In this section, we would give the proof of Proposition 1 for fixed constant \(\varepsilon >0\). We assume that \(w_{\varepsilon I}\rightarrow w_{I}\) in \(H^{1}\)-norm as \(\varepsilon \rightarrow 0\). For convenience, we use the notation w to represent the weak solutions of (1.4)–(1.6).
For the purpose of existence, we define the following approximate solution:
$$\begin{aligned} U^{(n)}(x,t)=\sum_{i=1}^{n} \mathbb{S}_{i}(t)w_{i}(x) \end{aligned}$$
for
$$\mathbb{S}_{i}(t)=\textstyle\begin{cases} 1, & t\in ((i-1)h, ih]; \\ 0, & \text{elsewhere} \end{cases}\displaystyle \quad \text{with } i=1, \dots , n. $$
For \(U^{(n)}\), we can establish the uniform estimates as follows.
Lemma 2
There is uniform constant
C
such that
$$\begin{aligned} & \bigl\Vert U^{(n)} \bigr\Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}\leq C, \end{aligned}$$
(3.1)
$$\begin{aligned} & \bigl\Vert \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle U^{(n)} \bigr\vert ^{p} \bigr\Vert _{L^{ \infty }(0, T; L^{1}(\Omega ))}\leq C, \end{aligned}$$
(3.2)
$$\begin{aligned} & \bigl\Vert U^{(n)} \bigr\Vert _{L^{\infty }(0,T; W_{0}^{2, p}(\Omega ))}\leq \frac{1}{\varepsilon ^{\frac{\alpha }{p}}}C. \end{aligned}$$
(3.3)
Proof
For any time \(t\in (0,T]\), there exists some interval \(((i-1)h,ih]\) such that \(t\in ((i-1)h,ih]\), and then \(\|U^{(n)}(x, t)\|^{2}_{L^{2}(\Omega )}=\|w_{i}(x)\|^{2}_{L^{2}( \Omega )} \leq C \). So we have (3.1). Besides, estimate (2.8) can give
$$\begin{aligned} \biggl(\frac{1}{p} \int _{\Omega }\varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle U^{(n)} \bigr\vert ^{p}{\,\mathrm{d}}x \biggr) (t) = \frac{1}{p} \int _{\Omega } \varrho _{\varepsilon }^{\alpha } \vert \triangle w_{i} \vert ^{p}{\,\mathrm{d}}x \leq \frac{1}{p} \int _{\Omega }\varrho _{\varepsilon }^{\alpha } \vert \triangle w_{0} \vert ^{p}{ \,\mathrm{d}}x \leq C. \end{aligned}$$
It implies (3.2)–(3.3). □
Now we introduce another approximate solution
$$\begin{aligned} V^{(n)}(x, t)=\sum_{i=1}^{n} \mathbb{S}_{i}(t) \bigl(\Theta _{i}(t)w_{i}(x)+ \bigl(1- \Theta _{i}(t)\bigr)w_{i-1}(x) \bigr) \end{aligned}$$
(3.4)
with
$$\begin{aligned} \Theta _{i}(t)=\textstyle\begin{cases} \frac{t}{h}-(i-1), & \text{if } t\in ((i-1)h, ih], \\ 0, & \text{otherwise }. \end{cases}\displaystyle \end{aligned}$$
For \(V^{(n)}\), we establish the estimates as follows.
