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Young measure solutions for a fourth-order wave equation with variable growth
Boundary Value Problems volume 2015, Article number: 123 (2015)
Abstract
In this paper, we study the existence of Young measure solutions to a fourth-order wave equation with variable exponent nonlinearity on a bounded domain. The asymptotic behavior of the Young measure solutions is also investigated by applying a lemma developed by Nakao.
1 Introduction
In this paper, we consider the initial boundary value problem of the following model:
where \(\Omega\subset\mathbb{R}^{N}\) (\(N\geq3\)) is a bounded domain with smooth boundary ∂Ω, \(0< T<\infty\) is a given constant, and \(Q_{T}=\Omega\times(0,T)\). The coefficient \(a:[0,\infty)\rightarrow (0,\infty)\) and the exponents \(p,q:\overline{\Omega}\rightarrow(1,\infty )\) are given continuous functions and \(f:Q_{T}\rightarrow\mathbb{R}\). PDEs with variable exponent growth conditions are usually called equations with nonstandard growth conditions. After Kováčik and Rákosník first discussed the variable exponent Lebesgue space \(L^{p(x)}\) and Sobolev space \(W^{k,p(x)}\) in [1], a lot of research has been done concerning these kinds of variable exponent spaces; see for example [2, 3] for the properties of such spaces and [4–13] for the applications of variable exponent spaces on partial differential equations. In [14] Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids [15] and nonlinear elastics [16–18]. Another field of application of equations with variable exponent growth conditions is image restoration [19].
We claim that the Young measure solutions of problem (1.1) can be approximated by the following problem with a viscosity term \(\varepsilon \Delta^{2}\frac{\partial u}{\partial t}\) (\(\varepsilon>0\)):
When \(p(x)\equiv2\) and the space dimension \(N=1\), problems of the type (1.2) are a class of essential fourth-order wave equations appearing in elastoplastic-microstructure models. They govern the longitudinal motion of an elastoplastic bar and antiplane shearing deformation; see [20]. For \(p(x)\equiv2\) and the multidimensional case, Chen and Yang [21] discussed the global existence, asymptotic behavior and blow-up of solutions to the initial boundary problem of the equation with weak damping term \(\frac{\partial u}{\partial t}\); see also Messaoudi [22] for wave equations with nonlinear damping. For the analysis of nonlinear second-order hyperbolic equations with damping, we refer to the seminal work of Lions and Strauss [23]; see also Friedman and Nec̆as [24], and Emmrich and Thalhammer [25, 26]. In recent years, hyperbolic equations with variable exponent growth conditions were studied by Antontsev in [27], Haehnle and Prohl in [28], Pinasco in [29]; see also Autuori et al. in [30, 31] for the Kirchhoff equations with \(p(x)\)-growth. It is to be noted here that the viscosity term \(\Delta^{2}\frac{\partial u}{\partial t}\) plays a key role in the proof of the global existence. The global existence results of weak solutions for second-order wave equations (even if \(p(x)\equiv \mathrm{constant}\neq2\)) without the viscosity term \(\Delta^{2}\frac{\partial u}{\partial t}\) have been found only in one space dimension; see DiPerna [32] and Shearer [33]. To the best of our knowledge, the equations without the viscosity term are studied only in [34–37]. In that work, the concept of Young measure solutions has been introduced and applied to dynamic problems and wave equations.
Thus motivated, in the present paper, we prove the global existence of Young measure solutions of problem (1.1), we first construct Young measure solutions as the limit of the sequence of solutions of problem (1.2). Then we give a decay estimate to the Young measure solutions of problem (1.1).
Our work is organized as follows. In Section 2, we give some necessary definitions and properties of variable exponent Lebesgue spaces and Sobolev spaces. In Section 3, we obtain the existence of weak solutions of problem (1.2) by Galerkin’s approximation method. In Section 4, under some conditions, from the sequence of solutions of problem (1.2) and some a priori estimates, we get the existence of Young measure solutions by letting \(\varepsilon\rightarrow0\). In Section 5, we investigate the decay property of Young measure solutions and get a decay rate estimate by using Nakao’s lemma.
2 Preliminaries
In this section, we first recall some necessary properties of variable exponent Lebesgue spaces and Sobolev spaces; see [1–3] for the details.
Let \(\Omega\subset\mathbb{R}^{N}\) be a domain. A measurable function \(p:\Omega\rightarrow[1,\infty)\) is called a variable exponent and we define \(p^{-}=\operatorname{ess}\inf_{x\in\Omega}p(x)\) and \(p^{+}=\operatorname{ess}\sup_{x\in\Omega}p(x)\). If \(p^{+}\) is finite, then the exponent p is said to be bounded. The variable exponent Lebesgue space is
with the Luxemburg norm
then \(L^{p(x)}(\Omega)\) is a Banach space, and when p is bounded, we have the following relations:
That is, if p is bounded, then norm convergence is equivalent to convergence with respect to the modular \(\rho_{p(x)}\). For a bounded exponent the dual space \((L^{p(x)}(\Omega))^{\prime}\) can be identified with \(L^{p^{\prime}(x)}(\Omega)\), where the conjugate exponent \(p^{\prime}(x)\) is defined by \(p^{\prime}(x)=\frac {p(x)}{p(x)-1}\) for each \(x\in\Omega\). If \(1< p^{-}\leq p^{+}<\infty\), then \(L^{p(x)}(\Omega)\) is separable and reflexive.
