Let us first introduce the concept of BV function according to Ref. [19].
Definition 2.1
Let \(\Omega\subset\mathbb {R}^{m}\) be an open set and let \(f\in L^{1}(\Omega)\). Define
$$\begin{aligned}& \int_{\Omega} \vert Df\vert =\sup \biggl\{ \int_{\Omega}f \operatorname {div}g\,dx: g=(g_{1}, g_{2}, \ldots, g_{N})\in C^{1}_{0} \bigl(\Omega; \mathbb{R}^{m} \bigr), \bigl\vert g(x) \bigr\vert \leq1, x\in\Omega \biggr\} , \end{aligned}$$
(2.1)
where \(\operatorname {div}g=\sum_{i=1}^{m}\frac{\partial g_{i}}{\partial x_{i}}\).
Definition 2.2
A function of \(f\in L^{1}(\Omega)\) is said to have a bounded variation in Ω if
$$\int_{\Omega} \vert Df\vert < \infty. $$
We define \(\operatorname {BV}(\Omega)\) as the space of all functions in \(L^{1}(\Omega)\) with bounded variation.
This is equivalent to the idea that the generalized derivatives of every function in \(\operatorname {BV}(\Omega)\) are regular measures on Ω. Under the norm
$$\Vert f\Vert _{\mathrm{BV}}=\Vert f\Vert _{L^{1}}+ \int_{\Omega} \vert Df\vert , $$
\(\operatorname {BV}(\Omega)\) is a Banach space.
Proposition 2.3
(Semicontinuity)
Let
\(\Omega \subseteq \mathbb{R}^{m}\)
be an open set and
\(\{f_{j}\}\)
a sequence of functions in
\(\operatorname {BV}(\Omega)\)
which converge in
\(L^{1}_{\mathit {loc}}(\Omega)\)
to a function
f. Then
$$\int_{\Omega} \vert Df\vert \leq\lim_{j\rightarrow\infty} \inf \int_{\Omega} \vert Df_{j}\vert . $$
Proposition 2.4
(Integration by part)
Let
$$C^{+}_{R}=\mathscr{B}(0,R)\times(0,R)= \mathscr{B}_{R}\times(0,R) $$
and
\(f\in \operatorname {BV}(C^{+}_{R})\). Then there exists a function
\(f^{+}\in L^{1}(\mathscr{B}_{R})\)
such that for
\(H_{m-1}\)-almost all
\(y\in \mathscr{B}_{R}\),
$$\lim_{\rho\rightarrow 0}\rho^{-m} \int_{C^{+}_{\rho}(y)} \bigl\vert f(z)-f^{+}(y) \bigr\vert \,dz=0. $$
Moreover, if
\(C_{R}=\mathscr{B}_{R}\times(-R, R)\), then for every
\(g\in C_{0}^{1}(C_{R}; \mathbb{R}^{m})\),
$$\begin{aligned}& \int_{C^{+}_{R}}f\operatorname {div}g\,dx=- \int_{C^{+}_{R}}\langle g, Df\rangle+ \int_{\mathscr{B}_{R}}f^{+}g\,dH_{m-1}, \end{aligned}$$
(2.2)
where
\(\mathscr{B}_{\rho}=\{x\in\mathbb{R}^{m};\vert x\vert <\rho \}\).
Remark 2.5
The function \(f^{+}\) is called the trace of f on \(\mathscr{B}_{R}\) and obviously
$$\begin{aligned}& f^{+}(y)=\lim_{\rho\rightarrow 0}\frac{1}{\vert C^{+}_{\rho}(y)\vert } \int_{C^{+}_{\rho}(y)}f(z)\,dz. \end{aligned}$$
(2.3)
In our paper, we consider the solution of equation (1.1) in \(\operatorname {BV}(Q_{T})\), where \(Q_{T}=\Omega\times(0,T)\), and the dimension of \(Q_{T}\) is \(m=N+1\).
Let \(\Gamma_{u}\) be the set of all jump points of \(u\in \operatorname {BV}(Q_{T})\), \(v=(v_{1}, v_{2}, \ldots, v_{N}, v_{N+1})\) be the normal of \(\Gamma_{u}\) at \(X=(x,t)\), \(u^{+}(X)\) and \(u^{-}(X)\) be the approximate limits of u at \(X\in\Gamma_{u}\) with respect to \((v,Y-X)>0\) and \((v,Y-X)<0\), respectively. For the continuous function \(p(u,x,t)\) and \(u\in \operatorname {BV}(Q_{T})\), define
$$ \widehat{p}(u,x,t)= \int_{0}^{1}p \bigl(\tau u^{+}+(1- \tau)u^{-},x,t \bigr)\,d\tau, $$
(2.4)
which is called the composite mean value of p. For a given t, we denote by \(\Gamma_{u}^{t}\), \(H^{t}\), \((v_{1}^{t}, \ldots,v_{N}^{t})\), and \(u_{\pm}^{t}\) all jump points of \(u(\cdot,t)\), the Hausdorff measure of \(\Gamma_{u}^{t}\), the unit normal vector of \(\Gamma_{u}^{t}\), and the asymptotic limit of \(u(\cdot,t)\), respectively. Moreover, if \(f(s)\in C^{1}( \mathbb{R})\), \(u\in \operatorname {BV}(Q_{T})\), then \(f(u)\in \operatorname {BV}(Q_{T})\) and
$$ \frac{\partial f(u)}{\partial x_{i}}=\widehat{f^{\prime }}(u)\frac{\partial u}{\partial x_{i}}, \quad i=1,2, \ldots, N. $$
(2.5)
Lemma 2.6
([20])
Assume that
\(\Omega\subset \mathbb{R}^{N}\)
is an open bounded set and let
\(f_{k}\), \(f\in L^{q}(\Omega)\), as
\(k\rightarrow\infty\), \(f_{k}\rightharpoonup f\)
weakly in
\(L^{q}(\Omega)\), \(1\leq q<\infty\). Then
$$ \lim_{k\rightarrow\infty}\inf \Vert f_{k}\Vert _{L^{q}(\Omega)}^{q}\geq \Vert f\Vert _{L^{q}(\Omega )}^{q}. $$
The solution of our problem will be obtained as a limit point of the family \(\{u_{\varepsilon}\}\) of solutions of the regularized problem
$$\begin{aligned}& \frac{\partial u}{\partial t} =\frac{\partial}{\partial x_{i}} \biggl(a^{ij}(u,x,t) \frac{\partial u}{\partial x_{j}} \biggr)+\varepsilon\Delta u+\frac {\partial b_{i}(u,x,t)}{\partial x_{i}},\quad \text{in } Q_{T}, \end{aligned}$$
(2.6)
with the compatible initial-boundary values (1.5)-(1.6).
