1 Introduction

In this article, we consider the initial-boundary problem of the following parabolic variational inequality:

$$ \left \{ \textstyle\begin{array}{l} {u_{t}} - u\operatorname{div} ( {{{\vert {\nabla u} \vert }^{p - 2}}\nabla u} ) - \gamma{ \vert {\nabla u} \vert ^{p}} \le0, \quad \mbox{in } {\Omega_{T}}, \\ {[ {{u_{t}} - u\operatorname{div} ( {{{\vert {\nabla u} \vert }^{p - 2}}\nabla u} ) - \gamma{{ \vert {\nabla u} \vert }^{p}}} ]} \cdot ( {u - {u_{0}}(x)} ) = 0,\quad \mbox{in } {\Omega _{T}}, \\ u(x,t) \le{u_{0}}(x),\quad \mbox{in } {\Omega_{T}}, \\ u(x,0) = {u_{0}}(x), \quad \mbox{in } \Omega, \\ u(x,t) = 0,\quad \mbox{on } \partial\Omega \times(0,T), \end{array}\displaystyle \right . $$
(1)

where \({\Omega_{T}} = \Omega \times(0,T)\), \(\Omega \subset{\mathbb {R}^{N}}\) is a bounded domain with appropriately smooth boundary ∂Ω, \(p \ge2\), \(\gamma > 0\), and \({u_{0}}(x)\) satisfies

$$ 0 \le{u_{0}} \in C(\bar{\Omega}) \cap W_{0}^{1,p}( \Omega). $$
(2)

Readers can refer to [1] and [2] for the motivation and references about the study of problem (1). The linear parabolic variational inequality problem

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial}{{\partial\tau}}V - \frac{1}{2}{\sigma^{2}}\frac {{{\partial^{2}}}}{{\partial{x^{2}}}}V - ( {r - \frac{1}{2}{\sigma ^{2}}} )\frac{\partial}{{\partial x}}V + rV \ge0,\quad \mbox{in } {\Omega_{T}}, \\ V \ge g(x),\quad \mbox{in } {\Omega_{T}}, \\ ( {\frac{\partial}{{\partial\tau}}V - \frac{1}{2}{\sigma ^{2}}\frac{{{\partial^{2}}}}{{\partial{x^{2}}}}V - ( {r - \frac {1}{2}{\sigma^{2}}} )\frac{\partial}{{\partial x}}V + rV} )(V - g(x)) = 0,\quad \mbox{in } {\Omega_{T}}, \\ V(t,x) = 0, \quad \mbox{on } \partial{\Omega_{T}}, \\ V(x,0) = g(x), \quad \mbox{in } \Omega, \end{array}\displaystyle \right . $$
(3)

is similar to (1). The existence of solutions to problem (3) was studied in a series of papers (see [3] and [4] and references therein). Here, r and σ are positive constant. In [5], the authors discussed a general case in which the linear parabolic operator with constant coefficients can be replaced by a quasi-linear one with integro-differential terms. Later, the authors in [6] extended the corresponding conclusions to the \(\mathbb{R}^{d}\)-values case in which the existence and uniqueness of solution to parabolic variational inequalities with integro-differential terms were proved by using the penalty method and the reduction method.

However, to the best of our knowledge, the existence of solutions to the variational inequality problem with the degenerate parabolic operators has not been studied. The purpose of this paper is to fill this gap.

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

$$ G(\lambda) = \left \{ \textstyle\begin{array}{l@{\quad}l} 0,& \lambda > 0, \\ {[0,+\infty )}, &\lambda = 0. \end{array}\displaystyle \right . $$
(4)

In addition, we define a function class for the solution as follows:

$$ B = \bigl\{ {u \in{L^{\infty}}({\Omega_{T}}) \cap {L^{p}}\bigl(0,T;W_{0}^{1,p}(\Omega)\bigr)} \bigr\} . $$
(5)

Based on the above basic knowledge, we define the weak solution of problem (1) below.

Definition 1

A pair \((u,\xi) \in B \times{L^{\infty}}({\Omega _{T}})\) is called a weak solution of problem (1), if

  1. (a)

    \(u(x,t) \le{u_{0}}(x)\),

  2. (b)

    \(u(x,0) = {u_{0}}(x)\),

  3. (c)

    \(\xi \in G(u -{u_{0}})\),

  4. (d)

    \(\forall\varphi \in C_{0}^{\infty}({\Omega_{T}})\),

    $$ \int_{\Omega_{T}} \bigl( { - u{\varphi_{t}} + u{{\vert { \nabla u} \vert }^{p - 2}}\nabla u\nabla\varphi + (1 - \gamma){{\vert { \nabla u} \vert }^{p}}\varphi} \bigr)\,\mathrm{d}x\,\mathrm{d}t + \int _{\Omega}{\xi \phi\,\mathrm{d}x}\, \mathrm{d}t = 0, $$
    (6)
  5. (e)

    \(\lim_{t \to\infty} \int_{\Omega}{ \vert {{u^{\mu}}(x,t) - u_{0}^{\mu}(x)} \vert \,\mathrm{d}x} = 0\) holds for some \(\mu > 0\).

In Section 2, we prove that for \(p \ge2\), \(\gamma \in(0,1)\), problem (1) admits a weak solution in the sense of Definition 1. We end the introduction by showing the following lemma which is used to prove our main results (see [7]).

Lemma 2

Let \(\theta \ge0\) and \(A(\eta) = {({\eta^{2}} + \theta )^{\frac{{P - 2}}{2}}}\eta\). Then

$$ \bigl[ {A(\eta) - A\bigl(\eta'\bigr)} \bigr] \cdot \bigl[ {\eta - \eta'} \bigr] \ge C{\bigl\vert {\eta - \eta'} \bigr\vert ^{p}} ,\quad \forall\eta,\eta' \in \mathbb{R}, $$
(7)

where C is a positive constant only depending on p.

2 The existence of weak solutions

This section is devoted to the proof of the existence of solutions to problem (1). We prove the following theorem.