In the variable exponent Lebesgue space, Hölder’s inequality is still valid; see [1], Theorem 2.1. For all \(u\in L^{p(x)}(\Omega)\), \(v\in L^{p^{\prime}(x)}(\Omega)\) with \(p(x)\in(1,\infty)\) the following inequality holds:
If \(0<|\Omega|<\infty\) and p, q are variable exponents such that \(p(x)\leq q(x)\) for each \(x\in\Omega\), then there exists a continuous embedding \(L^{p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)\).
Definition 2.1
(see [2])
We say that a bounded exponent \(p: \Omega\rightarrow\mathbb{R}\) is log-Hölder continuous if there is a constant \(C>0\) such that
for all points \(y, z\in\Omega\).
The variable exponent Sobolev space \(W^{k,p(x)}(\Omega)\) is defined as
and equipped with the norm
then the space \(W^{k,p(x)}(\Omega)\) is a Banach space. The space \(W^{k,p(x)}_{0}(\Omega)\) is defined as the closure of \(C^{\infty }_{0}(\Omega)\) with the above norm. If \(1< p^{-}\leq p^{+}<\infty\), then the space \(W^{k,p(x)}(\Omega)\) is separable and reflexive; If \(p:\Omega \rightarrow(1,\infty)\) is a bounded log-Hölder continuous function, then \(C_{0}^{\infty}(\Omega)\) is dense in \(W_{0}^{k,p(x)}(\Omega)\).
Theorem 2.1
(see [2])
Let \(\Omega\subset\mathbb{R}^{N}\) be a bounded domain and assume that \(p: \mathbb{R}^{N}\rightarrow(1, \infty)\) is a bounded log-Hölder continuous exponent such that \(p^{-}>1\), then for any \(u\in W^{1, p(x)}_{0}(\Omega)\) we have
where the constant c only depends on the dimension N, \(|\Omega|\) and the log-Hölder constant of p.
Theorem 2.2
(see [2])
Let Ω be a bounded domain with smooth boundary. Assume that \(p:\Omega\rightarrow(1,\infty)\) is a bounded log-Hölder continuous function with \(p^{+}<\frac{N}{k}\) and \(q:\Omega\rightarrow(1,\infty)\) is a bounded measurable function with \(q(x)\leq p^{*}=\frac {Np(x)}{N-kp(x)}\). Then there is a continuous embedding
where the embedding constant depends on \(|\Omega|\), N, \(q^{+}\) and the log-Hölder constant of p.
Theorem 2.3
(see [38])
Let \(\Omega\subset\mathbb{R}^{N}\) be a bounded domain with smooth boundary. Suppose that p is a bounded log-Hölder continuous functions in Ω, with \(p^{-}>1\). Then there exists a constant \(C>0\) depending only on N, Ω and the log-Hölder constant of p such that for each \(u\in W_{0}^{2,p(x)}(\Omega)\), the following inequality holds:
Proposition 2.1
Let Ω be a bounded domain in \(\mathbb{R}^{N}\) and let \(\{\omega_{i}\} _{i=1}^{\infty}\) be an orthogonal basis in \(L^{2}(\Omega)\). Then for any \(\varepsilon>0\), there exists a positive number \(N_{\varepsilon}\) such that
for all \(u\in W_{0}^{1,p}(\Omega)\) where \(2\leq p<\infty\).
The following theorem gives a relation between almost everywhere convergence and weak convergence.
Theorem 2.4
(see [7])
Let \(p: \Omega\rightarrow\mathbb{R}\) be a bounded log-Hölder continuous function with \(p^{-}>1\). If \(\{u_{n}\}^{\infty}_{n=1}\) is bounded in \(L^{p(x)}(Q_{T})\) and \(u_{n}\rightarrow u\) a.e. in \(Q_{T}\) as \(n\rightarrow\infty\), then there exists a subsequence of \(\{ u_{n}\}\) still denoted by \(\{u_{n}\}\) such that \(u_{n}\rightharpoonup u\) weakly in \(L^{p(x)}(Q_{T})\) as \(n\rightarrow\infty\).
Denote by \(C_{0}(\mathbb{R}^{N})\) (\(N\geq1\)) the closure of continuous functions in \(\mathbb{R}^{N}\) with compact support. The dual of \(C_{0}(\mathbb{R}^{N})\) can be identified with the space \(\mathcal {M}(\mathbb{R}^{N})\) of signed Radon measures with finite mass via the pairing
A map \(\mu:E\rightarrow\mathcal{M}(\mathbb{R}^{N})\) (\(E\subset\mathbb {R}^{N}\)) is called weak ∗ measurable if the functions \(x\rightarrow \langle\mu(x),f\rangle\) are measurable for all \(f\in C_{0}(\mathbb {R}^{N})\). We write \(\mu_{x}\) instead of \(\mu(x)\).
Theorem 2.5
(see [40], Theorem 3.1)
Let \(E\subset\mathbb{R}^{N}\) be a measurable set of finite measure and let \(z_{j}:E\rightarrow\mathbb{R}^{N}\) be a sequence of measurable functions. Then there exist a subsequence \(z_{j_{k}}\) and a weak ∗ measurable map \(\nu:E\rightarrow\mathcal{M}(\mathbb{R}^{N})\) such that
-
(i)
\(\nu_{x}\geq0\), \(\|\nu_{x}\|_{\mathcal{M}(\mathbb{R}^{N})}= \int_{\mathbb{R}^{d}}\,d\nu_{x}\leq1\), for a.e. \(x\in E\).
