TO A QUASILINEAR WAVE EQUATION SUBJECT TO INTEGRAL CONDITIONS

TO A QUASILINEAR WAVE EQUATION SUBJECT TO INTEGRAL CONDITIONS

ABDELFATAH BOUZIANI AND NABIL MERAZGA

Received 27 January 2004 and in revised form 12 February 2004

This paper presents a well-posedness result for an initial-boundary value problem with only integral conditions over the spatial domain for a one-dimensional quasilinear wave equation. The solution and some of its properties are obtained by means of a suitable application of the Rothe time-discretization method.

1. Introduction

Recently, the study of initial-boundary value problems for hyperbolic equations with boundary integral conditions has received considerable attention. This kind of condi- tions has many important applications. For instance, they appear in the case where a direct measurement quantity is impossible; however, their mean values are known.

In this paper, we deal with a class of quasilinear hyperbolic equations (Tis a positive constant):

∂²v

∂t^{2 −}

∂²v

∂x^{2 =}f

x,t,v,∂v

∂t

, (x,t)∈(0, 1)×[0,T], (1.1) subject to the initial conditions

v(x, 0)=v0(x), ∂v

∂t(x, 0)=v1(x), 0x1, (1.2) and the boundary integral conditions

₁

0v(x,t)dx=E(t), 0tT, 1

0xv(x,t)dx=G(t), 0tT,

(1.3)

where f,v0,v1,E, andGare sufficiently regular given functions.

Problems of this type were first introduced in [3], in which the first author proved the well-posedness of certain linear hyperbolic equations with integral condition(s). Later,

Copyright©2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:3 (2004) 211–235

2000 Mathematics Subject Classification: 35L05, 35D05, 35B45, 35B30 URL:http://dx.doi.org/10.1155/S1687183904401071

similar problems have been studied in [1,4,5,6,7,8,16,24,25] by using the energetic method, the Schauder fixed point theorem, Galerkin method, and the theory of charac- teristics. We refer the reader to [2,9,10,11,12,13,14,15,17,21,22,23,26] for other types of equations with integral conditions.

Differently to these works, in the present paper, we employ the Rothe time-discre- tization method to construct the solution. This method is a convenient tool for both the theoretical and numerical analyses of the stated problem. Indeed, in addition to giving the first step towards a fully discrete approximation scheme, it provides a constructive proof of the existence of a unique solution. We remark that the application of Rothe method to this nonlocal problem is made possible thanks to the use of the so-calledBouziani space, first introduced by the first author, see, for instance, [4,6,20].

Introducing a new unknown functionu(x,t)=v(x,t)−r(x,t), where

r(x,t)=62G(t)−E(t)x−23G(t)−2E(t), (1.4) problem (1.1)–(1.3) with inhomogeneous integral conditions (1.3) can be equivalently reduced to the problem of finding a functionusatisfying

∂²u

∂t^{2 −}

∂²u

∂x^{2 =}f

x,t,u,∂u

∂t

, (x,t)∈(0, 1)×I, (1.5) u(x, 0)=U0(x), ∂u

∂t(x, 0)=U1(x), 0x1, (1.6) 1

0u(x,t)dx=0, t∈I, (1.7)

1

0xu(x,t)dx=0, t∈I, (1.8)

where

I:=[0,T], f

x,t,u,∂u

∂t

:=f

x,t,u+r,∂u

∂t +∂r

∂t

−∂²r

∂t², U0(x) :=v0(x)−r(x, 0),

U1(x)=v1(x)−∂r

∂t(x, 0).

(1.9)

Hence, instead of looking forv, we simply look foru. The solution of problem (1.1)–(1.3) will be directly obtained by the relationv=u+r.

The paper is divided as follows. In Section 2, we present notations, definitions, as- sumptions, and some auxiliary results. Moreover, the concept of the required solution is stated, as well as the main result of the paper.Section 3is devoted to the construction of approximate solutions of problem (1.5)–(1.8) by solving the corresponding linearized time-discretized problems, while inSection 4, some a priori estimates for the approxima- tions are derived. We end the paper bySection 5where we prove the convergence of the method and the well-posedness of the investigated problem.

2. Preliminaries, notation, and main result

LetH²(0, 1) be the (real) second-order Sobolev space on (0, 1) with norm · H²(0,1)and let (·,·) and · be the usual inner product and the corresponding norm, respectively, inL²(0, 1). The nature of the boundary conditions (1.7) and (1.8) suggests introducing the following space:

V:=

φ∈L²(0, 1);

1

0φ(x)dx= 1

0xφ(x)dx=0

, (2.1)

which is clearly a Hilbert space for (·,·).

