Notations and Main Result

New York J. Math. (1998) 167{176.

Boundary Stabilization of a Hyperbolic Equation

with Viscosity

M. M. Cavalcanti

Abstract. This paper is concerned with the solvability and uniform stability of a hyperbolic equation with spatially varying coecients of viscosity and elasticity and boundary damping.

Contents

1. Introduction 167

2. Notations and Main Result 168

3. Solvability of strong and weak solutions 169

4. Asymptotic Behaviour 173

References 176

1.

Introduction

Let be a bounded domain of

R

ⁿ with C² boundary ; and let (;⁰;¹) be a partition of ;both parts with positive measure.

We consider the following hyperbolic problem:

@²y

@t^{2 ;r(}aij(x)^ry)^; @

@t^r(bij(x)^ry) =f in Q= (0¹) y= 0 on ¹= ;¹(0¹)

@@ya + @

@t @y

@b +(x)

@y

@t ^;g= 0 on ⁰= ;⁰(0¹) y(0) =y⁰ @y

@t^{(0) =}y¹ in (1.1)

where _@^@y_a (resp. _@^@y_b) is the outer cornormal derivative with respect to the matrix (a_ij) (resp. (b_ij)) dened by

@@ya =iaij @y

@xj

and= (¹n) denotes the exterior unit normal at ;.

Received June 30, 1997.

Mathematics Subject Classication. 35B40, 35L80.

Key words and phrases. elasticity, viscosity, boundary stabilization.

c1998StateUniversityofNewYork

ISSN1076-9803/98

167

In physical terms the entriesaijandbijare related to coecients of elasticity and viscosity, respectively. Without viscosity, i.e., ifbij = 0 and assuming thataij = 1, f = 0 andg= 0the problem (1.1) was studied by many authors: See J. P. Quinn and D. L. Russell 10], G. Chen 2, 3, 4], J. E. Lagnese 7, 8] and V. Komornik and E. Zuazua 6]. And when= 0, an asymptotic regularization procedure is proved by G. C. Hsiao and J. Sprekels 5]. Inspired by the above works we show solvability of strong and weak solutions to problem (1.1), and obtain boundary stabilization.

To obtain the existence of solutions we make use of Galerkin's aproximation.

However, as we are also interested in srong solutions we have some technical dif- culties wich lead us to transform the problem (1.1) into an equivalent one with zero initial data.

Stability problems with nonhomogeneous conditions require a special treatement because we don't have any information about the inuence of the inner products (f(t)y⁰(t))_L²⁽⁾ and (g(t)y⁰(t))_L^2(;0

)on the energy E(t_{) = 12}^Z

jy⁰(xt)^j2dx_{+ 12}^Z

aij(x)@y

@xi(xt) @y

@xj(xt)dx (1.2)

or about the sign of its derivativeE⁰(t):

To obtain the uniform decay we use the perturbed energy method. Our paper is organized as follows. In Section 2 we give notations and state our main result.

In Section 3 we prove solvability of strong and weak solutions of (1.1) while in Section 4 we obtain the boundary stabilization of solutions from Section 3.

2.

Notations and Main Result

We dene

(uv) =

Z

u(x)v(x)dx ^ju^j2=

Z

ju(x)^j2dx (uv)^;0 =

Z

;0

u(x)v(x)dx ^ju^j2;0 =

Z

;0

ju(x)^j2dx and letV be

V =^fv²H¹() v= 0 on ;^1g

which, equipped with the topology^jrj is a Hilbert subspace ofH¹().

In order to establish our main result, we make the following assumptions on the coecients:

aijbij ²C¹(): (2.1)

There exist positive constantsa⁰ andb⁰ such that a_ij =a_ji and for ²

R

^{n Xn}

ij⁼¹a_ijij a^0jj2 (2.2)

bij=bji and for ²

R

^{n Xn}

ij⁼¹bijij b^0jj2 (2.3)

²L¹() (x)0 a.e. on ;⁰ (x) = 0 ⁸x²;^0\;¹ (2.4)

and (x)^!0 wheneverx tends to a point x²;^0\;¹: This choice of the function was done in order to avoid eventual singularities.