-
(ii)
For all \(f\in C_{0}(\mathbb{R}^{N})\)
$$\begin{aligned} f(z_{j_{k}})\rightharpoonup\langle\nu_{x},f\rangle \quad\textit{weakly}*\textit{ in } L^{\infty}(E). \end{aligned}$$ -
(iii)
Let \(K\subset\mathbb{R}^{N}\) be compact. Then
$$\begin{aligned} \operatorname{supp}\nu_{x}\subset K \quad\textit{if } \operatorname{dist}(z_{j_{k}},K) \rightarrow 0 \textit{ in measure}. \end{aligned}$$ -
(iv)
Furthermore one has
$$\begin{aligned} \bigl(\mathrm{i}^{\prime}\bigr)\quad \|\nu_{x} \|_{\mathcal{M}(\mathbb{R}^{N})}=1, \quad\textit{for a.e. } x\in E \end{aligned}$$if and only if the sequence does not escape to infinity, i.e. if
$$\begin{aligned} \lim_{L\rightarrow\infty}\sup_{k}\bigl|\{x\in E:|z_{j_{k}}|\geq L\}\bigr|=0. \end{aligned}$$(2.1) -
(v)
If (i′) holds, if \(A\subset E\) is measurable and \(f\in C(\mathbb{R}^{N})\) and if \(f(z_{j_{k}})\) is relatively weakly compact in \(L^{1}(A)\), then \(f(z_{j_{k}})\rightharpoonup\langle\nu_{x},f\rangle\) weakly in \(L^{1}(A)\).
-
(vi)
If (i′) holds, then in (iii) one can replace ‘if’ by ‘if and only if’.
Remark 2.1
If for some \(s>0\) and all \(j\in\mathbb{N}\)
then (2.1) holds.
3 Existence of weak solutions of problem (1.2)
In this section, let \(\varepsilon\in(0,1)\) fixed, we prove the existence of weak solutions for problem (1.2). Our main hypotheses are the following:
-
(H1)
\(p,q:\overline{\Omega}\rightarrow(1,\infty)\) are two log-Hölder continuous functions satisfying
$$\begin{aligned} \max \biggl\{ 1,\frac{2N}{N+4} \biggr\} < p^{-}=\inf_{\overline{\Omega }}p(x) \leq p^{+}=\sup_{\overline{\Omega}}p(x)< \frac{N}{2} \end{aligned}$$and
$$\begin{aligned} 1< q^{-}=\inf_{\overline{\Omega}}q(x)\leq q(x)< \frac{Np(x)}{N-2p(x)}, \quad \mbox{for } x\in\overline{\Omega}. \end{aligned}$$Here \(\overline{\Omega}\) denotes the closure of Ω.
-
(H2)
\(a\in C([0,\infty))\) and there exists a constant \(a_{0}>0\) such that
$$\begin{aligned} a(s)\geq a_{0}>0, \quad\mbox{for } s\in[0,\infty). \end{aligned}$$ -
(H3)
\(u_{0}\in W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\), \(u_{1}\in L^{2}(\Omega)\), \(f\in L^{2}(Q_{T})\).
Definition 3.1
A function \(u_{\varepsilon}:Q_{T}\rightarrow\mathbb{R}\) is called a weak solution of (1.2), if
and
for all \(\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\) and \(\tau\in(0,T]\).
We choose a sequence \(\{\omega_{j}\}_{j=1}^{\infty}\subset C_{0}^{\infty}(\Omega )\) such that \(C_{0}^{\infty}(\Omega)\subset\overline{\bigcup_{n=1}^{\infty}V_{n}}^{C^{2}(\bar{\Omega})} \) and \(\{\omega_{j}\}_{j=1}^{\infty}\) is a complete orthogonal basis in \(L^{2}(\Omega)\), where \(V_{n}=\operatorname{span}\{ \omega_{1}, \omega_{2}, \ldots, \omega_{n}\}\); see [7, 39].
Since \(\bigcup_{n=1}^{\infty}V_{n}\) is dense in \(C^{2}(\bar{\Omega})\), we have the following lemma.
Lemma 3.1
(see [41])
For the function \(u_{0}\in W^{2,p(x)}(\Omega )\cap W_{0}^{2,2}(\Omega)\), there exists a sequence \(\psi_{n}\) with \(\psi _{n}\in V_{n}\) such that \(\psi_{n}\rightarrow u_{0}\) in \(W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\) as \(n\rightarrow\infty\).
