Behavior Of The Solution Of Evolutionary Type Equation In A Nonsmooth Domain

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2 (2), 2008, 159-165

©BEYKENT UNIVERSITY

BEHAVIOR OF THE SOLUTION OF

EVOLUTIONARY TYPE EQUATION IN A

NONSMOOTH DOMAIN

R.A.RASULOV

The Sheki Filial of the Institute of Teachers, Sheki, Az5, Republic of Azerbaijan, rasulovr@yahoo.com

Received: 11 July 2008, Accepted: 14 July 2008

ABSTRACT

In this study the estimations of the solution and its derivative with respect to t of the initial boundary value problem for the second order evolutionary type equation in a domain with non smooth boundary are obtained.

EVOLÜSYONAL TÜRDEN DİFERANSİYEL

DENKLEMİN DÜZGÜN OLMAYAN

BÖLGEDEKİ ÇÖZÜMÜNÜN DAVRANIŞI

ÖZET

Makalede ikinci basamaktan evolüsyonal türden denklem için yazılmış başlangıç sınır değer probleminin düzgün olmayan bölgedeki çözümü ve çözümün t 'e göre türevi için bir değerlendirme elde edilmiştir.

Key Words: Evoluationary type equation; Estimations in a nonsmooth domain

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1 INTRODUCTION

It is known that the theory of general boundary value problems for a hyperbolic equation or system of equations in cylindrical domain with smooth data has been developed sufficiently well. But, many problems of the mathematical physics are reduced to an evolutionary type equations. However, equations and system of equations of an evolutionary type in a domain with non smooth boundary has not been studied. It should be noted that the similar results for domain with smooth boundary has been obtained in [1].

0

Let W2U ( Qt ) be the Sobolev's space of the functions having the generalized

d2 0

dtdx

i

following formula

derivatives ^ ^ — g L2 (QT). Define the scalar product in W~2U (QT ) by the

{u,v)i~u(

ßr

) = JQ {u

t

v

t

+ VuVv + Vu

t

Vv

t

)dxdt.

T

We will consider the class of equation

( a.. ^ (i)

d

2

u

^ d . .du

y

— a

i

• (t, x ) —

dt , j=! dx

t

_V

dx. J J d 2 ( 3 ^

_du

dx

;

V

J J d

V — — bi J(t,x)—— + —(b(t,x)u)+a(t,x)u = f(t,x)

dtdx

i

dx

;

dt

with following initial and boundary conditions

u(0, x) = p(x), u

t

(0, x) = y(x), u i

3Q

= 0

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Here, QT = Q X ( 0 , T ) where QG Rn, n > 2 , T < ~ and has the

nonsmooth boundary, and the real value functions at J (t, x ) , bi J (t, x ) ,

(i = 1,2,..., n ) , a(t, x) and b(t, x) satisfy the conditions above

n (3)

a, j (t, x) =

aji

(t, x), y

l

|||

2

<

¿ a - , j (t, x ) | | j

<

Y$\, Y > 0,

i,

j=1 n

da (t x)

|G

R

n

, X ^ ' '

| |

>

0, a

hJ

(t,x)

G

C^

(QT),

a(t, x)

>

0,

i,

j=1 dt n

(3)

da(t, x)__ . , „1,0,

dt > 0, a(t,X)G C1XU(QT),

¿ b , J (t, xllJ > 0, £

D

-J

X)

| | > 0,

i, J=1 i, j = 1 at bu (t, x) G C1,,X1(QT ), b(t, x) > 0, b(t, x) G C1x0 (QT ). ( 4 ) In addition f (x, t) G L2 (QT ) .

The equation (1) is the general form of the equation

d 2

u

A

au

A ( 5 )

=

nA— + Au

dt2

3t

where, n

= const > 0

is a parameter. The eq.(5) describes the distribution of perturbation in viscous medium, the distribution of sound in viscous medium and etc.

Now, we will estimate the solution and its derivatives with respect to t in norm W21,1 (QT ) . We approximate the domain Q by domains Q£ with the

smooth boundary such that Q£ c Q , £ > 0 , dQE G Cc , and

lim

£

^

c Q£

= Q.

Let Q j

= Q

£

x[0,

T], and let u£ is generalized solution of the problem (1),(2) in domain Q£ .

In order to solve the problem (1),(2) we use the following auxiliary results.

Lemma 1. If f G Cc (Qt ), then the problem (1),(2) is a generalized solution

u (x, t) in QT and the following estimates are valid

ll

u(

x

,

1

W~21,1(QT ) <

C

JI

f

llL

2

(QT),

I k

( x

,

t

% % )

< C2

0

f ( x

,

t L2(QT )

+1

ft(x,11

(6)

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with some constants C1, C2 .

