In this section, let \((H_{1})\), \((H_{2})\) hold. We make the following additional assumption:
-
\((H_{2}^{\prime})\)
:
-
\(f_{1}\in C^{1}([0,1]\times \mathbb{R} _{+}\times \mathbb{R} ^{q+2})\).
We consider the following perturbed problem, where ε is a small parameter such that, \(\vert \varepsilon \vert \leq1\):
$$ (P_{\varepsilon}) \quad \textstyle\begin{cases} u_{tt}-u_{xx}=F_{\varepsilon}[u](x,t) ,\quad 0< x< 1 , 0< t< T , \\ u_{x}(0,t)-h_{0}u(0,t)=u_{x}(1,t)+h_{1}u(1,t)=0 , \\ u(x,0)=\tilde{u}_{0}(x) , \quad\quad u_{t}(x,0)= \tilde{u}_{1}(x) ,\end{cases} $$
where
$$ \textstyle\begin{cases} F_{\varepsilon}[u](x,t)=f[u](x,t)+ \varepsilon f_{1}[u](x,t) , \\ f[u](x,t)=f ( x,t,u(x,t),u(\eta_{1},t),\ldots,u(\eta _{q},t),u^{\prime }(t) ) , \\ f_{1}[u](x,t)=f_{1} ( x,t,u(x,t),u( \eta_{1},t),\ldots,u(\eta _{q},t),u^{\prime}(t) ) .\end{cases} $$
First, we note that if the functions f, \(f_{1}\) satisfy \((H_{2})\), \((H_{2}^{\prime})\), then the a priori estimates of the Galerkin approximation sequence \(\{u_{m}^{(k)}\}\) in the proof of Theorem 3.1 for Prob. (1.1)-(1.3) corresponding to \(f=F_{\varepsilon}[u]\), \(\vert \varepsilon \vert \leq1\), satisfy \(u_{m}^{(k)}\in W_{1}(M,T)\), where M, T are constants independent of ε. We also note that the positive constants M and T are chosen as in (3.22)-(3.23) with \(\vert f ( \cdot,0,\tilde{u}_{0},\tilde{u}_{0}(\eta_{1}),\ldots,\tilde{u}_{0}(\eta_{q}),\tilde{u}_{1} ) \vert \), \(K_{M}(f)\), stand for
$$ \begin{gathered} \bigl\vert f \bigl( \cdot,0,\tilde{u}_{0},\tilde{u}_{0}( \eta _{1}),\ldots,\tilde{u}_{0}( \eta_{q}),\tilde{u}_{1} \bigr) \bigr\vert + \bigl\vert f_{1} \bigl( \cdot,0,\tilde{u}_{0},\tilde{u}_{0}( \eta_{1}),\ldots ,\tilde{u}_{0}( \eta_{q}),\tilde{u}_{1} \bigr) \bigr\vert ,\\ K_{M}(f)+K_{M}(f_{1}), \end{gathered} $$
respectively.
Hence, the limit \(u_{\varepsilon}\) in suitable function spaces of the sequence \(\{u_{m}^{(k)}\}\) as \(k\rightarrow+\infty\), after \(m\rightarrow +\infty\), is a unique weak solution of the problem \((P_{\varepsilon})\) satisfying \(u_{\varepsilon}\in W_{1}(M,T)\).
Then we can prove, in a manner similar to the proof of Theorem 3.2, that the limit \(u_{0}\) in suitable function spaces of the family \(\{ u_{\varepsilon}\} \) as \(\varepsilon\rightarrow0\) is a unique weak solution of the problem \((P_{0})\) (corresponding to \(\varepsilon=0\)) satisfying \(u_{0}\in W_{1}(M,T)\).
We shall study the asymptotic expansion of the solution of the problem \((P_{\varepsilon})\) with respect to a small parameter ε.
We use the following notations. For a multi-index \(\alpha=(\alpha _{1},\ldots,\alpha_{N})\in \mathbb{Z} _{+}^{N}\), and \(x=(x_{1},\ldots,x_{N})\in \mathbb{R} ^{N}\), we put
$$ \textstyle\begin{cases} \vert \alpha \vert = \alpha_{1}+\cdots+\alpha_{N},\quad\quad \alpha!= \alpha_{1}!\cdots \alpha_{N}! , \\ \alpha, \beta\in \mathbb{Z} _{+}^{N}, \quad \alpha\leq\beta\quad \Longleftrightarrow\quad \alpha _{i}\leq \beta_{i}\quad \forall i=1,\ldots,N , \\ x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{N}^{\alpha_{N}} .\end{cases} $$
Next, we need the following lemma.