Proof
For \(u_{0}\in W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\), there exists a sequence \(\{v_{n}\}\) in \(C_{0}^{\infty}(\Omega)\) such that \(v_{n}\rightarrow u_{0}\) in \(W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\). Since \(\{v_{n}\}\subset C_{0}^{\infty}(\Omega)\subset\overline{\bigcup_{m=1}^{\infty}V_{m}}^{C^{2}(\bar{\Omega})}\), we can find a sequence \(\{v_{n}^{k}\}\subset\bigcup_{m=1}^{\infty}V_{m}\) such that, for each \(n\in\mathbb{N}\), we have \(v_{n}^{k}\rightarrow u_{n}\) in \(C^{2}(\overline{\Omega})\) as \(k\rightarrow\infty\). For \(\frac{1}{2^{n}}\), there exists \(k_{n}\geq1\) such that \(\|v_{n}^{k_{n}}-u_{n}\|_{C^{2}(\overline{\Omega})}\leq\frac{1}{2^{n}}\). Thus
That is, \(v_{n}^{k_{n}}\rightarrow u_{0}\) in \(W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\) as \(n\rightarrow\infty\). Denote \(u_{n}=v_{n}^{k_{n}}\). Since \(u_{n}\in\bigcup_{m=1}^{\infty}V_{m}\), there exists \(V_{m_{n}}\) such that \(u_{n}\in V_{m_{n}}\); without loss of generality, we assume that \(V_{m_{1}}\subset V_{m_{2}}\) as \(m_{1}\leq m_{2}\). We assume that \(m_{1}>1\) and define \(\psi_{n}\) as follows: \(\psi_{n}(x)=0\), \(n=1,\ldots,m_{1}-1\); \(\psi_{n}=u_{1}\), \(n=m_{1},\ldots,m_{2}-1\); \(\psi_{n}=u_{2}\), \(n=m_{2},\ldots,m_{3}-1;\ldots \) , then we obtain the sequence \(\{\psi_{n}\}\) and \(\psi_{n}\rightarrow u_{0}\) in \(W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\) as \(n\rightarrow \infty\). □
The existence of weak solutions of problem (1.1) is proved by Galerkin’s approximation. We shall find the sequence of approximate solutions in the form
The unknown functions \((\eta_{n}(t))_{j}\) are determined by ordinary differential equations in the following.
We first define a vector-valued function \(P_{n}(t,\mu,\nu): [0,T]\times \mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) as follows:
where \(\mu=(\mu_{1},\ldots,\mu_{n})\) and \(\nu=(\nu_{1},\ldots,\nu_{n})\). Now we consider the following Cauchy problem of second-order ordinary differential equations:
where \((U_{0n})_{i}=\int_{\Omega}\psi_{n}\omega_{i}\,dx\), \((U_{1n})_{i}=\int_{\Omega}\phi_{n}\omega_{i}\,dx\), \(F_{n}=\int_{\Omega}f_{n}\omega_{i}\,dx\), \(\psi_{n}\in V_{n}\), \(\phi_{n}\in V_{n}\), \(f_{n}\in C_{0}^{\infty}(Q_{T})\), and \(\psi_{n}\rightarrow u_{0}\) strongly in \(W^{2,p(x)}(\Omega)\cap W_{0}^{1,2}(\Omega)\) (\(\psi _{n}\) from Lemma 3.1), \(\phi_{n}\rightarrow u_{1}\) strongly in \(L^{2}(\Omega)\), \(f_{n}\rightarrow f\) strongly in \(L^{2}(Q_{T})\) (since \(C_{0}^{\infty}(Q_{T})\) is dense in \(L^{2}(Q_{T})\)).
Let \(\eta^{\prime}(t)=X(t)\), \(Y(t)= (\eta(t), X(t) )\), and \(H_{n}(t,Y)= (X,F_{n} -P_{n}(t,\eta,X) )\). Then the problem (3.1) is transformed into the following problem:
The assumption (H2) implies
From (3.2) and Young’s inequality, we obtain
Thus,
Gronwall’s inequality and \(f_{n}\rightarrow f\) strongly in \(L^{2}(Q_{T})\) imply
where C is a constant independent of n and ε. Thus, \(|Y-Y(0)|\leq2 \sqrt{C}\). We denote
where \(B (Y(0),2 \sqrt{C} )\) is the ball of radius \(2 \sqrt{C}\) with center at the point \(Y(0)\) in \(\mathbb{R}^{2n}\). From the definition of \(H(t,Y)\), \(H(t,Y)\) is continuous with respect to \((t,Y)\). By Peano’s theorem, we know that (3.2) admits a \(C^{1}\) solution on \([0, \tau_{n}]\), that is, (3.1) has a \(C^{2}\) solution on \([0,\tau_{n}]\) denoted by \(\eta_{n}^{1}(t)\). Let \(\eta(\tau_{n})\), \(\frac{\partial\eta (\tau_{n})}{\partial t}\) be the initial value of problem (3.1), then we can repeat the above process and get a \(C^{2}\) solution \(\eta_{n}^{2}(t)\) on \([\tau_{n}, 2\tau_{n}]\). Without loss of generality, we assume that \(T=[\frac{T}{\tau_{n}}]\tau_{n}+(\frac{T}{\tau_{n}})\tau_{n}\), \(0<(\frac{T}{\tau _{n}})<1\), where \([\frac{T}{\tau_{n}}]\) is the integer part of \(\frac {T}{\tau_{n}}\), \((\frac{T}{\tau_{n}})\) is the decimal part of \(\frac{T}{\tau _{n}}\). We can divide \([0, T]\) into \([(i-1)\tau_{n},i\tau_{n}]\), \(i=1,\ldots,L\), and \([L\tau_{n},T]\) where \(L=[\frac{T}{\tau_{n}}]\), then there exists a \(C^{2}\) solution \(\eta^{i}_{n}(t)\) in \([(i-1)\tau_{n},i\tau_{n}]\), \(i=1,\ldots,L\), and \(\eta^{L+1}_{n}(t)\) in \([L\tau_{n},T]\). Therefore, we get a solution \(\eta_{n}(t) \in C^{2}([0, T])\) defined by
Lemma 3.2
(A priori estimates)
The estimates
hold uniformly with respect to n.
Proof
By (3.4), we have
Further, integrating the inequality (3.3) with respect to t over \([0,T]\), we obtain
Moreover, for each \(t\in[0,T]\),
Thus, this lemma is proved □
By Lemma 3.2, we have the following.
