Existence for a nonlinear hyperbolic system

Nonlmear Printed

Anolysrs. Theory, m Great Brlfain

Methods

& App,rcarrons.

EXISTENCE

Vol

0362-546X/81:040341-13 SO2.00.0 0 1981 Pergamon Press Ltd

N-353.

5, No. 4, pp

FOR A NONLINEAR

HYPERBOLIC

SYSTEM

V. BARBU and G. MOROSANU Department of Mathematics, University of Ia& 6600 Ia$i, Romania (Receioed

4 February 1980)

Key words: maximal monotone operator, subdifferential, global boundary value conditions.

1. INTRODUCTION THE PURPOSE of this paper

nonlinear

hyperbolic

is to investigate differential system

the existence

and uniqueness

of solutions

to the

L$ - g +A(x, u) = p(x, t), (1.1)

c;- ; +

B(x, t’) = q(x, t),

for

O0) dr = 0.

(2.53)

graphs it follows by (2.48), (2.49), (2.50), (2.51) (2.52) (2.53)

350

V. BARBU AND

G. MOROSANU

that we may pass to the limit in q0,

r) E &(O, aq,

--Un(l,r)Ea

t)) t)

at

+ f(u,(L t))

to deduce that U, I‘ satisfy (1.3).

3.

Consider

the approximating

PROOF

OF

THEOREM

1

problems

L 2

- 2

+ A,(x, Un) = p(x, t),

C%

- 2

+ B,(x, Up) = q(x, t),

(3.1)

qx,

0) = u,(x),

UA(X,0) = co(x),

~~(0, r) E s(un(O, t)) -U,(l,

t)caL

for 0 d x < 1,

a.e.

0 < x d 1,

(3.3)

au (I t) at

A,(x,u)

(3.2)

t E IO, T[,

+ f(~#,

t)),

where A,(.x, .) and B,(x, .) are the Yosida approximations

qx,

O

(3.8)

Existence

for a nonlinear

hyperbolic

system

351

(3.8)

We notice that {f,}, {gj,}

are bounded

in L”(0, T; _L?(O,1)).

(3.9)

L,: IO, I[ xR* + ] - cc, a]

Let us now define the function

L,(x, r, s) = (P&, r) -

where

& + tqx3

gl + s),

(3.10)

$,(x, r) =

1 Uj,(x, r) dz

(3.11)

)’ :1,(x, z) dr,

(P#’ 4 =

s0

s0 and @(x, .) is the conjugate of ii(x, the system (3.8) can be written as

.) (see, e.g. [S, p. 521). Of course, L; also depends

@+) =aL,(x,uj.,t$,

on t. Then

(3.12)

where dL,(x, .) is the subdifferential of L,(x, .) (see [S, p. 531). By (3.10) and (3.12) and the definition of the subdifferential we infer that d L,(x, 0, -gA d $;(x,

u*) + cA(gn -

al:, + u*) + u, ax T ‘A

u*) + z

ax

i

au, ax + gi -

u*

for all ir* E R.

(u,c,),

If u* = a$,(x, 0) = B,(x, 0) then $;(x, u) = - Gn(x, 0) = 0. Hence

+ (x,uA,g) &(u,c,)

L,

uAgj. + IB(x,O)/IujJ, a.e.

d

E

IO, 1[.

1) such that

fro E H’(0,

We choose a function

x

q)(O) E S(t+)(O))? -u”,(l)

(3.14)

Ef(UO(l))

Using (3.3), (3.5), (3.6) and (3.14) one obtains

s,:&

(uAun)dx =

+2

la -(tin - Eo)(un - co) dx

+

s 0 ax

’

dG 0 (Uj. - vo) dx

S[o

(nA-zZo)+Cg~+CO$ldx

+ p 91 + 2

I

-

1

ICl+, PW),

(3.16)

352

V. BARBU AND G. MOROSANU

where p > 0 is arbitrary

and w = sign (gn + (au,/ax)).