Lemma 3
There is a constant
C
such that
$$\begin{aligned} \bigl\Vert V^{(n)}_{t} \bigr\Vert _{L^{2}(0, T; H^{1}(\Omega ))} + \bigl\Vert V^{(n)} \bigr\Vert _{L^{\infty }(0, T; W_{0}^{2, p}(\Omega ))}\leq C. \end{aligned}$$
Proof
By using \(\frac{\partial V^{(n)}}{\partial t}=\frac{1}{h}\sum_{i=1}^{n} \mathbb{S}_{i}(w_{i}-w_{i-1}) \) and (2.8), we have
$$\begin{aligned} \biggl\Vert \frac{\partial \nabla V^{(n)}}{\partial t} \biggr\Vert _{L^{2}(Q_{T})}^{2} = \frac{1}{h^{2}}\sum_{i=1}^{n}h \int _{\Omega }(\nabla w_{i}-\nabla w_{i-1})^{2}{ \,\mathrm{d}}x \leq C. \end{aligned}$$
For \(t\in [0, T]\), there is a positive integer i satisfying \(t\in ((i-1)h, ih]\). Thus, (2.7) gives
$$\begin{aligned} & \biggl( \int _{\Omega } \bigl\vert \triangle V^{(n)} \bigr\vert ^{p}{\,\mathrm{d}}x \biggr) (t) \\ & \quad = \int _{\Omega } \bigl\vert \bigl(\Theta _{i}(t) \triangle w_{i}(x)+\bigl(1-\Theta _{i}(t)\bigr) \triangle w_{i-1}(x)\bigr) \bigr\vert ^{p}{\,\mathrm{d}}x \\ &\quad \leq C_{1} \int _{\Omega } \bigl\vert \triangle w_{i}(x) \bigr\vert ^{p}{\,\mathrm{d}}x+C_{2} \int _{\Omega } \bigl\vert \triangle w_{i-1}(x) \bigr\vert ^{p}{\,\mathrm{d}}x \\ & \quad \leq C. \end{aligned}$$
It shows the estimate in \(L^{\infty }(0, T; W_{0}^{2, p}(\Omega ))\). □
Next we give the proof of Proposition 1.
Proof of Proposition 1
Lemma 2 can ensure the existence of a subsequence of \(U^{(n)}\) (we always take the same notation) and two functions \(w\in L^{\infty }(0,T; W_{0}^{2, p}(\Omega ))\) and \(v\in L^{\frac{p}{p-1}}(Q_{T})\) such that
$$\begin{aligned} & U^{(n)}\stackrel{*}{\rightharpoonup } w \quad \text{weakly* in } L^{\infty }\bigl(0,T; W_{0}^{2, p}(\Omega )\bigr), \\ & \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle U^{(n)} \bigr\vert ^{p-2}\triangle U^{(n)} \rightharpoonup \varrho _{\varepsilon }^{\alpha }v \quad \text{weakly in } L^{ \frac{p}{p-1}}(Q_{T}), \end{aligned}$$
as \(n\rightarrow \infty \). Besides, from Lemma 3, we can find a subsequence of \(V^{(n)}\) and a function ϖ such that
$$\begin{aligned} & \frac{\partial V^{(n)}}{\partial t}\rightharpoonup \frac{\partial \varpi }{\partial t} \quad \text{weakly in }L^{2}\bigl(0, T; H^{1}( \Omega )\bigr), \\ & V^{(n)}\stackrel{*}{\rightharpoonup }\varpi\quad \text{weakly* in }L^{ \infty }\bigl(0, T; W_{0}^{2, p}(\Omega )\bigr), \\ & V^{(n)}\rightarrow \varpi \quad \text{strongly in } L^{2} \bigl(0, T; H^{1}( \Omega )\bigr), \\ & V^{(n)}\rightarrow \varpi\quad \text{a.e. in } Q_{T}. \end{aligned}$$
On the other hand, for any \(\varphi \in C_{0}^{\infty }(Q_{T})\), we have
$$\begin{aligned} & \int _{0}^{T} \int _{\Omega } \bigl\vert \bigl(\nabla U^{(n)}-\nabla V^{(n)}\bigr) \bigr\vert ^{2}{ \,\mathrm{d}}x{\,\mathrm{d}}t \\ &\quad = \int _{0}^{T} \int _{\Omega } \Biggl\vert \sum_{i=1}^{n} \mathbb{S}_{i}(t) \bigl(1- \Theta _{i}(t)\bigr) (\nabla w_{k}-\nabla w_{k-1}) \Biggr\vert ^{2}{ \,\mathrm{d}}x{\,\mathrm{d}}t \\ & \quad \leq \sum_{i=1}^{n} \int _{(i-1)h}^{ih} \int _{\Omega } \bigl\vert (\nabla w_{i}- \nabla w_{i-1}) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ &\quad \leq CTh \rightarrow 0 \end{aligned}$$