Dening

a(uv) = ^Xn

ij⁼¹

Z

aij(x)@u

@x_i @v

@x_jdx ⁸uv²V (2.5)

b(uv) = ^Xn

ij⁼¹

Z

bij(x)@u

@xi @v

@xjdx ⁸uv²V (2.6)

from (2.1), (2.2) and (2.3) there exist positive constantsa⁰a¹b⁰andb¹such that a^0jru^j2a(uu)a^1jru^{j2 8}u²V

(2.7)

b^0jru^j2b(uu)b^1jru^{j2 8}u²V:

(2.8)

Now, we are in position to state our main result.

Theorem 2.1.

Let

y⁰y¹fg²V L²()L²(0¹L²())H¹(0¹L²(;⁰)):

Under the assumptions (2.1)-(2.3) and assuming that g(0) = 0, the problem (1.1) possesses a unique weak solution y: (0¹)^!

R

^{such that}

y²L¹(0¹V) y⁰²L¹(0¹L²()): (2.9)

Moreover, provided that for large t, the inequality

Z t

0

exp("

2s)^jf(s)^j2+^jpg(s)^j2;0ds t^r (2.10)

holds for some positive constants", , and r, we obtain the following energy decay E(t)Cexp(^;"

2t) ⁸t0 and ⁸"²(0"⁰] whereC and"⁰ are positive constants.

Remark 1.

The hypothesis (2.10) means that the map t^{7;!Z t}

0

exp("

2s)^jf(s)^j2+^jpg(s)^j2;0ds must be bounded by a polinomialP(t).

3.

Solvability of strong and weak solutions

In this section we are going to prove the existence of strong solutions of problem (1.1). Using density arguments we conclude the same for weak solutions. The exis- tence of solutions may be proven either by the Galerkin method or using semigroup arguments. We employ the Galerkin method.

We dene the Hilbert space

H=u²VAuBu²L²() (3.1)

whereAandB are the operators dened by A=^; @

@xi

aij(x) @

@xj

and B=^; @

@xi

bij(x) @

@xj

andH is endowed by the natural inner product

(uv)H= (uv)V + (AuAv) + (BuBv):

Let us consider

y⁰y¹²H (3.2)

satisfying the compatibility condition

@y⁰

@a +@y¹

@b +(x)^;y^1;g(0) = 0: (3.3)

In addition, let us assume that

f ²H¹(0¹L²()) g²H²(0¹L²(;⁰)): (3.4)

The variational formulation associated with problem (1.1) is given by (y⁰⁰(t)w) +a(y(t)w) +b(y(t)w) + (y⁰(t)w)^;0

= (f(t)w) + (g(t)w)^;0 ⁸w²V:

In order to obtain strong solutions and since we can not use a `special basis' (for instance, one formed by eigenfunctions) because of the boundary condition (y⁰(t)w)^;0, we need to derive the above expression with respect to t: But it lead us to technical diculties when we estimate y⁰⁰(0): To solve this problem we transform the boundary value problem (1.1) into an equivalent one with null initial data. In fact, considering the change of variables

v(xt) =y(xt)^;(xt) (3.5)

where

(xt) =y⁰(x) +ty¹(x) (xt)²0¹) we obtain the equivalent problem forv:

v^{0 0;r}(a_ij(x)^rv)^;r(b_ij(x)^rv⁰) =F in Q v= 0 on ¹

@@va +@v⁰

@b +v⁰=G on ¹ v(0) =v⁰(0) = 0

(3.6) where

F =f+^r(aij(x)^r) +^r(bij(x)^r0) (3.7)

G=g^; @

@a ^;@⁰

@b ^;0: (3.8)

If v is a solution of (3.6) in 0,T], theny=v+is a solution of (1.1) in the same interval. However, after two estimates we are going to prove later, we have that

jAv(t)^j2+^jrv⁰(t)^j2C(T) ⁸t²0T] and ⁸T >0: (3.9)

Thus, from (3.4) and (3.5) we have the same estimate obtained in (3.9) for the solution y: So, we can extend y to the whole interval 0¹) using the standard argument

Tmax=¹ or if Tmax<¹ then _t lim

!Tmax

;

jAv(t)^j2+^jrv⁰(t)^j2 =¹: Hence, it is sucient to prove that (3.6) has a solution in 0,T], which will be done by the Galerkin method.

