## Boundary Value Problems

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# Global solutions to a class of nonlinear damped wave operator equations

Boundary Value Problems20122012:42

DOI: 10.1186/1687-2770-2012-42

Accepted: 13 April 2012

Published: 13 April 2012

## Abstract

This study investigates the existence of global solutions to a class of nonlinear damped wave operator equations. Dividing the differential operator into two parts, variational and non-variational structure, we obtain the existence, uniformly bounded and regularity of solutions.

Mathematics Subject Classification 2000: 35L05; 35A01; 35L35.

### Keywords

nonlinear damped wave operator equations global solutions uniformly bounded regularity

## 1 Introduction

In recent years, there have been extensive studies on well-posedness of the following nonlinear variational wave equation with general data:
$\left\{\begin{array}{cc}{{\partial }_{t}}^{2}u-c\left(u\right){\partial }_{x}\left(c\left(u\right){\partial }_{x}u\right)=0\hfill & \mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\left(0,\infty \right)×\mathbf{R},\hfill \\ u{|}_{t=0}={u}_{0}\hfill & \mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\mathbf{R},\hfill \\ {\partial }_{t}u{|}_{t=0}={u}_{1}\hfill & \mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\mathbf{R},\hfill \end{array}\right\$
(1.1)
where c(·) is given smooth, bounded, and positive function with c'(·) ≥ 0 and c'(u0) > 0,u0 H1(R),u1(x) L2(R). Equation (1.1) appears naturally in the study for liquid crystals [14]. In addition, Chang et al. [5], Su [6] and Kian [7] discussed globally Lipschitz continuous solutions to a class one dimension quasilinear wave equations
$\left\{\begin{array}{c}{u}_{tt}-{\left(p\left(\rho \left(x\right),{u}_{x}\right)\right)}_{x}=\rho \left(x\right)h\left(\rho \left(x\right),u,{u}_{x}\right),\hfill \\ u\left(x,0\right)={u}_{0}\left(x\right),\hfill \\ {u}_{t}\left(x,0\right)={\omega }_{0}\left(x\right),\hfill \end{array}\right\$
(1.2)
where (x,t) R × R+, u0(x),ω0(x) R. Furthermore, Nishihara [8] and Hayashi [9] obtained the global solution to one dimension semilinear damped wave equation
$\left\{\begin{array}{c}{u}_{tt}+{u}_{t}-{u}_{xx}=f\left(u\right),\left(t,x\right)\in {\mathbf{R}}^{+}×{\mathbf{R}}^{+}\hfill \\ \left(u,{u}_{t}\right)\left(0,x\right)=\left({u}_{0},{u}_{1}\right)\left(x\right).\hfill \end{array}\right\$
(1.3)

Ikehata [10] and Vitillaro [11] proved global existence of solutions for semilinear damped wave equations in R N with noncompactly supported initial data or in the energy space, in where the nonlinear term f(u) = |u| p or f(u) = 0 is too special; some authors [1214] discussed the regularity of invariant sets in semilinear wave equation, but they didn't refer to any the initial value condition of it. Unfortunately, it is difficulty to classify a class wave operator equations, since the differential operator structure is too complex to identify whether have variational property. Our aim is to classify a class of nonlinear damped wave operator equations in order to research them more extensively and go beyond the results of [12].

In this article, we are interested in the existence of global solutions of the following nonlinear damped wave operator equations:
$\left\{\begin{array}{cc}\frac{{d}^{2}u}{d{t}^{2}}+k\frac{du}{dt}=G\left(u\right),\hfill & k>0\hfill \\ u\left(x,0\right)=\phi \left(x\right),\hfill \\ {u}_{t}\left(x,0\right)=\psi \left(x\right),\hfill \end{array}\right\$
(1.4)

where $G:{X}_{2}×{\mathbf{R}}^{+}\to {X}_{1}^{*}$ is a mapping, X2 X1, X1, X2 are Banach spaces and ${X}_{1}^{*}$ is the dual spaces of X1, R+ = [0, ∞), u = u(x,t). If k > 0, (1.4) is called damped wave equation. We obtain the existence, uniformly bounded and regularity of solutions by dividing the differential operator G(u) into two parts, variational and non-variational structure.