Lemma 4.1
Let
\(m, N\in \mathbb{N} \)
and
\(x=(x_{1},\ldots,x_{N})\in \mathbb{R} ^{N}\), \(\varepsilon\in \mathbb{R} \). Then
$$ \Biggl( \sum_{i=1}^{N}x_{i} \varepsilon^{i} \Biggr) ^{m}=\sum _{k=m}^{mN}P_{k}^{(m)}[N,x] \varepsilon^{k}, $$
(4.1)
where the coefficients
\(P_{k}^{(m)}[N,x]\), \(m\leq k\leq mN\)
depending on
\(x=(x_{1},\ldots,x_{N})\)
are defined by the formulas
$$ P_{k}^{(m)}[N,x]= \textstyle\begin{cases} x_{k} , & 1\leq k\leq N , m=1 , \\ {\sum}_{\alpha\in A_{k}^{(m)}(N)} \frac{m!}{\alpha !}x^{\alpha} , & m\leq k\leq mN , m\geq2 ,\end{cases} $$
(4.2)
where
\(A_{k}^{(m)}(N)=\{\alpha\in \mathbb{Z} _{+}^{N}: \vert \alpha \vert =m, \sum_{i=1}^{N}i\alpha _{i}=k\}\).
The proof of Lemma 4.1 is easy, hence we omit the details.
Now, we assume that
-
\((H_{2}^{(N)})\)
:
-
\(f\in C^{N+1}([0,1]\times \mathbb{R} _{+}\times \mathbb{R} ^{q+2})\), \(f_{1}\in C^{N}([0,1]\times \mathbb{R} _{+}\times \mathbb{R} ^{q+2})\).
Let \(u_{0}\) be a unique weak solution of the problem \(( P_{0} ) \) corresponding to \(\varepsilon=0\), i.e.,
$$ (P_{0})\quad \textstyle\begin{cases} u_{0}^{\prime\prime}- \Delta u_{0}=f[u_{0}]\equiv F_{0} , \quad 0< x< 1 , 0< t< T , \\ u_{0x}(0,t)-h_{0}u_{0}(0,t)=u_{0x}(1,t)+h_{1}u_{0}(1,t)=0 , \\ u_{0}(x,0)=\tilde{u}_{0}(x) , \quad\quad u_{0}^{\prime}(x,0)= \tilde {u}_{1}(x) , \\ u_{0}\in W_{1}(M,T) .\end{cases} $$
Let us consider the sequence of weak solutions \(u_{k}\), \(1\leq k\leq N\), defined by the following problems:
$$ (\tilde{P}_{k}) \quad \textstyle\begin{cases} u_{k}^{\prime\prime}-\Delta u_{k}=F_{k} ,\quad 0< x< 1 , 0< t< T , \\ u_{kx}(0,t)-h_{0}u_{k}(0,t)=u_{kx}(1,t)+h_{1}u_{k}(1,t)=0 , \\ u_{k}(x,0)=u_{k}^{\prime}(x,0)=0 , \\ u_{k}\in W_{1}(M,T) ,\end{cases} $$
where \(F_{k}\), \(1\leq k\leq N\), are defined by the formulas
$$ F_{k}= \textstyle\begin{cases} \bar{\Phi}_{1}[N,f]+f_{1}[u_{0}] , & k=1 , \\ \bar{\Phi}_{k}[N,f]+\bar{\Phi}_{k-1}[N-1,f_{1}] , & 2\leq k\leq N ,\end{cases} $$
(4.3)
with \(\bar{\Phi}_{k}[N,f]=\bar{\Phi}_{k}[N,f,u_{0},u_{0}^{\prime},\vec {u},\vec{u}^{\prime}]\), \(0\leq k\leq N\), are defined by the formulas
$$ \bar{\Phi}_{k}[N,f]= \textstyle\begin{cases} f[u_{0}] , & k=0 , \\ \sum_{1\leq \vert \gamma \vert \leq k}\frac {1}{\gamma!}D^{\gamma}f[u_{0}] \Psi_{k} [\gamma,N,\vec{u},\vec{u}^{\prime} ] , & 1\leq k \leq N ,\end{cases} $$
(4.4)
where
$$\begin{aligned} &\Psi_{k} \bigl[\gamma,N,\vec{u},\vec{u}^{\prime} \bigr] \\ &\quad =\sum_{\substack{ (\beta_{1},\ldots,\beta_{q+2})\in \widetilde{A}(\gamma,N), \\ \beta_{1}+\cdots+\beta_{q+2}=k}}P_{\beta _{1}}^{(\gamma _{1})} \bigl[N, \vec{u}(x,t) \bigr]P_{\beta_{2}}^{(\gamma_{2})} \bigl[N,\vec{u}(\eta _{1},t) \bigr]\cdots \\ &\quad\quad{}\times P_{\beta_{q+1}}^{(\gamma_{q+1})} \bigl[N,\vec{u}(\eta _{q},t) \bigr]P_{\beta_{q+2}}^{(\gamma_{q+2})} \bigl[N, \vec{u}^{\prime}(x,t) \bigr], \end{aligned}$$
(4.5)
with
$$ \textstyle\begin{cases} \widetilde{A}(\gamma,N)= \{( \beta_{1},\ldots,\beta_{q+2})\in \mathbb{Z} _{+}^{q+2}:\gamma_{i}\leq \beta_{i}\leq N\gamma_{i} , 1\leq i\leq q+2 \} , \\ \gamma=(\gamma_{1},\ldots,\gamma_{q+2})\in \mathbb{Z} _{+}^{q+2} , \quad 1 \leq \vert \gamma \vert \leq N ,\end{cases} $$
(4.6)
and \(\vec{u}(x,t)=(u_{1}(x,t),\ldots,u_{N}(x,t))\), \(\vec{u}^{\prime }(x,t)=(\dot{u}_{1}(x,t),\ldots,\dot{u}_{N}(x,t))\).
