Nonlinear hyperbolic-parabolic partial differential equation in noncylindrical domain

RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie U, Tomo XLIV (1995), pp. 135-146

N O N L I N E A R H Y P E R B O L I C - P A R A B O L I C P A R T I A L

D I F F E R E N T I A L E Q U A T I O N IN N O N C Y L I N D R I C A L D O M A I N

JORGE FERREIRA

In this paper we study the existence of weak solutions of the mixed problem for the nonlinear hyperbolic-parabolic equation

Kl(X)U" + K2(x)u' + A(t)u + H(u) = f in Q

u = 0 o n e

u(0) = uo, K1 (x)ut(O) = ~ U l (x)

where Q is a noncylindrical domain of R n+l with lateral boundary Z, {A(t); t > 0} is a family of operators of .~(Hd(f2), H-l(f2)) and H(s) is a continuous function which satisfies some appropriate conditions.

1. Introduction.

Let Q be a noncylindrical domain of R n x (0, T) with lateral

boundary E, and [2 be a bounded open set in R n, with smooth

boundary F such that

Q C f2 x (0, T).

Keywords. Noncylindrical domain, weak solutions, hyperbolic-parabolic equation.

136 JORGE FERREIRA

(1.1)

In Q consider the following mixed hyperbolic-parabolic problem:

where Kl(x) > O,

with 1 < p < oo. 5f(Hd(~2), n - l ( n ) ) the conditions

Kl(x)u" + K2(x)u' + A(t)u + H(u) = f in Q

u = 0 on ]3

u(O) = uo, Kl(x)u'(O) -- x / - K - ~ u l ( x )

1 K2(x) > 0 a.e. in g2 and e Le(~)

K2(x) {A(t); t >_ 0} is a family of operators of

and H(s) is a continuous function satisfying

f $

0 < I(s) < Clsl e+2 where I(s) = H(~)d~ (1.2) 0

IH(s)l _< Cls[ "+1, /9 e R

2 such that 0</9_< ~ if n >2 and 0<p <c~ if n= I or

n-2 n--2.

Linear and nonlinear wave equations in noncylindrical domains

have been treated by a number of authors, among them we can

mention J. L. Lions in [9] who introduced the so-called penaty

method to solve the problem of existence of solutions. Using this method, Medeiros [10] proved the existence of weak solution of the mixed problem for the equation

(1.3) u " - Au + H(u) = f in Q

for a wide class of functions H(u) such that uH(u) > 0 Cooper and Bardos [3] studied the existence and uniqueness of weak solutions of (1.3) in the case H(u) = lul~u(o~ _ 0) and E globally time-like; Cooper and Medeiros [2] included the above results in a general model

(1.4) u" - Au + H(u) = 0

where H(s) is a continuous function and s H ( s ) > O.

Inoue [7] succeeded in proving the existence of classical solutions of (1.3) for the case n = 3 and H ( u ) = (u') 3 when the

NONLINEAR HYPERBOLIC-PARABOLIC PARTIAL DIFFERENTIAL EQUATION .... 137

body is time-like at each point. Clark [1] proved the existence of weak solutions of the mixed problem for the equation

(1.5) Kl(x)u" + K2(x)u' + A(t)u + [ulPu : f in Q.

Mello [11] obtained weak solutions to the problem (1.1) in the case A ( t ) = - A and H(s) is a differentiable function such that

0 <_ I(s) <_ Clsl p+2 where l (s) = H(~)d~ 0

IH(s)l _< Clsl p+I

IH'(s)l ~ Clsl, p ~ I~

2 O < p < ~ if n > 2 and O < p < c ~ i f n = l or n = 2 .

n - 2 In this paper we study the mixed problem for (1.1) and obtain

existence of weak solutions under the hypothesis that the domain is monotone increasing.

2. Some terminology and assumptions.

By 9(~2) we denote the space of infinitely differentiable functions with compact support contained in ~ ; the inner product and norm in LZ(g2) and H01(f2) will be represented by (.), I" I and ((')), I1" ii, respectively.