## 2 Preliminaries

First we introduce a sequence of function spaces:
$\left\{\begin{array}{c}X\subset {H}_{2}\subset {X}_{2}\subset {X}_{1}\subset H,\hfill \\ {X}_{2}\subset {H}_{1}\subset H,\hfill \end{array}\right\$
(2.1)
where H, H1, H2 are Hilbert spaces, X is a linear space, X1, X2 are Banach spaces and all inclusions are dense embeddings. Suppose that
$\left\{\begin{array}{c}L:X\to {X}_{1}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{is}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{one}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{to}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{one}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{dense}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{linear}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{operator}},\hfill \\ {〈Lu,v〉}_{H}={〈u,v〉}_{H},\phantom{\rule{1em}{0ex}}\forall u,v\in X.\hfill \end{array}\right\$
(2.2)
In addition, the operator L has an eigenvalue sequence
$L{e}_{k}={\lambda }_{k}{e}_{k},\phantom{\rule{1em}{0ex}}\left(k=1,2,...\right)$
(2.3)
such that {e k } X is the common orthogonal basis of H and H2. We investigate the existence of global solutions of the Equation (1.4), so we need define its solution. Firstly, in Banach space X, introduce
${L}^{p}\left(\left(0,T\right),X\right)=\left\{u:\left(0,T\right)\to X|\underset{0}{\overset{T}{\int }}{∥u∥}^{p}dt<\infty \right\},$
where p = (p1, p2,..., p m ),p i ≥ 1(1 ≤ im),
${∥u∥}^{p}=\sum _{k=1}^{m}{\left|u\right|}_{k}^{{p}_{k}},$
where | · | k is semi-norm in X, and ${∥\cdot ∥}_{X}={\sum }_{i=1}^{m}{\left|\cdot \right|}_{i}$. Similarily, we can define
${W}^{1,p}\left(\left(0,T\right),X\right)=\left\{u:\left(0,T\right)\to X|u,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{u}^{\prime }\in {L}^{p}\left(\left(0,T\right),X\right)\right\}.$

Let ${L}_{\mathsf{\text{loc}}}^{p}\left(\left(0,\infty \right),X\right)=\left\{u\left(t\right)\in X|u\in {L}^{p}\left(\left(0,T\right),X\right),\forall T>0\right\}.$.

Definition 2.1. Set (φ, ψ) X2 × H1, $u\in {W}_{\mathsf{\text{loc}}}^{1,\infty }\left(\left(0,\infty \right),{H}_{1}\right)\bigcap {L}_{\mathsf{\text{loc}}}^{\infty }\left(\left(0,\infty \right),{X}_{2}\right)$ is called a globally weak solution of (1.4), if for v X1, it has
${〈{u}_{t},v〉}_{H}+k{〈u,v〉}_{H}=\underset{0}{\overset{t}{\int }}〈Gu,v〉dt+k{〈\phi ,v〉}_{H}+{〈\psi ,v〉}_{H}.$
(2.4)
Definition 2.2. Let Y1,Y2 be Banach spaces, the solution u(t, φ, ψ) of (1.4) is called uniformly bounded in Y1 × Y2, if for any bounded domain Ω1 × Ω2Y1 × Y2, there exists a constant C which only depends the domain Ω1 × Ω2, such that
${∥u∥}_{{Y}_{1}}+{∥{u}_{t}∥}_{{Y}_{2}}\le C,\phantom{\rule{1em}{0ex}}\forall \left(\phi ,\psi \right)\in {\mathrm{\Omega }}_{1}×{\mathrm{\Omega }}_{2}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}t\ge 0.$
Definition 2.3. A mapping $G:{X}_{2}\to {X}_{1}^{*}$ is called weakly continuous, if for any sequence {u n } X2, u n u0 in X2,
$\underset{n\to \infty }{\mathsf{\text{lim}}}〈G\left({u}_{n}\right),v〉=〈G\left({u}_{0}\right),v〉,\phantom{\rule{1em}{0ex}}\forall v\in {X}_{1}.$

Lemma 2.1. [15]Let H2, H be Hilbert spaces, and H2 H be a continuous embedding. Then there exists a orthonormal basis {e k } of H, and also is one orthogonal basis of H2.