as \(n\rightarrow \infty \) (i.e. \(h\rightarrow 0\)). It implies \(w=\varpi \) a.e. in \(Q_{T}\) and
$$\begin{aligned} & U^{(n)}\rightarrow w \quad \text{strongly in } L^{2}\bigl(0, T; H^{1}(\Omega )\bigr), \\ & U^{(n)}\rightarrow w \quad \text{a.e. in } Q_{T}. \end{aligned}$$
If we perform the limit \(n\rightarrow \infty \) in the expression
$$\begin{aligned} & \iint _{Q_{T}}\frac{\partial V^{(n)}}{\partial t}\varphi {\,\mathrm{d}}x{ \,\mathrm{d}}t+\gamma \iint _{Q_{T}}\nabla V^{(n)}_{t}\nabla \varphi {\,\mathrm{d}}x{ \,\mathrm{d}}t \\ &\quad {} + \iint _{Q_{T}}\varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle U^{(n)} \bigr\vert ^{p-2} \triangle U^{(n)} \triangle \varphi {\,\mathrm{d}}x{\,\mathrm{d}}t =0, \end{aligned}$$
(3.5)
then we have
$$\begin{aligned} \iint _{Q_{T}}\frac{\partial w}{\partial t}\varphi {\,\mathrm{d}}x{\,\mathrm{d}}t+ \gamma \iint _{Q_{T}} \nabla w_{t}\nabla \varphi \,\mathrm{d}x \,\mathrm{d}t + \iint _{Q_{T}} \varrho _{\varepsilon }^{\alpha }v\triangle \varphi {\,\mathrm{d}}x{\,\mathrm{d}}t=0 \end{aligned}$$
(3.6)
for any \(\varphi \in C_{0}^{\infty }(Q_{T})\).
The next job is to prove \(v=|\triangle w|^{p-2}\triangle w\). For each test function \(\psi \in C_{0}^{\infty }(0, T)\), we define \(\varphi =\psi w\) as the multiplier in (3.6) to get
$$\begin{aligned} & -\frac{1}{2} \iint _{Q_{T}}w^{2}\frac{{\,\mathrm{d}}\psi }{{\,\mathrm{d}}t}{\,\mathrm{d}}x{ \,\mathrm{d}}t-\frac{\gamma }{2} \iint _{Q_{T}} \vert \nabla w \vert ^{2} \frac{{\,\mathrm{d}}\psi }{{\,\mathrm{d}}t}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ & \quad {}+ \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha }v\triangle w{ \,\mathrm{d}}x{\,\mathrm{d}}t =0. \end{aligned}$$
(3.7)
In (2.4), we use \(\psi (t)w_{i}\) as the test function to give
$$\begin{aligned} & \frac{1}{h} \int _{\Omega }\psi w_{i}^{2}{\,\mathrm{d}}x+ \frac{\gamma }{h} \int _{\Omega }\psi \nabla w_{i}^{2}{ \,\mathrm{d}}x + \int _{\Omega }\psi \varrho _{\varepsilon }^{\alpha } \vert \triangle w_{i} \vert ^{p}{\,\mathrm{d}}x \\ &\quad =\frac{1}{h} \int _{\Omega }\psi w_{i-1}w_{i}{\,\mathrm{d}}x +\frac{\gamma }{h} \int _{\Omega }\psi \nabla w_{i-1}\nabla w_{i}{ \,\mathrm{d}}x \\ & \quad \leq \frac{1}{2h} \int _{\Omega }\psi w_{i}^{2}{\,\mathrm{d}}x+ \frac{1}{2h} \int _{\Omega }\psi w_{i-1}^{2}{\,\mathrm{d}}x+ \frac{\gamma }{2h} \int _{ \Omega }\psi \nabla w_{i}^{2}{ \,\mathrm{d}}x+\frac{\gamma }{2h} \int _{\Omega } \psi \nabla w_{i-1}^{2}{ \,\mathrm{d}}x. \end{aligned}$$
That becomes
