Existence and asymptotic expansion of the weak solution for a wave equation with nonlinear source containing nonlocal term

Abstract

In this paper, we consider the Robin problem for a wave equation with nonlinear source containing nonlocal term. Using the Faedo-Galerkin method and the linearization method for nonlinear term, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

1 Introduction

In this paper, we consider the Robin problem for a wave equation as follows:

$$\begin{aligned}& u_{tt}-u_{xx}=f \bigl( x,t,u(x,t),u(\eta_{1},t), \ldots,u(\eta _{q},t),u_{t}(x,t) \bigr) , \quad 0< x< 1, 0< t< T, \end{aligned}$$

(1.1)

$$\begin{aligned}& u_{x}(0,t)-h_{0}u(0,t)=u_{x}(1,t)+h_{1}u(1,t)=0, \end{aligned}$$

(1.2)

$$\begin{aligned}& u(x,0)=\tilde{u}_{0}(x), \quad\quad u_{t}(x,0)= \tilde{u}_{1}(x), \end{aligned}$$

(1.3)

where f, $\tilde{u}_{0}$, $\tilde{u}_{1}$ are given functions and $h_{0}, h_{1}\geq0$, $\eta_{1}, \eta_{2},\ldots,\eta_{q}$ are given constants with $h_{0}+h_{1}>0$, $0\leq\eta_{1}<\eta_{2}<\cdots<\eta _{q}\leq 1$.

In some special cases, when the nonlinear term has various forms, the following nonlinear wave equation

$$ u_{tt}-\Delta u=F(x,t,u,u_{t}), $$

(1.4)

where Δ is a Laplace operator, has been extensively studied by many authors, for example, we refer to [1–10] and the references given therein. In these works, many interesting results about existence, nonexistence, uniqueness, nonuniqueness, regularity, asymptotic behavior, asymptotic expansion, and decay of solutions were obtained.

In [2], Bergounioux considered Prob. (1.3)-(1.4) with the following boundary conditions:

$$ u_{x}(0,t)=P(t), \quad\quad u_{x}(1,t)+K_{1}u(1,t)+ \lambda_{1}u_{t}(1,t)=0, $$

(1.5)

where f, $\tilde{u}_{0}$, $\tilde{u}_{1}$ are given functions, $K_{1}$, $\lambda_{1}$ are given constants and the unknown $u(x,t)$ and the unknown boundary value $P(t)$ satisfy the following Cauchy problem for an ordinary differential equation:

$$ \textstyle\begin{cases} P^{\prime\prime}(t)+\omega^{2}P(t)=hu_{tt}(0,t) ,\quad 0< t< T , \\ P(0)=P_{0} , \quad\quad P^{\prime}(0)=P_{1} ,\end{cases} $$

(1.6)

where $\omega>0$, $h\geq0$, $P_{0}$, $P_{1}$ are given constants and K, λ are given nonnegative constants.

Prob. (1.4)-(1.6), with $F(x,t,u,u_{t})=f(x,t)-Ku-\lambda u_{t}$, describes the shock between a solid body and a linear viscoelastic bar resting on a viscoelastic base with linear elastic constraints at the side, constraints associated with a viscous frictional resistance.

With $F(x,t,u,u_{t})=f(x,t)-h(u_{t})$, Jokhadze, in [4], considered existence, uniqueness, and nonuniqueness, and nonexistence of a global classical solution for wave equations with nonlinear damping term.

In [5], the authors established the unique existence, stability, and asymptotic expansion of Prob. (1.3)-(1.4) with the nonlocal boundary conditions

$$ \textstyle\begin{cases} u_{x} ( 0,t ) =g_{0} ( t ) + \int _{0}^{t}k_{0} ( t-s ) u ( 0,s ) \,ds , \\ -u_{x} ( 1,t ) =g_{1} ( t ) + \int _{0}^{t}k_{1} ( t-s ) u ( 1,s ) \,ds ,\end{cases} $$

(1.7)

where $F(x,t,u,u_{t})=-\lambda u_{t}-f(u)$, with λ is a given constant and f, $g_{0}$, $g_{1}$, $k_{0}$, $k_{1}$ are given functions. The existence and exponential decay for a nonlinear wave equation with a nonlocal boundary condition were also proved in [9].

Beilin, see [1], investigated the existence and uniqueness of a generalized solution for the following wave equation with an integral nonlocal condition:

$$ \textstyle\begin{cases} u_{tt}-\Delta u+c(x,t)u=f(x,t) ,\quad (x,t)\in\Omega\times(0,T) , \\ \frac{\partial u}{\partial\eta}+ \int _{0}^{t} \int _{\Omega}k(x,\xi,\tau)u(\xi,\tau)\,d\xi \,d\tau=0 ,\quad (x,t)\in\partial \Omega\times(0,T) , \\ u(x,0)=\tilde{u}_{0}(x) , \quad\quad u_{t}(x,0)= \tilde{u}_{1}(x) , \quad x\in \Omega ,\end{cases} $$

(1.8)

where Ω is a bounded domain in $\mathbb{R} ^{N}$ with a smooth boundary, η is the unit outward normal on ∂Ω, f, $\tilde{u}_{0}$, $\tilde{u}_{1}$, $k(x,\xi ,\tau)$ are given functions. Nonlocal conditions come up when values of the function on the boundary are connected to values inside the domain. There are various types of nonlocal boundary conditions of integral form for hyperbolic, parabolic or elliptic equations, the ones were introduced in [1].