Proof. Let I : H2H be imbedded. According to assume I is a linear compact operator, we define the mapping A : H2H as follows
${〈Au,v〉}_{{H}_{2}}={〈Iu,v〉}_{H}={〈u,v〉}_{H},\phantom{\rule{1em}{0ex}}\forall v\in {H}_{2}.$
obviously, A : H2H2 is linear symmetrical compact operator and positive definite. Therefore, A has a complete eigenvalue sequence {λ k } and eigenvector sequence$\left\{{ẽ}_{k}\right\}\subset {H}_{2}$ such that
$A{ẽ}_{k}={\lambda }_{k}{ẽ}_{k},\phantom{\rule{1em}{0ex}}k=1,2,...,$
and $\left\{{ẽ}_{k}\right\}$ is orthogonal basis of H2. Hence
${〈{ẽ}_{i},{ẽ}_{j}〉}_{H}={〈A{ẽ}_{i},{ẽ}_{j}〉}_{{H}_{2}}={\lambda }_{i}{〈{ẽ}_{i},{ẽ}_{j}〉}_{{H}_{2}}=0,\phantom{\rule{1em}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}i\ne j.$

it implies $\left\{{ẽ}_{i}\right\}$ is also orthogonal sequence of H. Since H2 H is dense, $\left\{{ẽ}_{i}\right\}$ is also orthogonal sequence of H, so $\left\{{e}_{i}\right\}=\left\{{ẽ}_{i}/{∥{ẽ}_{i}∥}_{H}\right\}$ is norm orthogonal basis of H. The proof is completed.

Now, we introduce an important inequality

Lemma 2.2. [16] (Gronwall inequality) Let x(t), y(t), z(t) be real function on [a, b], where x(t) ≥ 0,atb, z(t) C[a, b], y(t) is differentiable on [a, b]. If the inequality as follows is hold
$z\left(t\right)\le y\left(t\right)+\underset{a}{\overset{t}{\int }}x\left(\tau \right)z\left(\tau \right)d\tau ,\phantom{\rule{1em}{0ex}}a\le t\le b,$
(2.5)
then
$z\left(t\right)\le y\left(a\right){e}^{{\int }_{a}^{t}x\left(s\right)ds}+\underset{a}{\overset{t}{\int }}{e}^{{\int }_{a}^{t}x\left(\tau \right)}\frac{dy}{ds}ds.$
(2.6)

## 3 Main results

Suppose that $G=A+B:{X}_{2}×{\mathbf{R}}^{+}\to {X}_{1}^{*}$. Throughout of this article, we assume that
1. (i)
There exists a function F C 1 : X 2R 1 such that
$〈Au,Lv〉=〈-DF\left(u\right),v〉,\phantom{\rule{1em}{0ex}}\forall u,v\in X$
(3.1)

2. (ii)
Function F is coercive, if
$F\left(u\right)\to \infty ⇔{∥u∥}_{{X}_{2}}\to \infty$
(3.2)

3. (iii)
B as follows
$\left|〈Bu,Lv〉\right|\le {C}_{1}F\left(u\right)+{C}_{2}{∥v∥}_{{H}_{1}}^{2},\phantom{\rule{1em}{0ex}}\forall u,v\in X$
(3.3)

for some $g\in {L}_{\mathsf{\text{loc}}}^{1}\left(0,\infty \right)$.

