# Study of weak solutions for degenerate parabolic inequalities with nonstandard conditions

## Abstract

In this paper, we study the degenerate parabolic variational inequalities in a bounded domain. By solving a series of penalty problems, the existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process.

## 1 Introduction

Let $$0 < T < \infty$$ and $$\Omega \subset {{\mathrm{{R}}}_{N}} (N \ge 2)$$ be a bounded simple domain with appropriately smooth boundary âˆ‚Î©. In this article, we consider the following quasilinear degenerate parabolic inequalities:

\begin{aligned} \textstyle\begin{cases} \min \{ Lu,u(x,0) - {u_{0}}\} = 0, & (x,t) \in {Q_{T}}, \\ u(x,t) = 0, & (x,t) \in {\Gamma _{T}}, \\ u(x,0) = {u_{0}}, & x \in \Omega, \end{cases}\displaystyle \end{aligned}
(1.1)

with

\begin{aligned} Lu = {u_{t}} - u{\mathrm{{div}}} \bigl(a(u){ \vert {\nabla u} \vert ^{p(x,t) - 2}} \nabla u \bigr) - \gamma { \vert {\nabla u} \vert ^{p(x,t)}} - f(x,t), \end{aligned}
(1.2)

where $${Q_{T}} = \Omega \times (0,T]$$, $${\mathrm{{a}}}(u) = {u^{\sigma }} + {d_{0}}$$, and $${\Gamma _{T}}$$ is the lateral boundary of cylinder $$Q_{T}$$.

In applications, Problem (1.1) arises in the model of American option pricing in the Blackâ€“Scholes models. We refer to [1â€“4] for the financial background of parabolic inequalities. Among them, the most interesting research topic is to construct different types of variational parabolic inequalities and analyze the existence and uniqueness for their solutions (see, for example, [3â€“10] and the references therein). In 2014, the authors in [5] discussed the problem

\begin{aligned} \textstyle\begin{cases} {u_{t}} - Lu - F(u,x,t) \ge 0 &\text{in } {Q_{T}}, \\ u(x,t) \ge {u_{0}}(x) &\text{in } \Omega, \\ ( {{u_{t}} - Lu - F(u,x,t)} ) \cdot ( {u(x,t) - {u_{0}}(x)} ) = 0 &\text{in } {Q_{T}}, \\ u(x,0) = {u_{0}}(x) &\text{in } \Omega, \\ u(x,t) = 0 &\text{on } {\Gamma _{T}}, \end{cases}\displaystyle \end{aligned}

with second-order elliptic operator

\begin{aligned} Lu = - u{\mathrm{{div}}} \bigl(a(u){ \vert {\nabla u} \vert ^{p(x,t) - 2}} \nabla u \bigr) - \gamma { \vert {\nabla u} \vert ^{p(x,t)}} - f(x,t). \end{aligned}

They proved the existence and uniqueness of a solution to this problem with some restrictions on $$u_{0}$$, F, and L. Later, the authors in [6, 7] extended the relative conclusions with the assumption that $$a(u)$$ is a constant, and $$p(x) =2$$. The authors discussed the existence and numerical algorithm of the solution.

To the best of our knowledge, the existence and uniqueness of this problem with the assumption that $$p(x,t)$$ are variables have been less studied. We cannot easily apply the method in [6, 7] to the case that $$p(x,t)$$ and $$a(u)$$ are not constants.

The aim of this paper is to study the existence and uniqueness of solutions for a degenerate parabolic variational inequality problem. Throughout the paper, we assume that the exponent $$p(x, t)$$ is continuous in $$Q_{T}$$ with a logarithmic module of a country:

\begin{aligned} 1 < {p^{-} } < p(x,t) < {p^{+} } < \infty, \end{aligned}
(1.3)

where $${p^{-} } = \mathop {\inf } _{(x,t) \in {Q_{T}}} p(x,t)$$ and $${p^{+} } = \mathop {\sup } _{(x,t) \in {Q_{T}}} p(x,t)$$.

The outline of this paper is as follows: In Sect.Â 2, we introduce the function spaces of Orliczâ€“Sobolev type, give the definition of the weak solution to the problem, and state our main theorems. In Sect.Â 3, we give some estimates of the penalty problem (approximating problem). SectionÂ 4 proves the existence and uniqueness of the solution obtained in Sect.Â 2.

## 2 The main results of weak solutions

In this section, we recall some useful definitions and known results, which can be found in [11â€“14]. Set

\begin{aligned} &{L^{p(x,t)}}({Q_{T}}) = \biggl\{ u(x,t)|u \text{ is measurable in } {Q_{T}},{A_{p(\cdot)}}(u) = \int { \int _{{Q_{T}}} {{{ \vert u \vert }^{p(x,t)}} {\, \mathrm{d}}x{\,\mathrm{d}}t} } \biggr\} ,\\ &{ \Vert u \Vert _{p(\cdot)}} = \inf \bigl\{ \lambda > 0,{A_{p(\cdot)}}(u/ \lambda ) \le 1 \bigr\} ,\\ &{V_{t}}(\Omega ) = \bigl\{ u| {u \in {L^{2}}(\Omega ) \cap W_{0}^{1,1}, \vert {\nabla u} \vert \in {L^{p(x,t)}}( \Omega )} \bigr\} ,\\ &{ \Vert u \Vert _{{V_{t}}(\Omega )}} = { \Vert u \Vert _{2, \Omega }} + { \Vert {\nabla u} \Vert _{p( \cdot,t),\Omega }},\\ &W({Q_{T}}) = \bigl\{ u:[0,T] \to {V_{t}}( \Omega )| {u \in {L^{2}}({Q_{T}}) \cap W_{0}^{1,1},} \\ &\vert {\nabla u} \vert \in {L^{p(x,t)}}({Q_{T}}),u = 0 \text{ on } {\Gamma _{T}}\bigr\} , \\ &{ \Vert u \Vert _{{W_{t}}({Q_{T}})}} = { \Vert u \Vert _{2,{Q_{T}}}} + { \Vert {\nabla u} \Vert _{p( \cdot,t),{Q_{T}}}} \end{aligned}

and denote by $$W'({Q_{T}})$$ the dual of $$W({Q_{T}})$$ with respect to the inner product in $${L^{2}}({Q_{T}})$$.

In the spirit of [3] and [4], we introduce the following maximal monotone graph

\begin{aligned} G(x) = \textstyle\begin{cases} 0,& x > 0, \\ \theta,& x =0, \end{cases}\displaystyle \end{aligned}
(2.1)

where $$\theta \in [0,M)$$ and M depends only on $$|u_{0}|_{\infty}$$.

The purpose of the paper is to obtain the existence and uniqueness of weak solutions of (1.1). Let $$B = W({Q_{T}}) \cap {L^{\infty }}(0,T;{L^{\infty }}(\Omega ))$$, and the weak solution is defined as:

### Definition 2.1

A pair is called a weak solution of problem (1.1), if (a) $$u(x,t) \ge {u_{0}}(x)$$, (b) $$u(x,0) = {u_{0}}(x)$$, (c) $$\xi \in G(u - {u_{0}})$$, (d) for every test function $$\phi \in Z \equiv \{ \eta (z):\eta \in W({Q_{T}}) \cap {L^{\infty }}(0,T;{L^{2}}( \Omega )),{\eta _{t}} \in W({Q_{T}})\}$$ and every $${t_{1}},{t_{2}} \in [0,T]$$ the following identity holds:

\begin{aligned} \begin{aligned} &\int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{u \cdot {\phi _{t}} - a(u){{ \vert { \nabla u} \vert }^{p(x,t) - 2}}\nabla u\nabla \phi - \bigl(a(u) - \gamma \bigr){{ \vert {\nabla u} \vert }^{p(x,t)}}\phi {\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &\quad{}+ \int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{ f(x,t)\phi + \xi \phi { \,\mathrm{d}}x{\, \mathrm{d}}t} } = \int _{\Omega }{u\phi {\,\mathrm{d}}x} \int _{{t_{1}}}^{{t_{2}}}. \end{aligned} \end{aligned}
(2.2)

Our main results are the following two theorems.