We represent by (!) a basis inH which is orthonormal in L (), by Vm the subspace ofH generated by the m-rst vectors!¹!m and dene

vm(t) =^Xm

i⁼¹gim(t)!i

(3.10)

wherevm(t) is the solution of the folowing Cauchy problem:

(3.11) (v⁰⁰_m(t)!_j) +a(v_m(t)!_j) +b(v_m⁰ (t)!_j) + (v_m⁰ (t)!_j)^;0

= (F(t)!j) + (G(t)!j)^;0 vm(0) =v_m⁰ (0) = 0 j= 1m:

The aproximate system is a normal one of ordinary dierential equations. It has a solution in 0tm): The extension of the solution on the whole interval 0,T] is a consequence of the rst estimate we are going to obtain below.

3.1.

A Priori Estimates.

3.1.1. The First Estimate. Multiplying both sides of (3.11) byg⁰_jm(t)summing over 1j mand considering (2.2), we obtain

12d dt

n

jv_m⁰ (t)^j2+a(vm(t)vm(t))^o+b(v⁰_m(t)v_m⁰ (t)) + ^pv_m⁰ (t)²

;0

(3.12)

= (F(t)v⁰_m(t)) + d

dt⁽G(t)vm(t))^;0

;(G⁰(t)vm(t))^;0

12^jF(t)^j2+ 12^jv⁰_m(t)^j2+ d

dt⁽G(t)vm(t))^;0+C⁰²

2 ^jG⁰(t)^j2;0+ 12^jrvm(t)^j2 whereC⁰ is a positive constant such that^jv^j;0 C^0jrv^j ,⁸v²V:

Integrating (3.12) over (0,t) 0< t < tm, taking into consideration (2.7) and (2.8) and noting thatvm(0) =v⁰_m(0) = 0, we get

12^jv⁰_m(t)^j2+a⁰

2 ^jrv_m(t)^j2+b^{0Z t}

0

jrv⁰_m(s)^j2ds+

Z t

0

pv⁰_m(s)²

;0

(3.13) ds

12^kF^k2_L²⁽⁰_TL²⁽⁾⁾+C⁰²

2 ^kG^0k2_L²⁽⁰_TL^2(;0))+ (G(t)v_m(t))^;0 + 12

Z t

0 n

jv⁰_m(s)^j2+^jrvm(s)^j2ods:

On the other hand, for ar arbitrary >0 we have (G(t)v_m(t))^;0 C⁰²

4 ^jG(t)^j2;0+^jrv_m(t)^j2: (3.14)

Combining (3.13), (3.14) and choosing >0 small enough, we obtain the rst estimate

jv⁰_m(t)^j2+^jrvm(t)^j2+

Z t

0

jrv_m⁰ (s)^j2ds+

Z t

0

pv⁰_m(s)²

;0

dsL¹ (3.15)

whereL¹is a positive constant independent ofm²

N

^andt20T]:

3.1.2. The Second Estimate. First of all we are going to estimatev_m(0) inL () norm. Takingt = 0 in (3.11) and taking into account thatvm(0) =v_m⁰ (0) = 0, it follows that

(v_m⁰⁰(0)!j) = (F(0)!j) + (G(0)!j)^;0: (3.16)

From (3.3) and (3.8) we have thatG(0) = 0, and from (3.16) we conclude

jv^{0 0}_m(0)^j2= (F(0)v_m⁰⁰(0)) which implies that

jv_m⁰⁰(0)^jL ⁸m²

N

(3.17)

whereLis a positive constant independent ofm²

N

^:

Now, taking the derivative of (3.11) with respect tot, we can write (3.18) (v⁰⁰⁰_m(t)!j) +a(v⁰_m(t)!j) +b(v⁰⁰_m(t)!j) + (v^{0 0}_m(t)!j)^;0

= (F⁰(t)!j) + (G⁰(t)!j)^;0: Multiplying both sides of (3.18) by g_jm⁰⁰ (t) and summing over 1 j m we obtain

(3.19) 12 d dt

n

jv⁰⁰_m(t)^j2+a(v_m⁰ (t)v_m⁰ (t))^o+b(v_m⁰⁰(t)v_m^{0 0}(t)) + ^pv_m^{0 0}(t)²

;

0

= (F⁰(t)v_m^{0 0}(t)) + d

dt⁽G⁰(t)v⁰_m(t))^;0;(G⁰⁰(t)v_m⁰ (t))^;0: Using arguments analogous to those considered in the rst estimate, observing that v_m⁰ (0) = 0, and taking into account (3.17), from (3.19) we obtain the second estimate

jv⁰⁰_m(t)^j2+^jrv_m⁰ (t)^j2+

Z t

0

jrv_m⁰⁰(s)^j2ds+

Z t

0

pv⁰⁰_m(s)²

;