Lemma 2.7
([3])
Let
\(u_{\varepsilon}\)
be the solution of equation (2.6) with initial-boundary values (1.5)-(1.6). If the assumptions of Theorem
1.2
are true, then
$$\begin{aligned}& \varepsilon \int_{\Sigma} \biggl\vert \frac{\partial u_{\varepsilon}}{\partial n} \biggr\vert \,d \sigma \leq c_{1}+c_{2} \biggl(\vert \nabla u_{\varepsilon} \vert _{L^{1}(\Omega)}+ \biggl\vert \frac{\partial u_{\varepsilon}}{\partial t} \biggr\vert _{L^{1}(\Omega)} \biggr), \end{aligned}$$
(2.7)
with constants
\(c_{i}\), \(i=1, 2\)
independent of
ε.
Under the assumptions of A, \(b_{i}\) and \(u_{0}\) in Theorem 1.2, it is well known that there is a classical solution \(u_{\varepsilon}\) of the initial-boundary values problem (2.6)-(1.5)-(1.6), e.g. one may refer to Chapter 8 of [21].
We need to make some estimates for \(u_{\varepsilon}\). First of all, by the maximum principle, we have
$$\vert u_{\varepsilon} \vert \leq \Vert u_{0}\Vert _{L^{\infty }} \leq c. $$
Second, let us make the BV estimates on \(u_{\varepsilon}\).
Theorem 2.8
Let
\(u_{\varepsilon}\)
be the solution of equation (2.6) with initial-boundary conditions (1.5)-(1.6). If the assumptions of Theorem
1.2
are true, then
$$\vert \operatorname {grad}u_{\varepsilon} \vert _{L^{1}(\Omega)}\leq c, $$
where
\(\vert \operatorname {grad}u\vert ^{2}=\sum_{i=1}^{N}\vert \frac{\partial u}{\partial x_{i}}\vert ^{2}+\vert \frac{\partial u}{\partial t}\vert ^{2}\), c
is independent of
ε, and independent of
t.
Proof
Differentiate (2.6) with respect to \(x_{s}\), \(s=1, 2, \ldots, N, N+1\), \(x_{N+1}=t\), and sum up for s after multiplying the resulting relation by \(u_{\varepsilon x_{s}}\frac{S_{\eta}(\vert \operatorname {grad}u_{\varepsilon} \vert )}{\vert \operatorname {grad}u_{\varepsilon} \vert }\). In the following, we simply denote \(u_{\varepsilon}\) by u. Integrating over Ω yields
$$\begin{aligned}& \int_{\Omega}\frac{\partial u_{x_{s}}}{\partial t}u_{x_{s}} \frac {S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert } \,dx \,dx =\frac{d}{dt} \int_{\Omega}I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr) \,dx, \end{aligned}$$
(2.8)
$$\begin{aligned}& \int_{\Omega}\frac{\partial}{\partial x_{s}} \biggl[\frac{\partial}{\partial x_{i}} \biggl(a^{ij}(u,x,t)\frac{\partial u}{\partial x_{j}} \biggr) \biggr]u_{x_{s}} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert }\,dx \\& \quad= \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(a^{ij}_{u}(u,x,t)u_{x_{j}}u_{x_{s}}+a^{ij}_{x_{s}}(u,x,t)u_{x_{j}} \bigr)u_{ x_{s}} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert }\,dx \\& \quad\quad{} + \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(a^{ij}(u,x,t)u_{x_{j}x_{s}} \bigr)u_{x_{s}} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert }\,dx, \end{aligned}$$
(2.9)
and, moreover, every term in the right-hand side of (2.9) can be handled as (2.10)-(2.12), respectively,
$$\begin{aligned}& \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(a^{ij}_{u}(u,x,t)u_{x_{j}}u_{x_{s}} \bigr)u_{ x_{s}} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert }\,dx \\& \quad= \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(a^{ij}_{u}(u,x,t)u_{x_{j}} \bigr) \bigl[\vert \operatorname {grad}u\vert S_{\eta } \bigl(\vert \operatorname {grad}u\vert \bigr)-I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr) \bigr]\,dx \\& \quad\quad{} - \int_{\Sigma}a^{ij}_{u}(u,x,t)u_{x_{i}}n_{j}I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr)\,d\sigma, \end{aligned}$$
(2.10)
$$\begin{aligned}& \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(a^{ij}_{x_{s}}(u,x,t)u_{x_{j}} \bigr)u_{x_{s}} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert }\,dx \\& \quad= \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(a^{ij}_{x_{s}}(u,x,t)u_{x_{j}} \bigr)\frac{\partial}{\partial\xi _{s}}I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr)\,dx \\& \quad=- \int_{\Sigma}a^{ij}_{x_{s}}(u,x,t)u_{x_{j}}n_{i} \frac{\partial }{\partial\xi_{s}}I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr)\,d\sigma \\& \quad\quad{} - \int_{\Omega}a^{ij}_{x_{s}}(u,x,t)u_{x_{j}} \frac{\partial ^{2}I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial\xi_{s}\,\partial\xi _{p}} u_{x_{p}x_{i}}\,dx, \end{aligned}$$
