On a Mixed Problem for a Constant Coefficient Second-Order System

Volume 2010, Article ID 526917,11pages doi:10.1155/2010/526917

Research Article

On a Mixed Problem for a Constant Coefficient

Second-Order System

Rita Cavazzoni

Via Millaures 12, 10146 Turin, Italy

Correspondence should be addressed to Rita Cavazzoni,[email protected]

Received 2 July 2010; Accepted 1 December 2010

Academic Editor: Peter W. Bates

Copyrightq2010 Rita Cavazzoni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The paper is devoted to the study of an initial boundary value problem for a linear second-order differential system with constant coefficients. The first part of the paper is concerned with the existence of the solution to a boundary value problem for the second-order differential system, in the stripΩARd−1×0, A, whereAis a suitable positive number. The result is proved by means

of the same procedure followed in a previous paper to study the related initial value problem. Subsequently, we consider a mixed problem for the second-order constant coefficient system, where the space variable varies inΩAand the time-variable belongs to the bounded interval0, T ,

withT sufficiently small in order that the operator satisfies suitable energy estimates. We obtain by superposition the existence of a solutionu ∈ L2 _{0, T× 0, A, H}3_Rd−1_{, by studying two}

related mixed problems, whose solutions exist due to the results proved for the Cauchy problem in a previous paper and for the boundary value problem in the first part of this paper.

1. Introduction

Consider the second-order linear differential operator

Q · λ∂2_t−S∂t− d

α1

Fα∂α d−1

α1 d−1

β1

Eα,β∂α∂βEd,d∂2d−G. 1.1

The coefficients of the operatorQsatisfy the following assumptions:

iλis a positive real number;

iifor allα, β1, . . . , d,S, Fα_{, G, E}α,β_{ared×dsymmetric matrices with real entries;}

iiifor everyv∈Cd_{,ReGv, v ≥cGv}2_{, wherec}

ivfor all v ∈ Cd_{,Sv, v ∈ R,F}d_{v, v ∈ R; in addition, there exist two positive} constantscSand cd such that for everyv ∈ Cd,Sv, v ≥ cSv2, andFdv, v ≥

cdv2_;

vfor every α 1, . . . , d, for all v ∈ Cd_,ReEα,α_{v, v ≥ cα,αv}2_{, with c}

α,α positive constant.

We will denote byxa point ofRd_{, byythe firstd−1 coordinates ofx, and bytthe} time variable.

LetAbe a positive real number, and denote byΩAthe subset ofRd,ΩARd−1×0, A. In the first section of the paper we will be concerned with the following boundary value problem

Q u Jx, t, x∈ΩA, t∈R,

Ed,d_{uy,0, tgy, t, y∈R}d−1_{, t∈R,} 1.2

whereuis the unknown vector-valued function, whereasJandgare given functions, which take values inRd_{and are defined inΩ}

A×RandRd−1×R, respectively.

Under suitable assumptions on the functionsJ andg and on the coefficients of the operatorQ, we will prove that, in the case where the positive real numberAis sufficiently small, there exists a functionu ∈ L2

locRd−1×R, H20, A ∩L2 0, A, H3Rd−1×R, which provides a solution to the boundary value problem 1.2. The existence of the solution is established by means of the techniques applied in 1to prove that the initial value problem for the system Q u J, admits a solution u ∈ L2 _{0, T, H}3_Rd_{: the main result of 1} states that if the assumptionsi–ivlisted above along with other suitable conditions are fulfilledseeProposition 3.1, then the adjoint operator ofQsatisfies a priori estimates, which allow proving the existence of the solution to the Cauchy problem, through the definition of a suitable functional and a duality argument. As we will explain below, the boundary value problem1.2can be regarded exactly as an initial value problem. For this reason, the result ofSection 2does not represent any significant advance with respect to the results proved in

1.

The main novelty of the paper is represented by the study of a mixed problem in the third section. The interest in this kind of problems relies on the fact that they appear frequently as physical models: mixed problems for second-order hyperbolic equations and systems of equations occur in the theory of sound to describe for instance the evolution of the air pressure inside a room where noise is produced, as well as in the electromagnetism to describe the evolution of the electromagnetic field in some region of spacethe system of Maxwell equations accounts for this kind of phenomenon.

