# On a parabolic equation related to the p-Laplacian

## Abstract

Consider a parabolic equation related to the p-Laplacian. If the diffusion coefficient of the equation is degenerate on the boundary, no matter we can define the trace of the solution on the boundary or not, by choosing a suitable test function, the stability of the solutions always can be established without a boundary condition.

## 1 Introduction and the main results

Consider an equation related to the p-Laplacian,

$${u_{t}} = \operatorname{div} \bigl({\rho^{\alpha}} {\vert { \nabla u} \vert ^{p - 2}}\nabla u\bigr) + f(u,x,t) ,\quad (x,t) \in{Q_{T}} = \Omega \times (0,T),$$
(1.1)

where Ω is a bounded domain in $$\mathbb{R}^{N}$$ with appropriately smooth boundary, $$\rho(x) = \operatorname{dist} (x,\partial\Omega)$$, $$p>1$$, $$\alpha >0$$. Yin and Wang  first studied the equation

$${u_{t}} = \operatorname{div}\bigl({\rho^{\alpha}} {\vert { \nabla u} \vert ^{p - 2}}\nabla u\bigr)$$
(1.2)

and showed that, when $$\alpha>p-1$$, the solution of the equation is completely controlled by the initial value.

The author studied in cooperation with Zhan and Yuan  the following equation:

$$u_{t} = \operatorname{div} \bigl(\rho^{\alpha} \vert \nabla u \vert ^{p - 2}\nabla u\bigr) + \sum _{i = 1}^{N} \frac{\partial b_{i}(u)}{\partial x_{i}} ,\quad (x,t) \in{Q_{T}},$$
(1.3)

and had shown that, to consider the well-posedness of equation (1.3), instead of the whole boundary condition

$$u(x,t) = 0,\quad (x,t) \in\partial\Omega\times (0,T),$$
(1.4)

only the partial boundary condition

$$u(x,t) = 0,\quad (x,t) \in\Sigma_{p} \times (0,T)$$
(1.5)

is necessary. Here, $$\Sigma _{p} \subseteq\partial\Omega$$ is just a portion of Ω, which is determined by the first order derivative term $$\frac{\partial b_{i}(u)}{\partial x_{i}}$$, $$i=1, 2, \ldots , N$$. Certainly, the initial value is always necessary,

$$u(x,0)=u_{0}(x).$$
(1.6)

In our paper, we will consider the well-posedness of the solutions of equation (1.1). First of all, we give some basic functional spaces. For every fixed $$t\in[0, T]$$, we introduce the Banach space

\begin{aligned}& V_{t}(\Omega) =\Bigl\{ u(x) : u(x)\in L^{2}(\Omega)\cap W^{1,1}_{0}(\Omega), \bigl|\nabla u(x)\bigr|^{p}\in L^{1} (\Omega)\Bigr\} ,\\& \|u\|_{V_{t}(\Omega)} = \|u\|_{2,\Omega} + \|\nabla u\|_{p,\Omega} , \end{aligned}

and denote by $$V'_{t}(\Omega)$$ its dual. By $$\mathbf{W}(Q_{T})$$ we denote the Banach space

$$\left \{ \textstyle\begin{array}{@{}l} \mathbf{W}(Q_{T}) = \{u : [0,T]\rightarrow V_{t}(\Omega)|u\in L^{2}(Q_{T}),|\nabla u|^{p} \in L^{1}(Q_{T}), u = 0\mbox{ on }\Gamma_{T}\}, \\ \|u\|_{\mathbf{W}(Q_{T})} = \|\nabla u\|_{p,Q_{T}} + \|u\|_{2,Q_{T}} . \end{array}\displaystyle \right .$$

$$\mathbf{W}'(Q_{T})$$ is the dual of $$\mathbf{W}(Q_{T})$$ (the space of linear functionals over $$\mathbf{W}(Q_{T})$$):

$$w\in\mathbf{W}'(Q_{T})\quad\Longleftrightarrow\quad \left \{ \textstyle\begin{array}{@{}l} w=w_{0}+\sum_{i=1}^{n}D_{i}w_{i},\quad w_{0}\in L^{2}(Q_{T}), w_{i}\in L^{p'}(Q_{T}),\\ \forall\phi\in\mathbf{W}(Q_{T}),\quad \ll w,\phi\gg=\int_{Q_{T}} (w_{0}\phi +\sum_{i}w_{i}D_{i}\phi )\,dz. \end{array}\displaystyle \right .$$