(2.11)
$$\begin{aligned}& \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(a^{ij}(u,x,t)u_{x_{j}x_{s}} \bigr)u_{x_{s}} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert }\,dx \\& \quad= \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(a^{ij}(u,x,t)u_{x_{j}x_{s}} \bigr)\frac{\partial}{\partial\xi _{s}}I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr)\,dx \\& \quad=- \int_{\Sigma}a^{ij}(u,x,t)u_{x_{i}x_{s}}n_{j} \frac{\partial }{\partial\xi_{s}}I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr)\,d\sigma \\& \quad\quad{} - \int_{\Omega}a^{ij}(u,x,t)\frac{\partial^{2}I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial\xi_{s}\,\partial\xi_{p}} u_{x_{s}x_{i}}u_{x_{p}x_{j}}\,dx, \end{aligned}$$
(2.12)
where \(\{n_{i}\}_{i=1}^{N}\) is the inner normal vector of Ω, \(\xi_{s}=u_{x_{s}}\). At the same time,
$$\begin{aligned}& \varepsilon \int_{\Omega}\Delta u_{x_{s}}u_{ x_{s}} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert }\,dx \\& \quad =-\varepsilon \int_{\Sigma}\frac{\partial I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial x_{i}}n_{i}\,d\sigma- \varepsilon \int_{\Omega}\frac{\partial^{2}I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial\xi_{s}\,\partial\xi_{p}} u_{x_{s}x_{i}}u_{x_{p}x_{i}} \, dx \end{aligned}$$
(2.13)
and
$$\begin{aligned}& \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl[b_{iu}(u,x,t)u_{x_{s}}+b_{ix_{s}}(u,x,t) \bigr]u_{ x_{s}}\frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert }\,dx \\& \quad= \int_{\Omega}\frac{\partial (b_{iu}(u,x,t)u_{x_{s}})}{\partial x_{i}}u_{ x_{s}} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert } \,dx + \int_{\Omega}\frac{\partial b_{ix_{s}}(u,x,t)}{\partial x_{i}}u_{ x_{s}} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert } \,dx, \end{aligned}$$
here we have \(b_{ix_{s}}(u,x,t)=\frac{\partial b_{i}(u,x,t)}{\partial x_{s}}\), \(b_{iu}(u,x,t)=\frac{\partial b_{i}(u,x,t)}{\partial u}\), and
$$\begin{aligned}& \int_{\Omega}\frac{\partial b_{iu}(u,x,t)u_{x_{s}}}{\partial x_{i}}u_{ x_{s}} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert } \,dx \\& \quad= \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(b_{iu}(u,x,t) \bigr) \vert \operatorname {grad}u\vert S_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr) \,dx + \int_{\Omega}b_{iu}(u,x,t)\frac{\partial I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial x_{i}}\,dx \\& \quad= \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(b_{iu}(u,x,t) \bigr) \bigl[\vert \operatorname {grad}u\vert S_{\eta} \bigl(\vert \operatorname {grad}u \vert \bigr)-I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr) \bigr]\,dx \\& \quad\quad{} - \int_{\Sigma}b_{iu}(u,x,t)I_{\eta} \bigl( \vert \operatorname {grad}u\vert \bigr)n_{i}\,d\sigma. \end{aligned}$$
(2.14)
From (2.8)-(2.14), by the assumption \(a^{ij}(0,x,t)=0\), and so
$$a^{ij}_{x_{s}}(0,x,t)=0,\quad(x,t)\in Q_{T}, $$
we have
$$\begin{aligned}& \frac{d}{dt} \int_{\Omega}I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr) \,dx \\& \quad= \int_{\Omega }\frac{\partial}{\partial x_{i}} \bigl(a^{ij}_{u}(u,x,t)u_{x_{j}} \bigr) \bigl[\vert \operatorname {grad}u\vert S_{\eta } \bigl(\vert \operatorname {grad}u\vert \bigr)-I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr) \bigr]\,dx \\& \quad\quad{} - \int_{\Omega}a^{ij}(u,x,t)\frac{\partial^{2}I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial\xi_{s}\,\partial\xi_{p}} u_{x_{s}x_{i}}u_{x_{p}x_{j}}\,dx-\varepsilon \int_{\Omega}\frac {\partial^{2}I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial\xi_{s}\, \partial\xi_{p}} u_{x_{s}x_{i}}u_{x_{p}x_{i}} \,dx \\& \quad\quad{}+ \int_{\Omega}\frac{\partial}{\partial x_{i}} \bigl(b_{iu}(u,x,t) \bigr) \bigl[\vert \operatorname {grad}u\vert S_{\eta} \bigl(\vert \operatorname {grad}u \vert \bigr)-I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr) \bigr]\,dx \\& \quad\quad{}- \int_{\Sigma}a^{ij}_{u}(u,x,t)u_{x_{i}}n_{j}I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr)\,d\sigma- \int_{\Sigma }a^{ij}_{x_{s}}(u,x,t)u_{x_{j}}u_{x_{s}}n_{i} \frac{S_{\eta}(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert }\,dx \\& \quad\quad{}- \int_{\Sigma}b_{iu}(u,x,t)I_{\eta} \bigl( \vert \operatorname {grad}u\vert \bigr)n_{i}\,d\sigma \\& \quad\quad{}- \int_{\Sigma}a^{ij}(u,x,t)\frac{\partial I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial x_{j}}n_{i} \,d\sigma -\varepsilon \int_{\Sigma}\frac{\partial I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial x_{i}}n_{i}\,d \sigma. \end{aligned}$$
(2.15)
Now, if we set
$$\left ( \textstyle\begin{array}{c} v_{1}^{i}\\ v_{2}^{i}\\ \vdots\\ v_{N+1}^{i} \end{array}\displaystyle \right ) =\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} q^{1 1}&q^{1 2}&\cdots&q_{1 N+1}\\ q^{2 1}&q^{2 2}&\cdots&q_{2 N+1}\\ \vdots\\ q^{N+1 1}&q^{N+1 2}&\cdots&q_{N=1 N+1} \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{c} u_{x_{1}x_{i}}\\ u_{x_{2}x_{i}}\\ \vdots\\ u_{x_{N+1}x_{i}} \end{array}\displaystyle \right ), $$
where \((q^{sp})\) is the square root of \((\frac{\partial ^{2}I_{\eta}}{\partial\xi_{s}\,\partial\xi_{p}} )\), then
$$\begin{aligned} \biggl\vert a^{ij}_{x_{s}}u_{x_{j}} \frac{\partial^{2}I_{\eta}}{\partial\xi _{s}\,\partial\xi_{p}}u_{x_{p}x_{i}} \biggr\vert =& \left \vert \bigl(a^{ij}_{x_{1}}u_{x_{j}},a^{ij}_{x_{2}}u_{x_{j}}, \ldots,a^{ij}_{x_{N+1}}u_{x_{j}} \bigr) \bigl(q^{sp} \bigr)\left ( \textstyle\begin{array}{c} v_{1}^{i}\\ v_{2}^{i}\\ \vdots\\ v_{N+1}^{i} \end{array}\displaystyle \right ) \right \vert = \bigl\vert a^{ij}_{x_{s}}u_{x_{j}}q^{sp}v_{p}^{i} \bigr\vert \\ \leq&\sum_{j=1}^{N} \Biggl[ \delta\sum _{s,p=1}^{N+1} \bigl(a^{ij}_{x_{s}}v_{p}^{i} \bigr)^{2}+\frac{1}{4\delta}\sum_{s,p=1}^{N+1} \bigl(q^{sp}u_{x_{j}} \bigr)^{2} \Biggr]. \end{aligned}$$
By the assumption
$$a^{ij}(u,x,t)\xi_{i}\xi_{j}-\delta\sum _{s=1}^{N+1}\sum_{j=1}^{N} \bigl(a^{ij}_{x_{s}}\xi_{j} \bigr)^{2} \geq0, $$
then
$$\begin{aligned}& \int_{\Omega} a^{ij}(u,x,t)u_{x_{s}x_{i}}u_{x_{p}x_{j}} \frac{\partial^{2} I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial\xi_{s}\,\partial\xi_{p}} \,dx - \int_{\Omega}a^{ij}_{x_{s}}(u,x,t)u_{x_{j}} \frac{\partial^{2} I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial \xi_{s}\,\partial\xi_{p}}u_{x_{p}x_{i}}\,dx \\& \quad\geq-\frac{1}{4\delta} \int_{\Omega}\sum_{s,p=1}^{N+1} \sum_{j=1}^{N} \bigl(q^{sp}u_{x_{j}} \bigr)^{2}\,dx\geq-c \int_{\Omega} \vert \operatorname {grad}u\vert ^{2} \,dx. \end{aligned}$$
(2.16)
We will use the fact that, on Σ, \(u=0\),
$$\begin{aligned}& -b_{iu}(0,x,t)\frac{\partial u}{\partial n}n_{i}=\varepsilon\Delta u+\frac{\partial}{\partial x_{i}} \biggl(a^{ij}(u,x,t)\frac{\partial u}{\partial x_{j}} \biggr)+ \frac{\partial b_{i}(u,x,t)}{\partial x_{i}}, \end{aligned}$$
(2.17)
to calculate the surface integrals in (2.15). Equation (2.17) involves the derivatives on the boundary; let us give some explanation in terms of the concept of the local coordinates. Let \(\delta_{0}>0\) be small enough that
$$E^{\delta_{0}}= \bigl\{ x\in\bar{\Omega}; \operatorname {dist}(x, \Sigma)\leq \delta_{0} \bigr\} \subset\bigcup_{\tau=1}^{n}V_{\tau}, $$
where \(V_{\tau}\) is a region, on which one can introduce local coordinates
$$y_{k}=F^{k}_{\tau}(x) \quad(k=1, 2, \ldots, N), y_{N}\vert_{\Sigma}=0, $$
with \(F^{k}_{\tau}\) appropriately smooth and \(F^{N}_{\tau}=F^{N}_{l}\), such that the \(y_{N}\)-axis coincides with the normal vector. Since the domain is bounded, there exists finite \(V_{\tau}\), \(\tau=1,2,\ldots, n\), such that \(\bigcup_{\tau=1}^{n}V_{\tau} \supset\Sigma\).
Using these local coordinates on \(V_{\tau}\), \(\tau=1,2,\ldots, n\), by elementary computations (refer to [3]), we obtain on \(\Sigma\cap V_{\tau}\),
$$\begin{aligned}& u_{x_{i}x_{j}}=\sum_{k=1}^{N}u_{y_{N}y_{k}}F^{N}_{x_{i}}F^{k}_{x_{j}} +\sum_{k=1}^{N-1}u_{y_{N}y_{k}}F^{N}_{x_{i}}F^{k}_{x_{j}}+ u_{y_{m}}F^{m}_{x_{i}x_{j}}. \end{aligned}$$
(2.18)
By this formula, what (2.17) means is clear.
