The nonlinear singular perturbation evolution equations are an important target in the mathematical, engineering mathematics and physical etc. circles. Many approximate methods were improved. Recently, many scholars did a great of work, such as de Jager et al. [1], Barbu et al. [2], Hovhannisyan et al. [3], Graef et al. [4], Barbu et al. [5], Bonfoh et al. [6], Faye et al. [7], Samusenko [8], Liu [9] and so on. Using the singular perturbation and other's theorys and methods the authoes also studied a class of nonlinear singular perturbation problems [10-24]. In this paper, using the special and simple method, we consider a class of the evolution equation.

Now we studied the following singular perturbation evolution equations initial-boundary value problem with two parameters

$\begin{eqnarray} & & \varepsilon^{2}\frac{\partial^{2}u}{\partial t^{2}}-\mu^{2}Lu=f(t, x, u), \ \ (t, x)\in(0, T_{0}]\times\Omega, \\ \end{eqnarray}$

(1.1)

$\begin{eqnarray} & & u=g(t, x), \ \ x\in\partial\Omega, \\ \end{eqnarray}$

(1.2)

$\begin{eqnarray} & & u|_{t=0}=h_{1}(x), \ \ \varepsilon\frac{\partial u}{\partial t}|_{t=0}=h_{2}(x), \ \ x\in\Omega, \end{eqnarray}$

(1.3)

where

$\begin{eqnarray*} & & L=\sum^{n}_{i, j=1}\alpha_{ij}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}} +\sum^{n}_{i=1}\beta_{i}\frac{\partial}{\partial x_{i}}, \\ & & \sum^{n}_{i, j=1}\alpha_{ij}\xi_{i}\xi_{j}\geq\lambda\sum^{n}_{i=1}\xi^{2}_{i}, \ \ \forall\xi_{i}\in\Re, \ \lambda>0, \end{eqnarray*}$

$\varepsilon$ and $\mu$ are small positive parameters, $x=(x_{1}, x_{1}, \cdots, x_{n})\in\Omega, \Omega$ is a bounded region in $\Re^n, \ \partial\Omega$ denotes boundary of $\Omega$ for class $C^{1+\alpha}$, where $\alpha\in(0, 1)$ is Hölder exponent, $T_{0}$ is a positive constant large enough, $f(t, x, u)$ is a disturbed term, $L$ signifies a uniformly elliptic operator.

Hypotheses that

$[H_{1}]$ $\sigma=\varepsilon/\mu$ as $\mu\rightarrow 0$;

$[H_{2}]$ $\alpha_{ij}, \beta_{i}$ with regard to $x$ are Hölder continuous, $g$ and $h_{i}$ are sufficiently smooth functions in correspondence ranges;

$[H_{3}]$ $f$ is a sufficiently smooth functions in correspondence ranges except $x_{0}\in\Omega$;

$[H_{4}]$ $f(t, x, u)\leq -c < 0, (x\neq x_{0})$, where $C>0$ is a constant and for $f(t, x, u)=0$, there exists a solution $U_{00}$, such as $\lim\limits_{x\rightarrow x_{0}}U(t, x)\neq U(t, x_{0})$.

Now we construct the outer solution of problems (1.1)--(1.3).

The reduced problem for the original problem is

$\begin{equation} f(t, x, u)=0. \end{equation}$

(2.1)

From hypotheses, there is a solution $U_{00}(t, x)\ (x\neq x_{0})$ to equation (2.1). And there is a $U_{00}(t, x)$ which satisfies $f(t, x_{0}, U_{00}(t.x_{0}))=0$.

Let the outer solution $U_{00}(t, x)$ to problems (1.1)--(1.3), and

$\begin{equation} U(t, x)\sim\sum^{\infty}_{i, j=0}U_{ij}(t, x)\varepsilon^{i}\mu^{j}. \end{equation}$

(2.2)

Substituting eq. (2.2) into eq. (1.1), developing the nonlinear term $f$ in $\varepsilon$, and $\mu$, and equating coefficients of the same powers of $\varepsilon^{i}\mu^{j}\ (i, j=0, 1, \cdots, i+j\neq 0)$, respectively. We can obtain $U_{ij}(t, x), \ i, j=0, 1, \cdots, i+j\neq 0$. Substituting $U_{00}(t, x)$ and $U_{ij}(t, x), \ i, j=0, 1, \cdots, i+j\neq 0$ into eq.(2.2), we obtain the outer solution $U(t, x)$ to the original problem. But it does not continue at $(t, x_{0})$ and it may not satisfy the boundary and initial conditions (1.2)--(1.3), so that we need to construct the pointed layer, boundary layer and initial layer corrective functions.