Theorem 3

Let \(p \ge2\) and \(\gamma \in(0,1)\). Under the assumption (2), problem (1) admits a weak solution u with \(\frac{{\partial{u^{\mu}}}}{{\partial t}} \in {L^{2}}({\Omega_{T}})\) where

$$ \mu = \frac{{\gamma p}}{{2(p - 1)}} + \frac{1}{2}. $$
(8)

To prove Theorem 3, let us consider the approximation problem

$$ \left \{ \textstyle\begin{array}{l} { ( {{u_{\varepsilon}}} )_{t}} - {u_{\varepsilon}}\operatorname {div} ( {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}}\nabla{u_{\varepsilon}}} ) - \gamma{ \vert {\nabla {u_{\varepsilon}}} \vert ^{p}} - {\beta_{\varepsilon}}({u_{0\varepsilon }} - {u_{\varepsilon}}) = 0,\quad \mbox{in } {\Omega_{T}}, \\ {u_{\varepsilon}}(x,0) = {u_{0\varepsilon}}(x) = {u_{0}}(x) + \varepsilon , \quad \mbox{on }\Omega, \\ {u_{\varepsilon}}(x,t) = 0,\quad \mbox{on }\partial\Omega \times(0,T), \end{array}\displaystyle \right . $$
(9)

where \({\beta_{\varepsilon}}( \cdot)\) is the penalty function satisfying

$$ \begin{aligned} &0 < \varepsilon \le1, \qquad {\beta_{\varepsilon}}(x) \in{C^{2}}(R),\qquad {\beta _{\varepsilon}}(x) \le0, \qquad { \beta_{\varepsilon}}(0) = - 1, \\ &{\beta'_{\varepsilon}}(x) \ge0,\qquad {\beta''_{\varepsilon}}(x) \le0,\qquad \lim_{\varepsilon \to0} {\beta_{\varepsilon}}(x) = \left \{ \textstyle\begin{array}{l@{\quad}l} 0,& x > 0, \\ - \infty, &x < 0. \end{array}\displaystyle \displaystyle \displaystyle \right . \end{aligned} $$
(10)

Definition 4

A nonnegative function \({u_{\varepsilon}}\) is called a weak solution of problem (9), if

  1. (a)

    \({u_{\varepsilon}} \in{L^{\infty}}({\Omega_{T}}) \cap {L^{p}}(0,T;W_{0}^{1,p}(\Omega))\),

  2. (b)

    \(\int_{\Omega_{T}} { ( {{u_{\varepsilon}}{\varphi_{t}} + {u_{\varepsilon}}{{ \vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}}\nabla{u_{\varepsilon}}\nabla\varphi + (1 - \gamma){{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\varphi - {\beta_{\varepsilon}}({u_{0\varepsilon}} - {u_{\varepsilon}})\varphi} )}{\,\mathrm{d}x\,\mathrm{d}t}= 0\) hold, for any \(\forall\varphi \in C_{0}^{\infty}({\Omega_{T}})\),

  3. (c)

    \(\lim_{t \to\infty} \int_{\Omega}{ \vert {u_{\varepsilon}^{\mu}(x,t) - u_{0\varepsilon}^{\mu}(x)} \vert \,\mathrm{d}x} = 0\) holds for some \(\mu > 0\).

According to the standard theory for parabolic equations [7], problem (9) admits a weak solution

$$ {u_{\varepsilon}} \in{L^{\infty}}({\Omega_{T}}) \cap {L^{p}}\bigl(0,T;W_{0}^{1,p}({\Omega_{T}}) \bigr) $$
(11)

in the sense of Definition 4, which satisfies

$$ { \bigl( {u_{\varepsilon}^{\mu}} \bigr)_{t}} \in{L_{2}}({\Omega_{T}}),\qquad \nabla {u_{\varepsilon}} \in{L_{p}}({\Omega_{T}}). $$
(12)

Further, it follows by the comparison principle and the maximum principle [8, 9] that

$$ \varepsilon \le{u_{\varepsilon}} \le{u_{0\varepsilon}} \le{ \vert {{u_{0}}} \vert _{\infty}} + \varepsilon,\qquad {u_{{\varepsilon_{1}}}} \le{u_{{\varepsilon_{2}}}} \quad \mbox{for } {\varepsilon_{1}} \le{\varepsilon_{2}}. $$
(13)

Moreover, from (12) and (13), we assert that there exists a subsequence ε (still denoted by ε) such that

$$\begin{aligned}& {u_{\varepsilon}} \to u \in{L^{p}}\bigl(0,T;W_{0}^{1,p}({ \Omega_{T}})\bigr) \quad \mbox{as } \varepsilon \to0, \end{aligned}$$
(14)
$$\begin{aligned}& {u_{\varepsilon}} \ge u \ge0\quad \mbox{for any } \varepsilon > 0. \end{aligned}$$
(15)

Lemma 5

Assume \(Q_{c}^{\varepsilon}= \{ {(x,t) \in{\Omega _{T}};{u_{\varepsilon}} \ge c,c > 0} \}\), \(Q_{c} = \{ {(x,t) \in{\Omega_{T}};u \ge c,c > 0} \}\) such that, as \(\varepsilon \to0\),

$$ \int_{Q_{c}^{\varepsilon}} {{{\vert {\nabla{u_{\varepsilon}} - \nabla u} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} \to0,\qquad \int _{{Q_{c}}} {{{\vert {\nabla {u_{\varepsilon}} - \nabla u} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} \to0. $$
(16)

Proof

Choosing \(\varphi = u_{\varepsilon}^{p - 2}(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}})\) as the test function in

$$ \int_{{\Omega_{T}}} { \bigl( {{u_{\varepsilon}} { \varphi_{t}} + {u_{\varepsilon}} {{\vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}}\nabla{u_{\varepsilon}}\nabla\varphi + (1 - \gamma){{ \vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\varphi - { \beta_{\varepsilon}}({u_{0\varepsilon}} - {u_{\varepsilon}})\varphi} \bigr)\, \mathrm{d}x\,\mathrm{d}t} = 0. $$
(17)