Moreover, by (2.17), the surface integrals in (2.15) can be rewritten as
$$\begin{aligned} S =&- \biggl[ \int_{\Sigma}b_{iu}(u,x,t)I_{\eta} \bigl( \vert \operatorname {grad}u\vert \bigr)n_{i}\,d\sigma+ \int_{\Sigma}a^{ij}(u,x,t)\frac{\partial I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial x_{j}}n_{i} \,d\sigma \\ &{}+\varepsilon \int_{\Sigma}\frac{\partial I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial x_{i}}n_{i}\,d\sigma+ \int_{\Sigma }a^{ij}_{u}(u,x,t)u_{x_{i}}n_{j}I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr)\,d\sigma \biggr] \\ =& \int_{\Sigma}b_{ix_{i}}(0,x,t)\frac{ I_{\eta}(\vert \operatorname {grad}u\vert )}{\frac{\partial u}{\partial n}} \, d \sigma- \varepsilon \int_{\Sigma} \biggl[\frac{\partial I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial x_{i}}n_{i}-\Delta u \frac{I_{\eta}(\vert \operatorname {grad}u\vert )}{\frac{\partial u}{\partial n}} \biggr]\,d\sigma \\ &{}+ \int_{\Sigma}a^{ij}(0,x,t) \biggl[\frac{\partial I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial x_{i}}n_{j}- u_{x_{i}x_{j}} \frac{I_{\eta}(\vert \operatorname {grad}u\vert )}{\frac{\partial u}{\partial n}} \biggr]\,d\sigma \\ &{}+ \int_{\Sigma}a_{x_{j}}^{ij}(0,x,t)u_{x_{j}} \frac{ I_{\eta}(\vert \operatorname {grad}u\vert )}{\frac{\partial u}{\partial n}} \, d\sigma+ \int_{\Sigma}a_{x_{s}}^{ij}(0,x,t)u_{x_{j}}u_{x_{s}} \frac{S_{\eta }(\vert \operatorname {grad}u\vert )}{\vert \operatorname {grad}u\vert }n_{i} \,d\sigma \\ =& \int_{\Sigma}b_{ix_{i}}(0,x,t)\frac{I_{\eta}(\vert \operatorname {grad}u\vert )}{\frac{\partial u}{\partial n}} \,d\sigma- \varepsilon \int_{\Sigma} \biggl[\frac{\partial I_{\eta}(\vert \operatorname {grad}u\vert )}{\partial x_{i}}n_{i}-\Delta u \frac{I_{\eta}(\vert \operatorname {grad}u\vert )}{\frac{\partial u}{\partial n}} \biggr]\,d\sigma. \end{aligned}$$
Since
$$u_{x_{N+1}}\vert_{\Sigma}=u_{t}\vert _{\Sigma}=0, $$
we have
$$\lim_{\eta\rightarrow 0}S= \int_{\Sigma}b_{ix_{i}}(0,x,t)\operatorname {sgn}\biggl( \frac{\partial u}{\partial n} \biggr)\,d\sigma+\varepsilon \int_{\Sigma} \operatorname {sgn}\biggl(\frac{\partial u}{\partial n} \biggr) (u_{x_{s}x_{i}}n_{i}n_{s}-\Delta u)\,d\sigma. $$
Noticing that
$$u_{x_{i}x_{j}}n_{j}n_{i}=\frac{\sum_{k=1}^{N}u_{y_{N}y_{k}}F^{N}_{x_{i}}F^{k}_{x_{j}}F^{N}_{x_{j}}F^{N}_{x_{i}}}{ \vert \operatorname {grad}F^{N}\vert ^{2}}+\sum _{k=1}^{N-1}u_{y_{N}y_{k}}F^{k}_{x_{i}}F^{N}_{x_{j}} +\frac{u_{y_{m}}F^{m}_{x_{i}x_{j}}F^{N}_{x_{j}}F^{N}_{x_{i}}}{\vert \operatorname {grad}F^{N}\vert ^{2}} $$
in which \(F^{k}=F^{k}_{\tau}\), by the fact that the normal vector is
$$\vec{n}= \biggl(\frac{\partial F^{N}}{\partial x_{1}},\ldots, \frac {\partial F^{N}}{\partial x_{N}} \biggr)=\operatorname {grad}F^{N}, $$
we have
$$u_{x_{i}x_{j}}n_{j}n_{i}-\Delta u=u_{y_{m}} \biggl(\frac{F^{m}_{x_{i}x_{j}}F^{N}_{x_{j}}F^{N}_{x_{i}}}{\vert \operatorname {grad}F^{N}\vert ^{2}}-F^{m}_{x_{i}x_{i}} \biggr). $$
Using Lemma 2.7, one is able to deduce that \(\lim_{\eta \rightarrow 0}S\) can be estimated by \(\vert \operatorname {grad}u\vert _{L_{1}(\Omega)}\).
Thus, letting \(\eta\rightarrow0\) in (2.15), and noticing that
$$\lim_{\eta\rightarrow 0} \bigl[\vert \operatorname {grad}u\vert S_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr)- I_{\eta} \bigl(\vert \operatorname {grad}u\vert \bigr) \bigr]=0, $$
using the fact of that \(\lim_{\eta\rightarrow0}S\) can be estimated by \(\vert \operatorname {grad}u\vert _{L_{1}(\Omega)}\), we have
$$\frac{d}{dt} \int_{\Omega} \vert \operatorname {grad}u\vert \,dx\leq c_{1}+c_{2} \int_{\Omega} \vert \operatorname {grad}u\vert \,dx, $$
