Degenerate Quasilinear Hyperbolic Equation with Strong Damping

in 

Funkcialaj Ekvacioj, 27 (1984), 125-145 

Degenerate Quasilinear Hyperbolic Equation 

with Strong Damping 

By 

Kenji NISHIHARA 

(Tokyo National Technical College, Japan) 

1. Introduction. 

Let $a(¥cdot)$ be a real valued $C^{1}[0,$ $¥mathrm{o}¥mathrm{o}$)-function satisfying 

(1.1) 

$a(s)¥geq 0$ for $s¥in[0,$ $¥infty)$ . 

We consider the initial-boundary value problem (IBVP) for degenerate quasilinear 

hyperbolic equations 

(1.2) 

$u_{tt}(x, t)-a(¥int_{¥Omega}|¥nabla u(y, t)|^{2}dy)¥Delta u(x, t)-¥Delta ll_{t}(x, t)=0$ 

the n-dimen- 

on where $¥Omega¥times(0, ¥infty)$ $¥Omega$ $¥partial¥Omega$ 

is a bounded domain with its boundary 

sional Euclidean space . When 

$R^{n}$ 

(1.3) 

$a(s)¥geq a_{0}>0$ for $s¥in[0,$ $¥infty)$ 

$a_{0}$ 

for some constant instead of (1.1), Eq. (1.2) has a classical solution with null 

Dirichlet boundary condition and suitable initial data, which is unique and decays 

$ t¥rightarrow¥infty$ 

exponentially as . 

Our aim in this paper is to show the existence, uniqueness, regularity and 

asymptotic behaviour of the solution to (1.2) under the condition (1.1). We shall 

remark about the asymptotic behaviour: in general the decay property of the solution 

to (1.2) cannot be expected for an arbitrary . In fact, let $a$ $s_{0}$ 

be a positive number 

such that $a(s_{0})=0$ and let $u_{0}(x)$ satisfy $u_{0}(x)¥in H^{¥infty}(¥Omega)$ , 

$u_{0}(x)|_{¥partial¥Omega}=0$ 

and 

$=s_{0}$ . Then $u(x, t)¥equiv u_{0}(x)$ is just a solution to (1.2) with initial conditions 

$¥int_{¥Omega}|¥nabla u_{0}(x)|^{2}dx$ 

and boundary condition 

$u(x, 0)=u_{0}(x)$ , $u_{t}(x, 0)=0$ 

$u(x, t)|_{¥partial¥Omega}=0$ 

. 

We note that such $u_{0}(x)$ ’s are infinitely many, and that the above example is appli-

126 K. NISHIHARA 

cable to (1.4), (1.5) appeared below. However, we can prove the decay properties 

of the solution to (1.2) with $a(s)=s$ . 

We mention about related results. The equation (1.2) arises in the study of the 

motion of an elastic string. The equation related to (1.2) are 

(1.4) 

$u_{tt}-a(¥int_{¥Omega}|¥nabla u(y, t)|^{2}dy)¥Delta u=0$ 

(1.5) 

$¥iota¥iota_{tt}-a(¥int_{¥Omega}|¥nabla u(y, t)|^{2}dy)¥Delta u+u_{t}=0$ 

(1.6) 

$u_{tt}-a(¥int_{¥Omega}|¥nabla u(y, t)|^{2}dy)¥Delta u+¥Delta^{2}u=0$ 

and etc., which are studied together with (1.3) by many authors; for (1.4) Dickey [5], 

[8], [9], Nishida [22], Menzala [17], Rodriguez [24], Greenberg and Hu [12], Pohozaev 

[23], Nishihara [20] (see also Lions [15]); for (1.5) Dickey [6], Brito [3], Yamada [27], 

Nishihara [21]; for (1.6) Ball [2], Dickey [7], Medeiros [16], Menzala [17], Yamaguchi 

[28] and others. The degenerate case is proposed in Lions [14], [15]. However, in 

our results the strongly damping term is necessary. 

The (IBVP)’s for the other (degenerate) quasilinear hyperbolic equations, such 

as 

(1.7) 

$u_{tt}-¥sigma(u_{x})_{x}-u_{xxt}=f$ , 

are investigated by many authors, Tsutsumi [26], Andrews [1], Greenberg, MacCamy 

and Mizel [11], Greenberg [10] and etc., while the periodic problems to those are 

also studied by Clement [4], Kakita [13], Sowunmi [25], Narazaki [19] and etc.. In a 

final section, we shall show the existence of periodic solutions to (1.2) with the 

periodic forcing term when $a(s)=s$ . 

Acknowledgment. The author should like to express his sincere gratitude to 

Professor Takao Kakita for his helpful suggestions and constant encouragment and 

also to Professor Masaru Yamaguchi for his kind advices. 

2. Formulation of tke problem and results. 

Let $H$ $||¥cdot||$ 

be a real, separable Hilbert space with norm and scalar product 

$(¥cdot, ¥cdot)$ 

. Let A be a linear operator in $H$ with domain dense in $H$ . We assume 

$D(¥mathrm{A})$ 

that A satisfies the following conditions: 

(HI) 

A is a self-adjoint, positive definite operator with discrete spectrum. 

$D(¥mathrm{A}^{r})$ 

Then the linear operator A $r(r>0)$ with domain 

condition 

is 

well-defined and the

$D(¥mathrm{A}^{r})$ 

Degenerate Quasilinear Hyperbolic Equation 127 

$i$ $D(¥mathrm{A}^{r^{¥prime}})$ $r>r^{¥prime}¥geq 0$ 

(H2) the inject on into , , is compact 

is assumed. It is also assumed that 

$a¥in C^{1}[0,$ $¥infty$ 

(H3) ) with $a(s)¥geq 0$ for $s¥geq 0$ . 

In the space $H$ we now consider the initial value problem to the equation 

(2. 1) 

$u^{¥prime¥prime}(t)+a(||¥mathrm{A}^{1/2}u(t)||^{2})¥mathrm{A}u(t)+Au^{¥prime}(t)=0$ 

with ’boundary’ condition 

(2.2) $u(t)$ , 

$u^{¥prime}(t)¥in D(¥mathrm{A})$ 

for any $t¥in[0,$ $¥infty)$ 

and initial data 

(2.3) $u(0)=u_{0}$ , 

$u^{¥prime}(0)=u_{1}$ 

, 

which is the abstract form of (IBVP) to (1.2). Here we denote $’=d_{J}/dt$ and $’’=$ 

$d^{2}/dt^{2}$ 

. 

Then 

$¥mathrm{v}¥mathrm{v}¥mathrm{e}$ 

have the followings: 

and 

Theorem 1 (Existence and uniqueness). 

We assume the conditions $(¥mathrm{H}1)-(¥mathrm{H}3)$ 

(H4) 

$u_{0}¥in D(¥mathrm{A})$ 

and $u_{1}¥in D(¥mathrm{A}^{1/2})$ . 

Then there exists a unique solution $u¥in C^{1}([0, ¥infty);H)$ of $(2.1)-(¥backslash 2.3)$ satisfying Au, 

$¥mathrm{A}^{1/2}u^{¥prime}¥in L^{¥infty}(0, ¥infty; H)$ 

and Au’, . 

$u^{¥prime¥prime}¥in L^{2}(0, ¥infty; H)$ 

Theorem 2 (Regularity). 

We assume 

$botf¥iota u_{0}$ $u_{1}$ 

(H3) and are 

belonged to $D(¥mathrm{A}^{3/2})$ . 