Then it is easy to see that

$$\begin{aligned}& \int_{{\Omega_{T}}} {\frac{{\partial{u_{\varepsilon}}}}{{\partial t}}u_{\varepsilon}^{p - 2} \bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\,\mathrm{d}t} \\& \quad = - \int_{{\Omega_{T}}} {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}}\nabla{u_{\varepsilon}}\nabla \bigl\{ {u_{\varepsilon}^{p - 1}\bigl(u_{\varepsilon}^{2} - { \varepsilon^{2}} - {u^{2}}\bigr)} \bigr\} \,\mathrm{d}x\, \mathrm{d}t} \\& \qquad {} + \gamma\int_{{\Omega_{T}}} {u_{\varepsilon}^{p - 2}{{ \vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\bigl(u_{\varepsilon}^{2} - {\varepsilon ^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\, \mathrm{d}t} \\& \qquad {} + \int_{{\Omega_{T}}} {u_{\varepsilon}^{p - 2} \bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr){\beta_{\varepsilon}}({u_{0\varepsilon}} - {u_{\varepsilon}})\,\mathrm{d}x\,\mathrm{d}t} \\& \quad = - \int_{{\Omega_{T}}} {u_{\varepsilon}^{p - 1}{{ \vert {\nabla {u_{\varepsilon}}} \vert }^{p - 2}}\nabla{u_{\varepsilon}} \nabla \bigl(u_{\varepsilon}^{2} - {u^{2}}\bigr)\,\mathrm{d}x \,\mathrm{d}t} \\& \qquad {} + \int_{{\Omega_{T}}} {u_{\varepsilon}^{p - 2} \bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr){\beta_{\varepsilon}}({u_{0\varepsilon}} - {u_{\varepsilon}})\,\mathrm{d}x\,\mathrm{d}t} \\& \qquad {} + (\gamma - p + 1)\int_{{\Omega_{T}}} {u_{\varepsilon}^{p - 2}{{ \vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\, \mathrm{d}t} . \end{aligned}$$
(18)

From (15) and the definition of \({\beta_{\varepsilon}}\), we derive

$$\begin{aligned}& \int_{{\Omega_{T}}} {u_{\varepsilon}^{p - 2}{{\vert {\nabla {u_{\varepsilon}}} \vert }^{p}}\bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\, \mathrm{d}t} \ge - {\varepsilon^{2}}\int_{{\Omega_{T}}} {u_{\varepsilon}^{p - 2}{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t}, \end{aligned}$$
(19)
$$\begin{aligned}& \int_{{\Omega_{T}}} {u_{\varepsilon}^{p - 2} \bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr){\beta_{\varepsilon}}({u_{0\varepsilon}} - {u_{\varepsilon}})\,\mathrm{d}x\,\mathrm{d}t} \\& \quad \le{ \bigl( {{{\vert {{u_{0}}} \vert }_{\infty}} + \varepsilon} \bigr)^{p - 2}}\biggl\vert {\int_{{\Omega_{T}}} { \bigl(u_{\varepsilon}^{2} - {\varepsilon ^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\,\mathrm{d}t} } \biggr\vert . \end{aligned}$$
(20)

Observing that\(\gamma - p + 1 < 0\), and combing (18), (19), and (20), we have

$$\begin{aligned}& \int_{{\Omega_{T}}} {\frac{{\partial{u_{\varepsilon}}}}{{\partial t}}u_{\varepsilon}^{p - 2} \bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\,\mathrm{d}t} \\& \quad \le - {2^{1 - p}}\int_{{\Omega_{T}}} {{{\bigl\vert { \nabla u_{\varepsilon}^{2}} \bigr\vert }^{p - 2}}\nabla u_{\varepsilon}^{2}\nabla\bigl(u_{\varepsilon}^{2} - {u^{2}}\bigr)\,\mathrm{d}x\,\mathrm{d}t} + p{\varepsilon^{2}} \int_{{\Omega_{T}}} {u_{\varepsilon}^{p - 2}{{\vert { \nabla{u_{\varepsilon}}} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} \\& \qquad {}+ { \bigl( {{{\vert {{u_{0}}} \vert }_{\infty}} + \varepsilon} \bigr)^{p - 2}}\biggl\vert {\int_{{\Omega_{T}}} { \bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\,\mathrm{d}t} } \biggr\vert . \end{aligned}$$
(21)

Note that \(\varepsilon \le{u_{\varepsilon}} \le{ \vert {{u_{0}}} \vert _{\infty}} + \varepsilon\). Thus, it follows by the trigonometrical inequality and the Hölder inequality that

$$\begin{aligned}& \int_{{\Omega_{T}}} {\frac{{\partial{u_{\varepsilon}}}}{{\partial t}}u_{\varepsilon}^{p - 2} \bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\,\mathrm{d}t} \\& \qquad {} + \int_{{\Omega_{T}}} { \bigl[ {{{\bigl\vert {\nabla u_{\varepsilon}^{2}} \bigr\vert }^{p - 2}}\nabla u_{\varepsilon}^{2} - {{\bigl\vert {\nabla u^{2}} \bigr\vert }^{p - 2}}\nabla u^{2}} \bigr]\nabla \bigl(u_{\varepsilon}^{2} - {u^{2}}\bigr)\,\mathrm{d}x\, \mathrm{d}t} \\& \quad \le - {2^{1 - p}}\int_{{\Omega_{T}}} {{{\bigl\vert { \nabla u^{2}} \bigr\vert }^{p - 2}}\nabla u^{2}\nabla \bigl(u_{\varepsilon}^{2} - {u^{2}}\bigr)\,\mathrm{d}x\, \mathrm{d}t} + p{\varepsilon^{2}}\int_{{\Omega_{T}}} {u_{\varepsilon}^{p - 2}{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} \\& \qquad {} + { \bigl( {{{\vert {{u_{0}}} \vert }_{\infty}} + \varepsilon } \bigr)^{p - 2}}\biggl\vert {\int_{{\Omega_{T}}} {\bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\,\mathrm{d}t} } \biggr\vert \\& \quad \le{2^{1 - p}} { \biggl( {\int_{{\Omega_{T}}} {{{\bigl\vert {\nabla u^{2}} \bigr\vert }^{p}}\,\mathrm{d}x\, \mathrm{d}t} } \biggr)^{\frac{{p - 1}}{p}}} { \biggl( {\int_{{\Omega_{T}}} {{{\bigl\vert {\nabla\bigl(u_{\varepsilon}^{2} - {u^{2}} \bigr)} \bigr\vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} } \biggr)^{\frac{1}{p}}} \\& \qquad {} + {\bigl(\vert {{u_{0}}} \vert + \varepsilon \bigr)^{p - 2}}\mu {\varepsilon^{2}}\int_{{\Omega_{T}}} {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\,\mathrm{d}x\, \mathrm{d}t} \\& \qquad {} + { \bigl( {{{\vert {{u_{0}}} \vert }_{\infty}} + \varepsilon } \bigr)^{p - 2}}\biggl\vert {\int_{{\Omega_{T}}} {\bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\,\mathrm{d}t} } \biggr\vert . \end{aligned}$$
(22)