### Theorem 2.1

Let us satisfy conditions (1.3). If the following conditions hold:

1. (H1)

$$\max \{ 1,\frac{{2N}}{{N + 1}}\} < {p^{-} } < N$$, $$2 \le \sigma < \frac{{2{p^{+} }}}{{{p^{+} } - 1}}$$, $$0 < \gamma < {d_{0}}$$, and

2. (H2)

$${u_{0}}(x) \ge 0, f \ge 0, { \Vert {{u_{0}}} \Vert _{ \infty,\Omega }} + \int _{0}^{T} {{{ \Vert {f(x,t)} \Vert }_{ \infty,\Omega }}{\,\mathrm{d}}t} + \vert \Omega \vert \cdot T = K(T) < \infty$$,

then Problem (1.1) has at least one weak solution in the sense of DefinitionÂ 2.1.

### Theorem 2.2

Suppose that the conditions in TheoremÂ 2.1are fulfilled and $$p^{+} \geq 2$$. Then, Problem (1.1) admits a unique solution in the sense of DefinitionÂ 2.1.

## 3 Penalty problems

In this section, we consider a family of auxiliary parabolic problems

\begin{aligned} \textstyle\begin{cases} {L_{\varepsilon }}{u_{\varepsilon }} + \beta ({u_{\varepsilon }} - {u_{0}}) = 0,& (x,t) \in {Q_{T}}, \\ {u_{\varepsilon }}(x,t) = \varepsilon, & (x,t) \in { \Gamma _{T}}, \\ {u_{\varepsilon }}(x,0) = {u_{0}} + \varepsilon, & x \in \Omega, \end{cases}\displaystyle \end{aligned}
(3.1)

with

\begin{aligned} {L_{\varepsilon }} {u_{\varepsilon }} = {u_{\varepsilon }}_{t} - {u_{ \varepsilon }} \cdot {\mathrm{{div}}} \bigl(a({u_{\varepsilon }}){ \vert { \nabla {u_{\varepsilon }}} \vert ^{p(x) - 2}}\nabla {u_{ \varepsilon }} \bigr) - \gamma { \vert {\nabla {u_{\varepsilon }}} \vert ^{p(x,t)}} - f(x,t), \end{aligned}
(3.2)

$${\beta _{\varepsilon }}( \cdot )$$ is the penalty function satisfying

\begin{aligned} \begin{aligned} &\varepsilon \in (0,1),\qquad{\beta _{\varepsilon }}( \cdot ) \in {C^{2}}({ \mathrm{{R}}}),\qquad{\beta _{\varepsilon }}(x) \le 0,\qquad{\beta _{\varepsilon }}(0) = - 1, \\ &{{\beta '}_{\varepsilon }}(0) \ge 0,\qquad{{\beta ''}_{\varepsilon }}(0) \ge 0,\qquad\mathop {\lim } _{x \to 0 + } \beta (x) = \textstyle\begin{cases} 0,&x > - 0, \\ - 1,&x = 0. \end{cases}\displaystyle \end{aligned} \end{aligned}
(3.3)

With a similar method as in [8], we may prove that the regularized problem has a unique weak solution

\begin{aligned} {u_{\varepsilon }}(x,t) \in W({Q_{T}}) \cap {L^{2}}({Q_{T}}),\quad{ \partial _{t}} {u_{\varepsilon }}(x,t) \in W\prime ({Q_{T}}), \end{aligned}

satisfying the following integral identities

\begin{aligned} \begin{aligned} &\int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{{u_{\varepsilon }} \cdot { \phi _{t}} - a({u_{\varepsilon }}){u_{\varepsilon }} {{ \vert { \nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{ \varepsilon }}\nabla \phi - \bigl(a({u_{\varepsilon }}) - \gamma \bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t)}}\phi {\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\quad= \int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{ \bigl( {{\beta _{ \varepsilon }}({u_{\varepsilon }} - {u_{0}}) - f(x,t)} \bigr)\phi { \,\mathrm{d}}x{\,\mathrm{d}}t} } + \int _{\Omega }{{u_{\varepsilon }}\phi {\,\mathrm{d}}x} \int _{{t_{1}}}^{{t_{2}}} {} \end{aligned} \end{aligned}
(3.4)

and

\begin{aligned} \begin{aligned} &\int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{{u_{\varepsilon }}_{t} \cdot \phi + a({u_{\varepsilon }}){u_{\varepsilon }} {{ \vert { \nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{ \varepsilon }}\nabla \phi + \bigl(a({u_{\varepsilon }}) - \gamma \bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t)}}\phi {\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\quad= \int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{ \bigl( {f(x,t) - { \beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr)\phi { \,\mathrm{d}}x{\,\mathrm{d}}t} }. \end{aligned} \end{aligned}
(3.5)

We start with two preliminary results that will be used several times below.

### Lemma 3.1

Let $$M(s) = { \vert s \vert ^{p(x,t) - 2}}s$$, then $$\forall \xi,\eta \in {{\mathrm{{R}}}^{N}}$$

\begin{aligned} \begin{aligned} & \bigl( {M(\xi ) - M(\eta )} \bigr) \cdot (\xi - \eta ) \\ &\quad\ge \textstyle\begin{cases} {2^{ - p(x,t)}}{ \vert {\xi - \eta } \vert ^{p(x,t)}},& 2 \le p(x,t) < \infty, \\ (p(x,t) - 1){ \vert {\xi - \eta } \vert ^{2}}{ ( {{{ \vert \xi \vert }^{p(x,t)}} + {{ \vert \eta \vert }^{p(x,t)}}} )^{ \frac{{p(x,t) - 2}}{{p(x,t)}}}},& 1 \le p(x,t) < 2. \end{cases}\displaystyle \end{aligned} \end{aligned}

### Proof

The proof can be found in [15].â€ƒâ–¡

### Lemma 3.2

(Comparison principle)

Assume $$2 < \sigma < \frac{{2{p^{+} }}}{{{p^{+} } - 1}}$$, $${p^{+} } \ge 2$$, u and v are in $$W({Q_{T}}) \cap {L^{\infty }}(0,T;{L^{ \infty }}(\Omega ))$$. If $${L_{\varepsilon }}u \ge {L_{\varepsilon }}v$$ in $$Q_{T}$$ and $$u(x,t) \le v(x,t)$$ on $$\partial {Q_{T}}$$, then $$u(x,t) \le v(x,t)$$ in $$Q_{T}$$.

### Proof

We argue by contradiction. Suppose $$u(x,t)$$ and $$v(x,t)$$ satisfy $${L_{\varepsilon }}u \ge {L_{\varepsilon }}v$$ in $${Q_{T}}$$ and there is a $$\delta > 0$$ such that for $$0 < \tau \le T$$, $$w = u - v$$ on the set

\begin{aligned} {\Omega _{\delta }} = \Omega \cap \bigl\{ {x:w(x,t) > \delta } \bigr\} \end{aligned}

and $$\mu ({\Omega _{\delta }}) > 0$$. Let

\begin{aligned} {F_{\varepsilon }}(\xi ) = \textstyle\begin{cases} \frac{1}{{\alpha - 1}}{\varepsilon ^{1 - \alpha }} - \frac{1}{{\alpha - 1}}{\xi ^{1 - \alpha }} &\text{if }\xi > \varepsilon, \\ 0 &\text{if }\xi \le \varepsilon, \end{cases}\displaystyle \end{aligned}
(3.6)

where $$\delta > {\mathrm{{2}}}\varepsilon > 0$$ and $$\alpha = \frac{\sigma }{2}$$. Let a test function $$\xi = {F_{\varepsilon }}(w) \in Z$$ in (3.4),

\begin{aligned} \begin{aligned} 0 \ge{}& \int { \int _{{Q_{T}}} {{w_{t}} {F_{\varepsilon }}(w) + a(v)v \bigl( { \bigl\vert \nabla u{|^{p(x,t) - 2}}\nabla u - \bigr\vert \nabla v{|^{p(x,t) - 2}} \nabla v} \bigr)\nabla {F_{\varepsilon }}(w){\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u)u - a(v)v\bigr]|\nabla u{|^{p(x,t) - 2}}\nabla u\nabla {F_{\varepsilon }}(w){\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u) - \gamma \bigr] \bigl\vert \nabla u{|^{p(x,t)}} - \bigl[a(v) - \gamma \bigr] \bigr\vert \nabla v{|^{p(x,t)}}} } \\ ={}& {J_{1}} + {J_{2}} + {J_{3}} + {J_{4}}, \end{aligned} \end{aligned}
(3.7)