Then, we have the following theorem.
Theorem 4.2
Let
\((H_{1})\)
and
\((H_{2}^{(N)})\)
hold. Then there exist constants
\(M>0\)
and
\(T>0\)
such that, for every
\(\varepsilon\in [-1,1]\), the problem
\((P_{\varepsilon})\)
has a unique weak solution
\(u_{\varepsilon }\in W_{1}(M,T)\)
satisfying the asymptotic estimation up to order
\(N+1\)
as follows:
$$ \Biggl\Vert u_{\varepsilon}-\sum_{k=0}^{N}u_{k} \varepsilon ^{k} \Biggr\Vert _{W_{1}(T)}\leq C_{T} \vert \varepsilon \vert ^{N+1}, $$
(4.7)
where the functions
\(u_{k}\), \(0\leq k\leq N\), are the weak solutions of the problems
\((P_{0})\), \((\tilde{P}_{k})\), \(1\leq k\leq N\), respectively, and
\(C_{T}\)
is a constant depending only on
N, T, f, \(f_{1}\), \(u_{k}\), \(0\leq k\leq N\).
Remark 4.1
By the fact that it is very difficult to find \(u_{\varepsilon} \) of the problem \((P_{\varepsilon})\), we try to search the weak solutions \(u_{k}\), \(0\leq k\leq N\), of the problems \((P_{0})\), \((\tilde{P}_{k})\). Clearly, they are found much more easily than \(u_{\varepsilon} \) and \(u_{\varepsilon}\) can be approximated by \(u_{k}\) via (4.7).
In order to prove Theorem 4.2, we need the following lemmas.
Lemma 4.3
Let
\(\bar{\Phi}_{k}[N,f]\), \(0\leq k\leq N\), be the functions defined by formulas (4.4)-(4.6). Put
\(h=\sum_{k=0}^{N}u_{k}\varepsilon^{k}\), then we have
$$ f[h]=\sum_{k=0}^{N}\bar{ \Phi}_{k}[N,f] \varepsilon^{k}+ \vert \varepsilon \vert ^{N+1} \hat{R}_{N} \bigl[f,u_{0},\vec{u},\vec {u}^{\prime }, \varepsilon \bigr], $$
(4.8)
with
\(\Vert \hat{R}_{N}[f,u_{0},\vec{u},\vec{u}^{\prime },\varepsilon] \Vert _{L^{\infty}(0,T;L^{2})}\leq C\), where
C
is a constant depending only on
N, T, f, \(u_{k}\), \(\dot{u}_{k}\), \(0\leq k\leq N\).