Then tfre unique solution $u$ in Theorem 1 satisfies $u¥in C^{2}([0, ¥infty);H)$ , A $3/2u$ , Au’, 

$¥mathrm{A}^{1/2}u^{¥prime¥prime}$ 

$¥in L^{¥infty}(0, ¥infty;H)$ and 

$¥mathrm{A}^{3/2}u^{¥prime}$ 

, Au’, . 

$u^{¥prime¥prime¥prime}¥in L^{2}(0, ¥infty; H)$ 

Theorem 3 (Regularity). If we assume 

$u_{0}¥in D(¥mathrm{A}^{k})$ 

(H2) $k$ 

and , $u_{1}¥in D(¥mathrm{A}^{k-1/2})$ $k¥geq 2$ , 

then the solution $u$ in Theorem 2 satisfies A $ku$ , A $k-1/2u’¥in L^{¥infty}(0, ¥infty;H)$ and A 

$(0, ¥infty;H)$ . 

$ku^{¥prime}¥in L^{2}$ 

The bounded solution of $(2.1)-(2.3)$ obtained above cannot be expected the 

decay properties, as indicated in Introduction, provided that there exists $s_{0}>0$ such 

that $a(s_{0})=0$ . But when $a(s)=s$ we have 

Theorem 4 (Decay). Let (HI), (H2) and (H5) be assumed and let $u$ be 

$t^{l}¥iota 

e$ 

unique

. 

and 

depending 

$00$ for some constant 

$a_{0}$ 

, then it is easily seen that the 

solution $u(t)$ to $(2.1)-(2.3)$ with its ’higher derivatives’ decays exponentially to zero 

$t$ 

as tends to infinity (cf. Nishihara [21], Yamada [27]). 

3. A priori estimates. 

We employ Galerkin’s procedure for the proofs. 

$¥mathrm{i}¥mathrm{m}¥mathrm{p}1_{1}^{¥tau}.¥mathrm{e}¥mathrm{s}$ 

$¥{¥lambda_{j}^{2}¥}$ 

The condition (HI) that A has an infinite sequence of eigenvalues 

with 

an eigen- 

and there exists a complete orthonormal system in $¥{w_{j}¥}$ $H$ , wach $w_{j}$ 

$¥lambda_{j}^{2}$ 

vector to For each $u¥in H$ , we have an expantion: 

being 

$u=¥sum_{j=1}^{¥infty}u_{j}w_{j}$, 

$u_{j}=(u, w_{j})$ 

with $||u||=(¥sum_{j=1}^{¥infty}u_{j}^{2})^{1/2}$ If $u¥in D(¥mathrm{A}^{k})$ , there holds 

A 

$ku=¥sum_{j=1}^{¥infty}¥lambda_{j}^{¥prime}k/l_{j}w_{j}$ 

with 

$||$ 

$ku||=(¥sum_{j=1}^{¥infty}1_{j}^{4k}u_{j}^{2})^{1/2}$ 

We now define Galerkin’s approximation $u_{m}(t)=¥sum_{j=1}^{m}g_{jm}(t)w_{j}$ 

the initial value problem for the system: 

as a solution to 

(3. 1) 

$(u_{m}^{¥prime¥prime}(t)+a(||¥mathrm{A}^{1/2}u_{m}(t)||^{2})¥mathrm{A}u_{m}(t)+Au_{m}^{J}(t), w)=0$ 

for any fixed $w¥in V^{m}$ , 

with data


(3.2) 

$u_{m}(0)=¥sum_{j=1}^{m}¥alpha_{j}w_{j}$? 

$u_{m}^{J}(0)=¥sum_{j=1}^{m}¥beta_{j}w_{j}$? 

$u_{0}$ 

$u_{1}$ 

strongly 

strongly as 

$ 

as 

$ 

m¥rightarrow¥infty$ 

m¥rightarrow¥infty$ 

, 

where is an $¥mathrm{m}$-dimensional vector space spanned by . Then we 

$V^{m}$ $¥{w_{1^{ }},--, w_{m}¥}$ 

obtain certain nonlinear ordinary differential system for the $g_{jm}$ ’s, which has a unique 

$¥mathrm{C}^{3}$-solution defined on some interval [0, 

$t_{m}$) 

since . 

$a¥in C^{1}[0,$ $¥infty)$ 

We estimate the under the condition (H4). 

$u_{m}$ 

(I) Putting in (3. 1) we have 

$w=¥mathit{2}u_{m}^{J}(t)$ 

(3.3) 

$¥frac{d}{dt}[||u_{m}^{¥prime}(t)||^{2}+¥hat{a}(||¥mathrm{A}^{1/2}u_{m}(t)||^{2})]+2||¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||^{2}=0$ 

where $¥hat{a}(s)=¥int_{0}^{s}a(¥tau)d¥tau$ . 

(3.2), we have 

Integrating both sides of (3.3) from 0 to $t(t


from which it follows that 

(3.7) 

$||¥mathrm{A}^{1/2}u_{m}(t)||^{2}¥leq C_{2}$ 

and 

(3.8) 

$¥int_{0}^{t}a$ 

$||$ 

( A ) . 

$1/2u_{m}(¥tau)||^{2}$ $||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2}d¥tau¥leq C$ 

From (3.4) and (3.7) we conclude that all the intervals [0, 

$t_{m}$) are extended to the 

$¥infty$ 

whole interva1.[0, ). 

$t$ 

(Ill) Taking $w=Au_{m}(t)$ in (3. 1) and integrating it over [0, ), we have 

(3.9) 

$a(||¥mathrm{A}^{1/2}u_{m}(t)||^{2})||$ $Au_{m}(t)||^{2}+¥frac{1}{2}¥frac{d}{dt}||$Au $m(f)||^{2}$ 

$=-¥frac{d}{dt}(¥mathrm{A}^{1/2}u_{m}^{¥prime}(t), ¥mathrm{A}^{1/2}u_{m}(t))+||¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||^{2}$ 

and 

$¥int_{0}^{t}a(||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2})||Au_{m}(¥tau)||^{2}d¥tau+¥frac{1}{2}||$ 

Au $m(f)||^{2}$ 

$¥leq¥frac{1}{2}||¥mathrm{A}u_{0}||^{2}+||u_{1}||||¥mathrm{A}u_{0}||+||u_{m}^{J}(t)||||¥mathrm{A}u_{m}(t)||+¥int_{0}^{t}||¥mathrm{A}^{1/2}u_{m}^{¥prime}(¥tau)||^{2}d¥tau$ 

$¥leq C+C_{1}||$ Au$m(f)||$ , 

the latter of which means 

(3. 10) 

$||$ 

Au$m(t)||¥leq C$ 

and 

(3. 11) 

$¥int_{0}^{¥infty}a$ 

$||$ 

( A ) $1/2u_{m}(¥tau)||^{2}$ $||$ 

Au . 

