- Research
- Open Access
- Published:
Existence and asymptotic expansion of the weak solution for a wave equation with nonlinear source containing nonlocal term
Boundary Value Problems volume 2017, Article number: 87 (2017)
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:
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
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:
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:
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
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:
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:
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:
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
associated with the mixed homogeneous boundary conditions
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:
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
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:
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
\(0< x<1\), \(0< t< T\), 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
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:
We put
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
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.,
for all \(u, v\in H^{1}\), where \(a_{1}=1+2h_{0}+2h_{1}\) and
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
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
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:
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:
for all \(w\in H^{1}\), a.e., \(t\in(0,T)\), together with the initial conditions
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:
where
For every \(T\in(0,T^{\ast}]\) and \(M>0\), we put
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
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
where
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
where the coefficients \(c_{mj}^{(k)}\) satisfy the system of linear differential equations
where \(F_{m}(x,t)\) is defined as in (3.3) and
The system of equations (3.5) can be rewritten in the form
It is not difficult to show that (3.7) has a unique solution \(c_{mj}^{(k)}(t)\) in \([0,T]\) as follows:
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
where
Next, by replacing \(w_{j}\) in (3.5)1 by \(-\frac{1}{\lambda _{j}} \Delta w_{j}\), we obtain that
similar to (3.5)1, it gives
where
Put
then we deduce from (3.9), (3.11), and (3.13) that
We estimate all terms on the right-hand side of (3.14) as follows:
We note that Eq. (3.5)1 can be written as follows:
Hence, after replacing \(w_{j}\) with \(\ddot{u}_{m}^{(k)}(t)\), we obtain
so
It follows from (3.14)-(3.17), (3.19) that
where
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
We choose \(T\in(0,T^{\ast}]\) such that
and
Finally, it follows from (3.20), (3.22), and (3.23) that
By using Gronwall’s lemma, we deduce from (3.25) that
for all \(t\in [0,T]\), for all m and k. Therefore, we have
Step 3. Limiting process. From (3.26), we deduce the existence of a subsequence of \(\{u_{m}^{(k)}\}\), still so denoted, such that
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
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]).
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
Taking \(w=w_{m}^{\prime}\) in (3.32)1, after integrating in t, we get
where
By \((H_{2})\) it is clear that
Hence
Using Gronwall’s lemma, we deduce from (3.36) that
where \(k_{T}\in(0,1)\) is defined as in (3.24), which implies that
It follows that \(\{u_{m}\}\) is a Cauchy sequence in \(W_{1}(T)\). Then there exists \(u\in W_{1}(T)\) such that
Note that \(u_{m}\in W_{1}(M,T)\), then there exists a subsequence \(\{u_{m_{j}}\}\) of \(\{u_{m}\} \)such that
We also note that
Hence, from (3.39) and (3.41), we obtain
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
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
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
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, \(\vert \varepsilon \vert \leq1\):
where
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
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
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
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
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.,
Let us consider the sequence of weak solutions \(u_{k}\), \(1\leq k\leq N\), defined by the following problems:
where \(F_{k}\), \(1\leq k\leq N\), are defined by the formulas
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
where
with
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:
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
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:
By using Taylor’s expansion of the function \(f[h]\) around the point
up to order \(N+1\), we obtain
where
By formula (4.1), we get
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
where
We deduce from (4.10), (4.13) that
where \(\bar{\Phi}_{k}[N,f]\), \(1\leq k\leq N\), are defined by (4.4)-(4.6) and
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
where
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
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
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:
Hence, we deduce from (4.8) and (4.21) that
where
Combining (4.3), (4.18), and (4.22) leads to
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
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
By multiplying two sides of (4.26) with \(v_{m}^{\prime}\) and after integration in t, we have
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
Estimating \(\bar{J}_{2}\). We note that
with \(M_{\ast}=(N+2)M\).
It follows from (4.29) that
Estimating \(\bar{J}_{3}\). Similarly,
Combining (4.27), (4.28), (4.30), and (4.31) leads to
By using Gronwall’s lemma, we deduce from (4.32) that
where
We can assume that
We require the following lemma whose proof is immediate.
