1. Introduction

The objective of the paper is to study the nonnegative weak solutions of nonlinear parabolic equation with the type

$u_{t}=\text{div}(\mid \nabla u^{m}\mid^{p-2}\nabla u^{m})-a(x)u^{mq_{1}}|\nabla u^{m}|^{p_1}+f_0(u^{m})\int_{\Omega}K(y)|u^{m}(y, t)|^{\beta}dy+g(x), \ \text{ in }S=\Omega\times (0, \infty),$

(1.1)

$u(x, 0)=u_{0}(x), \ \ x\in \Omega,$

(1.2)

$u(x, t)=0, \ \ (x, t)\in \partial \Omega\times (0, \infty),$

(1.3)

where $\Omega\subset \mathbb{R}^{N}$ is a bounded open domain with smooth boundary $\partial \Omega$, $\int_{\Omega}K(y)|u(y, t)|^{\beta}dy$ represents a nonlocal function dependent on spatial domain $\Omega$, $a(x)\geq0$ is a bounded function, $K(x)$ and $g(x)$ are bounded functions too, and $\nabla$ is the spatial gradient operator. We assume that $p>1, m>1$, $p_1\leq 2$, $p>2p_1$, $N\geq 1$,

$0\leq u_{0}^m(x)\in L^{q-1+\frac{1}{m}}(\Omega), q>1, \ |f_{0}(s)|\leq c|s|^{\frac{1}{m}}, \ s\in R^{1}=(-\infty, \infty).$

(1.4)

As usual, the here and after, the constants $c$ may be different from one to another. The equation with the type of (1.1) has been suggested as the mathematical model for a variety of problems in mechanics, physics and biology, which can be found in ^{[10,11,15,17]} et al. Equation (1.1) has been widely researched, whether it is linear or nonlinear, is uniformly parabolic or degenerate parabolic. In what follows, we only give a very roughly review.

If $a(x)=g(x)=f_0\equiv0$, the existence of nonnegative solution of the problem (1.1)-(1.3), defined in weak sense, is well established (see ^[10], ^[6] et al.).

If $g(x)=f\equiv0$, some special cases of equation (1.1) had been researched by Bertsh ^[3], Zhou ^[36] and Zhang ^[34] et al. For examples, the existence and the properties of the viscosity solution to the following equation are obtained in ^[3,36]

$u_t=u\triangle u-\gamma|\nabla u|^{2},$

(1.5)

where $\gamma$ is a positive constant. The existence and the properties of the viscosity solution to the following equation are obtained in ^[34]

$u_t=\triangle u-b(x)|u|^{q-1}|\nabla u|^{2},$

(1.6)

where $b(x)$ is a known function. The most important characteristic of the equation (1.5) or (1.6) lies in that, generally, the uniqueness of the solutions is not true, one can refer to ^{[4,9,29,34,36]} for the details. Thus, for the equation with the type of (1.1), one mainly concerns with the existence of the viscosity solution and the related properties such as the large time behavior, one can refer to ^[8,20,33,35] et al. for some progresses in the direction.

But if $p_{1}=0$, it is well-known the uniqueness of the solutions is true. Aassila ^[1] studied equation (1.1) when $p=2, m=1$ and proved the existence of solution by Schauder fixed point theorem, studied the convergence of the solution towards a steady state by using the point of view in dynamical systems. Cholewa and Dlotko ^[7], Teman ^[28] considered the following problem

$u_{t}-\text{div}(\mid \nabla u\mid^{p-2}\nabla u)+|u|^{\alpha}u=f_{0}(u)+g(x),$

(1.7)

and proved the existence of global attractor in $L^{2}$ which is in fact a bounded set in $W_{0}^{1, p}\bigcap L^{\alpha+2}$. Chen ^[20] studied the long time behavior of solutions for following equation

$u_{t}-\text{div}(\mid \nabla u\mid^{p-2}\nabla u)+a(x)|u|^{\alpha}u=f_0(u)\int_{\Omega}K(y)|u(y, t)|^{\beta}dy+g(x),$

(1.8)

and obtained the existence and $L^{p}$ estimate of the global attractor.

While the papers, first by Nakao-Chen ^[25] and later by Chen-Wang ^[6], had studies the global existence and the gradient estimate for the quasilinear parabolic equation of $m$-Laplacian type with a nonlinear convection term, the typical equations included in ^[6,25] are with the form as

$u_{t}=\text{div}(u^{r}|\nabla u|^{p-2}\nabla u)+\nabla A(u).$

(1.9)

In our paper, we will study the global solution of equation (1.1) with the initial value (1.2) and homogeneous boundary value (1.3) by the usual regularized method. The main techniques are inspired by ^[6,25]. However, due to the local and the nonlocal nonlinearity of the equation we considered, even to prove the initial value condition, we have to put some restrictions in the exponents of $m, p, p_1, q_1$. In particular, as we have said, instead of the nonlinear convection term $\nabla A(u)$ in equation (1.9), equation (1.1) contains the damping term $-a(x)u^{mq_{1}}|\nabla u^{mp_1}|$, the uniqueness of the solutions generally is not true. We can only prove the uniqueness of the solutions under the condition $p_1=0$. If $p_1\neq 0$ we only can prove the uniqueness of the viscosity solutions. At the same time, comparing with ^[5], since equation (1.1) is more complicated, how to get the estimate in the gradient term of the solution, and how to prove the continuity of the solution etc, become more difficult. A clear promotion lies in that we put not any restrictions in the derivative $f_0'(s)$ of the function $f_0(s)$, while it must satisfy that $|f_0'(s)|\leq c|s|^{r-1}$ in ^[5]. Other related works on equation (1.1), one can refer to the references ^{[2,14,16,18,19,22,24,27,30,31,32]} et al.

Now we quote the following definition.

Definition 1.1. A nonnegative function $u(x, t)$ is called a weak solution of (1.1)-(1.3) if $u$ satisfies

$(i)$

$u\in L^{\infty}_{\text{loc}}(0, \infty; L^{\infty}(\Omega)),$

(1.10)

$u_{t}\in L^{2}_{\text{loc}}(0, \infty; L^{2}(\Omega)), \ \ u^{m}\in L^{\infty}_{\text{loc}}(0, \infty; W^{1, p}_{0}(\Omega)),$

(1.11)

$(ii)$

$\iint_{S}\left[u\varphi_{t}-\mid \nabla u^{m}\mid^{p-2}\nabla u^{m}\cdot \nabla\varphi-a(x)u^{mq_{1}}|\nabla u^{m}|^{p_1} \varphi\right]dxdt +\iint_{S}\left[f_0(u^m)\int_{\Omega}K(y)|u^m(y, t)|^{\beta}dy+g(x)\right]\varphi dxdt=0, \ \forall \varphi \in C_{0}^{1}(S);$

(1.12)

$(iii)$

$\lim\limits_{t\rightarrow 0}\int_{\Omega}\mid u(x, t)-u_{0}(x)\mid dx=0.$

(1.13)

We are to get the solution of problem (1.1)-(1.3) by considering the regularized equation

$u_{t}=\text{div}((|\nabla u^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}\nabla u^{m})-a(x)u^{mq_{1}}|\nabla u^{m}|^{p_1}+f_0(u^{m})\int_{\Omega}K(y)|u^{m}(y, t)|^{\beta}dy+g(x),$

(1.14)

with the initial value (1.2) and the homogeneous boundary value (1.3). Here $0\leq u_{0k}(x)$ is a suitable smooth function such that $u_{0k}(x)\in L^{\infty}(\Omega)$, $\lim_{k\rightarrow \infty} \|u_{0k}^{m}\|_{q-1+\frac{1}{m}}=\|u_0^{m}\|_{q-1+\frac{1}{m}}$.

Definition 1.2. If $u_k$ is the solution of the initial boundary value problem of (1.14)-(1.2)-(1.3), $\lim_{k\rightarrow \infty}u_{k}=u$, a.e in $S$, $u$ is a weak solution of (1.1)-(1.3), then $u$ is said to be a viscosity solution.

We need some important lemmas in order to get our results.

Lemma 1.1. If $1\leq l < N$, $1+\beta\leq q$, $1\leq r\leq q\leq (1+\beta)Nl/(N-l)$, $u^{1+\beta}\in W^{1, l}(\Omega)$, then

$\|u\|_{q}\leq c^{1/(1+\beta)}\|u\|_{r}^{1-\theta}\|u^{1+\beta}\|^{\theta/(1+\beta)}_{1, l},$

(1.15)

where $\theta=(\beta+1)(r^{-1}-q^{-1})/(N^{-1}-l^{-1}+(\beta+1) r^{-1})$.

This lemma is a general version of Gagliardo-Nirenberg inequality, it is first proved by M. Nakao ^[23].

Lemma 1.2. Let $y(t)$ be a nonnegative function on $(0, T]$. If it satisfies

$y'(t)+At^{\lambda\theta-1}y^{1+\theta}(t)\leq Bt^{-k}y(t)+Ct^{-\delta}, 0<t\leq T,$

(1.16)

where $A, \theta>0$, $\lambda\theta\geq 1$, $B, C\geq 0, k\leq 1$, then

$y(t)\leq A^{-\frac{1}{\theta}}(2\lambda+2BT^{1-k})^{\frac{1}{\theta}}t^{-\lambda}+2C(\lambda+BT^{1-k})^{-1}t^{1-\delta}, 0<t\leq T.$

(1.17)

This lemma can be found in ^[26].

Lemma 1.3. Suppose $L_{1}\geq 1$, $r, R, M>0$, $\lambda_1>0$. For $n=2, 3, \cdots$, let

$L_n=RL_{n-1}-M, \ \theta_n=NR(1-L_{n-1}L_{n}^{-1})(N(R-1)+r)^{-1},$

$\beta_n=(L_n+M)\theta_{n}^{-1}-L_n, \ \lambda_n=(1+\lambda_{n-1}(\beta_n-M))\beta_{n}^{-1}.$

Then

$\lim\limits_{n\rightarrow\infty} \lambda_n=\frac{L_1\lambda_1r+N}{l_1+MN}.$

(1.18)

This lemma also was first proved in ^[25], then used in ^[6].