The norm in $$\mathbf{W}'(Q_{T})$$ is defined by

$$\|v\|_{\mathbf{W}'(Q_{T})}= \sup\bigl\{ \ll v,\phi\gg|\phi\in\mathbf{W(Q_{T})}, \| \phi\|_{\mathbf{W}(Q_{T})}\leq1\bigr\} .$$

### Definition 1.1

A function $$u(x,t)$$ is said to be a weak solution of equation (1.1) with the initial value (1.6), if

$$u \in L^{\infty}(Q_{T}),\qquad u_{t}\in \mathbf{W}'(Q_{T}), \qquad{\rho^{\alpha}} {\vert { \nabla u} \vert ^{p}} \in{L^{1}}({Q_{T}}),$$
(1.7)

and for any function $$\varphi \in C_{0}^{\infty}({Q_{T}})$$, we have

$$\iint_{{Q_{T}}} \bigl(-u\varphi_{t} + \rho^{\alpha} \vert {\nabla u} \vert ^{p - 2}\nabla u \cdot\nabla\varphi - f(u,x,t) \varphi\bigr)\,dx\,dt= 0.$$
(1.8)

The initial value, as usual, is satisfied in the sense that

$$\lim_{t\rightarrow0} \int_{\Omega}u(x,t)\phi(x)\,dx= \int_{\Omega }u_{0}(x)\phi(x)\,dx,\quad\forall\phi(x)\in C_{0}^{\infty}(\Omega).$$
(1.9)

We can easily obtain the existence of the weak solution.

### Theorem 1.2

Let us suppose $$1< p$$, $$0<\alpha$$, $$f(s,x,t)$$ is a Lipschitz function. If

$$u_{0}(x)\in L^{\infty}(\Omega),\quad \rho^{\alpha}\|\nabla u_{0}\|^{p}\in L^{1}(\Omega),$$
(1.10)

then equation (1.1) with initial value (1.6) has a weak solution u in the sense of Definition  1.1.

We mainly are concerned with the stability of the solutions. As in , due to the fact that the weak solution defined in our paper satisfies (1.7), when $$\alpha< p-1$$, we can define the trace of u on the boundary, while for $$\alpha\geq p-1$$, the obtained weak solution lacks the regularity to define the trace on the boundary. However, in the short paper, by choosing a suitable test function, we can obtain the stability of the weak solutions without the boundary condition only if $$\alpha>0$$. In other words, whether the weak solution is regular enough to define the trace on the boundary is not so important.

The main result of our paper is the following theorem.

### Theorem 1.3

Let u and v be two weak solutions of equation (1.1) with the different initial values $$u(x,0)=v(x,0)$$, respectively. If $$\alpha>0$$ and $$f(s,x,t)$$ is a Lipschitz function, then

$$\int_{\Omega}\bigl\vert u(x,t) - v(x,t) \bigr\vert \,dx \leq \int_{\Omega}\bigl\vert u_{0}(x) - v_{0}(x) \bigr\vert \,dx.$$
(1.11)

Comparing with , the greatest improvement lies in that we do not require any boundary condition, no matter that $$\alpha< p-1$$ or $$\alpha \geq p-1$$. At the same time, the nonlinear source term $$f(u,x,t)$$ adds the difficulty when we use the compact convergence theorem. The proof of the existence of the weak solution is quite different from that in . Moreover, we consider the following equation, which seems similar to our equation (1.1):

$${u_{t}} = \operatorname{div} \bigl({\vert {\nabla u} \vert ^{p - 2}}\nabla u\bigr) + f(u,x,t) ,\quad (x,t) \in{Q_{T}},$$
(1.12)

which has been studied thoroughly for a long time, one may refer to . Generally, to the growth order of u in $$f(u,x,t)$$ should be added some restrictions. Very recently, Benedikt et al.  have studied the equation

$${u_{t}} = \operatorname{div} \bigl({\vert {\nabla u} \vert ^{p - 2}}\nabla u\bigr) + q(x)u^{\gamma} ,\quad (x,t) \in{Q_{T}} ,$$
(1.13)

with $$0<\gamma<1$$, and shown that the uniqueness of the solutions of equation (1.13) is not true. From the short comment, one can see that the degeneracy of the coefficient $$\rho^{\alpha}$$ plays an important role in the well-posedness of the solutions, it even can eliminate the action from the source term $$f(u,x,t)$$. By the way, the author has been interested in the boundary value condition of a degenerate parabolic equation for some time, one may refer to .