$m(¥tau)||^{2}d¥tau¥leq C$ 

(IV) Taking $w=¥mathit{2}Au_{m}^{J}(t)$ in 

$(3.¥mathrm{t})$ 

, we have 

$¥frac{d}{dt}||¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||^{2}+2||$ $Au_{m}^{¥prime}(t)||^{2}=-2a$ 

( 

$||$ 

A $1/2u_{m}(t)||^{2}$ )($¥mathrm{A}u_{m}(t)$ 

, Au ) 

$m’(f)$ 

$¥leq a$ 

( 

$||$ 

A ) $1/2u_{m}(t)||^{2}$ $(a_{1}||Au_{m}(t)||^{2}+¥frac{1}{a_{1}}||$ $Au_{m}^{¥prime}(t)||^{2})$ 

that is, 

$¥leq a_{1}¥cdot a(||¥mathrm{A}^{1/2}u_{m}(t)||^{2})||$ 

Au $m(t)||^{2}+||Au_{m}^{¥prime}(t)||^{2}$ ,


(3.12) 

$¥frac{d}{dt}||¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||^{2}+||Au_{m}^{¥prime}(t)||^{2}¥leq 

a_{1}¥cdot a$ $||$ 

( A ) $1/2u_{m}(t)||^{2}$ $||$ 

Au $m(f)||^{2}$ 

where $a_{1}=¥max¥{a(s);0¥leq s¥leq C_{2}¥}$ . Then, integrating (3. 12) over $[0, t]$ , we have 

$||¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||^{2}+¥int_{0}^{t}||Au_{m}^{¥prime}(¥tau)||^{2}d¥tau¥leq||$A $1/2u_{1}||^{2}+a_{1}¥int_{0}^{¥infty}a(||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2})||$ 

Au $ m(¥tau)||^{2}d¥tau$ 

which gives by (3.11) 

(3. 13) 

$||¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||¥leq 

C$ 

and 

(3. 14) 

$¥int_{0}^{¥infty}||Au_{m}^{¥prime}(¥tau)||^{2}d¥tau¥leq 

(V) Putting $w=u_{m}^{¥prime¥prime}(t)$ in 

C$. 

(3. 1), we have 

(3. 15) 

$||u_{m}^{¥prime¥prime}(t)||^{2}+¥frac{1}{2}¥frac{d}{dt}||¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||^{2}=-¥frac{d}{dt}¥{a(||¥mathrm{A}^{1/2}u_{m}(t)||^{2})(¥mathrm{A}^{1/2}u_{m}(t), ¥mathrm{A}^{1/2}u_{m}^{J}(t))¥}$ 

$+2a^{¥prime}(||¥mathrm{A}^{1/2}u_{m}(t)||^{2})(¥mathrm{A}^{1/2}u_{m}(t), ¥mathrm{A}^{1/2}u_{m}^{J}(t))^{2}+a(||¥mathrm{A}^{1/2}u_{m}(t)||^{2})||¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||^{2}$ 

and 

$¥int_{0}^{t}||u_{m}^{¥prime J}(¥tau)||^{2}d_{T}+¥frac{1}{2}||¥mathrm{A}^{1/2}u_{m}^{J}(t)||^{2}$ 

$¥leq¥frac{1}{2}||¥mathrm{A}^{1/2}u_{1}||^{2}+a_{1}||¥mathrm{A}^{1/2}u_{0}||||$ 

A $1/2u_{1}||+a_{1}||$ A $1/2u_{m}(t)||||$ A $1/2u_{m}(t)||J$ 

$+2a_{2}¥int_{0}^{t}||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2}||¥mathrm{A}1/2u_{m}^{;}(¥tau)||^{2}d_{T}+a_{1}¥int_{0}^{t}||¥mathrm{A}^{1/2}u_{m}^{¥prime}(¥tau)||^{2}d¥tau$. 

where $a_{2}=¥max¥{|a^{¥prime}(s)|; 0¥leq s¥leq C_{2}¥}$ . Hence, by (3.5), (3.7) and (3.13), 

$¥int_{0}^{¥infty}||u_{m}^{¥prime¥prime}(¥tau)||^{2}d¥tau¥leq $¥langle$3. 

16) 

C$. 

Assuming (H5) we proceed a priori estimates. 

Similarly to (III), taking in $w=A^{¥mathit{2}}u_{m}(t)$ 

($¥mathrm{V}¥mathrm{D}$ 

(3.1) derives 

$||¥mathrm{A}^{3/2}u_{m}(t)||¥leq C$ 

(3. 17) 

and 

(3. 18) 

$¥int_{0}^{¥infty}a(||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2})||¥mathrm{A}^{3/2}u_{m}(¥tau)||^{2}d¥tau¥leq 

C$.


(3. 19) 

$||$ 

and 

(VII) 

Also, similar estimates to (IV) yield 

Au $mJ(t)||¥leq C$ 

(3.20) 

$¥int_{0}^{¥infty}||¥mathrm{A}^{3/2}u_{m}^{¥prime}(¥tau)||^{2}d¥tau¥leq 

C$ 

by virtue of (3. 18). 

Now we differentiate (3. 1) with respect to 

$t$ 

and have 

(3.21) 

$(u_{m}^{¥prime¥prime¥prime}(t)+a(|| 

¥mathrm{A}^{1/2}u_{m}(t)||^{2})¥mathrm{A}u_{m}^{J}(t)$ 

$+2a^{¥prime}(||¥mathrm{A}^{1/2}u_{m}(t)||^{2})(¥mathrm{A}^{1/2}u_{m}(t), ¥mathrm{A}^{1/2}u_{m}^{J}(t))¥mathrm{A}u_{m}(I)$ 

, $w)=0$ . 

(VIII) Putting 

$w=2¥mathrm{A}u_{m}^{¥prime¥prime}(t)$ 

in 

(3.21) we have 

$¥frac{d}{dt}$ $||¥mathrm{A}^{1/2}u_{m}^{¥prime J}(t)||^{2}+a(||$ 

[ A $Au_{m}^{¥prime}(t)||^{2}$] $1/2u_{m}(t)||^{2})||$ $+2||$ $Au_{m}^{¥prime¥prime}(t)||^{2}$ 

or 

$=2a^{¥prime}$ 

$||$ 

( A ) $1/2u_{m}(t)||^{2}$ $(¥mathrm{A}^{1/2}u_{m}(t), ¥mathrm{A}^{1/2}u_{m}^{J}(t))$ $||$ $Au_{m}^{¥prime}(t)||^{2}-2(¥mathrm{A}u_{m}(t)$ 

( , Au ) 

$m’’(t))$ 

(3.22) 

$¥frac{d}{dt}[||¥mathrm{A}^{1/2}u_{m}^{¥prime¥prime}(t)||^{2}+a(||¥mathrm{A}^{1/2}u_{m}(t)||^{2})||¥mathrm{A}u_{m}^{¥prime}(t)||^{2}]+||¥mathrm{A}u_{m}^{¥prime J}(t)||^{2}$ 

$¥leq C$ 

( 

$||$ 

$’$ 

$Au_{m}^{¥prime}(t)||^{2}+||$ 

A ) 

$1/2u_{m}(t)||^{2}$ 

by (3.7), (3.10) and (3. 13). Integrating (3.22) over $[0, t]$ leads to 

(3.23) 

$||$ 

$¥mathrm{A}^{1/2}u_{m}^{¥prime¥prime}(t)||¥leq C$ 

and 

(3.24) 

$¥int_{0}^{¥infty}||$ 

$Au_{m}^{¥prime¥prime}(¥tau)||^{2}d¥tau¥leq C$. 

Here 

$||¥mathrm{A}^{1/2}u_{m}^{¥prime¥prime}(0)||^{2}¥leq 

C$ 

follows from (3.1) and (H5). 

(IX) 

(3.21) with (3.5), (3.7), (3. [0, (3. 14) and (3.24) yields 

(3.25) 

$¥int_{0}^{¥infty}||u_{m}^{¥prime¥prime¥prime}(¥tau)||^{2}d_{T}¥leq 

C$ 

. 