In our paper, we assume that $p>1+\frac{1}{m}$, so equation (1.1) is a doubly degenerate parabolic equation. By considering the solution $u_{k}$ of the regularized problem (1.14)-(1.2)-(1.3) and using Moser iteration technique, we get $u_{k}$'s local bounded properties and the local bounded properties of the $L^{p}$-norm of the gradient $\nabla u_{k}$. By the compactness theorem, we get the existence of the viscosity solution of the diffusion equation itself. In details, we will prove the following theorems.

Theorem 1.1. It is supposed that $K$, $g$ are suitable smooth bounded functions, $a(x)\in C(\overline{\Omega})$ and exists $a_{0}>0$, such that $a(x)\geq a_{0}$ in $\Omega$, $f_0$ satisfies (1.4). If $p>1+\frac{1}{m}$, $u_{0}(x)\geq 0$,

$u_{0}^{m}(x)\in L^{q-1+\frac{1}{m}}(\Omega), 3>q>2-\frac{1}{m},$

(1.19)

$p_1\leq 2, \ 2p_1<p, \ \beta<\max\{p-1-\frac{1}{m}, q-1+\frac{1}{m}\},$

(1.20)

$\epsilon=\max\{\frac{mNq_1}{Nm(p-1)-N+mq}+\frac{p_1(m(p-1)+m-2)}{m(p-1)-1}, \frac{(\beta+m)N}{Nm(p-1)-N+mq}\}<1,$

(1.21)

then the problem (1.1)-(1.3) has a weak viscosity solution $u$, satisfying

$u^{m}\in L^{\infty}_{\rm loc}(0, \infty; L^{q+1-\frac{1}{m}}(\Omega))\bigcap L^{\infty}_{\text{loc}}(0, \infty; W_{0}^{1, p}(\Omega)),$

(1.22)

and

$\|u^{m}(t)\|_{\infty}\leq c(1+t^{-\lambda})(1+t)^{-1/(p-1-\frac{1}{m})}, t>0,$

(1.23)

where $\lambda= N(pq+(p-1-\frac{1}{m})N)^{-1}$. Moreover, if $p>2$, then

$\|\nabla u^{m}\|_{p}\leq c(1+t^{-\delta_{1}})(1+t)^{-\sigma}, t>0,$

(1.24)

where

$\delta_{1}=\max\{1+\frac{m-1}{m(p-1)-1}, \delta-1\}, \delta=\max\{\frac{m+1}{m}, 2\beta\},$

and

$\sigma=\frac{p[m(2q_1+1)-1]+mp_1}{[m(p-1)-1](p-p_1)}.$

Remark 1.1. The condition (1.21) is only used to prove (1.13). We conjecture that this condition can be weaken.

Theorem 1.2. Let $u$ be a nonnegative weak solution of problem (1.1)-(1.3). If $g(x)\leq 0$, $f_{0}'(s)\geq 0$, if $p>1+\frac{1}{m}$, $p_{1}+q_{1}>(p-1)$ then

${\rm supp}u(., s)\subset {\rm supp}u(., t),$

(1.25)

for all $s, t$ with $0 < s < t$.

---

2. The $L^{\infty}$ estimation of the solution

Instead of considering the regularized problem (1.14)-(1.2)-(1.3) directly as one deals with the case $m=1$, we have to consider the following approximate problem. For small $s>0$, we consider

$u_{t}=\text{div}((|\nabla u^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}\nabla u^{m})-a(x)u^{mq_{1}}|\nabla u^{m}|^{p_1}+f_0(u^m)\int_{\Omega}K(y)|u^{m}(y, t)|^{\beta}dy+g(x),$

(2.1)

$u(x, 0)=u_{0k}(x)+s, x\in \Omega,$

(2.2)

$u(x, t)=s, x\in \partial \Omega, t\geq 0,$

(2.3)

where $0\leq u_{0k}(x)$ is a suitable smooth function such that $u_{0k}(x)\in L^{\infty}(\Omega)$, $\lim_{k\rightarrow \infty} \|u_{0k}^{m}\|_{q-1+\frac{1}{m}}=\|u_0^{m}\|_{q-1+\frac{1}{m}}$.

Similar as the chapter 8 of ^[13], in which the existence of the initial boundary value problem of the quaslinear equation in the divergent form is obtained, by Leray-Schauder fixed point theory, using the condition $p_1\leq 2$, we know that problem (2.1)-(2.3) has a nonnegative classical solution $u_{ks}$, we omit the details here.

Let $s\rightarrow 0$. In a similar way as ^[33], we are able to prove that

$u_{ks}\rightarrow u_{k}, \ \text{in} \ \ C(S),$

$\nabla u_{ks}^{m}\rightharpoonup \nabla u_{k}^{m}, \ \text{in} \ \ L^{p}(S),$

$u_{kst}\rightharpoonup \nabla u_{kt}, \ \text{in} \ \ L^{2}(S),$

$|\nabla u^{m}_{ks}|^{p-2}\nabla u^{m}_{ksx_{i}}\rightharpoonup*\ |\nabla u^{m}_{k}|^{p-2}\nabla u^{m}_{kx_{i}}, \text{weakly star in}\ L_{\text{loc}}^{\infty}(0, \infty; L^{\frac{p}{p-1}}(\Omega)),$

and $u_{k}$ is the solution of equation (2.1) with the following initial boundary values

$u(x, 0)=u_{0k}(x), x\in \Omega,$

(2.4)

$u(x, t)=0, x\in \partial \Omega, t\geq 0.$

(2.5)

Lemma 2.1. Assume that

$(H_1)$ $a(x)\in C(\overline{\Omega})$ and exists $a_{0}>0$, such that $a(x)\geq a_{0}$ in $\Omega$;

$(H_2)$ $f_{0}(s)\in C(R^{1})$, $|f_{0}(s)|\leq K_{0}|s|^{\frac{1}{m}}$, for some $K_0>0$.

$(H_3)$ $g(x), K(x)\in L^{\infty}$.

In addition, $\beta+\frac{1}{m} < q_1$, $3>q\geq 2-\frac{1}{m}$, then $u^{m}_{k}\in L_{\text{loc}}^{\infty}(0, \infty; L^{q-1+\frac{1}{m}}(\Omega))$ and

$\|u^{m}_{k}\|_{q-1+\frac{1}{m}}\leq c(1+t)^{-\frac{1}{p-1-\frac{1}{m}}}, t\geq 0.$

(2.6)

Proof. In the proof what follows, we only denote $u_{k}$ as $u$ for simplicity. We only give the proof of the case $q>2-\frac{1}{m}$, if $q=2-\frac{1}{m}$, one can get the conclusion just a minor version. Let $A_{n}=(q-2) n^{3-q}, B_{n}=(3-q)n^{2-q}$, and

$f_{n}(s)=\left\{ \begin{array}{cc} s^{q-1}, \ \ {\rm if}\ \ s\geq \frac{1}{n}, \\ A_{n}s^{2}+B_{n}s, \ \ {\rm if}\ \ 0\leq s<\frac{1}{n}. \end{array} \right.$

Suppose that $n>k$, multiply (2.1) with $f_{n}(u^{m})$ and integrate it on $\Omega$. Since $f'(s)>0$, then we have

$\int_{\Omega}f_{n}(u^{m})\text{div}(|\nabla u^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}\nabla u^{m})dx=-\int_{\Omega}(|\nabla u^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}|\nabla u^{m}|^{2}f_{n}'(u^{m})dx$

$\leq-\int_{\Omega}|\nabla u^{m}|^{p}f_{n}'(u^{m})dx=-\int_{\Omega}\mid\nabla\int_{0}^{u^{m}}(f'_{n}(s))^{\frac{1}{p}}ds\mid^{p}dx,$

(2.7)

$-\int_{\Omega}a(x)f_{n}(u^{m})u^{mq_1}|\nabla u^{m}|^{p_1}dx\leq 0.$

(2.8)

Suppose that $\mid f_{0}(s)\mid\leq K_{0}s^{r}$. Then

$|\int_{\Omega\bigcap\{u^{m}\leq \frac{1}{n}\}}f_0(u^m)f_{n}(u^{m})\int_{\Omega}K(y)|u^m(y, t)|^{\beta}dydx|$

$\leq c(K) \int_{\Omega\bigcap\{u^{m}\leq \frac{1}{n}\}}u^{mr}(A_{n}u^{2m}+B_{n}u^{m})dx\int_{\Omega}|u|^{m\beta}dy$

$\leq c(K)n^{1-q-r}\int_{\Omega}|u|^{m\beta}dy\leq c(K)n^{1-q-r}\|u^{m}\|_{q-1+\frac{1}{m}}^{\beta}.$

If $r=\frac{1}{m}$,

$|\int_{\Omega\bigcap\{u^{m}> \frac{1}{n}\}}f_0(u^m)f_{n}(u^{m})\int_{\Omega}K(y)|u^m(y, t)|^{\beta}dydx|$

$\leq c(K)\int_{\Omega}u^{m(r+q-1)}dx\int_{\Omega}|u|^{m\beta}dy \leq c\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}+\beta},$

we have

$|\int_{\Omega}f_0(u^m)f_{n}(u^{m})\int_{\Omega}K(y)|u^m(y, t)|^{\beta}dydx|$

$\leq c\|u^{m}\|_{q-1+\frac{1}{m}}^{\beta}[n^{1-q-\frac{1}{m}}+\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}}].$

(2.9)

$|\int_{\Omega\bigcap\{u^{m}> \frac{1}{n}\}}f_{n}(u^{m})g(x)dx|\leq c(g)\int_{\Omega}u^{m(q-1)}dx\leq c(g)\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1}.$

(2.10)

From the above calculations, we have

$\int_{\Omega}f_{n}(u^{m})u_{t}dx+\int_{\Omega}\mid\nabla\int_{0}^{u^{m}}(f'_{n}(s))^{\frac{1}{p}}ds\mid^{p}dx\leq c\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}+\beta}+O(\frac{1}{n^{q-1}}),$

(2.11)

by Poincare inequality, we have

$\int_{\Omega}f_{n}(u^{m})u_{t}dx+c\int_{\Omega}\mid\int_{0}^{u^{m}}(f'_{n}(s))^{\frac{1}{p}}ds\mid^{p}dx\leq c\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}+\beta}+O(\frac{1}{n^{q-1}}).$

(2.12)

Let $n\rightarrow\infty$ in (2.12). We can deduce that

$\frac{d}{dt}\int_{\Omega}u^{m(q-1)+1}dx+c\int_{\Omega}u^{m[q-1+\frac{1}{m}+p-1-\frac{1}{m}]}dx\leq c\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}+\beta}.$

(2.13)

By Jessen inequality, from (2.13) we get

$\frac{d}{dt}\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}}+c\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}+p-1-\frac{1}{m}}\leq c\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}+\beta}.$

If

$\beta <p-1-\frac{1}{m}$

by young inequality,

$\frac{d}{dt}\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}}+c\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}+p-1-\frac{1}{m}}\leq c,$

then

$\|u^{m}\|_{q+1-\frac{1}{m}}\leq c(1+t)^{-\frac{1}{p-1-\frac{1}{m}}}.$

We get the desired result.