Our analysis requires the use of the so-called Bouziani spaceB¹2(0, 1) (see, e.g., [4,5]) defined as the completion of the spaceC0(0, 1) of real continuous functions with compact support in (0, 1), for the inner product

(u,v)_B2¹= 1

0xu· xv dx (2.2)

and the associated norm

v_B2¹=

(v,v)_B¹₂, (2.3)

where_xv:= 0^xv(ξ)dξfor every fixedx∈(0, 1). We recall that, for everyv∈L²(0, 1), the inequality

v²_B¹₂1

2v² (2.4)

holds, implying the continuity of the embeddingL²(0, 1)B¹2(0, 1).

Moreover, we will work in the standard functional spaces of the types C(I,X), C^0,1(I,X),L²(I,X), andL^∞(I,X), whereXis a Banach space, the main properties of which can be found in [19].

For a given functionw(x,t), the notationw(t) is automatically used for the same func- tion considered as an abstract function of the variablet∈Iinto some functional space on (0, 1). Strong or weak convergence is denoted by→or, respectively.

The Gronwall lemma in the following continuous and discrete forms will be very use- ful to us thereafter.

Lemma2.1. (i)Letx(t)0, and leth(t),y(t)be real integrable functions on the interval [a,b]. If

y(t)h(t) + _t

ax(τ)y(τ)dτ, ∀t∈[a,b], (2.5) then

y(t)h(t) + _t

ah(τ)x(τ) exp _t

τx(s)ds

dτ, ∀t∈[a,b]. (2.6)

In particular, ifx(τ)≡Cis a constant andh(τ)is nondecreasing, then

y(t)h(t)e^C(t−a), ∀t∈[a,b]. (2.7) (ii)Let{ai}be a sequence of real nonnegative numbers satisfying

aia+bh i k=1

ak, ∀i=1,..., (2.8)

wherea,b, andhare positive constants withh <1/b. Then ai a

1−bhexp

b(i−1)h 1−bh

, ∀i=1, 2,.... (2.9)

Proof. The proof is the same as that of [18, Lemma 1.3.19].

Throughout the paper, we will make the following assumptions:

(H1) f(t,w,p)∈L²(0, 1) for each (t,w,p)∈I ×V×V and the following Lipschitz condition:

f(t,w,p)−f(t,w,p)_B2¹l|t−t|+w−w_B¹₂+p−p_B¹₂

(2.10) is satisfied for allt,t∈Iand allw,w,p,p∈V, for some positive constantl;

(H2)U0,U1∈H²(0, 1);

(H3) the compatibility conditionU0,U1∈V, that is, concretely, 1

0U0(x)dx= 1

0xU0(x)dx=0, (2.11)

₁

0U1(x)dx= ₁

0xU1(x)dx=0. (2.12)

We look for a weak solution in the following sense.

Definition 2.2. A weak solution of problem (1.5)–(1.8) means a functionu:I→L²(0, 1) such that

(i)u∈C^0,1(I,V);

(ii)u has (a.e. in I) strong derivatives du/dt ∈L^∞(I,V)∩C^0,1(I,B2¹(0, 1)) and d²u/dt²∈L^∞(I,B¹2(0, 1));

(iii)u(0)=U0inVand (du/dt)(0)=U1inB2¹(0, 1);

(iv) the identity d²u

dt²(t),φ

B¹2

+u(t),φ= f

t,u(t),du dt(t)

,φ

B¹2

(2.13) holds for allφ∈V and a.e.t∈I.

Note that sinceu∈C^0,1(I,V) anddu/dt∈C^0,1(I,B2¹(0, 1)), condition (iii) makes sense, whereas assumption (H1), together with (i) and the fact that du/dt∈L^∞(I,V) and d²u/dt²∈L^∞(I,B2¹(0, 1)), implies that (2.13) is well defined. On the other hand, the ful- fillment of the integral conditions (1.7) and (1.8) is included in the fact thatu(t)∈V, for allt∈I.

The main result of the present paper reads as follows.