4. Proof of Theorem 1. 

By virtue of a priori estimates (3.10), (3.13), (3. 14) and (3.25), 

of with the properties 

$¥{u_{m}¥}$ 

subsequence 

$¥{u_{¥mu}¥}$ 

we may extract a

u^{¥prime}$ 

in 

u^{¥prime¥prime}$ 

in 


(4. 1) 

$u_{¥mu}¥rightarrow 

$u_{¥mu}^{¥prime}¥rightarrow 

(4.2) 

$u_{¥mu}^{¥prime¥prime}¥rightarrow 

(4.3) 

u$ in 

$¥mathrm{w}¥mathrm{e}¥mathrm{a}¥mathrm{k}1¥mathrm{y}^{*}$ 

$L^{¥infty}(0, ¥infty; D(¥mathrm{A}))$ 

$L^{¥infty}(0, ¥infty; D(¥mathrm{A}^{1/2}))¥cap L^{2}(0, ¥infty;D(¥mathrm{A}))$ 


$L^{2}(0, ¥infty; H)$ weakly 

and 

(4.4) 

$a(||¥mathrm{A}^{1/2}u_{¥mu}(¥cdot)||^{2})¥mathrm{A}u_{¥mu}$ 

for some $u$ and 

$¥chi$ 

. 

? 

in 

$L^{¥infty}(0, ¥infty; H)$ $¥mathrm{w}¥mathrm{e}¥mathrm{a}¥mathrm{k}1¥mathrm{y}^{*}$ 

We must show $¥chi_{=a}(||¥mathrm{A}^{1/2}u(¥cdot)||^{2})¥mathrm{A}u$ . For any $¥phi¥in C_{0}(0, ¥infty; H)$ and $T>0$ we 

have 

$¥int_{0}^{T}(¥chi-a(||¥mathrm{A}^{1/2}u(¥tau)||^{2})¥mathrm{A}u(¥tau), ¥phi)dt=¥int_{0}^{T}(¥chi-a(||¥mathrm{A}^{1/2}u_{¥mu}(¥tau)||^{2})¥mathrm{A}u_{¥mu}(¥tau), ¥phi)d¥tau$ 

(4. 5) 

$+¥int_{0}^{T}a$ 

( 

$||$ 

A $1/2u(¥tau)||^{2}$ )($¥mathrm{A}u_{¥mu}(¥tau)-$ 

Au(), 

$¥phi$) 

$ d¥tau$ 

$+¥int_{0}^{T}$ 

$(a(||¥mathrm{A}^{1/2}u_{¥mu}(¥tau)||^{2})-a(||¥mathrm{A}^{1/2}u(¥tau)||^{2}))(¥mathrm{A}u_{¥mu}(¥tau), ¥phi)d¥tau$. 

But, by the mean value theorem and by (4. 1) and (H2), 

$|$ 

the last term in $(4.5)|$ 

$¥leq C¥int_{0}^{T}a_{2}|$ 

($¥mathrm{A}^{1/2}u_{¥mu}(¥tau)-¥mathrm{A}^{1/2}u(¥tau)$ 

, 

A $1/2u_{¥mu}(¥tau)+¥mathrm{A}^{1/2}u(¥tau)$ ) $|d¥tau$ 

$¥leq C¥int_{0}^{T}||¥mathrm{A}^{1/2}u_{¥mu}(¥tau)-¥mathrm{A}^{1/2}u(¥tau)||d¥tau¥rightarrow 0$ 

as $¥mu¥rightarrow¥infty$ 

. 

The other integrals in right-hand side in (4.5) also tend to zero. The arbitrariness 

$¥phi$ 

of and $T$ implies . Hence we get (2.1) for almost all $¥chi_{=a}(||¥mathrm{A}^{1/2}u(¥cdot)||^{2})¥mathrm{A}u$ $t$ 

. The 

initial conditions (2.3) follow from (4.3) and the next Lemma 1. 

Lemma 1 (Ball [2]). Let $X$ be a Banach space. If $f¥in L^{2}(0, T;X)$ and 

$f^{¥prime}¥in L^{2}$ 

$(0, T;X)$ , then $f$ , possibly after redefinition on a set of measure zero, is continuous from 

$[0, T]$ to $X$ . 

Thus the existence part of Theorem 1 is completed. 

In order to prove the uniqueness part, let and be two solutions of $u$ $v$ $(2.1)-$ 

$(2.3)$ . Then $w¥equiv u-v$ satisfies 

(4. 6) 

and 

$w^{¥prime¥prime}(t)+a(||¥mathrm{A}^{1/2}u(t)||^{2})¥mathrm{A}w(t)+¥mathrm{A}w^{¥prime}(t)$ 

$=-$ [ $a(||¥mathrm{A}^{1/2}u(t)||^{2})-a(||$ A $1/2u(t)||^{2})$ ] 

$¥mathrm{A}v(t)$


(4.7) $w(0)=0$, $w^{¥prime}(0)=0$ . 

Applying , $2w^{¥prime}(t)+¥mathcal{T}w(t)$ $0


and 

(5.4) 

$u^{¥prime¥prime¥prime}¥in L^{2}(0, ¥infty;H)$ 

by (3.16)?(3.19) and (3.22)?(3.24). Combining Lemma 1 and $(5.1)-(54)$ implies 

$u¥in C^{2}(.[0, ¥infty);H)$ . 

Q.E.D. 

Proof of Theorem 3. Putting $w=¥mathrm{A}^{2k-1}u_{m}(t)$ , $k=2,3$ , 

$¥cdots$ 

, in (3.1) and integrating 

it from 0 to 

$t$, 

(5. 5) 

$||$ 

and 

we obtain 

A $ku_{m}(t)||¥leq C$ 

(5.6) 

$¥int_{0}^{¥infty}a(||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2})||$A 

$ku_{m}(¥tau)||^{2}d_{T}¥leq C$ 

by the similar way to (III). 

(5.7) 

$||$ 

and 

Then, similarly to (IV), we get 

A 

$k-1/2u_{m}^{¥prime}(t)||^{2}¥leq C$ 

(5.8) 

$¥int_{0}^{¥infty}||$ 

A 

$ku_{m}^{¥prime}(¥tau)||^{2}d¥tau¥leq C$ , 

using (5.6). Hence Theorem 3 follows from (5.5), (5.7) and (5.8) Q.E.D. 

6. Proof of Theorem 4. 

From Theorem 2 it suffices to obtain the apriori estimates $(2.5)-(2.9)$ to $u_{m}(t)_{>}$ 

which satisfies 

(6.1) 

$(u_{m}^{¥prime¥prime}(t)+||¥mathrm{A}^{1/2}u_{m}(t)||^{2}Aum(t)+Au_{m}^{J}(t), w)=0$ 

for any 

$w¥in V^{m}$ 

together with (3.2). Since $a(s)=s$, (3.5) and (3.8) give 

(6.2) 

$¥int_{0}^{t}||$ 

$¥mathrm{A}^{1/2}u^{J}(¥tau)||^{2}d¥tau¥leq C$ 

and 

(6.3) 

$¥int_{0}^{t}||¥mathrm{A}^{1/2}u(¥tau)||^{4}d¥tau¥leq 

C$ 

. 

(From now on we abbreviate the suffix $m$ 

for simplicity.) 