Lemma 2.2. If $p> 1+\frac{1}{m}$, $u_{k}$ is the solution of problem (2.1)-(2.4)-(2.5), then

$\|u^{m}_{k}\|_{\infty}\leq ct^{-\lambda}, \ 0<t\leq 1,$

(2.14)

$\|u^{m}_{k}\|_{\infty}\leq c(1+t)^{-\frac{1}{p-1-\frac{1}{m}}}, t\geq 1,$

(2.15)

where $\lambda=\frac{N}{(p-1-\frac{1}{m})N+qp}$.

Proof. Multiply (2.1) with $u^{m(l-1)}$, and integrate it on $\Omega$, then

$\int_{\Omega}u^{m(l-1)}u_{t}dx=\int_{\Omega}\text{div}(|\nabla u^{m}|+\frac{1}{k})^{\frac{p-2}{2}}\nabla u^{m})u^{m(l-1)}dx-\int_{\Omega}a(x)u^{mq_1}|\nabla u^{m}|^{p_1}u^{m(l-1)}dx$

$+\int_{\Omega}f_0(u^m)u^{m(l-1)}\int_{\Omega}K(y)|u^m(y, t)|^{\beta}dydx+\int_{\Omega}g(x)u^{m(l-1)}dx$

$=-(l-1)\int_{\Omega}(|\nabla u^{m}|+\frac{1}{k})^{\frac{p-2}{2}}|\nabla u^{m}|^{2}u^{m(l-2)}dx-\int_{\Omega}a(x)u^{mq_1}|\nabla u^{m}|^{p_1}u^{m(l-1)}dx$

$+\int_{\Omega}K(y)|u^m(y, t)|^{\beta}dy\int_{\Omega}f_0(u^m)u^{m(l-1)}dx+\int_{\Omega}g(x)u^{m(l-1)}dx$

$\leq-(l-1)\int_{\Omega}(|\nabla u^{m}|+\frac{1}{k})^{\frac{p-2}{2}}|\nabla u^{m}|^{2}u^{m(l-2)}dx$

$+c(K)\int_{\Omega}|u^m(y, t)|^{\beta}dy\int_{\Omega}u^{m(l-1)+1}dx+c(g)\int_{\Omega}u^{m(l-1)}dx,$

which deduces that

$\frac{d}{dt}\|u^{m}\|_{l-1+\frac{1}{m}}^{l-1+\frac{1}{m}}+c(l-1+\frac{1}{m})^{2-p}\int_{\Omega} \mid\nabla u^{m\frac{p+l-1+\frac{1}{m}-1-\frac{1}{m}}{p}}\mid^{p}dx\leq c\|u^{m}\|_{l-1+\frac{1}{m}}^{l-1+\frac{1}{m}}\|u^{m}\|_{q-1+\frac{1}{m}}^{q-1+\frac{1}{m}+\beta} +c\|u^{m}\|_{l-1+\frac{1}{m}}^{l-1}$

$\leq\|u^{m}\|_{l-1+\frac{1}{m}}^{l-1+\frac{1}{m}} +c\|u^{m}\|_{l-1+\frac{1}{m}}^{l-1}, ({\rm by}\ (2.6)).$

Set $L=l-1+\frac{1}{m}$. Then

$\frac{d}{dt}\|u^{m}\|_{L}^{L}+cL^{2-p}\int_{\Omega} \mid\nabla u^{m\frac{L+p-1-\frac{1}{m}}{p}}\mid^{p}dx\leq c\|u^{m}\|_{L}^{L+\beta} +c\|u^{m}\|_{L}^{L-\frac{1}{m}},$

(2.16)

where $c$ is a constant independent of $l$.

Now, if we choose $L_{1}=q-1+\frac{1}{m}, L_{n}=rL_{n-1}-(p-1-\frac{1}{m})$, $\theta_{n}=rN(1-L_{n-1}L_{n}^{-1})(p+N(r-1))^{-1}$, $\mu_{n}=(L_{n}+p-1-\frac{1}{m})\theta_{n}^{-1}-L_{n}$, $r>1+(p-1-\frac{1}{m})q^{-1}$, $n=2, 3, \cdots$, by Lemma 1.3, we have

$\|u^{m}\|_{L_{n}}\leq c^{p/(L_{n}+p-1-\frac{1}{m})}\|u^m\|_{L_{n-1}}^{1-\theta_{n}}\|\nabla u^{m(L_{n}+p-1-\frac{1}{m})/p}\|_{p}^{p\theta_{n}/(p-1-\frac{1}{m}+L_{n})}.$

(2.17)

If we choose $L=L_{n}$ in (2.16), by (2.17), we have

$\frac{d}{dt}\|u^{m}\|_{L_{n}}^{L_{n}}+c^{-p/\theta_{n}}L_{n}^{2-p}\| u^{m}\|_{L_{n}}^{L_{n}+\mu_{n}} \| u^{m}\|_{L_{n-1}}^{p-1-\frac{1}{m}-\mu_{n}}\leq c\|u^{m}\|_{L_{n}}^{L_{n}+\beta} +c\|u^{m}\|_{L_{n}}^{L_{n}-\frac{1}{m}}.\ 0<t\leq 1.$

(2.18)

We will prove that there exist two bounded sequences $\{\xi_{n}\}, \{\lambda_{n}\}$ such that

$\|u^{m}\|_{L_{n}}\leq \xi_{n}t^{-\lambda_{n}}, \ 0<t\leq1.$

(2.19)

Without loss of the generality, we may assume that $\|u^{m}\|_{L_{n}}\geq 1$. Otherwise, choosing $\xi_{n}\equiv 1$, (2.17) is true naturally. Thus, by (2.16), we have

$\frac{d}{dt}\|u^{m}\|_{L_{n}}^{L_{n}}+c^{-p/\theta_{n}}L_{n}^{2-p}\| u^{m}\|_{L_{n}}^{L_{n}+\mu_{n}} \| u^{m}\|_{L_{n-1}}^{p-1-\frac{1}{m}-\mu_{n}}\leq c\|u^{m}\|_{L_{n}}^{L_{n}+\beta}.\ 0<t\leq 1.$

If $n=1$, by Lemma 2.1, $\lambda_{1}=0, \xi_{1}=\sup_{t\geq 0}\|u^{m}(t)\|_{q-1+\frac{1}{m}}$ makes (2.19) sure. If (2.19) is true for $n-1$, from (2.18),

$\frac{d}{dt}\|u^{m}\|_{L_{n}}^{L_{n}}+c^{-p/\theta_{n}}L_{n}^{2-p}\|u^{m}\|_{L_{n}}^{L_{n}+\mu_{n}} \xi_{n-1}^{p-1-\frac{1}{m}-\mu_{n}}t^{-(p-1-\frac{1}{m}-\mu_{n})\lambda_{n-1}}\leq c\|u^{m}\|_{L_{n}}^{L_{n}+\beta}.\ 0<t\leq 1.$

(2.20)

we can choose

$\lambda_{n}=(\lambda_{n-1}(\mu_{n}-p+1+\frac{1}{m})+1)\mu_{n}^{-1}, \ \xi_{n}=\xi_{n-1}(c^{p/\theta_{n}}L_{n}^{p-1}\lambda_{n})^{1/\mu_{n}}, \ n=2, 3, \cdots,$

$\frac{d}{dt}\|u^{m}\|_{L_{n}}^{L_{n}}+c\|u^{m}\|_{L_{n}}^{L_{n}+\lambda_{n}}\leq c\|u^{m}\|_{L_{n}}^{L_{n}+\beta}.\ 0<t\leq 1.$

(2.21)

Suppose that

$\beta<\frac{N}{(p-1-\frac{1}{m})N+qp},$

(2.22)

and notice that as $n\rightarrow \infty$, $\lambda_{n}\rightarrow\lambda=\frac{N}{(p-1-\frac{1}{m})N+pq}$.

$\frac{d}{dt}\|u^{m}\|_{L_{n}}^{L_{n}}+c\|u^{m}\|_{L_{n}}^{L_{n}+\lambda_{n}}\leq 0.\ 0<t\leq 1.$

(2.23)

By Lemma 1.2 and (2.23), we know (2.19) is true.

Moreover, it is easy to see that $\{\xi_{n}\}$ is bounded. Thus, by Lemma 1.2, (2.14) is true.