Since \(\frac{\partial}{{\partial t}}u_{\varepsilon}^{\mu}\in {L^{2}}({\Omega_{T}})\), using the Hölder inequality, one derives

$$\begin{aligned}& \int_{{\Omega_{T}}} {\biggl\vert {\frac{{\partial u_{\varepsilon}^{p - 1}}}{{\partial t}}} \biggr\vert \,\mathrm{d}x\,\mathrm{d}t} \\& \quad = (p - 1)\int_{{\Omega _{T}}} {u_{\varepsilon}^{p - 2} \biggl\vert {\frac{{\partial u_{\varepsilon}}}{{\partial t}}} \biggr\vert \,\mathrm{d}x\,\mathrm{d}t} \\& \quad \le(p - 1){\bigl({\vert {{u_{0}}} \vert _{\infty}} + \varepsilon\bigr)^{p - 2}}\int_{{\Omega_{T}}} {\biggl\vert { \frac{{\partial u_{\varepsilon}}}{{\partial t}}} \biggr\vert \,\mathrm{d}x\,\mathrm{d}t} \\& \quad \le(p - 1){\bigl({\vert {{u_{0}}} \vert _{\infty}} + \varepsilon\bigr)^{p - 2}}\sqrt{\vert {{\Omega_{T}}} \vert } \sqrt{\int_{{\Omega_{T}}} {{{\biggl\vert {\frac{{\partial u_{\varepsilon}}}{{\partial t}}} \biggr\vert }^{2}}\,\mathrm{d}x\,\mathrm{d}t} } < \infty. \end{aligned}$$
(23)

From the above equation and (13), we may conclude that, as \(\varepsilon \to0\),

$$\begin{aligned}& \biggl\vert {\int_{{\Omega_{T}}} {\frac{{\partial{u_{\varepsilon}}}}{{\partial t}}u_{\varepsilon}^{p - 2} \bigl(u_{\varepsilon}^{2} - {\varepsilon ^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\,\mathrm{d}t} } \biggr\vert \\& \quad \le\frac{1}{{p - 1}}\int_{{\Omega_{T}}} {\biggl\vert { \frac{{\partial u_{\varepsilon}^{p - 1}}}{{\partial t}}\bigl(u_{\varepsilon}^{2} - {\varepsilon ^{2}} - {u^{2}}\bigr)} \biggr\vert \,\mathrm{d}x\, \mathrm{d}t} \\& \quad \le\frac{1}{{p - 1}}\sqrt{\int_{{\Omega_{T}}} {{{ \biggl( {\frac {{\partial u_{\varepsilon}^{p - 1}}}{{\partial t}}} \biggr)}^{2}}\,\mathrm{d}x\,\mathrm{d}t} } \cdot\sqrt{\int_{{\Omega_{T}}} {{{\bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr)}^{2}}\, \mathrm{d}x\,\mathrm{d}t} } \\& \quad \le\frac{1}{{p - 1}}{\varepsilon^{2}}\sqrt{\vert {{ \Omega_{T}}} \vert } \sqrt{\int_{{\Omega_{T}}} {{{ \biggl( {\frac{{\partial u_{\varepsilon}^{p - 1}}}{{\partial t}}} \biggr)}^{2}}\,\mathrm{d}x\,\mathrm{d}t} } \to0, \end{aligned}$$
(24)
$$\begin{aligned}& { \bigl( {{{\vert {{u_{0}}} \vert }_{\infty}} + \varepsilon} \bigr)^{p - 2}}\biggl\vert {\int_{{\Omega_{T}}} { \bigl(u_{\varepsilon}^{2} - {\varepsilon^{2}} - {u^{2}}\bigr)\,\mathrm{d}x\,\mathrm{d}t} } \biggr\vert \to0. \end{aligned}$$
(25)

Substituting (24) and (25) into (22) yields

$$ \limsup_{\varepsilon \to0} \int_{{\Omega_{T}}} { \bigl[ {{{\bigl\vert {\nabla u_{\varepsilon}^{2}} \bigr\vert }^{p - 2}} \nabla u_{\varepsilon}^{2} - {{\bigl\vert {\nabla u^{2}} \bigr\vert }^{p - 2}}\nabla u^{2}} \bigr]\nabla \bigl(u_{\varepsilon}^{2} - {u^{2}}\bigr)\,\mathrm{d}x\, \mathrm{d}t} \le0. $$
(26)

This and Lemma 2 lead to

$$ \lim_{\varepsilon \to0} \int_{{\Omega_{T}}} {{{\bigl\vert { \nabla u_{\varepsilon}^{2} - \nabla u^{2}} \bigr\vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} = 0. $$
(27)

Hence, in view of

$$\begin{aligned}& \nabla u_{\varepsilon}^{2} - \nabla{u^{2}} = 2{u_{\varepsilon}}\nabla {u_{\varepsilon}} - 2u\nabla u = 2{u_{\varepsilon}}( \nabla{u_{\varepsilon}} - \nabla u) + 2\nabla u ( {{u_{\varepsilon}} - u} ), \end{aligned}$$
(28)
$$\begin{aligned}& \nabla u_{\varepsilon}^{2} - \nabla{u^{2}} = 2{u_{\varepsilon}}\nabla {u_{\varepsilon}} - 2u\nabla u = 2u( \nabla{u_{\varepsilon}} - \nabla u) + 2\nabla{u_{\varepsilon}}({u_{\varepsilon}} - u), \end{aligned}$$
(29)