Theorem 3.1. Set$G:{X}_{2}×{\mathbf{R}}^{+}\to {X}_{1}^{*}$is weakly continuous, (φ, ψ) X2 × H1, then we obtain the results as follows:

(1) If G = A satisfies the assumption (i) and (ii), then there exists a globally weak solution of (1.4)
$u\in {W}_{\mathsf{\text{loc}}}^{1,\infty }\left(\left(0,\infty \right),{H}_{1}\bigcap \underset{\mathsf{\text{loc}}}{\overset{\infty }{L}}\left(\left(0,\infty \right),{X}_{2}\right)\right)$

and u is uniformly bounded in X2 × H1;

(2) If G = A + B satisfies the assumption (i), (ii) and (iii), then there exists a globally weak solution of (1.4)
$u\in {W}_{\mathsf{\text{loc}}}^{1,\infty }\left(\left(0,\infty \right),{H}_{1}\right)\bigcap \underset{\mathsf{\text{loc}}}{\overset{\infty }{L}}\left(\left(0,\infty \right),{X}_{2}\right);$
(3) Furthermore, if G = A + B satisfies
$\left|〈Gu,v〉\right|\le \frac{1}{2}{∥v∥}_{H}^{2}+CF\left(u\right)+g\left(t\right)$
(3.4)

for some$g\in {L}_{\mathsf{\text{loc}}}^{1}\left(0,\infty \right)$, then$u\in {W}_{\mathsf{\text{loc}}}^{2,2}\left(\left(0,\infty \right),H\right)$.

Proof. Let {e k } X be the public orthogonal basis of H and H2, satisfies (2.3).

Note
$\left\{\begin{array}{c}{X}_{n}=\left\{\sum _{i=1}^{n}{\alpha }_{i}{e}_{i}|{\alpha }_{i}\in {\mathbf{R}}^{1}\right\},\hfill \\ {\stackrel{̃}{X}}_{n}=\left\{\sum _{j=1}^{n}{\beta }_{j}\left(t\right){e}_{j}|{\beta }_{j}\in {C}^{2}\left[0,\infty \right)\right\}.\hfill \end{array}\right\$
(3.5)
From the assumption, we know $L{X}_{n}={X}_{n},L\stackrel{̃}{{X}_{n}}=\stackrel{̃}{{X}_{n}}$, apply the Galerkin method to make truncate in $\stackrel{̃}{{X}_{n}}$:
$\left\{\begin{array}{cc}\frac{{d}^{2}{u}_{i}}{d{t}^{2}}+k\frac{d{u}_{i}}{dt}=〈G\left({u}_{n}\right),{e}_{i}〉,\hfill & 1\le i\le n\hfill \\ {u}_{i}\left(x,0\right)={〈\phi ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{e}_{i}〉}_{H},\hfill \\ {{u}^{\prime }}_{i}\left(x,0\right)={〈\psi ,\phantom{\rule{2.77695pt}{0ex}}{e}_{i}〉}_{H}\hfill \end{array}\right\$
(3.6)
there exists ${u}_{n}={\sum }_{i=1}^{n}{u}_{i}\left(t\right){e}_{i}\in {C}^{2}\left(\left(0,\infty \right),{X}_{n}\right)$ for any $v\in \stackrel{̃}{{X}_{n}}$ satisfies
$\underset{0}{\overset{t}{\int }}{〈\frac{{d}^{2}{u}_{n}}{d{t}^{2}}+k\frac{d{u}_{n}}{dt},v〉}_{H}dt=\underset{0}{\overset{t}{\int }}〈G{u}_{n},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}v〉dt$
(3.7)
for any v X n , it yields that
${〈\frac{d{u}_{n}}{dt},v〉}_{H}+k{〈{u}_{n},v〉}_{H}=\underset{0}{\overset{t}{\int }}〈G{u}_{n},v〉dt+k{〈\phi ,v〉}_{H}+{〈\psi ,v〉}_{H}$
(3.8)
1. (1)
If $G=A,{u}_{n}\in \stackrel{̃}{{X}_{n}}$ substitute $v=\frac{d}{dt}L{u}_{n}$ into (3.7), we get
$\underset{0}{\overset{t}{\int }}{〈\frac{{d}^{2}{u}_{n}}{d{t}^{2}}+k\frac{d{u}_{n}}{dt},\frac{d}{dt}L{u}_{n}〉}_{{H}_{1}}dt=\underset{0}{\overset{t}{\int }}〈G{u}_{n},\phantom{\rule{2.77695pt}{0ex}}\frac{d}{dt}\phantom{\rule{2.77695pt}{0ex}}L{u}_{n}〉dt$