Proof of Lemma 4.3
(i) In the case of \(N=1\), the proof of (4.8) is easy, hence we omit the details, which we only prove with \(N\geq2\). Put \(h=u_{0}+\sum _{k=1}^{N}u_{k}\varepsilon^{k}\equiv u_{0}+h_{1}\), we rewrite as follows:
$$\begin{aligned} &f[h](x,t) \\ &\quad =f \bigl( x,t,h(x,t),h(\eta_{1},t),\ldots,h(\eta_{q},t), \dot {h}(x,t) \bigr) \\ &\quad =f \bigl(x,t,u_{0}(x,t)+h_{1}(x,t),u_{0}( \eta_{1},t)+h_{1}(\eta _{1},t),\ldots, \\ &\quad\quad{} u_{0}( \eta_{q},t)+h_{1}(\eta_{q},t),\dot {u}_{0}(x,t)+\dot{h}_{1}(x,t) \bigr). \end{aligned}$$
(4.9)
By using Taylor’s expansion of the function \(f[h]\) around the point
$$ {}[ u_{0}]\equiv \bigl(x,t,u_{0}(x,t),u_{0}( \eta_{1},t),\ldots ,u_{0}(\eta _{q},t), \dot{u}_{0}(x,t) \bigr) $$
up to order \(N+1\), we obtain
$$ \begin{aligned}[b] f[h]&=f[u_{0}]+\sum_{1\leq \vert \gamma \vert \leq N} \frac{1}{\gamma!}D^{\gamma}f[u_{0}]h_{1}^{\gamma_{1}}(x,t)h_{1}^{\gamma _{2}}( \eta_{1},t)\cdots \\ &\quad{}\times h_{1}^{\gamma_{q+1}}(\eta_{q},t)\dot {h}_{1}^{\gamma _{q+2}}(x,t)+R_{N}[f,u_{0},h_{1}], \end{aligned} $$
(4.10)
where
$$\begin{aligned} &R_{N}[f,u_{0},h_{1}] \\ &\quad =\sum_{ \vert \gamma \vert =N+1}\frac{N+1}{\gamma!}\int_{0}^{1}(1-\theta)^{N}D^{\gamma}f[u_{0}+ \theta h_{1}]h_{1}^{\gamma _{1}}(x,t)h_{1}^{\gamma_{2}}( \eta_{1},t)\cdots \\ &\quad \quad{}\times h_{1}^{\gamma_{q+1}}(\eta _{q},t) \dot{h}_{1}^{\gamma_{q+2}}(x,t)\,d\theta \\ &\quad = \vert \varepsilon \vert ^{N+1}R_{N}^{(1)}[f,u_{0},h_{1}, \varepsilon], \end{aligned}$$
(4.11)
$$\begin{aligned}& \gamma = (\gamma_{1},\ldots,\gamma_{q+2})\in \mathbb{Z} _{+}^{q+2}, \\& \vert \gamma \vert = \gamma_{1}+\cdots+\gamma_{q+2}, \\& \gamma! = \gamma_{1}!\cdots \gamma_{q+2}!, \\& D^{\gamma}f = D_{3}^{\gamma_{1}}D_{4}^{\gamma_{2}}\cdots D_{q+4}^{\gamma _{q+2}}f, \\& D^{\gamma}f[u_{0}] = D^{\gamma}f \bigl(x,t,u_{0}(x,t),u_{0}( \eta _{1},t),\ldots,u_{0}(\eta_{q},t), \dot{u}_{0}(x,t) \bigr). \end{aligned}$$
By formula (4.1), we get
$$\begin{aligned}& h_{1}^{\gamma_{1}}(x,t) = \Biggl( \sum _{k=1}^{N}u_{k}(x,t) \varepsilon^{k} \Biggr) ^{\gamma _{1}}=\sum_{k=\gamma_{1}}^{N\gamma_{1}}P_{k}^{(\gamma_{1})} \bigl[N, \vec{u}(x,t) \bigr]\varepsilon^{k}, \\& h_{1}^{\gamma_{2}}(\eta_{1},t) = \Biggl( \sum _{k=1}^{N}u_{k}(\eta _{1},t) \varepsilon^{k} \Biggr) ^{\gamma_{2}}=\sum _{k=\gamma _{2}}^{N\gamma_{2}}P_{k}^{(\gamma_{2})} \bigl[N, \vec{u}( \eta _{1},t) \bigr]\varepsilon ^{k}, \\& \vdots \\& h_{1}^{\gamma_{q+1}}(\eta_{q},t) = \Biggl( \sum _{k=1}^{N}u_{k}(\eta_{q},t) \varepsilon^{k} \Biggr) ^{\gamma _{q+1}}=\sum _{k=\gamma_{q+1}}^{N\gamma_{q+1}}P_{k}^{(\gamma _{q+1})} \bigl[N, \vec{u}( \eta_{q},t) \bigr]\varepsilon^{k}, \\& \dot{h}_{1}^{\gamma_{q+2}}(x,t) = \Biggl( \sum _{k=1}^{N} \dot{u}_{k}(x,t) \varepsilon^{k} \Biggr) ^{\gamma_{q+2}}=\sum _{k=\gamma _{q+2}}^{N\gamma_{q+2}}P_{k}^{(\gamma_{q+2})} \bigl[N, \vec{u}^{\prime }(x,t) \bigr]\varepsilon^{k}, \end{aligned}$$
(4.12)
where \(\vec{u}(x,t)=(u_{1}(x,t),\ldots,u_{N}(x,t))\), \(\vec{u}^{\prime }(x,t)=(\dot{u}_{1}(x,t),\ldots,\dot{u}_{N}(x,t))\).