If X is a Banach space we denote by Le(O, T; X), 1 < p < oo, the Banach space of vector-valued functions u : (0, T) ~ X which are measurable and Ilu(t)llx E LP(O, T) with the norm

1

E j0 IlullL,(0.r:x) = Ilu(t)ll~dt ,

and by L~(0, T; X) the Banach space of vector-valued functions u : (0, T) --+ X which are measurable and Ilu(t)llx ~ L~176 T) with the norm

Ilu IIL=(0,r:x) = ess sup Ilu(u)llx. 0 < t < T

138 JORGE FERREIRA

(H.1)

(H.2)

Let f2 be an open and bounded set of I~ n with smooth boundary

1", Q c f2 • (0, T) an open noncylindrical domain. We will use the

following notations: f2s = Q fq {t = s} for s > 0, f20 = Q (q {t = 0}

such that f 2 s # ~ for al s > 0, Fs = 0f2,,

Z = U F s 0<s<T

and 0Q = f20 u Z the boundary of Q.

Our assumptions on Q are:

~'2 t is monotone increasing, that is, f2~ C fl~ if t < s where f2~* is the projection of f2t on the hyperplane t = 0.

For each t e (0, T), f2t has the following property of regularity:

if u e Hd(f2) and u = 0 a.e. in f2\f2, then the restriction of u

to u~ belongs to H01(f2~).

In order to simplify the notation we will identify f2* with f2,.

We define Lq(O, T; Le(f2,)) as the space of functions

w E Lq(0, T; L P ( ~ ) ) such that w(x , t) = 0 a.e. in f2\f2, and a.e. in

(0, T). When 1 < q < cx~ we consider the norm 1

which agree with Ilwllt,(o.r:t,(n)); and when q = e~ we consider

II to IIL,(0.r;L~(a,)) = ess sup Ilw(t)IIL,(a,). O<t<T

On the same way we define Lq(O, T; Hd(f2t)) for 1 < q < o<~.

THEOREM 2.1. Let w �9 [2 x (0, T) --~ I~. Lebesgue measurable. Then w(x , t) = 0 a.e. in f2\f2, f o r almost all t ~ (0, T) i f and only i f w = 0 a.e. in [2 x (0, T ) \ Q .

Proof See [13].

Let us consider the following family of operators in

A(t) = -- aij (x, t) i , j= l ~


where aij = aji and aij ~ WI'~ T; L~176 for all i, j , = 1 . . . . . n.

We suppose that

n

( 2 . 1 ) E aij(X, t)~i~j >__/~(l~ll 2 + . . . + 1~12), i,j=l

for all (t, ~) e [0, T] x ]R n and a.e. in ~ , with /~ > 0 a constant.

If we denote by a(t, u, v) the family of bilinear forms H01(f2) x H0t(f2) associated with A(t) , we have

a(t,u,v)= ~-~ f aij(x, tl OU Ov i,j=l [2 3X----'7 3xj dx ,

in

which is symmetric. From (2.1) it follows that

(2.2) a(t, u, u) >_ ~llull 2, for all u ~ Hd(f2) and t ~ [0, T].

For each u , v E Hd(f2) fixed, we consider the h : (0, T) ---> IR be defined by h(t) = a(t , u, v).

Now, with a~j, --~ai; ~_ (L~(O, T; L ~ ( ~ ) ) ) 2 for all

i , j = 1 . . . . . n, it follows that there exist the derivative in distributional sense of h given

- ~ f 0 Ou Ov h'(t) = --~aij(x, t) - - - - dx i,j=l OX i OXj

function

and h'(t) belong to L~176 t).

We will denote h'(t) by a'(t, u, v).

For u, v ~ H01(f2,) we use

a ( t , u , v ) = ~ - ~ f , a i j ( x , t ) - - i,j=l

Ou Ov

3xi Oxj m d x .