Lemma 3.3
The estimate
holds uniformly with respect to n and ε.
Proof
By Theorem 2.3, we have
Thus, \(\|u_{n}\|_{L^{\infty}(0,T;W_{0}^{2,p(x)}(\Omega))}\leq C\). By Lemma 3.2, we obtain
Thus,
Similarly, \(\||u_{n}|^{q(x)-2}u_{n} \|_{L^{q^{\prime}(x)}(Q_{T})}\leq C\). Since \(a\in C([0,\infty))\) and \(\int_{\Omega}|u_{n}(x,t)|^{q(x)}\,dx\leq C\), we have
This lemma is proved. □
Theorem 3.1
Assume (H1)-(H3). Then for each \(\varepsilon\in(0,1)\) problem (1.2) has a weak solution.
Proof
By Lemma 3.2 and Lemma 3.3, there exist a subsequence of \(\{u_{n}\}\) (still denoted by \(\{u_{n}\}\)), \(u_{\varepsilon}\), ξ, η, and ζ such that
Since \(u_{n}\in L^{\infty}(0,T;W_{0}^{1,2}(\Omega))\) and \(\frac{\partial u_{n}}{\partial t}\in L^{2}(Q_{T})\), by the Lions-Aubin lemma, there exists a subsequence of \(\{u_{n}\}\) still denoted by \(\{u_{n}\}\) such that \(u_{n}\rightarrow u_{\varepsilon}\) strongly in \(L^{2}(Q_{T})\) and a.e. on \(Q_{T}\). Further, \(|u_{n}|^{q(x)-2}u_{n}\rightarrow|u_{\varepsilon}|^{q(x)-2}u_{\varepsilon}\) a.e. on \(Q_{T}\). In view of Theorem 2.4, we obtain \(\eta=|u_{\varepsilon}|^{q(x)-2}u_{\varepsilon}\).
Next, we prove that there exists a subsequence of \(\{u_{n}\}\) (still denoted by \(\{u_{n}\}\)) such that \(\frac{\partial u_{n}}{\partial t}\rightarrow\frac{\partial u_{\varepsilon}}{\partial t}\) strongly in \(L^{2}(Q_{T})\).
Since \((\eta^{\prime}_{n}(t))_{j}=\int_{\Omega}\frac{\partial u_{n}}{\partial t}\omega_{j}\,dx\), by Lemma 3.2, \((\eta^{\prime}_{n}(t))_{j}\) is uniformly bounded on \([0,T]\). For \(\forall0\leq t_{1}< t_{2}\leq T\), integrating (3.1) with respect to t from \(t_{1}\) to \(t_{2}\), we have
Hölder’s inequality, Lemma 3.2, and Lemma 3.3 imply
where \(Q^{t_{2}}_{t_{1}}=\Omega\times(t_{1},t_{2})\). Thus, the sequence \(\{(\eta _{n}(t))_{j}\}^{\infty}_{n=1}\) is uniformly bounded and equi-continuous for fixed j and arbitrary \(n\geq j\). By the Ascoli-Arzela theorem and the usual diagonal procedure, there exists a subsequence of \(\{(\eta_{n})_{j}\} \) still denoted by \(\{(\eta_{n})_{j}\}\) such that \((\eta_{n}(t))_{j}\) converges uniformly on \([0,T]\) to some continuous function \(\lambda^{\varepsilon}_{j}(t)\) for each fixed \(j=1,2,\ldots \) .
For \(r\leq n\) with \(r\in\mathbb{N}\), by Lemma 3.2, we have
Letting \(n\rightarrow\infty\), we get
Then letting \(r\rightarrow\infty\), we obtain
Set \(\overline{u}_{\varepsilon}(x,t)=\sum_{j=1}^{\infty}\lambda ^{\varepsilon}_{j}(t)\omega_{j}(x)\), then \(\sup_{0\leq t\leq T}\| \overline{u}_{\varepsilon}(x,t)\|_{L^{2}(\Omega)}\leq C(T)\) and, for each \(j\in\mathbb{N}\), we have
uniformly on \([0,T]\). For each \(\delta_{1}>0\) and \(\phi\in L^{2}(\Omega)\), by the completeness of \(\{\omega_{j}\}\), there exists a \(m_{0}>0\) such that \(\|\phi-\sum_{i=1}^{m_{0}}(\int_{\Omega}\phi\omega_{i}\,dx)\omega_{i}\| _{L^{2}(\Omega)}\leq\delta_{1}\). Thus,
For \(\delta_{1}>0\), by (3.5), there exists a \(M>0\) such that
By (3.6) and the Hölder inequality, we have
It follows from (3.7) and the arbitrariness of \(\delta_{1}\) that
uniformly on \([0,T]\) as \(n\rightarrow\infty\). For each \(\varphi\in C_{0}^{\infty}(Q_{T})\), by Lebesgue’s dominated convergence theorem, we obtain
Hence,
On the other hand, by integration by parts, we get
Letting \(n\rightarrow\infty\) in above equality, we have
Thus, we obtain \(\overline{u}= \frac{\partial u_{\varepsilon}}{\partial t}\). Moreover, for each \(j\in\mathbb{N}\), Lemma 3.2, and Lebesgue’s dominated convergence theorem yield
Thus, for \(\delta>0\), by Proposition 2.1, there exists a positive number \(N_{\delta}\) independent of n such that
A similar discussion to (3.7) shows that there is a \(\widetilde {M}(\delta)>0\) such that
Thus, \(\frac{\partial u_{n}}{\partial t}\rightarrow\frac{\partial u_{\varepsilon}}{\partial t}\) strongly in \(L^{2}(Q_{T})\). Further, there exists a subsequence of \(\{u_{n}\}\) still denoted by \(\{u_{n}\}\) such that \(\frac{\partial u_{n}}{\partial t}\rightarrow\frac{\partial u_{\varepsilon}}{\partial t}\) a.e. on \(Q_{T}\).