Hence, we deduce from (4.12), that
$$ \begin{aligned}[b] & h_{1}^{\gamma_{1}}(x,t)h_{1}^{\gamma_{2}}( \eta_{1},t)\cdots h_{1}^{\gamma _{q+1}}(\eta_{q},t) \dot{h}_{1}^{\gamma _{q+2}}(x,t) \\ &\quad =\sum_{k= \vert \gamma \vert }^{N} \Psi _{k} \bigl[\gamma,N,\vec{u},\vec{u}^{\prime} \bigr] \varepsilon ^{k}+\sum_{k=N+1}^{ \vert \gamma \vert N} \Psi _{k} \bigl[\gamma ,N,\vec{u},\vec{u}^{\prime} \bigr] \varepsilon^{k}, \end{aligned} $$
(4.13)
where
$$ \textstyle\begin{cases} \Psi_{k} [\gamma,N,\vec{u}, \vec{u}^{\prime} ] \\ \quad =\sum_{ (\beta_{1},\ldots,\beta_{q+2})\in\widetilde {A}(\gamma,N), \beta_{1}+\cdots+\beta_{q+2}=k}P_{\beta _{1}}^{(\gamma _{1})} [N, \vec{u}(x,t) ]P_{\beta_{2}}^{(\gamma_{2})} [N,\vec{u}(\eta _{1},t) ]\cdots \\ \quad \quad{}\times P_{\beta_{q+1}}^{(\gamma_{q+1})} [N,\vec{u}(\eta _{q},t) ]P_{\beta_{q+2}}^{(\gamma_{q+2})} [N, \vec{u}^{\prime}(x,t) ] , \\ \widetilde{A}(\gamma,N)= \{(\beta_{1},\ldots, \beta_{q+2})\in \mathbb{Z} _{+}^{q+2}: \gamma_{i}\leq\beta_{i}\leq N\gamma_{i} , 1\leq i\leq q+2 \} .\end{cases} $$
(4.14)
We deduce from (4.10), (4.13) that
$$\begin{aligned} f[h] =&f[u_{0}]+\sum _{1\leq \vert \gamma \vert \leq N} \frac{1}{\gamma!}D^{\gamma}f[u_{0}] \sum_{k= \vert \gamma \vert }^{N} \Psi_{k} \bigl[ \gamma,N,\vec{u}, \vec{u}^{\prime} \bigr]\varepsilon ^{k}+ \vert \varepsilon \vert ^{N+1}\hat{R}_{N} \bigl[f,u_{0}, \vec {u},\vec{u}^{\prime},\varepsilon \bigr] \\ =& f[u_{0}]+\sum_{k=1}^{N} \biggl( \sum_{1\leq \vert \gamma \vert \leq k}\frac{1}{\gamma!}D^{\gamma }f[u_{0}] \Psi_{k} \bigl[\gamma,N,\vec{u},\vec{u}^{\prime} \bigr] \biggr) \varepsilon ^{k}+ \vert \varepsilon \vert ^{N+1} \hat{R}_{N} \bigl[f,u_{0},\vec {u},\vec{u}^{\prime},\varepsilon \bigr] \\ =&f[u_{0}]+\sum_{k=1}^{N} \bar{ \Phi}_{k}[N,f]\varepsilon ^{k}+ \vert \varepsilon \vert ^{N+1}\hat{R}_{N} \bigl[f,u_{0},\vec {u},\vec{u}^{\prime},\varepsilon \bigr] , \end{aligned}$$
(4.15)
where \(\bar{\Phi}_{k}[N,f]\), \(1\leq k\leq N\), are defined by (4.4)-(4.6) and
$$ \begin{aligned}[b] &\vert \varepsilon \vert ^{N+1} \hat{R}_{N} \bigl[f,u_{0},\vec {u},\vec{u}^{\prime}, \varepsilon \bigr] \\ &\quad =\sum_{1\leq \vert \gamma \vert \leq N}\frac {1}{\gamma!}D^{\gamma}f[u_{0}] \sum_{k=N+1}^{ \vert \gamma \vert N}\Psi _{k} \bigl[ \gamma,N,\vec{u},\vec{u}^{\prime} \bigr]\varepsilon^{k}+ \vert \varepsilon \vert ^{N+1}R_{N}^{(1)}[f,u_{0},h_{1}, \varepsilon] .\end{aligned} $$
(4.16)
By the boundedness of the functions \(u_{k}\), \(\dot{u}_{k}\), \(1\leq k\leq N\), in the function space \(L^{\infty}(0,T; H^{1})\), we obtain from (4.11), (4.14), and (4.16) that \(\Vert \hat{R}_{N}[f,u_{0},\vec {u},\vec{u}^{\prime},\varepsilon] \Vert _{L^{\infty}(0,T;L^{2})} \leq C\), where C is a constant depending only on N, T, f, \(u_{k}\), \(\dot {u}_{k}\), \(1\leq k\leq N\). Thus, Lemma 4.3 is proved. □
Remark 4.2
Lemma 4.3 is a generalization of the formula contained in ([7], p.262, formula (4.38)) and then Lemma 4.4 follows. These lemmas are the key to establishing the asymptotic expansion of the weak solution \(u_{\varepsilon}\) of order \(N+1\) in a small parameter ε as below.