Multiplying (3.3) by $(1+t)$ and integrating it over $[0, t]$ , we have


$¥frac{d}{dt}[(1+t)(||u^{¥prime}(t)||^{2}+¥frac{1}{2}||¥mathrm{A}^{1/2}u(t)||^{4})]+2(1+t)||¥mathrm{A}^{1/2}u^{¥prime}(t)||^{2}$ 

$=||u^{j}(t)||^{2}+¥frac{1}{2}||¥mathrm{A}^{1/2}u(t)||^{4}$ 

and 

(6. 4) 

$(1+t)(||u^{¥prime}(t)||^{2}+¥frac{1}{2}||¥mathrm{A}^{1/2}u(t)||^{4})+2¥int_{0}^{t}(1+¥tau)||¥mathrm{A}^{1/2}u^{¥prime}(¥tau)||^{2}d¥tau$ 

$¥leq||u_{1}||^{2}+¥frac{1}{2}||u_{0}||^{4}+¥int_{0}^{t}(||u^{¥prime}(¥tau)||^{2}+¥frac{1}{2}||¥mathrm{A}^{1/2}u(¥tau)||^{4})d¥tau$ 

. 

By virtue of (6.2), (6.3) and (H5), (6.4) gives 

(6.3) 

$(1+t)||u^{¥prime}(t)||^{2}¥leq C$ 

(6.6) 

$(1+t)||¥mathrm{A}^{1/2}u(t)||^{4}¥leq C$ 

and 

(6.7) 

$¥int_{0}^{t}(1+¥tau)||¥mathrm{A}^{1/2}u^{¥prime}(¥tau)||^{2}d¥tau¥leqq C$. 

Multiplying (3.6) by $(1+t)^{1/2-¥epsilon}$ for small positive constant 

$¥epsilon$ 

, we have 

$(1+t)^{1/2-¥epsilon}||¥mathrm{A}^{1/2}u(t)||^{4}+¥frac{1}{2}¥frac{d}{dt}(1+t)^{1/2-¥epsilon}||$A $1/2u(t)||^{2}$ 

$=¥frac{1}{2}(1/2-¥epsilon)(1+t)^{-1/2-¥epsilon}||¥mathrm{A}^{1/2}u(t)||^{2}+(1+t)^{1/2-¥epsilon}||u^{¥prime}(t)||^{2}$ 

$-¥frac{d}{dt}(1+t)^{1/2-8}(u^{¥prime}(t), u(t))+(1/2-¥epsilon)(1+t)^{-1/2-¥epsilon}(u^{¥prime}(t), u(t))$ 

which yields, by integration over $[0, t]$ and by (6.2)?(6.3), $(6.5)-(6.7)$ , 

$¥int_{0}^{t}(1+¥tau)^{1/2-¥epsilon}||¥mathrm{A}^{1/2}u(¥tau)||^{4}d¥tau+¥frac{1}{2}(1+t)^{1/2-¥epsilon}||¥mathrm{A}^{1/2}u(t)||^{2}$ 

$¥leq¥frac{1}{2}||¥mathrm{A}^{1/2}u_{0}||^{2}+||u_{1}||||u_{0}||+(1+t)^{1/2-¥epsilon}||u^{¥prime}(t)||||u(t)||+¥frac{1}{4}¥int_{0}^{t}(1+¥tau)^{-1/2-¥epsilon}||¥mathrm{A}^{1/2}u(¥tau)||^{2}d¥tau$ 

$+¥int_{0}^{t}(1+¥tau)^{1/2-¥epsilon}||u^{¥prime}(¥tau)||^{2}d¥tau+¥frac{1}{2}¥int_{0}^{t}(1+¥tau)^{-1/2-¥epsilon}||u^{¥prime}(¥tau)||||u(¥tau)||d¥tau$ 

$¥leq C+¥{(1+t)||u^{¥prime}(t)||^{2}¥}^{1/2}||u(t)||+¥frac{1}{¥lambda_{1}^{2}}¥int_{0}^{t}(1+¥tau)||$A $1/2u(¥tau)||^{2}d¥tau’$ 

$+¥frac{1}{4}(¥int_{0}^{¥infty}(1+¥tau)^{-1-2¥epsilon}d¥tau)^{1/2}(¥int_{0}^{¥infty}||¥mathrm{A}^{1/2}u(¥tau)||^{4}d¥tau)^{1/2}$ 

$+¥frac{1}{2¥lambda_{1}}¥max||u(¥tau)||¥cdot(¥int_{0}^{¥infty}(1+¥tau)^{-1-2¥epsilon}d¥tau)^{1/2}(¥int_{0}^{¥infty}||¥mathrm{A}^{1/2}u^{¥prime}(¥tau)||^{2}d¥tau)^{1/2}¥leq C_{2¥epsilon}$ .

}¥mathrm{i}¥mathrm{v}¥mathrm{e}¥mathrm{s}$ 

$(2.5)$ 

}t]$ 

, 

Here we denote by 

$C_{¥epsilon}$ 

Degenerate Quasilinear Hyperbolic Equation 13” 

the constant depending only upon 

$¥int_{0}^{¥infty}(1+¥tau)^{-1-¥epsilon}d¥tau$ 

. 

Hence we obtain 

(6. 8) 

$¥int_{0}^{t}(1+¥tau)^{1/2-¥epsilon}||$ 

A $1/2u(¥tau)||^{4}d¥tau¥leq C_{2¥epsilon}$ . 

By induction we shall show 

$¥langle$6.9) 

$¥int_{0}^{t}(1+¥tau)^{1-¥epsilon}||¥mathrm{A}^{1/2}u(¥tau)||^{4}d¥tau¥leq 

C_{¥epsilon/2}$ 

(6. 10) 

$(1+t)^{1-¥epsilon}||¥mathrm{A}^{1/2}u(t)||^{2}¥leq C_{¥epsilon/2}$, 

the latter of which 

$¥mathrm{g}_{¥backslash 

. For $N¥geq 1$ we assume 

$(6.11)_{N}$ 

$¥int_{0}^{t}(1+¥tau)^{1-2^{-N}-¥epsilon/2}||¥mathrm{A}^{1/2}u(¥tau)||^{4}d¥tau¥leq C_{¥epsilon/2}$ 

$(6.12)_{N}$ $(1+t)^{1- 2-¥epsilon/2}-N||$ $¥mathrm{A}^{1/2}u(t)||^{2}¥leq C_{¥epsilon/2}$ 

. 

Note that and are valid from (6.8) and (6.6). Multiplying (3.6) by 

$(6.11)_{1}$ $(6.12)_{1}$ 

$(1+t)^{1-2-¥epsilon/2}-N-1$ and integrating it over 

$[¥mathrm{O}_{¥backslash 

we get 

$(1+t)^{1-2-¥epsilon/2}-N-1||¥mathrm{A}^{1/2}u(t)||^{4}+¥frac{1}{2}¥frac{d}{dt}(1+t)^{1- 2-¥epsilon/2}-N-1||¥mathrm{A}^{1/2}u(t)||^{2}$ 

$=¥frac{1}{2}(1-2^{-N-1}-¥epsilon/2)(1+t)^{-2-¥epsilon/2}-N-1||¥mathrm{A}^{1/2}u(t)||^{2}$ 

$+(1+t)^{1-2-¥epsilon/2}-N-1||u^{¥prime}(t)||^{2}-¥frac{d}{dt}(1+t)^{1-2-¥epsilon/}-N-12(u^{¥prime}(t), u(t))$ 

and 

$+(1-2^{-N-1}-¥epsilon/2)(1+t)^{-2-¥epsilon/2}-N-1(u^{¥prime}(t), u(t))$ 

$¥int_{0}^{t}(1+¥tau)^{1-N-1}-2-¥epsilon/2||$ A $1/2u(¥tau)||^{4}d_{T}+¥frac{1}{2}(1+t)^{1-N-1}-2-¥epsilon/2||$ A $1/2u(t)||^{¥mathrm{z}}$ 

