1. Introduction

Consider the following Euler-Poisson system for the bipolar hydrodynamical model of semi-conductor devices:

$ $$\begin{equation} \label{1.1} \left\{\begin{array}{lll} n_{1t}+j_{1x} = 0, \\ j_{1t}+(\frac{j_{1}^2}{n_{1}}+p(n_{1}))_x = n_{1}E-j_1, \\ n_{2t}+j_{2x} = 0, \\ j_{2t}+(\frac{j_{2}^2}{n_{2}}+q(n_{2}))_x = -n_{2}E-j_2, \\ E_x = n_{1}-n_{2}-D(x), \end{array}\right. \end{equation}$$ $

(1)

in the region $ \Omega = {\bf (0, 1)}\times R_{+}$. In this paper, $ n_{1}(x, t)$, $ n_{2}(x, t)$, $ j_{1}(x, t)$, $ j_{2}(x, t)$ and $ E(x, t)$ represent the electron density, the hole density, the electron current density, the hole current density and the electric field, respectively. In this note, we assume that the $p$ and $q$ satisfy the $\gamma$-law:$p(n_1) = n_{1}^{2}$ and $q(n_2) = n_{2}^{2}$ ($\gamma = 2$), which denote the pressures of the electrons and the holes. The function $D(x)$, called the doping profile, stands for the density of impurities in semiconductor devices.

For system $(1)$, the initial conditions are

$ n_{i}(x, 0) = n_{i0}(x) \ge 0, \;\; j_{i}(x, 0) = j_{i0}(x), \;\; i = 1, 2, $

(2)

and the boundary conditions at $x = 0$ and $x = 1$ are

$ j_{i}(0, t) = j_{i}(1, t) = 0, \;\; i = 1, 2, \;\; E(0, t) = 0. $

(3)

So, we can get the compatibility condition

$ j_{i0}(0) = j_{i0}(1) = 0, \;\; i = 1, 2. $

(4)

Moreover, in this paper, we assume the doping profile $D(x)$ satisfies

$ D(x)\in C[0, 1]~ {\rm{and}}~ D^* = \mathop {{\rm{sup}}}\limits_x D(x)\ge \mathop {{\rm{inf}}}\limits_x D(x) = D_*. $

(5)

Now, the definition of entropy solution to problem $(1)-(4)$ is given. We consider the locally bounded measurable functions $ n_{1}(x, t)$, $ j_{1}(x, t)$, $ n_{2}(x, t)$, $ j_{2}(x, t)$, $ E(x, t)$, where $ E(x, t)$ is continuous in $x$, a.e. in $t$.

Definition 1.1. The vector function $ (n_{1}, n_{2}, j_{1}, j_{2}, E)$ is a weak solution of problem $(1)-(4)$, if it satisfies the equation $(1)$ in the distributional sense, verifies the restriction $(2)$ and $(3)$. Furthermore, a weak solution of system $(1)-(4)$ is called an entropy solution if it satisfies the entropy inequality

$ \eta_{et}+ q_{ex}+\frac{j_{1}^2}{n_{1}}+\frac{j_{2}^2}{n_{2}}-j_{1}E+j_{2}E \leq0, $

(6)

in the sense of distribution. And the $(\eta_{e}, q_{e})$ are mechanical entropy-entropy flux pair which satisfy

$ \left\{ $$\begin{array}{lll} \eta_{e}(n_1, n_2, j_1, j_2) = \frac{j_{1}^2}{2n_{1}}+n_{1}^2+\frac{j_{2}^2}{2n_{2}}+n_{2}^2, \\\\ q_{e}(n_1, n_2, j_1, j_2) = \frac{j_{1}^3}{2n_{1}^2}+2n_{1}j_{1}+\frac{j_{2}^3}{2n_{2}^2}+2n_{2}j_{2}.\\ \end{array}$$ \right. $

(7)

For bipolar hydrodynamic model, the studies on the existence of solutions and the large time behavior as well as relaxation-time limit have been extensively carried out, for example, see ^{[1][2][3][4][5][6]} etc. Now, we make it into a semilinear ODE about the potential and the pressures with the exponent $\gamma = 2$. We can get the existence, uniqueness and some bounded estimates of the steady solution. Then, using a technical energy method and a entropy dissipation estimate, we present a framework for the large time behavior of bounded weak entropy solutions with vacuum. It is shown that the weak solutions converge to the stationary solutions in $L^2$ norm with exponential decay rate.

