Formation of singularities in solutions of a wave equation

Applied Mathematics Letters 12 (1999) 23-28

PERGAMON

Applied Mathematics Letters

F o r m a t i o n of S i n g u l a r i t i e s in Solutions of a Wave Equation M. A. SALIM Department of Mathematics King Fahd University of Petroleum and Minerals Dharan 31261, Saudi Arabia messaoud©kfupm, edu. sa

(Received and accepted August 1998) Communicated by M. Slemrod A b s t r a c t - - W e prove a blow up result for the equation wtt(x, t) = a(x)~o(wx(x,t))w,x(x, t), which can be taken as a model for a transverse motion of a string with nonconstant density. (~) 1999 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - W a v e equation, Smooth solutions, Singularities, Nonlinear response, Global solutions.

1. I N T R O D U C T I O N The aim of this paper is to study the existence and nonexistence of classical solutions to the one-dimensional nonlinear equation of the form

w,,(x, t) =

t),

(1)

where x E I (bounded or nonbounded interval), t _> 0. This equation can be regarded as a model for a transverse motion of a nonhomogeneous vibrating string (the density is a function of x). By assuming that a(x)>O, VxeI, ~o(~)>0, V ~ • R , (2) equation (1) is strictly hyperbolic. Generally, classical solutions of problems associated with (1) develop singularities in finite time, if the elastic response function ~o is 'genuinely' nonlinear. Many authors studied 'initial' boundary value problems associated to (1), with a(x) =- 1, and proved results concerning existence and formation of singularities. Lax [1] and MacCamy and Mizel [2] showed that classical solutions break down in finite time even for smooth and small initial data. In his work, Lax assumed that ~t does not change sign, whereas MacCamy and Mizel allowed ~o' to change sign. They also showed, under appropriate conditions on ~o, that intervals of x can exist, in which the solution must exist for all time t even though it breaks down for values x outside these intervals. The author would like to express his deep thanks to KFUPM for its sincere support. 0893-9659/99/$ - see front matter (~) 1999 Elsevier Science Ltd. All rights reserved PII: S0893-9659(99)00027-0

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24

M.A. SALIM

In the dissipative case, the situation is different. For initial data small and smooth enough, the effect of the damping term dominates the nonlinear elastic response and global solutions can be obtained (see [3]). However, for large initial data the nonlinearity in the elastic response takes over and classical solutions may develop singularities in finite time. These results have been established by several authors (see [4-6]). It is interesting to mention that nonlinear hyperbolic systems, of which equation (1) with a = 1 is a special case, have attracted the attention of many authors, and several results concerning global existence and blow up have been established (see [6-10]). This work will be divided into two parts. In the first part we state, without proof, a local existence theorem. In the second part, we state and prove our main blow up result. 2. L O C A L

EXISTENCE

In this section, we state a local existence theorem. The proof is omitted. It can be easily established by either using a classical energy argument [11], or applying the nonlinear semigroup theory presented in [12]. We set u(~,t) := ~ , ( x , t ) ,

, ( x , t ) := ~ ( x , t )

and substitute in (1) to get the system

u,(~, t) = a(x)v(,(x, t)),~(~, t), vt(z,t) = u,(x,t),

z • R,

t > o.

(3)

We consider (3) together with the initial data u ( z , o ) = uo(z),

v ( z , o ) = vo(x),

• •R.

(4)

In order to state the local existence result, we make the following hypotheses. (H1) a • wl'+c~(R),

~ • C2(R).

(H2) a(x) > a,

~(~) > a,

x, ~ • R,

a > O.

PROPOSITION. Assume that (H1), (H2) hold and let Uo, vo in H2(R) be given. Then the initial value problem (3),(4) has a unique local solution (u, v) on a maximal time interval [0,T) such that u,v • C ([0, T); H2(R)) n C 1 ([O,T);HI(R)). (5) REMARK 2.1. The Sobolev embedding theorem implies that u, v are C 1 functions on R x [0, T). Hence (u, v) is a classical solution. REMARK 2.2. I f ~ is a C k+l function and Uo, Vo E Hk(R), then u ( . , t ) , v ( . , t ) E H~(R), k >_ 1. REMARK 2.3. A similar result holds if ~ were depending on both u and v. 3. F O R M A T I O N

OF

SINGULARITIES

In this section, we state and prove our main result. We first start with establishing uniform upper bounds on the solution (u, v) in terms of the initial data.