## 2 The proof of Theorem 1.2

By  and , we have the following lemma.

### Lemma 2.1

Let $$q\geq1$$. If $$u_{\varepsilon}\in L^{\infty }(0,T;L^{2}(\Omega))\cap\mathbf{W}(Q_{T})$$, $$\| u_{\varepsilon t}\| _{\mathbf{W}'(Q_{T})}\leq c$$, $$\|\nabla(|u_{\varepsilon }|^{q-1}\times u_{\varepsilon})\|_{p,Q_{T}}\leq c$$, then there is a subsequence of $$\{u_{\varepsilon}\}$$ which is relatively compact in $$L^{s}(Q_{T})$$ with $$s\in(1,\infty)$$.

To study equation (1.1), since f is Lipschitz, without loss the generality, we may assume that $$f\in C^{1}$$. We consider, as e.g. in  by Ragusa, the associated regularized problem

\begin{aligned}& u_{\varepsilon t} - \operatorname{div} \bigl(\rho^{\alpha}_{\varepsilon}\bigl(\vert \nabla u_{\varepsilon} \vert ^{2}+\varepsilon \bigr)^{\frac{p - 2}{2}}\nabla {u_{\varepsilon}}\bigr) -f(u_{\varepsilon}, x, t) = 0,\quad (x,t)\in{Q_{T}}, \end{aligned}
(2.1)
\begin{aligned}& {u_{\varepsilon}}(x,t) = 0,\quad (x,t) \in\partial\Omega \times (0,T), \end{aligned}
(2.2)
\begin{aligned}& {u_{\varepsilon}}(x,0) = {u_{0\varepsilon}}(x),\quad x\in\Omega, \end{aligned}
(2.3)

where $${\rho_{\varepsilon}} = \rho\ast\delta_{\varepsilon}+ \varepsilon$$, $$\varepsilon > 0$$, $$\delta_{\varepsilon}$$ is the mollifier as usual, $${u_{\varepsilon ,0}} \in{C^{\infty}_{0} }(\Omega)$$ and $$\rho_{\varepsilon}^{\alpha}{ \vert {\nabla{u_{\varepsilon,0}}} \vert ^{p}}\in {L^{1}}(\Omega)$$ is uniformly bounded, and $${u_{\varepsilon,0}}$$ converges to $$u_{0}$$ in $$W_{0}^{1,p}(\Omega)$$. It is well known that the above problem has a unique classical solution .

### Lemma 2.2

There is a subsequence of $${u_{\varepsilon}}$$ (we still denote it as $${u_{\varepsilon}}$$), which converges to a weak solution u of equation (1.1) with the initial value (1.5).

### Proof

By the maximum principle, there is a constant c only dependent on $${\Vert {{u_{0}}} \Vert _{{L^{\infty}}(\Omega)}}$$ but independent on ε, such that

$${\Vert {{u_{\varepsilon}}} \Vert _{{L^{\infty}}({Q_{T}})}} \leqslant c.$$
(2.4)

Multiplying (2.1) by $$u_{\varepsilon}$$ and integrating it over $$Q_{T}$$, we get

\begin{aligned} &\frac{1}{2} \int_{\Omega}u_{\varepsilon}^{2}\,dx + \iint_{{Q_{T}}} \rho _{\varepsilon}^{\alpha}\bigl(|\nabla u_{\varepsilon}|^{2}+\varepsilon\bigr)^{\frac {p-2}{2}}|\nabla u_{\varepsilon}|^{2}\,dx\,dt+ \iint_{{Q_{T}}} u_{\varepsilon}f(u_{\varepsilon},x,t)\,dx\,dt \\ &\quad= \frac{1}{2} \int_{\Omega}u_{0}^{2}\,dx. \end{aligned}
(2.5)

By (2.4), (2.5), and the assumption that f is $$C^{1}$$, we have

$$\frac{1}{2} \int_{\Omega}{u_{\varepsilon}^{2}\,{d}x} + \iint_{{Q_{T}}} {\rho_{\varepsilon}^{\alpha}\bigl(|\nabla u_{\varepsilon}|^{2}+\varepsilon\bigr)^{\frac {p-2}{2}}|\nabla u_{\varepsilon}|^{2}\,dx\,dt} \leqslant c.$$
(2.6)