$¥leq¥frac{1}{2}||¥mathrm{A}^{1/2}u_{0}||^{2}+||u_{1}||||u_{0}||+¥frac{1}{2}(¥int_{0}^{¥infty}(1+¥tau)^{-1-¥epsilon/2}d¥tau)^{1/2}$ 

$¥times(¥int_{0}^{¥infty}(1+¥tau)^{-1- 2-¥epsilon/2}-N||¥mathrm{A}^{1/2}u(¥tau)||^{4}d¥tau)^{1/2}+¥frac{1}{¥lambda_{1}^{2}}¥int_{0}^{¥infty}(1+¥tau)||¥mathrm{A}^{1/2}u^{¥prime}(¥tau)||^{2}d¥tau$ 

$+¥frac{1}{¥lambda_{1}}¥{(1+t)||u^{¥prime}(t)||^{2}¥}^{1/2}¥{(1+t)^{1-2-¥epsilon/2}-N||¥mathrm{A}^{1/2}u(t)||^{2}¥}^{1/2}$ 

$+¥frac{1}{¥lambda_{1}}¥sup_{¥tau}||u(¥tau)||¥cdot(¥int_{0}^{¥infty}(1+¥tau)^{-1-¥epsilon}d¥tau)^{1/2}(¥int_{0}^{¥infty}(1+¥tau)||¥mathrm{A}^{1/2}u^{¥prime}(¥tau)||^{2}d¥tau)^{1/2}¥leq C_{¥epsilon/2}$.


This shows $(6.11)_{N+1}$ and $(6.12)_{N+1}$ . Thus we obtain (6.9) and (6. 10) by selecting $N$ 

$2^{-N}

$||$ 

Au 


(6. 16) 

$(1+t)^{3/2-¥epsilon}||$ A $1/2u(t)||¥leq C_{¥epsilon/12}’$ . 

(6. 16) gives (2.7). 

In (3.21), putting $w=2(1+t)^{4-¥epsilon}u^{¥prime¥prime}(t)$ and integrating it from 0 to 

$t$ 

, we obtain 

$¥frac{d}{dt}$ 

[ A ] $(1+t)^{4-¥epsilon}(||u^{¥prime¥prime}(t)||^{2}+||$ $1/2u(t)||^{2}||¥mathrm{A}^{1/2}u^{¥prime}(t)||^{2})$ $+2(1+t)^{4-¥epsilon}||$ A $l/2u(t)||^{2}’’$ 

$=(4-¥epsilon)(1+t)^{3-6}(||u^{¥prime¥prime}(t)||^{2}+||¥mathrm{A}^{1/2}u(t)||^{2}||¥mathrm{A}^{1/2}u^{¥prime}(t)||^{2})$ 

$+2(1+t)^{4-¥epsilon}$ (Au(t), 

$u^{¥prime}(t)$ 

$(|| ¥mathrm{A}^{1/2}u^{¥prime}(t)||^{2}-2(¥mathrm{A}u(t), u^{¥prime¥prime}(t)))$ 

) 

and 

$(1+t)^{4-¥epsilon}[||u^{¥prime¥prime}(t)||^{2}+||¥mathrm{A}^{1/2}u(t)||^{2}||¥mathrm{A}^{1/2}u^{¥prime}(t)||^{2}]+2¥int_{0}^{t}(1+¥tau)^{4-¥epsilon}||¥mathrm{A}^{1/2}u^{¥prime¥prime}(¥tau)||^{2}d¥tau$ 

$¥leq||u^{¥prime¥prime}(0)||^{2}+||¥mathrm{A}^{1/2}u_{0}||^{2}||¥mathrm{A}^{1/2}u_{1}||^{2}+4¥int_{0}^{¥infty}(1+¥tau)^{3-¥epsilon}||u^{¥prime¥prime}(¥tau)||^{2}d¥tau$ 

$+4¥sup_{¥tau}$ { $(1+¥tau)^{1-¥epsilon/2}||$ A $1/2u(¥tau)||^{2}$ } 

$¥int_{0}^{¥infty}(1+¥tau)^{2-¥epsilon/2}||$ 

A $1/2u(¥tau)||^{2}d¥tau’$ 

$+2[¥sup_{¥tau}¥{(1+¥tau)^{1-2¥epsilon/3}||¥mathrm{A}^{1/2}u^{¥prime}(¥tau)||^{2}¥}]^{1/2}¥sup_{¥tau}${ 

A } 

$(1+¥tau)^{3/2-¥epsilon/3}||$ $1/2u(¥tau)||$ 

$’$ 

$¥times¥int_{0}^{¥infty}(1+¥tau)^{2-¥epsilon/3}||$ 

A $1/2u(¥tau)||^{2}d¥tau+4¥sup_{¥tau}${ 

$(1’+¥tau)^{1-¥epsilon/6}||$ A } 

$1/2u(¥tau)||^{2}$ 

$¥times¥sup_{¥tau}¥{(1+¥tau)^{3/2-¥epsilon/6}||¥mathrm{A}^{1/2}u^{¥prime}(¥tau)||¥}(¥int_{0}^{¥infty}(1+¥tau)^{-1-¥epsilon¥prime 3}d¥tau)^{1/2}$ 

$¥times(¥int_{0}^{t}(1+¥tau)^{4-¥epsilon}||¥mathrm{A}^{1/2}u^{¥prime¥prime}(¥tau)||^{2}d¥tau)^{1/2}$ 

$¥leq C_{¥epsilon/9}+C_{¥epsilon/18}(¥int_{0}^{t}(1+¥tau)^{4-¥epsilon}||¥mathrm{A}^{1/2}u^{¥prime¥prime}(¥tau)||^{2}d¥tau)^{1/2}$ 

which gives (2.8), (2.9). 

Finally we show (2.6). 

In fact, by the Cauchy-Schwarz inequality 

$||^{4}=(¥sum¥lambda_{j}^{4}g_{jm}^{2})^{2}¥leq(¥sum¥lambda_{j}^{2}g_{jm}^{2})(¥sum¥lambda_{j}^{6}g_{jm}^{2})=||$ A $1/2u||^{2}||¥mathrm{A}^{3/2}u||^{2}$ 

and hence, by (6.9) and (3.17), 

$(1+t)^{1-¥epsilon}||u(t)$ $||^{4}¥leq(1+t)^{1-¥epsilon}||¥mathrm{A}^{1/2}u(t)||^{2}¥cdot||$ $¥mathrm{A}^{3/2}u(t)||^{2}¥leq C_{¥epsilon/2}$ 

. 

Thus the proof of Theorem 4 is completed. 

$¥mathrm{Q}.¥mathrm{E}$.D. 

