} \rho_{x_{i}}\biggl[\biggl(|\nabla\Psi_{k}|^{2}+ \frac {1}{n}\biggr)^{\frac{p(x)-2}{2}}\Psi_{k,x_{i}}\biggr] \\& \qquad {}-\biggl( \rho^{\alpha}+\frac{1}{n}\biggr)\biggl[\biggl(|\nabla \Psi_{k}|^{2}+\frac {1}{n}\biggr)^{\frac{p(x)-2}{2}} \Psi_{k,x_{i}}\biggr]_{x_{i}} \\& \quad =-CMe^{-k(|x-y|-1)}e^{t}+\gamma-\alpha \rho^{\alpha-1} \rho_{x_{i}}\biggl[\biggl(|kCM\eta_{k}|^{2}+ \frac{1}{n}\biggr)^{\frac {p(x)-2}{2}}kCM\eta_{k}\frac{x_{i}-y_{i}}{|x-y|} \biggr] \\& \qquad {}-\biggl(\rho^{\alpha}+\frac{1}{n}\biggr)\biggl\{ -G_{n}(x)k^{2}CM\eta_{k} +\biggl[(kCM \eta_{k})^{2}+\frac{1}{n}\biggr]^{\frac{p(x)-2}{2}}kCM \eta _{k}\biggl(-k+\frac{N-1}{|x-y|}\biggr)\biggr\} \\& \qquad {}-\ln\biggl[(KCM\eta_{k})^{2}+\frac{1}{n} \biggr]p_{x_{i}}(x)\cdot\frac {x_{i}-y_{i}}{|x-y|}KCM\eta_{k} \\& \quad \geq-CMe^{-k(|x-y|-1)}e^{t}+\gamma-\alpha \rho^{\alpha-1} \biggl(|kCM\eta_{k}|^{2}+\frac{1}{n} \biggr)^{\frac{p(x)}{2}} \\& \qquad {}+\biggl(\rho^{\alpha}+\frac{1}{n}\biggr)G_{n}k^{2}CM \eta_{k} +k\biggl(\rho^{\alpha}+\frac{1}{n}\biggr) \biggl(|kCM\eta_{k}|^{2}+\frac{1}{n} \biggr)^{\frac{p(x)-2}{2}}kCM\eta_{k} \\& \qquad {} -(N-1) \biggl(\rho^{\alpha}+ \frac{1}{n}\biggr) \biggl(|kCM\eta_{k}|^{2}+ \frac {1}{n}\biggr)^{\frac{p(x)-2}{2}} \\& \qquad {}-\ln\biggl[(KCM\eta_{k})^{2}+\frac{1}{n} \biggr]p_{x_{i}}(x)\cdot\frac {x_{i}-y_{i}}{|x-y|}KCM\eta_{k}, \end{aligned}$$
where we have used the facts that \(|\rho_{x_{i}}|=1\), \(\alpha\geq1\).
Clearly, if we choose k large enough, then we have
$$\Psi_{k,t}-\operatorname{div}\biggl[\biggl(\rho^{\alpha}+ \frac{1}{n}\biggr) \biggl(|\nabla\Psi _{k}|^{2}+ \frac{1}{n}\biggr)^{\frac{p(x)-2}{2}}\nabla\Psi_{k}\biggr]\geq 0. $$
It follows by the comparison theorem that the solution of the problem (5.6)-(5.8) v, \(v\leq\Psi_{k}\) in \(\aleph_{k}\). In particular, \(\forall 0< x_{N}<\frac{1}{k}\), we have
$$\begin{aligned} u(0,0,\ldots,0_{N-1},x_{N},0) =&\lim_{n\rightarrow\infty} v(0,0,\ldots,0_{N-1},x_{N},0) \\ \leq&\Psi_{k}(0,0,\ldots,0_{N-1},x_{N},0) \\ =&CM\bigl(1-e^{-kx_{N}}\bigr)\leq kCMx_{N}. \end{aligned}$$
Therefore there exists a constant k depending only upon N, such that
$$u(x,t)\leq k M \operatorname{dist}(x,\partial\Omega), $$
for all \(x\in\Omega\) such that \(\rho(x)\leq\frac{1}{k}\). On the other hand, if \(\rho(x)>\frac{1}{k}\), we have
$$u(x,t)\leq M\leq kC M \rho(x). $$
Thus (5.5) holds in both cases. □
Estimates below and near Ω: Let u be a nonnegative bounded viscous solution of (1.1) in the sense of Definition 5.1,
$$u\leq M, $$
for some \(M>0\). For \(r>0\) let
$$\begin{aligned}& \Omega_{r}\equiv\bigl\{ x\in\Omega\mid d(x,\partial\Omega)\geq r\bigr\} , \\& \Omega_{r,t}\equiv\Omega_{r}\times[s,t], \quad \forall s< t \leq T \end{aligned}$$
and
$$\mu(r)\equiv\inf_{(x,\tau)\in\Omega_{r,t}}u(x,\tau). $$
For \(0< s< t< T\), let
$$r(M,s,t)=\rho_{0}\min\{1,\sqrt{t-s}\}, $$
where the constant \(\rho_{0}\) makes the inclusion \(\Omega\cap\{|x|<\rho_{0}\}\subset\{x_{N}>0\}\) true as before.