Let \(u=u_{\varepsilon}\in W_{1}(M,T)\) be the unique weak solution of the problem \((P_{\varepsilon})\). Then \(v=u_{\varepsilon }-\sum_{k=0}^{N}u_{k}\varepsilon^{k}\equiv u_{\varepsilon}-h\) satisfies the problem
$$ \textstyle\begin{cases} v^{\prime\prime}-\Delta v=f[v+h]-f[h]+ \varepsilon ( f_{1}[v+h]-f_{1}[h] ) \\ \hspace{38pt}{} +E_{\varepsilon}(x,t) , \quad 0< x< 1 , 0< t< T , \\ v_{x}(0,t)-h_{0}v(0,t)=v_{x}(1,t)+h_{1}v(1,t)=0 , \\ v(x,0)=v^{\prime}(x,0)=0 ,\end{cases} $$
(4.17)
where
$$ E_{\varepsilon}(x,t)=f[h]-f[u_{0}]+\varepsilon f_{1}[h]- \sum_{k=1}^{N}F_{k} \varepsilon^{k}, $$
(4.18)
and \(F_{k}\), \(1\leq k\leq N\), are defined by formulas (4.3).
Then we have the following lemma.
Lemma 4.4
Let
\((H_{1})\)
and
\((H_{2}^{(N)})\)
hold. Then there exists a constant
\(C_{\ast}\)
such that
$$ \Vert E_{\varepsilon} \Vert _{L^{\infty}(0,T;L^{2})}\leq C_{\ast } \vert \varepsilon \vert ^{N+1}, $$
(4.19)
where
\(C_{\ast}\)
is a constant depending only on
N, T, f, \(f_{1}\), \(u_{k}\), \(0\leq k\leq N\).
Proof of Lemma 4.4
In the case of \(N=1\), the proof of Lemma 4.4 is easy, hence we omit the details, which we only prove with \(N\geq2\).
By using formula (4.8) for the function \(f_{1}[h]\), we obtain
$$ f_{1}[h]=f_{1}[u_{0}]+\sum _{k=1}^{N-1} \bar{\Phi}_{k}[N-1,f_{1}]\varepsilon^{k}+ \vert \varepsilon \vert ^{N} \hat{R}_{N-1} \bigl[f_{1},u_{0},\vec{u}, \vec{u}^{\prime},\varepsilon \bigr], $$
(4.20)
where \(\Vert \hat{R}_{N-1}[f_{1},u_{0},\vec{u},\vec{u}^{\prime },\varepsilon] \Vert _{L^{\infty}(0,T;L^{2})}\leq C\), with C is a constant depending only on N, T, \(f_{1}\), \(u_{k}\), \(0\leq k\leq N\).