For small enough $$\lambda>0$$, let $$\Omega_{\lambda}=\{x\in\Omega: \operatorname{dist}(x,\partial\Omega)>\lambda\}$$. Since $$p>1$$, by (2.6),

$$\int_{0}^{T} \int_{\Omega_{\lambda}}|\nabla u_{\varepsilon}|\,dx\,dt\leq c \biggl( \int_{0}^{T} \int_{\Omega_{\lambda}}|\nabla u_{\varepsilon }|^{p}\,dx\,dt \biggr)^{\frac{1}{p}}\leq c(\lambda).$$
(2.7)

Now, for any $$v\in\mathbf{W}(Q_{T})$$, $$\|v\|_{W(Q_{T})}=1$$,

$$\langle u_{\varepsilon t}, v\rangle=- \iint_{Q_{T}}\rho_{\varepsilon}^{\alpha}\bigl(|\nabla u_{\varepsilon}|^{2}+\varepsilon\bigr)^{\frac{p(x)-2}{2}}\nabla u_{\varepsilon}\cdot\nabla v \,dx\,dt+ \iint_{Q_{T}}vf(u_{\varepsilon},x,t)\,dx\,dt,$$

by the Young inequality, we can show that

$$\bigl|\langle u_{\varepsilon t}, v\rangle\bigr|\leq c \biggl[ \iint_{{Q_{T}}} \rho_{\varepsilon }^{\alpha}|\nabla u_{\varepsilon}|^{p}\,dx\,dt+ \iint_{{Q_{T}}} \bigl(|v|^{p}+|\nabla v|^{p} \bigr)\,dx\,dt \biggr]\leq c,$$

then

$$\| u_{\varepsilon t}\|_{\mathbf{W}'(Q_{T})}\leq c.$$
(2.8)

Now, let $$\varphi\in C_{0}^{1}(\Omega)$$, $$0\leq\varphi\leq1$$, such that

$$\varphi|_{\Omega_{2\lambda}}=1,\qquad \varphi|_{\Omega\setminus\Omega _{\lambda}}=0.$$

Then

$$\bigl|\bigl\langle (\varphi u_{\varepsilon})_{t}, v\bigr\rangle \bigr|= \bigl|\langle \varphi u_{\varepsilon t}, v\rangle\bigr|\leq\bigl|\langle u_{\varepsilon t}, v\rangle\bigr|,$$

we have

\begin{aligned}& \bigl\| \bigl(\varphi(x) u\bigr)_{\varepsilon t}\bigr\| _{\mathbf{W}'(Q_{T})}\leq\| u_{\varepsilon t}\|_{\mathbf{W}'(Q_{T})}\leq c, \end{aligned}
(2.9)
\begin{aligned}& \iint_{Q_{T}}\bigl|\nabla(\varphi u_{\varepsilon})\bigr|^{p}\,dx\,dt \leq c(\lambda ) \biggl(1+ \int_{0}^{T} \int_{\Omega_{\lambda}}|\nabla u_{\varepsilon}|^{p} \,dx\,dt\biggr) \leq c(\lambda), \end{aligned}
(2.10)

and so

$$\bigl\| \nabla\bigl(\varphi u_{\varepsilon}\bigr)\bigr\| _{p,Q_{T}}\leq c(\lambda).$$
(2.11)

By Lemma 2.1, $$\varphi u_{\varepsilon}$$ is relatively compact in $$L^{s}(Q_{T})$$ with $$s\in(1,\infty)$$. Then $$\varphi u_{\varepsilon }\rightarrow\varphi u$$ a.e. in $$Q_{T}$$. In particular, due to the arbitrariness of λ, $$u_{\varepsilon}\rightarrow u$$ a.e. in $$Q_{T}$$.

Hence, by (2.4), (2.6), (2.8), there exist a function u and an n-dimensional vector function $$\overrightarrow{\zeta}= ({\zeta_{1}}, \ldots,{\zeta_{n}})$$ satisfying

$$u \in L^{\infty}(Q_{T}),\qquad u_{t}\in \mathbf{W}'(Q_{T}), \qquad \vert {\overrightarrow{\zeta}} \vert \in{L^{\frac{p}{{p - 1}}}}({Q_{T}}),$$

and

\begin{aligned}& {u_{\varepsilon}} \rightharpoonup * u, \quad\mbox{in } {L^{\infty}(Q_{T})},\\& {\nabla u_{\varepsilon}} \rightharpoonup\nabla u \quad\mbox{in } L_{\mathrm{loc}}^{p}(Q_{T}),\\& \rho_{\varepsilon}^{\alpha}{ \vert {\nabla{u_{\varepsilon}}} \vert ^{p - 2}}\nabla{u_{\varepsilon}} \rightharpoonup\overrightarrow{\zeta}\quad\mbox{in } {L^{\frac{p}{p-1}}}({Q_{T}}). \end{aligned}

