Existence, uniqueness and stabilization for a nonlinear plate system with nonlinear damping

A NNALES DE LA FACULTÉ DES SCIENCES DE T OULOUSE

C ^ARLOS F ^REDERICO V ASCONCELLOS

L UCIA M ARIA T EIXEIRA

Existence, uniqueness and stabilization for a nonlinear plate system with nonlinear damping

Annales de la faculté des sciences de Toulouse 6

^e

série, tome 8, n

^o

1 (1999), p. 173-193

<http://www.numdam.org/item?id=AFST_1999_6_8_1_173_0>

© Université Paul Sabatier, 1999, tous droits réservés.

L’accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions).

Toute utilisation commerciale ou impression systématique est constitu- tive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

- 173 -

Existence, uniqueness ^and stabilization for

a

nonlinear plate system

with nonlinear damping(*)

CARLOS FREDERICO

VASCONCELLOS(1)

and LUCIA MARIA

TEIXEIRA(1)

Annales de la Faculte des Sciences de Toulouse Vol. VIII, ^nO 1, 1999

On etudie 1’existence et 1’unicite des solutions globales ^et

on etablit l’amortissement polynomial ^de l’énergie ^{pour un} systeme ^de plaque ^{non lineaire}

où n est un ouvert borne de N > 3, a frontiere reguliere ^{r et on}

considere la fonction _cp une fois continument différentiable avec cp(s) ^{> 0,}

pour ^{tout s} > 0 et g ^{est continue et non} decroissante.

ABSTRACT. - We study the existence and uniqueness ^of global ^solu-

tions and the polynomial decay ^{of the} energy for the following ^nonlinear plate system with nonlinear damping

where 11 is an open subset of 3, with smooth boundary ^{r and} So is

a continuously differentiable real valued function which satisfies 0, for all s > 0 and g ^a continuous nondecreasing real function.

(*) Recu ^{le 27} janvier 1997, accepté ^{le 1°r} juillet ¹⁹⁹⁸

(1) Instituto de Matemática - UFRJ, ^{PO Box} 68530, ^{CEP 21945-970} Rio de Janeiro RJ (Brazil)

1. Introduction

In this _paper we consider the

following

^nonlinear

plate

system ^with nonlinear

damping:

where H is a bounded _open subset of

IIBN (N 3) having

^{a smooth}

boundary

^{80 =}

F and p

^{is a}

continuously

differentiable real valued

function,

which satisfies

0,

^for

all s >

0. We also

consider g

^{a continuous}

nondecreasing

function and we use the notation

The

following problem

in dimension N ⁼

1,

^{is a}

general

mathematical formulation of ^a

problem arising

^{in the}

dynamic buckling

^{of a}

hinged

extensible beam under an axial force

(Eisley [3]

^and

Dickey [2]).

It is

interesting

^to observe that the

nonlinearity

^{is an}

approximation by averaging

of the classical von Karman model

(Nayfeh-

Mook

[19]).

The existence of solutions for the Problem

(1.2),

ⁱⁿ the one-dimensional

case was studied

by Dickey [2].

^Pohozaev

[20]

considered the existence of solutions of the

corresponding hyperbolic

systems ⁱⁿ

which,

ⁱⁿ

particular,

the fourth order

term 02u

is

dropped.

^In

higher dimensions, regular unique

solutions were obtained

by

^Medeiros

[15].

When the

equation

^of

(1.2)

^{does not include the term}

^02u

^{and the}

dimension N ⁼

1,

^this system describes the vibrations of an elastic stretched

string.

^{This case was studied}

by

^Lions

[14],

^when

)O(s)

^{> mo >} 0 and under the same

assumption,

^local

strong

^{solutions were obtained}

by

^Rivera

[22].