by the well-known Gronwall lemma, we have
$$\begin{aligned}& \int_{\Omega} \vert \operatorname {grad}u\vert \,dx\,dt\leq c. \end{aligned}$$
(2.19)
By (2.19), it is easy to show that
$$\begin{aligned}& \iint_{Q_{T}}a^{ij}(u,x,t) u_{x_{i}}u_{x_{j}} \,dx\,dt \leq c. \end{aligned}$$
(2.20)
□
Now we put back the solution of equation (2.6) as \(u_{\varepsilon}\). Then by (2.19)-(2.20), there exist a subsequence \(\{ u_{\varepsilon _{n}}\}\) of \(u_{\varepsilon}\) and a function \(u\in \operatorname {BV}(Q_{T})\cap L^{\infty }(Q_{T})\) such that \(u_{\varepsilon_{n}}\rightarrow u\) a.e. on \(Q_{T}\), we can simply denote this subsequence as \(\{\varepsilon\}\) itself; there exist functions \(g^{i}\in L^{2}(Q_{T})\) and a subsequence of \(\{\varepsilon\}\), such that, when \(\varepsilon\rightarrow0\),
$$\begin{aligned}& \widehat{\gamma^{ij}}\frac{\partial u_{\varepsilon}}{\partial x_{j}}\rightharpoonup g^{i}, \quad\text{in } L^{2}(Q_{T}). \end{aligned}$$
Proof of Theorem 1.2
We now prove that u is a generalized solution of (1.1)-(1.5)-(1.13). Let \(\varphi\in C^{2}(\overline{Q_{T}})\), \(\varphi_{1}\geq 0\), \(\operatorname {supp}\varphi\subset\overline{\Omega}\times(0,T)\), \(\nabla \varphi_{1}\vert_{\Omega}=0\), and \(\{n_{i}\}\) be the inner normal vector of Ω. Multiply equation (2.6) by \(\varphi_{1}S_{\eta}(u_{\varepsilon}-k)\), and integrate over \(Q_{T}\), to obtain
$$\begin{aligned}& \iint_{Q_{T}}\frac{\partial u_{\varepsilon}}{\partial t}\varphi_{1}S_{\eta}(u_{\varepsilon }-k) \,dx\,dt \\& \quad= \iint_{Q_{T}}\frac{\partial}{\partial x_{i}} \biggl(a^{ij}(u_{\varepsilon}) \frac{\partial u_{\varepsilon}}{\partial x_{j}} \biggr)\varphi_{1}S_{\eta }(u_{\varepsilon}-k) \,dx\,dt \\& \quad\quad{} +\varepsilon \iint_{Q_{T}}\Delta u_{\varepsilon}\varphi_{1}S_{\eta }(u_{\varepsilon}-k) \,dx\,dt + \iint_{Q_{T}}\frac{\partial b_{i}(u_{\varepsilon},x,t)}{\partial x_{i}}\varphi_{1}S_{\eta }(u_{\varepsilon}-k) \,dx\,dt. \end{aligned}$$
(2.21)
Let us calculate every term in (2.21) by the partial integration method. We have
$$\begin{aligned}& \iint_{Q_{T}}\frac{\partial u_{\varepsilon}}{\partial t}\varphi_{1}S_{\eta}(u_{\varepsilon }-k) \,dx\,dt=- \iint_{Q_{T}}I_{\eta}(u_{\varepsilon}-k)\varphi _{1t}\,dx\,dt, \end{aligned}$$
(2.22)
$$\begin{aligned}& \varepsilon \iint_{Q_{T}}\Delta u_{\varepsilon}\varphi_{1}S_{\eta}(u_{\epsilon }-k) \,dx\,dt \\& \quad=- \varepsilon \int_{0}^{T} \int_{\Sigma}\nabla u_{\varepsilon}\cdot\vec{n} \varphi_{1}S_{\eta}(u_{\epsilon}-k)\,dt\,d\sigma \\& \quad\quad{} -\varepsilon \iint_{Q_{T}}\nabla u_{\varepsilon} \bigl[S_{\eta }(u_{\varepsilon}-k) \nabla\varphi_{1}+\varphi_{1}S_{\eta}^{\prime }(u_{\varepsilon}-k) \nabla u_{\varepsilon} \bigr]\,dx\,dt \\& \quad=\varepsilon S_{\eta}(k) \int_{0}^{T} \int_{\Sigma}\nabla u_{\varepsilon}\cdot\vec{n} \varphi_{1}\,dt\,d\sigma-\varepsilon \iint_{Q_{T}}\nabla u_{\varepsilon }S_{\eta}(u_{\varepsilon}-k) \nabla\varphi_{1}\,dx\,dt \\& \quad\quad{} -\varepsilon \iint_{Q_{T}}\vert \nabla u_{\varepsilon }\vert ^{2}S_{\eta}^{\prime}(u_{\varepsilon}-k) \varphi_{1}\,dx\,dt, \end{aligned}$$
(2.23)
$$\begin{aligned}& \iint_{Q_{T}}\frac{\partial}{\partial x_{i}} \biggl(a^{ij}(u_{\varepsilon},x,t) \frac{\partial u_{\varepsilon}}{\partial x_{j}} \biggr)\varphi_{1}S_{\eta }(u_{\varepsilon}-k) \,dx\,dt \\& \quad=S_{\eta }(k) \int_{0}^{T} \int_{\Sigma}a^{ij}(u_{\varepsilon},x,t) \frac {\partial u_{\varepsilon}}{\partial x_{j}}n_{i}\varphi_{1}\,dt\,d\sigma \\& \quad\quad {}- \iint_{Q_{T}}a^{ij}(u_{\varepsilon},x,t) \frac {\partial u_{\varepsilon}}{\partial x_{j}} \bigl[S_{\eta}(u_{\varepsilon}-k) \varphi_{1x_{i}} +\varphi_{1}S_{\eta}^{\prime}(u_{\varepsilon }-k)u_{\varepsilon x_{i}} \bigr]\,dx\,dt \\& \quad=S_{\eta}(k) \int_{0}^{T} \int_{\Sigma}a^{ij}(u_{\varepsilon },x,t) \frac{\partial u_{\varepsilon}}{\partial x_{j}}n_{i}\varphi_{1}\,dt\,d\sigma- \iint_{Q_{T}}a^{ij}(u_{\varepsilon},x,t) \frac{\partial u_{\varepsilon}}{\partial x_{j}}S_{\eta}(u_{\varepsilon}-k) \varphi _{1x_{i}} \,dx\,dt \\& \quad\quad{} - \iint_{Q_{T}}a^{ij}(u_{\varepsilon},x,t)u_{\varepsilon x_{i} }u_{\varepsilon x_{j}}S_{\eta}^{\prime}(u_{\varepsilon}-k) \varphi_{1}\,dx\,dt, \end{aligned}$$