where $${Q_{T,\varepsilon }} = \{ (x,t) \in {Q_{T}}|w > \varepsilon \}$$,

\begin{aligned} &{J_{1}} = \int { \int _{{Q_{T}}} {{w_{t}} {F_{\varepsilon }}(w){\, \mathrm{d}}x{ \,\mathrm{d}}t} },\\ &{J_{2}} = \int { \int _{{Q_{T}}} {a(v)v \bigl( { \bigl\vert \nabla u{|^{p(x,t) - 2}} \nabla u - \bigr\vert \nabla v{|^{p(x,t) - 2}}\nabla v} \bigr)\nabla w{\,\mathrm{d}}x{ \,\mathrm{d}}t} },\\ &{J_{3}} = \int { \int _{{Q_{T}}} { \bigl[a(u)u - a(v)v \bigr]|\nabla u{|^{p(x,t) - 2}} \nabla u\nabla {F_{\varepsilon }}(w){\,\mathrm{d}}x{\, \mathrm{d}}t} },\\ &{J_{4}} = \int { \int _{{Q_{T}}} { \bigl[a(u) - \gamma \bigr] \bigl\vert \nabla u{|^{p(x,t)}} - \bigl[a(v) - \gamma \bigr] \bigr\vert \nabla v{|^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} }. \end{aligned}

Now, let $${t_{0}} = \inf \{ t \in (0,\tau ]:w > \varepsilon \}$$, then we estimate $$J_{1}$$ as follows

\begin{aligned} \begin{aligned} {J_{1}}& = \int { \int _{{Q_{T}}} {{w_{t}} {F_{\varepsilon }}(w){\, \mathrm{d}}x{ \,\mathrm{d}}t} } = \int _{\Omega }\biggl( \int _{0}^{{t_{0}}} {{w_{t}} {F_{ \varepsilon }}(w){\,\mathrm{d}}t} + \int _{0}^{{t_{0}}} {{w_{t}} {F_{ \varepsilon }}(w){\,\mathrm{d}}t}\biggr)\,\mathrm{d}x \\ &\ge \int _{\Omega }{ \int _{\varepsilon }^{w} {{F_{ \varepsilon }}({\mathrm{{s}}}){ \,\mathrm{d}}s} } {\,\mathrm{d}}x \ge \int _{{\Omega _{ \delta }}} { \int _{\varepsilon }^{w} {{F_{\varepsilon }}({\mathrm{{s}}}){ \,\mathrm{d}}s} } {\,\mathrm{d}}x. \end{aligned} \end{aligned}
(3.8)

Let us first consider the case $${p^{-} } \ge 2$$. By virtue of the first inequality of LemmaÂ 3.1, we obtain

\begin{aligned} \begin{aligned} {J_{2}} &= \int { \int _{{Q_{T}}} {a(v)v\bigl({{ \vert {\nabla u} \vert }^{p(x,t) - 2}}\nabla u - {{ \vert {\nabla v} \vert }^{p(x,t) - 2}}\nabla v\bigr) \nabla w{\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\ge \int { \int _{{Q_{T}}} {a(v)v \cdot {w^{ - \alpha }} {2^{ - p(x,t)}} {{ \vert {\nabla w} \vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &= {2^{ - {p^{+} }}} \int { \int _{{Q_{T}}} {a(v)v \cdot {w^{ - \alpha }} {{ \vert {\nabla w} \vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} } > 0. \end{aligned} \end{aligned}
(3.9)

Noting that $$\frac{{p(x,t)}}{{p(x,t) - 1}} \ge \frac{{p + }}{{p + - 1}} \ge { \sigma ^{2}} = \alpha > 1$$ and applying Youngâ€™s inequality, we may estimate the integrand of $$J_{3}$$ in the following way

\begin{aligned} &\bigl\vert {\bigl[a(u)u - a(v)v \bigr]{w^{ - \alpha }} {{ \vert {\nabla w} \vert }^{p(x,t) - 2}}\nabla u \nabla w} \bigr\vert \\ \begin{aligned} &\quad= \biggl\vert { \biggl[ {(\delta + 1)w \int _{0}^{1} {{{\bigl(\theta u + (1 - \theta )v \bigr)}^{\sigma }}\,\mathrm{d}\theta } + {d_{0}}(u - v)} \biggr]{w^{ - \alpha }} {{ \vert {\nabla u} \vert }^{p(x,t) - 2}}\nabla u \nabla w} \biggr\vert \\ &\quad\le \frac{C}{{{w^{\alpha }}}} \biggl[ {\frac{{a(v)v}}{C}{{ \vert { \nabla w} \vert }^{p(x,t)}} + {C_{1}}\bigl(\delta,{d_{0}},K,{p^{\pm }} \bigr){{ \vert w \vert }^{p'(x,t)}} {{ \vert {\nabla u} \vert }^{p(x,t)}}} \biggr] \end{aligned} \\ &\quad\le \frac{{a(v)v}}{{{2^{{p^{+} } - 1}}{w^{\alpha }}}}{ \vert { \nabla w} \vert ^{p(x,t)}} + {C_{1}}\bigl(\delta,{d_{0}},K,{p^{\pm }}\bigr){ \vert u \vert ^{p'(x,t)}}. \end{aligned}
(3.10)

Substituting (3.10) into $$J_{3}$$ and combining it with $$J_{2}$$, we obtain

\begin{aligned} {J_{3}} \le \frac{1}{2}{J_{2}} + C \int { \int _{{Q_{T}}} {{{ \vert { \nabla u} \vert }^{p(x,t) - 2}} {\, \mathrm{d}}x{\,\mathrm{d}}t} }. \end{aligned}
(3.11)

Recall that $$0 < \gamma \le {d_{0}}$$, $$u \in W({Q_{T}}) \cap {L^{\infty }}(0,T; {L^{\infty }}( \Omega ))$$. Then, we have

\begin{aligned} {J_{4}} \le \int { \int _{{Q_{T}}} {{u^{\sigma }}|\nabla u{|^{p(x,t)}} { \, \mathrm{d}}x{\,\mathrm{d}}t} } \le C \int { \int _{{Q_{T}}} {|\nabla u{|^{p(x,t)}} { \,\mathrm{d}}x{\, \mathrm{d}}t} }, \end{aligned}
(3.12)

where C is a positive constant. Thus, we insert the above estimates (3.8), (3.9), (3.11), and (3.12) into (3.7) and dropping the nonnegative terms, we arrive at

\begin{aligned} (\delta - 2\varepsilon ) \bigl(1 - {2^{1 - \alpha }} \bigr){\varepsilon ^{1 - \alpha }}\mu (\Omega ) < C. \end{aligned}
(3.13)

Secondly, we consider the case $$1 < {p^{-} } \le p(x,t) < 2, {p^{+} } \ge 2$$. According to the second inequality of LemmaÂ 3.1, it is easily seen that the following inequalities hold

\begin{aligned} \begin{aligned} & \bigl\vert {\bigl[a(u)u - a(v)v \bigr]{w^{ - \alpha }} {{ \vert {\nabla w} \vert }^{p(x,t) - 2}}\nabla u \nabla w} \bigr\vert \\ &\quad= \biggl\vert { \biggl[ {(\delta + 1)w \int _{0}^{1} {{{\bigl(\theta u + (1 - \theta )v \bigr)}^{\sigma }}\,\mathrm{d}\theta } + {d_{0}}(u - v)} \biggr]|w{|^{2 - \alpha }} {{ \vert {\nabla u} \vert }^{p(x,t) - 2}}\nabla u \nabla w} \biggr\vert \\ &\quad\le \frac{{a(v)v({p^{ - 1}} - 1)}}{{2{w^{\alpha }}}}{\bigl( \vert { \nabla u} \vert + \vert {\nabla v} \vert \bigr)^{p(x,t)}} { \vert { \nabla {\mathrm{{w}}}} \vert ^{2}} + {C_{1}}\bigl(\delta,{d_{0}},K,{p^{\pm }} \bigr){ \vert w \vert ^{2 - \alpha }} {\bigl( \vert {\nabla u} \vert + \vert {\nabla v} \vert \bigr)^{p(x,t)}} \\ &\quad\le \frac{{a(v)v({p^{ - 1}} - 1)}}{{2{w^{\alpha }}}}{\bigl( \vert { \nabla u} \vert + \vert {\nabla v} \vert \bigr)^{p(x,t)}} { \vert { \nabla {\mathrm{{w}}}} \vert ^{2}} + {C_{1}}\bigl(\delta,{d_{0}},K,{p^{\pm }} \bigr){\bigl( \vert {\nabla u} \vert + \vert {\nabla v} \vert \bigr)^{p(x,t)}}. \end{aligned} \end{aligned}