In recent years, some close forms of Eq. (1.4), with power-type nonlinearities containing integer power-type, fractional power-type or variable exponent, have been paid attention to by many researchers [3, 11–13]. Benaissa and Messaoudi, in [3], considered the following problem:

$$ \textstyle\begin{cases} u_{tt}-\Delta u+g(u_{t})+f(u)=0 ,\quad (x,t)\in\Omega\times(0,T) , \\ u=0 , \quad x\in\partial\Omega , t\geq0 , \\ u(x,0)=\tilde{u}_{0}(x) , \quad\quad u_{t}(x,0)= \tilde{u}_{1}(x) , \quad x\in \Omega ,\end{cases} $$

(1.9)

where $f(u)=-b \vert u \vert ^{p-2}u$, $g(u_{t})=a ( 1+ \vert u_{t} \vert ^{m-2}u_{t} )$, $a, b>0$, $m, p>2$ and Ω is a bounded domain in $\mathbb{R} ^{N}$ with a smooth boundary ∂Ω. The authors showed that for suitably chosen initial data, (1.9) possesses a global weak solution, which decays exponentially even if $m>2$. The proof of global existence is based on the use of the potential well theory. In [11], Bhattarai proved the existence and stability of solitary-wave solutions of a system of 2-coupled nonlinear Schrödinger equations with power-type nonlinearities. By using variational methods, Repovs̆, in [13], established several existence results for Schrödinger-type equations containing Laplace-type operators with variable exponent. Moreover, by using the fractional homotopy analysis transform method, Kumar [12] proposed a modified and simple algorithm for fractional modeling arising in unidirectional propagation of long wave in dispersive media.

In [14], the authors considered a one-dimensional nonlocal nonlinear strongly damped wave equation with dynamical boundary conditions. In other words, they looked to the following problem:

$$ \textstyle\begin{cases} u_{tt}-u_{xx}-\alpha u_{txx}+\varepsilon f ( u(1,t),\frac {u_{t}(1,t)}{\sqrt{\varepsilon}} ) =0 , \\ u(0,t)=0 , \\ u_{tt}(1,t)=-\varepsilon [ u_{x}(1,t)+\alpha u_{tx}(1,t)+ru_{t}(1,t) ] -\varepsilon f ( u(1,t),\frac{u_{t}(1,t)}{\sqrt{\varepsilon }} ) ,\end{cases} $$

(1.10)

with $0< x<1$, $t>0$, $\alpha, r>0$, and $\varepsilon>0$. Prob. (1.10) models a spring-mass-damper system, where the term $\varepsilon f ( u(1,t),\frac{u_{t}(1,t)}{\sqrt{\varepsilon}} )$ represents a control acceleration at $x=1$. By using the invariant manifold theory, the authors proved that for small values of the parameter ε, the solution of (1.10) attracted to a two-dimensional invariant manifold.

In [6], Long and Diem studied Prob. (1.3)-(1.4) with the nonlinear term of the form

$$ f ( x,t,u,u_{x},u_{t} ) +\varepsilon g ( x,t,u,u_{x},u_{t} ) , $$

(1.11)

associated with the mixed homogeneous boundary conditions

$$ u_{x}(0,t)-h_{0}u(0,t)=u_{x}(1,t)+h_{1}u(1,t)=0. $$

(1.12)

In the case of $f\in C^{2} ( [ 0,1 ] \times [ 0,\infty )\times \mathbb{R} ^{3} )$ and $g\in C^{1} ( [ 0,1 ] \times [ 0,\infty)\times \mathbb{R} ^{3} )$, an asymptotic expansion of order 2 in ε is obtained for ε sufficiently small.

We consider the following wave equation with the source containing nonlocal term:

$$ u_{tt}-u_{xx}=F \biggl( x,t,u(x,t), \int _{0}^{1}g \bigl(u(y,t) \bigr)\,dy \biggr) ,\quad 0< x< 1, 0< t< T, $$

(1.13)

where F, g are given continuous functions. Then, if the function $u(x,t)$ is continuous in x, the integral $\int _{0}^{1}g(u(y,t))\,dy$ can be approximated by its Riemann sum

$$ \int _{0}^{1}g \bigl(u(y,t) \bigr)\,dy \thickapprox \sum_{i=1}^{q} \frac{1}{q}g \bigl(u( \eta_{i},t) \bigr), $$

(1.14)

with q is large enough and $\eta_{i}=i/q$, $i=1,2,\ldots,q$.

Therefore, the nonlinear term in (1.1) can be considered as an approximation of the one that appeared in (1.13) as follows:

$$ F \biggl( x,t,u(x,t), \int _{0}^{1}g \bigl(u(y,t) \bigr)\,dy \biggr) \thickapprox F \Biggl( x,t,u(x,t),\sum_{i=1}^{q} \frac{1}{q}g \bigl(u(\eta _{i},t) \bigr) \Biggr) . $$

(1.15)

The approximation given in (1.15) and the aforementioned works lead to the ideas to study the existence and asymptotic expansion for the Robin problem for a wave equation with nonlinear source containing nonlocal term (1.1)-(1.3). The paper consists of four sections. In Section 2, we present some preliminaries. In Section 3, we associate with Prob. (1.1)-(1.3) a linear recurrent sequence which is bounded in a suitable space of functions. The existence of a local weak solution and the uniqueness are proved by using the Faedo-Galerkin method and the weak compact method. In Section 4, we establish an asymptotic expansion of a weak solution $u_{\varepsilon}(x,t)$ of order $N+1$ in a small parameter ε for the equation

$$\begin{aligned} u_{tt}-u_{xx} =&f \bigl( x,t,u(x,t),u(\eta_{1},t), \ldots,u(\eta _{q},t),u_{t}(x,t) \bigr) \\ &{}+\varepsilon f_{1} \bigl( x,t,u(x,t),u(\eta_{1},t), \ldots,u(\eta _{q},t),u_{t}(x,t) \bigr) , \end{aligned}$$

(1.16)

$0< x<1$, $0< t< T$, associated with (1.2), (1.3). The results obtained here may be considered as a relative generalization of the results obtained in [2, 4–6, 9], and [10].