7. Periodic solution. 

In this section we seek the $¥omega$-periodic solution $u(t)$ to the equation 

(7. 1) 

$u^{¥prime¥prime}(t)+||¥mathrm{A}^{1/2}u(t1||^{2}Au(t)+Au^{¥prime}(t)=f(t)$ in $H$

degree 

, 


where : $f$ $(-¥infty, ¥infty)¥rightarrow H$ 

is an $¥omega$-periodic function. When $X$ is a Banach space with 

$||¥cdot||_{X}$ 

norm , $L^{p}(¥omega; X)$ means the space of functions $f(t)¥in X$ 

$ t¥in$ 

for each $(-¥infty, ¥infty)$ 

and periodic, with period 

$¥omega$ 

, such that , equipped with norm 

$¥int_{0}^{¥omega}||f(t)||_{X}^{p}dt

$¥epsilon||$ 


(7.5) 

$¥int_{¥omega}||¥mathrm{A}^{1/2}u_{m}^{J}(¥tau)||^{2}d¥tau¥leq¥frac{1}{¥lambda_{1}^{2}}¥int_{¥omega}||f(¥tau)||^{2}d¥tau¥equiv 

C_{3}$ 

by integration of (7.4) over $[t, t+¥omega]$ $¥int_{t}^{t+¥omega}=¥int_{¥omega}$ 

and (7.3), where we denote for 

simplicity. (In this section several constants $C$ $C_{i}$ $t$ 

and are independent of , $m$ and 

$¥epsilon.)$ 

If we put $w=u_{m}$ in (7.2), then we have 

$¥frac{d}{dt}(u_{m}^{¥prime}(t), u_{m}(t))-||u_{m}^{¥prime}(t)||^{2}+¥epsilon||¥mathrm{A}^{1/2}u_{m}(t)||^{2}+||¥mathrm{A}^{1/2}u_{m}(t)||^{4}$ 

$+¥frac{1}{2}¥frac{d}{dt}||¥mathrm{A}^{1/2}u_{m}(t)||^{2}=(f(t), u_{m}(t))$ 

and 

$¥int_{¥omega}||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{4}d¥tau¥leq C_{3}/¥lambda_{1}^{2}+¥frac{1}{2¥lambda_{1}}¥int_{¥omega}||f(¥tau)||^{2}d¥tau+¥frac{¥vee¥overline{¥omega}}{2¥lambda_{1}}(¥int_{¥omega}||$ 

A $1/2l_{m}^{¥prime}(¥tau)||^{4}d¥tau)^{1/2}$ 

and hence 

(7.6) 

$¥int_{¥omega}||$ 

A 

$1/2u_{m}(¥tau)||^{4}d¥tau¥leq C$ . 

By virtue of (7.5) and (7.6) there exists 

$t_{0}¥in[0, ¥omega]$ 

such 

that 

$||u_{m}^{J}(t_{0})||$ 

, 

$||$ 

$¥mathrm{A}^{1/2}u_{m}(t_{0})||¥leq C$ 

from which and (7.4) it follows that, for $t¥in[t_{0}, t_{0}+¥omega]$ , 

$||u_{m}^{¥prime}(t)||^{2}+¥epsilon||¥mathrm{A}^{1/2}u_{m}(t)||^{2}+¥frac{1}{2}||$ 

A $ 1/2u_{m}(t)||^{4}+¥int_{t¥mathrm{o}}^{t}||¥mathrm{A}^{1/2}u_{m}^{J}(¥tau)||^{2}d¥tau$ 

$¥leq||u_{m}^{¥prime}(t_{0})||^{2}+¥epsilon||¥mathrm{A}^{1/2}u_{m}(t_{0})||^{2}+¥frac{1}{2}||¥mathrm{A}^{1/2}u_{m}(t_{0})||^{4}+¥frac{1}{¥lambda_{1}^{2}}¥int_{t¥mathrm{o}}^{t}||f(¥tau)||^{2}d¥tau$ , 

that is, 

(7.7) 

$||u_{m}^{j}(t)||¥leq C$ 

and 

$||¥mathrm{A}^{1/2}u_{m}(t)||^{2}¥leq C_{4}$ 

. 

Taking $w=Au_{m}$ in (7.2) and integrating it over $[t, t+¥omega]$ , 

we have 

(7. 8) 

Au$m(t)||^{2}+||$ A $1/2u_{m}(t)||^{2}||$ Au 

$m(f)||^{2}+¥frac{1}{2}¥frac{d}{dt}||$ $Au_{m}(t)||^{2}$ 

$=(¥mathrm{A}^{1/2}f(t), ¥mathrm{A}^{1/2}u_{m}(t))+||¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||^{2}-¥frac{d}{dt}(¥mathrm{A}^{1/2}u_{m}^{¥prime}(t), ¥mathrm{A}^{1/2}u_{m}(t))$ 

and

Au 

to 


(7.9) 

$¥int_{¥omega}||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2}||$ 

Au 

$m(¥tau)||^{2}d¥tau¥leq¥int_{¥omega}||¥mathrm{A}^{1/2}u_{m}^{J}(¥tau)||^{2}d_{T}+¥int_{¥omega}(¥mathrm{A}^{1/2}f(¥tau), ¥mathrm{A}^{1/2}u_{m}(¥tau))d¥tau¥leq C$ 

by (7.5) and (7.6). Putting 

$¥mathrm{w}=2¥mathrm{A}¥mathrm{u}_{m}^{J}$ 

in 

(7.2) we have 

$¥frac{d}{dt}$ $||¥mathrm{A}^{1/2}u_{m}^{J}(t)||^{2}+¥epsilon||$ 

( Au ) $m(t)||^{2}$ $Au_{m}^{¥prime}(t)||^{2}$ 

$+2||$ 

$=2$ ( $f(t)$ , Au $m’(f)$ ) $-2||¥mathrm{A}^{1/2}u_{m}(t)||^{2}$ (Au $m(t)$ , Au $m’(f)$ ) 

$¥leq||¥mathrm{A}^{1/2}f(t)||^{2}+||$ 

A $1/2u_{m}^{;}(t)||^{2}+||¥mathrm{A}^{1/2}u_{m}(t)||^{2}(C_{4}||$ 

Au $m(t)||^{2}+¥frac{1}{C_{4}}||Au_{m}^{¥prime}(t)||^{2})$ 

and 

(7. 10) 

$¥frac{d}{dt}$ $||¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||^{2}+¥epsilon||$ 

( Au ) $m(f)||^{2}$ $+||$ $Au_{m}^{J}(t)||^{2}$ 

$¥leq||¥mathrm{A}^{1/2}f(t)||^{2}+||$ 

A $1/2u_{m}^{¥prime}(t)||^{2}+C_{4}||¥mathrm{A}^{1/2}u_{m}(t)||^{2}||$ Au $m(f)||^{2}$ . 

Then, integrating (7. 10) over 

$[t, t+¥omega]$ and 

using (7.7), (7.9), we have 

(7. 11) 


$Au_{m}^{J}(¥tau)||^{2}d¥tau¥leq C$ 

$||$ 

which implies for some . Integrating (7.10) from $mJ(t_{1})||¥leq C$ $t_{1}¥in[0, ¥omega]$ $t_{1}$ 

$[t_{1}, t_{1}+¥omega]$ 

, we obtain 

$t$ 

$ t¥in$ 

, 

(7. 12) 

$||$ 

Putting 

$¥mathrm{w}=¥mathrm{A}^{2}¥mathrm{u}_{m}$ 

in 

$¥mathrm{A}^{1/2}u_{m}^{¥prime}(t)||¥leq C$ 

. 

(7.2) and integrating it over $[t, t+¥omega]$ , we get 

$¥epsilon¥int_{¥omega}||¥mathrm{A}^{3/2}u_{m}(¥tau)||^{2}d¥tau+¥int_{¥omega}||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2}||$A $ 3/2u_{m}(¥tau)||^{2}d¥tau$ 

(7. 13) 

$¥leq¥frac{1}{2}¥int_{¥omega}||¥mathrm{A}f(¥tau)||^{2}d_{T}+¥frac{1}{2}¥int_{¥omega}||$ 

A $ll_{m}(¥tau)||^{2}d¥tau+¥int_{¥omega}||$ $ Au_{m}^{¥prime}(¥tau)||^{2}d¥tau$ . 

By virtue of the Cauchy-Schwarz inequality it holds 

Hence 

$||$ 

Au . 