The existence results, stated both for the initial value problem in 1 and for the boundary value problem inSection 2of this paper, turn out to be the backbone in proving the existence of the solution to the following mixed problem in the strip ΩA, as the time variabletbelongs to the bounded interval 0, T, whereTis a suitable positive real number:

Q w Jx, t, x∈ΩA, t∈ 0, T,

Ed,dwy,0, tgy, t, y∈Rd−1, t∈ 0, T,

wx,0 hx, x∈ΩA.

We will assume that the vector-valued function J belongs to the space L2 _{0, A,}

H2_Rd−1_{×R, while, as for the initial data, we suppose thatg∈H}3_Rd−1_×Randh∈H3_Rd_. Let us notice that in the problem1.3an initial value for the unknown vector fielduand a Dirichlet boundary condition only are prescribed. Due to the a priori estimates that we will derive for the operatorQ, it is not required in problem1.3, in contrast with classical mixed problems for hyperbolic second-order systems, that the first-order derivatives of usatisfy a prescribed condition at the boundary of the domain ΩA×0, T . This lack of information about the initial value of the first-order derivatives results in a possible nonuniqueness of the solution to1.3. The existence result ofSection 3will be achieved by means of the definition of two mixed problems related to1.3: the existence of the solution to the former will be established similarly to the result obtained in 1, while the latter will be studied like the boundary value problem considered inSection 2. Subsequently, thanks to the linearity of the operatorQ, a solution to1.3belonging to the space L2 _{0, T× 0, A, H}3_Rd−1_{will be} determined by superposition of the solutions to the preliminary mixed problems.

2. Boundary Value Problem

By adopting the same strategy of 1to prove the existence of the solution to the initial value problem, let us determine the existence of a solution to1.2through a duality argument, by proving energy estimates.

Let us denote byQ_{the adjoint operator ofQ:}

Q u λ∂2_tuS∂tu

d

α1

Fα∂αu d−1

α1 d−1

β1

Eα,β∂α∂βuEd,d∂2du−Gu. 2.1

For allα, β1, . . . , d−1, letCG_{, C}S_{, C}α_{, C}α,β_{be the norms of the matricesG, S, F}α_{, E}α,β_, respectively.

Proposition 2.1. Consider the operator Q defined in1.1 and the corresponding adjointQ_{. Let}

the conditions (i)–(iv), listed in the Introduction, be fulfilled. In addition, assume the sums ηd

cd/2−Cd,d/2,η1cG−CS/2− _d−1

α1Cα/2−1/2,η2 λ−CS/2, and for everyα1, . . . , d−1,

ηα cα,α−Cα/2−d−_{β /}1_α,β1Cα,β to be positive real numbers. Moreover, denote byδthe sumδ _d−1

α1Cα/2CG/2CS/2Cd1/2,and suppose that, as long as the positive numberAis sufficiently

small,min{η1, ηα, η2:α1, . . . , d−1} −δeδA>0.

Define the linear space

ΣA

φ:Rd×R−→Rd :φ∈C2Rd1,

supp φ compact subset ofRd−1×− ∞, A×R; ∀y,∀t, φy,0, t0.

2.2

Then, for all functionsφ∈ΣA, the following estimates hold:

_φ 2

L2 _0,A,H−2_Rd−1_×R≤C1 Q

∂dφ0 2_H−3_Rd−1_×R≤C2eδA Q

φ 2_L2 _0,A,H−3_Rd−1_×R, 2.4

whereC1, C2are suitable positive constants.

Proof. Consider a vector functionφ∈C2_Rd1_{, with compact support in the subsetR}d−1_×− ∞, A×R. Applying the Fourier transform with respect to both the tangential variabley

and the time variablet, we can obtain a priori estimates, which, by substitutingxdfortand carrying out similar calculations, turn out to be like the estimates obtained in 1for the initial value problem. Subsequently, assuming the functionφbelongs toΣA, we deduce the a priori estimates2.3and2.4for the adjoint operatorQ_.

Due to2.3 and2.4, by means of a duality argument, we can prove the existence result for the solution to the boundary value problem1.2.

Proposition 2.2. Consider the boundary value problem 1.2, and let the

assumptions of Proposition 2.1 be satisfied. Furthermore, suppose that J ∈

L2 _{0, A, H}2_Rd−1 _{× R and g ∈ H}3_Rd−1 _{× R. Then, the boundary value}

problem 1.2 has a solution u ∈ L2

locRd−1 × R, H20, A ∩ L2 0, A,

H3Rd−1×R.