Set up a local coordinate system $(\rho, \phi)$ near $x_{0}\in\Omega$. Define the coordinate of every point $Q$ in the neighborhood of $x_{0}$ with the following way: the coordinate $\rho(\leq\rho_{0})$ is the distance from the point $Q$ to $x_{0}$, where $\rho_{0}$ is small enough. The $\phi=(\phi_{1}, \phi_{2}, \cdots, \phi_{n-1})$ is a nonsingular coordinate.

In the neighborhood of $x_{0}: (0\leq\rho\leq\rho_{0})\in\Omega$,

$\begin{equation} L=a_{nn}\frac{\partial^{2}}{\partial\rho^{2}} +\sum^{n-1}_{i=1}a_{ni}\frac{\partial^{2}}{\partial\rho\partial \phi_{i}} +\sum^{n-1}_{i, j=1}a_{ij}\frac{\partial^{2}}{\partial \phi_{i}\partial \phi_{j}} +b_{n}\frac{\partial}{\partial\rho}+\sum^{n-1}_{i=1}b_{i}\frac{\partial}{\partial \phi_{i}}, \end{equation}$

(3.1)

where

$\begin{eqnarray*} & & a_{nn}=\sum^{n}_{i, j=1}\alpha_{ij}\frac{\partial\rho}{\partial x_{i}} \frac{\partial\rho}{\partial x_{j}}, \ \ a_{ni}=2\sum^{n}_{j, k=1}\alpha_{jk}\frac{\partial\rho}{\partial x_{j}} \frac{\partial\phi_{i}}{\partial x_{k}}, \ \ a_{ij}=\sum^{n}_{k, l=1}\alpha_{kl}\frac{\partial\phi_{i}}{\partial x_{k}} \frac{\partial\phi_{j}}{\partial x_{l}}, \\ & & b_{n}=\sum^{n}_{i, j=1}\alpha_{ij}\frac{\partial^{2}\rho}{\partial x_{i}\partial x_{j}}, \ \ b_{i}=\sum^{n}_{i, j=1}\alpha_{ij} \frac{\partial^{2}_{i}\phi_{i}}{\partial x_{i}\partial x_{j}}. \end{eqnarray*}$

We lead into the variables of multiple scales^[1] on $0\le \rho \le {{\rho }_{0}}\subset \Omega $:

$$ \widetilde{\sigma}=\frac{h(\rho, \phi)}{\mu}, \ \ \widetilde{\rho}=\rho\ \ \widetilde{\phi}=\phi, $$

where $h(\rho, \phi)$ is a function to be determined. For convenience, we still substitute $\rho, \phi$ for $\widetilde{\rho}, \widetilde{\phi}$ below respectively. From eq.(3.1), we have

$\begin{equation} L=\frac{1}{\mu^{2}}K_{0}+\frac{1}{\mu}K_{1}+K_{2}, \end{equation}$

(3.2)

while

$$K_{0}=a_{nn}h^{2}_{\rho}\frac{\partial^{2}}{\partial\rho^{2}}$$

and $K_{1}, K_{2}$ are determined operators and their constructions are omitted.

Let $h_{\rho}=\sqrt{1/a_{nn}}$ and the solution $u$ of original problems (1.1)--(1.3) be

$\begin{equation} u=U(t, x)+V_{1}(t, \rho, \phi), \end{equation}$

(3.3)

where $V_{1}$ is a pointed layer corrective term. And

$\begin{equation} V_{1}\sim\sum^{\infty}_{i, j=0}v_{1ij}(t, \rho, \phi)\sigma^{i}\mu^{j}. \end{equation}$

(3.4)

Substituting eqs.(3.1)--(3.4) into eq.(1.1), expanding nonlinear terms in $\sigma$ and $\mu$, and equating the coefficients of like powers of $\sigma^{i}\mu^{j}$, respectively, for $i, j=0, 1, \cdots$, we obtain