To prove (2.15), we set $\tau=\log(1+t), t\geq 1$, $w(\tau)=(1+t)^{\frac{1}{p-1-\frac{1}{m}}}u^{m}(t)$. By (2.16), we have

$\frac{d}{d\tau}\|w(\tau)\|_{L}^{L}+cL^{2-p}\|\nabla w^{\frac{L+p-1-\frac{1}{m}}{p}}\|_{p}^{p}\leq \frac{L}{p-1-\frac{1}{m}}\|w(\tau)\|_{L}^{L}+c\|w(\tau)\|_{L}^{L+\beta}, \ \tau\geq \log 2.$

(2.21)

By the lemma 3.1 in ^[24], we can get (2.15), we omit details here.

---

3. The $L^{\infty}$ estimation of the gradient

Lemma 3.1. If $p> \max\{2, 1+\frac{1}{m}\}$, $u_{k}$ is the solution of problem (2.1)-(2.4)-(2.5), then

$\|\nabla u^{m}_{k}\|_{p}\leq ct^{-(1+\frac{m-1}{m(p-1)-1})}+ct^{1-\delta}, \ 0<t\leq 1,$

(3.1)

$\|\nabla u^{m}_{k}\|_{p}\leq c(1+t)^{-\frac{p(m(2q_1+1)-1)+mp_1}{(m(p-1)-1)(p-p_1)}}, t\geq 1.$

(3.2)

Here $\delta =\max\{\frac{m-1}{m}, 2\beta\}$.

Proof. Multiply (2.1) with $u^{m}_{t}$, and integrate it on $\Omega$, then

$m\int_{\Omega}u^{m-1}(u_{t})^{2}dx=\int_{\Omega}\text{div}((|\nabla u^{m}|^2+\frac{1}{k})^{\frac{p-2}{2}}\nabla u^{m})u_{t}^{m}dx-\int_{\Omega}a(x)u^{mq_1}|\nabla u^{m}|^{p_1}u_{t}^{m}dx$

$+\int_{\Omega}f_0(u^m)u^{m}_{t}dx\int_{\Omega}K(y)|u^{m}(y, t)|^{\beta}dy+\int_{\Omega}g(x)u^{m}_{t}dx .$

(3.3)

$\int_{\Omega}\text{div}((|\nabla u^{m}|^2+\frac{1}{k})^{\frac{p-2}{2}}\nabla u^{m})u_{t}^{m}dx=-\int_{\Omega}(|\nabla u^{m}|+\frac{1}{k})^{\frac{p-2}{2}}\nabla u^{m}\nabla u^{m}_{t}dx$

$=-\frac{1}{2}\int_{\Omega}(|\nabla u^{m}|^2+\frac{1}{k})^{\frac{p-2}{2}}|\nabla u^{m}|^{2}_{t}dx$

$=-\frac{1}{2}\int_{\Omega}\frac{d}{dt}\int_{0}^{|\nabla u^{m}|^{2}}(s+\frac{1}{k})^{\frac{p-2}{2}}dsdx=-\frac{1}{2}\frac{d}{dt}\Gamma_{k}(|\nabla u^{m}|^{2}),$

(3.4)

where we define that

$\Gamma_{k}(|\nabla u^{m}|^{2})=\int_{\Omega}\int_{0}^{|\nabla u^{m}|^{2}}(s+\frac{1}{k})^{\frac{p-2}{2}}dsdx.$

At the same time,

$|-a(x)u^{mq_1}|\nabla u^{m}|^{p_1}u_{t}^{m}dx| \leq \frac{m}{2}\int_{\Omega}u^{m-1}(u_{t})^{2}dx+c\int_{\Omega}|u^{m}|^{2q_1+\frac{m-1}{m}}|\nabla u^{m}|^{2p_1}dx.$

(3.5)

By Lemma 2.1, using Young inequality and H$\ddot{o}$lder inequality,

$|\int_{\Omega}f_0(u^m)u^{m}_{t}dx\int_{\Omega}K(y)|u^{m}(y, t)|^{\beta}dy|$

$\leq c(\varepsilon\int_{\Omega}u^{m-1}(u_{t})^{2}dx+c\int_{\Omega}u^{m+1}dx)\|u^{m}\|_{q-1+\frac{1}{m}}^{\beta}$

$\leq c\varepsilon\int_{\Omega}u^{m-1}(u_{t})^{2}dx+c\int_{\Omega}u^{m+1}dx$

$|\int_{\Omega}g(x)u^{m}_{t}dx|\leq\varepsilon\int_{\Omega}u^{m-1}(u_{t})^{2}dx+c\int_{\Omega}u^{m-1}dx.$

By (3.3)-(3.5), we have

$\int_{\Omega}u^{m-1}(u_{t})^{2}dx+\frac{1}{m}\frac{d}{dt}\Gamma_{k}(|\nabla u^{m}|^{2})\leq c\int_{\Omega}|u^{m}|^{2q_1+\frac{m-1}{m}}|\nabla u^{m}|^{2p_1}dx+c\int_{\Omega}u^{m+1}dx+c\int_{\Omega}u^{m-1}dx.$

(3.6)

Multiply (2.1) with $u^{m}$, and integrate it on $\Omega$, then

$\frac{1}{m+1}\int_{\Omega}\frac{d}{dt}u^{m+1}dx=\int_{\Omega}{\rm div}((|\nabla u^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}\nabla u^{m})u^{m}dx-\int_{\Omega}a(x)u^{mq_1}|\nabla u^{m}|^{p_1}u^{m}dx$

$+\int_{\Omega}f_0(u^m)u^{m}\int_{\Omega}K(y)|u^m(y, t)|^{\beta}dydx+\int_{\Omega}g(x)u^{m}dx$

$=-\int_{\Omega}(|\nabla u^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}|\nabla u^{m}|^{2}dx-\int_{\Omega}a(x)u^{mq_1}|\nabla u^{m}|^{p_1}u^{m}dx$

$+\int_{\Omega}f_0(u^m)u^{m}\int_{\Omega}K(y)|u^m(y, t)|^{\beta}dydx+\int_{\Omega}g(x)u^{m}dx$

and

$\Gamma_{k}(|\nabla u^{m}|^{2})\leq\int_{\Omega}(|\nabla u^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}|\nabla u^{m}|^{2}dx$

$=-\frac{1}{m+1}\int_{\Omega}\frac{d}{dt}u^{m+1}dx-\int_{\Omega}a(x)u^{mq_1}|\nabla u^{m}|^{p_1}u^{m}dx+\int_{\Omega}f_0(u^m)u^{m}\int_{\Omega}K(y)|u^m(y, t)|^{\beta}dydx+\int_{\Omega}g(x)u^{m}dx$

$\leq\frac{1}{m+1} \|u^{\frac{m+1}{2}}\|_{2}\|u^{\frac{m-1}{2}}u_{t}\|_{2} +c(K)\int_{\Omega}|u^m(y, t)|^{\beta}dy\int_{\Omega}u^{m+1}dx+c(g)\int_{\Omega}u^{m}dx,$

so

$\frac{1}{m}\frac{d}{dt}\Gamma_{k}(|\nabla u^{m}|^{2})+(m+1)^{2}\|u^{\frac{m+1}{2}}\|_{2}^{-2}\Gamma_{k}^{2}(|\nabla u^{m}|^{2})$

$\leq c\int_{\Omega}|u^{m}|^{2q_1+\frac{m-1}{m}}|\nabla u^{m}|^{2p_1}dx+c\int_{\Omega}u^{m+1}dx+c\int_{\Omega}u^{m-1}dx$

$+c\|u^{\frac{m+1}{2}}\|_{2}^{-2}(\int_{\Omega}|u^m(y, t)|^{\beta}dy\int_{\Omega}u^{m+1}dx+\int_{\Omega}u^{m}dx)^{2}$

$\leq c\int_{\Omega}|u^{m}|^{2q_1+\frac{m-1}{m}}|\nabla u^{m}|^{2p_1}dx+c\int_{\Omega}u^{m+1}dx+c\int_{\Omega}u^{m-1}dx$

$+c(\int_{\Omega}|u^m(y, t)|^{\beta}dy)^{2}\int_{\Omega}u^{m+1}dx+c\|u^{\frac{m+1}{2}}\|_{2}^{2\frac{m}{m+1}-2}.$

(3.7)

Setting $2\gamma=2q_1+1-\frac{1}{m}$, for $\forall a\in[0, 2\gamma]$, if we notice that $p>2p_1$, then we have

$\int_{\Omega}|u^{m}|^{2a}|\nabla u^{m}|^{2p_1}dx\leq\|u^{m}(t)\|_{\infty}^{a}\left(\int_{\Omega}|u^{m}|^{\frac{(2\gamma-a)p}{p-2p_1}}dx\right)^{\frac{p-2p_1}{p}}\|\nabla u^{m}\|_{p}^{2p_1}.$

(3.8)

If $2\gamma\geq (p-2p_1)(N+1)/N$, let $a=(2\gamma-(p-2p_1)(1+\frac{q}{N}))^{+}$. By Lemma 1.3,

$\left(\int_{\Omega}|u^{m}|^{\frac{(2\gamma-a)p}{p-2p_1}}dx\right)^{\frac{p-2p_1}{p}} \leq c\|u^{m}(t)\|_{s}^{(2\gamma-a)(1-\theta)}\|\nabla u^{m}\|_{p}^{p-2p_1},$

(3.9)

where $\theta=(s^{-1}-(1-\frac{2p_1}{p})(2\gamma-a)^{-1})/(N^{-1}-p^{-1}+s^{-1})$, and $s=(2\gamma-p+2p_1-a)N/(p-2p_1)$ when $2\gamma \geq (p-2p_1)(1+q/N)$, $s=q$ when $(p-2p_1)(1+N^{-1})\leq2\gamma \leq (p-2p_1)(1+q/N)$. By Lemma 2.1 and Lemma 2.2, from (3.8), we have

$\int_{\Omega}|u^{m}|^{2a}|\nabla u^{m}|^{2p_1}dx\leq ct^{-\lambda a}\|\nabla u^{m}\|_{p}^{p}\leq ct^{-\lambda a}\Gamma_{k}(|\nabla u^{m}|^{2}).\ 0<t\leq 1.$

(3.10)

