ON A SHOCK PROBLEM INVOLVING A NONLINEAR VISCOELASTIC BAR NGUYEN THANH LONG, ALAIN PHAM NGOC DINH, ppt

Problem 1.1–1.2 describes the shockbetween a solid body and a nonlinear viscoelastic bar resting on a viscoelastic base withnonlinear elastic constraints at the side, constraints associa

A NONLINEAR VISCOELASTIC BAR

NGUYEN THANH LONG, ALAIN PHAM NGOC DINH, AND TRAN NGOC DIEM

Received 3 August 2004 and in revised form 23 December 2004

We treat an initial boundary value problem for a nonlinear wave equationutt − uxx+

K | u | α u + λ | ut | β ut = f (x, t) in the domain 0 < x < 1, 0 < t < T The boundary condition

at the boundary pointx =0 of the domain for a solutionu involves a time convolution

term of the boundary value ofu at x =0, whereas the boundary condition at the otherboundary point is of the formux(1,t) + K1u(1, t) + λ1ut(1,t) =0 withK1 andλ1givennonnegative constants We prove existence of a unique solution of such a problem inclassical Sobolev spaces The proof is based on a Galerkin-type approximation, variousenergy estimates, and compactness arguments In the case ofα = β =0, the regularity ofsolutions is studied also Finally, we obtain an asymptotic expansion of the solution (u, P)

of this problem up to orderN + 1 in two small parameters K, λ.

• u0,u1,f are given functions,

• K, K1,α, β, λ and λ1≥0 are given constants

and the unknown functionu(x, t) and the unknown boundary value P(t) satisfy the

fol-lowing Cauchy problem for ordinary differential equation

P //(t) + ω2P(t) = hutt(0,t), 0< t < T,

Copyright©2006 Hindawi Publishing Corporation

Boundary Value Problems 2005:3 (2005) 337–358

DOI: 10.1155/BVP.2005.337

338 On a shock problem involving a nonlinear viscoelastic bar

whereω > 0, h ≥0,P0,P1 are given constants Problem (1.1)–(1.2) describes the shockbetween a solid body and a nonlinear viscoelastic bar resting on a viscoelastic base withnonlinear elastic constraints at the side, constraints associated with a viscous frictionalresistance

In [1], An and Trieu studied a special case of problem (1.1)–(1.2) withα = β =0 andf ,

u0,u1andP0vanishing, associated with the homogeneous boundary conditionu(1, t) =0instead of (1.1)3being a mathematical model describing the shock of a rigid body and alinear visoelastic bar resting on a rigid base

From (1.2), solving the equation ordinary differential of second order, we get

which we will do henceforth

In [9,10], Dinh and Long studied problem (1.1)1,2,4and (1.5) with Dirichlet boundarycondition at boundary pointx =1 in [10] extending an earlier result of theirs fork =0

The integral in (1.6)3is a boundary condition which includes the memory effect Here,

byu we denote the displacement and by G the relaxation function The function µ ∈

Wloc1,∞(R +) withµ(t) ≥ µ0> 0 and µ /(t) ≤0 for allt ≥0 Frictional dissipative boundarycondition for the wave equation was studied by several authors, see for example [4,5,6,

11,16,17,18,19] and the references therein In these works, existence of solutions andexponential stabilization were proved for linear and for nonlinear equations In contrastwith the large literature for frictional dissipative, for boundary condition with memory,

we have only a few works as for example [12,13,14]

Applying the Volterra’s inverse operator, Santos [15] transformed (1.6)3into

is devoted to the study of the regularity of the solutionu Finally, inSection 4we obtain

an asymptotic expansion of the solution (u, P) of the problem (1.1), (1.5) up to order

N + 1 in two small parameters K, λ The results obtained here may be considered as

gen-eralizations of those in An and Trieu [1] and in Long and Dinh [2,3,8,9,10]

2 The existence and uniqueness theorem

Put Ω=(0, 1),Q T =Ω×(0,T), T > 0 We omit the definitions of the usual function

spaces:C m(Ω), L p(Ω) and W m,p

Ωand denoteW m,p = W m,p(Ω), L p = W0,p(Ω) and

H m = W m,2(Ω), 1≤ p ≤ ∞,m ∈ IN The norm in L2is denoted by ·  Also, we denote

by·,·the scalar product inL2or the dual pairing between continuous linear functionalsand elements of a function space, by  ·  X the norm of a Banach spaceX, by X / itsdual space, and byL p(0,T; X), 1 ≤ p ≤ ∞the Banach space of real measurable functions

At last, denoteu(t) = u(x, t), u /(t) = u t(t) =(∂u/∂t)(x, t), u //(t) = u tt(t) =(∂2u/∂t2)(x, t),

u(r)(t) =(∂ r u/∂t r)(x, t), ux(t) =(∂u/∂x)(x, t), uxx(t) =(∂2u/∂x2)(x, t).