2 Preliminaries

Put $\Omega=(0,1)$. We will omit the definitions of the usual function spaces and denote them by $L^{p}= L^{p}(\Omega)$, $H^{m}=H^{m} ( \Omega )$. Let $\langle\cdot,\cdot\rangle$ be either the scalar product in $L^{2}$ or the dual pairing of a continuous linear functional and an element of a function space. The notation $\Vert \cdot \Vert$ stands for the norm in $L^{2}$, and we denote by $\Vert \cdot \Vert _{X}$ the norm in the Banach space X. We call $X^{\prime}$ the dual space of X. We denote $L^{p}(0,T;X)$, $1\leq p\leq\infty$ the Banach space of real functions $u:(0,T)\rightarrow X$ measurable, such that $\Vert u \Vert _{L^{p}(0,T;X)}<+\infty$, with

$$ \Vert u \Vert _{L^{p}(0,T;X)}= \textstyle\begin{cases} ( \int_{0}^{T} \Vert u(t) \Vert _{X}^{p}\,dt ) ^{1/p} , & \text{if } 1\leq p< \infty , \\ {\operatorname{ess}\sup}_{0< t< T} \Vert u(t) \Vert _{X} , & \text{if } p=\infty .\end{cases} $$

Let $u(t)$, $u^{\prime}(t)=u_{t}(t)=\dot{u}(t)$, $u^{\prime\prime }(t)=u_{tt}(t)=\ddot{u}(t)$, $u_{x}(t)=\bigtriangledown u(t)$, $u_{xx}(t)=\Delta u(t)$, denote $u(x,t)$, $\frac{\partial u}{\partial t}(x,t)$, $\frac{\partial^{2}u}{\partial t^{2}}(x,t)$, $\frac{\partial u}{\partial x}(x,t)$, $\frac{\partial^{2}u}{\partial x^{2}}(x,t)$, respectively.

With $f\in C^{k}([0,1]\times \mathbb{R} _{+}\times \mathbb{R} ^{q+2})$, $f=f(x,t,y_{1},\ldots,y_{q+2})$, we put $D_{1}f=\frac {\partial f}{\partial x}$, $D_{2}f=\frac{\partial f}{\partial t}$, $D_{i+2}f=\frac{\partial f}{\partial y_{i}}$ with $i=1,\ldots,q+2$, and $D^{\alpha }f=D_{1}^{\alpha_{1}}\cdots D_{q+4}^{\alpha_{q+4}}f$, $\alpha=(\alpha _{1},\ldots, \alpha_{q+4})\in \mathbb{Z} _{+}^{q+4}$, $\vert \alpha \vert =\alpha_{1}+\cdots+\alpha _{q+4}=k$, $D^{(0,\ldots,0)}f=f$.

On $H^{1}$, we shall use the following norm:

$$ \Vert v \Vert _{H^{1}}= \bigl( \Vert v \Vert ^{2}+ \Vert v_{x} \Vert ^{2} \bigr) ^{1/2}. $$

We put

$$ a(u,v)= \int _{0}^{1}u_{x}(x)v_{x}(x) \,dx+h_{0}u(0)v(0)+h_{1}u(1)v(1), \quad u,v\in H^{1}. $$

(2.1)

We have the following lemmas, the proofs of which are straightforward, hence we omit the details.

Lemma 2.1

[15], Theorem 8.8, pp.212-213

The imbedding $H^{1}\hookrightarrow C^{0}(\overline{\Omega})$ is compact and

$$ \Vert v \Vert _{C^{0}(\overline{\Omega})}\leq\sqrt{2} \Vert v \Vert _{H^{1}}\quad \textit{for all } v\in H^{1}. $$

(2.2)

Lemma 2.2

Let $h_{0}, h_{1}\geq0$, with $h_{0}+h_{1}>0$. Then the symmetric bilinear form $a(\cdot ,\cdot)$ defined by (2.1) is continuous on $H^{1}\times H^{1}$ and coercive on $H^{1}$, i.e.,

$$ \begin{gathered} \mathrm{(i)}\quad \bigl\vert a(u,v) \bigr\vert \leq a_{1} \Vert u \Vert _{H^{1}} \Vert v \Vert _{H^{1}} , \\ \mathrm{(ii)}\quad a(v,v)\geq a_{0} \Vert v \Vert _{H^{1}}^{2} ,\end{gathered} $$

(2.3)

for all $u, v\in H^{1}$, where $a_{1}=1+2h_{0}+2h_{1}$ and

$$ a_{0}=\frac{1}{4}\min \bigl\{ 1, \max\{h_{0}, h_{1}\} \bigr\} . $$

(2.4)

Remark 2.1

It follows from (2.3) that on $H^{1}$, $v\longmapsto \Vert v \Vert _{H^{1}}$, $v\longmapsto \Vert v \Vert _{a}=\sqrt{a(v,v)}$ are two equivalent norms satisfying

$$ \sqrt{a_{0}} \Vert v \Vert _{H^{1}}\leq \Vert v \Vert _{a}\leq\sqrt{a_{1}} \Vert v \Vert _{H^{1}},\quad \forall v\in H^{1}. $$

(2.5)

Lemma 2.3

Let $h_{0}\geq0$. Then there exists the Hilbert orthonormal base $\{\widetilde{w}_{j}\}$ of $L^{2}$ consisting of the eigenfunctions $\widetilde{w}_{j}$ corresponding to the eigenvalue $\lambda_{j}$ such that

$$ \textstyle\begin{cases} 0< \lambda_{1}\leq \lambda_{2}\leq\cdots\leq\lambda_{j}\leq\cdots,\quad\quad \lim _{j\rightarrow+\infty} \lambda_{j}=+\infty , \\ a(\widetilde{w}_{j},v)=\lambda_{j}\langle \widetilde{w}_{j},v\rangle \quad \textit{for all } v\in H^{1} , j=1,2 ,\ldots. \end{cases} $$

Furthermore, the sequence $\{\widetilde{w}_{j}/\sqrt{\lambda _{j}}\}$ is also a Hilbert orthonormal base of $H^{1}$ with respect to the scalar product $a(\cdot,\cdot)$.