$\begin{eqnarray} & & K_{10}v_{100}=0, \ \ (\rho, \phi)\in(0\leq\rho\leq\rho_{0}), \\ \end{eqnarray}$

(3.5)

$\begin{eqnarray} & & v_{100}|_{\rho=0}=-U_{00}(t, x_{0}), \\ \end{eqnarray}$

(3.6)

$\begin{eqnarray} & & K_{10}v_{1ij}=G_{ij}, \ \ (\rho, \phi)\in(0\leq\rho\leq\rho_{0}), \ \ i, j=0, 1, \cdots, \ \ i+j\neq 0, \\ \end{eqnarray}$

(3.7)

$\begin{eqnarray} & & v_{1ij}|_{\rho=0}=-U_{ij}(t, x_{0}), \ \ i, j=0, 1, \cdots, \ \ i+j\neq 0, \end{eqnarray}$

(3.8)

where $G_{ij}\ (i, j=0, 1, \cdots, \ \ i+j\neq 0)$ are determined functions. From problems (3.5)--(3.6), we can have $v_{100}$, From $v_{100}$ and eqs.(3.7)--(3.8), we can obtain solutions $v_{1ij}\ (i, j=0, 1, \cdots, \ \ i+j\neq 0)$, successively.

From the hypotheses, it is easy to see that $v_{1ij}\ (i, j=0, 1, \cdots)$ possesses boundary layer behavior

$\begin{equation} v_{1ij}=O(\exp(-\delta_{ij}\frac{\rho}{\sigma})), \ \ i=0, 1, \cdots, \end{equation}$

(3.9)

where $\delta_{ij}>0\ (i, j=0, 1, \cdots)$ are constants.

Let $\overline{v}_{1ij}=\psi(\rho)v_{1ij}$, where $\psi(\rho)$ is a sufficiently smooth function in $0\leq\rho\leq\rho_{0}$, and satisfies

$\begin{equation*}\psi(\rho)= \left\{ \begin{aligned} & 1, ~~~~ 0\leq\rho\leq(1/3)\rho_{0}, \\ & 0, ~~~~ \rho\geq(2/3)\rho_{0}. \end{aligned} \right. \end{equation*}$

For convenience, we still substitute $v_{1ij}$ for $\overline{v}_{1ij}$ below. Then from eq. (3.4), we have the pointed layer corrective term $V_{1}$ near $(0\leq\rho\leq\rho_{0})\subset\Omega$.

Now we set up a local coordinate system $(\overline{\rho}, \overline{\phi})$ in the neighborhood near $\partial\Omega: 0\leq\overline{\rho}\leq\overline{\rho}_{0}$ as Ref. [9], where $\overline{\phi}= (\overline{\phi}_{1}, \overline{\phi}_{2}, \cdots, \overline{\phi}_{n-1}).$ In the neighborhood of $\partial\Omega: 0\leq\overline{\rho}\leq\overline{\rho}_{0}$,

$\begin{equation} L=\overline{a}_{nn}\frac{\partial^{2}}{\partial\overline{\rho}^{2}} +\sum^{n-1}_{i=1}\overline{a}_{ni} \frac{\partial^{2}}{\partial\overline{\rho}\partial\overline{ \phi}_{i}} +\sum^{n-1}_{i, j=1}\overline{a}_{ij} \frac{\partial^{2}}{\partial \overline{\phi}_{i}\partial \overline{\phi}_{j}} +\overline{b}_{n}\frac{\partial}{\partial\overline{\rho}} +\sum^{n-1}_{i=1}\overline{b}_{i}\frac{\partial}{\partial \overline{\phi}_{i}}, \end{equation}$