Proof. The result can be proved by means of the same tools used to establish the existence result for the initial value problem in 1. For the sake of completeness, let us sketch the proof. Because of the a priori estimate2.3, the operatorQis one-to-one on the linear space

ΣA. Letφ∈ΣA, and define the following linear functional on the spaceQΣA:

LQ_φ A

0

J, φ_H2_,H−2dxd−

g, ∂xdφ0

H3_,H−3. 2.5

The functionalLturns out to be well defined, sinceQ_{is injective onΣ}

A. Moreover,L is continuous with respect to the norm of the spaceL2 0, A, H−3Rd−1×R, because of the energy estimates proved inProposition 2.1. Due to the Hahn-Banach Theorem, the functional Lcan be extended to the spaceL2 _{0, A, H}−3_Rd−1_{×R. Let us denote byMthis functional.} By the Riesz Theorem, there exists a function u, which belongs to the dual space

L2 _{0, A, H}3_Rd−1 _{× R, so that for every v ∈ L}2 _{0, A, H}−3_Rd−1 _{× R, Mv A}

0 u, vH3_,H−3dxd. In particular, in the case whereφ∈ΣA, _A

0

u, Qφ_H3_,H−3dxd _A

0

J, φ_H2_,H−2dxd−

g, ∂xdφ0

H3_,H−3. 2.6

Hence, since for everyφ∈C∞₀ Rd−1×0, A ×R

A

0

u, Qφ_H3_,H−3dxd

A

0

J, φ_H2_,H−2dxd, 2.7

let us extend the functions u and J by zero outside the interval 0, A. By means of an approximation argument, we construct a sequence of smooth functionsvn ∈C∞Rd×R∩

L2_{R, H}3_Rd−1_{×R, so thatv}

nnturns out to be convergent to the functionu, with respect to the norm of the spaceL2_{R, H}3_Rd−1_{×R. We define the approximating sequence in such} a way for alln ∈ N, vn vanishes outside a compact neighbourhood of 0, A, for example,

−1, A1.

Moreover, let us denote byP the differential operatorP · λ∂2

t −S∂t− _d−1

α1Fα∂α _d−1

α1 _d−1

β1Eα,β∂α∂β−G. Therefore,Q · Ed,d∂d∂d −Fd∂d P ·. Set for alln ∈ N,Jn

Ed,d_∂

d∂dvn−Fd∂dvnP vn. Letφ∈C∞0Rd×−1, A . Thus,

A1

−1

Rd−1_×R

Ed,d∂2_dvn−Fd∂dvn, φ

dy dt dxd

A1

−1

Rd−1_×R

−P vn Jn, φ

dy dt dxd.

2.8

Since the sequencevnnis convergent to the functionuwith respect to the norm of

L2_{R, H}3_Rd−1_{×R, integrating by parts, we obtain}

lim n→ ∞

A1

−1

Rd−1_×R

Jn, φ

dy dt dxd

_A1

−1

Rd−1_×R

P u, φEd,d_{u, ∂}2 dφ

Fd_{u, ∂} dφ

dy dt dxd

_A1

−1

Rd−1_×R

J, φdy dt dxd.

2.9

As a result, the sequence of functionsJnn is weakly convergent to the functionJ in the spaceL2_Rd−1_{× −1, A1×R. Hence, the sequenceJ}

nn turns out to be bounded in

L2_Rd−1_{× −1, A1×R.}

Let us consider again the systemEd,d_∂

d∂dvn−Fd∂dvn−P vn Jn, for everyn∈N. By means of integration on the interval −1, a, witha < A1, we obtain

Ed,d∂dvna−Fdvna a

−1

−P vn Jndσ. 2.10

Let us estimate theL2_{-norm of the r.h.s. of2.10. We deduce that}

_A1

−1

Rd−1_×R

a

−1

−P vn Jndσ 2

dy dt da≤A22const. 2.11

Since there exists a suitable constant such that, for alln∈N, Fd_vnL2

Rd−1_×−1,A1×R≤

const., the sequence∂dvnnturns out to be bounded inL2Rd−1× −1, A1×R. Thus, the sequence of functions∂2

xdvnnalso turns out to be bounded inL

Letα 1, . . . , d−1. Differentiating with respect toxα or totboth members of2.10, we have

Ed,d_∂

α∂dvna−Fd∂αvna _a

−1

∂α−P vn Jndσ. 2.12

The sequence∂αvnnis convergent to∂αuinL2Rd−1×−1, A1 ×R.