(2.24)
and
$$\begin{aligned}& - \iint_{Q_{T}}a^{ij}(u_{\varepsilon},x,t) \frac{\partial u_{\varepsilon}}{\partial x_{j}}S_{\eta}(u_{\varepsilon}-k) \varphi _{1x_{i}} \,dx\,dt \\& \quad= \iint_{Q_{T}} \int_{k}^{u_{\varepsilon }}a^{ij}_{x_{j}}(s,x,t)S_{\eta}(s-k) \,ds \varphi_{1x_{i}}\,dx\,dt \\& \quad\quad{} + \iint_{Q_{T}}A^{ij}_{\eta}(u_{\varepsilon},x,t,k) \varphi_{1x_{i}x_{j}}\,dx\,dt+ \int_{0}^{T} \int_{\Sigma} A^{ij}_{\eta}(u_{\varepsilon},k) \varphi_{1x_{i}}n_{j}\,dt\,d\sigma, \end{aligned}$$
(2.25)
$$\begin{aligned}& \iint_{Q_{T}}\frac{\partial b_{i}(u_{\varepsilon },x,t)}{\partial x_{i}}\varphi_{1}S_{\eta}(u_{\varepsilon}-k) \,dx\,dt \\& \quad=- \int_{0}^{T} \int_{\Sigma} \bigl[b_{i}(u_{\varepsilon },x,t)-b(k,x,t) \bigr]n_{i}\varphi_{1}S_{\eta}(u_{\varepsilon}-k)\,dt \,d \sigma \\& \quad\quad{} - \iint_{Q_{T}} \bigl[b_{i}(u_{\varepsilon },x,t)-b_{i}(k,x,t) \bigr] \biggl[\frac{\partial\varphi_{1}}{\partial x_{i}}S_{\eta }(u_{\varepsilon}-k) + \varphi_{1}S_{\eta}^{\prime }(u_{\varepsilon}-k) \frac{\partial u_{\varepsilon}}{\partial x_{i}} \biggr]\,dx\,dt \\& \quad=-S_{\eta}(k) \int_{0}^{T} \int_{\Sigma}\varphi_{1} \bigl[b_{i}(0,x,t)-b_{i}(k,x,t) \bigr]n_{i}\,d\sigma\,dt \\& \quad\quad{} - \iint_{Q_{T}}B_{\eta}^{i}(u_{\varepsilon},x,t,k) \varphi_{1x_{i}}\,dx\,dt. \end{aligned}$$
(2.26)
From (2.21)-(2.26), we have
$$\begin{aligned}& \iint_{Q_{T}}I_{\eta}(u_{\varepsilon}-k)\varphi _{1t}\,dx\,dt+ \iint_{Q_{T}}A^{ij}_{\eta}(u_{\varepsilon },x,t,k) \varphi_{1x_{i}x_{j}}\,dx\,dt- \iint_{Q_{T}}B_{\eta}^{i}(u_{\varepsilon },x,t,k) \varphi_{1x_{i}}\,dx\,dt \\& \quad{} -\varepsilon \iint_{Q_{T}}\nabla u_{\varepsilon}\cdot\nabla \varphi_{1}S_{\eta}(u_{\varepsilon }-k)\,dx\,dt-\varepsilon \iint_{Q_{T}}\vert \nabla u_{\varepsilon} \vert ^{2}S_{\eta}^{\prime }(u_{\varepsilon }-k) \varphi_{1}\,dx\,dt \\& \quad{} - \iint_{Q_{T}}a^{ij}(u_{\varepsilon},x,t) u_{\varepsilon x_{i}}u_{\varepsilon x_{j}}S_{\eta}^{\prime}(u_{\varepsilon }-k) \varphi_{1}\,dx\,dt \\& \quad{} + \iint_{Q_{T}} \int_{k}^{u_{\varepsilon }}a^{ij}_{x_{j}}(s,x,t)S_{\eta}(s-k) \,ds \varphi_{1x_{i}}\,dx\,dt \\& \quad{} +S_{\eta}(k) \int_{0}^{T} \int_{\Sigma}\!\frac{\partial}{\partial x_{i}}\bigl(a^{ij}(u_{\varepsilon},x,t)\bigr)n_{i}u_{\varepsilon x_{j}} \varphi_{1}\,dt\,d\sigma+ S_{\eta}(k) \int_{0}^{T} \int_{\Sigma}A^{ij}_{\eta}(0,x,t,k) \varphi_{1x_{i}} n_{j}\,dt\,d\sigma \\& \quad{} +\varepsilon S_{\eta}(k) \int_{0}^{T} \int_{\Sigma}\nabla u_{\varepsilon}\cdot\vec{n} \varphi_{1}\,dt\,d\sigma+S_{\eta }(k) \int_{0}^{T} \int_{\Sigma_{1\eta k}} \bigl[b_{i}(0,x,t)-b_{i}(k,x,t) \bigr]n_{i}\varphi_{1}\,dt\,d\sigma \\& \quad{} +S_{\eta}(k) \int_{0}^{T} \int_{\Sigma_{2\eta k}} \bigl[b_{i}(0,x,t)-b_{i}(k,x,t) \bigr]n_{i}\varphi_{1}\,dt\,d\sigma=0. \end{aligned}$$
(2.27)
Taking \(\varphi_{2}\in C^{2}(\bar{Q}_{T})\), \(\varphi_{1}\vert _{\partial\Omega\times[0,T]}=\varphi_{2}\vert_{\partial \Omega\times[0,T]}\), \(\operatorname {supp}\varphi_{2}\subset \bar{\Omega}\times(0,T)\),