Theorem 2.3. Under assumptions (H1), (H2), and (H3), problem (1.5)–(1.8) admits a unique weak solutionu, in the sense ofDefinition 2.2, that depends continuously upon the data f,U0, andU1. Moreover, the following convergence statements hold:

uⁿ−→u inC(I,V), with convergence orderO 1

n^1/2

, δuⁿ−→du

dt inCI,B¹2(0, 1), d

dtδuⁿ d²u

dt² inL²I,B¹2(0, 1),

(2.14)

asn→∞, where the sequences{uⁿ}_nand{δuⁿ}_nare defined in (3.18) and (3.19), respectively.

3. Construction of an approximate solution

Letnbe an arbitrary positive integer, and let{tj}ⁿ_j=1be the uniform partition ofI,tj=jhn

withh_n=T/n. Successively, for j=1,...,n, we solve the linear stationary boundary value problem

uj−2uj−1+uj−2

h²_n ⁻ d²uj

dx^{2 =} f_j, x∈(0, 1), (3.1) 1

0u_j(x)dx=0, (3.2)

1

0xuj(x)dx=0, (3.3)

where

fj:= f

tj,uj−1,uj−1−uj−2

hn

, (3.4)

starting from

u−1(x)=U0(x)−h_nU1(x), u0(x)=U0(x), x∈(0, 1). (3.5) Lemma3.1. For eachn∈N^∗and eachj=1,...,n, problem (3.1)j–(3.3)_jadmits a unique solutionuj∈H²(0, 1).

Proof. We use induction on j. For this, suppose thatuj−1 anduj−2 are already known and that they belong toH²(0, 1), then f_j∈L²(0, 1). From the classical theory of linear ordinary differential equations with constant coefficients, the general solution of (3.1)_j

which can be written in the form d²uj

dx^{2 −} 1

h²_nuj=−2uj−1+uj−2

h²_n ⁻fj (3.6)

is given by

u_j(x)=k1(x) cosh x

h_n+k2(x) sinh x

h_n, x∈(0, 1), (3.7) wherek1andk2are two functions ofxsatisfying the linear algebraic system

dk1

dx(x) cosh x h_n+dk2

dx (x) sinh x h_n⁼0, dk1

dx(x) sinh x h_n+dk2

dx(x) cosh x

h_n ⁼h_nF_j(x),

(3.8)

with

Fj:=−2uj−1+uj−2

h²_n ⁻fj. (3.9)

Since the determinant of (3.8) is

∆=cosh² x

hn−sinh² x

hn=1, (3.10)

then

dk1

dx(x)=

0 sinh x

h_n hnFj(x) cosh x

h_n

= −hnFj(x) sinh x h_n,

dk2

dx(x)=

cosh x

h_n 0

sinh x

h_n hnFj(x)

=hnFj(x) cosh x h_n,

(3.11)

that is,

k1(x)= −hn

_x

0Fj(ξ) sinh ξ

hndξ+λ1, k2(x)=h_n

_x

0F_j(ξ) cosh ξ

h_ndξ+λ2,

(3.12)

withλ1andλ2two arbitrary real constants. Inserting (3.12) into (3.7), we get uj(x)=hn

_x

0Fj(ξ) sinhx−ξ

hn dξ+λ1cosh x

hn+λ2sinh x

hn. (3.13)

Obviously, the functionu_j will be a solution to problem (3.1)_j–(3.3)_j if and only if the pair (λ1,λ2) is selected in such a manner that conditions (3.2)_jand (3.3)_jhold, that is,

λ1

₁

0cosh x hndx+λ2

₁

0sinh x

hndx= −hn

₁

0

_x

0Fj(ξ) sinhx−ξ hn dξ dx, λ1

1

0xcosh x h_ndx+λ2

1

0xsinh x

h_ndx= −h_n 1

0

_x

0xF_j(ξ) sinhx−ξ h_n dξ dx.

(3.14)

An easy computation shows that (λ1,λ2) is the solution of the linear algebraic system

λ1sinh 1 hn+λ2

cosh 1

hn−1

= − 1

0

_x

0Fj(ξ) sinhx−ξ hn dξ dx, λ1

sinh 1

hn−hncosh 1 hn+hn

+λ2

cosh 1

hn−hnsinh 1 hn

= − 1

0

_x

0 xF_j(ξ) sinhx−ξ h_n dξ dx,

(3.15)

whose determinant is

Dh_n=2h_n−2h_ncosh 1

h_n+ sinh 1 h_n

=2 sinh 1 2hn

cosh 1

2hn−2h_nsinh 1 2hn

.