Due to the convergence of vnn to u in L2R, H3Rd−1 × R, the sequence

a

−1P vndσnturns out to converge to a

−1P udσinL2Rd−1×−1, A1 ×R. Furthermore, the sequence ∂α

a

−1P vndσn is weakly convergent in L

2_Rd−1_×− 1, A1 ×R. Therefore, it is bounded in the spaceL2_Rd−1_{×−1, A1 ×R. Similarly, the} sequence₋a₁∂αJndσ_n also turns out to be bounded in L2Rd−1×−1, A1 ×R. Hence,

∂α∂dvnnis bounded inL2Rd−1×−1, A1 ×R. Since the sequences of functions∂2dvnn and∂α∂dvnnare bounded in L2Rd−1×−1, A1 ×R, the sequence∂dvnnturns out to satisfy the assumptions of the Riesz-Fr´echet-Kolmogorov theorem.

Thus the function u admits a first-order weak derivative with respect to xd in

L2

locRd−1× − 1, A 1 ×R. Therefore the function u belongs to the space L2locRd−1 ×

R, H1_{0, A ∩L}2 _{0, A, H}3_Rd−1_×R.

If we introduce a new variable, the system1.2may be reduced to a first-order system with respect to the variablexd. Let us denote byUthe vector functionU u, Ed,d∂duTand byC1andC2the following differential operators

C1− d−1

α1 d−1

β1

Eα,β∂α∂β−λ∂2t d−1

α1

Fα∂αS∂tG; C2FdEd,d− 1

. 2.13

Thus, the system1.2can be rewritten as

∂d

u

Ed,d_∂ du

−

⎛

⎝0d Ed,d− 1

C1 C2 ⎞ ⎠

u

Ed,d_∂ du

0d

J

. 2.14

By settingC0dEd,d−1_I C1 C2

andF0d_J , the system1.2becomes∂dUCUF. Because of the regularity properties of the functionu,C1uandC2uturn out to belong toL2 _{0, A, L}2

locRd−1×R.

Multiplying both members of the system by any function of the space

C∞₀Rd−1_{×0, A ×R, we prove that the vector functionUhas a weak partial derivative with} respect to the variablexd. Thusu∈L2_locRd−1×R, H20, A ∩L2 0, A, H3Rd−1×Rand

Q u J, a.e. inRd−1_{×0, A ×R.} Sinceubelongs toL2

locRd−1×R, H20, A , the traces ofuand∂duare well-defined on the hyperplane xd 0, and belong to L2_locRd−1 × R, H3/20, A and L2_locRd−1 ×

Let us consider a functionφ∈C∞₀ Rd−1_{×− ∞, A ×R, which, in a neighbourhood of}

xd0 has the formφy, xd, t xdψy, t, withψ ∈C₀∞Rd−1×R. Thus,

LQ_{φ A}

0

J, φ_H2_,H−2dxd−

g, ∂xdφ0

H3_,H−3

_A

0

u, Q_φ

H3_,H−3dxd. 2.15

Integrating by parts,

Rd−1

A

0

R

Q u, φdy dxddt−

Rd−1

R

Ed,du0, ψdy dt

Rd−1

_A

0

R

J, φdy dxddt−

Rd−1

R

g, ψdy dt.

2.16

Hence, we obtainEd,d_{u0 g, a.e. inR}d−1_×R.

3. Mixed Problem

This section deals with the study of the initial boundary value problem1.3. We will prove the existence of the solution after solving two auxiliary problems: first we will determine the solution of an initial value problem, by means of the techniques developed in 1; next, we will find the solution of a suitable boundary value problem, in accordance with the results stated in the previous section. Since the operatorQis linear, the solution to the mixed problem

1.3will be determined by superposition. As a matter of fact, both auxiliary problems are mixed problems, but, as we will explain below, the solution of the former will be found as in the case of initial value problems, whereas the latter may be studied in the framework of boundary value problems.

