The effect of boundary damping for the quasilinear wave equation

JOURNAL

OF DIFFERENTIAL

52, 66-75 (1984)

EQUATIONS

The Effect of Boundary Damping for the Quasilinear Wave Equation J. M. GREENBERG* Departmem

of Mathematics,

Ohio Stale

University,

Columbus,

Ohio

43210

AND

LI TA TSIEN Department

of Mathematics,

Fudan

University,

Shanghai,

China

Received July 12, 1982; revised September 8, 1982

1, INTRODUCTION In this note we examine the effect of boundary damping on smooth solutions of the quasilinear wave equation a~

av

at

ax

-0

and

au

a44 = o

at--z-

’

(1.1)

We assumethat u(0) = 0

and

u’(u) > 0.

(1.2)

We seek solutions in the domain 0 < x < L and 0 < 1 which satisfy the boundary conditions o(u(0, t)) - rv(0, t) = 0,

0 < r < 00,

and

v(L, t) = 0.

(1.3)

The latter condition implies that the bar or string modelled by (1.1) has its end x = L fixed, while the former implies that the end x = 0 is connected to some sort of viscous dash pot. It should be noted that smooth solutions of (1.1~( 1.3) are weakly dissipative, that is, the following energy identity holds: (x, t) dx = -rv*(O,

t) < 0.

(1.4)

* Research was partially supported by a grant from the National Science Foundation. 66 0022.0396184 $3.00 Copyright All rights

0 1984 by Academx Press, Inc. of reproduction in any form reserved.

QUASILINEAR

Our principal initial condition

WAVE

result deals with the solution of (1.1~( 1.3) satisfying

lim (u, u>(x,0 = (u,, u,>(x),

O
r-o+ and is summarized THEOREM

1.

the compatibility

67

EQUATION

the

(1.5)

in If the data (u,, v,) are small in the C’ norm and satisfy conditions v;(o) - r-u;(o) = 0,

~(uo(O>> - rv,(O) = 0, u,(L)

= 0,

and

u;(L) = 0,

(1.6)

then the problem (l.l), (1.2), (1.3), and (1.5) has a unique smooth solution which decays to zero exponentially (in time) in the C’ norm.’ We feel this result partially answers a question posed to Li by C. Bardos as to why actual strings do not develop singularities in their derivatives and, hence, shocks. A dissipative-type boundary condition is typically what is exerted when a string is excited by shaking one end and, as our result shows, such a condition prevents shock formation, at least for small data. As to the effect of interior damping on smooth solutions, see [ 1, 21.

2.

PROOF

OF THEOREM

1

In order to establish Theorem 1 it is convenient to introduce the Riemann invariants

It is a simple matter to show that smooth solutions of (1.1) satisfy g+C(j?-a)$=0

and

@ Z-C@-a)g=O.

(2.2)

The function y-’ C(y) is defined by C(Y) = VmwK I For precise

conditions

on the data see inequalities

(2.3) (2.24)

and (2.27).

68

GREENBERG

and y+ U(y) is defined implicitly

The boundary

condition

AND

LI

by

at x = 0 takes the form

a( U@ - a)(O, t)) - rp + a)(O, t) = 0, while the condition

(2.5)

at x = L becomes P(L, t) = -a(L,

t).

P-6)

Moreover, the boundary condition (2.5) may be written as 40, f) = F(P(O, q>,

(2.7)

where dF

F(0) = 0,

C@-F(P))-r

and

dp=C@-Fp))+r’

(2.8)

for anyO
B Ef Cl’*@ - a) p,

(2.9)

satisfy ++C@-a)$-c&3--a)A’=O,

(2.10)

-$--C(j?-a)$--q(p-a)B*=O,

(2.11)

AtO, l> = -

;;;

I;;;;

; ;; (0, t) B(0, t),

(2.12)

and B(L,

t) = A (L, 0,

(2.13)

where 4(Y) dAfC’(Y>lC”‘(Y).

(2.14)

Finally, the initial conditions for u and u translate into initial conditions

QUASILINEAR

WAVE

69

EQUATION

for a and p which in the sequel we shall call a,, and /I,,. Moreover, given a,, and PO, initial conditions for A and B are readily computed from (2.9). In the sequel we shall refer to these as A,, and B,. The compatibility conditions (1.6) imply compatibility conditions for aO, PO, A,, and B, ; that is, that the following identities hold:

- F(P,P))) - r> aoF=Wo(O>>,A”(o)= - (WOW (C&m- F@)(O))) + r) B”(o)T(2*15) PO(L)= -a,(L), Throughout the remainder additional notation: a =

and

of this section we shall employ the following

SUP la,(x>l,

P=

O
A=

B,(L) =A,&).

o;ywL IrRo(x)I, ,

SUP lAo(x

,

(2.16)

3 = o;w,

O
\

IBo(x>l.