Substituting the above inequality into $$J_{3}$$, we obtain

\begin{aligned} {J_{3}} \le \frac{1}{2}{J_{2}} + C \int { \int _{{Q_{T}}} {{{ \bigl( \vert { \nabla u} \vert + \vert { \nabla v} \vert \bigr)}^{p(x,t) - 2}} {\,\mathrm{d}}x{ \,\mathrm{d}}t} }. \end{aligned}
(3.14)

Similar to the case $${p^{-} } \ge 2$$, estimate (3.13) still holds using (3.14) instead of (3.11). Note that $$\mathop {{\mathrm{{lim}}}} _{ \varepsilon \to 0} (\delta - {\mathrm{{ 2}}}\varepsilon )({\mathrm{{1 }}} - {{\mathrm{{2}}}^{{\mathrm{{1}}} - \alpha }}){\varepsilon ^{{\mathrm{{1}}} - \alpha }}\mu ({\Omega _{\delta }}) = + \infty$$, we obtain a contradiction. This means $$\mu ({\Omega _{\delta }}) = 0$$ and $$w \le 0$$ a.e. in $${{\mathrm{{Q}}}_{\tau }}$$.â€ƒâ–¡

### Lemma 3.3

Let $${u_{\varepsilon }}$$ be weak solutions of (3.1). Then,

\begin{aligned} &{u_{0\varepsilon }} \le {u_{\varepsilon }} \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon, \end{aligned}
(3.15)
\begin{aligned} &{u_{{\varepsilon _{1}}}} \le {u_{{\varepsilon _{2}}}} \quad\textit{for } { \varepsilon _{1}} \le {\varepsilon _{2}}, \end{aligned}
(3.16)

where $$|{u_{0}}{|_{\infty }} = \mathop {\sup } _{x \in \Omega } |{u_{0}}(x)|$$.

### Proof

First, we prove $${u_{\varepsilon }} \ge {u_{0\varepsilon }}$$ by contradiction. Assume $${u_{\varepsilon }} \le {u_{0\varepsilon }}$$ in $$Q_{T}^{0}$$, $$Q_{T}^{0} \subset Q_{T}$$. Noting $${u_{\varepsilon }} \ge {u_{0\varepsilon }}$$ on $$\partial {Q_{T}}$$, we may assume that $${u_{\varepsilon }} = {u_{0\varepsilon }}$$ on $$\partial {Q_{T}}$$. With (3.1) and letting $$t = 0$$, it is easy to see that

\begin{aligned} &L{u_{0,\varepsilon }} = - {\beta _{\varepsilon }}({u_{0,\varepsilon }} - {u_{0,\varepsilon }}) = 1, \end{aligned}
(3.17)
\begin{aligned} &L{u_{\varepsilon }} = - {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0, \varepsilon }}) \le 1. \end{aligned}
(3.18)

From LemmaÂ 3.2, we arrive at

\begin{aligned} {u_{\varepsilon }}(x,t) \ge {u_{0,\varepsilon }}(x) \quad\text{for any } (x,t) \in {Q_{T}}. \end{aligned}
(3.19)

Secondly, we pay attention to $${u_{\varepsilon }}(t,x) \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon$$. Applying the definition of $${\beta _{\varepsilon }}( \cdot )$$, we have that

\begin{aligned} L \bigl({ \vert {{u_{0}}} \vert _{\infty }} + \varepsilon \bigr) = 0,\qquad L{u_{ \varepsilon }} = - {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0, \varepsilon }}) \ge 0. \end{aligned}
(3.20)

From (3.20), we obtain

\begin{aligned} {u_{\varepsilon }}(t,x) \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon \quad\text{on } \partial \Omega \times (0, \mathrm{T}) \end{aligned}
(3.21)

and $${u_{\varepsilon }}(t,x) \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon$$ in Î©. Thus, combining (3.20) and (3.21) and repeating LemmaÂ 3.2, we have

\begin{aligned} {u_{\varepsilon }}(t,x) \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon\quad \text{in } {Q_{T}}. \end{aligned}
(3.22)

Thirdly, we aim to prove (3.16). From (3.1),

\begin{aligned} &L{u_{{\varepsilon _{1}}}} = {\beta _{{\varepsilon _{1}}}}({u_{{ \varepsilon _{1}}}} - {u_{0,{\varepsilon _{1}}}}), \end{aligned}
(3.23)
\begin{aligned} &L{u_{{\varepsilon _{2}}}} = {\beta _{{\varepsilon _{2}}}}({u_{{ \varepsilon _{2}}}} - {u_{0,{\varepsilon _{2}}}}). \end{aligned}
(3.24)

It follows by $${\varepsilon _{1}} \le {\varepsilon _{2}}$$ and the definition of $${\beta _{\varepsilon }}( \cdot )$$ that

\begin{aligned} \begin{aligned} &L{u_{0,{\varepsilon _{2}}}} + {\beta _{{\varepsilon _{1}}}}({u_{{ \varepsilon _{2}}}} - {u_{0,\varepsilon }}) \\ &\quad= {\beta _{{\varepsilon _{2}}}}({u_{{\varepsilon _{2}}}} - {u_{0, \varepsilon }}) - {\beta _{{\varepsilon _{1}}}}({u_{{\varepsilon _{1}}}} - {u_{0,\varepsilon }}) \\ &\quad= {\beta _{{\varepsilon _{2}}}}({u_{{\varepsilon _{2}}}} - {u_{0, \varepsilon }}) - {\beta _{{\varepsilon _{1}}}}({u_{{\varepsilon _{2}}}} - {u_{0,\varepsilon }}) \ge 0. \end{aligned} \end{aligned}
(3.25)

Thus, combining the initial and boundary conditions in (3.1) can be proved by LemmaÂ 3.2.â€ƒâ–¡

To prove this theorem, we need the following lemmas.

### Lemma 3.4

The solution of problem (3.1) satisfies the estimate

\begin{aligned} { \Vert {{u_{\varepsilon }}} \Vert _{\infty,{Q_{T}}}} \le { \Vert {{u_{0}}} \Vert _{\infty,\Omega }} + \int _{0}^{T} {{{ \bigl\Vert {f(x,t)} \bigr\Vert }_{\infty,\Omega }} {\,\mathrm{d}}t} + \vert \Omega \vert \cdot T = K(T) < \infty. \end{aligned}

### Proof

Let us introduce the following function

\begin{aligned} {u_{\varepsilon,M}} = \textstyle\begin{cases} M & \text{if }{u_{\varepsilon }} > M, \\ {u_{\varepsilon }}& \text{if } \vert {{u_{\varepsilon }}} \vert < M, \\ - M & \text{if }{u_{\varepsilon }} < - M. \end{cases}\displaystyle \end{aligned}
(3.26)