$$\begin{aligned} &\frac{1}{2h} \int _{\Omega }\psi w_{i}^{2}{\,\mathrm{d}}x- \frac{1}{2h} \int _{ \Omega }\psi w_{i-1}^{2}{\,\mathrm{d}}x + \frac{\gamma }{2h} \int _{\Omega }\psi \nabla w_{i}^{2}{ \,\mathrm{d}}x-\frac{\gamma }{2h} \int _{\Omega }\psi \nabla w_{i-1}^{2}{ \,\mathrm{d}}x \\ & \quad {}+ \int _{\Omega }\psi \varrho _{\varepsilon }^{\alpha } \vert \triangle w_{i} \vert ^{p}{ \,\mathrm{d}}x \leq 0. \end{aligned}$$
By introducing the notation \(\tilde{U}^{(n)}(x,t)=\sum_{i=1}^{n}\mathbb{S}_{i}(t)w_{i-1}(x) \), we have
$$\begin{aligned} & \frac{1}{2h} \iint _{Q_{T}}\psi \bigl\vert U^{(n)} \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t- \frac{1}{2h} \iint _{Q_{T}}\psi \bigl\vert \tilde{U}^{(n)} \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t + \frac{\gamma }{2h} \iint _{Q_{T}}\psi \bigl\vert \nabla U^{(n)} \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ &\qquad {} -\frac{\gamma }{2h} \iint _{Q_{T}}\psi \bigl\vert \nabla \tilde{U}^{(n)} \bigr\vert ^{2}{ \,\mathrm{d}}x{\,\mathrm{d}}t + \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle U^{(n)} \bigr\vert ^{p}{\,\mathrm{d}}x{ \,\mathrm{d}}t \\ & \quad \leq 0. \end{aligned}$$
(3.8)
For any function \(\varphi _{1}\in C_{0}^{\infty }(Q_{T})\), we can seek two constants \(t_{1}\) and \(t_{2}\) with \(0< t_{1}< t_{2}< T\) such that \(\operatorname{supp}\varphi _{1}, \operatorname{supp}\triangle \varphi _{1}\subset (t_{1}, t_{2})\times \Omega \). Meanwhile, we redefine ψ as \(\psi \equiv 1\) on \((t_{1}, t_{2})\) and \(\psi \equiv 0\) on \([0, h)\cup (T-h, T]\) for small h (\(h< t_{1}\) and \(T-h>t_{2}\)). Now a direct computation gives
Similarly, one has
$$\begin{aligned} & \int _{0}^{T}\psi \bigl\vert \tilde{\nabla U}^{(n)} \bigr\vert ^{2}{\,\mathrm{d}}t = \int _{0}^{T-h} \psi (t+h) \bigl\vert \nabla U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}t. \end{aligned}$$
Therefore, we have
$$\begin{aligned} & \frac{1}{2h} \iint _{Q_{T}}\psi \bigl\vert U^{(n)} \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t- \frac{1}{2h} \iint _{Q_{T}}\psi \bigl\vert \tilde{U}^{(n)} \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ & \qquad {}+\frac{\gamma }{2h} \iint _{Q_{T}}\psi \bigl\vert \nabla U^{(n)} \bigr\vert ^{2}{\,\mathrm{d}}x{ \,\mathrm{d}}t-\frac{\gamma }{2h} \iint _{Q_{T}}\psi \bigl\vert \nabla \tilde{U}^{(n)} \bigr\vert ^{2}{ \,\mathrm{d}}x{\,\mathrm{d}}t \\ &\quad =\frac{1}{2h} \int _{T-h}^{T} \int _{\Omega }\psi (t) \bigl\vert U^{(n)}(x, t) \bigr\vert ^{2}{ \,\mathrm{d}}x{\,\mathrm{d}}t+\frac{1}{2h} \int _{0}^{T-h} \int _{\Omega }\psi (t) \bigl\vert U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ & \qquad {}-\frac{1}{2h} \int _{0}^{T-h} \int _{\Omega }\psi (t+h) \bigl\vert U^{(n)}(x, t) \bigr\vert ^{2}{ \,\mathrm{d}}x{\,\mathrm{d}}t \\ &\qquad {} +\frac{\gamma }{2h} \int _{T-h}^{T} \int _{\Omega }\psi (t) \bigl\vert \nabla