At the same time, if we choose $q=2$ in Lemma 2.1, we have

$\|u^{m}\|_{1+\frac{1}{m}}=\left(\int_{\Omega}u^{m+1}dx\right)^{\frac{m}{m+1}}\leq c(1+t)^{-(p-1-\frac{m}{m+1})^{-1}}\leq c,$

and

$\int_{\Omega}u^{m-1}dx\leq ct^{\frac{m-1}{m}\lambda}, \ \|u^{\frac{m+1}{2}}\|_{2}^{2}=\int_{\Omega}u^{m+1}dx\leq c.$

(3.11)

By (3.7) and Lemma 2.2, we have

$\Gamma_{k}'(t)+ct^{\frac{m+1}{m(p-1)-1}}\Gamma_{k}^{2}(t)\leq ct^{-\lambda a}\Gamma_{k}(t)+c(t^{-\lambda\frac{m-1}{m}}+t^{-2\beta\lambda}), \ 0<t\leq 1.$

(3.12)

If $2\gamma < (p-2p_1)(N+1)/N$ and $p-2p_1\leq 2a\leq 2\gamma$,

$\int_{\Omega}|u^{m}|^{2a}|\nabla u^{m}|^{2p_1}dx\leq c\|\nabla u^{m}\|_{1}^{2a(1-\theta)}\|\nabla u^{m}\|_{p}^{2a\theta+2p_1}\leq c \|\nabla u^{m}\|_{p}^{p}\leq c\Gamma_{k}(|\nabla u^{m}|^{2}).\ 0<t\leq 1.$

(3.13)

If $2\gamma < (p-2p_1)(N+1)/N$ and $p-2p_1\geq 2a\geq 0$,

$\int_{\Omega}|u^{m}|^{2a}|\nabla u^{m}|^{2}dx\leq c(1+\|\nabla u^{m}\|_{p}^{p})\leq c(1+\Gamma_{k}(|\nabla u^{m}|^{2})).\ 0<t\leq 1.$

(3.14)

(3.13) and (3.14) imply that (3.12) is still true when $2\gamma < (p-2p_1)(N+1)/N$. Using Lemma 1.2,

$\Gamma_{k}(t)\leq ct^{-(1+\frac{m-1}{m(p-1)-1})}+ct^{1-\delta}, \ 0<t\leq 1,$

where $\delta =\max\{\frac{m+1}{m}, 2\beta\}$. Then (3.1) is true. Now, we will prove (3.2). For $t\geq 1$, by (2.15)

$\int_{\Omega}|u^{m}|^{2a}|\nabla u^{m}|^{2p_1}dx\leq c\|\nabla u^{m}\|_{p}^{2}\|u^{m}(t)\|_{2\gamma p/p-2p_1}^{2\gamma}\leq c(1+t)^{-2\gamma/(p-1-\frac{1}{m})}\|\nabla u^{m}\|_{p}^{2p_1}.\ t\geq 1.$

(3.15)

$\Gamma_{k}(|\nabla u^{m}|^{2})=\int_{0}^{|\nabla u^{m}|^{2}}(s^{2}+\frac{1}{k})^{\frac{p-2}{2}}ds\leq c \|\nabla u^{m}\|_{p}^{p}=c(\|\nabla u^{m}\|_{p}^{2p_1})^{\frac{p}{2p_1}}, \ t\geq 1.$

(3.16)

$\|u^{\frac{m+1}{2}}\|^{2}_{2}=\left(\int_{\Omega}u^{m+1}dx\right)^{2}\leq c(1+t)^{-(p-1-\frac{1}{m})^{-1}}, \ t\geq 1.$

(3.17)

by (3.7), using (3.15)-(3.17)

$\Gamma_{k}'(t)+c(1+t)^{-(p-1-\frac{1}{m})^{-1}}\Gamma_{k}^{2}(t)\leq c(1+t)^{2\gamma/(p-1-\frac{1}{m})}(\Gamma_{k}(t))^{\frac{2p_1}{p}}$

$+c\int_{\Omega}u^{m+1}dx+c\int_{\Omega}u^{m-1}dx +c(\int_{\Omega}|u^m(y, t)|^{\beta}dy)^{2}\int_{\Omega}u^{m+1}dx+c\|u^{\frac{m+1}{2}}\|_{2}^{2(m-1)},$

by Young inequality,

$\Gamma_{k}'(t)+c(1+t)^{-(p-1-\frac{1}{m})^{-1}}\Gamma_{k}^{2}(t)\leq c(1+t)^{\frac{-m(2\gamma p+p_1)}{(m(p-1)-1)(p-p_1)}}+c(1+t)^{-\frac{m(m+1)}{m(p-1)-1}}$

$=c(1+t)^{-\frac{p(m(2q_1+1)-1)+mp_1}{(m(p-1)-1)(p-p_1)}}+c(1+t)^{-\frac{m(m+1)}{m(p-1)-1}},$

which means (3.2) is true.

Lemma 3.2. If $p> 1+\frac{1}{m}$, $u_{k}$ is the solution of problem (2.1) -(2.4)-(2.5), then

$\int_{t}^{T}\int_{\Omega}u^{m-1}_{k}(u_{kt})^{2}dxds\leq ct^{-(1+\frac{m-1}{m(p-1)-1})}+ct^{-(\lambda \gamma+\frac{m-1}{m(p-1)-1})}+ct^{-\frac{1+m}{m}\lambda}, \ 0<t\leq T.$

(3.18)

Proof. From (3.6), (3.10) and (2.14), we have

$\int_{\Omega}u^{m-1}(u_{t})^{2}dx+\frac{1}{m}\frac{d}{dt}\Gamma_{k}(|\nabla u^{m}|^{2})\leq c\int_{\Omega}|u^{m}|^{2q_1+\frac{m-1}{m}}|\nabla u^{m}|^{2p_1}dx+c\int_{\Omega}u^{m+1}dx+c\int_{\Omega}u^{m-1}dx$

$\int_{t}^{T}\int_{\Omega}u^{m-1}(u_{t})^{2}dxds\leq \Gamma_{k}(t)+c\int_{t}^{T}\int_{\Omega}|u^{m}|^{2q_1+\frac{m-1}{m}}|\nabla u^{m}|^{2p_1}dxds+c\int_{t}^{T}\int_{\Omega}u^{m+1}dx$

$\leq\Gamma_{k}(t)+c\int_{t}^{T}s^{-\lambda(2q_1+\frac{m-1}{m})}\Gamma_{k}(s)ds+c\int_{t}^{T}\int_{\Omega}u^{m+1}dx$

$\leq ct^{-(1+\frac{m-1}{m(p-1)-1})}+ct^{-(\lambda \gamma+\frac{m-1}{m(p-1)-1})}+ct^{-\frac{1+m}{m}\lambda}.$

(3.19)

---

4. The proof of Theorem 1.1

The proof of Theorem 1.1 from Lemma 2.1, Lemma 2.2, Lemma 3.1 and Lemma 3.2, using the compactness theory (cf ^[21]), there is a sequence (still denoted it as $\{u_{k}\}$) of $\{u_{k}\}$ such that when $k\rightarrow \infty$, $u_{k}\rightarrow u, \ a.e. \ {\rm in}\ S$ and so

$\lim\limits_{k\rightarrow \infty}f_0(u_{k}^{m})\int_{\Omega}K(y)|u_{k}^{m}(y, t)|^{\beta}dy= f_0(u^{m})\int_{\Omega}K(y)|u^{m}(y, t)|^{\beta}dy.$

Moreover, we have

$u_{k}\rightharpoonup \ u, \text{weakly* star in}\ L^{\infty}_{\rm loc}(0, \infty; L^{m(q-1)+1}(\Omega)),$

(4.1)

$u_{kt}\rightharpoonup u_{t}, \text{ weakly in} \ L^{2}(0, \infty; L^{2}(\Omega)), \nabla u^{m}_{k}\rightharpoonup \nabla u^{m}, \ \text{weakly in} \ L^{p}_{{\rm loc}}(0, \infty; L^{p}(\Omega))$

(4.2)

$|\nabla u^{m}_{k}|^{p-2} u^{m}_{kx_{i}}\rightharpoonup\ \chi_{i}, \text{weakly* in}\ L_{\text{loc}}^{\infty}(0, \infty; L^{\frac{p}{p-1}}(\Omega)),$

(4.3)

$a(x)u_{k}^{mq_1}|\nabla u^{m}_{k}|^{p_1}\rightharpoonup\ \nu, \text{weakly* in}\ L_{\text{loc}}^{\infty}(0, \infty; L^{\frac{p}{p_1}}(\Omega)),$

(4.4)

where $\chi=\{\chi_{i}: 1\leq i\leq N\}$ and every $\chi_{i}$ is a function in $L_{loc}^{\infty}(0, T; L^{\frac{p}{p-1}}(\Omega))$, $\nu \in L_{loc}^{\infty}(0, \infty; L^{\frac{p}{p_1}}(\Omega))$. (4.1) and (4.2) are clearly true.

In what follows, we only need to prove that

$\chi=|\nabla u^{m}|^{p-2}\nabla u^{m}, \ \ {\rm in}\ L_{{\rm loc}}^{\infty}(0, \infty; L^{\frac{p}{p-1}}(\Omega)).$

(4.5)

and

$\nu=a(x)u^{mq_1}|\nabla u^{m}|^{p_1}, \ \ {\rm in} \ L_{{\rm loc}}^{\infty}(0, \infty; L^{\frac{p}{p_1}}(\Omega)).$

(4.6)

It is easy to know that

$\iint_{S}\left(u\varphi_{t}-\chi\cdot \nabla\varphi-\nu \varphi+ f_0(u^{m})\int_{\Omega}K(y)|u^{m}(y, t)|^{\beta}dy\varphi+g(x)\varphi\right)dxdt=0, \ \forall \varphi \in C_{0}^{\infty}(S),$

(4.7)

so, if we can prove that

$\iint_{S}\mid \nabla u^{m}\mid^{p-2}\nabla u^{m}\cdot \nabla\varphi dxdt=\iint_{S}\chi\cdot \nabla\varphi dxdt, \ \forall \varphi \in C_0^{1}(S);$

(4.8)

$\iint_{S}a(x)u^{mq_1}|\nabla u^{m}|^{p_1}\varphi dxdt=\iint_{S}\nu \varphi dxdt, \ \ \forall \varphi \in C_0^{1}(S);$

(4.9)

then (4.5), (4.6) and (1.12) are true.