The organization of this paper is as follows. In Section 2, the existence, uniqueness and some bounded estimates of stationary solutions are given. we present a framework for the large time behavior of bounded weak entropy solutions with vacuum in Section 3.

---

2. Steady solutions

In this part, we will prove the existence and uniqueness of steady solution to problem $(1)-(4)$. Moreover, we can obtain some important estimates on the steady solution $(N_1, N_2, \mathcal{E})$.

The steady equation of $(1)-(4)$ is as following

$ \left\{ $$\begin{array}{ll} &J_{1} = J_{2} = 0, \\ &2N_{1}N_{1x} = N_{1}{\mathcal{E}}, \\ &2N_{2}N_{2x} = -N_{2}{\mathcal{E}}, \\ &{\mathcal{E}}_x = N_{1}-N_{2}-D(x), \\ \end{array}$$ \right. $

(8)

and the boundary condition

$ {\mathcal{E}}(0) = 0. $

(9)

We only concern the classical solutions in the region where the density

$ \mathop {{\rm{inf}}}\limits_x {N_1}>0~~~ {\rm {and}}~~~ \mathop {{\rm{inf}}}\limits_x {N_2}>0. $

(10)

hold.

Now, we introduce a new variation $\Phi(x)$, and make $\Phi'(x)$: = ${\mathcal{E}}(x)$. To eliminate the additive constants, we set $\int_0^1{\Phi(x)}dx = 0$. Then (2.1) turns into

$ \left\{ $$\begin{array}{ll} 2N_{1x} = \Phi_x, \\ 2N_{2x} = -\Phi_x, \\ \Phi_{xx} = N_{1}-N_{2}-D(x). \end{array}$$ \right. $

(11)

Obviously, $(11)_{1}$ and $(11)_{2}$ indicate

$ \left\{ $$\begin{array}{ll} N_1(x) = \frac{1}{2}\Phi(x)+C_1, \\ N_2(x) = -\frac{1}{2}\Phi(x)+C_2, \\ \Phi_{xx}(x) = \frac{1}{2}\Phi(x)+C_1+ \frac{1}{2}\Phi(x)-C_2-D(x). \end{array}$$ \right. $

(12)

where $C_{1}$ and $C_{2}$ are two unknown positive constants. To calculate these two constants, we suppose^*

^*Using the conservation of the total charge: integrating $(1)_1$ and $(1)_3$ from 0 to 1

$ \big{(}\int_0^1 n_idx\big{)}_t = -\int_0^1 j_{ix}dx = 0, ~~{\rm{for}}~~ i = 1, 2, $

we see this assumption is right.

$ \int_0^1\bigg{(}n_i(x, 0)-N_i(x)\bigg{)}dx = 0 ~~{\rm{for}}~~ i = 1, 2, $

(13)

then

$ \bar{n}_1: = \int_0^1n_1(x, 0)dx = \int_{0}^{1}N_1(x)dx = \int_{0}^{1}\big{(}\frac{\Phi(x)}{2}+C_1\big{)}dx = C_1, \\ \bar{n}_2: = \int_0^1n_2(x, 0)dx = \int_{0}^{1}N_2(x)dx = \int_{0}^{1}\big{(}-\frac{\Phi(x)}{2}+C_2\big{)}dx = C_2. $

(14)

Substituting $(14)$ into $(12)_3$, we have

$ \Phi_{xx} = \Phi(x)+\bar{n}_1-\bar{n}_2-D(x). $

(15)

Clearly, we can prove the existence and uniqueness of solutions to $(15)$ with the Neumann boundary condition

$ \Phi_x(0) = \Phi_x(1) = 0. $

(16)

Integrate$(15)$ from $x = 0$ to $x = 1$, we get

$ \bar{n}_1-\bar{n}_2 = \int_{0}^{1}D(x)dx. $

(17)