3. Existence of weak solutions.

2(F+D THEOREM 3.1. Let be uo E H~)(f20), ul E L2(~o) and f E L p-,

(0, T; L ~-' (f2,)). Assume that H satisfies the conditions (1.2) and

I 4 0 JORGE FERREIRA

a(t, u, v) be as in (2.2). Suppose that the functions K1, K2 satisfy:

(3.1) Kl(x) > 0 a.e. in f2,

(3.2) K2 (x) > 0 a.e. in ~2 and K2(x)

- - E L P ( ~ ) with lp < cx~,

(3.3) K1, K2 ~ L ~ ( ~ ) .

Then under the assumptions (H.1) and (H.2), function u defined in Q such that

u 6 L~(0 , T;H01(f2,)), (3.4)

u' E L p+, (0, T; L p+l (f2,)) = L p+t (Q),

there exists a

(3.5) v / ~ l u ' ~ L~(0 , T; L2(g2t)), Klu" ~ L2(0, T; H-l(g2,)) ,

(3.6) Kl(X)U" "k- K 2 (x )u ' -}- A(t)u + H(u) = f in L2(0, T; H-I(~,)),

(3.7) u(O) = Uo, Kl(x)u'(O) = ~ - - ~ ) u l in f2 o

Proof of the Existence. We observe that by (3.4) and (3.5), the initial conditions make sense. To prove the theorem we consider rio, fil by zero outside of f20, and M 6 L~(f2 • (0, T)) defined by

j 1 in f2 • (0, T)\Q, M(x, t)

I 0 i n Q ,

It is clear that fi0 6 Hd(g2), t71 ~ L2(~2). For each e > 0, we consider the problem:

1 Kl~(x)u" + Kz(x)u' e + a(t)u~ + --Mu'~ + H(ue) = f

E

(pE) uE(0) = rio

=


where K1E = K1 + e.

The Proof of Theorem following:

3.1 will be a consequence of the

THEOREM 3.2. For each e > 0 there exists a function defined in f2 x (0, T) such that:

(3.8) uE ~ L~176 T; Hol(f2)), u' ~ L ,§ (f2 x (0, T))),

U e ,

(3.9) z oo tt 2 V~IEuEL (0, T; Lz(Q)), K1EueL (0, T; H- t ( f2) ) ,

(3.10)

and

(3.11)

1 K,e(x)u~ + K2(x)u'~ + A(t)uE + --Mu'e + H(ue) = f

E

in L2(0, T; H - I ( ~ ) )

uE(O) = tto, Kic(x)u'e(O) = ~-~lE(X)fil.

Proof of the Theorem 3.2. Let (wv)vsN be a basis of H d (f2) and V., = [wl . . . . . Win] the subspace generated by the m first vectors of the basis (wv). For m ~ N consider the function U.m " [0, t.,,,) ~ V,. as solutions of the following system:

p(c,.,)

(KIE(X)U'~, w) + (K2(x)U'~m, w) + a(t, U~m, w)+

1 + --(MU'em, w) + (H(u~m), w) = ( f , w) VW E Vm

8

u~., (0) = u0,. ~ tT0 strongly in H d (f2), uo,. ~ Vm

fil U'.m(O) = Ul,. -'+ ~ strongly in L2(f2), Ulm ~ V,.

The system ( W ' ' ) has solution in [0, t~m), 0 < t~,. < T. The "a priori" estimates which shall be obtained, permit us to extend the approximate solutions u~m to the interval [0, T) and also pass to the limit as m ~ oo.

1 4 2 JORGE F I ~ I R A

In fact, setting w = 2u',m(t) in (per,) and integrating form 0 to t < t,m we have

(3.12)

1 , f ' + h/- U;mt2ds+ "0

+ Ilu"llZ+ T ~lM(u")l dxds+

s ( + 2 1 (u,m)dx <_ C + (f, u'em)ds + C 0 "0

II u~,~ llZds.