For \(\forall\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\), we can choose a sequence \(\varphi_{k}\in C^{1}(0,T;V_{k})\) such that \(\varphi_{k}\rightarrow \varphi\) in \(C^{1,2}(Q_{T})\). Here for \(v\in C^{1,2}(Q_{T})\) equipped with the norm \(\|v\|=\sup_{|\alpha|\leq2,(x,t)\in\overline {Q_{T}}} \{| D^{\alpha}v|, |\frac{\partial v}{\partial t}| \}\). For \(\forall\tau\in(0,T]\), we have
where \(Q_{\tau}=\Omega\times(0,\tau)\). Replacing \(\omega_{i}\) in (3.1) by \(\varphi_{k}\), we obtain
Thus, we have
Furthermore, for any \(\psi(x)\in C_{0}^{\infty}(\Omega)\), we get
as \(\tau\rightarrow0\). Similarly, for \(t_{0}\in[0,T]\), we have
Furthermore, we obtain \(\frac{\partial u_{\varepsilon}(x,0)}{\partial t}=u_{1}\). Since \(u_{\varepsilon}\in L^{\infty}(0,T; W_{0}^{2,2}(\Omega))\) and \(\frac{\partial u_{\varepsilon}}{\partial t}\in L^{2}(0,T; W_{0}^{2,2}(\Omega ))\), we can assume that \(u_{\varepsilon}\in C(0,T;W_{0}^{2,2}(\Omega))\). Lemma 3.3 and the embedding \(W_{0}^{1,2}(\Omega)\hookrightarrow L^{2}(\Omega )\) imply that \(\int_{\Omega}u_{n}^{2}(x,T)\,dx\leq C(T)\). Thus, there exist a subsequence of \(\{u_{n}\}\) still denoted by \(\{u_{n}\}\) and a function \(\widehat{u}\) such that \(u_{n}(x,T)\rightharpoonup\widehat{u}\) weakly in \(L^{2}(\Omega)\). For each \(\varphi\in C_{0}^{\infty}(\Omega)\) and \(\eta\in C^{1}([0,T])\), we have
Letting \(n\rightarrow\infty\), we get
By integration by parts, we have
Choosing \(\eta(T)=1\), \(\eta(0)=0\) or \(\eta(T)=0\), \(\eta(0)=1\), we obtain \(\widehat{u}=u_{\varepsilon}(x,T)\) and \(u_{\varepsilon}(x,0)=u_{0}(x)\) for \(x\in\Omega\). Similarly, we can prove that \(\Delta u_{\varepsilon}(x,0)=\Delta u_{0}\), \(\Delta u_{n}(x,T)\rightharpoonup\Delta u_{\varepsilon}(x,T)\) weakly in \(L^{2}(\Omega)\) (up to a subsequence) and
Further, by the compact embedding \(W_{0}^{1,2}(\Omega)\hookrightarrow \hookrightarrow L^{2}(\Omega)\), we get \(u_{n}(x,T)\rightarrow u_{\varepsilon}(x,T)\) strongly in \(L^{2}(\Omega)\).
Taking \(\varphi=u_{k}\) in (3.9) and letting \(k\rightarrow\infty\), we get
Multiplying (3.1) by \((\eta_{n})_{j}\) and summing up j from 1 to n, then integrating with respect to t over \([0,T]\), we have
Thus,
It follows that
Following the ideas of [4], we set \(Q_{1}=\{(x,t)\in Q_{T}: p(x)\geq2\}\) and \(Q_{2}=\{(x,t)\in Q_{T}: 1< p(x)<2\}\), then, as \(n\rightarrow\infty\),
and
Therefore, we obtain \(\Delta u_{n}\rightarrow\Delta u_{\varepsilon}\) strongly in \(L^{p(x)}(Q_{\tau})\). Thus, there exists a subsequence of \(\{u_{n}\}\) still denoted by \(\{u_{n}\} \) such that \(\Delta u_{n}\rightarrow\Delta u_{\varepsilon}\) a.e. on \(Q_{T}\). Further,
In view of Theorem 2.4, we get \(\xi=|\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\). Similarly, we can prove that \(u_{n}\rightarrow u_{\varepsilon}\) strongly in \(L^{q(x)}(Q_{T})\). Thus, there exists a subsequence of \(\{u_{n}\}\) still denoted \(\{u_{n}\}\) such that
Furthermore, we have
Thus, \(a(\int_{\Omega}|u_{n}|^{q(x)}\,dx)\frac{\partial u_{n}}{\partial t}\rightarrow a(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx)\frac{\partial u_{\varepsilon}}{\partial t}\) a.e. on \(Q_{T}\). By Theorem 2.4, we obtain \(\zeta=a(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx)\frac{\partial u_{\varepsilon}}{\partial t}\). It follows from (3.9) that the theorem is proved. □
Remark 3.1
Obviously, in this section, the two inequalities in (H2) can be replaced by \(1< p^{-}\leq p^{+}<\infty\) and \(1< q^{-}\leq q^{+}<\infty\), respectively.