Menzala

[17]

solved this

problem

^{when Q}

Vasconcellos-Teixeira

[25] ^proved

the existence and

uniqueness

^of

global strong

solutions and the

exponential decay

^{of the} energy for the

following

nonlinear

damped hyperbolic problem:

The Problem

(1.3),

without the term

g(u),

^{was solved}

by

^Medeiros-

Miranda

[16]

^{and the}

exponential decay

^{for the} energy ^was

proved.

We would like to make a few comments on the related literature.

Lagnese [8]

considered the

modelling

of nonlinear

plates

and studied the stabilization for these

systems.

^We also refer to the book

by

^J.

Lagnese [9]

^{for a}

^systematic

study

of the stabilization of

plate systems.

For a

study

^on

stability

for nonlinear waves and beams with

damping

^on

the

boundary,

^we

refer,

^for

example,

^{to the works} of Lasiecka

[12],

^Lasiecka-

Triggiani [13]

^and

Lagnese-Leugering [10],

^Kormonik

[7]

and Zuazua

[27].

The von Karman

system

^was

studied,

^for

example, by

Horn-Lasiecka

[6],

where the existence and

uniqueness

^of

global

solutions for the

system

^with nonlinear

boundary dissipation

^was

proved.

^{In Horn}

[5]

and Puel-Tucsnak

[21]

^the

^boundary

stabilization for the

system

^{was showed.}

Recently, Bisognin,

^{et al.}

[1]

and Menzala-Zuazua

[18] ^proved

existence of

global

solutions and the

exponential stability

^{for von Karman}

system

^of

plates,

ⁱⁿ

the _presence of thermal effects.

The aim of this work is to

study

^the

global

existence of solutions and the stabilization of

(1.1).

The existence and

uniqueness

of solutions is

proved

^{in Section}

2,

^{where no} restriction is

placed

^{on the}

growth

^{of the}

damping

term g,

namely, g

^is

only

assumed to be a continuous

nondecreasing

function with

g(0) =

^0.

There,

we use Galerkin methods and

since g

^{does not} have further

restrictions,

we need

special

^{estimates to} obtain the convergence of nonlinear terms and

we must also consider the dimension

N

^{3. This is not} relevant from a

physical point

^{of view since the}

system (1.1)

^{is a} model built for N ⁼ 2.

We can find in Lan et al.

[11]

the existence and

uniqueness

^of

global

solutions for

(1.1),

^{when the}

damping g

^{is a very}

particular function, namely, g(s) =

^{with 0 a 1.}

Therefore, they only

consider nonlinearities

going

^at

infinity

^{in a sublinear way, so, in this case, it is} easy to show that the function _g is continuous from

L2(S2)

^into

L2(O).This

^{is not the case in}

the context we are

working

^on.

In Section

3,

^we

study

the stabilization of

(1.1)

^{when the}

damping g

satisfies convenient

assumptions

^to control its

growth.

Let us consider the _energy

where u is a

regular

solution of

(1.1)

^and

We can prove that

and

therefore,

^the energy is a

decreasing

function of time.

Now,

^to prove the uniform

polynomial decay

^{of the} energy

(1.4)

^we

perturbe

^{it with a} convenient

Lyapunov

^function. We show that the resultant

perturbed

energy is

equivalent

^{to the} energy

(1.4).

^The

origin

of this method can be found in Zuazua

[25] (see

^also

[26]).

Finally,

in Section 4 some remarks about Problem

(1.1)

will be made.

2. The existence theorem

Let H be the

L2(S2)

^space with the usual

norm 1

^.

~.

^{W is} the Sobolev space normed

by

^{= and V means the} space with the norm

Ilullv -

Let g :

^{II8 -> R be a} continuous function such that

We

consider p

^E

C1 ~( 0 , +00); R)

^{such that}

For each T > 0 we put

Q

^{= 52 x}

(0, T).

THEOREM 2.1.- Under the above

assumptions, if (up, ul)

^{E V x Hand}

for

every T >

0,

^{there exists an}

unique

^{u :}

~ 0 , T ~ --r

^{H such that}

Proof.