Suppose $\Phi(x)$ attains its maximum in $x_0\in [0, 1]$, then we get $\Phi_{xx}(x_0) \le 0$^† and

^† If $x_0\in (0, 1)$, then $\Phi_x(x_0) = 0, $ $\Phi_{xx}(x_0) \le 0$ clearly. If $x_0 = 0$ or $x_0 = 1$, the Taylor expansion

$ \Phi(x) = \Phi(x_0)+\Phi'(x_0)(x-x_0)+{{\Phi''}(x_0)\over 2}(x-x_0)^2+o(x-x_0)^2, $

the boundary condition ($16$) indicates ${\Phi''}(x_0) \le 0$.

$ \Phi(x_0)+\bar{n}_1-\bar{n}_2-D(x_0)\leq0 . $

So we get

$ \Phi(x_0)\leq D^{*}+\bar{n}_2-\bar{n}_1 . $

(18)

Similarly, if $\Phi $ attains its minimum in $x_1\in [0, 1]$, we obtain

$ \Phi(x_1) \ge D_{*}+\bar{n}_2-\bar{n}_1 . $

(19)

Moreover, from $(12), (14), (15), (18)$, and $(19)$, we have

$ \frac{D_*+\bar{n}_2+\bar{n}_1}{2}\leq N_1(x)\leq \frac{D^*+\bar{n}_2+\bar{n}_1}{2}, \\\\ \frac{-D^*+\bar{n}_2+\bar{n}_1}{2}\leq N_2(x)\leq \frac{-D_*+\bar{n}_2+\bar{n}_1}{2}, $

(20)

$ D_* \le (N_1-N_2)(x) \le D^* ~~{\rm {for~any }}~~ x\in [0, 1]. $

(21)

Above that, the theorem of existence and uniqueness of steady equation is given.

Theorem 2.1. Assume that $(5)$ holds, then problem $(8)$, $(9)$ has an unique solution $(N_1, N_2, {\mathcal{E}})$, such that for any $x\in [0, 1]$

$ n_* \le N_1(x) \le n^*, ~~ n_* \le N_2(x) \le n^*, $

(22)

and

$ D_* \le (N_1-N_2)(x) \le D^*, $

(23)

satisfy, where

$ n^*: = \max\bigg{\{}{\frac{D^*+\bar{n}_2+\bar{n}_1}{2}, \frac{-D_*+\bar{n}_2+\bar{n}_1}{2}}\bigg{\}}, \\\\ n_*: = \min\bigg{\{}{\frac{D_*+\bar{n}_2+\bar{n}_1}{2}, \frac{-D^*+\bar{n}_2+\bar{n}_1}{2}}\bigg{\}}, $

(24)

$ \bar{n}_{1} $, $ \bar{n}_{2} $ are defined in $(14)$.

---

3. Large time behavior

Now, our aim is to prove the weak-entropy solution of $(1)-(4)$ convergences to corresponding stationary solution in $L^2$ norm with exponential decay rate. For this purpose, we introduce the relative entropy-entropy flux pair:

$ $$\begin{eqnarray*} \eta^*(x, t)& = & \sum\limits_{i = 1}^2\bigg{(}{\frac{j_i^2}{2n_i}}+n_i^2-N_i^2-2N_i(n_i - N_i)\bigg{)}(x, t)\\ & = &\bigg{(}\eta_{e}-\sum\limits_{i = 1}^2{Q_i}\bigg{)}(x, t) \ge 0, \end{eqnarray*}$$ $

(25)

$ $$\begin{eqnarray*} q^*(x, t)& = &\sum\limits_{i = 1}^2\bigg{(}{\frac{j_i^3}{2n_i^2}}+2n_ij_i-2N_ij_i\bigg{)}(x, t)\\\\ & = & \bigg{(}q_e-\sum\limits_{i = 1}^2 P_i\bigg{)}(x, t), \end{eqnarray*}$$ $

(26)

where

$Q_i = {N_i^2+2N_i(n_i - N_i)}, \;\; P_i = 2N_ij_i, $

$\eta_e$ and $q_e$ are the entropy-entropy flux pair defined in $(1.7)$.

The following theorem is our main result in section 3.

