Existence of an Anti-periodic Solution for the Quasilinear Wave Equation with Viscosity

204, 754]764 Ž1996.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

0465

Existence of an Anti-periodic Solution for the Quasilinear Wave Equation with Viscosity Mitsuhiro Nakao Graduate School of Mathematics, Kyushu Uni¨ ersity, Ropponmatsu, Fukuoka 810, Japan Submitted by Howard A. Le¨ ine Received October 30, 1995

We prove the existence of a strong anti-periodic solution for the quasilinear wave equation with viscosity u t t y div  s Ž < =u < 2 . =u 4 y D u t s f Ž x, t .

in V = R

under the Dirichlet boundary condition uŽ t .< ­ V s 0, where V is a bounded domain in R N with the boundary ­ V and s Ž ¨ 2 . is a function like 1r 1 q ¨ 2 . Q 1996

'

Academic Press, Inc.

1. INTRODUCTION In this paper we are concerned with the existence of an anti-periodic solution to the problem u t t y div  s Ž < =u < 2 . =u4 y D u t s f Ž x, t .

in V = R

Ž 1.1.

with the boundary condition uŽ x, t .< ­ V s 0, where V is a bounded domain in R N with a smooth, say C 3 class, boundary ­ V and s s s Ž ¨ 2 . is a positive function like 1r '1 q ¨ 2 . Here, a function g Ž x, t . is called v anti-periodic in t if and only if g Ž x, t q v . s yg Ž x, t .

for any t g R

Ž 1.2.

Žcf. H. Okochi w16x.. Needless to say, an v anti-periodic function is 2 v periodic in t. 754 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

QUASILINEAR WAVE EQUATIONS

755

Equation Ž1.1. in one space dimension N s 1 was introduced by J. Greenberg w6x and J. Greenberg, R. Maccamy, and V. Mizel w7x as a model of a modified quasilinear wave equation which has global smooth solutions for large data. Since then, many authors have investigated the global existence as well as the asymptotic behaviour of solutions to this and related equations Žsee M. Tsutsumi w20x, G. Webb w21x, Y. Ebihara w2x, G. Andrews w1x, Y. Yamada w22x, H. Engler w3x, A. Friedman and J. Necas w5x, H. Pecher w17x, S. Kawashima and Y. Shibata w9x, T. Kobayashi, H. Pecher, and Y. Shibata w10x, K. Mizohata and S. Ukai w11x, M. Nakao and T. Nanbu w14x, M. Nakao w12, 13x, and the references cited in these papers.. Concerning the existence of periodic solution C. Sowunmi w18x treated the one space dimensional case N s 1 to prove that if f Ž x, t . is v periodic in t, the problem Ž1.1. has an v periodic strong solution under the assumption

e1 G s Ž ¨ 2 . G e0 ) 0

and

s Ž ¨ 2 . y 2 < s X < ¨ 2 G e 0 ) 0 Ž 1.3.

with some constants e 0 , e 1. When N s 1 it is easy to see that the strong solution becomes smoother if f Ž x, t . is so. Quite recently we have generalized in w13x the main result of Sowunmi to the case N G 2, that is, we have proved the existence of a strong periodic solution to the problem Ž1.1. in general dimensions. Here, we mean by ‘‘strong solution’’ a generalized solution uŽ t . such that every term appearing in Eq. Ž1.1. is continuous as an L2 Ž V . valued function. For the arguments in w18, 13x the uniform ellipticity condition Ž1.3. is used essentially. This condition, however, excludes the most important example

s Ž < =u < 2 . s 1r 1 q < =u < 2 .

'

Ž 1.4.

In another recent paper w15x we have shown that if we restrict ourselves to the antiperiodic solutions we can overcome the difficulty and the result of Sowunmi w18x holds for a class of s Ž ¨ 2 . including Ž1.4.. That is, we have proved that when N s 1, the problem Ž1.1. admits an v anti-periodic strong or smooth solution if f Ž x, t . is v anti-periodic in t and smooth in an appropriate sense. The object of this paper is to extend the main result of w15x to the general case N G 2. That is, we shall prove for a class of s including Ž1.4. that the problem Ž1.1. admits an v anti-periodic strong solution uŽ t . if f Ž x, t . is v anti-periodic in t and appropriately smooth.