By (4.20), we rewrite \(\varepsilon f_{1}[h]\) as follows:
$$ \varepsilon f_{1}[h]=\varepsilon f_{1}[u_{0}]+ \sum_{k=2}^{N}\bar {\Phi }_{k-1}[N-1,f_{1}] \varepsilon^{k}+\varepsilon \vert \varepsilon \vert ^{N} \hat{R}_{N-1} \bigl[f_{1},u_{0},\vec{u}, \vec{u}^{\prime },\varepsilon \bigr]. $$
(4.21)
Hence, we deduce from (4.8) and (4.21) that
$$\begin{aligned}& f[h]-f[u_{0}]+\varepsilon f_{1}[h] \\& \quad = \bigl( f_{1}[u_{0}]+\bar{\Phi}_{1}[N,f] \bigr) \varepsilon +\sum_{k=2}^{N} \bigl( \bar{ \Phi}_{k}[N,f]+\bar{\Phi}_{k-1}[N-1,f_{1}] \bigr) \varepsilon^{k} \\& \quad\quad{} + \vert \varepsilon \vert ^{N+1}\tilde {R}_{N} \bigl[f,f_{1},u_{0},\vec{u},\vec{u}^{\prime}, \varepsilon \bigr], \end{aligned}$$
(4.22)
where
$$ \vert \varepsilon \vert ^{N+1}\tilde {R}_{N} \bigl[f,f_{1},u_{0},\vec{u},\vec{u}^{\prime}, \varepsilon \bigr]= \vert \varepsilon \vert ^{N+1}\hat{R}_{N} \bigl[f,u_{0},\vec{u},\vec{u}^{\prime}, \varepsilon \bigr]+\varepsilon \vert \varepsilon \vert ^{N} \hat{R}_{N-1} \bigl[f_{1},u_{0},\vec {u},\vec{u}^{\prime},\varepsilon \bigr]. $$
(4.23)
Combining (4.3), (4.18), and (4.22) leads to
$$ E_{\varepsilon}(x,t)= \vert \varepsilon \vert ^{N+1}\tilde{R} _{N} \bigl[f,f_{1},u_{0},\vec{u}, \vec{u}^{\prime},\varepsilon \bigr]. $$
(4.24)
By the boundedness of the functions \(u_{k}\), \(u_{k}^{\prime}\), \(0\leq k\leq N\), in the function space \(L^{\infty}(0,T; H^{1})\), we obtain from (4.8), (4.20), (4.23), and (4.24) that
$$ \Vert E_{\varepsilon} \Vert _{L^{\infty}(0,T;L^{2})}\leq C_{\ast } \vert \varepsilon \vert ^{N+1}, $$
(4.25)
where \(C_{\ast}\) is a constant depending only on N, T, f, \(f_{1}\), \(u_{k}\), \(u_{k}^{\prime}\), \(0\leq k\leq N\).
The proof of Lemma 4.4 is complete. □
Proof of Theorem 4.2
Consider the sequence \(\{v_{m}\}\) defined by
$$ \textstyle\begin{cases} v_{0}\equiv0 , \\ v_{m}^{\prime\prime}-\Delta v_{m}=f[v_{m-1}+h]-f[h]+ \varepsilon ( f_{1}[v_{m-1}+h]-f_{1}[h] ) \\ \hspace{47pt}{}+E_{\varepsilon}(x,t) , \quad 0< x< 1 , 0< t< T , \\ v_{mx}(0,t)-h_{0}v_{m}(0,t)=v_{mx}(1,t)+h_{1}v_{m}(1,t)=0 , \\ v_{m}(x,0)=v_{m}^{\prime}(x,0)=0 , \quad m \geq1 .\end{cases} $$
(4.26)
By multiplying two sides of (4.26) with \(v_{m}^{\prime}\) and after integration in t, we have
$$\begin{aligned} Z_{m}(t) =&2 \int _{0}^{t} \bigl\langle E_{\varepsilon }(s),v_{m}^{\prime }(s) \bigr\rangle \,ds+2 \int _{0}^{t} \bigl\langle f[v_{m-1}+h]-f[h],v_{m}^{\prime}(s) \bigr\rangle \,ds \\ &{}+2\varepsilon \int _{0}^{t} \bigl\langle f_{1}[v_{m-1}+h]-f_{1}[h],v_{m}^{\prime}(s) \bigr\rangle \,ds \\ =&\bar{J}_{1}+\bar{J}_{2}+\bar{J}_{3}, \end{aligned}$$
(4.27)
where \(Z_{m}(t)= \Vert v_{m}^{\prime}(t) \Vert ^{2}+ \Vert v_{m}(t) \Vert _{a}^{2}\).
We estimate the integrals on the right-hand side of (4.27) as follows.
Estimating
\(\bar{J}_{1}\). By using Lemma 4.4, we deduce that
$$ \bar{J}_{1}=2 \int _{0}^{t} \bigl\langle E_{\varepsilon }(s),v_{m}^{\prime }(s) \bigr\rangle \,ds\leq TC_{\ast}^{2} \vert \varepsilon \vert ^{2N+2}+ \int _{0}^{t}Z_{m}(s)\,ds. $$
(4.28)
Estimating
\(\bar{J}_{2}\). We note that
$$ \bigl\Vert f[v_{m-1}+h]-f[h] \bigr\Vert \leq\sqrt{2}(q+1)K_{M_{\ast }}(f) \Vert v_{m-1} \Vert _{W_{1}(T)}, $$
(4.29)
with \(M_{\ast}=(N+2)M\).