Theorem 3.1(Large time behavior) Suppose $(n_1, n_2, j_1, j_2, E)(x, t)$ be any weak entropy solution of problem $(1.1)-(1.4)$ satisfying

$ 2(2D^*-\bar{n}_1-\bar{n}_2)<(n_1-n_2)(x, t)< 2(2D_*+\bar{n}_1+\bar{n}_2), $

(27)

for a.e. $x\in [0, 1]$ and $t>0$. $(N_1, N_2, {\mathcal{E}})(x)$ is its stationary solution obtained in Theorem 2.1. If

$ \int_0^1 \eta^*(x, 0)dx<\infty, ~~ \int_0^1\bigg{(}n_i(s, 0)-N_i(s)\bigg{)}ds = 0, $

(28)

then for any $t>0$, we have

$ \int_0^1[j_1^2+j_2^2+(E-{\mathcal{E}})^2+(n_1-N_1)^2+(n_2-N_2)^2](x, t)dx\\\\ \;\;\;\; \le C_0e^{-\tilde{C}_0t}\int_0^1 \eta^*(x, 0)dx. $

(29)

holds for some positive constant $C_0$ and $\tilde{C}_0$ .

Proof. We set

$ y_i(x, t) = -\int_0^{x}\bigg{(}n_i(s, t)-N_i(s)\bigg{)}ds, ~~~~ i = 1, 2, ~ x\in[0, 1], ~ t>0. $

(30)

Clearly, $y_i(i = 1, 2)$ is absolutely continuous in $x$ for a.e. $t>0$. And

$ y_{ix} = -(n_i-N_i), \;\;\;\; y_{it} = j_i, \\\\ y_2-y_1 = E-{\mathcal{E}}, \;\;\;\; y_i(0, t) = y_i(1, t) = 0, $

(31)

following (1.1), (2.1), and ($2.1$). From $(1.1)_2$ and $(2.1)_2$, we get $y_1$ satisfies the equation

$ y_{1tt}+(\frac{y_{1t}^2}{n_1})_x-y_{1xx}+y_{1t} = {n_1}E-N_{1}\mathcal{E}. $

(32)

Multiplying $y_1$ with $(32)$ and integrating over $(0, 1)$^‡, we have

^‡For weak solutions, ($1$) satisfies in the sense of distribution. We choose test function $\varphi_n(x, t)\in C_0^\infty \bigg{(}(0, 1)\times [0, T)\bigg{)}$ and let $\varphi_n(x, t)\to y_i(x, t)$ as $n\to +\infty $ for $i = 1, 2$.

$ $$\begin{array}{ll} \frac{d}{dt} \int_0^1(y_1y_{1t}+\frac12y_1^2)~dx-\int_0^1 (\frac{y_{1t}^2}{n_1})y_{1x}~dx-\int_0^1(n_1^2-N_1^2)y_{1x}dx-\int_0^1 y_{1t}^2~ dx\\\\ = \int_0^1 (N_1(y_2-y_1)y_1+\frac{E_x}{2} y_1^2) dx. \end{array}$$ $

(33)

In above calculation, we have used the integration by part. Similarly, from $(1.1)_4$ and $(2.1)_3$, we get

$ $$\begin{array}{ll} \frac{d}{dt} \int_0^1(y_2y_{2t}+\frac12y_2^2)~dx-\int_0^1 (\frac{y_{2t}^2}{n_2})y_{2x}~dx -\int_0^1(n_2^2-N_2^2)y_{2x}~ dx-\int_0^1 y_{2t}^2 ~dx\\\\ = -\int_0^1( N_2(y_2-y_1)y_2+\frac{E_x}{2} y_2^2)~ dx. \end{array}$$ $

(34)

Add ($33$) and ($34$), we have

$ $$\begin{array}{ll} \frac{d}{dt} \int_0^1(y_1y_{1t}+\frac12y_1^2+y_2y_{2t}+\frac12y_2^2)~dx-\int_0^1(n_1^2-N_1^2)y_{1x}dx-\int_0^1(n_2^2-N_2^2)y_{2x}~ dx \\\\ = \int_0^1 \bigg{(}(\frac{y_{1t}^2}{n_1})y_{1x}~+(\frac{y_{2t}^2}{n_2})y_{2x}\bigg{)}~ dx+\int_0^1 (y_{1t}^2+y_{2t}^2)~dx\\\\ +\int_0^1\bigg{(}N_1(y_2-y_1)y_1+\frac{E_x}{2} y_1^2 -N_2(y_2-y_1)y_2-\frac{E_x}{2} y_2^2\bigg{)}~ dx. \end{array}$$ $