756

MITSUHIRO NAKAO

2. STATEMENT OF RESULTS The function spaces we use are all familiar and we omit the defintiion of them. But, we note that if X is a Banach space the set of X valued v anit-periodic continuous functions on R is denoted by C Ž v ; X .. Similar notation will be employed freely. We make the following asusmptions on s . HYPOTHESIS 1.

s Ž?. belongs to C 2 Žw0, `.. and satisfies

Ž1. 0 F s Ž ¨ 2 . F k 1 - ` and < s X Ž ¨ 2 .< F k 1 - `, and Ž2. there exists p ) 0 such that s y 2 < s X < ¨ 2 G k 2 s q 2 < s X < ¨ 2 4 p for some k 1 , k 2 ) 0. Remark. Without loss of generality we may assume p ) 2 in Hypothesis 1Ž2.. When s Ž ¨ 2 . s 1r '1 q ¨ 2 we see s Ž ¨ 2 . y 2 < s X Ž ¨ 2 .< ¨ 2 s Ž1 q X ¨ 2 .y3 r2 and s q 2 < s < ¨ 2 G 2Ž1 q ¨ 2 .y1 r2 and hence we can take p s 3. Hypothesis 1Ž2. will play an important role in our argument, which is different from the case N s 1 Žcf. w15x.. Our results read as follows. THEOREM 1. Let f g L2 Ž v ; L2 Ž V ... Then, under the Hypothesis 1Ž1. and Ž2., the problem Ž1.1. admits an v anti-periodic solution uŽ t . in the class W 2, 2 Ž v ; L2 Ž V . . l W 1, 2 Ž v ; H2 l H10 Ž V . . , and the estimates sup 5 =u Ž t . 5 2 q 5 u t Ž t . 5 2 q t

½

v

H0 5 =u Ž t . 5 t

2

5

dt F CM02

Ž 2.1.

and sup  5 D u Ž t . 5 2 q 5 =u t Ž t . 5 2 4 q t

v

H0 Ž 5 D u Ž t . 5

2

t

q 5 u t t Ž t . 5 2 . dt F CM02

Ž 2.2. hold, where we set M02 s

v

H0 5 f Ž t . 5

2

dt.

The solution uŽ t . in Theorem 1 is not ‘‘strong’’ in the sense that the terms appearing in Ž1.1. may not be continuous as L2 Ž V . valued functions.

757

QUASILINEAR WAVE EQUATIONS

THEOREM 2. Let f g W 1, 2 Ž v ; L2 Ž V ... Then, under the Hypothesis 1Ž1. and Ž2., the problem Ž1.1. admits an v anti-periodic solution uŽ t . in the class C 2 Ž v ; L2 Ž V . . l C 1 Ž v ; H2 Ž V . l H10 Ž V . . l W 2, 2 Ž v ; H10 Ž V . . , and, in addition to the estimates Ž2.1., Ž2.2. in Theorem 1, we ha¨ e sup Ž 5 u t t Ž t . 5 q 5 D u t Ž t . 5 . q t

v

H0 5 =u

tt

Ž t . 5 2 dt F C Ž M02 q M12 . , Ž 2.3.

where we set M12 s

v

H0 5 f Ž t . 5

2

t

dt.

For the proofs of the theorems we use the following elementary lemma concerning anti-periodic functions. Let u g W 1, 1 Ž v ; X ., X being a Banach space. Then, the

LEMMA 1. estimate

sup 5 u Ž t . 5 X F t

v

H0

d dt

uŽ t .

dt

Ž 2.4.

X

holds. The proof is easy and omitted Žcf. Haraux w8x.. 3. A Priori Estimates In this section we derive a priori estimates for an assumed v anti-periodic solution uŽ t . for the problem Ž1. which has a sufficient regularity as required. We begin with: For a solution uŽ t . the estimate Ž2.1. in Theorem 1 holds.

LEMMA 2.

Proof. Multiplying Eq. Ž1.1. by u t we see 1 d 2 dt

½

5 ut Ž t . 5 2 q

HVG Ž t . dx

5

where we set G Ž t . s H0< =uŽ t .< s Žh . dh. 2

q 5 =u t Ž t . 5 2 s Ž f Ž t . , u t Ž t . . ,

Ž 3.1.

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MITSUHIRO NAKAO

Integrating the above over w0, 2 v x we have, by use of the anti-periodicity, v

H0 5 =u Ž t . 5

2

t

v

dt s

H0

F

ž

Ž f Ž t . , u t Ž t . . dt 1r2

v

H0 5 f Ž t . 5

2

dt

/ ž

1r2

v

H0 5 u Ž t . 5 t

2

/

which implies, by Poincare’s lemma, v

H0 5 =u Ž t . 5

2

t

dt F CM02 .