^- First of all we establish the existence. Let T > 0 be fixed and denote

by Vm

^the space

generated by

w2, ..., where the set

{wm;

^{is a} "basis" of V. Since

(uo, U1)

^{E V x H we obtain}

Then for all mEN there exists

Tm

T and

um(t) = fj(t)wj

^defined

for t E

[0, Tm)

such that the

following

^holds:

where

( ~ I ^{. )}

^denotes the usual inner

product

ⁱⁿ

H,

^A

= -A,

uom ⁼

ajwj, ^{u1m =}

03A3mj=1 bj wj

^{and V’ is the dual} space of V.

Remark 2.1.2014

Since N ~ 3,

^we have that V is

compactly

embedded in

hence, by (2.1),

^{for each t > 0 and}

_{m E N,} g u~ t

^E

L°° S2

^C

H C

L1 (S2).

^{l ~n l}

)) ( )

We need a

priori

^estimates to extend the solution of the system

(2.9)

^to

the whole

interval [0 , T].

Replacing v

ⁱⁿ

(2.9) ^by 2um(t)

^and

integrating

^{from 0 to}

^Tm

^{we have:}

Since and ^are bounded _sequences in H and V

respectively, then, by (2.1)

^and

(2.2)

there exists a constant K > 0

independent

^{of m}

and t,

^such

that,

^{for all m ~ N and t}

>

⁰

Now,

^{we need the}

following

^lemmas.

LEMMA 2.1.- There exists

h’1

^> 0 such that

R’1 for

all mEN.

Proof

^{- If we define}

then,

^from

(2.1):

Hence, by (2.13),

^{we have}

which concludes the lemma.

By ~( ~ )

^we denote the

Lebesgue

^{measure in}

o

LEMMA 2.2.- The _sequences

{ua~.,L( ~ )~

^and

~u",,(~)}

^are

^Cauchy

^se-

quences in

C ([

⁰

, T ; H)

^and

C([

⁰

^{, T ;} ; V)

^,

respectively.

Proof.

^{- Let}

m2 >

ml be two natural numbers and

Then, by (2.9), (t)

⁼ um2

(t) - (t)

^satisfies

If we

take,

ⁱⁿ

(2.14), v

^{= 2w’ then}

where

Now, by (2.11)

and the fact

that p

^{is a function of class}

C1,

^{we have}

constants C > 0 and K > 0 such that

Hence,

^from

(2.1), (2.15)

^and

(2.16)

^{we obtain}

Since

it

follows, by (2,2), (2.10)

^and

(2.11),

^that

Replacing (2.18)

ⁱⁿ

(2.17),

^{we have}

So,

On the other

hand, by (2.2)

^and

(2.8)

Then, by (2.2)

^and

(2.19)

^{we prove} the Lemma 2.2. 0

By

Lemma 2.2 there exists u :

[0, T] -

V such that

So

(2.20)

^{and the}

continuity of p:

LEMMA 2.3.- For each T >

0, g(ut)

^E

L1(Q)

^and

h’1,

^,

where

Iil

^{is obtained in Lemma 2.1.}

Proof . - By (2.1)

^and

(2.21)

^{there exists a}

subsequence {u;,}

^of

{u,~",,}

such that

Hence, by (2.13)

and Fatou’s lemma we have

Now, using (2.24),

^the

^proof

^follows

^similarly

^as Lemma 2.1. ~ LEMMA 2.4.

g(uv) --~ g(ut)

ⁱⁿ

V’).

Proof.

^{- To} prove this Lemma we need the

following

^Theorem.

THEOREM

(Strauss [24]).