In order to prove u satisfies equation (1.1), we notice that, for any function $$\varphi \in C_{0}^{\infty}({Q_{T}})$$,

$$\iint_{{Q_{T}}}\bigl[-u_{\varepsilon}\varphi_{t} + \rho_{\varepsilon}^{\alpha}\bigl(|\nabla u_{\varepsilon}|^{2}+ \varepsilon\bigr)^{\frac {p-2}{2}}\nabla{u_{\varepsilon}} \cdot\nabla \varphi-f({u_{\varepsilon}},x,t) \varphi\bigr]\,dx\,dt = 0$$
(2.12)

and $$u_{\varepsilon} \rightarrow u$$ is almost everywhere convergent, so $$f(u_{\varepsilon},x,t)\rightarrow f(u,x,t)$$ is true. Then

$$\iint_{{Q_{T}}} \biggl(\frac{\partial u}{\partial t}\varphi + \vec{\varsigma}\cdot\nabla\varphi + f(u,x,t)\varphi\biggr)\,dx\,dt= 0.$$
(2.13)

Now, similar to  or , we can prove

$$\iint_{Q_{T}} \rho^{\alpha} \vert \nabla u \vert ^{p - 2}\nabla u \cdot \nabla\varphi \,dx \,dt = \iint_{Q_{T}} \overrightarrow{\zeta}\cdot\nabla\varphi \,dx \,dt$$
(2.14)

for any function $$\varphi \in C_{0}^{\infty}({Q_{T}})$$, we omit the details here. Thus u satisfies equation (1.1).

Finally, we can prove (1.9) as in , we also omit the details here. Then Lemma 2.2 is proved. □

Theorem 1.2 is a direct corollary of Lemma 2.2.

## 3 The stability of the solutions

As we have said before, by choosing a suitable test function, we can prove the stability of the solutions without any boundary value condition only if $$\alpha>0$$.

### Proof of Theorem 1.3

For any given positive integer n, let $${g_{n}}(s)$$ be an odd function, and

$${g_{n}}(s) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} 1,&s > \frac{1}{n}, \\ {n^{2}}{s^{2}}{{\text{e}}^{1 - {n^{2}}{s^{2}}}},&s \leqslant\frac{1}{n}. \end{array}\displaystyle \right .$$
(3.1)

Clearly,

$$\lim_{n\rightarrow0}g_{n}(s)=\operatorname{sgn}(s), \quad s\in(- \infty, +\infty ),$$
(3.2)

and

$$0\leq g_{n}'(s)\leq\frac{c}{s},\quad 0< s< \frac{1}{n},$$
(3.3)

where c is independent of n.

Let $$\beta\leq\frac{\alpha}{p}$$ and

$$\phi(x)=\rho^{\beta}(x).$$
(3.4)

By taking the limit, we can choose $${g_{n}}(\phi(u - v))$$ as the test function, then

\begin{aligned} &\int_{\Omega} g_{n}\bigl(\phi(u - v)\bigr) \frac{\partial(u - v)}{\partial t}\,dx \\ &\quad{}+\int_{\Omega} \rho^{\alpha}\bigl(\vert \nabla u \vert ^{p- 2}\nabla u - \vert \nabla v \vert ^{p- 2}\nabla v\bigr) \cdot\phi\nabla(u - v)g'_{n} \,dx \\ &\quad{}+\int_{\Omega} \rho^{\alpha}\bigl(\vert \nabla u \vert ^{p - 2}\nabla u - \vert \nabla v \vert ^{p- 2}\nabla v\bigr) \cdot\nabla\phi (u - v)g'_{n} \,dx \\ &\quad{}+\int_{\Omega} \bigl(f(u,x,t) - f(v,x,t)\bigr){g_{n}} \bigl(\phi(u - v)\bigr) = 0. \end{aligned}
(3.5)