combine condition (2.2) with (3.1), we get
$\begin{array}{c}\underset{0}{\overset{t}{\int }}\underset{\mathrm{\Omega }}{\int }\frac{{d}^{2}{u}_{n}}{d{t}^{2}}\frac{d{u}_{n}}{dt}dxdt+\underset{0}{\overset{t}{\int }}\underset{\mathrm{\Omega }}{\int }k\frac{d{u}_{n}}{dt}\frac{d{u}_{n}}{dt}dxdt+\underset{0}{\overset{t}{\int }}DF\left({u}_{n}\right)\frac{d{u}_{n}}{dt}dxdt=0\\ \underset{0}{\overset{t}{\int }}\frac{1}{2}\frac{d}{dt}{∥\frac{d{u}_{n}}{dt}∥}_{{H}_{1}}^{2}dt+k\underset{0}{\overset{t}{\int }}{∥\frac{d{u}_{n}}{dt}∥}_{{H}_{1}}^{2}dt+\underset{0}{\overset{t}{\int }}\frac{d}{dt}F\left({u}_{n}\right)dt=0\\ \frac{1}{2}{∥\frac{d{u}_{n}}{dt}∥}_{{H}_{1}}^{2}-\frac{1}{2}{∥{\psi }_{n}∥}_{{H}_{1}}^{2}+k\underset{0}{\overset{t}{\int }}{∥\frac{d{u}_{n}}{dt}∥}_{{H}_{1}}^{2}dt+F\left({u}_{n}\right)-F\left({\phi }_{n}\right)=0\end{array}$
consequently, we get
$F\left({u}_{n}\right)+\frac{1}{2}{∥{{u}^{\prime }}_{n}∥}_{{H}_{1}}^{2}+k\underset{0}{\overset{t}{\int }}{∥{{u}^{\prime }}_{n}∥}_{{H}_{1}}^{2}dt=F\left({u}_{n}\right)+\frac{1}{2}{∥{\psi }_{n}∥}_{{H}_{1}}^{2}.$
(3.9)
Assume φ H2, combine(2.2)with(2.3), we know {e n } is also the orthogonal basis of H1, then φ n φ in H2, ψ n ψ in H1, owing to H2 X2 is embedded, so
$\left\{\begin{array}{c}{\phi }_{n}\to \phi \phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}{X}_{2}\hfill \\ {\psi }_{n}\to \psi \phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}{X}_{1}\hfill \end{array}\right\$
(3.10)
due to the condition (3.6), from (3.9)and (3.10) we easily know
$\left\{{u}_{n}\right\}\subset {W}_{\mathsf{\text{loc}}}^{1,\infty }\left(\left(0,\infty \right),{H}_{1}\right)\bigcap \underset{\mathsf{\text{loc}}}{\overset{\infty }{L}}\left(\left(0,\infty \right),{X}_{2}\right)\phantom{\rule{2.77695pt}{0ex}}is\phantom{\rule{2.77695pt}{0ex}}bounded.$
consequently, assume that
${u}_{n}⇀{u}_{0}\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}{W}_{\mathsf{\text{loc}}}^{1,\infty }\left(\left(0,\infty \right),{H}_{1}\right)\bigcap \underset{\mathsf{\text{loc}}}{\overset{\infty }{L}}\left(\left(0,\infty \right),{X}_{2}\right)\phantom{\rule{2.77695pt}{0ex}}a.e.\phantom{\rule{2.77695pt}{0ex}}t>0$
i.e. u n u0in X2a.e. t > 0, and G is weakly continuous, so
$\underset{n\to \infty }{\mathsf{\text{lim}}}〈G{u}_{n},v〉=〈G{u}_{0},v〉.$
By (3.8), we have
$\begin{array}{cc}\hfill \underset{n\to \infty }{\mathsf{\text{lim}}}\left[{〈\frac{d{u}_{n}}{dt}〉}_{H}+k{〈{u}_{n},v〉}_{H}\right]& =\underset{n\to \infty }{\mathsf{\text{lim}}}\underset{0}{\overset{t}{\int }}〈G{u}_{n,}v〉dt+k{〈\phi ,v〉}_{H}+{〈\psi ,v〉}_{H}\hfill \\ \hfill {〈\frac{d{u}_{0}}{dt},v〉}_{H}+k{〈{u}_{0},v〉}_{H}& =\underset{0}{\overset{t}{\int }}〈G{u}_{0},v〉dt+k{〈\phi ,v〉}_{H}+{〈\psi ,v〉}_{H}\hfill \end{array}$
it indicates for any $v\in {\bigcup }_{n=1}^{\infty }{X}_{n}\subset {X}_{2}$, it holds. Hence, for any v X2, we have
${〈\frac{d{u}_{0}}{dt},v〉}_{H}+k{〈{u}_{0},v〉}_{H}=\underset{0}{\overset{t}{\int }}〈G{u}_{0},v〉dt+k{〈\phi ,v〉}_{H}+{〈\psi ,v〉}_{H}.$
(3.11)