(4.1)

where

$\begin{eqnarray*} & & \overline{a}_{nn}=\sum^{n}_{i, j=1}\alpha_{ij}\frac{\partial\overline{\rho}}{\partial x_{i}} \frac{\partial\overline{\rho}}{\partial x_{j}}, \ \ \overline{a}_{ni}=2\sum^{n}_{j, k=1}\alpha_{jk}\frac{\partial\overline{\rho}}{\partial x_{j}} \frac{\partial\overline{\phi_{i}}}{\partial x_{k}}, \ \ \overline{a}_{ij}=\sum^{n}_{k, l=1}\alpha_{kl}\frac{\partial\overline{\phi}_{i}}{\partial x_{k}} \frac{\partial\overline{\phi_{j}}}{\partial x_{l}}, \\ & & \overline{b}_{n}=\sum^{n}_{i, j=1}\alpha_{ij} \frac{\partial^{2}\overline{\rho}}{\partial x_{i}\partial x_{j}}, \ \ \overline{b}_{i}=\sum^{n}_{i, j=1}\alpha_{ij} \frac{\partial^{2}_{i}\overline{\phi}_{i}}{\partial x_{i}\partial x_{j}}. \end{eqnarray*}$

We lead into the variables of multiple scales [1] on $(0\leq\overline{\rho}\leq \overline{\rho}_{0})\subset\Omega$:

$$ \overline{\sigma}=\frac{\overline{h}(\overline{\rho}, \overline{\phi})}{\mu}, \ \ \widetilde{\rho}=\overline{\rho}\ \ \widetilde{\phi}=\overline{\phi}, $$

where $\overline{h}(\overline{\rho}, \overline{\phi})$ is a function to be determined. For convenience, we still substitute $\overline{\rho}, \overline{\phi}$ for $\widetilde{\rho}, \widetilde{\phi}$ below respectively. From (4.1), we have

$\begin{equation} L=\frac{1}{\mu^{2}}\overline{K}_{0}+\frac{1}{\mu}\overline{K}_{1}+\overline{K}_{2}, \end{equation}$

(4.2)

while

$$\overline{K}_{0}=\overline{a}_{nn}\overline{h}^{2}_{\overline{\rho}} \frac{\partial^{2}}{\partial\overline{\sigma}^{2}}$$

and $\overline{K}_{1}, \overline{K}_{2}$ are determined operators and their constructions are omitted too.

Let $\overline{h}_{\overline{\rho}}=\sqrt{1/\overline{a}_{nn}}$ and the solution $u$ of original problems (1.1)--(1.3) be

$\begin{equation} u=U+V_{2}, \end{equation}$

(4.3)

where $V_{2}$ is a boundary layer corrective term. And

$\begin{equation} V_{2}\sim\sum^{\infty}_{i, j=0}v_{2ij}(t, \overline{\rho}, \overline{\phi})\varepsilon^{i}\sigma^{j}. \end{equation}$

(4.4)

Substituting eq.(4.4) into eqs.(1.1) and (1.2), expanding nonlinear terms in $\varepsilon$ and $\sigma$, and equating the coefficients of like powers of $\varepsilon^{i}\sigma^{j}\ (i, j=0, 1, \cdots)$. And we obtain

$\begin{eqnarray} & & \overline{K}_{0}v_{200}=0, \ \ (\overline{\rho}, \overline{\phi})\in(0\leq\overline{\rho}\leq\overline{\rho}_{0}), \\ \end{eqnarray}$

(4.5)

$\begin{eqnarray} & & v_{200}|_{\overline{\rho}=0}=g(t, x)-U_{00}(t, x), \\ \end{eqnarray}$

(4.6)

$\begin{eqnarray} & & K_{0}v_{2ij}=\overline{G}_{ij}, \ \ (\overline{\rho}, \overline{\phi})\in(0\leq\overline{\rho}\leq\overline{\rho}_{0}), \ \ i, j=0, 1, \cdots, \ \ i+j\neq 0, \\ \end{eqnarray}$

(4.7)

$\begin{eqnarray} & & v_{2ij}|_{\overline{\rho}=0}=-U_{ij}(t, x_{0}), \ \ i, j=0, 1, \cdots, \ \ i+j\neq 0, \end{eqnarray}$

(4.8)

where $\overline{G}_{ij}\ (i, j=0, 1, \cdots, \ i+j\neq 0)$ are determined functions successively, their constructions are omitted too.

From problems (4.5)--(4.6), we can have $v_{200}$. And from eqs. (4.7), (4.8), we can obtain solutions $v_{2ij}\ (i=0, 1, \cdots, \ i+j\neq 0)$ successively. Substituting into eq. (4.4), we obtain the boundary layer corrective function $V_{2}$ for the original boundary value problems (1.1)--(1.3).