Now, we estimate u below, near the boundary Ω.

Theorem 5.3

If the hypothesis of Theorem  5.2 is true, then \(\forall0< s< t< T\), \(\forall x\in\Omega\), \(\rho(x)\leq r(M,s,t)\), the inequality
$$ u(x,t)\geq\mu\bigl(r(M,s,t)\bigr)\rho(x) $$
(5.10)
holds.

Proof

Fix \((x_{0},t_{0})\in\partial\Omega\times(s,T)\) and let \(\mu_{0}\equiv\mu(r(M,s,t_{0}))\). After flattening of Ω near \(x_{0}\), we may assume that \((x_{0},t_{0})\equiv(0,0)\) as before. Introduce the point
$$\overline {y}\equiv\biggl(0,0,\ldots, 0_{N-1},1+\frac{1}{k}\biggr), $$
and the domain
$$\overline {\aleph}_{kn}=\biggl\{ (x,t):0< x_{N}< \frac{1}{k},1< |x-\overline {y}|< 1+\frac {1}{k},-s_{n}\leq t \leq0\biggr\} \subset B_{\rho_{0}}^{+}\times[-s,0], $$
where
$$\frac{1}{k}=r(M,s,t_{0})=\rho_{0}\min\{1,\sqrt{s} \}. $$
Consider \(-s<-s_{n}<t\leq0\),
$$\begin{aligned}& \frac{\partial v}{\partial t}=\operatorname{div}\biggl[\biggl(\rho^{\alpha}+ \frac{1}{n}\biggr) \biggl(|\nabla v|^{2}+\frac{1}{n} \biggr)^{\frac{p(x)-2}{2}}\nabla v\biggr],\quad (x,t) \in \overline {\aleph}_{kn}, \end{aligned}$$
(5.11)
$$\begin{aligned}& v_{n}(x,-s_{n})=u_{n}(x,-s_{n}), \quad (x, s_{n})\in \overline {\aleph}_{kn}, \end{aligned}$$
(5.12)
$$\begin{aligned}& v(x,t)=\frac{1}{n}+u_{n}(x,t), \quad (x,t)\in\partial \overline {B}_{k}\times [-s_{n}, 0], \end{aligned}$$
(5.13)
where \(u_{n}\) is the nonnegative solution of the problem (5.1)-(5.3), \(s_{n}n\) is small enough, and \(\overline {B}_{k}=\{x: 0< x_{N}<\frac{1}{k}, 1<|x-y|<1+\frac{1}{k}\}\). Also by the comparison theorem ([40], p.119), we have
$$ v\geq u_{n}. $$
(5.14)
Consider the function
$$\overline {\eta}_{k}(x,t)=e^{-k(|x-\overline {y}|-1)}e^{\frac{t}{s}} $$
and construct the barrier
$$\overline {\Psi}_{k}(x,t)=\mu_{0}\bigl(\overline { \eta}_{k}(x,t)-e^{-1}\bigr)_{+}-\gamma t, $$
where \(\gamma=\gamma(s, \mu_{0}, k)\) is a large enough constant to be chosen later. Let us show that \(v\geq \overline {\Psi}_{k}\) on the parabolic boundary of \(\overline {\aleph}_{k}\). On the portion \(\{|x-\overline {y}|=1+\frac{1}{k}\}\times[-s_{n},0]\) we have \(\overline {\Psi}_{k}=-\gamma t\leq\gamma s_{n}\leq\frac{1}{n}\leq v\). On the portion lying on the hyperplane \(\{x_{N}=\frac{1}{k}\}\) one checks that \(\overline {\Psi}_{k}\leq\mu_{0}\leq u\leq v\). On the bottom of \(\overline {\aleph}_{k}\), we have