^{Let S2 C}

^RN

^{be an open set with}

^fi-

nite

Lebesgue

measure, and a sequence

of

measurable

functions,

w"

Let

fv

^{: II$ --~}

R,

^{v E} ~ be such that

(a) ^fv,

^{v E N is bounded on} bounded subsets

of R;

(b~ ^fv

^{o wv is} measurable and

(c ) ^/~/

^{o uJ" - v a. e. in S2.}

Then v E

L1(S2)

^and

From the

assumptions

on g,

(2.13)

^and

(2.23),

^it

follows, by

^{the above}

theorem,

^that

So,

^since

L1 (0)

^is

continuously

^{embedded on}

V~,

^we obtain the Lemma 2.4. 0 To _prove

(2.6),

^we first observe

that, by (2.20)

Fixing j :::; 1)

^and

integrating (2.9)

^{from 0 to t T we have}

Using (2.20), (2.21), (2.22), (2.27)

^{and the}

^density

^{of the set}

{u~ ;

in

V,

^{we can pass to the limit as v - +00 to obtain}

and so we can obtain

(2.6).

Finally, by (2.8), (2.20)

^and

(2.21)

^{we have}

(2.7).

⁰

We end

by proving

^the

uniqueness.

^{Consider u, v :}

[0 , T ^{) -~}

^H

^satisfying (2.3)-(2.7)

^{and define}

w(t)

⁼

u(t) - v(t),

^{t E}

[ 0 , T].

Then, by (2.6)

^and

(2.7)

^{we have}

We take s

E ~ [0, T]

^{fixed and we define}

Then

z(t)

^E for each t

E [0, T]

^{and moreover}

From

(2.28)

^{we obtain}

For t s

by (2.29)

^and

(2.31),

^{we have:}

From

(2.30)

^and

(2.31)

^{we obtain:}

We claim that

In

fact,

^{let X be the set}

Then, by (2.29)

^{we have 0 in X.}

So,

^from

(2.31)

^{and mean value theorem we obtain}

z(x,t)

^{0 in X.}

Therefore, since g

^{is a}

nondecreasing

^{function we have}

In a similar _way we can prove the above

inequality

^if

(x, t) belongs

^to

Now, using (2.32)

^and

(2.33)

^{we have}

By (2.3),

^the

assumptions

on 03C6 and

(2.30),

^we have that there exists C > 0 such that

Taking

y ⁼

max{6c2T, 1/3T},

^{we obtain}

So, using

^Gronwall

Inequality,

^we conclude the

uniqueness.

⁰

COROLLARY ^2.1. Under the

hypothesis of

Theorem ,~.1 there exists

an

unique

^{u E}

L°°{0, V)

^{such that}

utt +

~2u -

Au + ^{= 0 in}

V’

^{a.e. with}

respect

^{to t}

3. Stabilization

This section deals with the

decay

^{of the} energy

(1.4)

associated ^{with the}

system (1.1).

^{For this we need} additional

assumptions

about the functions

~ and _g.

Let _{So E}

C1 ([0 , +oo) ; II8)

be such that

The

function g

^satisfies

(2.1)

and there exist _p >

0,

^{A > 0 and} co ^> 0 such that

Remark 3.1. ^- From

(3.1~

^we

^obtain,

for each Iz’ >

0,

^{a constant} mK ^> 0 such that

Remark 3.2.- From

(3.2)

^and

(3.4)

^{we have p > a.}

THEOREM 3.1.- Under the above

hypothesis, if (up, u1)

^E

^{V x H,}

^then:

.

if

p ⁼ a ⁼ 1 there _{exists y} > 0 such that

.

if 03BB

^{> 1 and} p > 1 there exists b > 0 such that

.

if

^{~ 1 there} exists ~ ^> 0 such that

Proof.

^- It is sufficient to obtain

(3.5), (3.6)

^and

(3.7)

^{for the}

approxi-

mated solutions

um(t),

because the convergences showed in the

proof

^{of the}

Theorem 2.1

imply

the above result for the solution u. From now on,

by simplicity,

^we will denote

by

^u.

We

observe,

from the definition

of E(t),

^that

and _so,

by (2.1)

Let

p(t)

⁼

f~ u(x, t) ut(x, t) dx,

^{since V is}

continuously

embedded in

H,

there exists _ci > 0 such that

Now,

^{we consider two}

separated

^cases.