The function $$u_{\varepsilon,M}^{2k - 1}$$, with $$k \in N$$, can be chosen as a test function in (3.4). Let $${t_{2} } = t + h,{t_{1}} = t$$ in (3.4), with $$t,t + h \in ( 0, T )$$. Then,

\begin{aligned} \begin{aligned} &\frac{1}{{2k}} \int _{t}^{t + h} {\frac{{\mathrm{d}}}{{{\mathrm{d}}t}} \biggl( { \int _{\Omega }{u_{\varepsilon,M}^{2k}{\,\mathrm{d}}x} } \biggr)} {\,\mathrm{d}}t \\ &\qquad{}+ (2k - 1) \int _{t}^{t + h} { \int _{\Omega }{{a_{\varepsilon,M}}({u_{ \varepsilon,M}})u_{\varepsilon,M}^{2(k - 1)}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &\qquad{}+ \int _{t}^{t + h} { \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{ \varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{\varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x{ \,\mathrm{d}}t} } \\ &\quad= \int _{t}^{t + h} { \int _{\Omega }{ \bigl( {f(x,t) - {\beta _{ \varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr) \cdot u_{ \varepsilon,M}^{2k - 1}{\, \mathrm{d}}x{\,\mathrm{d}}t} }. \end{aligned} \end{aligned}
(3.27)

Dividing the last equality by h, letting $$h \to 0$$, and applying Lebesgueâ€™s dominated convergence theorem, we have that

\begin{aligned} \begin{aligned} &\frac{1}{{2k}}\frac{{\mathrm{d}}}{{{\mathrm{d}}t}} \int _{\Omega }{u_{ \varepsilon,M}^{2k}{\,\mathrm{d}}x} + (2k - 1) \int _{\Omega }{{a_{ \varepsilon,M}}({u_{\varepsilon,M}})u_{\varepsilon,M}^{2(k - 1)}{{ \vert {\nabla {u_{\varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\qquad{}+ \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} \\ &\qquad{}+ \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr]u_{\varepsilon,M}^{2k}{{ \vert {\nabla {u_{\varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\quad= \int _{\Omega }{ \bigl( {f(x,t) - {\beta _{\varepsilon }}({u_{ \varepsilon }} - {u_{0}})} \bigr) \cdot u_{\varepsilon,M}^{2k - 1}{ \, \mathrm{d}}x}. \end{aligned} \end{aligned}
(3.28)

By Holderâ€™s inequality, we have

\begin{aligned} \biggl\vert { \int _{\Omega }{ \bigl( {f(x,t) - {\beta _{\varepsilon }}({u_{ \varepsilon }} - {u_{0}})} \bigr) \cdot u_{\varepsilon,M}^{2k - 1}{ \, \mathrm{d}}x} } \biggr\vert \le \bigl\Vert {u_{\varepsilon,M}} \bigr\Vert _{2k, \Omega }^{2k - 1} \cdot \bigl\Vert {f( \cdot,t) - { \beta _{ \varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr\Vert _{2k,\Omega }. \end{aligned}
(3.29)

Using Minkowskiâ€™s inequality, we arrive at

\begin{aligned} \bigl\Vert {f( \cdot,t) - {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr\Vert _{2k,\Omega } \le \bigl\Vert {f( \cdot,t)} \bigr\Vert _{2k, \Omega } + \bigl\Vert {{\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr\Vert _{2k,\Omega }. \end{aligned}

From (3.15) and the definition of $${\beta _{\varepsilon }}( \cdot )$$, we have that

\begin{aligned} \bigl\Vert {f( \cdot,t) - {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr\Vert _{2k,\Omega } \le \bigl\Vert {f( \cdot,t)} \bigr\Vert _{2k, \Omega } + \vert \Omega \vert . \end{aligned}
(3.30)

Recall that $$0 < \gamma < {d_{0}}$$. Then, we use LemmaÂ 3.1 to find

\begin{aligned} \int _{\Omega }{ \bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{\varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} \ge 0. \end{aligned}
(3.31)

Substituting (3.29) and (3.30) into (3.28), we arrive at the inequality

\begin{aligned} \begin{aligned} &\bigl\Vert {u_{\varepsilon,M}} \bigr\Vert _{2k,\Omega }^{2k - 1} \frac{{\mathrm{d}}}{{{\mathrm{d}}t}} \bigl\Vert {u_{\varepsilon,M}} \bigr\Vert _{2k,\Omega } + (2k - 1) \int _{\Omega }{{a_{\varepsilon,M}}({u_{ \varepsilon,M}})u_{\varepsilon,M}^{2(k - 1)}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\qquad{}+ \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\quad\le \bigl\Vert {u_{\varepsilon,M}} \bigr\Vert _{2k,\Omega }^{2k - 1} \cdot \bigl\Vert {f( \cdot,t)} \bigr\Vert _{2k,\Omega } + \bigl\Vert {u_{ \varepsilon,M}} \bigr\Vert _{2k,\Omega }^{2k - 1} \cdot \vert \Omega \vert . \end{aligned} \end{aligned}
(3.32)

Integrating over $$(0, t)$$ in (3.32) and dropping the nonnegative term (3.31), we arrive at

\begin{aligned} \bigl\Vert {u_{\varepsilon,M}} \bigr\Vert _{2k,\Omega } \le \bigl\Vert {u_{\varepsilon,M}( \cdot,0)} \bigr\Vert _{2k,\Omega } + \int _{0}^{T} { \bigl\Vert {f( \cdot,t)} \bigr\Vert _{2k,\Omega }{\,\mathrm{d}}t} + \vert \Omega \vert \cdot T,\quad\forall k \in N. \end{aligned}

Then, as $$k \to \infty$$, we have that

\begin{aligned} \Vert {u_{\varepsilon,M}} \Vert _{\infty,\Omega } \le \bigl\Vert {u_{\varepsilon,M}( \cdot,0)} \bigr\Vert _{\infty,\Omega } + \int _{0}^{T} { \bigl\Vert {f( \cdot,t)} \bigr\Vert _{\infty,\Omega }{ \,\mathrm{d}}t} + \vert \Omega \vert \cdot T = K(T). \end{aligned}
(3.33)

If we choose $$M > K(T)$$, then

\begin{aligned} {u_{\varepsilon,M}}( \cdot,t) \le \sup \bigl\vert {{u_{\varepsilon,M}}( \cdot,t)} \bigr\vert \le K ( T ) < M \end{aligned}

and therefore $${u_{\varepsilon,M}}( \cdot,t) = {u_{\varepsilon }}( \cdot,t)$$.â€ƒâ–¡

### Lemma 3.5

The solution of problem (3.1) satisfies the estimates

\begin{aligned} &\int { \int _{{Q_{T}}} {a \bigl(u_{\varepsilon } \bigr){{ \bigl\vert {\nabla u_{ \varepsilon }} \bigr\vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} } \le K(T){ \vert \Omega \vert ^{\frac{1}{2}}}, \end{aligned}
(3.34)
\begin{aligned} &{d_{0}} \int { \int _{{Q_{T}}} {{{ \bigl\vert {\nabla u_{\varepsilon }} \bigr\vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} } \le K(T){ \vert \Omega \vert ^{\frac{1}{2}}}, \end{aligned}
(3.35)
\begin{aligned} &\int { \int _{{Q_{T}}} {u_{\varepsilon }^{\sigma }{{ \bigl\vert { \nabla u_{ \varepsilon }} \bigr\vert }^{p(x,t)}} {\, \mathrm{d}}x{\,\mathrm{d}}t} } \le K(T){ \vert \Omega \vert ^{\frac{1}{2}}}. \end{aligned}
(3.36)

### Proof

To prove LemmaÂ 3.5, we proceed as in the proof of LemmaÂ 3.4, and in (3.27) we take $$k=1$$. We then obtain

\begin{aligned} \begin{aligned} &\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}\bigl( \bigl\Vert {u_{\varepsilon }( \cdot ,t)} \bigr\Vert _{2,\Omega } \bigr) + \int _{\Omega }{{a_{\varepsilon,M}}({u_{ \varepsilon }}){{ \bigl\vert {\nabla u_{\varepsilon }} \bigr\vert }^{p(x,t)}} { \,\mathrm{d}}x} \\ &\qquad{}+ \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\quad\le \bigl\Vert {f - {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr\Vert _{2,\Omega }. \end{aligned} \end{aligned}