From the hypotheses, it is easy to see that $v_{2ij}\ (i, j=0, 1, \cdots)$ possesses boundary layer behavior

$\begin{equation} v_{2ij}=O(\exp(-\overline{\delta}_{ij}\frac{\overline{\rho}}{\overline{\sigma}})), \ \ i, j=0, 1, \cdots, \end{equation}$

(4.9)

where $\overline{\delta}_{ij}>0\ (i, j=0, 1, \cdots)$ are constants.

Let $\overline{v}_{2ij}=\overline{\psi}(\overline{\rho})v_{2ij}$, where $\overline{\psi}(\overline{\rho})$ is a sufficiently smooth function in $0\leq\overline{\rho}\leq\overline{\rho}_{0}$, and satisfies

$\begin{equation*} \overline{\psi}(\overline{\rho})=\left\{ \begin{aligned} & 1, ~~~~0\leq\overline{\rho}\leq(1/3)\overline{\rho}_{0}, \\ & 0, ~~~~ \overline{\rho}\geq(2/3)\overline{\rho}_{0}. \end{aligned} \right. \end{equation*}$

For convenience, we still substitute $v_{2ij}$ for $\overline{v}_{2ij}$ below. Then from eq. (4.4) we have the boundary layer corrective term $V_{2}$ near $(0\leq\overline{\rho}\leq\overline{\rho}_{0})$.

The solution $u$ of original problems (1.1)--(1.3) be

$\begin{equation} u=U+V_{1}+V_{2}+W, \end{equation}$

(5.1)

where $W$ is an initial layer corrective term. Substituting eq. (5.1) into eqs. (1.1)--(1.3), we have

$\begin{eqnarray} & & \varepsilon^{2}W_{tt}-\mu^{2}LW=f(t, x, U+V_{1}+V_{2}+W)\nonumber\\ & & -f(t, x, U+V_{1}+V_{2})-\mu^{2}L(U+V_{1}+V_{2}), \\ \end{eqnarray}$

(5.2)

$\begin{eqnarray} & & W|_{x\in\partial\Omega}=(g(t, x)-U(t, x)-V_{2}(t, x))_{x\in\partial\Omega}, \\ \end{eqnarray}$

(5.3)

$\begin{eqnarray} & & W|_{t=0}=h_{1}(x)-U(0, x)-V_{1}(0, x)-V_{2}(0, x), \ \ x\in\Omega, \\ \end{eqnarray}$

(5.4)

$\begin{eqnarray} & & \varepsilon\frac{\partial W}{\partial t}|_{t=0}= h_{2}(x)-\varepsilon(\frac{\partial U(0, x)}{\partial t} -\frac{\partial V_{1}(0, x)}{\partial t}-\frac{\partial V_{2}(0, x)}{\partial t}), \ \ x\in\Omega. \end{eqnarray}$

(5.5)

We lead into a stretched variable [1, 2]: $\tau=t/\varepsilon$ and let

$\begin{equation} W\sim\sum^{\infty}_{i, j=0}w_{ij}(\tau, x)\varepsilon^{i}\mu^{j}. \end{equation}$

(5.6)

Substituting eqs. (2.2), (3.4), (4.4) and (5.6) into eqs. (5.2)--(5.5), expanding nonlinear terms in $\varepsilon$ and $\mu$, and equating the coefficients of like powers of $\varepsilon^{i}\mu^{j}$, respectively, for $i, j=0, 1, \cdots$, we obtain

$\begin{eqnarray} & & (w_{00})_{\tau\tau}=f(0, x, U_{00}+v_{100}+v_{200}+w_{00})-f(0, x, U_{00}+v_{100}+v_{200}), \\ \end{eqnarray}$

(5.7)

$\begin{eqnarray} & & w_{00}|_{x\in\partial\Omega}=0, \\ \end{eqnarray}$

(5.8)

$\begin{eqnarray} & & w_{00}|_{\tau=0}=h_{1}(x)-U_{00}(0, x)-v_{100}(0, x)-v_{200}(0, x), \ \ x\in\Omega, \\ \end{eqnarray}$

(5.9)

$\begin{eqnarray} & & \frac{\partial w_{00}}{\partial \tau}|_{\tau=0}=0, \ \ x\in\Omega.\\ \end{eqnarray}$