(3.16)

Note thatD(hn) does not vanish for anyhn>0, indeed equationD(hn)=0 is equivalent to the equation cosh(1/2h_n)−2h_nsinh(1/2h_n)=0, that is, to the equation tanh(1/2h_n)= 1/2hn which clearly has no solution. Therefore, for all hn>0, system (3.15) admits a unique solution (λ1,λ2)∈R², which means that problem (3.1)_j–(3.3)_j is uniquely solv- able, and it is obvious thatu_j∈H²(0, 1) sinceF_j∈L²(0, 1).

Now, we introduce the notations

δuj:=uj−uj−1

hn , j=0,...,n, δ²uj:=δuj−δuj−1

hn =uj−2uj−1+uj−2

h²_n , j=1,...,n,

(3.17)

and construct the Rothe functionuⁿ:I→H²(0, 1)∩V by setting uⁿ(t)=u_j−1+δu_jt−t_j−1

, t∈

t_j−1,t_j, j=1,...,n, (3.18)

and the following auxiliary functions:

δuⁿ(t)=δuj−1+δ²uj t−tj−1

, t∈ tj−1,tj

, j=1,...,n, (3.19) uⁿ(t)=



u_j fort∈

t_j−1,t_j, j=1,...,n, U0 fort∈

−hn, 0, (3.20)

δuⁿ(t)=



δu_j fort∈

t_j−1,t_j, j=1,...,n, U1 fort∈

−hn, 0. (3.21)

We expect that the limitu:=lim_n→∞uⁿexists in a suitable sense, and that is the desired weak solution to our problem (1.5)–(1.8). The demonstration of this fact requires some a priori estimates whose derivation is the subject of the following section.

4. A priori estimates for the approximations

In what follows,cdenote generic positive constants which are not necessarily the same at any two places.

Lemma4.1. There existc >0andn0∈N^∗such that

ujc, (4.1)

δujc, (4.2)

δ²uj

B2¹c, (4.3)

for allj=1,...,nand allnn0.

Proof. To derive these estimates, we need to write problem (3.1)_j–(3.3)_j in a weak for- mulation.

Letφbe an arbitrary function from the spaceV defined in (2.1). One can easily find that

_x

0(x−ξ)φ(ξ)dξ= ²_xφ, ∀x∈(0, 1), (4.4) where

²_xφ:= _{x ξ}φ=

_x

0dξ _ξ

0φ(η)dη. (4.5)

This implies that ²1φ=

1

0(1−ξ)φ(ξ)dξ= 1

0φ(ξ)dξ− 1

0ξφ(ξ)dξ=0. (4.6) Next, we multiply, for all j=1,...,n, (3.1)jby²_xφand integrate over (0, 1) to get

₁

0δ²uj(x)²_xφ dx− ₁

0

d²uj

dx²(x)²_xφ dx= ₁

0 fj(x)²_xφ dx. (4.7)

Here, we used the notations (3.17). Performing some standard integrations by parts for each term in (4.7) and invoking (4.6), we obtain

1

0δ²u_j(x)²_xφ dx= 1

0

d dx

_x

δ²u_j²_xφ dx

= _x

δ²u_j²_xφ

x=1 x=0−

1 0_x

δ²u_jxφ dx

= −

δ²uj,φ_B¹₂, ₁

0

d²uj

dx² (x)²_xφ dx=duj

dx(x)²_xφ

x=1 x=0−

₁

0

duj

dx(x)xφ dx

= − ₁

0

duj

dx (x)xφ dx

= −uj(x)xφ

x=1 x=0+

₁

0uj(x)φ(x)dx

= uj,φ, 1

0 fj(x)²_xφ dx= 1

0

d dx

xfj

²_xφ dx

= xfj²_xφ

x=1 x=0−

1

0xfjxφ dx

= − fj,φ_B2¹,

(4.8)

so that (4.7) becomes finally

δ²u_j,φ_B¹₂+u_j,φ=

f_j,φ_B¹₂, ∀φ∈V,∀j=1,...,n. (4.9) Now, fori=2,...,j, we take the difference of the relations (4.9)i−(4.9)_i−1, tested with φ=δ²ui=(δui−δui−1)/hn which belongs toV in view of (3.2)_i −(3.3)_i, (3.2)_i−1 − (3.3)_i−1, and (H3). We have