Consequently, u0 is a globally weak solution of (1.4).

Furthermore, by (3.9) and (3.10), for any R > 0, there exists a constant C such that if
${∥\phi ∥}_{{X}_{2}}+{∥\psi ∥}_{{H}_{1}}\le R$
(3.12)
then the weak solution u(t, φ, ψ) of (1.4) satisfies
${∥u\left(t,\phi ,\psi \right)∥}_{{X}_{2}}+{∥{u}_{t}\left(t,\phi ,\psi \right)∥}_{{H}_{1}}\le C.\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall t\ge 0$
(3.13)
Assume (φ,ψ) X2 × H1 satisfies (3.12), by H2 X2 is dense. May fix φ n H2 such that
${∥{\phi }_{n}∥}_{{X}_{2}}+{∥\psi ∥}_{{H}_{1}}\le R,\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{\mathsf{\text{lim}}}{\phi }_{n}=\phi \phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}{X}_{2}$

by (3.13), the solution {u(t, φ n , ψ)} of (1.4) is bounded in ${W}_{\mathsf{\text{loc}}}^{1,\infty }\left(\left(0,\infty \right),{H}_{1}\right)\bigcap {L}_{\mathsf{\text{loc}}}^{\infty }\left(\left(0,\infty \right),{X}_{2}\right)$ a.e. t > 0.

Therefore, assume u(t, φ n , ψ) u in ${W}_{\mathsf{\text{loc}}}^{1,\infty }\left(\left(0,\infty \right),{H}_{1}\right)\bigcap {L}_{\mathsf{\text{loc}}}^{\infty }\left(\left(0,\infty \right),{X}_{2}\right)$ then u(t) is a weak solution of (1.4), it satisfies uniformly bounded of (3.13). So the conclusion (1) is proved.
1. (2)
If $G=A+B,{u}_{n}\in \stackrel{̃}{{X}_{n}}$, substitute $v=\frac{d}{dt}L{u}_{n}$ into (3.7), we get
$\begin{array}{l}\phantom{\rule{1em}{0ex}}\underset{0}{\overset{t}{\int }}\left[{〈\frac{{d}^{2}{u}_{n}}{d{t}^{2}},\frac{d}{dt}L{u}_{n}〉}_{{H}_{1}}\right]+k{〈\frac{d{u}_{n}}{dt},\frac{d}{dt}L{u}_{n1}〉}_{{H}_{1}}dt\\ =\underset{0}{\overset{t}{\int }}\left[〈A{u}_{n},\frac{d{u}_{n}}{dt}〉+〈B{u}_{n},\frac{d{u}_{n}}{dt}〉\right]dt\end{array}$