$m||^{2}¥leq||¥mathrm{A}^{1/2}u_{m}||||¥mathrm{A}^{3/2}u_{m}||$ 


$Au_{m}(¥tau)||^{4}d¥tau¥leq¥frac{1}{2}¥int_{¥omega}||¥mathrm{A}f(¥tau)||^{2}d¥tau+¥frac{¥vee¥overline{¥omega}}{2}(¥int_{¥omega}||$ $Au_{m}(¥tau)||^{4}d¥tau)^{1/2}+¥int_{¥omega}||$ $ Au_{m}^{¥prime}(¥tau)||^{2}d¥tau$ 

and 

(7. 14) 


Au 

$m(¥tau)||^{4}d¥tau¥leq C$ .

Au 

Au 

It also follows from (7. 13) that 


(7. 15) 

$¥int_{¥omega}¥epsilon||¥mathrm{A}^{3/2}u_{m}(¥tau)||^{2}d¥tau+¥int_{¥omega}||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2}||$ 

$¥mathrm{A}^{¥mathrm{S}/2}u_{m}(¥tau)||^{2}d¥tau¥leq C$. 

From (7.14) there exists such that $t_{2}¥in[0, ¥omega]$ $||$ 

$Au_{m}(t_{2})||¥leq C$ . If we consider (7.8) as 

$||$ 

the first order differential equation in with the initial value $m(f)||^{2}$ $||$ 

$Au_{m}(t_{2})||^{2}$ 

at 

$t$ 

$t=t_{2}$ $ t_{2}¥leq t¥leq t_{2}+¥omega$ 

, then we have for , , 

$||$ 

$Au_{m}(t)||^{2}=e^{-P(t)}$ [ 

$||$ $m(t_{¥mathrm{z}})||^{2}-2(¥mathrm{A}^{1/2}u_{m}^{¥prime}(t_{2}),$ $¥mathrm{A}^{1/2}u_{m}(t_{2}))$ 

Au ] 

$+2(¥mathrm{A}^{1/2}u_{m}^{J}(t), ¥mathrm{A}^{1/2}u_{m}(t))+2e^{-P(b)}¥int_{t_{2}}^{t}e^{P(¥tau)}¥{||$ 

A $1/2u_{m}(¥tau)||^{2}’$ 

$(¥mathrm{A}^{1/2}u_{m}^{¥prime}(¥tau), ¥mathrm{A}^{1/2}u_{m}(¥tau))||$ 

?2 A 

$¥iota/2u_{m}(¥tau)||^{2}+(¥mathrm{A}^{1/2}f(¥tau), ¥mathrm{A}^{1/2}u_{m}(¥tau))¥}d¥tau$ 

where $P(t)=2¥int_{t_{2}}^{t}$ $(¥epsilon+||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2})d¥tau$ . The estimates obtained above gives 

(7. 16) 

$||$ 

Next, if we replace $w=2¥mathrm{A}^{2}u_{m}^{J}$ 

Au$m(t)||¥leq C$. 

in (7.2), then we have 

(7. 17) 

$¥frac{d}{dt}(|| Au_{m}^{¥prime}(t)||^{2}+¥epsilon||¥mathrm{A}^{3/2}u_{m}(t)||^{2})+2||¥mathrm{A}^{3/2}u_{m}^{J}(t)||^{2}$ 

$=2$ 

($¥mathrm{A}f(t)$ 

, Au $m’(t)$ ) 

$-2||¥mathrm{A}^{1/2}u_{m}(t)||^{2}(¥mathrm{A}^{3/2}u_{m}(t), ¥mathrm{A}^{3/2}u_{m}^{¥prime}(t))$ 

and 

2 

$¥int_{¥omega}||¥mathrm{A}^{3/2}u_{m}^{J}(¥tau)||^{2}d_{T}¥leq 

C_{4}¥int_{¥omega}||¥mathrm{A}^{1/2}u_{m}(¥tau)||^{2}||¥mathrm{A}^{3/2}u_{m}(¥tau)||^{2}d_{T}+¥int_{¥omega}(||¥mathrm{A}f(¥tau)||^{2}+||Au_{m}^{¥prime}(¥tau)||^{2})d¥tau$ 

. 

By (7. 15) 

$¥int_{¥omega}||¥mathrm{A}^{3/2}u_{m}^{¥prime}(¥tau)||^{2}d¥tau¥leq C$ 

which means with (7.15) that there is some such that $t_{3}¥in[0, ¥omega]$ $¥epsilon||$ 

$||¥mathrm{A}^{3/2}u_{m}^{J}(t_{3})||¥leq C$ 

. Then, integrating (7. 17) over , $[t_{3}, t]$ $t¥in[t_{3}, t_{3}+¥omega]$ 

, we have 

$¥mathrm{A}^{3/2}u_{m}(t_{3})||+$ 

(7. 18) 

$||$ 

(7. 19) 

$||u_{m}^{¥prime¥prime}(t)||¥leq 

Finally putting $w=u_{m}^{JJ}$ 

$¥prime m(t)||¥leq C$ . 

in (7.2) then we have 

C$ 

since . 

$f¥in L^{¥infty}$ $(¥omega; H)$ $.¥mathrm{n}¥mathrm{d}(7.19)$ 

We now pass $u_{m,¥epsilon}$ to the limit. From (7.7), (7. 12), (7. 16), (7. 18) a we 

$¥{u_{¥nu,¥delta}¥}$ 

may extract a subsequence of such that for some and $u$ $¥chi$ 

$¥{u_{m,¥epsilon}¥}$

$u_{¥nu,¥delta}^{J}$? 

$u_{¥nu,¥delta}^{¥prime¥prime}$? 

$u^{¥prime}$ 

in 

$u^{¥prime¥prime}$ 

in 


$u_{¥nu,¥delta}$ 

? 

$u$ 

in 

$L^{¥infty}(¥omega; D(¥mathrm{A}))$ 


$L^{¥infty}(¥omega;D(¥mathrm{A}))$ 


$L^{¥infty}(¥omega;H)$ 


and 

$||¥mathrm{A}^{1/2}u_{¥nu,¥delta}(¥cdot)||^{2}Au_{¥nu,¥delta}$ ? $¥chi$ in 

$L^{¥infty}(¥omega;H)$ 


$¥delta¥rightarrow 0$ 

$ p¥rightarrow¥infty$ $¥chi_{=}||¥mathrm{A}^{1/2}u(¥cdot)||^{2}Au$ 

as and . is shown by the compact method as in the 

proof of Theorem 1. Hence this is the solution to (7. 1). 

$u$ 

Thus the proof is completed 

Q.E.D. 

We shall state the regularity theorem without proof. 

In addition to the assumptions in Theorem 5, is as- 

$f^{¥prime}¥in L^{¥infty}(¥omega; H)$ 

Then the solution $u(t)$ satisfies 

sumed. 

Theorem 

$¥epsilon$. 

$u¥in C^{1}(¥omega;D(¥mathrm{A}))¥cap C^{2}(¥omega; D(¥mathrm{A}^{1/2}))$ 

. 

Moreover, if we assume $f^{(j)}¥in L^{¥infty}(¥omega;D(¥mathrm{A}^{k}))$ , $j$ , $k=0,1,2$ ,?, then we have 

$u¥in C^{¥infty}(¥omega; D(¥mathrm{A}^{k^{¥prime}}))$ 

, $k^{¥prime}=0,1,2$ ,?. 

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nuna adreso: 

Tokyo National Technical College 

Hachioji, Tokyo 

Japan 

(Ricevita la 23-an de majo, 1983)

Degenerate Quasilinear Hyperbolic Equation with Strong Damping

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