On the other hand, we also have $\widetilde{w}_{j}$ satisfying the following boundary value problem:

$$ \textstyle\begin{cases} -\Delta\widetilde{w}_{j}= \lambda_{j}\widetilde{w}_{j} , \quad \textit{in } (0,1) , \\ \widetilde{w}_{jx}(0)-h_{0}\widetilde{w}_{j}(0)= \widetilde {w}_{jx}(1)+h_{1}\widetilde{w}_{j}(1)=0 ,\quad \widetilde{w}_{j}\in C^{\infty}(\overline { \Omega}) .\end{cases} $$

The proof of Lemma 2.3 can be found in ([16], p.87, Theorem 7.7), with $H=L^{2}$ and $V=H^{1}$, $a(\cdot,\cdot)$ as defined by (2.1).

Remark 2.2

The weak formulation of the initial-boundary value problem (1.1)-(1.3) can be given in the following manner: Find $u\in\widetilde{W}=\{u\in L^{\infty}(0,T;H^{2}):u_{t}\in L^{\infty }(0,T;H^{1}), u_{tt}\in L^{\infty}(0,T;L^{2})\}$ such that u satisfies the following variational equation:

$$ \bigl\langle u_{tt}(t),w \bigr\rangle +a \bigl(u(t),w \bigr)= \bigl\langle f \bigl( \cdot ,t,u(t),u(\eta_{1},t),\ldots,u( \eta_{q},t),u_{t}(t) \bigr) ,w \bigr\rangle $$

(2.6)

for all $w\in H^{1}$, a.e., $t\in(0,T)$, together with the initial conditions

$$ u(0)=\tilde{u}_{0}, \quad\quad u_{t}(0)=\tilde{u}_{1}. $$

(2.7)

3 The existence and uniqueness

We make the following assumptions:

$(H_{1})$ :

$(\tilde{u}_{0},\tilde{u}_{1})\in H^{2}\times H^{1}$, $\tilde{u}_{0x}(0)-h_{0}\tilde{u}_{0}(0)=\tilde{u}_{0x}(1)+h_{1}\tilde {u}_{0}(1)=0$;

$(H_{2})$ :

$f\in C^{1}([0,1]\times \mathbb{R} _{+}\times \mathbb{R} ^{q+2})$.

Fix$T^{\ast}>0$. For each $M>0$ given, we set the constant $K_{M}(f)$ as follows:

$$ K_{M}(f)=\sum_{i=1}^{q+4}K_{0}(M,D_{i}f), $$

where

$$ \textstyle\begin{cases} K_{0}(M,f)= {\sup }_{(x,t,y_{1}, \ldots,y_{q+2})\in A_{1}(M)} \vert f(x,t,y_{1},\ldots,y_{q+2}) \vert , \\ A_{1}(M)=[0,1]\times [0,T^{\ast} ]\times [-\sqrt {2}M,\sqrt{2}M]^{q+2} .\end{cases} $$

For every $T\in(0,T^{\ast}]$ and $M>0$, we put

$$ \textstyle\begin{cases} W(M,T)= \{v\in L^{\infty} (0,T;H^{2} ):v_{t}\in L^{\infty } (0,T;H^{1} ), v_{tt}\in L^{2}(Q_{T}) , \\ \hspace{50pt} \text{with } \max \{ \Vert v \Vert _{L^{\infty}(0,T;H^{2})} , \Vert v_{t} \Vert _{L^{\infty }(0,T;H^{1})} , \Vert v_{tt} \Vert _{L^{2}(Q_{T})} \}\leq M \} , \\ W_{1}(M,T)= \{v\in W(M,T):v_{tt}\in L^{\infty} (0,T;L^{2} ) \} ,\end{cases} $$

in which $Q_{T}=\Omega\times(0,T)$.

Now, we establish the recurrent sequence $\{u_{m}\}$. The first term is chosen as $u_{0}\equiv\tilde{u}_{0}$, suppose that

$$ u_{m-1}\in W_{1}(M,T), $$

(3.1)

we associate Prob. (1.1)-(1.3) with the following problem.

Find $u_{m}\in W_{1}(M,T)$ ($m\geq1$) satisfying the linear variational problem

$$ \textstyle\begin{cases} \langle u_{m}^{\prime\prime}(t),w \rangle+a (u_{m}(t),w )= \langle F_{m}(t),w \rangle , \quad \forall w\in H^{1} , \\ u_{m}(0)=\tilde{u}_{0} , \quad\quad u_{m}^{\prime}(0)= \tilde{u}_{1} ,\end{cases} $$

(3.2)

where

$$\begin{aligned} F_{m}(x,t) =&f[u_{m-1}](x,t) \\ =&f \bigl( x,t,u_{m-1}(x,t),u_{m-1}(\eta_{1},t), \ldots,u_{m-1}(\eta _{q},t),u_{m-1}^{\prime}(x,t) \bigr) . \end{aligned}$$

(3.3)

Then we have the following theorem.

Theorem 3.1

Let $( H_{1} )$, $( H_{2} )$ hold. Then there exist positive constants $M, T>0$ such that, for $u_{0}\equiv\tilde{u}_{0}$, there exists a recurrent sequence $\{u_{m}\}\subset W_{1}(M,T)$ defined by (3.1)-(3.3).