Ž 3.2.

Applying Lemma 1 we have further sup 5 =u Ž t . 5 F CM0 .

Ž 3.3.

t

From Ž3.2. there exists tU such that 5 u t Ž tU .5 F CM0 and hence, by Ž2.5. and Ž3.3., sup 5 u t Ž t . 5 2 q

½

t

HVG Ž t .

F 5 u t Ž tU . 5 2 q

5

HVG Ž t

U

v

. dx q H Ž f Ž t . , u t Ž t . . dt F CM02 . Ž 3.4. 0

The following two lemmas are keys in our argument. LEMMA 3. We ha¨ e for a solution uŽ t ., v

X

H0 HV  s y 2 ¬ s < =u < 4 < D 2

2

u < 2 dx dt F Ce M02 q e sup 5 D u Ž t . 5 2 Ž 3.5. t

for arbitrarily small e ) 0, where < D 2 u < 2 denotes Ý i, j < ­ 2 ur­ x i ­ x j < 2 and Ce is a constant depending on e . Proof. Multiplying the equation by yDu and integrating we have v

H0 HV div  s Ž < =u Ž t . < . =u Ž t . 4 D u Ž t . dx dt 2

s

v

H0 5 =u Ž t . 5 t

2

dt y

v

H0

Ž f Ž t . , D u Ž t . . dt

F CM02 q M0'v sup 5 2 D u Ž t . 5 F Ce M02 q e sup 5 D u Ž t . 5 . Ž 3.6. t

t

759

QUASILINEAR WAVE EQUATIONS

Here, by integration by parts Žcf. w4, 11, 12x. we see

HV div  s Ž < =u < s

HV

½

s

2

. =u4 D u dx ­ 2u

­ xi ­ x j

q Ž N y 1. G

q 2s X

H­ V

­ 2u

X

2

­u

5

­ xk ­ xi ­ xk ­ xi

Ž s q 2 s X < =u < 2 .

HV Ž s y 2 < s < < =u < q Ž N y 1.

­u

­u

­ 2u ­ xi ­ x j

dx

2

H Ž x . dS

­n

. < D 2 u Ž t . < 2 dx X

H­ V Ž s q 2 s < =u <

2

.

­u

2

H Ž x . dS,

­n

Ž 3.7.

where H Ž x . denotes the mean curvature of ­ V at x with respect to the outward normal. By the assumption Ž1. of Hypothesis 1 and a standard trace theorem we see

Ž N y 1. H

­V

FC

H­ V

X

Ž s q 2 s < =u < 2 . ­u ­n

­u ­n

2

H Ž x . dS

2

dS

F C 5 =u 5 2Ž1y u . 5 D u 5 2 u ,

1r2 - u - 1,

F Ce M02 q e 5 D u 5 2 . Thus, Ž3.5. is proved. LEMMA 4. Let uŽ t . be an appropriate smooth v anti-periodic solution of the problem Ž1.1.. Then, under the same assumption as in Lemma 3, we ha¨ e sup Ž 5 D u Ž t . 5 2 q 5 =u t Ž t . 5 2 . q t

v

H0 5 D u Ž t . 5 t

2

dt q

v

H0 5 u

tt

Ž t . 5 2 dt F CM02 . Ž 3.8.

Proof. Multiplying the equation by yDu t and integrating we have v

H0 5 D u Ž t . 5 t

2

dt F

v

v

H0 HV
t

760

MITSUHIRO NAKAO

and hence, v

H0 5 D u Ž t . 5

2

t

dt F I q CM02 ,

Ž 3.9.

where we set Is

v

X

H0 HV Ž s < D u < q 2 < s < < =u <

2

2

< D 2 u < . dx dt.

Here, by the assumption Ž2. in Hypothesis 1, we see v

X

2 2

< D 2 u < 2 dx dt

X

2 2

< D 2 u < 4r p ¬ < D 2 u < 2Ž py2.r p dx dt

IF

H0 HV Ž s q 2 < s < < =u <

s

H0 HV Ž s q 2 < s < < =u <

F

v

½

v

. .

X

H0 HV Ž s q 2 < s < < =u <

FC

v

½H H 0

V

2

p

.