(35)

Since

$ $$\begin{array}{ll} \int_0^1\bigg{(}N_1(y_2-y_1)y_1+\frac{E_x}{2} y_1^2 -N_2(y_2-y_1)y_2-\frac{E_x}{2} y_2^2\bigg{)}~ dx\\\\ = \int_0^1 \frac{n_1-N_1-n_2+N_2-D(x)}{2}y_1^2 dx+\int_0^1 \frac{n_2-N_2-n_1+N_1+D(x)}{2}y_2^2 dx\\\\ -\int_0^1\frac{N_1+N_2}{2}(y_1-y_2)^2 dx, \end{array}$$ $

(36)

then, from $(31)_1$ and $(36)$ we get

$ $$\begin{array}{ll} \frac{d}{dt} \int_0^1(y_1y_{1t}+\frac12y_1^2+y_2y_{2t}+\frac12y_2^2)~dx+\int_0^1(N_1+n_1)y_{1x}^2\\\\ \;\;\;\; +\int_0^1(N_2+n_2)y_{2x}^2 dx+\int_0^1{{{N_1+N_2}\over 2} (y_1-y_2)^2}dx\\\\ = \int_0^1 \bigg{(}(\frac{y_{1t}^2}{n_1})y_{1x}+(\frac{y_{2t}^2}{n_2})y_{2x}\bigg{)}~ dx+\int_0^1 (y_{1t}^2+y_{2t}^2) ~dx\\\\ \;\; \;\; +\int_0^1\bigg{(} \frac{n_1-N_1-n_2+N_2-D(x)}{2}y_1^2 +\frac{n_2-N_2-n_1+N_1+D(x)}{2}y_2^2 \bigg{)}dx. \end{array}$$ $

(37)

Moreover, since

$ |y_i(x)| = |\int_0^xy_{is}(s)ds| \le x^{1\over 2}(\int_0^x y_{is}^2ds)^{1\over 2} \le x^{1\over 2}(\int_0^1 y_{is}^2ds)^{1\over 2}, \;\; x\in [0, 1], $

(38)

we can obtain

$ \|y_i\|^2_{L^2} = \int_0^1|y_i|^2dx \le {1\over 2}\|y_{ix}\|_{L^2}^2, $

(39)

verifies for $i = 1, 2$. If the weak solutions $n_1(x, t)$ and $n_2(x, t)$ satisfy $(27)$ then

$ \label{3.16}\mathop {{\rm{inf}}}\limits_x \{{N_1+n_1\}}>\mathop {{\rm{sup}}}\limits_x \bigg{\{}{\frac{n_1-N_1-n_2+N_2-D(x)}{4}}\bigg{\}}, $

(40)

and

$ \mathop {{\rm{inf}}}\limits_x \{{N_2+n_2\}}>\mathop {{\rm{sup}}}\limits_x \bigg{\{}{\frac{n_2-N_2-n_1+N_1+D(x)}{4}}\bigg{\}}, $

(41)

hold, where we have used the assumption $(5)$ and the estimate $(23)$.

Following ($39$), ($40$) and ($41$), we have

$ \int_0^1\frac{n_1-N_1-n_2+N_2-D(x)}{2}y_1^2dx<\int_0^1(N_1+n_1)y_{1x}^2dx, $

(42)

and

$ \int_0^1\frac{n_2-N_2-n_1+N_1+D(x)}{2}y_2^2dx<\int_0^1(N_2+n_2)y_{2x}^2dx. $

(43)

Thus (36), (42), and (43) indicate there is a positive constant $\beta>0$, such that