Proof

The proof consists of several steps.

Step 1. The Faedo-Galerkin approximation (introduced by Lions [17]). Consider the basis $\{w_{j}\}$ for $H^{1}$ as in Lemma 2.3. Put

$$ u_{m}^{(k)}(t)=\sum_{j=1}^{k}c_{mj}^{(k)}(t)w_{j}, $$

(3.4)

where the coefficients $c_{mj}^{(k)}$ satisfy the system of linear differential equations

$$ \textstyle\begin{cases} \langle\ddot{u}_{m}^{(k)}(t),w_{j} \rangle +a (u_{m}^{(k)}(t),w_{j} )= \langle F_{m}(t),w_{j} \rangle , \quad 1\leq j\leq k , \\ u_{m}^{(k)}(0)=\tilde{u}_{0k} ,\quad\quad \dot{u}_{m}^{(k)}(0)=\tilde{u}_{1k} ,\end{cases} $$

(3.5)

where $F_{m}(x,t)$ is defined as in (3.3) and

$$ \textstyle\begin{cases} \tilde{u}_{0k}=\sum_{j=1}^{k}\alpha_{j}^{(k)}w_{j} \rightarrow\tilde{u}_{0} \quad \text{strongly in } H^{2} , \\ \tilde{u}_{1k}=\sum_{j=1}^{k} \beta_{j}^{(k)}w_{j}\rightarrow \tilde{u}_{1} \quad \text{strongly in } H^{1} .\end{cases} $$

(3.6)

The system of equations (3.5) can be rewritten in the form

$$ \textstyle\begin{cases} \ddot{c}_{mj}^{(k)}(t)+ \lambda_{j}c_{mj}^{(k)}(t)= \langle F_{m}(t),w_{j} \rangle , \\ c_{m}^{(k)}(0)=\alpha_{j}^{(k)} ,\quad\quad \dot{c}_{mj}^{(k)}(0)=\beta_{j}^{(k)} , \quad 1 \leq j\leq k .\end{cases} $$

(3.7)

It is not difficult to show that (3.7) has a unique solution $c_{mj}^{(k)}(t)$ in $[0,T]$ as follows:

$$\begin{aligned} c_{mj}^{(k)}(t) =&\alpha_{j}^{(k)}\cos( \sqrt{\lambda_{j}}t)+\beta _{j}^{(k)} \frac{\sin(\sqrt{\lambda_{j}}t)}{\sqrt{\lambda_{j}}} \\ &{}+ \int _{0}^{t}\frac{\sin(\sqrt{\lambda_{j}}(t-s))}{\sqrt{\lambda_{j}}} \bigl\langle F_{m}(s),w_{j} \bigr\rangle \,ds, \\ &0 \leq t\leq T, 1\leq j\leq k. \end{aligned}$$

(3.8)

Therefore, (3.5) has a unique solution $u_{m}^{(k)}(t)$ in $[0,T]$.

Step 2. A priori estimates. First, for all $j=1,\ldots, k$, multiplying (3.5)₁ by $\dot{c}_{mj}^{(k)}(t)$, summing on j, and integrating with respect to the time variable from 0 to t, we have

$$ X_{m}^{(k)}(t)=X_{m}^{(k)}(0)+2 \int _{0}^{t} \bigl\langle F_{m}(s), \dot{u}_{m}^{(k)}(s) \bigr\rangle \,ds, $$

(3.9)

where

$$ X_{m}^{(k)}(t)= \bigl\Vert \dot{u}_{m}^{(k)}(t) \bigr\Vert ^{2}+ \bigl\Vert u_{m}^{(k)}(t) \bigr\Vert _{a}^{2}. $$

(3.10)

Next, by replacing $w_{j}$ in (3.5)₁ by $-\frac{1}{\lambda _{j}} \Delta w_{j}$, we obtain that

$$ a \bigl(\ddot{u}_{m}^{(k)}(t),w_{j} \bigr)+ \bigl\langle \Delta u_{m}^{(k)}(t),\Delta w_{j} \bigr\rangle = \bigl\langle F_{m}(t),-\Delta w_{j} \bigr\rangle ,\quad 1 \leq j\leq k, $$

similar to (3.5)₁, it gives

$$\begin{aligned} Y_{m}^{(k)}(t) =&Y_{m}^{(k)}(0)+2 \bigl\langle F_{m}(0),\Delta\tilde{u} _{0k} \bigr\rangle -2 \bigl\langle F_{m}(t),\triangle u_{m}^{(k)}(t) \bigr\rangle \\ &{}+2 \int _{0}^{t} \bigl\langle F_{m}^{\prime}(s), \triangle u_{m}^{(k)}(s) \bigr\rangle \,ds, \end{aligned}$$

(3.11)

where

$$ Y_{m}^{(k)}(t)= \bigl\Vert \dot{u}_{m}^{(k)}(t) \bigr\Vert _{a}^{2}+ \bigl\Vert \Delta u_{m}^{(k)}(t) \bigr\Vert ^{2}. $$

(3.12)

Put

$$\begin{aligned} S_{m}^{(k)}(t) =&X_{m}^{(k)}(t)+Y_{m}^{(k)}(t)+ \int _{0}^{t} \bigl\Vert \ddot {u}_{m}^{(k)}(s) \bigr\Vert ^{2}\,ds \\ =& \bigl\Vert \dot{u}_{m}^{(k)}(t) \bigr\Vert ^{2}+ \bigl\Vert \dot{u}_{m}^{(k)}(t) \bigr\Vert _{a}^{2}+ \bigl\Vert u_{m}^{(k)}(t) \bigr\Vert _{a}^{2}+ \bigl\Vert \Delta u_{m}^{(k)}(t) \bigr\Vert ^{2}+ \int _{0}^{t} \bigl\Vert \ddot{u}_{m}^{(k)}(s) \bigr\Vert ^{2}\,ds, \end{aligned}$$