3.1

Case 03BB ~

¹

LEMMA 3.1.2014 There exist

positive

constants c2, c3 and _c4 such that

Proof.-

^{We have}

p’(t)

^{= +}

(u(t) I

^so

^replacing

ⁱⁿ

(2.9)

v

by u(t),

^{we obtain}

By (2.11)

^{and Remark}

3.1,

^{there exists} c2 ⁼

E 0 , h’ ~ }

^such

that

On the other

hand,

^{it follows}

by

Remark 2.1 that there exists C > 0 such that

and _so, for c =

Hence

by (3.4), a > 1,

Then, by (2.11),

^{we have}

Now, replacing (3.13)

^and

(3.14)

ⁱⁿ

(3.12)

^and

using (3.8)

^{we obtain}

(3.11)

where _c3 ⁼

K C

^and c4 ⁼ 1 +

C2 c2/2.

Let we define ⁼

~E(t)~ ~p-1~~2p(t),

^{then we have}

Since

E~(t) 0,

^{it follows}

by (3.9)

^and

(3.10):

So, replacing (3.11)

^and

(3.16)

ⁱⁿ

(3.15)

^{we obtain}

where _{c5 =}

(ci (p - 1)/2

⁺

^{~p 1~~}

^{> 0.}

For each ê >

0,

^{we consider}

Remark 3.3.2014 We can conclude that there exists _~o > 0 such that

(i)

^{If we consider} p = 1

(then A

⁼

1)

^we

have, by (3.2), (3.3)

^and

(3.17)

if we take

then

So, by

Remark 3.3 we have

(3.5).

(ii)

^{If we consider p > 1 we}

^{have, by} (3.17)

^and

(3.18)

By (3.2)

^and

(3.3),

there exists _{c7 >}

0,

^which

depends

^{of Q and}

E(0),

^such

that .

then, by (3.8)

^and

(3.19)

Now, using Young’s inequality, with

^> 0 such that

(p+1)/(p-1)(c4

⁺

1/2) c6/2,

^{we obtain}

Taking

it

follows, by

^{Remark 3.3 that}

So,

Now, (3.6)

follows from Remark 3.3 and

(3.21).

^.

3.2 Case A 1

In this _case, the remark 3.2

implies

^that p + 1 - 2A > 0.

Let us define

~)

⁼

(E~))~~~~).

Then, by (3.8), (3.9)

^and

(3.10):

where

It follows

by (3.12)

^and

(3.13)

On the other

hand, by (3.2)

^and

(3.4)

there exists a

positive constant,

a,

which

depends

of S2 and _co such that

Replacing

^{the above}

inequality

ⁱⁿ

(3.23)

^and

using (2.11)

^and

(3.8),

^{we have:}

Taking

^{c9 =}

c3 ~E(0)~ ~p+1-2a)/2a

^and

^replacing

the above

inequality

ⁱⁿ

(3.22):

On the other

hand, by Young’s Inequality

^and

(3.8)

We choose

/3

^> 0 such that

for _y >

0,

^{such that}

Now,

^since p -f-

I/A

^> p I ¹

using (2.10)

^and

(3.20)

there exists R > 0 such that

Taking

and

replacing (3.25), (3.26)

^and

(3.27)

ⁱⁿ

(3.24)

it follows that

where _c12 ⁼ _Cs + Cg + cio + cu.

For each 6; >

0,

^{we define}

Then there exists _~o > 0 such that

Taking

the derivative of

(3.29), using (3.28)

^and

(3.30)

^we deduce that

So,

t >

0,

⁰

~

eo.

Finally (3.7)

follows from

(3.30)

and the above

inequality.

⁰

4. Final remarks If we consider the

inhomogeneous system

where

f belongs

^to

L2(0, T ; H),

^{we can prove,}

using

^{the same}

approach

^as

in Theorem

2.1,

the existence of solutions

satisfying (2.3)-(2.7).