combine the condition (2.2) and (3.1), we have
$\begin{array}{l}\phantom{\rule{1em}{0ex}}\underset{0}{\overset{t}{\int }}\underset{\Omega }{\int }\frac{{d}^{2}{u}_{n}}{d{t}^{2}}\frac{d{u}_{n}}{dt}dxdt+k\underset{0}{\overset{t}{\int }}\underset{\Omega }{\int }\frac{d{u}_{n}}{d{t}^{2}}\frac{d{u}_{n}}{dt}dxdt+\underset{0}{\overset{t}{\int }}〈DF\left({u}_{n}\right)\frac{d{u}_{n}}{dt}〉dt\phantom{\rule{2em}{0ex}}\\ =\underset{0}{\overset{t}{\int }}〈B{u}_{n},\frac{d{u}_{n}}{dt}〉dt\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\underset{0}{\overset{t}{\int }}\frac{1}{2}\frac{d}{dt}{∥\frac{d{u}_{n}}{dt}∥}_{{H}_{1}}^{2}dt+k\underset{0}{\overset{t}{\int }}{∥\frac{d{u}_{n}}{dt}∥}_{{H}_{1}}^{2}dt+\underset{0}{\overset{t}{\int }}\frac{d}{dt}F\left({u}_{n}\right)dt\phantom{\rule{2em}{0ex}}\\ =\underset{0}{\overset{t}{\int }}〈B{u}_{n},\frac{d{u}_{n}}{dt}〉dt\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\frac{1}{2}{∥{{u}^{\prime }}_{n}∥}_{{H}_{1}^{2}}-\frac{1}{2}{∥{\psi }_{n}∥}_{{H}_{1}}^{2}+k\underset{0}{\overset{t}{\int }}{∥{{u}^{\prime }}_{n}∥}_{{H}_{1}}^{2}dt+F\left({u}_{n}\right)+F\left({\phi }_{n}\right)\phantom{\rule{2em}{0ex}}\\ =\underset{0}{\overset{t}{\int }}〈B{u}_{n},\frac{d{u}_{n}}{dt}〉dt\phantom{\rule{2em}{0ex}}\end{array}$
consequently, we have
$F\left({u}_{n}\right)+\frac{1}{2}{∥{{u}^{\prime }}_{n}∥}_{{H}_{1}}^{2}+k\underset{0}{\overset{t}{\int }}{∥{{u}^{\prime }}_{n}∥}_{{H}_{1}}^{2}dt=\underset{0}{\overset{t}{\int }}〈B{u}_{n},\frac{d{u}_{n}}{dt}〉dt+F\left({\phi }_{n}\right)+\frac{1}{2}{∥{\psi }_{n}∥}_{{H}_{1}}^{2}$
(3.14)
by the condition (3.3),(3.14)implies
$F\left({u}_{n}\right)+\frac{1}{2}{∥{{u}^{\prime }}_{n}∥}_{{H}_{1}}^{2}\le C\underset{0}{\overset{t}{\int }}\left[F\left({u}_{n}\right)+\frac{1}{2}{∥{{u}^{\prime }}_{n}∥}_{{H}_{1}}^{2}\right]dt+f\left(t\right)$
(3.15)

where $f\left(t\right)={\int }_{0}^{t}g\left(\tau \right)dt+\frac{1}{2}{∥\psi ∥}_{{H}_{1}}^{2}+{\mathsf{\text{sup}}}_{n}F\left({\phi }_{n}\right).$.