(3.13)

then we deduce from (3.9), (3.11), and (3.13) that

$$\begin{aligned} S_{m}^{(k)}(t) =& S_{m}^{(k)}(0)+2 \bigl\langle F_{m}(0),\triangle\tilde{u}_{0k} \bigr\rangle +2 \int _{0}^{t} \bigl\langle F_{m}(s), \dot{u}_{m}^{(k)}(s) \bigr\rangle \,ds \\ &{} -2 \bigl\langle F_{m}(t),\triangle u_{m}^{(k)}(t) \bigr\rangle +2 \int _{0}^{t} \bigl\langle F_{m}^{\prime}(s), \triangle u_{m}^{(k)}(s) \bigr\rangle \,ds+ \int _{0}^{t} \bigl\Vert \ddot{u}_{m}^{(k)}(s) \bigr\Vert ^{2}\,ds \\ \equiv& S_{m}^{(k)}(0)+2 \bigl\langle F_{m}(0), \Delta\tilde{u}_{0k} \bigr\rangle +\sum _{j=1}^{4}I_{j}. \end{aligned}$$

(3.14)

We estimate all terms on the right-hand side of (3.14) as follows:

$$\begin{aligned}& I_{1}=2 \int _{0}^{t} \bigl\langle F_{m}(s), \dot{u}_{m}^{(k)}(s) \bigr\rangle \,ds\leq TK_{M}^{2}(f)+ \int _{0}^{t}S_{m}^{(k)}(s)\,ds; \end{aligned}$$

(3.15)

$$\begin{aligned}& \begin{aligned}[b] I_{2} &=-2 \bigl\langle F_{m}(t),\triangle u_{m}^{(k)}(t) \bigr\rangle \\ &\leq 4 \bigl\Vert F_{m}(0) \bigr\Vert ^{2}+4T \bigl( \sqrt{T} \bigl[ 1+(q+1)\sqrt{2}M \bigr] ^{2}+M \bigr) ^{2}K_{M}^{2}(f)+\frac{1}{2}S_{m}^{(k)}(t); \end{aligned} \end{aligned}$$

(3.16)

$$\begin{aligned}& \begin{aligned}[b] I_{3} &= 2 \int _{0}^{t} \bigl\langle F_{m}^{\prime}(s), \triangle u_{m}^{(k)}(s) \bigr\rangle \,ds \\ &\leq 2T \bigl( \bigl[ 1+(q+1)\sqrt{2}M \bigr] ^{2}+M^{2} \bigr) K_{M}^{2}(f)+ \int _{0}^{t}S_{m}^{(k)}(s)\,ds. \end{aligned} \end{aligned}$$

(3.17)

We note that Eq. (3.5)₁ can be written as follows:

$$ \bigl\langle \ddot{u}_{m}^{(k)}(t),w_{j} \bigr\rangle - \bigl\langle \Delta u_{m}^{(k)}(t),w_{j} \bigr\rangle = \bigl\langle F_{m}(t),w_{j} \bigr\rangle , \quad 1\leq j\leq k. $$

(3.18)

Hence, after replacing $w_{j}$ with $\ddot{u}_{m}^{(k)}(t)$, we obtain

$$\begin{aligned} \bigl\Vert \ddot{u}_{m}^{(k)}(t) \bigr\Vert ^{2} =& \bigl\langle \Delta u_{m}^{(k)}(t), \ddot{u}_{m}^{(k)}(t) \bigr\rangle + \bigl\langle F_{m}(t),\ddot{u}_{m}^{(k)}(t) \bigr\rangle \\ \leq& \bigl[ \bigl\Vert \Delta u_{m}^{(k)}(t) \bigr\Vert + \bigl\Vert F_{m}(t) \bigr\Vert \bigr] \bigl\Vert \ddot{u}_{m}^{(k)}(t) \bigr\Vert \\ \leq& \bigl[ \bigl\Vert \Delta u_{m}^{(k)}(t) \bigr\Vert + \bigl\Vert F_{m}(t) \bigr\Vert \bigr] ^{2}, \end{aligned}$$

so

$$\begin{aligned} I_{4} =& \int _{0}^{t} \bigl\Vert \ddot{u}_{m}^{(k)}(s) \bigr\Vert ^{2}\,ds\leq2 \int _{0}^{t} \bigl\Vert \Delta u_{m}^{(k)}(s) \bigr\Vert ^{2}\,ds+2 \int _{0}^{t} \bigl\Vert F_{m}(s) \bigr\Vert ^{2}\,ds \\ \leq&2 \int _{0}^{t}S_{m}^{(k)}(s) \,ds+2TK_{M}^{2}(f). \end{aligned}$$

(3.19)

It follows from (3.14)-(3.17), (3.19) that

$$ S_{m}^{(k)}(t)\leq D_{0}^{(k)}(f, \tilde{u}_{0},\tilde{u}_{1},\tilde {u}_{0k},\tilde{u}_{1k})+D_{1}(M,T)+8 \int _{0}^{t}S_{m}^{(k)}(s)\,ds, $$

(3.20)

where

$$ \textstyle\begin{cases} D_{0}^{(k)}(f, \tilde{u}_{0},\tilde{u}_{1},\tilde{u}_{0k}, \tilde{u}_{1k})=2S_{m}^{(k)}(0)+4 \langle F_{m}(0),\Delta\tilde{u}_{0k} \rangle+8 \Vert F_{m}(0) \Vert ^{2} , \\ D_{1}(M,T) =2 [ 3+4 ( \sqrt{T} [ 1+(q+1)\sqrt{2}M ] ^{2}+M ) ^{2} \\ \hspace{46pt}{}+2 ( [ 1+(q+1)\sqrt{2}M ] ^{2}+M^{2} ) ] TK_{M}^{2}(f) .\end{cases} $$