2rp

< D 2 u < 2 dx dt

5 ½ 5 ½H H

Ž s y 2 < s X < < =u < 2 . < D 2 u < 2 dx dt

F C Ce M02 q e sup 5 D u Ž t . 5 2

½

t

v

H0 HV< D

2rp

v

V

0

2rp

5 ž sup 5 D uŽ t . 5 / 2

2

u < 2 dx dt

Ž py2 .rp

5 5

< D 2 u < 2 dx dt

Ž py2 .rp

Ž py2 .rp

,

Ž 3.10.

t

where we have used Ž3.5. at the last stage. Since, by the anti-periodicity, sup t 5 D uŽ t .5 F C Ž H0v 5 D u t Ž s .5 2 ds .1r2 the inequality Ž3.10. together with Ž3.9. implies v

H0 5 D u Ž t . 5

2

dt F CM02

Ž 3.11.

sup 5 D u Ž t . 5 2 F CM02 .

Ž 3.12.

t

and t

Returning to the equation and using the Hypothesis 1Ž1., we have v

H0 5 u

tt

5 2 dt F C

v

H0 5 D u Ž t . 5

F CM02 .

2

dt q

v

H0 Ž 5 D u 5 t

2

q 5 f Ž t . 5 2 . dt

761

QUASILINEAR WAVE EQUATIONS

Finally, to show the boundedness of 5 =u t Ž t .5 we note that there exists tU from Ž3.2. such that 5 =u t Ž tU .5 2 F CM02 . Then, multiplying the equation by u t t and integrating we easily see sup 5 =u t Ž t . 5 2 F 5 =u t Ž tU . 5 2 q c'I t

(H

v

5 u t t 5 2 dt F CM02 . Ž 3.13.

0

The a priori estimates in Lemma 4 will be sufficient for the proof of Theorem 1. To prove Theorem 2 we prepare the following. LEMMA 5. Let f g W 1, 2 Ž v ; L2 Ž V ... Then, under the same assumptions as in Theorem 2, we ha¨ e v

H0 5 =u

tt

Ž t . 5 2 dt q sup Ž 5 u t t Ž t . 5 2 q 5 D u t Ž t . 5 2 . F C Ž M02 q M12 . Ž 3.14. t

for an assumed smooth solution uŽ t .. Proof. Differentiating the equation in t we have u t t t y div  s =u t q 2 s X = ? =u t =u4 y D u t t s f t Ž t . .

Ž 3.15.

Multiplying Ž2.19. by u t t and integrating we have v

H0 5 =u

tt

5 2 dt F

v

X

H0 HV  Ž s q 2 < s < < =u <

2

. < =u t < < =u t t < dx q < f t < < =u t t < 4 dx dt

and v

H0 5 =u

v

tt

Ž t . 5 2 dt F CH 5 =u t 5 2 dt q CM12 F C Ž M02 q M12 . ' C12 . Ž 3.16. 0

Further, we see from Ž3.16. that there exists tU g w0, v x such that 5 u t t Ž tU . 5 2 F C 5 =u t t Ž tU . 5 2 F CC12 and hence, by Eq. Ž3.15., sup 5 u t t Ž t . 5 2 F 5 u t t Ž tU . 5 2 q t

v

H0 5 f 5 5 u

q FC

t

v

H0

v

X

H0 HV Ž s q 2 < s < < =u <

tt

2

. < =u t < < =u t t < dx dt

5 dt

Ž 5 =u t 5 < =u t t 5 q 5 f t Ž t . 5 5 u t t Ž t . 5 . dt F CC12 .

762

MITSUHIRO NAKAO

Returning to the original equation we have 5 D u t Ž t . 5 F 5 u t t Ž t . 5 q C 5 D u Ž t . 5 q 5 f Ž t . 5 F CC1 . Now, the derivation of the desired a priori estimates is finished.

4. PROOF OF THEOREMS 1 AND 2 Let  f j 4 be the set of normalized eigenfunctions of yD in V under the homogeneous Dirichlet boundary condition, which is of course a basis of H10 . Setting m

um Ž t . s

Ý lmj Ž t . f j js1

Ž .4 m we determine  l m j t js1 , m s 1, 2, . . . , by the anti-periodic solutions of the system of the ordinary differential equations