0

dsL² (3.20)

whereL²is a positive constant independent ofm²

N

^andt20T]:

Due to estimates (3.15) and (3.20) we can extract a subsequence^fv^gof^fvm^g

such that

v* v weak star in L¹(0TV) (3.21)

v⁰ * v⁰ weak star in L¹(0TV) (3.22)

v⁰⁰* v⁰⁰ weak in L²(0TV) (3.23)

v^{0 0}* v^{0 0} weak star in L¹(0TL²()) (3.24)

v⁰* v⁰ weak in L²(0TL²(;⁰)): (3.25)

The above convergences are sucient to pass to the limit.

3.2.

Uniqueness.

Supose we have two solutions y and ^y of problem (1.1). Then z=y^;y^satises

(z⁰⁰(t)w) +a(z(t)w) +b(z⁰(t)w) + (z⁰(t)w)^;0 = 0 ⁸w²V (3.26)

z(0) =z⁰(0) = 0:

Takingw= 2z(t) in (3.26) we obtain dtd

n

jz⁰(t)^j2+a(z(t)z(t))^o+ 2b(z⁰(t)z⁰(t)) + 2 ^pz⁰(t)²

;0

= 0: (3.27)

Integrating (3.27) over (0,t) we obtain

jz⁰(t)^j2+a^0jrz(t)^j2+ 2b^{0Z t}

0

jrz⁰(s)^j2ds+ 2

Z t

0

pz⁰(s)²

;0

ds= 0 which implies that^jrz(t)^j=^jz⁰(t)^j= 0:This completes the proof.

3.3.

Solvability of weak solutions.

We have just proved the existence of solu- tions of the problem (1.1) when y⁰ andy¹ are smooth. By density arguments we conclude the same for weak solutions. However, the principal diculty is due to the existence of a sequence of initial data which satises the hypothesis of compatibility (3.3). For this end and sinceg(0) = 0 giveny⁰y¹² V L²() it is sucient to consider

y⁰²D(A) =

u²V Au²L²() and @u

@a = 0 on ;⁰

such that

y^0!y⁰ in V

and y¹²D(B)^\H⁰¹() such that y^1!y¹ in L²():

The uniqueness of weak solutions requires a regularization procedure and can be obtained using the standard method of Visik-Ladyshenskaya, cf. J. L. Lions 9]

Chapter 3, Section 8.2.

4.

Asymptotic Behaviour

In this section we obtain the uniform decay of the energy given in (1.2) for strong solutions, since the same occurs for weak solutions using standard density arguments.

The derivative of the energy given by (1.2) is E⁰(t) =^;b(y⁰(t)y⁰(t))^{; p}y⁰(t)²

;

0

+ (f(t)y⁰(t)) + (g(t)y⁰(t))^;0: (4.1)

Letand be positive constants such that

jv^j2jrv^{j2 8}v²V (4.2)

and

pv²

;

0

^jrv^{j2 8}v²V:

(4.3)

For an arbitrary" >0 we dene the perturbed energy E"(t) =E(t) +"(t) (4.4)

where

(t) =

Z

y⁰ydx:

(4.5)

Proposition 4.1.

There existsC¹>0 such that

jE"(t)^;E(t)^j"C¹E(t) ⁸t0 and ⁸" >0:

Proof.

From (2.7), (4.2) and (4.5) we obtain

j(t)^j1

2^jy^0j2_{+ 12}a^;10 a(yy)^;a^;10 + 1 E(t): (4.6)

If we deneC¹=a^;10 + 1then from (4.4) and (4.6) we can write

jE_"(t)^;E(t)^j="^j(t)^j"C¹E(t): This concludes the proof.

Proposition 4.2.

There existC²=C²(") and "¹ positive constants such that E_"⁰(t)^;"E(t) +C^2jf(t)^j2+^jpg(t)^j2;0

⁸t0 and ⁸"²(0"¹]:

Proof.