(5.10)

$\begin{eqnarray} & & (w_{ij})_{\tau\tau}=\overline{G}_{ij}, \ \ i, j=0, 1, \cdots, \ i+j\neq 0, \\ \end{eqnarray}$

(5.11)

$\begin{eqnarray} & & w_{ij}|_{x\in\partial\Omega}=h_{2}(x), \ \ i, j=0, 1, \cdots, \ i+j\neq 0, \\ \end{eqnarray}$

(5.12)

$\begin{eqnarray} & & w_{ij}|_{\tau=0}=-U_{ij}(0, x)-v_{1ij}(0, x)-v_{2ij}(0, x), \nonumber\\ & & x\in\Omega, \ \ i, j=0, 1, \cdots, \ i+j\neq 0, \\ \end{eqnarray}$

(5.13)

$\begin{eqnarray} & & \frac{\partial w_{ij}}{\partial \tau}|_{\tau=0}=-\frac{\partial U_{(i-1)j}(0, x)}{\partial t} -\frac{\partial v_{1(i-1)j}(0, x)}{\partial t}-\frac{\partial v_{2(i-1)j}(0, x)}{\partial t}, \nonumber\\ & & x\in\Omega, \ \ i, j=0, 1, \cdots, \ \ i+j\neq 0, \end{eqnarray}$

(5.14)

where $\overline{G}_{ij}\ (i, j=0, 1, \cdots, \ i+j\neq 0)$ are determined functions. From problems (5.7)--(5.10), we can have $w_{00}$, From $w_{00}$ and eqs. (5.11)--(5.14), we can obtain solutions $w_{ij} (i, j=0, 1, \cdots, \ i+j\neq 0)$ successively.

From the hypotheses, it is easy to see that $w_{ij}\ (i, j=0, 1, \cdots)$ possesses initial layer behavior

$\begin{equation} w_{ij}=O(\exp(-\widetilde{\delta}_{ij}\frac{t}{\varepsilon})), \ \ i=0, 1, \cdots, \end{equation}$

(5.15)

where $\widetilde{\delta}_{ij}>0\ (i, j=0, 1, \cdots)$ are constants.

Then from eq. (5.15) we have the initial corrective term $W$.

From eq. (5.1), thus we obtain the formal asymptotic expansion of solution $u$ for the nonlinear singular perturbation evolution equations initial-boundary value problems (1.1)--(1.3) with two parameters

$\begin{eqnarray} u & \sim & U_{00}+\sum^{\infty}_{i, j=0, 1, i+j\neq 0}U_{ij}\varepsilon^{i}\mu^{j} +\sum^{\infty}_{i, j=0}(v_{1ij}(t, x)+v_{2ij}(t, x))\sigma^{i}\mu^{j}\nonumber\\ & & +\sum^{\infty}_{i, j=0}w_{ij}(\tau, x)\varepsilon^{i}\mu^{j}, \ \ 0<\varepsilon, \mu, \sigma\ll 1. \end{eqnarray}$

(5.16)

Now we prove that this expansion (5.16) is a uniformly valid in $\Omega$ and we have the following theorem

Theorem Under hypotheses $[H_{1}]-[H_{4}]$, then there exists a solution $u(t, x)$ of the nonlinear singular perturbation evolution equation initial-boundary value problems (1.1)--(1.3) with two parameters and holds the uniformly valid asymptotic expansion (5.16) for $\varepsilon$ and $\mu$ in $(t, x)\in[0, T_{0}]\times\overline{\Omega}$.

Proof  We now get the remainder term $R(t, x)$ of the initial-boundary value problems (1.1)--(1.3). Let

$\begin{equation} u(t, x)=\overline{u}(t, x)+R(t, x), \end{equation}$

(6.1)

where

$\begin{eqnarray*} \overline{u} & \sim & U_{00}+\sum^{m}_{i, j=0, i+j\neq 0}U_{ij}\varepsilon^{i}\mu^{j} +\sum^{m}_{i, j=0}(v_{1ij}(t, x)+v_{2ij}(t, x))\sigma^{i}\mu^{j}\\ & & +\sum^{m}_{i, j=0}w_{ij}(\tau, x)\varepsilon^{i}\mu^{j}. \end{eqnarray*}$