Further, we make the following assumptions:

(H0)α ≥0,β ≥0,K ≥0,λ ≥0,

(H )h ≥0,K ≥0,K +h > 0 and λ > 0,

340 On a shock problem involving a nonlinear viscoelastic bar

(H2)u0∈ H2andu1∈ H1,

(H3) f , f t ∈ L2(0,T; L2),

(H4)k ∈ H1(0,T) ∩ W2,1(0,T),

(H5)g ∈ H2(0,T).

Then we have the following theorem

Theorem 2.1 Let assumptions (H0)–(H5) be satisfied Then there exists a unique weak solution u of problem ( 1.1 ), ( 1.5 ) such that

Proof of Theorem 2.1 The proof consists of Steps1–5

Step 1 (Galerkin approximation) Let { wj }be an enumeration of a basis ofH2 We findthe approximate solution of problem (1.1), (1.5) in the form

equation of (2.5)1byc / m j, summing up with respect to j and afterwards integrating with

respect to the time variable from 0 tot, we get

whereC1is a constant independent ofm Using the inequality 2ab ≤ εa2+ (1/ε)b2for all

a, b ∈ Rand for allε > 0, it follows that

342 On a shock problem involving a nonlinear viscoelastic bar

Sm(t) ≤ M(1)T +M(2)T

t

0Sm(τ)dτ, 0≤ t ≤ Tm ≤ T, (2.19)which implies by Gronwall’s lemma

grating with respect to the time variable from 0 tot, after some rearrangements we get

344 On a shock problem involving a nonlinear viscoelastic bar

Integrating by parts in the integrals of the right-hand side of (2.22), we get

whereC3> 0 is a constant depending on u0,u1,f , K, λ only.

On the other hand, it follows from (2.11)–(2.13) that

Then, by means of (2.13), (2.20), and (2.29) we deduce that

and from here and (2.22)–(2.28) we obtain

Xm(t) ≤ C2+C2+ k /(0) u2m(0,t) + 2 g /(t)u / m(0,t) + 2 k(0)um(0,t)u / m(0,t)

346 On a shock problem involving a nonlinear viscoelastic bar

≤ C2+C2+ k /(0) C2MT+ 4 g /(t) C0  2

+1

4Xm(t)+ 4k2(0)C 4MT+1

Xm(t) ≤ M T(3)+M T(4)

t

0Xm(τ)dτ ∀ t ∈[0,T], (2.33)where

andM T(3)is a constant depending onT, f , g, k, C2,C3,C0, andM T only By Gronwall’s

lemma we deduce that

Step 4 (limiting process) From (2.7), (2.20), (2.23), (2.35), and (2.36)1–3we deduce theexistence of a subsequence of{(u m,P m,Q m)}, still also so denoted, such that

um α um −→ | u | α u strongly inL2

QT

348 On a shock problem involving a nonlinear viscoelastic bar

Similarly, we can also obtain from (2.29), (2.35), (2.39)2and inequality (2.42) withα = β,

Passing to the limit in (2.5)1,4–5, by (2.38)1,2,4 and (2.40)–(2.41) and (2.45) we haveu

satisfying the problem

Henceu ∈ L ∞(0,T; H2) and the existence proof is completed

Step 5 (uniqueness of the solution) Let (u i,P i),i =1, 2 be two weak solutions of problem(1.1), (1.5) such that

350 On a shock problem involving a nonlinear viscoelastic bar

(2.58)

By Gronwall’s lemma, we deduce thatZ ≡0 andTheorem 2.1is completely proved

3 Regularity of solutions

In this section, we study the regularity of solution of problem (1.1), (1.5) corresponding

toα = β =0 From here, we assume that (h, K, K1,λ, λ1) satisfy assumptions (H0), (H1).Henceforth, we will impose the following stronger assumptions:

(H[1]1 )u0∈ H3andu1∈ H2,

(H[1]2 ) f , ft, ftt ∈ L2(0,T; L2) and f ( ·, 0)∈ H1,

(H[1]3 )g ∈ H3(0,T),

(H[1]4 )k ∈ H2(0,T).