$$\overline {\Psi}_{k}|_{t={-s_{n}}}\leq\mu_{0} \bigl(e^{-k(|x-\overline {y}|-1)}-1\bigr)_{+}+\gamma s_{n}\leq0. $$
By direct calculation
$$ \overline {\Psi}_{k,x_{j}}=-k\mu_{0}\overline {\eta}_{k} \frac{x_{j}-y_{j}}{|x-y|},\qquad \frac{\partial \overline {\Psi}}{\partial t}=\mu_{0}\overline {\eta}_{k} \frac{1}{s_{n}}-\gamma $$
and
$$\begin{aligned}& \overline{\Psi}_{k,x_{l}x_{j}}=k^{2}\mu_{0}\overline{\eta }_{k}\frac{(x_{j}-\overline{y}_{j})(x_{l}-\overline {y}_{l})}{\vert x-\overline{y}\vert ^{2}} -k\mu_{0}\overline{ \eta}_{k}\frac{1}{\vert x-\overline{y}\vert ^{2}}\biggl[\delta _{jl}\vert x- \overline{y}\vert -\frac{(x_{j}-\overline{y}_{j})(x_{l}-\overline {y}_{l})}{\vert x-\overline{y}\vert }\biggr], \\& \overline{\Psi}_{k,x_{i}x_{i}}=k^{2}\mu_{0}\overline{\eta }_{k}-k\mu_{0}\overline{\eta}_{k} \frac{1}{\vert x-\overline {y}\vert ^{2}}\bigl[N\vert x-\overline{y}\vert -\vert x-\overline{y} \vert \bigr] \\& \hphantom{\overline{\Psi}_{k,x_{i}x_{i}}}=k\mu_{0}\overline{\eta}_{k}\biggl(k- \frac{N-1}{\vert x-\overline{y}\vert }\biggr), \\& {\biggl[\biggl(\vert \nabla \overline{\Psi}_{k}\vert ^{2}+\frac{1}{n}\biggr)^{\frac{p(x)-2}{2}}\overline { \Psi}_{k,x_{i}}\biggr]_{x_{i}}} \\& \quad =\frac{p(x)-2}{2}\biggl(\vert \nabla \overline{\Psi}_{k} \vert ^{2}+\frac{1}{n}\biggr)^{\frac{p(x)-4}{2}}2\overline { \Psi}_{k,x_{l}}\overline{\Psi}_{k,x_{l}x_{i}}\overline{ \Psi}_{k,x_{i}} \\& \qquad {}+\biggl(\vert \nabla \overline{\Psi}_{k}\vert ^{2}+\frac{1}{n}\biggr)^{\frac{p(x)-2}{2}}\overline { \Psi}_{k,x_{i}x_{i}}-\ln\biggl(\vert \nabla \overline{\Psi}_{k} \vert ^{2}+\frac{1}{n}\biggr)\nabla p(x)\cdot\nabla \overline{ \Psi}_{k} \\& \quad =\bigl(p(x)-2\bigr)\biggl[(k\mu_{0}\overline{ \eta}_{k})^{2}+\frac{1}{n}\biggr]^{\frac {p(x)-4}{2}}(k \mu_{0}\overline{\eta}_{k})^{2} \frac{(x_{i}-\overline{y}_{i})(x_{l}-\overline{y}_{l})}{\vert x-\overline {y}\vert ^{2}} \\& \qquad {}\cdot\biggl\{ k^{2}\mu_{0}\overline{ \eta}_{k}\frac{(x_{i}-\overline {y}_{i})(x_{l}-\overline{y}_{l})}{\vert x-\overline{y}\vert ^{2}} +\frac{k\mu_{0}\overline{\eta}_{k}}{\vert x-\overline{y}\vert ^{2}} \biggl[ \delta_{il}\vert x-\overline{y}\vert -\frac{(x_{j}-\overline {y}_{j})(x_{l}-\overline{y}_{l})}{\vert