First, for any $\psi\in C^{\infty}_{0}(S)$, $0\leq\psi\leq 1$; $v^{m}\in L^{p}_{\text{loc}}(0, T; W^{1, p}_{0}(\Omega))$, we have

$\iint_{S}\psi(\mid \nabla u_{k}^{m}\mid^{p-2}\nabla u_{k}^{m}-\mid \nabla v^{m}\mid^{p-2}\nabla v^{m})\cdot\nabla(u_{k}^{m}-v^{m})dxdt\geq 0,$

(4.10)

If we multiply with $u^{m}_{k}\psi$ on two sides of (2.1), then we have

$\iint_{S}\psi\left(|\nabla u^{m}_{k}|^{2}+\frac{1}{k}\right)^{\frac{p-2}{2}}|\nabla u^{m}_{k}|^{2}dxdt=\frac{1}{m+1}\iint_{S}\psi_{t}u^{m+1}_{k}dxdt-\iint_{S}u^{m}_{k}\left(|\nabla u^{m}_{k}|^{2}+\frac{1}{k}\right)^{\frac{p-2}{2}}\nabla u^{m}_{k}\cdot\nabla\psi dxdt$

$-\iint_{S}a(x)u_{k}^{m(q_1+1)}|\nabla u^{m}_{k}|^{p_1}\psi dxdt+\iint_{S}[ f_0(u_{k}^{m})\int_{\Omega}K(y)|u_{k}^{m}(y, t)|^{\beta}dy+g(x)]u^{m}_{k}\psi dxdt.$

(4.11)

Noticing that when $1 < p < 2$,

$|\nabla u^{m}_{k}|^{2}\geq (|\nabla u_{k}^{m}|^{2}+\frac{1}{k})^{\frac{p}{2}}-(\frac{1}{k})^{\frac{p}{2}},$

$(|\nabla u_{k}^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}|\nabla u^{m}_{k}|\leq (|\nabla u_{k}^{m}|^{2}+\frac{1}{k})^{\frac{p-1}{2}},$

and when $p\geq 2$,

$(|\nabla u_{k}^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}|\nabla u_{k}^{m}|^{2}\geq|\nabla u_{k}^{m}|^{p},$

$(|\nabla u_{k}^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}|\nabla u_{k}^{m}|\leq(|\nabla u_{k}^{m}|^{p-1}+1),$

by (4.10), (4.11), we have

$\frac{1}{m+1}\iint_{S}\psi_{t}u^{m+1}_{k}dxdt-\iint_{S}u^{m}_{k}\left(|\nabla u^{m}_{k}|^{2}+\frac{1}{k}\right)^{\frac{p-2}{2}}\nabla u^{m}_{k}\cdot\nabla\psi dxdt$

$-\iint_{S}a(x)u_{k}^{m(q_1+1)}|\nabla u^{m}_{k}|^{p_1}\psi dxdt +(\frac{1}{k})^{\frac{p-2}{2}}\text{mes}\Omega$

$+\iint_{S}[ f_0(u_{k}^{m})\int_{\Omega}K(y)|u_{k}^{m}(y, t)|^{\beta}dy+g(x)]u^{m}_{k}\psi dxdt$

$-\iint_{S}\psi|\nabla u_{k}^{m}|^{p-2}\nabla u_{k}^{m}\cdot\nabla v^{m}dxdt-\iint_{S}\psi|\nabla v^{m}|^{p-2}\nabla v^{m}\cdot \nabla(u_{k}^{m}-v^{m})dxdt\geq 0.$

(4.12)

Since

$\left(|\nabla u^{m}_{k}|^{2}+\frac{1}{k}\right)^{\frac{p-2}{2}}\nabla u^{m}_{k}=|\nabla u_{k}^{m}|^{p-2}\nabla u_{k}^{m}+\frac{p-2}{2k}\int_{0}^{1}(|\nabla u_{k}^{m}|^{2}+\frac{s}{k})^{\frac{p-4}{2}}ds\nabla u^{m}_{k},$

and

$\lim\limits_{k\rightarrow\infty}\iint_{S}\int_{0}^{1}(|\nabla u_{k}^{m}|^{2}+\frac{s}{k})^{\frac{p-4}{2}}ds\nabla u^{m}_{k}\cdot\nabla\psi u^{m}_{k}dxdt=0,$

if we let $k\rightarrow\infty$ in (4.12), we have

$\frac{1}{m+1}\iint_{S}\psi_{t}u^{m+1}dxdt -\iint_{S}u^m\nu\psi dxdt-\iint_{S}u^m\chi\nabla\psi dxdt$

$-\iint_{S}\psi \chi\cdot\nabla v^{m}dxdt-\iint_{S}\psi|\nabla v^{m}|^{p-2}\nabla v^{m}\cdot \nabla(u^{m}-v^{m})dxdt$

$+\iint_{S}[ f_0(u^{m})\int_{\Omega}K(y)|u^{m}(y, t)|^{\beta}dy+g(x)]u^{m}\psi dxdt \geq 0.$

(4.13)

Now, we choose $\varphi=\psi u^{m}$ in (4.7),

$\frac{1}{m+1}\iint_{S}\psi_{t}u^{m+1}dxdt-\iint_{S}u^m\nu\psi dxdt -\iint_{S}\chi\cdot\nabla \psi u^m dxdt$

$+\iint_{S}[f_0(u^{m})\int_{\Omega}K(y)|u^{m}(y, t)|^{\beta}dy+g(x)]\psi u^{m}dxdt =\iint_{S}\psi \chi\cdot\nabla u^m dxdt.$

From this formula and (4.13), we have

$\iint_{S}\psi(\chi-|\nabla v^{m}|^{p-2}\nabla v^{m})\cdot \nabla(u^{m}-v^{m})dxdt\geq 0.$

(4.14)

Let $v^{m}=u^{m}-\lambda\varphi$, $\lambda\geq 0, \varphi\in C^{\infty}_{0}(S)$. Then

$\iint_{S}\psi(\chi_{i}-|\nabla(u^{m}-\lambda\varphi)|^{p-2}(u^{m}-\lambda\varphi)_{x_{i}})dxdt\geq 0.$

Let $\lambda\rightarrow 0$. We obtain

$\iint_{S}\psi(\chi_{i}-|\nabla u^{m}|^{p-2}u^{m}_{x_{i}})dxdt\geq0, \forall \varphi\in C^{\infty}_{0}(S).$

Moreover, if we choose $\lambda\leq 0$, we are able to get

$\iint_{S}\psi(\chi_{i}-|\nabla u^{m}|^{p-2}u^{m}_{x_{i}})dxdt\leq0, \forall \varphi\in C^{\infty}_{0}(S).$

Now, if we choose $\psi$ such that $\text{supp}\varphi\subset\text{supp}\psi$, and on $\text{supp}\varphi$, $\psi=1$, then we can get (4.8).

By a process of limitation, we can choose the test function $\varphi$ in (4.8) as $u^m$, then we have

$\lim\limits_{k\rightarrow 0}\iint_{S}|\nabla u_{k}^{m}|^{p}dxdt=\iint_{S}\chi\cdot \nabla u^m dxdt=\iint_{S}|\nabla u^{m}|^{p}dxdt.$

(4.15)

Due to (1.20), $2p_{1} < p$, then by H$\ddot{o}$lder inequality, we have

$\lim\limits_{k\rightarrow 0}\iint_{S}|\nabla u_{k}^{m}|^{p_1}dxdt=\iint_{S}\chi\cdot \nabla u^m dxdt=\iint_{S}|\nabla u^{m}|^{p_1}dxdt.$

(4.16)

By a refinement of Fatou's lemma, the theorem 1.4.1 in ^[12], we are easy to prove (4.9), and so (1.12) is true.

Secondly, we are to prove (1.13).

For small $r>0$, denote $\Omega_{r}=\{x\in\Omega: \text{dist}(x, \partial \Omega)\leq r\}$. For any $\eta>0$, let

$\text{sgn}_{\eta}(s)=\left\{\begin{array}{ccc} 1, \ \ &\ \ {\rm if}\ \ s>\eta, \\ \frac{s}{\eta}, \ \ &\ \ {\rm if}\ \ |s|\leq\eta, \\ -1, \ \ &\ \ {\rm if}\ \ s<-\eta. \end{array} \right.$

For any given small $r>0$, large enough $k, l$, we declare that

$\int_{\Omega_{2r}}|u_{k}(x, t)-u_{l}(x, t)|dx\leq \int_{\Omega_{r}}|u_{k}(x, 0)-u_{l}(x, 0)|dx+c_{r}(t),$

(4.17)

where $c_{r}(t)$ is independent of $k, l$, and $\lim_{t\rightarrow 0}c_{r}(t)=0$. By (2.1)

$\int_{0}^{t}\int_{\Omega_{r}}\varphi (u_{kt}-u_{lt})dxd\tau+\int_{0}^{t}\int_{\Omega_{r}}\nabla \varphi [(|\nabla u_{k}^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}\nabla u_{k}^{m}-(|\nabla u_{l}^{m}|^{2}+\frac{1}{l})^{\frac{p-2}{2}}\nabla u_{l}^{m}]dxd\tau$

$+\int_{0}^{t}\int_{\Omega_{r}}a(x)(u^{mq_1}_{k}|\nabla u_{k}^{m}|^{p_1}-u^{mq_1}_{l}|\nabla u_{l}^{m}|^{p_1})\varphi dxd\tau$

$+\int_{0}^{t}\int_{\Omega_{r}}[f_0(u^{m}_{k})\int_{\Omega}K(y)|u^{m}_{k}(y, t)|^{\beta}dy -f_0(u^{m}_{l})\int_{\Omega}K(y)|u^{m}_{l}(y, t)|^{\beta}dy]\varphi dxd\tau =0,$