Thus

\begin{aligned} &\lim_{n\rightarrow\infty} \int_{\Omega}g_{n}\bigl(\phi(u - v)\bigr) \frac{\partial (u - v)}{\partial t}\,dx \\ &\quad= \int_{\Omega}\operatorname{sgn}\bigl(\phi(u - v)\bigr) \frac{\partial(u - v)}{\partial t}\,dx \\ &\quad= \int_{\Omega}\operatorname{sgn}(u - v)\frac{\partial(u - v)}{\partial t}\,dx = \frac{d}{dt}\Vert u - v \Vert _{1}, \end{aligned}
(3.6)
\begin{aligned} &\int_{\Omega} \rho^{\alpha}\bigl(\vert \nabla u \vert ^{p- 2}\nabla u - \vert \nabla v \vert ^{p - 2}\nabla v\bigr) \cdot\nabla(u - v)g'_{n} \phi(x)\,dx \geq0, \end{aligned}
(3.7)
\begin{aligned} &\biggl|\int_{\Omega} \rho^{\alpha}\bigl(\vert \nabla u \vert ^{p- 2}\nabla u - \vert \nabla v \vert ^{p - 2}\nabla v\bigr) \cdot\nabla\phi (u - v)g'_{n}\bigl(\phi(u-v)\bigr)\,dx\biggr| \\ &\quad\leq c \int_{\{x:\rho^{\beta}(u-v)< \frac{1}{n}\}}\bigl|\rho^{\frac{\alpha }{p}}\rho^{\frac{\alpha(p-1)}{p}}\bigl( \vert \nabla u \vert ^{p- 2}\nabla u - \vert \nabla v\vert ^{p- 2}\nabla v\bigr)\bigr|\frac{1}{\phi}\,dx \\ &\quad\leq c \biggl( \int_{\{x:\rho^{\beta}(u-v)< \frac{1}{n}\}} \frac{\rho ^{\alpha}}{\rho^{p\beta}}\,dx \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \int_{\{x:\rho^{\beta}(u-v)< \frac{1}{n}\}} \bigl|\rho^{\frac{\alpha (p-1)}{p}}\bigl(\vert \nabla u \vert ^{p- 2}\nabla u - \vert \nabla v\vert ^{p- 2}\nabla v\bigr) \bigr|^{\frac{p}{p-1}}\,dx \biggr)^{\frac{p-1}{p}}. \end{aligned}
(3.8)

Let $$n\rightarrow\infty$$. If $$\{ x \in\Omega:\rho^{\beta}|u - v| = 0\}$$ is a set with 0 measure, then

$$\biggl( \int_{\{x:\rho^{\beta}(u-v)=0\}} \frac{\rho^{\alpha}}{\rho^{p\beta }}\,dx \biggr)^{\frac{1}{p}}=0.$$

If $$\{ x \in\Omega:\rho^{\beta}|u - v| = 0\}$$ is a set with positive measure, then

$$\biggl( \int_{\{x:\rho^{\beta}(u-v)=0\}} \bigl|\rho^{\frac{\alpha (p-1)}{p}}\bigl(\vert \nabla u \vert ^{p- 2}\nabla u - \vert \nabla v\vert ^{p- 2}\nabla v\bigr) \bigr|^{\frac{p}{p-1}}\,dx \biggr)^{\frac{p-1}{p}}=0.$$

Both cases lead to the right hand side of (3.8) going to 0 as $$n\rightarrow\infty$$. Meanwhile,

\begin{aligned} &\lim_{n\rightarrow\infty}\biggl| \int_{{\Omega}} {\bigl(f(u,x,t) - f(v,x,t)\bigr){g_{n}} \bigl(\phi(u - v)\bigr)\,dx} \biggr|\\ &\quad=\biggl|\int_{{\Omega}} \bigl(f(u,x,t) - f(v,x,t)\bigr)\operatorname{sgn} \bigl(\phi(u - v)\bigr)\,dx \biggr|\\ &\quad=\biggl| \int _{{\Omega}} \bigl(f(u,x,t) - f(v,x,t)\bigr)\operatorname{sgn}(u - v)\,dx \biggr|\\ &\quad\leq c \int_{{\Omega}} |u-v|\,dx. \end{aligned}

Now, let $$n\rightarrow\infty$$ in (3.5). Then

$$\frac{{d}}{{{d}t}}{\Vert {u - v} \Vert _{1}} \leqslant c{\Vert {u - v} \Vert _{1}}.$$

It implies that

$$\int_{\Omega}{\bigl\vert {u(x,t) - v(x,t)} \bigr\vert { d}x} \leqslant c(T) \int_{\Omega}{ \vert {{u_{0}} - {v_{0}}} \vert \,{d}x}.$$

Theorem 1.3 is proved. □

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## Acknowledgements

The paper is supported by NSF of China (no. 11371297) and supported by NSF of Fujian Province (no. 2015J01592), China.

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Correspondence to Huashui Zhan. 