we derive

$$\begin{aligned}& {2^{p}}\int_{{\Omega_{T}}} {u_{\varepsilon}^{p}{{ \vert {\nabla {u_{\varepsilon}} - \nabla u} \vert }^{p}}\, \mathrm{d}x\,\mathrm{d}t} \\& \quad \le{2^{p}}\int_{{\Omega_{T}}} {{{\bigl\vert {\nabla u_{\varepsilon}^{2} - \nabla{u^{2}}} \bigr\vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} + {4^{p}}\int _{{\Omega_{T}}} {{{\vert {\nabla u} \vert }^{p}} {{\vert {{u_{\varepsilon}} - u} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} \to0 \quad (\varepsilon \to0), \end{aligned}$$
(30)
$$\begin{aligned}& {2^{p}}\int_{{\Omega_{T}}} {{u^{p}} {{\vert { \nabla{u_{\varepsilon}} - \nabla u} \vert }^{p}}\,\mathrm{d}x\, \mathrm{d}t} \\& \quad \le{2^{p}}\int_{{\Omega_{T}}} {{{\bigl\vert {\nabla u_{\varepsilon}^{2} - \nabla{u^{2}}} \bigr\vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} + {4^{p}}\int _{{\Omega_{T}}} {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}} {{\vert {{u_{\varepsilon}} - u} \vert }^{p}}\, \mathrm{d}x\,\mathrm{d}t} \to0\quad (\varepsilon \to0). \end{aligned}$$
(31)

Note that \(Q_{c}^{\varepsilon}\subset{\Omega_{T}}\), \(Q_{c} \subset {\Omega_{T}}\). Then it is easy to see that as \(\varepsilon \to0\),

$$\begin{aligned}& {c^{p}}\int_{Q_{c}^{\varepsilon}} {{{\vert {\nabla{u_{\varepsilon}} - \nabla u} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} \le\int _{{\Omega_{T}}} {u_{\varepsilon}^{p}{{\vert { \nabla{u_{\varepsilon}} - \nabla u} \vert }^{p}}\,\mathrm{d}x\, \mathrm{d}t} \to0, \end{aligned}$$
(32)
$$\begin{aligned}& {c^{p}}\int_{Q_{c}} {{{\vert {\nabla{u_{\varepsilon}} - \nabla u} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} \le\int _{{\Omega_{T}}} {{u^{p}} {{\vert {\nabla {u_{\varepsilon}} - \nabla u} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} \to0. \end{aligned}$$
(33)

Thus, the lemma is proved. □

Lemma 6

The solution of (9) satisfies

$$ \int_{{\Omega_{T}}} {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}}u_{\varepsilon}^{ - \alpha}\,\mathrm{d}x\,\mathrm{d}t} \le C, $$
(34)

where C is independent of ε, \(\alpha \in[0,1 - \gamma)\).

Proof

Multiply (9) by \(u_{\varepsilon}^{ - \alpha}\) and integrate both sides of the equation over \({\Omega_{T}}\). After integrating by parts, we obtain

$$\begin{aligned}& \int_{{\Omega_{T}}} {\frac{{\partial{u_{\varepsilon}}}}{{\partial t}}u_{\varepsilon}^{ - \alpha} \,\mathrm{d}x\,\mathrm{d}t} \\& \quad = \int_{{\Omega_{T}}} {u_{\varepsilon}^{1 - \alpha} \operatorname {div} \bigl\{ {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}}\nabla{u_{\varepsilon}}} \bigr\} +\gamma u_{\varepsilon}^{ - \alpha}{{ \vert {\nabla{u_{\varepsilon}}} \vert }^{p}} + u_{\varepsilon}^{ - \alpha}{\beta_{\varepsilon}}({u_{0\varepsilon}} - {u_{\varepsilon}})\, \mathrm{d}x\,\mathrm{d}t} \\& \quad = \int_{0}^{T} {\mathrm{d}t\int _{\partial\Omega} { \biggl\{ {u_{\varepsilon}^{1 - \alpha}{{ \vert { \nabla{u_{\varepsilon}}} \vert }^{p - 2}}\frac {{\partial{u_{\varepsilon}}}}{{\partial\nu}}} \biggr\} \, \mathrm{d}x} } - (1 - \alpha - \gamma)\int_{{\Omega_{T}}} {u_{\varepsilon}^{ - \alpha }{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} \\& \qquad {}+ \int_{{\Omega_{T}}} {u_{\varepsilon}^{ - \alpha}{ \beta_{\varepsilon}} ( {{u_{0\varepsilon}} - {u_{\varepsilon}}} )\,\mathrm{d}x \,\mathrm{d}t} , \end{aligned}$$
(35)

where ν denotes the outward normal to \(\partial\Omega \times(0,T)\).

Further, putting together (10) and (13) implies

$$ \int_{{\Omega_{T}}} {u_{\varepsilon}^{ - \alpha}{ \beta_{\varepsilon}} ( {{u_{0\varepsilon}} - {u_{\varepsilon}}} )\,\mathrm{d}x \,\mathrm{d}t} \le0. $$
(36)

Since \({u_{\varepsilon}} \ge\varepsilon\), we have

$$ \frac{{\partial{u_{\varepsilon}}}}{{\partial\nu}} \le0,\quad \mbox{on } \partial\Omega \times(0,T). $$
(37)

This leads to

$$ \int_{0}^{T} {\int_{\partial\Omega} { \biggl\{ {u_{\varepsilon}^{1 - \alpha }{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}}\frac{{\partial {u_{\varepsilon}}}}{{\partial\nu}}} \biggr\} \,\mathrm{d}x\,\mathrm{d}t} } \le 0. $$
(38)

Now, we drop the non-positive terms (36) and (38) in (35) to get

$$ \int_{{\Omega_{T}}} {\frac{{\partial{u_{\varepsilon}}}}{{\partial t}}u_{\varepsilon}^{ - \alpha} \,\mathrm{d}x\,\mathrm{d}t} \le - (1 - \alpha - \gamma)\int_{{\Omega_{T}}} {u_{\varepsilon}^{ - \alpha}{{ \vert {\nabla {u_{\varepsilon}}} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} . $$
(39)

Clearly, using integration by parts, we derive

$$ \int_{{\Omega_{T}}} {\frac{{\partial{u_{\varepsilon}}}}{{\partial t}}u_{\varepsilon}^{ - \alpha} \,\mathrm{d}x\,\mathrm{d}t} = \frac{1}{{1 - \alpha}}\int_{\Omega}{u_{\varepsilon}^{1 - \alpha}(x,T) - u_{\varepsilon}^{1 - \alpha}(x,0)\, \mathrm{d}x} . $$
(40)