(4.18)

for $\forall \varphi \in L^{p}(0, T; W_{0}^{1, p}(\Omega))$. Suppose that $\xi(x)\in C_{0}^{1}(\Omega_{r})$ such that

$0\leq\xi\leq 1; \ \ \xi\mid_{\Omega_{2r}}=1,$

and choose $\varphi=\xi\text{sgn}_{\eta}(u_{k}^{m}-u_{l}^{m})$ in (4.18), then

$\int_{0}^{t}\int_{\Omega_{r}} \xi\text{sgn}_{\eta}(u_{k}^{m}-u_{l}^{m}) (u_{kt}-u_{lt})dxd\tau$

$+\int_{0}^{t}\int_{\Omega_{r}} [(|\nabla u_{k}^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}\nabla u_{k}^{m}-(x|\nabla u_{l}^{m}|^{2}+\frac{1}{l})^{\frac{p-2}{2}}\nabla u_{l}^{m}]\nabla\xi \text{sgn}_{\eta}(u_{k}^{m}-u_{l}^{m}) dxd\tau$

$+\int_{0}^{t}\int_{\Omega_{r}} [(|\nabla u_{k}^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}\nabla u_{k}^{m}-(x|\nabla u_{l}^{m}|^{2}+\frac{1}{l})^{\frac{p-2}{2}}\nabla u_{l}^{m}]\nabla(u_{k}^{m}-u_{l}^{m})\xi \text{sgn}'_{\eta}(u_{k}^{m}-u_{l}^{m}) dxd\tau$

$+\int_{0}^{t}\int_{\Omega_{r}}a(x)(u^{mq_1}_{k}|\nabla u_{k}^{m}|^{p_1}-u^{mq_1}_{l}|\nabla u_{l}^{m}|^{p_1})\xi\text{sgn}_{\eta}(u_{k}^{m}-u_{l}^{m}) dxd\tau$

$+\int_{0}^{t}\int_{\Omega_{r}}[f_0(u^{m}_{k})\int_{\Omega}K(y)|u^{m}_{k}(y, t)|^{\beta}dy -f_0(u^{m}_{l})\int_{\Omega}K(y)|u^{m}_{l}(y, t)|^{\beta}dy]\xi\text{sgn}_{\eta}(u_{k}^{m}-u_{l}^{m}) dxd\tau \leq0.$

(4.19)

If we notice that the third term in the left hand side on (4.19) is nonnegative when $\eta\rightarrow 0$, then we have

$\lim\limits_{\eta\rightarrow 0}\int_{0}^{t}\int_{\Omega_{r}} \xi\text{sgn}_{\eta}(u_{k}^{m}-u_{l}^{m}) (u_{kt}-u_{lt})dxd\tau$

$+\lim\limits_{\eta\rightarrow 0}\int_{0}^{t}\int_{\Omega_{r}}[(|\nabla u_{k}^{m}|^{2}+\frac{1}{k})^{\frac{p-2}{2}}\nabla u_{k}^{m}-(|\nabla u_{l}^{m}|^{2}+\frac{1}{l})^{\frac{p-2}{2}}\nabla u_{l}^{m}]\nabla\xi \text{sgn}_{\eta}(u_{k}^{m}-u_{l}^{m}) dxd\tau$

$+\lim\limits_{\eta\rightarrow 0}\int_{0}^{t}\int_{\Omega_{r}}a(x)(u^{mq_1}_{k}|\nabla u_{k}^{m}|^{p_1}-u^{mq_1}_{l}|\nabla u_{l}^{m}|^{p_1})\xi\text{sgn}_{\eta}(u_{k}^{m}-u_{l}^{m}) dxd\tau$

$+\lim\limits_{\eta\rightarrow 0}\int_{0}^{t}\int_{\Omega_{r}}[f_0(u^{m}_{k})\int_{\Omega}K(y)|u^{m}_{k}(y, t)|^{\beta}dy -f_0(u^{m}_{l})\int_{\Omega}K(y)|u^{m}_{l}(y, t)|^{\beta}dy]\xi\text{sgn}_{\eta}(u_{k}^{m}-u_{l}^{m}) dxd\tau =0.$

(4.20)

At the same time,

$\lim\limits_{\eta\rightarrow 0}\int_{0}^{t}\int_{\Omega_{r}} \xi\text{sgn}_{\eta}(u_{k}^{m}-u_{l}^{m}) (u_{kt}-u_{lt})dxd\tau=\int_{0}^{t}\int_{\Omega_{r}}\xi\text{sgn}(u_{k}^{m}-u_{l}^{m}) (u_{kt}-u_{lt})dxd\tau$

$=\int_{0}^{t}\int_{\Omega_{r}}\xi\text{sgn}(u_{k}-u_{l}) (u_{kt}-u_{lt})dxd\tau$

$\lim\limits_{\eta\rightarrow 0}\int_{0}^{t}\int_{\Omega_{r}} \xi\text{sgn}_{\eta}(u_{k}-u_{l}) (u_{kt}-u_{lt})dxd\tau=\lim\limits_{\eta\rightarrow 0}\int_{0}^{t}\int_{\Omega_{r}} \xi(\int_{0}^{u_{k}-u_{l}}\text{sgn}_{\eta}(s)ds)_{\tau}dxd\tau$

$=\lim\limits_{\eta\rightarrow 0}\int_{0}^{t}\int_{\Omega_{r}} \xi\int_{0}^{u_{k}-u_{l}}\text{sgn}_{\eta}(s)ds\mid_{0}^{t}dx =\int_{\Omega_{r}}\xi|u_{k}-u_{l}|dx-\int_{\Omega_{r}}\xi|u_{0k}-u_{0l}|dx.$

(4.21)

By (4.20) (4.21), we have

$\int_{\Omega_{2r}}\xi|u_{k}-u_{l}|dx\leq\int_{\Omega_{r}}|u_{0k}-u_{0l}|dx +c\int_{0}^{t}\int_{\Omega_{r}}[(|\nabla u_{k}^{m}|^{2}+\frac{1}{k})^{\frac{p-1}{2}}+(|\nabla u_{l}^{m}|^{2}+\frac{1}{l})^{\frac{p-1}{2}}]dxd\tau$

$+\int_{0}^{t}\int_{\Omega_{r}}a(x)|u^{mq_1}_{k}|\nabla u_{k}^{m}|^{p_1}-u^{mq_1}_{l}|\nabla u_{l}^{m}|^{p_1}| dxd\tau$

$\int_{0}^{t}\int_{\Omega_{r}}|f_0(u^{m}_{k})\int_{\Omega}K(y)|u^{m}_{k}(y, t)|^{\beta}dy -f_0(u^{m}_{l})\int_{\Omega}K(y)|u^{m}_{l}(y, t)|^{\beta}dy|dxd\tau.$

(4.22)

By Lemma 2.2 and Lemma 3.1, if $0 < t\leq 1$,

$\int_{0}^{t}\int_{\Omega_{r}}a(x)|u^{mq_1}_{k}|\nabla u_{k}^{m}|^{p_1}-u^{mq_1}_{1}|\nabla u_{1}^{m}|^{p_1}| dxd\tau\leq c\int_{0}^{t}\int_{\Omega_{r}} t^{-\epsilon}dxd\tau,$

which means (4.17) is true. Here

$\epsilon=\max\{\frac{mNq_1}{Nm(p-1)-N+mq}+\frac{p_1(m(p-1)+m-2)}{m(p-1)-1}, \frac{(\beta+m)N}{Nm(p-1)-N+mq}\}<1$

Now, for any given small $r$, if $k, l$ are large enough, by (4.17), we have

$\int_{\Omega_{2r}}|u(x, t)-u_{0}(x)|dx\leq \int_{\Omega_{r}}|u(x, t)-u_{k}(x, t)|dx+\int_{\Omega_{2r}}|u_{0k}(x)-u_{0l}(x)|dx$

$+\int_{\Omega_{2r}}|u_{l}(x, t)-u_{0l}(x)|dx+\int_{\Omega_{2r}}|u_{0l}(x)-u_{0}(x)|dx$

letting $t\rightarrow 0$, we get (1.13).

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5. The uniqueness of the solutions and the proof of Theorem 1.2

As we have said in the introduction, the uniqueness of the solutions of problem (1.1)-(1.3) is not true generally. But it is not difficult to prove the following theorems.

Theorem 5.1. Let $u_1, u_2$ be the two solutions of the problem (1.1)-(1.3) with the different initial values $u_{01}(x), u_{02}(x)$ respectively. If $(\frac{1}{m}+\beta-2)\setminus q_{1} < 1$ and

$p_1=0,$

(5.1)

then

$\int_{\Omega}|u_{1}(x, t)-u_{2}(x, t)|dx\leq \int_{\Omega}|u_{01}(x)-u_{02}(x)|dx, \ \forall t\geq 0.$

(5.2)

Proof. Let $u_1(t), u_2(t)$ be two solutions of equation (1.1). Let $v_1=u_{1}^{m}(t), v_2= u_{2}^{m}(t)$. Denote $w(t)=v_1^{\frac{1}{m}}(t)-u_2^{\frac{1}{m}}(t)$, $v(t)=v_{1}(t)-v_{2}(t)$. Then $w(t), v_1(t), v_2(t)$ satisfy that

$w'(t)-[{\rm div}(\|\nabla v_{1}\|^{p-2}\nabla v_{1})-{\rm div}(\|\nabla v_{2}\|^{p-2}\nabla v_{2})+a(x)(v_{1}^{q_1}-v_{2}^{q_1})$

$=f_{0}(v_{1})\int_{\Omega}K(y)|v_{1}|^{\beta}dy-f_{0}(v_{2})\int_{\Omega}K(y)|v_{2}|^{\beta}dy.$

(5.3)

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

$g_{n}(s)=\left\{\begin{array}{cc} 1, \ \ &\ \ {\rm if}\ \ s>\frac{1}{n}, \\ n^{2}s^{2}e^{1-n^{2}s^{2}}, \ \ &\ \ {\rm if}\ \ s\leq\frac{1}{n}. \end{array} \right.$

Clearly, when $|s|\geq n^{-1}$, $g'_{n}(s)=0$; when $|s|\leq n^{-1}$, $0\leq g'_{n}(s)=6s^{-1}$.