From (3.13), (3.15) and (3.16) one gets

PS:lg,+~ldxgSIpiS(x,pw)ldx+const.[ a.e. A similar

1 +/O’(I~l+~~i)dx],

t ~10, T[. reasoning

(3.17)

leads to an analogous

au. au. 2 $

estimate

arc bounded

for au,/ax.

in P(0,

Therefore

we may infer that

T: I!tO, 1)).

(3.18)

Since

we may conclude

that jqx,

r)J < const.,

Odx, P,(. , g

bounded By a standard ;

device involving

llu,(t)- UpOIl+ ;

T; L?(O, l)),

(3.20)

in Z?(O, T; I?(O, 1)).

(3.21)

(3.1), (3.2), (3.3) one obtains IIt’,

u,(t)J12 d

-

-

l ss

[(A,k Un)-

t

0

qxvqk

- up)

0

up))]d-x dt d -

+ (B;,(x, [,A) - Bp(x, u,)(ul -

Is0 0 - n&x, Up))(jLA,JX-,~~1- PA&

up)) + (B,b, cn) - Bu(x, uJ)(AB,(x, ui)

- ,+(.Y, c,)) dx dt.

(3.22)

From (3.20) and (3.22) it follows that {Us}, (0,)

are Cauchy

and denote by u and v their limits. We notice that by (3.4), au/at, au/& E P(0, du, ~ at

e

au, au

at'

at’dt

(3.23)

T; I_?(O,1)) and

weakly-star

Next by (3.20) and (3.21) it follows that

in the weak star topology

in C([O, T]; ,!?(O, 1))

of L”(0, T; L?(O, 1)).

in L”(0, T; I?(O, 1)).

(3.24)

Existence

In

for a nonlinear

hyperbolic

system

353

order to prove that (u, u) satisfies (1.1) it suffkes to show that w(x, t) = A(x, u(x, t)),

W(x, t) = B(x, t’(x, t))

a.e.

on Q.

(3.25)

We can pass to the limit in A,(x, uJ(uI sQ

- z) dx dt 2

‘pn(x, z) dx dt,

cpl(x, uI) dx dt -

for every z E L”(Q)

sQ

sQ

to obtain w(u - z) dx dt 2 sQ

cp(x, U) dx dt -

cp(x, z) dx dt

sQ

for every z E L”(Q),

sQ

where r cp(x, r) =

A@, s) ds,

s0 which together with the similar one corresponding to B implies (3.25) (see [8, p. 621). As in the proof of Lemma 3 it follows that (u, v) satisfies (1.2) and (1.3). Finally, (1.9) is a consequence of (3.19). Thus the proof of Theorem 1 is complete. REFERENCES 1. BR.;YTON R. K., Nonlinear oscillations in a distributed network, Quart. appl. Math. 24, 289-301 (1966). 7_. BRAYTON R. K. & MIRANKER W. L., A stability theory for nonlinear mixed initial boundary-value problems, Archs ration. Mech. Analysis 17, 358-376 (1964). Quart. appl. Math. 22, l-33 (1964). 3. BRAYTON R. K. & MOSER J. K., A theory of nonlinear networks. 4. CCMXE K. L. & KRUMME D. W., Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations, J. math. Analysis Applic. 24, 372-387 (1968). 5. RASVAN V., Stabilitatea AbsolutZ a Sistemelor Automate cu Intirziere. Editura Academiei, Bucharest (1974). problems for a class of hyperbolic systems, Rev. Roumaine Math. Pures Appl. 6. BARBU V., Nonlinear boundary-value 22, 155-168 (1977). problem of hyperbolic type, Nonlinear 7. BARBU V. & VRABIE I. I., An existence result for a nonlinear boundary-value Analysis 1,373-382 (1977). 8. BA&U V., Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff, Leyden (1976). 9. BREZ~~ H., Operateurs Maximaux Monotones et Semi-groupes de Contractions dam les Espaces de Hilbert. North Holland, Amsterdam (1973).