$ $$\begin{array}{ll} \frac{d}{dt} \int_0^1(y_1y_{1t}+\frac12y_1^2+y_2y_{2t}+\frac12y_2^2 )~dx+\beta\int_0^1 (y^2_{1x}+y^2_{2x}) dx+\int_0^1{{{N_1+N_2}\over 2} (y_1-y_2)^2}dx\\\\ \le \int_0^1 \bigg{(}(\frac{y_{1t}^2}{n_1})y_{1x}+(\frac{y_{2t}^2}{n_2})y_{2x}\bigg{)}~ dx+\int_0^1 (y_{1t}^2+y_{2t}^2) ~dx\\\\ = \int_0^1 \bigg{(} N_1 {{{y_{1t}^2}\over {n_1}}}+ N_2 {{{y_{2t}^2}\over {n_2}}}\bigg{)}~ dx. \end{array}$$ $

(44)

In view of the entropy inequality ($6$), and the definition of $\eta^*$ and $q^*$ in ($25$) and ($26$), the following inequality holds in the sense of distribution.

$ $$\begin{array}{lll}\label{3.21} \eta_{et}+ q_{ex}+\frac{j_{1}^2}{n_{1}}+\frac{j_{2}^2}{n_{2}}-j_{1}E+j_{2}E\\\\ = \eta^*_t+\sum\limits_{i = 1}^2{ Q_{it}}+q^*_x+\sum\limits_{i = 1}^2 P_{ix} +\frac{j_{1}^2}{n_{1}}+\frac{j_{2}^2}{n_{2}}-j_{1}E+j_{2}E\\\\ = \eta^*_t+q^*_x+\frac{j_{1}^2}{n_{1}}+\frac{j_{2}^2}{n_{2}}-j_1E+j_2E+j_1{\mathcal{E}}-j_2\mathcal{E}\\\\ \leq 0. \end{array}$$ $

(45)

Since

$ -j_1E+j_2E+j_1{\mathcal{E}}-j_2{\mathcal{E}} = (E-{\mathcal{E}})(j_2-j_1) = (y_2-y_1)(y_{2t}-y_{1t}), $

(46)

then ($44$) turns into

$ \eta^*_t+q^*_x+\frac{y_{1t}^2}{n_{1}}+\frac{y_{2t}^2}{n_{2}}+(y_2-y_1)(y_{2t}-y_{1t}) \le 0. $

(47)

We use the theory of divergence-measure fields, then

$ {d\over{dt}}\int_0^1 (\eta^*+{1\over 2}(y_2-y_1)^2)dx+\int_0^1 ( \frac{y_{1t}^2}{n_1} +\frac{y_{2t}^2}{n_2}) ~dx~ \leq 0, $

(48)

where we use the fact

$ \int_0^1 q^*_x~dx~ = 0. $

(49)

Let $\lambda>2+2n^*>0$. Then, we multiply $(48)$ by $\lambda$ and add the result to $(44)$ to get

$ {d\over{dt}}\int_0^1 (\lambda \eta^*+{\lambda\over 2}(y_2-y_1)^2 + y_1y_{1t}+\frac12y_1^2+y_2y_{2t}+\frac12y_2^2)dx+\beta\int_0^1(y_{1x}^2+y_{2x}^2)dx\\\\ \;\;\;\; +\int_0^1{{N_1+N_2}\over{2}}(y_1-y_2)^2dx+\int_0^1\bigg{(}(\lambda-N_1) \frac{y_{1t}^2}{n_1} +(\lambda-N_2)\frac{y_{2t}^2}{n_2}\bigg{)} dx\leq 0. $

(50)

Using the estimate ($22$) in Theorem 2.1. and the Poinc${\rm{\acute{a}}}$re inequality ($39$), we have

$ {d\over{dt}}\int_0^1 (\lambda \eta^*+{\lambda\over 2}(y_2-y_1)^2 + y_1y_{1t}+\frac12y_1^2+y_2y_{2t}+\frac12y_2^2) dx+{\beta\over 2}\int_0^1(y_{1x}^2+y_{2x}^2)dx\\\\ \;\;\;\; +{\beta\over 2}\int_0^1(y_{1}^2+y_{2}^2)dx+n_*\int_0^1(y_1-y_2)^2dx+\int_0^1\bigg{(} \frac{y_{1t}^2}{n_1} +\frac{y_{2t}^2}{n_2}\bigg{)} dx\leq 0. $