We have

(3.13) fo t 1 f t (f, uem)ds < C + -~ o

Using (3.13) and Gronwall 's inequality we have the following estimate:

(3.14)

1 , 12 1 f0' r /~ -~lv/-K~u,,, + -~ IV/-~2u'~,.12ds + ~ d s + -~-IlU~m[[2+

I + [M(u'em)12dxds + 2 l(Uem)dx < C ~2

where C is a positive constant independent of m e e. 2p

Since ~ > 1 it follows that p + l

# (3.15) Ilu~ll ~ ~ C

L e+l (f~x(0,T))

where C is a positive constant independent of m and e.

Observe that the estimates (3.14) and (3.15) implies that there exists a subsequence of (U~m), which we still denote by (u~,,), and


a function u, such that

u~m ---> u~, weakly - star in L~176 T; H~(~2)),

' ' L ( f 2 ) ) , u~m ---> u~, weakly in L ~+, (0, T;

(3.16) x/-~l~U'~m ~ v/--~l~u'~, weakly - star in L~176 T; L2(f2)),

v/~2U'~m ~ v/-~2u'~, weakly in L2(0, T; L2(f2)),

Mu'~m ~ -'-=Mu'~, weakly in L2(0, T; L2(~)),

From (3.16)1, (3.16)2 and since the injection of Hd(~) in

L P§ (f2) is connpact, we have, after passing to a subsequence, that U,m convergs to u, almost everyvvhere in ~2 • (0, T), then,

(3.17) H(uem ) ----> H(u~) a.e; in f2 x (0, T),

and we have

(3.18) IIH(U~m)IIL~(O,W:L~<~)) ~ C

where C is a positive constant independent of m e e.

From (3.17), (3.18) and Lions [9], Ch.I, Lemma conclude that:

(3.19) H(u~m) ---> H(u~) weakly in L2(0, T; L2(f2)).

1.3, we

It follows from (3.16), (3.18), that we can pass to the limit in the approximate system (p~.m) obtaining:

1 Kl~(x)u~ + K2u' e + A(t)ue + --Mu'~ -t- H(ue) = f

in L2(0, T; H-1([2)).

There is no difficulty to verify that the initial conditions are satisfied.

Proof of Theorem 3.1. Observe that the estimates obtained are independent of e too. Therefore, by the same argument used to obtain u~ from U,m, which is the solution of (PD we can pass to

1 4 4 JOR~E ~

the limit when e goes to zero in ue, or a subsequence, obtaining a function u independent of e and m, such that:

u~ ~ * u in L~(0, T; H~(f2)),

u t ~ u' in L p+, (0, T; L p+, (~)),

(3.20) v/-~l~u'E ~ * v/~lu ' in L~(O, T; L2(f2)),

vrk-~u'~ ~ v/-~u' in L2(0, T; L2(~)),

H(uD ~ H(u) in L2(0, T; LZ(g2)),

(3.21)

From estimate (3.14), we have

1 " - - - - , 2 l f0/ 2~.~ line 1,2+ -~h/ Kl~u~l + -~ Iv/-~2u'~12ds +

/ + [ M(u'c)[Zdxds + 2 l(uE)dx < C ~2

where C is a positive constant independent of e.

From (3.21) we obtain that

for fn lM(u '~) ,Zdxdt<Ce,

therefore,

Mu'~ --+ 0 strongly in L2(0, T; L2(f2))

as e --+ 0. Since that the injection of L2(0, T; L2(f2)) Ll(0, T; L~(~)) is continuous we obtain

(3.22) Mu' e --+ 0 in LI(0, T; Ll(f2)).

in

From (3.20)2, since - - 2p

p + l > 1 we conclude that:

u' C --+ u' in LI(0, T; Ll(f2)),

and therefore,

(3.23) Mu' e ~ Mu' in LI(0, T; Ll(f2)).


From (3.22) and (3.23) we conclude that:

(3.24) Mu' = 0 a.e. f2 • (0, T).

But, M = 1 in ~ x (0, T ) \ Q , therefore u' = 0 a.e. in g2 x (0, T ) \ Q . Applying the Theorem 2.1, we obtain that u ' = 0 a.e. in f2\g2, for almost all t ~ (0, T). As u e L~176 T; Hd(f l ) ) we have that

u L (0, r; Hal(f2,)).