It follows from (4.29) that
$$ \begin{aligned}[b] \bar{J}_{2} &\leq 2 \int _{0}^{t} \bigl\Vert f[v_{m-1}+h]-f[h] \bigr\Vert \bigl\Vert v_{m}^{\prime}(s) \bigr\Vert \,ds \\ &\leq 2T(q+1)^{2}K_{M_{\ast}}^{2}(f) \Vert v_{m-1} \Vert _{W_{1}(T)}^{2}+ \int _{0}^{t}Z_{m}(s)\,ds. \end{aligned} $$
(4.30)
Estimating
\(\bar{J}_{3}\). Similarly,
$$\begin{aligned} \bar{J}_{3} \leq&2 \int _{0}^{t} \bigl\Vert f_{1}[v_{m-1}+h]-f_{1}[h] \bigr\Vert \bigl\Vert v_{m}^{\prime }(s) \bigr\Vert \,ds \\ \leq&2T(q+1)^{2}K_{M_{\ast}}^{2}(f_{1}) \Vert v_{m-1} \Vert _{W_{1}(T)}^{2}+ \int _{0}^{t}Z_{m}(s)\,ds. \end{aligned}$$
(4.31)
Combining (4.27), (4.28), (4.30), and (4.31) leads to
$$\begin{aligned} Z_{m}(t) \leq&2T(q+1)^{2} \bigl[ K_{M_{\ast}}^{2}(f)+K_{M_{\ast }}^{2}(f_{1}) \bigr] \Vert v_{m-1} \Vert _{W_{1}(T)}^{2} \\ &{}+TC_{\ast}^{2} \vert \varepsilon \vert ^{2N+2}+3 \int _{0}^{t}Z_{m}(s)\,ds. \end{aligned}$$
(4.32)
By using Gronwall’s lemma, we deduce from (4.32) that
$$ \Vert v_{m} \Vert _{W_{1}(T)}\leq\sigma_{T} \Vert v_{m-1} \Vert _{W_{1}(T)}+\delta_{T}( \varepsilon),\quad \text{for all }m\geq1, $$
(4.33)
where
$$\begin{aligned}& \sigma_{T} = \sqrt{2}(q+1) \biggl( 1+\frac{1}{\sqrt{a_{0}}} \biggr) \sqrt {K_{M_{\ast}}^{2}(f)+K_{M_{\ast}}^{2}(f_{1})} \sqrt{Te^{3T}}, \\& \delta_{T}(\varepsilon) = C_{\ast} \biggl( 1+ \frac{1}{\sqrt{a_{0}}} \biggr) \sqrt{Te^{3T}} \vert \varepsilon \vert ^{N+1}. \end{aligned}$$
We can assume that
$$ \sigma_{T}< 1,\quad \text{with the suitable constant }T>0. $$
(4.34)
We require the following lemma whose proof is immediate.
Lemma 4.5
Let the sequence
\(\{\gamma_{m}\}\)
satisfy
$$ \gamma_{m}\leq\sigma\gamma_{m-1}+\delta\quad \textit{for all }m \geq 1,\quad\quad \gamma_{0}=0, $$
(4.35)
where
\(0\leq\sigma<1\), \(\delta\geq0\)
are the given constants. Then
$$ \gamma_{m}\leq\delta/(1-\sigma)\quad \textit{for all } m\geq 1. $$
(4.36)
Applying Lemma 4.5 with \(\gamma_{m}= \Vert v_{m} \Vert _{W_{1}(T)}\), \(\sigma=\sigma_{T}<1\), \(\delta=\delta_{T}(\varepsilon )\), it follows from (4.36) that
$$ \Vert v_{m} \Vert _{W_{1}(T)}\leq\frac{\delta _{T}(\varepsilon)}{1-\sigma_{T}}=C_{T} \vert \varepsilon \vert ^{N+1}, $$
(4.37)
where \(C_{T}\) is a constant depending only on T.
On the other hand, the linear recurrent sequence \(\{v_{m}\}\) defined by (4.26) converges strongly in the space \(W_{1}(T)\) to the solution v of problem (4.17). Hence, letting \(m\rightarrow+\infty\) in (4.37), we get
$$ \Vert v \Vert _{W_{1}(T)}\leq C_{T} \vert \varepsilon \vert ^{N+1}. $$
(4.38)
This implies (4.7). The proof of Theorem 4.2 is complete. □