Let us define the first problem as follows:

Q v 0, x∈ΩA, t∈ 0, T,

Ed,dvy,0, t0, y∈Rd−1, t∈ 0, T,

vx,0 hx, x∈ΩA,

3.1

whereh∈H3_Rd_.

We consider the Cauchy problem

Q v 0, x∈Rd, t∈ 0, T,

vx,0 hx, x∈Rd,

and determine the solution by means of a duality argument through the procedure followed in 1. For this purpose, we have to assume conditions on the coefficients of the operatorQ

in order for energy estimates to be satisfied. Furthermore, let us define the linear space

FTφ:Rd×R−→Rd:φ∈C2Rd−1× 0,∞ × 0, T,

supp φ compact subset ofRd−1× 0,∞ ×− ∞, T; ∀x, φx,0 0;

3.3

and quote from 1the following result.

Proposition 3.1. Consider the operatorQdefined in1.1and the corresponding adjointQ_{. Assume}

the conditions (i)–(iv) listed in the Introduction to be fulfilled. In addition, let the sumscS/2−λ/2,

C1cG−

_d

α1Cα/2−1/2, and for everyα1, . . . , d, C2minα1,...,d{cα,α−Cα/2− _d

β /α,β1Cα,β},

be positive real numbers. Moreover, we denote byCthe sumCd_α1Cα/2CG/2CS1/2and supposemin{C1, C2} −Ce CT is positive, provided thatTis small enough.

Then, the operatorQ_{satisfies the following estimates:}

afor everyu∈ FT,

u2_L2 _0,T,H−2_Rdλ∂tu2_L2 _0,T,H−3_Rd≤c1Q u2_L2 _0,T,H−3_Rd; 3.4

bfor allu∈ FT,

∂tu02_H−3_Rd≤c2eCT Q u2_L2 _0,T,H−3_Rd, 3.5

withc1, c2being positive constants that are independent ofT.

By taking into account the energy estimates of Proposition 3.1, we establish the following existence result.

Proposition 3.2. Consider the initial boundary value problem 3.1, and let the assumptions of Proposition 3.1 be satisfied. If the function h ∈ H3_Rd_{, then the problem 3.1 has a solution}

v∈L2 _{0, T, H}3_Rd_∩L2

locRd, H20, T .

Proof. Let us define onQ_{FTthe linear functional}

ΛQφ−λh, ∂tφ0

H3_,H−3, 3.6

whereφ∈ FT.

Through the procedure followed in 1, we can prove there exists a function v ∈ L2

locRd, H20, T ∩L2 0, T, H3Rd, such that for everyφ ∈ FT, _T

0 v, Q φH3_,H−3dt

In addition, due to the results proved in 1,Q v 0, a.e.x ∈ Rd_{, t ∈0, T , and}

vx,0 hx, a.e.x∈Rd_.

Moreover, since for all t ∈ 0, T, the function v·, t ∈ H3_Ω

A, the trace of v on the boundary ofΩA belongs toL2 0, T, H3−1/2Rd−1. Let us determine the trace ofvon

xd 0. Letψ be a function of the space C2Rd−1 × 0,∞ × 0, T, such that suppψis a compact subset ofRd−1_{× 0, A ×0, T . Therefore,ψ ∈ FT and}T

0 v, Q ψH3_,H−3dt 0.

By integrating by parts,

_T

0

ΩA

Q v, ψdt dx

− _T

0

Rd−1

Ed,dv0, ∂d ψ0

Fdv0, ψ0−Ed,d∂dv0, ψ0

dy dt0.

3.7

Thus,

_T

0

Rd−1

Ed,dv0, ∂d ψ0

Fdv0, ψ0−Ed,d∂dv0, ψ0

dy dt0. 3.8

Consider a vector function ψ, which, as long as xd is nonnegative and sufficiently small, has the formψy, xd, t xdχy, t, withχ∈ C₀∞Rd−1×0, T . Hence, we deduce by means of a standard argument thatEd,d_{v0 0, a.e. inR}d−1_{×0, T .}

Let us define now the second auxiliary initial boundary value problem in order to obtain by superposition a solution to1.3:

Q w J, x∈ΩA, t >0,

Ed,d_{wy,0, tg, y∈R}d−1_{, t >0,}

wx,0 0, x∈ΩA.

3.9

We assume thatJ∈L2 _{0, A, H}2_Rd−1_{×R, whereasg∈H}3_Rd−1_×R.