Multiplying (5.3) with $g_{n}(v_{1}-v_2)$ and integrating on $\Omega$, we have

$\int_{\Omega}g_{n}(v)w'(t)dx+\int_{\Omega}[|\nabla |v_{1}|^{p-2}|\nabla v_{1}-|\nabla |v_{2}|^{p-2}|\nabla v_{2}]\nabla(v_{1}-v_2)g_{n}'(v)dx+\int_{\Omega}a(x)(v_{1}^{q_1}-v_{2}^{q_1})g_{n}(v)dx$

$=\int_{\Omega}g_{n}(v) [f_{0}(v_{1})\int_{\Omega}K(y)|v_{1}|^{\beta}dy-f_{0}(v_{2})\int_{\Omega}K(y)|v_{2}|^{\beta}dy]dx.$

(5.4)

Moreover,

$\lim\limits_{n\rightarrow \infty}\int_{\Omega}g_{n}(v)w'(t)dx=\frac{d}{dt}\|w(t)\|_{1},$

$\int_{\Omega}[|\nabla |v_{1}|^{p-2}|\nabla v_{1}-|\nabla |v_{2}|^{p-2}|\nabla v_{2}]\nabla(v_{1}-v_2)g_{n}'(v)dx\geq 0,$

$\int_{\Omega}a(x)(v_{1}^{q_1}-v_{2}^{q_1})g_{n}(v)dx\geq 0,$

$|\int_{\Omega}g_{n}(v) [f_{0}(v_{1})\int_{\Omega}K(y)|v_{1}|^{\beta}dy-f_{0}(v_2)\int_{\Omega}K(y)|v_{2}|^{\beta}dy]dx|$

$\leq|\int_{\Omega}K(y)|v_{1}|^{\beta}dy\int_{\Omega}[f_{0}(v_{1})-f_{0}(v_{2})]dx| +c|\int_{\Omega}f_{0}(v_{2})dx||\int_{\Omega}\int_{v_{2}}^{v_{1}}s^{\beta-1}dsdy|$

$\leq c\|w(t)\|_{1}\|v_{1}\|^{\beta}_{\beta}+c\|v_{2}\|_{1}\int_{\Omega}|\xi|^{\beta-1}|v(t)|dx,$

where $\xi\in [v_{1}, v_{2}]$.

So

$\frac{d}{dt}\|w(t)\|_{1}\leq c\|w(t)\|_{1}\|v_{1}\|^{\beta}_{\beta}+c\|v_{2}\|_{1}(\|v_{1}\|^{\beta}_{\beta}+\|v_{2}\|^{\beta}_{\beta}),$

(5.5)

By using (1.23)-(1.24) of Theorem 1.1 to (5.5), letting $n\rightarrow \infty$. By Gronwall's inequality, for any given $T>0$, we can deduce that

$\|w(t)\|_{1}\equiv 0, 0\leq t\leq T.$

(5.6)

Another aim of the section is to prove the uniqueness of the viscosity solution of problem (1.1)-(1.3)

Theorem 5.2. Suppose that $a(x)$ and $K(x)$ are bounded functions. If $u(x, t)\in L^{\infty}(S)$, $|\nabla u|\leq c$ in addition, $2\geq p_{1}\geq 1$, then the viscosity solution of (1.1)-(1.3) is unique.

Proof. Let $u, v$ be the two viscosity solutions of (1.1)-(1.3). Then there are two sequences $\{u_{k}\}$ and $\{v_{l}\}$, which are the solutions of problem (1.14)-(1.2)-(1.3), such that

$\lim\limits_{k\rightarrow \infty} u_{k}=u, \ \ \lim\limits_{l\rightarrow \infty} v_{l}=v, \ {\rm a.e. in } \ S.$

(5.7)

Clearly, since $u(x, t), v(x, t)\in L^{\infty}(S)$, we may assume

$\|u_{k}\|_{\infty}\leq c, \ \|v_{l}\|_{\infty}\leq c.$

(5.8)

Let

$w =u_k-v_l, \ \ w_{1}=u^{m}_k-v^{m}_l.$

Then

$w_{t}=\left(a_{ij}(x, t)w_{1x_{j}}\right)_{x_{i}}+b(x, t, w, \nabla w), (x, t)\in \Omega\times (0, \infty)$

(5.9)

$w(x, 0)=u_{0k}(x)-v_{0l}(x), \ x\in \Omega$

(5.10)

$w(x, t)=0, \ (x, t)\in \partial \Omega\times (0, \infty),$

(5.11)

where

$a_{ij}(x, t)=\int_{0}^{1}{\left| s\nabla u_{k}^{m}+(1-s)\nabla v_{l}^{m}\right|}^{p-2}ds\cdot\delta_{ij}$

$+\int_{0}^{1}(p-2)\left|s\nabla u_{k}^{m}+(1-s)\nabla v_{l}^{m}\right| ^{p-4}(su_{kx_{i}}^{m}+(1-s)v_{lx_{i}}^{m})(su_{kx_{j}}^{m}+(1-s)v_{lx_{j}}^{m})ds,$

and since $p_1\geq 1$, using the convexity of the function $s^{p_1}$, by (5.8), we have

$b(x, t, w, \nabla w)= a(x)[u^{mq_1}_k|\nabla u^{m}_k|^{p_1}-v^{mq_1}_l|\nabla v^{m}_l|^{p_1}]$

$+f_0(u^{m}_k)\int_{\Omega}K(y)|u^{m}_k(y, t)|^{\beta}dy-f_0(v^{m}_l)\int_{\Omega}K(y)|v^{m}_l(y, t)|^{\beta}dy,$

$|b(x, t, w, \nabla w)|\leq c|\nabla (u^{m}_k-v^{m}_l)|^{p_1}\leq c|\nabla w|^{p_{1}}\leq c|\nabla w|^{2}+c.$

By the chapter 8 of ^[13], we know that

$\|u_{k}(x, t)-v_{l}(x, t)\|_{\infty}\leq c\|u_{0k}-v_{0l}\|_{\infty}.$

Let $k, l\rightarrow \infty$, we know that the uniqueness of the viscosity solution (1.1)-(1.3) is true.

Suppose that the viscosity solution of problem (1.1)-(1.3) is unique in what follows. Then, by considering the regularized problem (1.14)-(1.2)-(1.3), we easily get the following Theorem 5.3, and Theorem 1.2 is a simple corollary of Theorem 5.3.

Theorem 5.3. Let $u$ be a weak solution of problem (1.1)-(1.3). If $v$ satisfies

$v_{t}\geq\text{div}(\mid \nabla v^{m}\mid^{p-2}\nabla v^{m})-a(x)v^{mq_1}|\nabla v^{m}|^{p_1}$

$+f_0(v^{m})\int_{\Omega}K(y)|v^{m}(y, t)|^{\beta}dy+g(x)\ \text{ in }S=\Omega\times (0, \infty),$

(5.12)

$v(x, 0)\geq u_{0}(x), \ \ x\in \Omega,$

(5.13)

$v(x, t)=0, \ \ (x, t)\in \partial \Omega\times (0, \infty),$

(5.14)

then

$u(x, t)\geq v(x, t), \ \forall (x, t)\in S.$

(5.15)

Now, let

$v(x, t)=u_{kr}(x, t)=ru_{k}(x, r^{m(p-1)-1}t), \;\ r\in (0, 1).$

Then

$v_{t}(x, t) =\text{div}(\mid Dv^{m}\mid ^{p-2}Dv^{m}) -a(x)r^{m(p-1-q_{1}-p_{1})}v^{mq_{1}}\mid Dv^{m}\mid^{p_{1}}$

$+r^{m[p-1-\beta]}f_{0}(r^{-m}v^{m})\int_{\Omega}K(y)|v^{m}|^{\beta}dy+r^{m(p-1)}g(x), (x, t)\in \Omega\times(0, \infty)$

(5.16)

$v(x, 0) =ru_{k}(x, 0), x\in \Omega,$

(5.17)

$v(x, t)=0, \ (x, t)\in \partial \Omega\times(0, \infty).$

(5.18)

Noticing that $g(x)\leq 0$, $f_{0}(r^{-m}v^{m})\geq f_{0}(v^{m})$, and

$p_{1}+q_{1}<p-1, p-1-\beta<0, \ 0<r<1,$

which implies that

$r^{m(p-1-q_{1}-p_{1})}<1, \ r^{m[p-1-\beta]}>1,$

$v_{t}(x, t)\geq\text{div}(\mid Dv^{m}\mid ^{p-2}Dv^{m}) -a(x)v^{q_{1}m}\mid Dv^{m}\mid^{p_{1}} +f_{0}(v^{m})\int_{\Omega}K(y)|v^{m}|^{\beta}dy+g(x),$

using the argument similar to that in the proof Lemma 3.5 of ^[35], we can prove

$u_{k}\geq u_{kr}.$

It follows that

$\frac{u_{k}(x, r^{m(p-1)-1}t)-u_{k}(x, t)}{(r^{m(p-1)-1}-1)t}$

$\geq \frac{r-1}{(1-r^{m(p-1)-1})t}u_{k}(x, r^{m(p-1)-1}t).$

Letting $r\rightarrow 1$, we get

$u_{kt}\geq -\frac{u_{k}}{(m(p-1)-1)t}.$

(5.19)

By (5.19), we can easily get Theorem 1.2.

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Acknowledgement

The paper is supported by Natural Science Foundation of Fujian province in China (No: 2015J01592), supported by Science Foundation of Xiamen University of Technology.

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Conflict of Interest

All authors declare no conflicts of interest in this paper.

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