Ž uYm Ž t . , f j . q ž s Ž < =u m < 2 . =u m , =f j / q Ž =uXm Ž t . , =f j . s Ž f Ž t . , f j . , Ž 4.1. j s 1, 2, . . . , m. Clearly, the a priori estimate in Lemma 2 can be applied to u mŽ t . and hence, the existence of the solutions  l mj Ž t .4 follows from a standard argument Žcf. Haraux w8x.. Further, we see that all the estimates in the lemmas in the previous section are valid for u s u m . From Lemmas 2 and 4 we can prove by a standard compactness argument that there exists a solution uŽ t . of the problem Ž1.1. in the sense of Theorem 1 Žcf. w12, 13x.. Further, under the assumptions of Theorem 2, we get the estimates in Lemma 5 for u s u m and conclude the existence of a solution uŽ t . such that u t t g L` Ž v ; L2 . l L2 Ž v ; H10 . ,

u t g L` Ž v ; H2 l H10 . ,

and

u g C Ž v ; H2 l H10 . . Since u t g C Ž v ; H10 . and u g C Ž v ; H2 l H10 . we easily see from the equation that u t t g C Ž v ; Hy1 . .

763

QUASILINEAR WAVE EQUATIONS

Since L2 is compactly imbedded in Hy1 it holds in general that L` Ž v ; L2 . l C Ž v ; Hy1 . ; Cw Ž v ; L2 . , and hence, u t t is weakly continuous as an L2 valued function. Using this weak continuity and an argument as in Strauss w19x we have the identity t

5 ut t Ž t . 5 2 q 2

X

Hs HV  s =u q 2 s Ž =u ? =u . 4 =u Ž t . dx dt

t

Hs 5 =u

q

t

tt

t

t

Ž t . 5 2 dt

s 5 ut t Ž s . 5 2 q

t

Hs Ž f Ž t . , u t

tt

Ž t . . dt

for all t G s. ŽNote that this identity follows formally by multiplying Eq. Ž2.19. by u t t and integrating.. From this we conclude that 5 u t t Ž t .5 is continuous and hence, taking into account the weak continuity, u t t g C Ž v ; L2 .. Finally, returning to Eq. Ž1.1. we obtain D u t g C Ž v ; L2 ., i.e., u g 1Ž C v ; H2 l H10 .. The proof of Theorem 2 is complete. ACKNOWLEDGMENT The author thanks the referees for their careful reading of the manuscript and useful comments.

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MITSUHIRO NAKAO

9. S. Kawashima and Y. Shibata, Global existence and exponential stability of small solutions to nonlinear viscoelasticity, Comm. Math. Phys. 148 Ž1992., 189]208. 10. T. Kobayashi, H. Pecher, and Y. Shibata, On a global in time existence theorem of smooth solutions to nonlinear wave equation with viscosity, Math. Ann. 296 Ž1993., 215]234. 11. K. Mizohata and S. Ukai, The global existence of small amplitude solutions to the nonlinear acoustic wave equation, J. Math. Kyoto Uni¨ . 33 Ž1993., 505]522. 12. M. Nakao, Energy decay for the quasilinear wave equation with viscosity, Math. Z. 219 Ž1995., 289]299. 13. M. Nakao, On strong solutions of the quasilinear wave equation with viscosity, Ad¨ . Math. Sci. App., 6 Ž1996., 267]278. 14. M. Nakao and T. Nanbu, Existence of global Žbounded. solutions for some nonlinear evolution equations of second order, Math. Rep. Kyushu Uni¨ . 10 Ž1975., 67]75. 15. M. Nakao and H. Okochi, Anti-periodic solution for u t t y Ž s Ž u x .. x . y u x x t s f Ž x, t ., J. Math. Anal. Appl. 197 Ž1996., 796]809. 16. H. Okochi, On the existence of anti-periodic solutions to nonlinear evolution equation associated with odd subdifferential operators, J. Funct. Anal. 91 Ž1990., 771]783. 17. H. Pecher, On global regular solutions of third order partial differential equations, J. Math. Anal. Appl. 73 Ž1980., 279]299. 18. C. Sowunmi, On the existence of periodic solutions of the equation r u t t y Ž s Ž u x .. x y l u x t x y f s 0, Rend. Istit. Mat. Uni¨ . Trieste 8 Ž1975., 58]68. 19. W. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math. 19 Ž1966., 543]551. 20. M. Tsutsumi, Some nonlinear evolution equations of second order, Proc. Japan Acad. 47 Ž1970., 950]955. 21. G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math. 32 Ž1980., 631]643. 22. Y. Yamada, Some remarks on the equation yt t y s Ž y x . y x x y y x t x s f, Osaka J. Math. 17 Ž1980., 303]323.