4 Existence of Young measure solutions for problem (1.2)
In this section, from the sequence of approximate solutions \(\{ u_{\varepsilon}\}_{0<\varepsilon<1}\) of problem (1.2), we shall prove that the limit of \(u_{\varepsilon}\) (as \(\varepsilon\rightarrow0^{+}\)) is a Young measure solution of problem (1.1).
Definition 4.1
A pair \((u,\nu)\) is called a Young measure solution of problem (1.1) if
and
for all \(\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\) and \(\tau\in(0,T]\).
Theorem 4.1
Under conditions (H1)-(H3), problem (1.1) has a Young measure solution.
Proof
For each \(\tau\in(0,T]\) and \(\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\), we have by (3.9)
Since the constant in Lemma 3.2 is independent of n and ε, by the convergence of \(u_{n}\) and \(\frac{\partial u_{n}}{\partial t}\) in Section 3, we have
for a.e \(t\in[0,T]\) and
Similarly, by Lemma 3.3, we have
Thus, there exists a subsequence of \(\{u_{\varepsilon}\}_{0<\varepsilon <1}\) still denoted by \(\{u_{\varepsilon}\}_{0<\varepsilon<1}\) such that
Since \(p^{-}>\max\{1,\frac{2N}{N+4}\}\), the embedding \(W_{0}^{2,p(x)}(\Omega )\hookrightarrow L^{2}(\Omega)\) is compact. Further, as \(u_{\varepsilon}\in L^{\infty}(0,T;W_{0}^{2,p(x)}(\Omega))\) and \(\frac{\partial u_{\varepsilon}}{\partial t}\in L^{\infty}(0,T;L^{2}(\Omega))\), by the Lions-Aubin lemma, there exists a subsequence of \(\{u_{\varepsilon}\}_{0<\varepsilon<1}\) still denoted by \(\{u_{\varepsilon}\}_{0<\varepsilon<1}\) such that \(u_{\varepsilon}\rightarrow u\) strongly in \(L^{2}(Q_{T})\) and a.e. on \(Q_{T}\). Thus, \(|u_{\varepsilon}|^{q(x)-2}u_{\varepsilon}\rightarrow|u|^{q(x)-2}u\) a.e. on \(Q_{T}\). In view of Theorem 2.4, we obtain \(\alpha =|u|^{q(x)-2}u\). By assumption (H1), we have \(\mu=\inf_{\overline {\Omega}}(\frac{Np(x)}{N-2p(x)}-q(x))>0\). For each measurable subset \(U\subset Q_{T}\) with \(|U|\leq1\), by Hölder’s inequality, Theorem 2.2, and Theorem 2.3, we obtain
Thus, the sequence \(\{|u_{\varepsilon}-u|^{q(x)}\}_{0<\varepsilon<1}\) is equi-integrable on \(L^{1}(Q_{T})\). The Vitali convergence theorem implies that \(\int_{Q_{T}}|u_{\varepsilon}-u|^{q(x)}\,dx\,dt\rightarrow0\), that is to say, we obtain \(u_{\varepsilon}\rightarrow u\) strongly in \(L^{q(x)}(Q_{T})\). Thus, there exists a subsequence of \(\{u_{\varepsilon}\} _{0<\varepsilon<1}\) still labeled by \(\{u_{\varepsilon}\}_{0<\varepsilon <1}\) such that
Furthermore,
Hence, we find by the continuity of a that \(a(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx)\rightarrow a(\int_{\Omega}|u|^{q(x)}\,dx)\) for a.e. \(t\in[0,T]\). Since \(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx\leq C\) for a.e. \(t\in[0,T]\) and \(a\in C([0,\infty))\), for each \(\varphi\in L^{2}(Q_{T})\), by Lebesgue’s dominated convergence theorem, we have
Further, by the weak convergence of \(\frac{\partial u_{\varepsilon}}{\partial t}\) in \(L^{2}(Q_{T})\), we get
Thus, \(a(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx)\frac{\partial u_{\varepsilon}}{\partial t}\rightharpoonup a(\int_{\Omega}|u|^{q(x)}\,dx)\frac{\partial u}{\partial t}\) weakly in \(L^{2}(Q_{T})\). The uniqueness of the limit implies that \(\beta=a(\int_{\Omega}|u|^{q(x)}\,dx)\frac{\partial u}{\partial t}\).
Finally, we prove that the sequence \(\{\Delta u_{\varepsilon}\} _{0<\varepsilon<1}\) generates a Young measure \(\{\nu_{x,t}\}_{x,t}\) such that
Following Theorem 2.5, we first verify that the Young measure \(\nu _{x,t}\) generated by the sequence \(\{\Delta u_{\varepsilon}\} _{0<\varepsilon<1}\) is a probability measure for a.e. \((x,t)\in Q_{T}\). Indeed, for \(s\leq p^{-}\), we have
It follows from (iv) in Theorem 2.5 that \(\nu_{x,t}\) is a probability measure. Set \(H(x,A)=|A|^{p(x)-2}A\). Next, we prove that the sequence \(\{ H(x,\Delta u_{\varepsilon})\}_{\varepsilon}\) is weakly relatively compact in \(L^{1}(Q_{T})\). It is clear that \(\{H(x,\Delta u_{\varepsilon})\} _{\varepsilon}\) will be weakly relatively compact in \(L^{1}(Q_{T})\), if we prove that \(\{H(x,\Delta u_{\varepsilon})\}_{\varepsilon}\) is uniformly bounded and equi-integrable on \(L^{1}(Q_{T})\); see [42], Proposition 1.3. Indeed, for each measurable subset \(U\subset Q_{T}\) with \(|U|\leq1\), by (4.2) and Hölder’s inequality, we have
Thus, the sequence \(\{H(x,\Delta u_{\varepsilon})\}_{\varepsilon}\) is equi-integrable. Similarly, the sequence \(\{H(x,\Delta u_{\varepsilon})\} _{\varepsilon}\) is uniformly bounded on \(L^{1}(Q_{T})\). Therefore, the convergence property (4.3) holds.