$$\begin{aligned}& S_{\eta }(k) \int_{0}^{T} \int_{\Sigma}a^{ij}(u_{\varepsilon},x,t)n_{i} \frac {\partial u_{\varepsilon}}{\partial x_{j}}\varphi_{1}\,dt\,d\sigma+\varepsilon S_{\eta}(k) \int_{0}^{T} \int_{\Sigma}\nabla u_{\varepsilon}\cdot\vec{n} \varphi_{1}\,dt\,d\sigma \\& \quad=S_{\eta}(k) \biggl\{ -\varepsilon \iint_{Q_{T}}\frac{\partial u_{\varepsilon}}{\partial x_{i}}\frac{\partial\varphi _{2}}{\partial x_{i}}\,dx\,dt - \iint_{Q_{T}} a^{ij}(u_{\varepsilon},x,t) \frac{\partial u_{\varepsilon}}{\partial x_{j}}\varphi_{2x_{i}}\,dx\,dt \\& \quad\quad{} + \iint_{Q_{T}}\frac{\partial b_{i}(0,x,t)}{\partial x_{i}}\varphi_{2}\,dx\,dt - \iint_{Q_{T}} \bigl(b_{i}(u_{\varepsilon },x,t)-b_{i}(0,x,t) \bigr)\frac{\partial\varphi _{2}}{\partial x_{i}}\,dx\,dt \\& \quad\quad{} + \iint_{Q_{T}}u_{\varepsilon }\frac{\partial\varphi_{2}}{\partial t}\,dx\,dt- \int_{0}^{T} \int_{\Sigma} \bigl[b_{i}(0,x,t)-b_{i}(0,x,t) \bigr]n_{i}\varphi_{2}\,dt\,d\sigma \biggr\} , \end{aligned}$$
(2.28)
$$\begin{aligned}& \iint_{Q_{T}}a^{ij}(u_{\varepsilon},x,t)\varphi _{2x_{i}}\frac{\partial u_{\varepsilon}}{\partial x_{j}}\,dx\,dt \\& \quad=- \int_{0}^{T} \int_{\Sigma }A^{ij}(0,x,t)\varphi_{2x_{i}}n_{j} \,dt\,d \sigma \\& \quad\quad{} - \iint_{Q_{T}}A^{ij}(u_{\varepsilon},x,t) \varphi_{2x_{i}x_{j}}\,dx\,dt- \iint_{Q_{T}} \int_{0}^{u_{\varepsilon}}a^{ij}_{x_{j}}(s,x,t) \,ds \varphi_{2x_{i}}\,dx\,dt \\& \quad =- \iint_{Q_{T}}A^{ij}(u_{\varepsilon},x,t) \varphi_{2x_{i}x_{j}}\,dx\,dt- \iint_{Q_{T}} \int_{0}^{u_{\varepsilon}}a^{ij}_{x_{j}}(s,x,t) \,ds \varphi_{2x_{i}}\,dx\,dt . \end{aligned}$$
(2.29)
For \(\nabla\varphi_{1}\vert_{\Sigma}=0\), and by \(a^{ij}(0,x,t)=0\), from (2.27)-(2.29), we have
$$\begin{aligned}& \iint_{Q_{T}}I_{\eta}(u_{\varepsilon}-k)\varphi _{1t}\,dx\,dt+ \iint_{Q_{T}}A^{ij}_{\eta}(u_{\varepsilon },x,t,k) \varphi_{1x_{i}x_{j}} \,dx\,dt- \iint_{Q_{T}}B_{\eta}^{i}(u_{\varepsilon },x,t,k) \varphi_{1x_{i}}\,dx\,dt \\& \quad{} + \iint_{Q_{T}} \int_{0}^{u_{\varepsilon }}a^{ij}_{x_{j}}(s,x,t)S_{\eta}(s-k) \,ds \varphi_{1x_{i}}\,dx\,dt \\& \quad{} +S_{\eta}(k) \biggl[-\varepsilon \iint_{Q_{T}}\frac{\partial u_{\varepsilon}}{\partial x_{i}}\frac{\partial\varphi_{2}}{\partial x_{i}}\,dx\,dt + \iint_{Q_{T}}A^{ij}(u_{\varepsilon},x,t)\varphi _{2x_{i}x_{j}}\,dx\,dt \\& \quad{} + \iint_{Q_{T}}\frac{\partial b_{i}(0,x,t)}{\partial x_{i}}\varphi_{2}\,dx\,dt \\& \quad{} - \iint_{Q_{T}} \bigl[b_{i}(u_{\varepsilon },x,t)-b_{i}(0,x,t) \bigr]\frac{\partial\varphi_{2}}{\partial x_{i}}\,dx\,dt+ \iint_{Q_{T}}u_{\varepsilon}\frac{\partial \varphi_{2}}{\partial t}\,dx\,dt \biggr] \\& \quad{}+S_{\eta}(k) \iint_{Q_{T}} \int_{0}^{u_{\varepsilon }}a^{ij}_{x_{j}}(s,x,t) \,ds \varphi_{2x_{i}}\,dx\,dt \\& \quad{}-\varepsilon \iint_{Q_{T}}\nabla u_{\varepsilon}\cdot\nabla \varphi_{1}S_{\eta}(u_{\varepsilon }-k)\,dx\,dt- \iint_{Q_{T}}a^{ij}(u_{\varepsilon},x,t) u_{\varepsilon x_{i}}u_{\varepsilon x_{j}}S_{\eta}^{\prime}(u_{\varepsilon}-k) \varphi_{1}\,dx\,dt \\& \quad{} +S_{\eta }(k) \int_{0}^{T} \int_{\Sigma_{1\eta k}} \bigl[(b_{i}(0)-b_{i}(k) \bigr]n_{i}\varphi_{1}\,dt\,d\sigma\geq0. \end{aligned}$$
(2.30)
By Lemma 2.6,
$$\begin{aligned}& \lim\inf_{\varepsilon\rightarrow0} \iint_{Q_{T}}S_{\eta}^{\prime}(u_{\varepsilon }-k)a^{ij}(u_{\varepsilon},x,t) \frac{\partial u_{\varepsilon }}{\partial x_{i}}\frac{\partial u_{\varepsilon}}{\partial x_{j}}\varphi_{1}\, dx\,dt \\& \quad \geq\sum_{i=1}^{N} \iint_{Q_{T}} \bigl\vert g^{i} \bigr\vert ^{2}S_{\eta}^{\prime}(u-k) \varphi_{1}\,dx \,dt. \end{aligned}$$
(2.31)
Let \(\varepsilon\rightarrow0\) in (2.30). By (2.31), we get (1.16) and (1.17) is naturally concealed in the limiting process.
The proof of (1.18) is similar to that in [15, 22], we omit the details here. □