It should be noted that the _presence of the nonlinear

damping term g

in

(1.1),

without restriction on its

growth,

contributes to some technical difficulties at the level of the existence

theory. Therefore,

^{we needed to}

consider the dimension

N

³

(Remark 2.1).

To _prove the Theorem

2.1,

in dimension

N >

^{4 we}

consider,

^for

instance,

g(s)

^{= where}

We can also consider the sublinear _case, ^as it was mentioned in Section

1,

that is

g(s) =

^{s, 0 q 1. In both cases, the}

proof

^{of the}

Theorem 2.1 follows in the same way.

The Remark 3.1 still holds if ^we

replace

^the

assumption

^on

(3.1), by

^the

following:

It is

important

^to observe that the

assumptions (3.2)

^and

(3.3)

^{on the}

damping g

ensure,

respectively,

^its

coercivity

^{at the}

origin

^{and at the}

infinity.

On the other

hand,

^the

assumption (3.4)

controls the

growth

^of g. The

decay

order of the _energy

(1.4) depends only

^{on the} constants p and

A,

^i. e., it

depends

^on the behaviour

of g

^{at the}

origin.

The estimates on the

decay

^{rates of the} energy, obtained from Theo-

rem

3.1,

^are

probably optimal,

^{but this}

question

^{has not been}

proved

yet

even in the

simplest

^case of the semilinear

damped

^wave

equation.

Finally,

^we claim that the Theorem 3.1 can be

proved,

^if

N > 4, using

the same

method, provided

solutions do exist. In

fact,

^{it is sufhcient to}

make the

following

additional

hypothesis

^on g :

Acknowledgements

The authors thank E. Zuazua for his

important suggestions

^{and constru-}

tive comments.

References

[1]

^BISOGNIN

(E.),

^BISOGNIN

(V),

^MENZALA

(G.

P.) and ZUAZUA

(E.)

^{.- On} exponential stability ^{for von Kármán} equations ^{in the} presence of thermal effects, Math. Methods in the Applied ^{Science 21}

(1998),

pp. 393-416.

[2] ^DICKEY

(R. W.) .

^- Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl. ²⁹

(1970),

pp. 443-454.

[3]

^EISLEY (J.

G.) .

²⁰¹⁴ Nonlinear vibrations of beams and rectangular plates, ^Z. Angew

Math. Phys. ¹⁵

(1964),

pp. 167-175.

[4] ^HARAUX

(A.) .

^- Semilinear Hyperbolic Problems in Bounded Domains, ^Math.

reports 3, ^Part 1, Harwood Academic Publishers, Gordon &#x26; Breach (1987).

[5]

^HORN

(M. A.) .2014

Nonlinear boundary stabilization of a ^{von Kármán} plate equation,

Lecture Notes in Pure and Appl. ^{Math. 152}

(1994),

^{pp. 581-604.}

[6]

^HORN

(M. A.)

and LASIECKA

(I.) .

^- Global existence and uniqueness ^of regular

solutions to the dynamic ^{von Kármán} system ^{with nonlinear} boundary dissipation,

Lecture Notes in Pure and Appl. ^{Math. 165}

(1994),

pp. 99-119.

[7]

^KOMORNIK

(V.) .

^{2014 Exact} Controllabilty and Stabilization. The Multiplier Method, RAM, Masson and John Willey

(1994).

[8] ^LAGNESE (J.) ^.- Modelling and stabilization of nonlinear plates, Int. Ser. of Numerical Math. 100 (1991), pp. 247-264.

[9] ^LAGNESE (J.) ^{.2014 Boundary} Stabilization of Thin Plates, SIAM Studies in Applied

Mathematics. Philadelphia

(1989).

[10] ^LAGNESE (J.) and LEUGERING (G.) .²⁰¹⁴ Uniform stabilization of a nonlinear beam

by ^nonlinear boundary feedback, ^{J. Diff.} Eqs ⁹¹ (1991), pp. 355-388.