Lemma 4.5
Let the sequence \(\{\gamma_{m}\}\) satisfy
where \(0\leq\sigma<1\), \(\delta\geq0\) are the given constants. Then
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
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
This implies (4.7). The proof of Theorem 4.2 is complete.  □
References
Beilin, SA: On a mixed nonlocal problem for a wave equation. Electron. J. Differ. Equ. 2006, 103 (2006)
Bergounioux, M, Long, NT, Dinh, APN: Mathematical model for a shock problem involving a linear viscoelastic bar. Nonlinear Anal. 43, 547-561 (2001)
Benaissa, A, Messaoudi, SA: Exponential decay of solutions of a nonlinearly damped wave equation. Nonlinear Differ. Equ. Appl. 12, 391-399 (2005)
Jokhadze, O: The global Cauchy problem for wave equations with nonlinear damping term. Differ. Equ. 50(1), 57-65 (2014)
Ngoc, LTP, Hang, LNK, Long, NT: On a nonlinear wave equation associated with the boundary conditions involving convolution. Nonlinear Anal. TMA 70(11), 3943-3965 (2009)
Long, NT, Diem, TN: On the nonlinear wave equation \(u_{tt}-u_{xx}=f(x, t, u, u_{x}, u_{t})\) associated with the mixed homogeneous conditions. Nonlinear Anal. TMA 29(11), 1217-1230 (1997)
Long, NT: On the nonlinear wave equation \(u_{tt}-B(t, \Vert u \Vert ^{2}, \Vert u_{x} \Vert ^{2})u_{xx} =f(x, t, u, u_{x}, u_{t}, \Vert u \Vert ^{2}, \Vert u_{x} \Vert ^{2})\) associated with the mixed homogeneous conditions. J. Math. Anal. Appl. 306(1), 243-268 (2005)
Long, NT, Ngoc, LTP: On a nonlinear wave equation with boundary conditions of two-point type. J. Math. Anal. Appl. 385(2), 1070-1093 (2012)
Ngoc, LTP, Long, NT: Existence and exponential decay for a nonlinear wave equation with a nonlocal boundary condition. Commun. Pure Appl. Anal. 12(5), 2001-2029 (2013)
Truong, LX, Ngoc, LTP, Dinh, APN, Long, NT: The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions. Nonlinear Anal., Real World Appl. 11(1-2), 1289-1303 (2010)
Bhattarai, S: Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities. Adv. Nonlinear Anal. 4(2), 73-90 (2015)
Kumar, S: Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Adv. Nonlinear Anal. 5(4), 383-394 (2016). doi:10.1515/anona-2013-0033
Repovs̆, D: Stationary waves of Schrödinger-type equations with variable exponent. Anal. Appl. 13(6), 645-661 (2015). doi:10.1142/S0219530514500420
Pellicer, M, Solà -Morales, J: Spectral analysis and limit behaviours in a spring-mass system. Commun. Pure Appl. Math. 7(3), 563-577 (2008)
Brezis, H: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)
Showater, RE: Hilbert space methods for partial differential equations. Electron. J. Differ. Equ. Monograph 01 (1994)
Lions, JL: Quelques méthodes de résolution des problèmes aux limites nonlinéaires. Dunod, Paris (1969)
Acknowledgements
The authors wish to thank the referee for their valuable criticism and suggestions, leading to the present improved version of our paper. There are no funding sources supporting this work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Nhan, N.H., Ngoc, L.T.P. & Long, N.T. Existence and asymptotic expansion of the weak solution for a wave equation with nonlinear source containing nonlocal term. Bound Value Probl 2017, 87 (2017). https://doi.org/10.1186/s13661-017-0818-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-017-0818-5
MSC
- 35L20
- 35L70
- 35Q72
Keywords
- Faedo-Galerkin method
- linear recurrent sequence
- Robin conditions
- asymptotic expansion in a small parameter