(3.21)

By means of the convergence in (3.6), we can deduce the existence of a constant $M>0$ independent of k and m such that

$$ D_{0}^{(k)}(f,\tilde{u}_{0},\tilde{u}_{1}, \tilde{u}_{0k},\tilde {u}_{1k})\leq \frac{1}{2}M^{2} \quad \text{for all }m,k\in \mathbb{N} . $$

(3.22)

We choose $T\in(0,T^{\ast}]$ such that

$$ \biggl( \frac{1}{2}M^{2}+D_{1}(M,T) \biggr) e^{8T}\leq M^{2} $$

(3.23)

and

$$ k_{T}= \biggl( 1+\frac{1}{\sqrt{a_{0}}} \biggr) \sqrt{2Te^{T}}(q+1)K_{M}(f)< 1. $$

(3.24)

Finally, it follows from (3.20), (3.22), and (3.23) that

$$ S_{m}^{(k)}(t)\leq M^{2}e^{-8T}+8 \int _{0}^{t}S_{m}^{(k)}(s)\,ds. $$

(3.25)

By using Gronwall’s lemma, we deduce from (3.25) that

$$ S_{m}^{(k)}(t)\leq M^{2}e^{-8T}e^{8t} \leq M^{2} $$

(3.26)

for all $t\in [0,T]$, for all m and k. Therefore, we have

$$ u_{m}^{(k)}\in W(M,T)\quad \text{for all }m\text{ and }k. $$

(3.27)

Step 3. Limiting process. From (3.26), we deduce the existence of a subsequence of $\{u_{m}^{(k)}\}$, still so denoted, such that

$$ \textstyle\begin{cases} u_{m}^{(k)}\rightarrow u_{m} \quad \text{in } L^{\infty } (0,T;H^{2} )\text{ weak}^{*}, \\ \dot{u}_{m}^{(k)}\rightarrow u_{m}^{\prime} \quad \text{in } L^{\infty } (0,T;H^{1} )\text{ weak}^{*}, \\ \ddot{u}_{m}^{(k)}\rightarrow u_{m}^{\prime\prime} \quad \text{in } L^{2}(Q_{T})\text{ weak}, \\ u_{m}\in W(M,T) . \end{cases} $$

(3.28)

Passing to limit in (3.5), we have $u_{m}$ satisfying (3.2), (3.3) in $L^{2}(0,T)$. On the other hand, it follows from (3.2)₁ and (3.28)₄ that $u_{m}^{\prime\prime}=\Delta u_{m}+F_{m}\in L^{\infty}(0,T;L^{2})$, hence $u_{m}\in W_{1}(M,T)$ and the proof of Theorem 3.1 is complete. □

We use the result given in Theorem 3.1 and the compact imbedding theorems to prove the existence and uniqueness of a weak solution of Prob. (1.1)-(1.3). Hence, we get the main result in this section as follows.

Theorem 3.2

Let $(H_{1})$, $(H_{2})$ hold. Then

• (i) Prob. (1.1)-(1.3) has a unique weak solution $u\in W_{1}(M,T)$, where the constants $M>0$ and $T>0$ are chosen as in Theorem  3.1.

• (ii) The recurrent sequence $\{u_{m}\}$ defined by (3.1)-(3.3) converges to the solution u of Prob. (1.1)-(1.3) strongly in the space

  $$ W_{1}(T)= \bigl\{ v\in L^{\infty} \bigl(0,T;H^{1} \bigr):v^{\prime}\in L^{\infty } \bigl(0,T;L^{2} \bigr) \bigr\} . $$

  (3.29)

Furthermore, we also have the estimation

$$ \Vert u_{m}-u \Vert _{W_{1}(T)}\leq C_{T}k_{T}^{m}\quad \textit{for all } m\in \mathbb{N}, $$

(3.30)

where the constant $k_{T}\in [0,1)$ is defined as in (3.24) and $C_{T}$ is a constant depending only on T, $h_{0}$, $h_{1}$, f, $\tilde{u}_{0}$, $\tilde{u}_{1}$, and $k_{T}$.

Proof

(a) Existence of the solution. First, we note that $W_{1}(T)$ is a Banach space with respect to the norm (see Lions [17]).

$$ \Vert v \Vert _{W_{1}(T)}= \Vert v \Vert _{L^{\infty }(0,T;H^{1})}+ \bigl\Vert v^{\prime} \bigr\Vert _{L^{\infty}(0,T;L^{2})}. $$

(3.31)

We shall prove that $\{u_{m}\}$ is a Cauchy sequence in $W_{1}(T)$. Let $w_{m}=u_{m+1}-u_{m}$. Then $w_{m}$ satisfies the variational problem

$$ \textstyle\begin{cases} \langle w_{m}^{\prime\prime}(t),w \rangle+a (w_{m}(t),w )= \langle F_{m+1}(t)-F_{m}(t),w \rangle , \quad \forall w\in H^{1} , \\ w_{m}(0)=w_{m}^{\prime}(0)=0 .\end{cases} $$

(3.32)

Taking $w=w_{m}^{\prime}$ in (3.32)₁, after integrating in t, we get

$$ Z_{m}(t)=2 \int _{0}^{t} \bigl\langle F_{m+1}(s)-F_{m}(s),w_{m}^{\prime}(s) \bigr\rangle \,ds, $$

(3.33)

where

$$ Z_{m}(t)= \bigl\Vert w_{m}^{\prime}(t) \bigr\Vert ^{2}+ \bigl\Vert w_{m}(t) \bigr\Vert _{a}^{2}. $$

(3.34)