x-\overline{y}\vert }\biggr] \biggr\} \\& \qquad {}+\biggl[(k\mu_{0}\overline{\eta}_{k})^{2}+ \frac{1}{n}\biggr]^{\frac {p(x)-2}{2}}k\mu_{0}\overline{ \eta}_{k}\biggl(k-\frac{N-1}{\vert x-\overline{y}\vert }\biggr) \\& \qquad {}-\ln\biggl[(k \mu_{0}\overline{\eta}_{k})^{2}+ \frac{1}{n}\biggr] p_{x_{i}}(x)\frac{x_{i}-y_{i}}{\vert x-y\vert }k\mu_{0} \overline{\eta}_{k} \\& \quad =\overline{G}_{n}(x)k\mu_{0}\overline{ \eta}_{k}\biggl[k-\frac {1}{\vert x-\overline{y}\vert ^{2}}\bigl(\vert x-\overline{y} \vert -\vert x-\overline{y}\vert \bigr)\biggr] \\& \qquad {}+\biggl[(k\mu_{0} \overline{\eta}_{k})^{2}+\frac{1}{n} \biggr]^{\frac {p(x)-2}{2}}k\mu_{0}\overline{\eta}_{k}\biggl(k- \frac{N-1}{\vert x-\overline{y}\vert }\biggr) \\& \qquad {}-\ln\biggl[(k\mu_{0}\overline{\eta}_{k})^{2}+ \frac{1}{n}\biggr] p_{x_{i}}(x)\frac{x_{i}-y_{i}}{\vert x-y\vert }k\mu_{0} \overline{\eta}_{k} \\& \quad =\overline{G}_{n}(x)k^{2}\mu_{0} \overline{\eta}_{k} +\biggl[(k\mu_{0}\eta_{k})^{2}+ \frac{1}{n}\biggr]^{\frac{p(x)-2}{2}}k\mu _{0}\overline{ \eta}_{k}\biggl(k-\frac{N-1}{\vert x-y\vert }\biggr) \\& \qquad {}-\ln\biggl[(k\mu_{0}\overline{\eta}_{k})^{2}+ \frac{1}{n}\biggr]p_{x_{i}}(x)\frac {x_{i}-y_{i}}{\vert x-y\vert }k\mu_{0} \overline{\eta}_{k}, \end{aligned}$$
where \(\overline{G}_{n}(x)=(p(x)-2)[(k\mu_{0}\overline{\eta }_{k})^{2}+\frac{1}{n}]^{\frac{p(x)-4}{2}}(k\mu_{0}\overline{\eta }_{k})^{2}\),
$$\begin{aligned}& \overline{\Psi}_{k,t}-\operatorname{div}\biggl[\biggl( \rho^{\alpha}+\frac {1}{n}\biggr) \biggl(|\nabla\overline{ \Psi}_{k}|^{2}+\frac{1}{n}\biggr)^{\frac {p(x)-2}{2}} \nabla\overline{\Psi}_{k}\biggr] \\& \quad =\mu_{0}e^{-k(|x-y|-1)}e^{\frac{t}{s_{n}}}\frac{1}{s_{n}}- \gamma -\alpha \rho^{\alpha-1}\rho_{x_{i}}\biggl[\biggl(|\nabla \overline{\Psi}_{k}|^{2}+\frac {1}{n} \biggr)^{\frac{p-2}{2}}\overline{\Psi}_{k,x_{i}}\biggr] \\& \qquad {}-\biggl( \rho^{\alpha}+\frac{1}{n}\biggr)\biggl[\biggl(|\nabla\overline{\Psi }_{k}|^{2}+\frac{1}{n}\biggr)^{\frac{p-2}{2}} \overline{\Psi}_{k,x_{i}}\biggr]_{x_{i}} \\& \quad =\mu_{0}e^{-k(|x-y|-1)}e^{\frac{t}{s_{n}}}\frac{1}{s_{n}}- \gamma -\alpha \rho^{\alpha-1}\rho_{x_{i}}\biggl[\biggl(|k \mu_{0}\overline{\eta}_{k}|^{2} + \frac{1}{n}\biggr)^{\frac{p(x)-2}{2}}k\mu_{0}\overline{ \eta}_{k}\frac {x_{i}-\overline{y}_{i}}{|x-\overline{y}|}\biggr] \\& \qquad {}-\biggl(\rho^{\alpha}+\frac{1}{n}\biggr)\biggl\{ \overline{G}_{n}(x)k^{2}\mu _{0}\overline{ \eta}_{k} +\biggl[(k\mu_{0}\overline{\eta}_{k})^{2}+ \frac{1}{n}\biggr]^{\frac {p(x)-2}{2}}k\mu_{0}\overline{ \eta}_{k}\biggl(k-\frac{N-1}{|x-\overline {y}|}\biggr)\biggr\} \\& \qquad {}-\ln\biggl[(k\mu_{0}\overline{\eta}_{k})^{2}+ \frac{1}{n}\biggr] p_{x_{i}}(x)\frac{x_{i}-y_{i}}{|x-y|}k\mu_{0} \overline{\eta}_{k} \\& \quad \leq\mu_{0}e^{-k(|x-y|-1)}e^{\frac{t}{s_{n}}}\frac{1}{s_{n}}- \gamma +\alpha \rho^{\alpha-1}\biggl(|k\mu_{0}\eta_{k}|^{2}+ \frac{1}{n}\biggr)^{\frac{p(x)}{2}} \\& \qquad {}-\biggl(\rho^{\alpha}+\frac{1}{n}\biggr) \overline{G}_{n}k^{2}\mu_{0}\overline { \eta}_{k} -k\biggl(\rho^{\alpha}+\frac{1}{n}\biggr)|k \mu_{0}\overline{\eta}_{k}|^{p(x)-1} \\& \qquad {}+(N-1) \biggl( \rho^{\alpha}+\frac{1}{n}\biggr) \biggl(|k\mu_{0} \eta_{k}|^{2}+\frac {1}{n}\biggr)^{\frac{p(x)-2}{2}} \\& \qquad {}-\ln\biggl[(k\mu_{0}\overline{\eta}_{k})^{2}+ \frac{1}{n}\biggr] p_{x_{i}}(x)\frac{x_{i}-y_{i}}{|x-y|}k\mu_{0} \overline{\eta}_{k}, \end{aligned}$$
where we have used the fact that \(|\rho_{x_{i}}|=1\) too.
Clearly, if we choose γ large enough, then we have
$$\Psi_{k,t}-\operatorname{div}\biggl[\biggl(\rho^{\alpha}+ \frac{1}{n}\biggr) \biggl(|\nabla\Psi _{k}|^{2}+ \frac{1}{n}\biggr)^{\frac{p(x)-2}{2}}\nabla\Psi_{k}\biggr]\leq 0. $$
It follows from the comparison theorem that \(v\geq \overline {\Psi}_{k}\) in \(\overline {\aleph}_{k}\). In particular, \(\forall0< x_{N}<\frac{1}{k}\),
$$ v(0,0,\ldots,0_{N-1},x_{N},0)\geq\frac{\mu _{0}}{e} \bigl(e^{kx_{N}}-1\bigr)\geq\frac{\mu_{0}k}{e}x_{N}. $$
Let \(n\rightarrow\infty\). We have
$$ u(x,t)\geq\frac{k}{e}\mu\bigl(r(M,s,t)\bigr)\rho(x)\geq\mu\bigl(r(M,s,t) \bigr)\rho(x). $$
Theorem 5.3 is proved. □

Declarations

Acknowledgements

The author sincerely thanks the reviewers for their good modifying opinions on my paper. The paper is supported by NSF of China (No. 11371297), supported by NSF of Fujian Province (No. 2015J01592), supported by SF of Xiamen University of Technology, China.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Applied Mathematics, Xiamen University of Technology

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