Formally differentiating problem (1.1) with respect to time and lettingu=  ut andP= P /

we are led to consider the solution u of problem ( Q):

Letu0,u1, f , g, k satisfy assumptions (H[1]1 )–(H[1]4 ) Thenu 0,u 1, f , g, k satisfy assump-

tions (H1)–(H4) and byTheorem 2.1for problem (Q) there exists a unique weak solution

It follows from (3.3)–(3.4) that

We then have the following theorem

Theorem 3.1 Let α = β = 0 and let assumptions (H0), (H1) and (H[1]1 )–(H[1]4 ) hold Then there exists a unique weak solution (u, P) of problem ( 1.1 ), ( 1.5 ) satisfying ( 3.5 ).

Similarly, formally differentiating problem (1.1) with respect to time up to orderr

and lettingu[r] = ∂ r u/∂t r andP[r] = d r P/dt r we are led to consider the solutionu[r]ofproblem (Q[r]):

(H[4r])k ∈ H r+1(0,T), r ≥1

Thenu[0r],u[1r], f[r],g[r],k satisfy (H1)–(H4) Applying againTheorem 2.1for problem(Q[r]), there exists a unique weak solutionu[r] satisfying (2.2) and the inclusion from

352 On a shock problem involving a nonlinear viscoelastic bar

Remark 2.2, that is, such that

We then have the following theorem

Theorem 3.2 Let α = β = 0 and let assumptions (H1) and (H[1r] )–(H[4r] ) hold Then there exists a unique weak solution (u, P) of problem ( 1.1 ), ( 1.5 ) satisfying ( 3.9 ) and

4 Asymptotic expansion of solutions

In this section, we assume thatα = β =0 and (h, K1,λ1, f , g, k) satisfy the assumptions

t

k(t − s)u(0, s)ds.

(QK,λ )

Let (u0,0,P0,0) be a unique weak solution of problem (Q 0,0) as inTheorem 2.1, sponding to (K, λ) =(0, 0), that is,

Let (u, P) =(u K,λ,P K,λ) be a unique weak solution of problem (QK,λ) Then (v, R), with

354 On a shock problem involving a nonlinear viscoelastic bar

satisfies the problem

Then, we have the following lemma

Lemma 4.1 Let α = β = 0 and let assumptions (H1)–(H5) be satisfied Then

Proof By the boundedness of the functions uγ1−1,γ2,u / γ1,γ2−1, (γ1,γ2)∈ Z2

K γ1λ γ2=
K2γ1/(N+1) λ2γ2/(N+1) (N+1)/2

≤
K2+λ2  (N+1)/2

for all (γ1,γ2)∈ Z2,γ1+γ2= N + 1.

Finally, by the estimates (4.7), (4.8), we deduce that (4.5) holds, with

Next, we obtain the following theorem

Theorem 4.2 Let α = β = 0 and let assumptions (H1)–(H5) be satisfied Then, for every

K ≥ 0, λ ≥ 0, problem ( QK,λ ) has a unique weak solution (u, P) =(uK,λ, PK,λ ) satisfying the

asymptotic estimations up to order N + 1 as follows

356 On a shock problem involving a nonlinear viscoelastic bar

where the constantsC, C 0are defined by (2.11), (2.13), respectively Then, we prove, in amanner similar to the above part, that

On the other hand, it follows from (4.3)5, (4.20), that

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Nguyen Thanh Long: Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University HoChiMinh City, 227 Nguyen Van Cu Street, Dist.5, HoChiM- inh City, Vietnam

E-mail address:longnt@hcmc.netnam.vn

Alain Pham Ngoc Dinh: Laboratoire de Math´ematiques et Applications, physique Math´ematique d’Orl´eans (MAPMO), UMR 6628, Bˆatiment de Math´ematiques, Universit´e d’Orl´eans, BP 6759 Orl´eans Cedex 2, France

E-mail address:alpham@worldonline.fr

Tran Ngoc Diem: Department of Mathematics and Computer Science, University of Natural ence, Vietnam National University HoChiMinh City, 227 Nguyen Van Cu Street, Dist.5, HoChiMinh City, Vietnam

Sci-E-mail address:minhducfactory@yahoo.com