This and (39) lead to

$$ \int_{{\Omega_{T}}} {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}}u_{\varepsilon}^{ - \alpha}\,\mathrm{d}x\,\mathrm{d}t} \le \frac{1}{{(1 - \alpha - \gamma)(1 - \alpha)}}\int_{\Omega}{u_{\varepsilon}^{ - \alpha}(x,0) \,\mathrm{d}x} \le{C_{1}}, $$
(41)

where \({C_{1}} > 0\) depending on α, γ, Ω, and \(\vert {{u_{0}}} \vert \). Hence, the proof is completed. □

Lemma 7

As \(\varepsilon \to0\), we have

$$\begin{aligned}& \int_{{\Omega_{T}}} {\bigl\vert {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}} - {{\vert {\nabla u} \vert }^{p}}} \bigr\vert \,\mathrm{d}x\,\mathrm{d}t} \to 0, \end{aligned}$$
(42)
$$\begin{aligned}& \int_{{\Omega_{T}}} {\bigl\vert {{u_{\varepsilon}} {{\vert { \nabla {u_{\varepsilon}}} \vert }^{p - 2}}\nabla{u_{\varepsilon}} - u{{ \vert {\nabla u} \vert }^{p}}\nabla u} \bigr\vert \,\mathrm{d}x\, \mathrm{d}t} \to0, \end{aligned}$$
(43)
$$\begin{aligned}& {\beta_{\varepsilon}}({u_{0\varepsilon}} - {u_{\varepsilon}}) \to\xi \in G({u_{0}} - u). \end{aligned}$$
(44)

Proof

Let \({\chi_{\eta}}\) and \(\chi_{\eta}^{(\varepsilon)}\) be the characteristic functions of \(\{ {(x,t) \in{\Omega_{T}};u(x,t) < \eta} \}\) and \(\{ (x,t) \in{\Omega_{T}};{u_{\varepsilon}}(x,t) < \eta \}\), respectively. Since \({u_{\varepsilon}} \to u\), as \(\varepsilon \to0\), \({\chi_{\eta}} \le\chi_{\eta}^{(\varepsilon)}\), we have

$$\begin{aligned}& \int_{{\Omega_{T}}} {\bigl\vert {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}} - {{\vert {\nabla u} \vert }^{p}}} \bigr\vert \,\mathrm{d}x\,\mathrm{d}t} \\& \quad \le\int_{{\Omega_{T}}} {\bigl\vert {{{\vert {\nabla u} \vert }^{p}} {\chi _{\eta}} - {{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\chi_{\eta}^{(\varepsilon)}} \bigr\vert \, \mathrm{d}x\,\mathrm{d}t} + \int_{{\Omega_{T}}} {\bigl\vert {{{\vert {\nabla u} \vert }^{p}}(1 - {\chi_{\eta}}) - {{\vert { \nabla{u_{\varepsilon}}} \vert }^{p}}\bigl(1 - \chi_{\eta}^{(\varepsilon )} \bigr)} \bigr\vert \,\mathrm{d}x\,\mathrm{d}t} \\& \quad \le\int_{{\Omega_{T}}} {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\chi_{\eta}^{(\varepsilon)}\,\mathrm{d}x\, \mathrm{d}t} + \int_{{\Omega _{T}}} {{{\vert {\nabla u} \vert }^{p}} {\chi_{\eta}}\,\mathrm{d}x\,\mathrm{d}t} \\& \qquad {}+ \int_{{\Omega_{T}}} {{{\vert {\nabla u} \vert }^{p}}\bigl(\chi_{\eta}^{(\varepsilon)} - {\chi _{\eta}} \bigr)\,\mathrm{d}x\,\mathrm{d}t} + \int_{{\Omega_{T}}} {\bigl\vert {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}} - {{\vert { \nabla u} \vert }^{p}}} \bigr\vert \bigl(1 - \chi_{\eta}^{(\varepsilon)} \bigr)\,\mathrm{d}x\,\mathrm{d}t} \\& \quad = {H_{1}} + {H_{2}} + {H_{3}} + {H_{4}}. \end{aligned}$$
(45)

Taking \(\alpha = (1 - \gamma)/2\) in Lemma 6 one obtains

$$\begin{aligned} \begin{aligned}[b] {H_{1}} &= \int_{{\Omega_{T}}} {{{\vert { \nabla{u_{\varepsilon}}} \vert }^{p}}\frac{{u_{\varepsilon}^{\alpha}}}{{u_{\varepsilon}^{\alpha}}}\chi _{\eta}^{(\varepsilon)}\,\mathrm{d}x\,\mathrm{d}t} \\ &\le{\eta^{\alpha}}\int_{{\Omega_{T}}} {{{\vert {\nabla {u_{\varepsilon}}} \vert }^{p}}u_{\varepsilon}^{ - \alpha} \chi_{\eta}^{(\varepsilon)}\,\mathrm{d}x\,\mathrm{d}t} \\ &\le{\eta^{\alpha}}\int_{{\Omega_{T}}} {{{\vert {\nabla {u_{\varepsilon}}} \vert }^{p}}u_{\varepsilon}^{ - \alpha}\, \mathrm{d}x\,\mathrm{d}t} \le C{\eta^{\alpha}} \to0 \quad (\eta \to0). \end{aligned} \end{aligned}$$
(46)

Applying Lemma 5, (46), and the fact that \({\chi_{\eta}} \le\chi_{\eta}^{(\varepsilon)}\) implies

$$\begin{aligned}& {H_{2}} \le\int_{{\Omega_{T}}} {\chi_{\eta}^{(\varepsilon)}{{ \vert {\nabla u} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} \\& \hphantom{{H_{2}}} \le\int_{{\Omega_{T}}} {\chi_{\eta}^{(\varepsilon)}{{ \vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\,\mathrm{d}x\, \mathrm{d}t} + \int_{{\Omega_{T}}} {\chi_{\eta}^{(\varepsilon)}{{ \vert {\nabla {u_{\varepsilon}} - \nabla u} \vert }^{p}}\, \mathrm{d}x\,\mathrm{d}t} \to0\quad (\eta \to0), \end{aligned}$$
(47)
$$\begin{aligned}& {H_{4}} \to0 \quad (\eta \to0). \end{aligned}$$
(48)

For fixed \(\eta > 0\), \(\chi_{\eta}^{(\varepsilon)} \to{\chi_{\eta}}\) (\(\varepsilon \to0\)) a.e. in \({\Omega_{T}}\), so

$$ {H_{3}} \to0\quad (\eta \to0). $$
(49)

Putting together (46), (47), (48), and (49), we have

$$ \int_{{\Omega_{T}}} {\bigl\vert {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}} - {{\vert {\nabla u} \vert }^{p}}} \bigr\vert \,\mathrm{d}x\,\mathrm{d}t} \to 0 \quad (\eta \to0). $$
(50)

Thus (42) holds.