Solutions of a Wave Equation

25

LEMMA. Assume (H1) and (H2) hold. Then for any e > O, there exists 6 > 0 such that given any uo, vo in H2(R) satisfying

lu0(x)l < 5,

Ivo(x)l < 6,

Vx E R,

(6)

the solution (3) obeys

u(z,t)l < e,

Iv(x,t)l < e,

Vx • R,

t • [0,T).

(7)

PROOF. We define the quantities

u(~,t)

~(x, t) := - -

+ A(v(x, t)), (8)

s(x,t) := -u(x,t) -

A(v(x,t)),

and the differential operators 1

0

p(x,t) ot 1

0

0

ox'

(9)

0

p(z, t) ot + ~ ' where

A(z) =

~d~,

p(x, t) = ~/a(x)v(v(x, t)).

(10)

Straightforward computations yield a !

oVr = o~% = - ~ ( r

+ ~).

(11)

We then define the nonegative functions

n ( t ) := m a x l r ( x , t ) l , x

S(t)

:= maxls(x,t)l,

t e [0,T).

(12)

x

These maxima are attained since r and s die at infinity; consequently for each t E [0, T), there exist ~, • E ]~ such that

R(t) ----Ir (:~,t)l,

S(t) = Is (~,t)l.

(13)

Also by the definitions of R and S, we have

R(t - h) > tr(~ + h p ( ~ , t ) , t - h)l, S(t - h) >_ Is(~ - hp(yz, t ) , t - h)l,

0
(14)

We subtract (14) from (13), divide by h, and let h go to zero to arrive at la'(~)l R(t) < -4v/_ -~ v/~(v(~, t))(R(t)) + S(t)) (15) la'(~)l S(t) < 4V/~ x/~(v(~, t))(n(t)) + S(t)),

for almost every t E [0, T). By setting M :-- max ~

(e given in the lemma)

26

M.A. SALIM

and using (H1), (H2), and (15), we get d (R(t) + S(t)) < k M ( R ( t ) + S(t)),

k>0,

(16)

for almost every t E [0, T), provided that Iv(x,t)l

<

(17)

~.

Therefore, (16) and Gronwall's inequality yield

(R(t) + S(t)) < (R(0) + S(0))e kMr,

(is)

for any t E [0,T), provided that (12) holds. We then choose 5 > 0 small enough so that

(R(0) + S(0))e kMr 2a

< -. 2

(19)

Thus, we conclude that if v satisfies (12), it satisfies, in fact,

Iv(=,t)l < ~, by (8), (10), (lS), and (H2). Therefore, by continuity, (7) is established. REMARK 3.1. We can obtain the same result if ~(~) > a > 0 holds only on a neighborhood of zero. THEOREM. Assume that (H1) and (H2) hold. Assume further that a" e L°°(R),

qo'(0) < 0.

(20)

Then we can choose initial data uo, vo in H2(R) such that the solution (3) blows up in finite time.

PROOF. We take a t-partial derivative of (11) to get at O~'rt = - l a2 l / 2 ~ - l / 2 ~ ' v t r t - "~a (rt + st).

(21)

By using vra "r u, = -5- ( , + =,),

(22)

rt - st 2---~-

v, -

and substituting in (21), we obtain

o:~, = -

al/2cff al/2cp t a' 4~ ~ + - - ~ - - ~ , s , - ~ ( ~ , + s,).

(23)

We then set ?(x, t) := e (a(=)/4)~'(~(='t))

w ( = , t) := 7(=, t)r,(z, t),

(24)

and substitute in (23) to arrive at O~-W =

al/2~tw2 4"r~

al/2~tat 16

at w

To handle the last term in (25), we first note that St = --al/2(flOt v

-

~

at"/st w

-

--(j

.