(51)

Now, we consider $ \eta^* $ in ($25$). Clearly

$ n_i^2-N_i^2-2N_i(n_i-N_i), $

(52)

is the quadratic remainder of the Taylor expansion of the function $n_i^{2}$ around $N_i>n_*>0$ for $i = 1, 2$. And then, there exist two positive constants $C_1$ and $C_2$ such that

$ C_1y_{ix}^2 \le n_i^2-N_i^2-2N_i(n_i-N_i) \le C_2y_{ix}^2. $

(53)

Making $C_3 = \min\{C_1, {{1\over 2}}\}$ and $C_4 = \max\{C_2, {{1\over 2}\}}$, then we get

$ C_3({{y_{1t}^2}\over {n_1}}+{y_{2t}^2\over {n_2}}+y_{1x}^2+y_{2x}^2) \leq \eta^* \leq C_4({{y_{1t}^2}\over {n_1}}+{y_{2t}^2\over {n_2}}+y_{1x}^2+y_{2x}^2). $

(54)

Let

$ F(x, t) = \lambda \eta^*+{\lambda\over 2}(y_2-y_1)^2 + y_1y_{1t}+\frac12y_1^2+y_2y_{2t}+\frac12y_2^2, $

then there exist positive constants $C_5$, $C_6$, and $C_7$, depending on $\lambda, n_*, \beta$, such that

$ \int_0^1F(x, t)dx = \int_0^1[\lambda \eta^*+{\lambda\over 2}(y_2-y_1)^2 + y_1y_{1t}+\frac12y_1^2+y_2y_{2t}+\frac12y_2^2]dx \\\\ \leq C_5\int_0^1[({y_{1t}^2\over {n_1}} +{y_{2t}^2\over {n_2}})+ n_*(y_2-y_1)^2+ {\beta\over 2}(y_{1x}^2 +y_{2x}^2)~ + {\beta\over 2}(y_{1}^2 +y_{2}^2)]dx\\\\ \le C_6 \int_0^1\eta^*dx, $

(55)

and

$ 0<C_7\int_0^1[({y_{1t}^2\over {n_1}} +{y_{2t}^2\over {n_2}})+ n_*(y_2-y_1)^2+ {\beta\over 2}(y_{1x}^2 +y_{2x}^2)~ + {\beta\over 2}(y_{1}^2 +y_{2}^2)]dx\\\\ \leq \int_0^1[\lambda \eta^*+{\lambda\over 2}(y_2-y_1)^2 + y_1y_{1t}+\frac12y_1^2+y_2y_{2t}+\frac12y_2^2]dx = \int_0^1F(x, t)dx. $

(56)

Then

$ {d\over{dt}}\int_0^1 F(x, t) ~dx + {1\over {C_5}}\int_0^1 F(x, t)dx \leq 0, $

(57)

and

$ \int_0^1[({y_{1t}^2\over {n_1}} +{y_{2t}^2\over {n_2}})+ n_*(y_2-y_1)^2+ {\beta\over 2}(y_{1x}^2 +y_{2x}^2)~ + {\beta\over 2}(y_{1}^2 +y_{2}^2)]dx\\ \le{1\over {C_7}}\int_0^1F(x, t)dx \le {1\over {C_7}}e^{-{{t}\over {C_5}}}\int_0^1F(x, 0)dx\\ \le C_8e^{-{t\over {C_5}}}\int_0^1\eta^*(x, 0)dx. $

(58)

are given, following the Growall inequality and the estimates ($55$) and ($56$). Up to now, we finish the proof of Theorem 3.1.

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Acknowledgments

In the process of the selected topic and write a paper, I get the guidance from my tutor: Huimin Yu. In the teaching process, my tutor helps me develop thinking carefully. The spirit of meticulous and the rigorous attitude of my tutor gives me a lot of help. Gratitude to my tutor is unable to express in words. And this paper supported in part by Shandong Provincial Natural Science Foundation (Grant No. ZR2015AM001).

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Conflict of interest

The author declare no conflicts of interest in this paper.

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