,

The existence and uniquenessof a C’ solution to (2.2), (2.6), and (2.7) which is defined locally in time is easily established. To show that this solution may be continued to all of t > 0 it suffices to obtain a priori estimates for a, ,L?,A, and B. We start with estimates for the Riemann invariants a and p. Since a is constant on the forward characteristics 1= C@ - a), and /3 is constant on the backward characteristics 1= -C(j? - a), and since a and /I satisfy (2.6) and (2.7) with the function F(e) satisfying (2.8), we have the following bounds for a and /I:

Ia@,t>l< max(& P,

and

IP(x,[>I< max(K P).

(2.17)

Our next task is to show that if 2 and B are small enough (see (2.16)), then A and B remain bounded independently of t, and in fact, decay to zero at an exponential rate. Again, this result, once established,will guarantee the existence and uniquenessof a C’ solution to (2.2), (2.6), and (2.7) on all of t > 0. In the sequelwe shall let the numbers {t+, Jk=, and ( TZkJk=, be the time ordinates, where the forward characteristic eminating from x = 0 at t = 0 (and appropriately reflected at the boundaries x = L and x = 0) intersects the lines x = L and x = 0, respectively. Similarly, we shall let the numbers {T,,- , }k= I and {t,,},=, be the time ordinates, where the backward characteristic emanating from x = L at t = 0 (and appropriately reflected at the boundaries x = 0 and x = L) intersects the lines x = 0 and x = L, respectively. We also take to = To = 0. (See Fig. 1.) We let t E (Tk-, , Tk] and through the point (0, t) we construct the

70

GREENBERG

AND

FIGURE

LI

1

backward characteristic (backwards in time) until it hits x = L at some time v < t. Through the point (L, q) we construct the forward characteristic (again backwards in time) until it hits x = 0 at some time t < q satisfying r E (Tke3, 7’,-,] (Fig. 2). A simple calculation using the differential equations (2.10) and (2.11) and the boundary conditions (2.12) and (2.13) shows the following relation between A(0, t) and A(0, r):

A(0,r>

a$ - a)@,l) - r> A(o7 t, = -

C(/3 - a)(O, I) + r)

1 - A(0, r) j-;((q/C),

+ (q/C),)

dx ’ (2.18)

In the last formula q is the function defined in (2.14), and C is the sound speed defined in (2.3). By the notation (q/C), and (q/C), we mean that the quantity q/C is evaluated on the forward and backward characteristics respectively. Moreover, we have chosen to parameterize these curves by x rather than t. An immediate consequence of (2.18) and the a priori bounds (2.17) is that for any k > 3, 2 < PW~k-2 k’ 1 - tik-,ML’

(2.19)

t

t FIGURE

2

k-3 k-4

QUASILINEAR

WAVE

71

EQUATION

where M=

max

$

lyl S2 max(G,d)

iu(r>=

(2.20)

(Y>Y

ii

II

C(Y)- r < 1 I ’

max IYI
O
(2.21)

and for any k = 1, 2,..., (2.22) The same arguments yield the estimates A < &-I* “1-imL

and

Am2<

ru(r) 2 I-2xML’

(2.23)

where A and B are the upper bounds for the data A, and B, defined in (2.16). It now follows from (2.19) and (2.23) and the inequality 0 <,u(r) < 1 that if -(2.24) max(A, B) < (1 - ,u)/2ML, then for any k < 1, -xk < max(A, B), and for any O
(2.25)

and t>O, IA(x,t)l

G

2 max(A, B) 1 +~

Similar estimates are obtained for B(L, t) and B(x, t). Moreover, L satisfying 0 <,u(r) < 1 < 1 the quantities x and B satisfy max(A, B) < (,I - ,u)/2hbZL, then (2.19) and (2.23) imply that for any k > 1, --AZk < II k max(A, B) and AZk- r < Lk max(A, B). Since the numbers {T,},=,

(2.26) if for some (2.27)

(2.28)

satisfy kL -
kL cmin

(2.29)

72

GREENBERG

AND

LI

where

(2.3 1) we find (2.28) implies that -IA (0, t)l < max(A, B) ~~((‘m~n”~) log*jf.