Next we prove (43). It follows by the trigonometrical inequality that

$$\begin{aligned}& \int_{{\Omega_{T}}} {\bigl\vert {{u_{\varepsilon}} {{\vert { \nabla {u_{\varepsilon}}} \vert }^{p - 2}}\nabla{u_{\varepsilon}} - u{{ \vert {\nabla u} \vert }^{p - 2}}\nabla u} \bigr\vert \,\mathrm{d}x\, \mathrm{d}t} \\& \quad \le\int_{{\Omega_{T}}} {\vert {{u_{\varepsilon}} - u} \vert {{\vert {\nabla{u_{\varepsilon}}} \vert }^{p - 1}}\,\mathrm{d}x\, \mathrm{d}t} + \int_{{\Omega_{T}}} {u{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}}\vert {\nabla{u_{\varepsilon}} - \nabla u} \vert \, \mathrm{d}x\,\mathrm{d}t} \\& \qquad {}+ \int_{{\Omega_{T}}} {u\vert {\nabla u} \vert \cdot\bigl\vert {{{ \vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}} - {{ \vert {\nabla u} \vert }^{p - 2}}} \bigr\vert \,\mathrm{d}x\, \mathrm{d}t} \\& \quad = {H_{5}} + {H_{6}} + {H_{7}}. \end{aligned}$$
(51)

Using the Hölder inequality and Lemma 6, we obtain

$$ {H_{5}} \le C{ \biggl( {\int_{{\Omega_{T}}} {{{\vert {{u_{\varepsilon}} - u} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} } \biggr)^{\frac{1}{p}}} \to0\quad \mbox{as } \varepsilon \to0. $$
(52)

With the inequality \(\vert {{a^{r}} - {b^{r}}} \vert \le{ \vert {a - b} \vert ^{r}}\) (\(r \in[0,1]\), \(a,b \ge0\)), the Hölder inequality, and (42), we have

$$\begin{aligned} {H_{7}} =& \int_{{\Omega_{T}}} {u\vert {\nabla u} \vert \cdot\bigl\vert {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}} - {{\vert {\nabla u} \vert }^{p - 2}}} \bigr\vert \,\mathrm{d}x\, \mathrm{d}t} \\ =& \int_{{\Omega_{T}}} {u\vert {\nabla u} \vert \cdot \bigl( {{{ \bigl({{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}} \bigr)}^{\frac{{p - 2}}{p}}} - {{\bigl({{\vert {\nabla u} \vert }^{p}} \bigr)}^{\frac{{p - 2}}{p}}}} \bigr)\,\mathrm{d}x\,\mathrm{d}t} \\ \le& C\int_{{\Omega_{T}}} {\vert {\nabla u} \vert \cdot {{\bigl\vert {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}} - {{\vert {\nabla u} \vert }^{p}}} \bigr\vert }^{\frac{{p - 2}}{p}}}\,\mathrm{d}x \,\mathrm{d}t} \\ \le& C{ \biggl( {\int_{{\Omega_{T}}} {\vert {\nabla u} \vert \cdot \bigl\vert {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}} - {{ \vert {\nabla u} \vert }^{p}}} \bigr\vert \,\mathrm{d}x\, \mathrm{d}t} } \biggr)^{\frac{{p - 2}}{p}}} \\ &{}\times { \biggl( {\int_{{\Omega_{T}}} {{{\vert {\nabla u} \vert }^{\frac{p}{2}}}\,\mathrm{d}x\,\mathrm{d}t} } \biggr)^{\frac{2}{p}}} \to0\quad (\varepsilon \to0). \end{aligned}$$
(53)

Finally, we estimate \({H_{6}}\). Again by using trigonometrical inequality, we arrive at

$$\begin{aligned} {H_{6}} =& \int_{{\Omega_{T}}} {u{{\vert { \nabla{u_{\varepsilon}}} \vert }^{p - 2}} \cdot \vert { \nabla{u_{\varepsilon}} - \nabla u} \vert {\chi_{\eta}}\,\mathrm{d}x\, \mathrm{d}t} \\ &{}+ \int_{{\Omega_{T}}} {u{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}} \cdot \vert {\nabla {u_{\varepsilon}} - \nabla u} \vert (1 - {\chi_{\eta}})\, \mathrm{d}x\,\mathrm{d}t} \\ \le&\eta\int_{{\Omega_{T}}} {{{\vert {\nabla {u_{\varepsilon}}} \vert }^{p - 2}} \cdot \vert {\nabla{u_{\varepsilon}} - \nabla u} \vert {\chi_{\rho}}\,\mathrm{d}x\,\mathrm{d}t} \\ &{} + C \biggl( {\int_{{\Omega_{T}}} {{{\vert {\nabla {u_{\varepsilon}}} \vert }^{(p - 2)p/(p - 1)}}}\,\mathrm{d}x\,\mathrm{d}t} \biggr)^{p/(p-1)} \biggl( {\int_{{\Omega_{T}}} {{{\vert { \nabla{u_{\varepsilon}} - \nabla u} \vert }^{p}}(1 - {\chi _{\rho}})\, \mathrm{d}x\,\mathrm{d}t} } \biggr)^{1/p} \\ =& \eta\int_{{\Omega_{T}}} {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p - 2}} \cdot \vert {\nabla{u_{\varepsilon}} - \nabla u} \vert {\chi_{\rho}}\,\mathrm{d}x\,\mathrm{d}t} \\ &{}+ C \biggl( {\int _{{\Omega_{T}}} {{{\vert {\nabla{u_{\varepsilon}} - \nabla u} \vert }^{p}}(1 - {\chi _{\rho}})\, \mathrm{d}x\,\mathrm{d}t} } \biggr)^{1/p} \\ =& {H_{8}} + {H_{9}}. \end{aligned}$$
(54)