Therefore, integrating in time over $$(0,t)$$, $$\forall t \in (0,T)$$,

\begin{aligned} \begin{aligned} & \bigl\Vert {u_{\varepsilon }( \cdot,t)} \bigr\Vert _{2,\Omega } + \int _{0}^{t} { \int _{\Omega }{{a_{\varepsilon,M}}({u_{ \varepsilon }}){{ \vert { \nabla u_{\varepsilon }} \vert }^{p(x,t)}} { \,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\qquad{}+ \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\quad\le \int _{0}^{T} { \bigl\Vert {f - {\beta _{\varepsilon }}({u_{ \varepsilon }} - {u_{0}})} \bigr\Vert _{2,\Omega }{\,\mathrm{d}}t} \end{aligned} \end{aligned}

and since the first and third terms on the left-hand side are nonnegative and recalling the L2-norm

\begin{aligned} \int { \int _{{Q_{T}}} {a({u_{\varepsilon }}){{ \bigl\vert {\nabla u_{ \varepsilon }} \bigr\vert }^{p(x,t)}} {\,\mathrm{d}}x{\, \mathrm{d}}t} } \le K(T){ \vert \Omega \vert ^{\frac{1}{2}}}. \end{aligned}
(3.37)

From this we obtain (3.34). Since $$a({u_{\varepsilon }}) \ge {d_{0}}$$, $${a_{\varepsilon,M}}({u_{ \varepsilon }}) \ge u_{\varepsilon }^{\sigma }$$, (3.35), and (3.35) are immediate consequences of (3.34).â€ƒâ–¡

### Lemma 3.6

The solution of problem (3.1) satisfies the estimate

\begin{aligned} \bigl\Vert {u_{\varepsilon t}} \bigr\Vert _{W'({Q_{T}})} \le C \bigl( \sigma,{p^{\pm }},K(T), \vert \Omega \vert \bigr). \end{aligned}

### Proof

From identity (3.5), we obtain

\begin{aligned} \begin{aligned} &\int { \int _{{Q_{T}}} {u_{\varepsilon t}\xi {\,\mathrm{d}}x{ \,\mathrm{d}}t} } \\ &\quad= - \int { \int _{{Q_{T}}} {a({u_{\varepsilon }}){{ \vert {\nabla {u_{ \varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }} \nabla \xi {\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\qquad{}- \int { \int _{{Q_{T}}} {\bigl[a({u_{\varepsilon }}) - \gamma \bigr]{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t)}}\xi { \,\mathrm{d}}x{ \,\mathrm{d}}t} } + \int { \int _{{Q_{T}}} {f \cdot \xi {\,\mathrm{d}}x{ \,\mathrm{d}}t} } \\ &\quad= - {A_{1}} - {A_{1}} + {A_{1}}, \end{aligned} \end{aligned}
(3.38)

where

\begin{aligned} &{A_{1}} = \int { \int _{{Q_{T}}} {a({u_{\varepsilon }}){{ \vert { \nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{ \varepsilon }}\nabla \xi { \,\mathrm{d}}x{\,\mathrm{d}}t} },\\ &{A_{2}} = \int { \int _{{Q_{T}}} { \bigl[a({u_{\varepsilon }}) - \gamma \bigr]{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t)}}\xi {\,\mathrm{d}}x{ \,\mathrm{d}}t} },\qquad {A_{3}} = \int { \int _{{Q_{T}}} {f \cdot \xi {\,\mathrm{d}}x{ \,\mathrm{d}}t} }. \end{aligned}

First, we pay attention to $$A_{1}$$. Using Holder inequalities we obtain

\begin{aligned} \begin{aligned} \vert {{A_{1}}} \vert & \le \int _{0}^{t} { \int _{\Omega }{a({u_{ \varepsilon }}){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 1}} \vert {\nabla \xi } \vert {\, \mathrm{d}}x{\,\mathrm{d}}t} } \\ &\le 2{ \bigl\Vert {a({u_{\varepsilon }}){{ \vert {\nabla {u_{ \varepsilon }}} \vert }^{p(x,t) - 1}}} \bigr\Vert _{p'(x,t)}} { \Vert {\nabla \xi } \Vert _{p(x,t)}}. \end{aligned} \end{aligned}

When $$\int _{0}^{t} {\int _{\Omega }{{{ ( {a({u_{\varepsilon }}){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 1}}} )}^{ \frac{{p(x,t)}}{{p(x,t) - 1}}}}{\,\mathrm{d}}x{\,\mathrm{d}}t} } \ge 1$$, we arrive at

\begin{aligned} \vert {{A_{1}}} \vert \le 2{ \biggl( { \int _{0}^{t} { \int _{ \Omega }{{{ \bigl( {a({u_{\varepsilon }}){{ \vert {\nabla {u_{ \varepsilon }}} \vert }^{p(x,t) - 1}}} \bigr)}^{ \frac{{p(x,t)}}{{p(x,t) - 1}}}} {\, \mathrm{d}}x{\,\mathrm{d}}t} } } \biggr)^{ \frac{1}{{{{p'}^{+} }}}}} \cdot { \Vert {\nabla \xi } \Vert _{p(x,t)}}. \end{aligned}
(3.39)

Moreover, when $$\int _{0}^{t} {\int _{\Omega }{{{ ( {a({u_{\varepsilon }}){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 1}}} )}^{ \frac{{p(x,t)}}{{p(x,t) - 1}}}}{\,\mathrm{d}}x{\,\mathrm{d}}t} } < 1$$, we obtain

\begin{aligned} \vert {{A_{1}}} \vert \le 2{ \biggl( { \int _{0}^{t} { \int _{ \Omega }{{{ \bigl( {a({u_{\varepsilon }}){{ \vert {\nabla {u_{ \varepsilon }}} \vert }^{p(x,t) - 1}}} \bigr)}^{ \frac{{p(x,t)}}{{p(x,t) - 1}}}} {\, \mathrm{d}}x{\,\mathrm{d}}t} } } \biggr)^{ \frac{1}{{{{p'}^{-} }}}}} \cdot { \Vert {\nabla \xi } \Vert _{p(x,t)}}. \end{aligned}
(3.40)

Combining (3.39) and (3.40), and using LemmaÂ 3.5, we arrive at

\begin{aligned} \vert {{A_{1}}} \vert \le { \bigl( {2 \bigl[{{ \bigl({K^{2}}(T) + 1 \bigr)}^{ \sigma /2}} + {d_{0}} \bigr]} \bigr)^{\frac{1}{{{p^{\pm }} - 1}}}} {K^{2}}(T) \vert \Omega \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}}. \end{aligned}
(3.41)

Secondly, we calculate $$A_{2}$$ and $$A_{3}$$. Following a similar procedure as (3.41), we obtain

\begin{aligned} &\begin{aligned} \vert {{A_{2}}} \vert \le{}& 2{\bigl[{\bigl({K^{2}}(T) + 1\bigr)^{\sigma /2}} + {d_{0}}\bigr]^{ \frac{1}{{{p^{\pm }} - 1}}}} {K^{2}}(T) \vert \Omega \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}} \\ &{}+ 2{\gamma ^{\frac{1}{{{p^{\pm }} - 1}}}} {K^{2}}(T) \vert \Omega \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}}, \end{aligned} \end{aligned}
(3.42)
\begin{aligned} &\vert {{A_{3}}} \vert \le 2{ \vert f \vert _{\infty }} \vert T \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}}. \end{aligned}
(3.43)

Substituting (3.41), (3.42), and (3.43) into (3.38), we conclude that

\begin{aligned} \begin{aligned} \int { \int _{{Q_{T}}} {u_{\varepsilon t}\xi {\,\mathrm{d}}x{ \,\mathrm{d}}t} } \le {}&4{\bigl[{\bigl({K^{2}}(T) + 1\bigr)^{\sigma /2}} + {d_{0}}\bigr]^{ \frac{1}{{{p^{\pm }} - 1}}}} {K^{2}}(T) \vert \Omega \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}} \\ &{}+ 2{\gamma ^{\frac{1}{{{p^{\pm }} - 1}}}} {K^{2}}(T) \vert \Omega \vert \cdot + 2{ \vert f \vert _{\infty }} \vert T \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}}. \end{aligned} \end{aligned}