by Gronwall inequality [Lemma(2.2)], from (3.15) we easily know:
$F\left({u}_{n}\right)+\frac{1}{2}{∥{{u}^{\prime }}_{n}∥}_{{H}_{1}}^{2}\le f\left(0\right){e}^{{C}^{t}}+\underset{0}{\overset{t}{\int }}{f}^{\prime }\left(\tau \right){e}^{C\left(t-\tau \right)}d\tau$
(3.16)
it implies that, for any 0 < T < ∞
$\left\{{u}_{n}\right\}\subset {W}^{1,\infty }\left(\left(0,T\right),{X}_{2}\right)\bigcap {L}^{\infty }\left(\left(0,T\right),{X}_{2}\right)\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{is}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{bounded}}.$
now, use the same way as (1), we can obtain the result (2).
1. (3)
If the condition (3.4) is hold, ${u}_{n}\in \stackrel{̃}{{X}_{n}}$, substitute $v=\frac{{d}^{2}u}{d{t}^{2}}$ into (3.7), we can get
$\underset{0}{\overset{t}{\int }}\left[{〈\frac{{d}^{2}{u}_{n}}{d{t}^{2}},\frac{{d}^{2}{u}_{n}}{d{t}^{2}}〉}_{H}+k{〈\frac{d{u}_{n}}{dt},\frac{{d}^{2}{u}_{n}}{d{t}^{2}}〉}_{H}\right]dt=\underset{0}{\overset{t}{\int }}〈G{u}_{n},\frac{{d}^{2}{u}_{n}}{d{t}^{2}}〉dt$

then
$\begin{array}{l}\phantom{\rule{1em}{0ex}}\underset{0}{\overset{t}{\int }}{〈\frac{{d}^{2}{u}_{n}}{d{t}^{2}},\frac{{d}^{2}{u}_{n}}{d{t}^{2}}〉}_{H}dt+\frac{k}{2}\underset{0}{\overset{t}{\int }}\frac{d}{dt}{∥{{u}^{\prime }}_{n}\left(t\right)∥}_{H}^{2}dt\phantom{\rule{2em}{0ex}}\\ \le \underset{0}{\overset{t}{\int }}\left[\frac{1}{2}{∥{{u}^{″}}_{n}\left(t\right)∥}_{H}^{2}+CF\left({u}_{n}\right)+g\left(t\right)\right]dt\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\underset{0}{\overset{t}{\int }}{〈\frac{{d}^{2}{u}_{n}}{d{t}^{2}},\frac{{d}^{2}{u}_{n}}{d{t}^{2}}〉}_{H}dt+\frac{k}{2}{∥{{u}^{\prime }}_{n}∥}_{H}^{2}\phantom{\rule{2em}{0ex}}\\ \le \frac{k}{2}{∥{\psi }_{n}∥}_{H}^{2}+\underset{0}{\overset{t}{\int }}\left[\frac{1}{2}{∥\frac{{d}^{2}{u}_{n}}{d{t}^{2}}∥}_{H}^{2}+CF\left({u}_{n}\right)+g\left(\tau \right)\right]d\tau \phantom{\rule{2em}{0ex}}\end{array}$
by (3.16), it implies that
$\underset{0}{\int }t{∥\frac{{d}^{2}{u}_{n}}{d{t}^{2}}∥}_{H}^{2}d\tau \le C,\phantom{\rule{1em}{0ex}}\left(C>0\right)$
consequently, for any 0 < T < ∞
$\left\{{u}_{n}\right\}\subset {W}^{2,2}\left(\left(0,T\right),H\right)\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{is}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{bounded}}.$

it implies that u W2,2((0,T), H), the main theorem (3.1) has been proved.

## Declarations

### Acknowledgements

The author is very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. Foundation item: the National Natural Science Foundation of China (No. 10971148).

## Authors’ Affiliations

(1)
Yangtze Center of Mathematics, Sichuan University
(2)
College of Mathematics and Software Science, Sichuan Normal University

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