(2.32)

The exponential decay of the boundary data given by (2.32) together with the differential equations (2.10) and (2.11) and the boundary condition (2.13) imply that A and B decay at the same exponential rate, uniformly on O 0. If we employ the same geometric construction used to relate A(0, t) to A(0, r) (see Fig. 2), we readily obtain the identity a(% 0 = q-c@

(2.33)

r)),

where again t E (Tk-i, r,] and r E (TkP3, TkP2]. Moreover, the identity (2.33), when combined with the properties of F(.) (see (2.8)) and the definition of p(r) (see (2.21)) imply that for any k > 3,

where for any k > 1, a, =

(2.35)

sup I a(& 01. Tk-,St
Since 6, Q P(r) P

and

62


where E and p are defined in (2.16) we see immediately a,, < P k(r) max(K I3 Inequalities

and

(2.36)

4

that for any k > 1,

a,,- i < pk(r) max(E, p).

(2.37) and (2.29) then yield the boundary

(2.37)

estimate

Ia(0, t)l < max(E, @) e((Cmin’2L)log@jt;

(2.38)

QUASILINEAR

73

WAVE EQUATION

and this, together with (2.2), (2.6), and (2.7), implies that a and /I decay at the sameexponential rate, uniformly on 0
3. CONCLUDING

REMARKS

We conclude with a number of remarks about the behavior of our solution for special values of r. First, it should be noted that our results fail when r = +co. This case corresponds to the boundary condition ~(0, t) = 0 and here it is well known that the solution breaks down in finite time. For details we refer the reader to Klainerman and Majda [3], Lax [4], and MacCamy and Mizel [5]. The case r = 0 is also worthy of note. This corresponds to the boundary condition a(u(0, t)) = 0 or, equivalently, ~(0, t) = 0. In this case Greenberg [6] has shown that the problem has a C’ solution for small data when the function y + C(y) satisfies

for some constant 6. The solutions obtained do not decay. The last special case of interest is when r = C(0). When the system is linear, that is, when C(y) = C(O), this particular choice of r guaranteesthat both a and /I are identically zero for all t > 2L/C(O). When C(y) f C(0) and, in particular, when C’(0) # 0, this choice of r, together with the arguments leading to (2.34), implies that for k > 3, (3.1) provided only that a,_, 1C’(O)/C(O)] < 1 < co for some calculable constant A > 0. Moreover, the norms E, and c, satisfy

E < C’(O)p 1’ I C(0)I

and

6, <

C’(O) I-C(O) I a

-2

,

(3.2)

provided (C’(O)/C(O)] max(& Ir> < 1 < 00. Inequalities (3.1) and (3.2) then imply that (3.3)

74

GREENBERG AND LI

and

(3.4) and (3.3) and (3.4) when combined with (2.29) yield the boundary estimate (3.5) This estimate is valid as long as -C’(O) max(E, P, C(O)

< min(A, 1)

(3.6)

and implies that the Riemann invariants a and /I decay at a super exponential rate uniformly on 0 < x < L. Our last remark deals with solutions of the initial boundary value problem

@

and

OO,

,+C,(a,@=O,

(3.7)

fi:+(a, Pm, t> gf (0, PO)(X),

O
(3.8)

and and

P& 4 = g(a(L t)),

f > 0,

C,(a, P>< 0 < Cl@, PI.

(3.9)

(3.10)

THEOREM 2. Suppose the functions a,, &,, f, and g are C’ and satisfy the compatibility conditions

a,(O) =fCoo(0))T C,(adO), POP>) a,,(O) = C2(4Q

MO>> f’(PdO>)

Po,(O>~ (3.11)

P,(L) = g(adL)), and C2(ao(L>, PdL))P&)

= C,(adL),

P,(L)) g’(adL))

aox(L).

QUASILINEAR

75

WAVE EQUATION

Further, supposethat f and g satisfy and

If’P)g’P)l

< 1.

Then, the initial value problem defined bJJ (3.7~(3.9) has a unique C’ solution provided a0 and /I,, are small in the C’ norm. Moreover, these solutions decay to zero in the C’ norm at an exponential rate. Theorem 2 follows from arguments which are essentially identical to those used to establish Theorem 1.

REFERENCES 1. T. NISHIDA, Global smooth solutions for the second-order quasilinear wave equation with first-order dissipation, preprint. 2. L. HSIAO AND T. T. LI, Global smooth solutions of Cauchy problems for a class of quasilinear hyperbolic systems, Rapp. Rech. ZNRIA, 83 (1981); Chinese Ann. Math., in press. 3. S. KLAINERMAN AND A. MAJDA, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure Math. 33 (1980), 241-263. 4. P. D. LAX, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5 (1964), 61 l-6 13. 5. R. C. MACCAMY AND V. J. MIZEL, Existence and nonexistence in the large of solutions of quasilinear wave equations, Arch. Rational Mech. Anal. 25 (1967), 299-320. 6. J. M. GREENBERG, “Smooth and Time Periodic Solutions to the Quasi-linear Wave Equation,” Arch. Rational Mech. Anal. 60 (1975), 29-50.