Using eqs. (2.2), (3.9), (4.9), (5.15), (6.1), we obtain

$\begin{eqnarray*} F[R] & \equiv & \varepsilon^{2}\frac{\partial^{2}R}{\partial t^{2}}-\mu^{2}LR -f(t, x, \overline{u}+R)+f(t, x, \overline{u})\\ & = & O(\lambda^{m+1})\ \ x\in\Omega, \ \lambda=\max(\varepsilon, \mu, \sigma), \\ R & = & O(\lambda^{m+1})\ \ x\in\partial\Omega, \ \lambda=\max(\varepsilon, \mu, \sigma), \\ R|_{t=0} & = & O(\lambda^{m+1})\ \ x\in\Omega, \ \lambda=\max(\varepsilon, \mu, \sigma), \\ \varepsilon\frac{\partial R}{\partial t}|_{t=0} & = & O(\lambda^{m+1})\ \ x\in\Omega, \ \lambda=\max(\varepsilon, \mu, \sigma). \end{eqnarray*}$

The linearized differential operator $\overline{L}$ reads

$$ \overline{L}[p]=\varepsilon^{2}\frac{\partial^{2}p}{\partial t^{2}} -\mu^{2}L[p], $$

and therefore

$$ \Psi[p]\equiv f[p]-\overline{L}[p]=f(t, x, \overline{u})-f(t, x, (\overline{u}+p)) +f_{u}(t, x, (\overline{u}+p))p. $$

For fixed $\varepsilon, \mu$, the normed linear space $N$ is chosen as

$$ N=\{p|p\in C^{2}((0, T_{0}]\times\Omega), \ p|_{\partial\Omega}=g, \ p|_{t=0}=h_{1}, \ p_{t}|_{t=0}=h_{12}\} $$

with norm

$$ \|p\|=\max_{t\in(0.T_{0}], x\in\Omega}|p|, $$

and the Banach space $B$ as

$$ B=\{q|q\in C((0, T_{0}]\times\Omega)\} $$

with norm

$$ \|q\|=\max_{t\in(0.T_{0}], x\in\Omega}|q|. $$

From the hypotheses we may show that the condition

$$ \|L^{-1}[g]\|\leq l^{-1}\|g\|, \ \ \forall g\in B $$

of the fixed point theorem [1, 2] is fulfilled where $l^{-1}$ is independent of $\varepsilon$ and $\mu$, i.e., $L^{-1}$ is continuous. The Lipschitz condition of the fixed point theorem become

$\begin{eqnarray*} & & \|\Psi[p_{2}]-\Psi[p_{1}]\|\\ & < & C_{1}\max_{t\in(0, T_{0}], x\in\Omega}\{(|p|_{1}+|p|_{2})|p_{2}-p_{1}|\} +C_{2}\max_{t\in(0, T_{0}], x\in\Omega}\{|p^{2}|\cdot|p_{2}-p_{1}|\}\\ & < & C_{3}t\|p_{2}-p_{1}\|, \end{eqnarray*}$

where $C_{1}, C_{2}$ and $C_{3}$ are constants independent of $\varepsilon$ and $\mu$, this inequality is valid for all $p_{1}, p_{2}$ in a ball $K_{N}(r)$ with $\|r\|\leq 1$. Finally, we obtain the result that the remainder term exists and moreover

$$ \max_{t\in(0, T_{0}], x\in\Omega}|R(t, x)|=O(\lambda^{m+1}), \ \ \lambda=\max(\varepsilon, \mu, \sigma). $$

From eq. (6.1), we have

$\begin{eqnarray*} \overline{u} & \equiv & U_{00}+\sum^{m}_{i, j=0, i+j\neq 0}U_{ij}\varepsilon^{i}\mu^{j} +\sum^{m}_{i, j=0}(v_{1ij}(t, x)+v_{2ij}(t, x))\sigma^{i}\mu^{j}\\ & & +\sum^{m}_{i, j=0}w_{ij}(\tau, x)\varepsilon^{i}\mu^{j}+O(\lambda^{m+1}), \ \ 0<\varepsilon, \mu, \sigma., \ \ \lambda=\max(\varepsilon, \mu, \sigma). \end{eqnarray*}$

The proof of the theorem is completed.