(25)

Solutions of a Wave Equation

27

and then introduce the function

v(~,t)

f(x, t); =

fdO

e(a(x)/4)~(¢) ~(~) d¢.

Direct calculations yield

_a~.),st 4a

al a-ll2") ' -_ =

4

~oi v

= 10 t (a'a-1/2f) - 1at2a-1/2 fov e(a(z)/4)~(~)~02(~)d~

(26)

14(a"a-l12-1a'2a-3/2). By substituting in (25), we have

O~ W =

al/2~'W2 + A W + B + O;4"),~

( a 'a-112 ) f 4 '

(27)

where A and B are functions depending continuously and only on a, a ~, a", and v. By setting

F:= W

ala-1/2 - - f 4

and substituting in (27), we get

OtF=-~

al/2~fll ( 4"y~

a~al/2 \2 ( F + -----4---f) + f +

a,a_l/2 \ ~

f ) + S.

(28)

By applying Young's inequality to the first and the second term, we obtain

0t F >

al/2cfl'F 2 + C(x, t) 8~

(29)

where C is a function depending on f, f2, ~, T~, ~, a, a ~, and a". We choose 5 > 0 so small that max

(-al/2¢(v(x't))

>

\87( ,t) oCvCx, t ) ) ) and

max IC(x,t)l < C.

(30)

OtF > 13F2 - C.

(31)

X,t

Therefore, (29) yields

By choosing initial data small in L ~ norm so that (30) is satisfied with derivatives large enough, the quadratic term in (31) blows up in finite time. REMARK 3.2. The same result can be obtained if ~'(0) < 0 is replaced by ~t(0) > 0. REMARK 3.3. The calculations show that the larger the derivatives are, the shorter the time of existence of the solution is.

28

M.A. SALIM

REFERENCES 1. P.D. Lax, Development of singularities in solutions of nonlinear hyperbolic partial differential equations, J. Math. Physics 5, 611-613, (1964). 2. R.C. MacCamy and V.J. Mizel, Existence and nonexistence in the large solutions of quasilinear wave equations, Arch. Rational Mech. Anal. 25, 299-320, (1967). 3. T. Nishida, Global smooth solutions for the second order quasilinear wave equations with first order dissipation (unpublished note), (1975). 4. W. Kosinsky, Gradient catastrophe in the solutions of nonconservative hyperbolic systems, J. Math. Anal. 61, 672-688, (1977). 5. S.A. Messaoudi, Formation of singularities in heat propagation guided by second sound, J.D.E. 130, 92-99, (1996). 6. M. Slemrod, Instability of steady shearing flows in nonlinear viscoelastic fluid, Arch. Rational Mech. Anal. 68, 211-225, (1978). 7. T.-T. Li, Y. Zhou and D.-X. Kong, Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Analysis, 28 (1), 299-1332, (1997). 8. Liu Fagui and Yiang Zejiang, Formation of singularities of singularities of solutions for Cauchy problem of quasilinear hyperbolic systems with dissipative terms, J. Partial Diff. Eqns. 5 (1), 79-89, (1992). 9. L.T. Ping, Development of singularities in nonlinear waves for quasilinear hyperbolic partial differential equations, J. Diff. Eqns. 33, 92-111, (1997). 10. T. Nishida, Nonlinear Hyperbolic Systems and Related Topics in Fluid Dynamics, Department de Mathematicques, Publications Mathematiques D'orsay 78-02, Paris Sud, (1978). 11. C.M. Dafermos and W.J. Hrusa, Energy methods for quasilinear hyperbolic initial boundary value problems. Applications to elastodymamics, Arch. Rational Mech. Anal. 87, 267-292, (1985). 12. T.J.R. Hughes, T. Kato and J.E. Maarsden, Well-posed quasilinear second order hyperbolic systems with applications to nonlinear elastodynamic and general relativity, Arch.Rational Mech. Anal. 63, 273-294, (1977). 13. B.L. Keyfltz and H.C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72, 219-241, (1980).