The existence of the solution to3.9can be proved by means of the duality argument and a procedure similar to the previous problem.

Proposition 3.3. Consider the initial boundary value problem 3.9, and let the assumptions of Proposition 2.1be satisfied. If J ∈ L2 _{0, A, H}2_Rd−1_{×Rand g ∈ H}3_Rd−1_{×R, then there}

exists a functionw ∈ L2 _{0, A, H}3_Rd−1 _×R∩L2

locRd−1×R, H20, A , which provides a

solution to3.9.

Proof. We consider the boundary value problem

Q w J, x∈ΩA, t >0,

Ed,dwy,0, tg, y∈Rd−1, t >0.

Since the operatorQsatisfies the assumptions ofProposition 2.1, energy estimates can be proved for the adjoint operatorQ_{. Let us define the linear space}

GA φ:Rd_×R−→Rd_:φ∈C2_Rd−1_{× 0, A× 0,∞ ,}

supp φ compact subset ofRd−1× 0, A × 0,∞ ; ∀y, ∀t, φy,0, t0.

3.11

For everyφ∈ GA, consider the following functional:

ΔQφ

_A

0

J, φ_H2_,H−2dxd−

g, ∂dφ0

H3_,H−3. 3.12

As proved in the previous section, the functional turns out to be well defined and continuous as a consequence of the energy estimates. Furthermore, the functional can be extended to the space L2 _{0, A, H}−3_Rd−1 _{×R, and there exists a function w ∈}

L2 _{0, A, H}3_Rd−1 _{×R, so that for everyφ ∈ GA,}A

0 J, φH2_,H−2dxd− g, ∂dφ0H3_,H−3

_A

0 w, Q φH2_,H−2dxd. After studying the regularity properties of the function w as in the previous section and in 1, we can prove that w ∈ L2_locRd−1 × R, H20, A ∩ L2 _{0, A, H}3_Rd−1 _{×R, Q w J, a.e. in R}d−1_{×0, A ×R, and w satisfies the boundary} condition.

The functionw will be a solution to the mixed problem3.9after proving that the initial condition is satisfied. First of all, we have to remark that, sincew∈L2 _{0, A, H}3_Rd−1_×

R, for all xd ∈ 0, A, the trace of w·, xd,· on Rd−1 × {0} turns out to belong to the spaceH2Rd−1. Thus,w·,0∈L2 0, A, H2Rd−1. Consider a functionφ ∈ GA, such that suppφ ⊂Rd−1_{×0, A × 0,∞ . Hence,}A

0 J, φH2_,H−2dxd _A

0 w, Q φH2_,H−2dxd. By integrating by parts, we have

_A

0

Rd−1

_∞

0

Q w, φdt dy dxd− _A

0

Rd−1

λw0, ∂tφ0

dy dxd

A

0

Rd−1

λ∂tw0, φ0

dy dxd

A

0

Rd−1

_∞

0

J, φdt dy dxd,

3.13

whence

A

0

Rd−1

λw0, ∂tφ0

dxddy−

A

0

Rd−1

λ∂tw0, φ0

dxddy0. 3.14

By means of a suitable choice of the function φ, we prove that wx,0 0, a.e. in

Rd−1_{×0, A . Therefore, the functionwturns out to be a solution to3.9.}

Finally, both the solutionvto the auxiliary problem3.1and the solutionw to3.9

belong to the spaceL2 _{0, T× 0, A, H}3_Rd−1_{. Denote byuthe sumuwv. The function}

derivatives with respect totandxd, which belong toL2_locRd−1×0, A ×0, T . Hence,uturns out to be a solution to the initial boundary value problem2.3. To avoid inconsistencies in the auxiliary mixed problems3.1and3.9as well as in1.3, we have to require that the data

gandhsatisfy compatibility conditions: ifgandhare smooth functions up to the boundary, we assume that, for everyy∈Rd−1,hy,0 gy,0 0.

Let us state now the main result.

Theorem 3.4. Consider the initial boundary value problem 1.3. Suppose that the hypotheses of Propositions 3.2and 3.3are satisfied. IfJ ∈ L2 0, A, H2Rd−1×R,g ∈ H3Rd−1 ×R, and h∈H3_Rd_{, then there exists a functionu∈L}2 _{0, T× 0, A, H}3_Rd−1_{, which provides a solution}

to1.3.

References