By $(H_{2})$ it is clear that

$$\begin{aligned} \bigl\Vert F_{m+1}(t)-F_{m}(t) \bigr\Vert \leq&K_{M}(f) \bigl[ \sqrt {2}(q+1) \bigl\Vert w_{m-1}(t) \bigr\Vert _{H^{1}}+ \bigl\Vert w_{m-1}^{\prime }(t) \bigr\Vert \bigr] \\ \leq&\sqrt{2}(q+1)K_{M}(f) \Vert w_{m-1} \Vert _{W_{1}(T)}. \end{aligned}$$

(3.35)

Hence

$$ Z_{m}(t)\leq2T(q+1)^{2}K_{M}^{2}(f) \Vert w_{m-1} \Vert _{W_{1}(T)}^{2}+ \int _{0}^{t}Z_{m}(s)\,ds. $$

(3.36)

Using Gronwall’s lemma, we deduce from (3.36) that

$$ \Vert w_{m} \Vert _{W_{1}(T)}\leq k_{T} \Vert w_{m-1} \Vert _{W_{1}(T)} \quad \forall m\in \mathbb{N} , $$

(3.37)

where $k_{T}\in(0,1)$ is defined as in (3.24), which implies that

$$ \Vert u_{m}-u_{m+p} \Vert _{W_{1}(T)}\leq \Vert u_{0}-u_{1} \Vert _{W_{1}(T)}(1-k_{T})^{-1}k_{T}^{m}\quad \forall m,p\in \mathbb{N} . $$

(3.38)

It follows that $\{u_{m}\}$ is a Cauchy sequence in $W_{1}(T)$. Then there exists $u\in W_{1}(T)$ such that

$$ u_{m}\rightarrow u\quad \text{strongly in }W_{1}(T). $$

(3.39)

Note that $u_{m}\in W_{1}(M,T)$, then there exists a subsequence $\{u_{m_{j}}\}$ of $\{u_{m}\}$such that

$$ \textstyle\begin{cases} u_{m_{j}}\rightarrow u \quad \text{in } L^{\infty} (0,T;H^{2} )\text{ weak}^{*}, \\ u_{m_{j}}^{\prime}\rightarrow u^{\prime} \quad \text{in } L^{\infty } (0,T;H^{1} )\text{ weak}^{*}, \\ u_{m_{j}}^{\prime\prime}\rightarrow u^{\prime\prime} \quad \text{in } L^{2}(Q_{T})\text{ weak}, \\ u\in W(M,T) . \end{cases} $$

(3.40)

We also note that

$$\begin{aligned} & \bigl\Vert F_{m}-f \bigl( \cdot,t,u(t),u(\eta_{1},t), \ldots,u(\eta _{q},t),u^{\prime}(t) \bigr) \bigr\Vert _{L^{\infty}(0,T;L^{2})} \\ &\quad \leq \sqrt{2}(q+1)K_{M}(f) \Vert u_{m-1}-u \Vert _{W_{1}(T)}. \end{aligned}$$

(3.41)

Hence, from (3.39) and (3.41), we obtain

$$ F_{m}\rightarrow f \bigl( \cdot,t,u(t),u(\eta_{1},t), \ldots,u(\eta _{q},t),u^{\prime}(t) \bigr) \quad \text{strongly in }L^{\infty} \bigl(0,T;L^{2} \bigr). $$

(3.42)

Finally, passing to limit in (3.2)-(3.3) as $m=m_{j}\rightarrow \infty$, it implies from (3.39), (3.40)_1,3, and (3.42) that there exists $u\in W(M,T)$ satisfying (2.6), (2.7).

On the other hand, from assumption $(H_{2})$ we obtain from (2.6), (3.40)₄, and (3.42) that

$$ u^{\prime\prime}=u_{xx}+f \bigl( \cdot,t,u(t),u(\eta_{1},t), \ldots ,u(\eta _{q},t),u^{\prime}(t) \bigr) \in L^{\infty} \bigl(0,T;L^{2} \bigr), $$

(3.43)

thus we have $u\in W_{1}(M,T)$. The existence proof is completed.

(b) Uniqueness of the solution. Let $u_{1}, u_{2}\in W_{1}(M,T)$ be two different weak solutions of Prob. (1.1)-(1.3). Then $u=u_{1}-u_{2}$ satisfies the variational problem

$$ \textstyle\begin{cases} \langle u^{\prime\prime}(t),w \rangle+a (u(t),w )= \langle F_{1}(t)-F_{2}(t),w \rangle , \quad \forall w\in H^{1} , \\ u(0)=u^{\prime}(0)=0 ,\end{cases} $$

(3.44)

where $F_{i}(x,t)=f ( x,t,u_{i}(x,t),u_{i}(\eta_{1},t),\ldots ,u_{i}(\eta _{q},t),u_{i}^{\prime}(x,t) )$, $i=1,2$.

We take $w=u^{\prime}$ in (3.44)₁ and integrate in t to get

$$ \bigl\Vert u^{\prime}(t) \bigr\Vert ^{2}+ \bigl\Vert u(t) \bigr\Vert _{a}^{2}\leq\sqrt{\frac{2}{a_{0}}}(q+1)K_{M}(f) \int _{0}^{t} \bigl( \bigl\Vert u^{\prime}(s) \bigr\Vert ^{2}+ \bigl\Vert u(s) \bigr\Vert _{a}^{2} \bigr) \,ds. $$

(3.45)

Using Gronwall’s lemma, it follows that $\Vert u^{\prime }(t) \Vert ^{2}+ \Vert u(t) \Vert _{a}^{2}\equiv0$, i.e., $u_{1}\equiv u_{2}$.

So (i) is proved and (ii) follows. Theorem 3.2 is proved completely. □

4 Asymptotic expansion of the solution with respect to a small parameter

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.  □