δ²ui−δ²ui−1,δ²ui

B2¹+δui,δui−δui−1

=

fi−fi−1,δ²ui

B2¹, (4.10) then, using the identity

2v,v−w= v²− w²+v−w² (4.11) and its analog for (·,·)_B¹₂, it follows that

δ²ui²

B¹2−δ²ui−1²

B2¹+δ²ui−δ²ui−1²

B2¹+δui²

−δui−1²+δui−δui−1²=2fi−fi−1,δ²ui

B¹2, (4.12) hence, omitting the third and last terms in the left-hand side, we get

δ²ui²

B2¹+δui²δ²ui−1²

B¹2+δui−1²+ 2fi−fi−1

B¹2

δ²ui

B2¹. (4.13)

We sum up these inequalities and obtain δ²uj²

B2¹+δuj²δ²u1²

B¹2+δu1²+ 2 j i=2

fi−fi−1

B¹2

δ²ui

B¹2, (4.14) hence, thanks to the Cauchy inequality

2ab1

εa²+εb², ∀a,b∈R,∀ε∈R^∗+, (4.15) we can write, forε=hn,

δ²uj²

B¹2+δuj²δ²u1²

B2¹+δu1²+ 1 hn

j i=2

fi−fi−1²

B2¹+hn

j i=2

δ²ui²

B¹2. (4.16) To majorize_i^j₌₂f_i−f_i−1²_B¹₂, we remark that

f_i−f_i−1²

B2¹=ft_i,u_i−1,δu_i−1

−ft_i−1,u_i−2,δu_i−2²

B¹2

l²h_n+u_i−1−u_i−2

B¹2+δu_i−1−δu_i−2

B¹2

2

=l²h²_n1 +δui−1

B¹2+δ²ui−1

B¹2

2

3l²h²_n1 +δu_i−1²

B¹2+δ²u_i−1²

B¹2

, i=2,...,j.

(4.17)

Summing up fori=2,...,j, we may arrive at j

i=2

fi−fi−1²

B¹23l²(j−1)h²_n+ 3l²h²_n j i=2

δui−1²

B¹2+δ²ui−1²

B¹2

(4.18)

or

j i=2

f_i−f_i−1²

B2¹3l²(j−1)h²_n+ 3l²h²_n

j−1

i=1

δ²u_i²_B¹₂+δu_i²_B2¹. (4.19)

To estimateδ²u1²_B¹

2+δu1², we test the relation (4.9)1withφ=δ²u1=(δu1−δu0)/h_n

=(δu1−U1)/hnwhich is an element ofV owing to (3.2)1–(3.3)1and assumption (H3).

We have

δ²u1²

B¹2+ u1

hn,δu1−U1

= f1,δ²u1

B¹2 (4.20)

or

δ²u1²

B¹2+δu1,δu1−U1

= f1,δ²u1

B2¹− U0,δ²u1

. (4.21)

But

U0,δ²u1

= 1

0U0(x)d dx

xδ²u1

dx

=U0(x)xδ²u1

^x=1

x=0− 1

0

dU0

dx (x)xδ²u1dx

= − 1

0

dU0

dx (x)xδ²u1dx,

(4.22)

and since

_x d²U0

dx²

=dU0

dx (x)−dU0

dx (0), ∀x∈(0, 1), (4.23) we get, due to (4.6),

U0,δ²u1

= − 1

0_x d²U0

dx²

_xδ²u1dx−dU0

dx (0)²1δ²u1

= − 1

0x

d²U0

dx²

xδ²u1dx

= − d²U0

dx² ,δ²u1

B¹2

,

(4.24)

in light of which (4.21) becomes δ²u1²

B¹2+δu1,δu1−U1

=

f1+d²U0

dx² ,δ²u1

B¹2

. (4.25)

Therefore, δ²u1²

B¹2+1

2δu1²−1

2U1²+1

2δu1−U1²f1+d²U0

dx²

B¹2

δ²u1

B¹2, (4.26) hence,

2δ²u1²

B2¹+δu1²U1²+ 2f1+d²U0

dx²

B¹2

δ²u1

B¹2

U1²+f1+d²U0

dx^{2 2}

B2¹

+δ²u1²

B¹2

U1²+ 2 f1²

B¹2+d²U0

dx^{2 2}

B¹2

+δ²u1²

B¹2,

(4.27)

from which it follows that δ²u1²

B2¹+δu1²U1²+ 2

c1+d²U0

dx^{2 2}

B¹2

, (4.28)