By the same argument it can be show (3.4) and (3.5).

Now, we consider the equation of Theorem 3.1:

K " w) + (K:u'~, w) + a(t, u~, w)+ l e U c ,

(3.25) 1 + - - ( M u ' E, w) + (n(ue) , w) = ( f , w)

E

in ~ ' (0 , T) for all w ~ Hd(f2).

Therefore, restricting the equation (3.25) on Q, and with the functions M satisfy Mu' E = 0 we have that

(K1E)u'~, w) + (K2u'e, to) + a(t, ue, w)+ (3.26)

+ (H(ue), w) : ( f , w)

in ~ ' ( O , T) for all w 6 Hal(g2,).

It follows from (3.20), that we can pass to the limit, e ~ 0, in the equation (3.26) thus obtaining (3.6).

In order to check up that u ( 0 ) = u0 it is sufficient to use (3.20)1 and (3.20)2 followed by an integration by parts. By the same argument, using (3.20)3, it can be shown that K l (x )u' ( O) = q/--~l (x )u l .

Acknowledgment.

I take this opportunity to express my gratitude to Professors L.A. Medeiros and Nirzi G. de Andrade for their valuable suggestions in the course of preparation of this paper.

146 JORGE FERREIRA

REFERENCES

[1] Clark M.R., Uma Equagao Hiperb61ica-Parab6lica Abstrata Nao Linear: Existencia e Unicidade de Solu96es Fracas. Uma Equaqao Hiperb61ica- Parab61ica em Domfnio N~o Cilindrico, Doctoral Thesis, IM/UFRJ, Rio de Janeiro, Brasil, 1988.

[2] Cooper J., Medeiros L.A., The Cauchy problem for nonlinear wave equations in domains with moving boundary, Annali della Scuola Normale Superiore di Pisa, Vol. XXVI, Tasc. IV, 829-838.

[3] Cooper J., Bardos C., A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., (42) pp. 29-60.

[4] Ferreira J., Pereira D.C., Existence of global weak solutions or an equation of nonlinear vibrations, Boletim da Sociedade Brasileira Paranaense de Matem~itica, U (02), 1990, pp. 70-90.

[5] Ferreira J., Pereira D.C., On a nonlinear degenerate evolution equation with strong damping, International Journal of Math. and Mathematical Science], 1 (03), 1992, pp. 543-552.

[6] Ferreira J., On weak solutions of semi-linear hyperbolic-parabolic equations, International Journal of Math. and Mathematical Sciencel, (to appear).

[7] Inove A., Sur Du + u 3 = f dans un domaine noncylindrique, J. Math. Anal. Appl., (46), pp. 777-819.

[8] Lions J.L., Une remarque sur les problOmes d'evolutions nonlineaires dans les domaines noncylindrique, Rev. Romaine Purel Appl. Math., (9), pp. 11-18.

[9] Lions J.L., Quelques M~thodes R~solution des Problkmes aux Limites Non Lindaires, Dunod, Paris, 1969.

[10] Medeiros L.A., Nonlinear wave equations in domains with variable boundary, Arch. Rational Mech. Anal., (47), pp. 47-58.

[11] Mello E., SolugOes Fracas de um Problema Hiperb6lico-Parab61ico Nao Linear, Doctoral Thesis, IM/UFRJ, Rio de Janeiro, Brasil, 1983.

[12] Nakao M., Narazaki T., Existence and decay of solutions of some nonlinear wave equations in noncylindrical domains, Math. Rep., XI-2, pp. 117-125.

[13] Rabello T.N., Decaimento da Energia de um Sistema de EquafOes Hiperb6licas Nao Lineares num Domfnio Nao Cilindrico, Doctoral Thesis, ITA, S. J. dos Campos, Brasil, 1990.

Pervenulo il 22 seuembre 1993, in forma modifieata il 26 seuembre 1994.

Universidade Estadual de Maring6 Departamento de Matemdtica

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Nonlinear hyperbolic-parabolic partial differential equation in noncylindrical domain