First of all we must estimate ⁰(t) in terms of E(t): Taking the derivative of(t) given in (4.5) and replacingy⁰⁰ by^r(a_ij(x)^ry) +^r(b_ij^ry⁰) +f in the expression obtained it follows that

⁰(t) =

Z

r(a_ij(x)^ry)y dx+

Z

r(b_ij^ry⁰)y dx+

Z

fy dx+

Z

jy^0j2dx:

(4.7)

On the other hand, from Gauss' theorem and taking into account that

@@ya +@y⁰

@b =(g^;y⁰) on ;⁰ we obtain

(4.8)

Z

r(aij(x)^ry)y dx+

Z

r(bij^ry⁰)y dx

=^;a(y(t)y(t))^;b(y⁰(t)y(t))^;

Z

;0

y⁰y d; +

Z

;0

gy d;: Replacing (4.8) in (4.7) adding and subtracting the term^Rjy^0j2dxin (4.7) we get ⁰(t) =^;2E(t)^;b(y⁰(t)y(t)) + 2^Z

jy^0j2dx^;Z

;

0

y⁰y d; +^Z

fy dx+^Z

;

0

gy d;: (4.9)

Now, from (2.7), (4.2), (4.3) and (4.9), using for an arbitrary >0 the inequality ab ^a42 +b² we can write

⁰(t)^;(2^;8)E(t) +

2+^jjb^jj2a^;10 4

j ry⁰(t)^j2 (4.10)

+a^;10 4

py⁰(t)²

;0

+a^;10 4

jf(t)^j2+ ^pg(t)²

;0

where

jjb^jj= ^Xn

ij⁼¹

jjbij^jjL¹⁽⁾:

Choosing= ⁸ from (4.10) we have

⁰(t)^;E(t) +M^1jry⁰(t)^j2+M^{2 p}y⁰(t)²

;0

+M³

jf(t)^j2+ ^pg(t)²

;0

(4.11) where

M¹= 2^;+^jjb^jj2a^;10 M²= 2a^;10 and M³= 2a^;10 : Thus, combining (2.8), (4.1), (4.4) and (4.11), we conclude

E⁰_"(t) =E⁰(t) +"⁰(t)

;b^0jry⁰(t)^{j2; p}y⁰(t)²

;0

+ (f(t)y⁰(t)) + (g(t)y⁰(t))^;0

;"E(t) +"M^1jry⁰(t)^j2+"M^{2 p}y⁰(t)²

;0

+"M³

jf(t)^j2+ ^pg(t)²

;0

which implies

E_"⁰(t)^;(b^0;"(M¹+ 1))^jry⁰(t)^j2;(1^;"(M²+ 1)) ^py⁰(t)²

;0

(4.12)

;"E(t) +

4" ⁺"M^3jf(t)^j2+

1

4"⁺"M^3jg(t)^j2;0: Dening

"¹=min b⁰

M¹+ 1M²¹+ 1

and choosing"²(0"¹] it follows that E_"⁰(t)^;"E(t) +C²(")

jf(t)^j2+ ^pg(t)²

;

0

which concludes the proof.

Proof of the exponential decay.

We dene

"⁰=min^f"¹₂C^11g

and let us consider"²(0"⁰]:From Proposition 4.1 we have (1^;C¹")E(t)E_"(t)(1 +C¹")E(t) (4.13)

Since"1=2C¹,

12E(t)E"(t)3

2E(t)2E(t) ⁸t0 (4.14)

and therefore

;"E(t)^;"

2E"(t): (4.15)

Hence, from (4.15) and considering Proposition 4.2 we obtain E_"⁰(t)^;"

2E_"(t) +C^2jf(t)^j2+^jpg(t)^j2;0: Consequently

dtd

E"(t)exp("

2t)C^2jf(t)^j2+^jpg(t)^j2;0exp("

2t):

Integrating the above inequality over 0,t] we get E"(t)exp(^;"

2t)E"(0) +C²exp(^;"

2t)

Z t

0

exp("

2s)^jf(s)^j2+^jpg(s)^j2;0 and taking into consideration (4.14) we see that

E(t)

3E(0) + 2C^{2Z t}

0

exp("

2s)^jf(t)^j2+^jpg(t)^j2;0

exp(^;"

2t): (4.16)

Combining (4.16) with the assumption (2.10) we prove the desired decay and nish the proof of Theorem 2.1.

References

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Control and Optimization,¹⁹(1981), 114{122, MR 82i:93007a.

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6] V. Komornik and E. Zuazua,A direct method for boundary stabilization of the wave equation, J. Math. Pures et Appl.⁶⁹(1990), 33{54, MR 92c:35020.

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Departamento de Matematica, Universidade Estadual de Maringa, 87020-900, Mar- inga - PR, Brasil.

[email protected]

This paper is available via http://nyjm.albany.edu:8000/j/1998/4-11.html.