U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t+\frac{\gamma }{2h} \int _{0}^{T-h} \int _{\Omega } \psi (t) \bigl\vert \nabla U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ & \qquad {}-\frac{\gamma }{2h} \int _{0}^{T-h} \int _{\Omega }\psi (t+h) \bigl\vert \nabla U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ &\quad =-\frac{1}{2} \int _{0}^{T-h} \int _{\Omega } \frac{\psi (t+h)-\psi (t)}{h} \bigl\vert U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ &\qquad {} -\frac{\gamma }{2} \int _{0}^{T-h} \int _{\Omega } \frac{\psi (t+h)-\psi (t)}{h} \bigl\vert \nabla U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t. \end{aligned}$$
(3.8) implies
$$\begin{aligned} & -\frac{1}{2} \int _{0}^{T-h} \int _{\Omega } \frac{\psi (t+h)-\psi (t)}{h} \bigl\vert U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ & \qquad {}-\frac{\gamma }{2} \int _{0}^{T-h} \int _{\Omega } \frac{\psi (t+h)-\psi (t)}{h} \bigl\vert \nabla U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t + \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle U^{(n)} \bigr\vert ^{p}{ \,\mathrm{d}}x{ \,\mathrm{d}}t \\ & \quad \leq 0. \end{aligned}$$
(3.9)
If we choose \(\zeta =\triangle U^{(n)}\) and \(\eta =\triangle (w-\lambda \varphi _{1}))\) with \(\lambda >0\) and \(\varphi _{1}\in C_{0}^{\infty }(Q_{T})\) in (2.12), then we have
$$\begin{aligned} & \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle U^{(n)} \bigr\vert ^{p}{ \,\mathrm{d}}x{ \,\mathrm{d}}t \\ &\quad \geq - \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle (w- \lambda \varphi _{1}) \bigr\vert ^{p}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ &\qquad {} + \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle U^{(n)} \bigr\vert ^{p-2} \triangle U^{(n)}\triangle (w-\lambda \varphi _{1}){\,\mathrm{d}}x{ \,\mathrm{d}}t \\ &\qquad {} + \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle (w- \lambda \varphi _{1}) \bigr\vert ^{p-2} \triangle (w-\lambda \varphi _{1}) \triangle U^{(n)}{ \,\mathrm{d}}x{\,\mathrm{d}}t. \end{aligned}$$
(3.10)
Use (3.9) and (3.10) to get
$$\begin{aligned} & -\frac{1}{2} \int _{0}^{T-h} \int _{\Omega } \frac{\psi (t+h)-\psi (t)}{h} \bigl\vert U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ &\qquad {} -\frac{\gamma }{2} \int _{0}^{T-h} \int _{\Omega } \frac{\psi (t+h)-\psi (t)}{h} \bigl\vert \nabla U^{(n)}(x, t) \bigr\vert ^{2}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ &\qquad {} - \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle (w- \lambda \varphi _{1}) \bigr\vert ^{p}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ & \qquad {}+ \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle U^{(n)} \bigr\vert ^{p-2} \triangle U^{(n)}\triangle (w-\lambda \varphi _{1}){\,\mathrm{d}}x{ \,\mathrm{d}}t \\ & \qquad {}+ \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle (w- \lambda \varphi _{1}) \bigr\vert ^{p-2} \triangle (w-\lambda \varphi _{1}) \triangle U^{(n)}{ \,\mathrm{d}}x{\,\mathrm{d}}t \\ & \quad \leq 0. \end{aligned}$$