For all \(\delta > 0\) and \(\varepsilon \in(0,1)\), let η be small enough and use Lemma 5 such that, as \(\varepsilon \to0\),

$$\begin{aligned} {H_{8}} &\le{ \biggl( {\int_{{\Omega_{T}}} {{{\vert { \nabla{u_{\varepsilon}}} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} } \biggr)^{\frac{{p - 2}}{p}}} { \biggl( {\int_{{\Omega_{T}}} {{{\vert { \nabla{u_{\varepsilon}} - \nabla u} \vert }^{\frac{p}{2}}} \chi_{\rho}^{\frac{2}{p}}\,\mathrm{d}x\,\mathrm{d}t} } \biggr)^{\frac{2}{p}}} \\ &\le{ \vert {{\Omega_{T}}} \vert ^{\frac{1}{p}}} { \biggl( {\int _{{\Omega_{T}}} {{{\vert {\nabla{u_{\varepsilon}}} \vert }^{p}}\,\mathrm{d}x\,\mathrm{d}t} } \biggr)^{\frac{{p - 2}}{p}}} { \biggl( { \int_{{\Omega _{T}}} {{{\vert {\nabla{u_{\varepsilon}} - \nabla u} \vert }^{p}}\chi _{\rho}\,\mathrm{d}x\,\mathrm{d}t} } \biggr)^{\frac{1}{p}}} \to0. \end{aligned}$$
(55)

Clearly, for fixed \(\eta > 0\), using Lemma 5, we have

$$ {H_{9}} \to0\quad \mbox{as } \varepsilon \to0. $$
(56)

Substituting (55) and (56) into (54), we obtain

$$ {H_{6}} \to0\quad \mbox{as } \varepsilon \to0 . $$
(57)

Hence, (43) is proved by putting together (52), (53), and (57).

Finally we prove (44). Using (13) and the definition of \({\beta _{\varepsilon}}\), we have

$$ {\beta_{\varepsilon}}({u_{\varepsilon}} - {u_{0\varepsilon}}) \to\xi\quad \mbox{as } \varepsilon \to0 . $$
(58)

Now we prove \(\xi \in G({u_{0}} - u)\). According to the definition of \(G( \cdot)\), we only need to prove that if \(u({x_{0}},{t_{0}}) < {u_{0}}({x_{0}})\), \(\xi({x_{0}},{t_{0}}) = 0\). In fact, if \(u({x_{0}},{t_{0}}) < {u_{0}}(x)\), there exist a constant \(\lambda > 0\) and a δ neighborhood \({B_{\delta}}({x_{0}},{t_{0}})\) such that, if ε is small enough, we have

$$ {u_{\varepsilon}}(x,t) \le{u_{0\varepsilon}}(x) - \lambda,\quad \forall (x,t) \in{B_{\delta}}({x_{0}},{t_{0}}). $$
(59)

Thus, if ε is small enough, we have

$$ 0 \ge{\beta_{\varepsilon}}({u_{0\varepsilon}} - {u_{\varepsilon}}) \ge { \beta_{\varepsilon}}(\lambda) = 0, \quad \forall(x,t) \in{B_{\delta}}({x_{0}},{t_{0}}). $$
(60)

Furthermore, it follows by \(\varepsilon \to0\) that

$$ \xi(x,t) = 0,\quad \forall(x,t) \in{B_{\delta}}({x_{0}},{t_{0}}). $$
(61)

Hence, (44) holds, and the proof of Lemma 7 is completed. □

With Lemma 7 and (14), it is easy to check that u satisfies (c) and (d) in Definition 1. Moreover, applying (13), it is clear that

$$ u(x,t) \le{u_{0}}(x), \quad \mbox{in } {\Omega_{T}}, \qquad u(x,0) = {u_{0}}(x), \quad \mbox{in } \Omega. $$
(62)

Thus (a) and (b) hold.

To show the existence of problem (1), we only need to prove that (e) holds. Define

$$ I = \int_{\Omega}{\bigl\vert {u_{\varepsilon}^{\mu}- u_{0\varepsilon}^{\mu}} \bigr\vert \,\mathrm{d}x}. $$
(63)

Applying the Hölder inequality twice, we obtain

$$\begin{aligned} I &= \int_{\Omega}{\bigl\vert {u_{\varepsilon}^{\mu}(x,t) - u_{0\varepsilon }^{\mu}(x)} \bigr\vert \,\mathrm{d}x} = \int _{\Omega}{\biggl\vert {\int_{0}^{t} {\frac {\partial}{{\partial s}}} u_{\varepsilon}^{\mu}\,\mathrm{d}x} \biggr\vert \,\mathrm{d}x} \\ &\le\sqrt{t} \int_{\Omega}{\biggl\vert {\sqrt{\int _{0}^{t} {{{ \biggl( {\frac{{\partial u_{\varepsilon}^{\mu}}}{{\partial s}}} \biggr)}^{2}}} \,\mathrm{d}t} } \biggr\vert \,\mathrm{d}x} \le{ \vert \Omega \vert ^{\frac {1}{2}}}\sqrt{t} \int_{\Omega}{\int _{0}^{t} {{{ \biggl( {\frac{{\partial u_{\varepsilon}^{\mu}}}{{\partial s}}} \biggr)}^{2}}} \,\mathrm{d}t\,\mathrm{d}x} \\ &\le\sqrt{t} {\vert \Omega \vert ^{\frac{1}{2}}}\,\mathrm{d}x\,\mathrm{d}t. \end{aligned}$$
(64)

It follows by (12) that

$$ \int_{\Omega}{\bigl\vert {u_{\varepsilon}^{\mu}(x,t) - u_{0\varepsilon}^{\mu}(x)} \bigr\vert \,\mathrm{d}x} \le C\sqrt{t}, $$
(65)

where C is independent of ε. Using (14) and letting \(\varepsilon \to0\) yields

$$ \int_{\Omega}{\bigl\vert {u^{\mu}(x,t) - u_{0}^{\mu}(x)} \bigr\vert \,\mathrm{d}x} \le C\sqrt{t}. $$
(66)

So

$$ \int_{\Omega}{\bigl\vert {u^{\mu}(x,t) - u_{0}^{\mu}(x)} \bigr\vert \,\mathrm{d}x} \to0 \quad \mbox{as } t \to0. $$
(67)

Thus the proof of existence is completed.