Then, we obtain LemmaÂ 3.6.â€ƒâ–¡

## 4 Proof of the main results

In this section, we are ready to prove TheoremÂ 2.1 and TheoremÂ 2.2. From (3.15), LemmaÂ 3.5, and LemmaÂ 3.6, we see that $${u_{\varepsilon}}$$ is bounded and increasing in Îµ, which implies the existence of a function u, such that, as $$\varepsilon \to 0$$

\begin{aligned} &{u_{\varepsilon }} \to u \quad\text{a.e. in } {\Omega _{T}} , \end{aligned}
(4.1)
\begin{aligned} &\nabla {u_{\varepsilon }} \to \nabla u\quad \text{weakly in } {L^{p(x,t)}}({Q_{T}}), \end{aligned}
(4.2)
\begin{aligned} &\frac{\partial }{{\partial t}}{u_{\varepsilon }} \to \frac{\partial }{{\partial t}}u \quad\text{weakly in }W'({Q_{T}}), \end{aligned}
(4.3)
\begin{aligned} &{\mathrm{{a}}}({u_{\varepsilon }}){ \vert {\nabla {u_{\varepsilon }}} \vert ^{p(x,t) - 2}} {D_{i}} {u_{\varepsilon }} \to {A_{i}}(x,t)\quad \text{weakly in } {L^{p'(x,t)}}({Q_{T}}), \end{aligned}
(4.4)
\begin{aligned} &{ \vert {\nabla {u_{\varepsilon }}} \vert ^{p(x,t) - 2}} {D_{i}} {u_{ \varepsilon }} \to {W_{i}}(x,t) \quad\text{weakly in } {L^{p'(x,t)}}({Q_{T}}), \end{aligned}
(4.5)

for some functions $$u \in W({Q_{T}})$$, $${A_{i}}(x,t) \in {L^{p'(x,t)}}({Q_{T}})$$, $${W_{i}}(x,t) \in {L^{p'(x,t)}}({Q_{T}})$$.

### Lemma 4.1

For almost all $$(x,t) \in {Q_{T}}$$,

\begin{aligned} {A_{i}}(x,t) = a(u) {W_{i}}(x,t),\quad i = 1,2, \ldots,N. \end{aligned}

### Proof

In (4.4) and (4.5), letting $$\varepsilon \to 0$$, we have that

\begin{aligned} &\int { \int _{{Q_{T}}} {a(u_{\varepsilon }){{ \vert {\nabla {u_{ \varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }} \nabla \xi {\,\mathrm{d}}x{\,\mathrm{d}}t} } = \sum_{i} { \int { \int _{{Q_{T}}} {{A_{i}}(x,t) \cdot {D_{i}} \xi {\,\mathrm{d}}x{\,\mathrm{d}}t} } }, \end{aligned}
(4.6)
\begin{aligned} &\int { \int _{{Q_{T}}} {{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }}\nabla \xi {\,\mathrm{d}}x{\, \mathrm{d}}t} } = \sum_{i} { \int { \int _{{Q_{T}}} {{W_{i}}(x,t) \cdot {D_{i}} \xi {\,\mathrm{d}}x{\,\mathrm{d}}t} } }. \end{aligned}
(4.7)

By Lebesgueâ€™s dominated convergence theorem, we have

\begin{aligned} \mathop {\lim } _{\varepsilon \to 0} \int { \int _{{Q_{T}}} { \bigl[a \bigl(u_{ \varepsilon } \bigr) - a(u) \bigr]{A_{i}}(x,t) \cdot {D_{i}}\xi {\, \mathrm{d}}x{\,\mathrm{d}}t} } = 0. \end{aligned}
(4.8)

Hence, we have

\begin{aligned} \begin{aligned} &\mathop {\lim } _{\varepsilon \to 0} \int { \int _{{Q_{T}}} {a\bigl(u_{ \varepsilon }\bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}} {D_{i}} {u_{\varepsilon }} {D_{i}}\xi - a(u){W_{i}}(x,t){D_{i}} \xi {\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\quad= \mathop {\lim } _{\varepsilon \to 0} \int { \int _{{Q_{T}}} {\bigl[a\bigl(u_{ \varepsilon }\bigr) - a(u)\bigr] \cdot {{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}} {D_{i}} {u_{\varepsilon }} {D_{i}}\xi { \,\mathrm{d}}x{ \,\mathrm{d}}t} } \\ &\qquad{}+ \mathop {\lim } _{\varepsilon \to 0} \int { \int _{{Q_{T}}} {a(u) \bigl( {{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}} {D_{i}} {u_{\varepsilon }} - {W_{i}}(x,t)} \bigr){D_{i}}\xi {\,\mathrm{d}}x{ \, \mathrm{d}}t} } = 0. \end{aligned} \end{aligned}

This completes the proof of LemmaÂ 4.1.â€ƒâ–¡

### Lemma 4.2

For almost all $$(x,t) \in {Q_{T}}$$,

\begin{aligned} {W_{i}}(x,t) = |\nabla u{|^{p(x,t) - 2}} {D_{i}}u,\quad i = 1,2, \ldots,N. \end{aligned}

### Proof

In (3.5), choosing $$\xi = \Phi \cdot ({u_{\varepsilon }} - u)$$ with $$\Phi \in W({Q_{T}})$$, $$\Phi \ge 0$$, we have

\begin{aligned} \begin{aligned} &\int { \int _{{Q_{T}}} {{\partial _{t}}u_{\varepsilon }\cdot (u_{ \varepsilon }- u) \cdot \Phi + \Phi \cdot a\bigl(u_{\varepsilon } \bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{ \varepsilon }}\nabla ({u_{\varepsilon }} - u){\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\quad{}+ \int { \int _{{Q_{T}}} {\bigl(u_{\varepsilon } - u\bigr) \cdot a\bigl(u_{ \varepsilon }\bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }}\nabla \Phi - f(x,t) (u_{\varepsilon }- u){ \,\mathrm{d}}x{\,\mathrm{d}}t} } = 0. \end{aligned} \end{aligned}

It follows that

\begin{aligned} \mathop {\lim } _{\varepsilon \to \infty } \int { \int _{{Q_{T}}} {\Phi \cdot a \bigl(u_{\varepsilon } \bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }}\nabla ({u_{ \varepsilon }} - u){\,\mathrm{d}}x{\,\mathrm{d}}t} } = 0. \end{aligned}
(4.9)

On the other hand, from $${u_{\varepsilon }},u \in {L^{\infty }}(Q_{T})$$, $$|\nabla u| \in {L^{p(x,t)}}({Q_{T}})$$, we obtain

\begin{aligned} &\mathop {\lim } _{\varepsilon \to \infty } \int { \int _{{Q_{T}}} {\Phi \cdot a(u){{ \vert {\nabla {u}} \vert }^{p(x,t) - 2}} \nabla {u}\nabla ({u_{\varepsilon }} - u){\,\mathrm{d}}x{\,\mathrm{d}}t} } = 0, \end{aligned}
(4.10)
\begin{aligned} &\mathop {\lim } _{\varepsilon \to \infty } \int { \int _{{Q_{T}}} {\Phi \cdot \bigl[a \bigl(u_{\varepsilon } \bigr) - a(u) \bigr]{{ \vert {\nabla {u}} \vert }^{p(x,t) - 2}} \nabla {u}\nabla ({u_{\varepsilon }} - u){ \, \mathrm{d}}x{\, \mathrm{d}}t} } = 0. \end{aligned}
(4.11)

Note that

\begin{aligned} \begin{aligned} 0 \le{}& \bigl( {{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }} - {{ \vert {\nabla u} \vert }^{p(x,t) - 2}}\nabla u} \bigr) \cdot ( {\nabla {u_{\varepsilon }} - \nabla u} ) \\ \le{}& \frac{1}{{{d_{0}}}} \bigl[ {a\bigl(u_{\varepsilon }\bigr){{ \vert { \nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{ \varepsilon }} + \bigl[a\bigl(u_{\varepsilon }\bigr) - a(u) \bigr] \cdot {{ \vert { \nabla u} \vert }^{p(x,t) - 2}}\nabla u} \bigr] \cdot ( { \nabla {u_{\varepsilon }} - \nabla u} ) \\ &{}- \frac{1}{{{d_{0}}}}a(u){ \vert {\nabla u} \vert ^{p(x,t) - 2}}\nabla u \cdot ( {\nabla {u_{\varepsilon }} - \nabla u} ). \end{aligned} \end{aligned}
(4.12)