[11] ^LAN (H. B.), ^THANH (L. T.), ^LONG (N. T.), ^BANG (N. T.), ^CUONG (T. L.) ^and

MINH (T. N.) . ²⁰¹⁴ On the nonlinear vibrations equation ^{with a} coefficient containing

an integral, Comp. Maths Math. Phys. 33, n° ⁹ (1993), pp. 1171-1178.

[12] ^LASIECKA (I.) .^- Stabilization of wave and plate-like equations with nonlinear

dissipation ^{on the} boundary, ^{J. Diff.} Eqs ⁷⁹ (1989), pp. 340-381.

- 193 -

[13] ^LASIECKA (I.) ^{and TRIGGIANI} (R.) .²⁰¹⁴ Uniform stabilization of the wave equation

with Dirichlet feedback control without geometrical conditions, Appl. ^Math. Optim.

25

(1992),

pp. 189-224.

[14]

^LIONS (J.-L.) . ^{- On Some Questions in Boundary} Value Problems of Mathemat- ical Physics, ^in: International Symposium ^{on Continuum Mechanics} and Partial Differential Equations, North-Holland

(1978).

[15]

^MEDEIROS (L. A.) .^{2014 On a new} class of nonlinear wave equations, J. Math. Anal.

and Appl. ⁶⁹

(1979),

pp. 252-262.

[16]

^MEDEIROS

(L. A.)

and MIRANDA

(M.) .

^{2014 On a nonlinear wave} equations ^with damping, ^Revista Matematica de la Universidad Complutense ^de Madrid 3, n° ²

(1990), pp. 213-231.

[17]

^MENZALA

(G.

P.) . ^- On classical solutions of a quasi-linear hyperbolic equations,

Nonlinear Anal. 3 (1979), pp. 613-627.

[18]

^MENZALA

(G.

P.) and ZUAZUA

(E.). .2014

Explicit exponential decay ^{rates for}

solutions of von Kármán system of thermoelastic plates, Differential and Integral Equation ¹¹

(1998),

pp. 755-770.

[19]

^NAYFEH

(A.)

^{and MOOK} (D.

T.) .

^{2014 Nonlinear} Oscillations, ^John Willey

(1979).

[20] ^POHOZAEV (S. L.) . ^{2014 On a class of} quasilinear hyperbolic equations, ^{Math. Sb.}

25

(1975),

pp. 145-158.

[21]

^PUEL (J.-P.) and TUCSNAK

(M.)

^{.2014 Boundary} stabilization of the von Kármán

equations, SIAM J. Control and Opt. ³³

(1995),

pp. 255-273.

[22] ^RIVERA (P.

H.) .

^{- On local strong} solutions of a nonlinear partial differential

equation, Applicable Analysis ¹⁰

(1980),

pp. 93-104.

[23]

^STRAUSS

(W.) .

^{2014 The} Energy ^{Method in} Nonlinear Partial Differential Equations, IMPA

(1969).

[24] ^STRAUSS

(W.) .

²⁰¹⁴ Weak solutions of semilinear hyperbolic equations, ^{Anais da}

Acad. Bras. Ciências 42 n° 4

(1970).

[25] VASCONCELLOS (C. F.) ^{and TEIXEIRA} (L.

M.) .

²⁰¹⁴ Strong solution ^and exponential decay ^{for a nonlinear} hyperbolic equation, Applicable Analysis ⁵¹ (1993), ^{pp. 155-}

173.

[26]

^ZUAZUA

(E.) .

^{2014 Stability and decay for a} class of nonlinear hyperbolic problems, Asymptotic Analysis ¹

(1988),

pp. 141-185.

[27] ^ZUAZUA (E.) . ^- Uniform stabilization of the wave equation by ^nonlinear boundary feedback, ^{SIAM J.Control} Optim. 28, n° ² (1990), pp. 466-477.

[28]

^ZUAZUA (E.) . ^- Controlabilidad Exact a _y Estabilización de la Ecuación de Ondas, Textos de Métodos Matemáticos 23, ^IM-UFRJ

(1992).