The estimate (4.2) implies
From the same procedures as in Section 3, we can prove there exists a subsequence of \(\{u_{\varepsilon}\}_{0<\varepsilon<1}\) still denoted by \(\{u_{\varepsilon}\}_{0<\varepsilon<1}\) such that \(\frac{\partial u_{\varepsilon}(x,t)}{\partial t}\rightharpoonup\frac{\partial u(x,t)}{\partial t}\) weakly in \(L^{2}(\Omega)\) uniformly on \([0,T]\). Taking \(\varepsilon\rightarrow0\) in Definition 4.1, we obtain
for all \(\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\) and \(\tau\in(0,T]\). □
By Theorem 4.1, we have the following corollary.
Corollary 4.1
Suppose that \(f(x,t)\equiv0\) and (H1) and (H2) are satisfied. Then for a given \(u_{0}\in W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\), there exist a function \(u:\Omega\times(0,\infty)\rightarrow\mathbb{R}\) and a Young measure \(\nu_{x,t}\) such that for \(\forall T>0\),
and
for all \(\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\) and \(\tau\in(0,T]\).
5 Energy decay of Young measure solutions
In this section, we give the decay estimates of weak solutions obtained by Corollary 4.1. First we give a lemma by Nakao [43]
Lemma 5.1
(see [43])
Let \(\Psi:(0,\infty)\rightarrow\mathbb{R}\) be a bounded nonnegative function. If there exist two constants \(\alpha>0\) and \(\beta\geq0\) such that
then there exist positive constants C and γ such that
Theorem 5.1
Let \(p^{-}\geq2\). Then there exist constants \(C, \gamma>0\) such that the weak solutions obtained by Corollary 4.1 have the following estimates: If \(p^{+}=2\), then
If \(p^{+}>2\), then
Proof
We define
The definition of \(I_{n}(t)\) and equality (3.1) imply \(I_{n}(t)\) is nonnegative and uniformly bounded. We assume that \(I_{n}(t)\leq M\), \(M>0\) is a constant. For \(\forall t>0\) fixed, it follows from (3.9) and (H2) that
This implies \(I_{n}(t)\) is a nonincreasing function. Putting \(J^{2}_{n}(t)=I_{n}(t)-I_{n}(t+1)\) and integrating (5.1) over \((t,t+1)\), we get
By the mean value theorem and (5.2), there exist \(t_{1}\in[t,t+\frac {1}{3}]\) and \(t_{2}\in[t+\frac{2}{3},t+1]\) such that
From (3.1), we have
Integrating (5.4) from \(t_{1}\) to \(t_{2}\), we obtain
The Hölder inequality, (5.3), Theorem 2.2, Theorem 2.3, and \(I_{n}(t)\) being decreasing imply
Here the third inequality in (5.6) is obtained by
Similarly,
From the assumption (H2), Hölder’s inequality, the second inequality in (5.2), Theorem 2.2, Theorem 2.3, and the boundedness of \(I_{n}\), we have
Gathering (5.5) with (5.6)-(5.8), we obtain
Thus,
By \(I_{n}(t+1)\leq3\int_{t_{1}}^{t_{2}}I_{n}(\tau)\,d\tau\) and \(I_{n}(t+1)=I_{n}(t)-J_{n}^{2}(t)\), we have
Further, Young’s inequality yields
Now we divide the proof in two cases: \(p^{+}=2\) and \(p^{+}>2\). We consider the case \(p^{+}=2\) first. By the boundedness of \(J_{n}(t)\), we have \(I_{n}(t)\leq C_{9}J_{n}^{2}(t)\). Since \(I_{n}(t)\) is nonincreasing, by Lemma 5.1, there exist constants \(C>0\) and \(\gamma>0\) such that
Letting \(n\rightarrow\infty\) in the above inequality, we arrive at
Thus,
Since \(\frac{\partial u_{\varepsilon}(x,t)}{\partial t}\rightharpoonup \frac{\partial u(x,t)}{\partial t}\) weakly in \(L^{2}(\Omega)\) uniformly on \([0,T]\) (\(\forall T>0\)) and \(u_{\varepsilon}\rightarrow u\) strongly in \(L^{q(x)}(\Omega)\) for a.e. \(t\in[0,T]\), we obtain
It remains to consider the case \(p^{+}>2\). It follows from (5.9) that \(I_{n}(t)\leq C_{10}(J_{n}(t))^{\frac{p^{+}}{p^{+}-1}}\). Employing Lemma 5.1, we obtain
Then letting \(n\rightarrow\infty\), we deduce
Finally, letting \(\varepsilon\rightarrow0\), we conclude
Hence the theorem is proved. □
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Xiang, M. Young measure solutions for a fourth-order wave equation with variable growth. Bound Value Probl 2015, 123 (2015). https://doi.org/10.1186/s13661-015-0386-5
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DOI: https://doi.org/10.1186/s13661-015-0386-5