By (4.9)â€“(4.12), we obtain

\begin{aligned} \mathop {\lim } _{\varepsilon \to \infty } \int { \int _{{Q_{T}}} { \bigl( {{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}} \nabla {u_{\varepsilon }} - {{ \vert {\nabla u} \vert }^{p(x,t) - 2}} \nabla u} \bigr) \cdot ( {\nabla {u_{\varepsilon }} - \nabla u} ){\,\mathrm{d}}x{\,\mathrm{d}}t} } = 0. \end{aligned}
(4.13)

Then, the proof of LemmaÂ 4.2 is complete.â€ƒâ–¡

### Lemma 4.3

As $$\varepsilon \to 0$$, we have

\begin{aligned} {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}}) \to \xi \in G(u - {u_{0}}). \end{aligned}
(4.14)

### Proof

Using (3.15) and the definition of $${\beta _{\varepsilon }}$$, we have

\begin{aligned} {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}}) \to \xi \quad \text{as } \varepsilon \to 0. \end{aligned}

Now, we prove $$\xi \in G(u - {u_{0}})$$. According to the definition of $$G( \cdot )$$, we only need to prove that if $$u({x_{0}}, {t_{0}}) > {u_{0}}({x_{0}})$$,

\begin{aligned} \xi ({x_{0}}, {t_{0}}) = 0. \end{aligned}

In fact, if $$u({x_{0}},{t_{0}}) > {u_{0}}({x_{0}})$$, there exist a constant $$\lambda > 0$$ and a Î´-neighborhood $${B_{\delta }}({x_{0}},{t_{0}})$$ such that if Îµ is small enough, we have

\begin{aligned} {u_{\varepsilon }}(x,t) \ge {u_{0}}(x) + \lambda,\quad\forall (x, t) \in {B_{\delta }}({x_{0}},{t_{0}}). \end{aligned}

Thus, if Îµ is small enough, we have

\begin{aligned} 0 \ge {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}}) \ge { \beta _{\varepsilon }}(\lambda ) = 0, \quad\forall (x, t) \in {B_{ \delta }}({x_{0}}, {t_{0}}). \end{aligned}

Furthermore, it follows by $$\varepsilon \rightarrow 0$$ that

\begin{aligned} \xi (x, t) = 0, \quad\forall (x, t) \in {B_{\delta }}({x_{0}}, {t_{0}}). \end{aligned}

Hence, (4.13) holds, and the proof of LemmaÂ 4.3 is complete.â€ƒâ–¡

### The proof of TheoremÂ 2.1.

Applying (3.15), (3.16), and LemmaÂ 4.3, it is clear that

\begin{aligned} u(x,t) \le {u_{0}}(x), \quad\text{in } {\Omega _{T}},\qquad u(x,0) = {u_{0}}(x), \quad\text{in } \Omega,\qquad\xi \in G(u - {u_{0}}), \end{aligned}

thus (a), (b), and (c) hold. The remaining arguments of the existence part are the same as those of TheoremÂ 2.1 in [8] by a standard limiting process. Thus, we omit the details.â€ƒâ–¡

### The proof of TheoremÂ 2.2

We argue by contradiction. Suppose $$(u,{\xi _{1}})$$ and $$(v,{\xi _{2}})$$ are two nonnegative weak solutions of problem (1.1). Define $$w = u - v$$,

\begin{aligned} F(w) = \textstyle\begin{cases} - \frac{1}{{\alpha - 1}}{w^{1 - \alpha }}& \text{if }w > 0, \\ 0 &\text{if }w \le 0, \end{cases}\displaystyle \end{aligned}
(4.15)

and let $$\xi = {F_{\varepsilon }}(w) \in Z$$ be a test function in (3.4),

\begin{aligned} \begin{aligned} 0 \ge{}& \int { \int _{{Q_{T}}} {{w_{t}}F(w) + a(v)v \bigl( { \bigl\vert \nabla u{|^{p(x,t) - 2}}\nabla u - \bigr\vert \nabla v{|^{p(x,t) - 2}}\nabla v} \bigr)\nabla {F_{ \varepsilon }}(w){\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u)u - a(v)v\bigr]|\nabla u{|^{p(x,t) - 2}} \nabla u\nabla {F_{\varepsilon }}(w){\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u) - \gamma \bigr] \bigl( { \bigl\vert \nabla u{|^{p(x,t)}} - \bigr\vert \nabla v{|^{p(x,t)}}} \bigr){\, \mathrm{d}}x{\,\mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u) - a(v)\bigr]|\nabla u{|^{p(x,t)}} { \,\mathrm{d}}x{ \,\mathrm{d}}t} } - \int { \int _{{Q_{T}}} {({\xi _{1}} - {\xi _{2}})F(w){ \,\mathrm{d}}x{\,\mathrm{d}}t} }. \end{aligned} \end{aligned}
(4.16)

Now, we prove

\begin{aligned} \int _{\Omega }{({\xi _{1}} - {\xi _{2}})F(w){ \,\mathrm{d}}x{\,\mathrm{d}}t} \le 0. \end{aligned}
(4.17)

On the one hand, if $${u_{1}}(x,t) > {u_{2}}(x,t)$$, then using (3.16) yields

\begin{aligned} {u_{1}}(x,t) > {u_{0}}(x). \end{aligned}
(4.18)

From (2.1) and (4.18), it is easy to see that

\begin{aligned} {\xi _{1}} = 0 < {\xi _{2}}. \end{aligned}
(4.19)

Combining (4.18) and (4.19) and the fact that $$\alpha = \frac{1}{2}\sigma > 1$$, (4.16) is obtained.

On the other hand, if $${u_{1}}(x,t) < {u_{2}}(x,t)$$, it is easy to see that $$F(w) = 0$$. Equation (4.16) still holds.

Using (4.16) in (4.15) and dropping the nonnegative term, (4.15) becomes

\begin{aligned} \begin{aligned} 0 \ge{}& \int { \int _{{Q_{T}}} {{w_{t}}F(w) + a(v)v \bigl( { \bigl\vert \nabla u{|^{p(x,t) - 2}}\nabla u - \bigr\vert \nabla v{|^{p(x,t) - 2}}\nabla v} \bigr)\nabla {F_{ \varepsilon }}(w){\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u)u - a(v)v\bigr]|\nabla u{|^{p(x,t) - 2}} \nabla u\nabla {F_{\varepsilon }}(w){\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u) - \gamma \bigr] \bigl( { \bigl\vert \nabla u{|^{p(x,t)}} - \bigr\vert \nabla v{|^{p(x,t)}}} \bigr){\, \mathrm{d}}x{\,\mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u) - a(v)\bigr]|\nabla u{|^{p(x,t)}} { \,\mathrm{d}}x{ \,\mathrm{d}}t} }. \end{aligned} \end{aligned}

By the above inequality and combining the initial and boundary conditions in (1.1), the uniqueness of the solution can be proved following the similar proof of (3.7)â€“(3.14).â€ƒâ–¡

## 5 Conclusion

In this paper, an initial Dirichlet problem of degenerate parabolic variational inequalities in the following form

\begin{aligned} \textstyle\begin{cases} \min \{ Lu,u(x,0) - {u_{0}}\} = 0, & (x,t) \in {Q_{T}}, \\ u(x,t) = 0, & (x,t) \in {\Gamma _{T}}, \\ u(x,0) = {u_{0}}, &x \in \Omega, \end{cases}\displaystyle \end{aligned}

is studied. The existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process. The localization property of weak solutions is also discussed.

Not applicable.

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## Acknowledgements

The authors are sincerely grateful to the reviewer and the Associate Editor who handled the paper for their valuable comments.

## Funding

This work was supported by the Guizhou Provincial Education Foundation of Youth Science and Technology Talent Development (No. [2016]168), the Foundation of Guizhou Minzu University (NO.GZMUSK[2021]ZK02) and the Doctoral Project of Guizhou Education University (NO.2021BS037).

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### Contributions

YDS was a major contributor in writing the manuscript. TW performed the validation and formal analysis. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Tao Wu.

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Sun, Y., Wu, T. Study of weak solutions for degenerate parabolic inequalities with nonstandard conditions. J Inequal Appl 2022, 141 (2022). https://doi.org/10.1186/s13660-022-02872-3