(3.11)
By letting \(n\rightarrow \infty \) (i.e. \(h\rightarrow 0\)), we have
$$\begin{aligned} & -\frac{1}{2} \int _{0}^{T} \int _{\Omega }w^{2} \frac{{\,\mathrm{d}}\psi }{{\,\mathrm{d}}t}{\,\mathrm{d}}x{ \,\mathrm{d}}t -\frac{\gamma }{2} \int _{0}^{T} \int _{\Omega } \vert \nabla w \vert ^{2} \frac{{\,\mathrm{d}}\psi }{{\,\mathrm{d}}t}{\,\mathrm{d}}x{\,\mathrm{d}}t \\ &\qquad {} - \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle (w- \lambda \varphi _{1}) \bigr\vert ^{p}{\,\mathrm{d}}x{\,\mathrm{d}}t + \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha }v\triangle (w-\lambda \varphi _{1}){ \,\mathrm{d}}x{\,\mathrm{d}}t \\ & \qquad {}+ \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle (w- \lambda \varphi _{1}) \bigr\vert ^{p-2} \triangle (w-\lambda \varphi _{1}) \triangle w{\,\mathrm{d}}x{\,\mathrm{d}}t \\ &\quad \leq 0. \end{aligned}$$
(3.12)
Apply (3.7) and (3.12) to get
$$\begin{aligned} & - \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle (w- \lambda \varphi _{1}) \bigr\vert ^{p}{\,\mathrm{d}}x{\,\mathrm{d}}t -\lambda \iint _{Q_{T}} \psi \varrho _{\varepsilon }^{\alpha }v \triangle \varphi _{1}{\,\mathrm{d}}x{ \,\mathrm{d}}t \\ & \qquad {}+ \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl\vert \triangle (w- \lambda \varphi _{1}) \bigr\vert ^{p-2} \triangle (w-\lambda \varphi _{1}) \triangle w{\,\mathrm{d}}x{\,\mathrm{d}}t \\ & \quad \leq 0. \end{aligned}$$
Therefore, we have
$$\begin{aligned} \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl[ \bigl\vert \triangle (w- \lambda \varphi _{1}) \bigr\vert ^{p-2}\triangle (w-\lambda \varphi _{1})-v \bigr] \triangle \varphi _{1}{\,\mathrm{d}}x{\,\mathrm{d}}t \leq 0. \end{aligned}$$
We pass to the limit \(\lambda \rightarrow 0\) to get
$$\begin{aligned} \iint _{Q_{T}}\psi \varrho _{\varepsilon }^{\alpha } \bigl[ \vert \triangle w \vert ^{p-2} \triangle w-v \bigr]\triangle \varphi _{1}{\,\mathrm{d}}x{\,\mathrm{d}}t \leq 0. \end{aligned}$$
Finally, the arbitrariness of \(\varphi _{1}\) and ψ implies \(v=|\triangle w|^{p-2}\triangle w \text{ a.e. in } Q_{T} \). Thus, (3.6) becomes
$$\begin{aligned} & \iint _{Q_{T}}\frac{\partial w}{\partial t}\varphi {\,\mathrm{d}}x{\,\mathrm{d}}t+ \gamma \iint _{Q_{T}} \nabla w_{t} \nabla \varphi { \,\mathrm{d}}x{\,\mathrm{d}}t + \iint _{Q_{T}}\varrho _{\varepsilon }^{\alpha } \vert \triangle w \vert ^{p-2} \triangle w\triangle \varphi {\,\mathrm{d}}x{ \,\mathrm{d}}t \\ &\quad =0. \end{aligned}$$
(3.13)
For other estimates in Proposition 1, we may apply J. Simon’s lemma (see [15]) and Sobolev’s embedding theorem (see [1] and [5]), and so we omit the details. The uniqueness can be shown as the corresponding proof of Lemma 1. □