Proof. If we consider the problem (1),(2) in Q , then for the solution of this £

(4)

l k (

x

, Oil

#2

uf

Q

T>

-

c j 4

2 (

j

U

« 'IWU(Qj ) -

C

4 (ll

f

'

1L2 (Qj ) +1 ¿ L (Q

Then the sequence u£(x, t) has a weak limit in W2U (QT) and, we assume

that for £ ^ 0 the u0 (x, t) is a weak limit of the sequence u£ (x, t) in

W21'1 (QT ). The function will be a generalized solution of the problem (1),(2)

in QT satisfying the estimates (6),(7).

Lemma 2. If

f (x,t), f

t

(x,t)e L

2

(Q

T

),

then the problem (1),(2) is a

generalized solution u(x, t) e W21'1 (QT ) and the following estimates are

valid

l|u(x,

t

iW

2

U(QT)

-

Q |

f (

x,

t)

|L2(QT),

( 8 )

ll

u

t

( x

,

t I W~U(QT

)

- C6

(ll

f (x,11L2

Q )+l

FT(x,t )

l

L2 QT ) )

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with some constants C5, C6 .

Proof. Let fh be the average function of the left part of the equation (1), and

uh is solution of the equation (1) responding to fh . The convergence of uh

to u should appear from the estimates (6),(7), and for u (x, t) is valid the inequality (8),(9).

We denote the intersection of the cylinder QT with hyperplane t = const by

.

Assume that the boundary of domain ^ consists of the intersection of two infinitely smooth (n — 1) dimensional hypersurfaces along some infinitly smooth (n — 2) dimensional manifold ae C™ under nonzero angle y ( p ) .

d2

Fix

p e ax[0,T]

and reduce the operator . to canonical

^J dxdx. _'

J

(5)

that these results carry a local character, that these are valid in the case when the DQ consists of finite number surfaces C ™ , in pairs intersecting along manifolds of the class C™ .

Note that at the neighborhood U p of the any point p of manifold

S = ax [0, T ] it is possible to introduce a local system of coordinates in the

special form. If the neighborhood U p is small enough, then there is infinity

differentiable transform translating the domain Q n U to angle. Therefore, for the sake of generality it is possible to assume that at some neighborhood of the point p the boundary surface is the form 0 < ^xf + x?, = r < ^ ,

0 < p < 5 = const, here p is polar angle in plane ( xl, x2) ,

I x ! < - , (i = 3,4,...,n).

Let Vk ( Qt ) is a space of the functions having generalized derivatives with 2

respect to all variables until k. order inclusively in QT with norm

KK (QT ) 2 k

JJL 2>

T s=0 2s—2k d s

u

dtS0 dx'dz2

Here, the function P ( t ) G C ( Q t ) is positive everywhere except on the

manifold S and any neighborhood S coinciding with r( x) .

Theorem 1. Assume that

d

' f

( x

:

t

;

z )

e ^

1

(Qt ), 0 < i < i, f = 0

dtldz2 1

at some neighborhood of the

{(x, t), x

e dQ,

t = 0}, 0 <a<

2n,

a^n, k

+1

^

mn

a

m > 0

set

is

integer. Let for any pair . and ( m2, s2) , here mt > 0, st > 0 are integers,

7m.'

^

7IM,' 71m'

such that '- + st — k — 1 < 0, is fulfilled '- + s1 ^ '- + s2.

a a a

2 2

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Then the generalized solution of the problem (1),(2) has the form

(( x, t )u (x, z, t) =

mn+s (10)

Z(x, z)r

a

ln

q

rKj (z, t

j

(p, z, t) + ( ( x , z)v(x, z, t),

j

where

((x, z)

is infinitely differentiable function, with

supp ((x, z) c V

p

and the summation is introduced with respect to multi-index

mn

J = (m, S, q, p ) with

a

+ S < k +1,

0 <

q < q0(m, s),

0 < p < p0( m , s), m > 0 , s > 0 , (m, s, p , q) are integer numbers. Here,

q0 (m,0) = 0 , p0 (m,0) = 1, and for all functions Kj (z, t), v ( x , z, t) the

following estimate i—1

Z

j = 0 d jK:

dt

J1

d

J2

1 i

< C10

Z

i—1 k —2

+ Z Z

H (QT) J-»'-» J—k—2

d

J

+'K

:

dt

11

dz

2

dx'

(11) L2 (QT ) j = 0

d

j

f

dtj dz2 vk (QT ) 2

is valid, here, Q = ( Q n U ) x [ 0 , T], C10 is constant which depends on

domain QT only, and the functions n j ( p , z, t) are infinitely differentiable

with respect to all variables.

Proof. It is obvious that u(t, x) is the generalized solution of the elliptical type equations from space W / ( Q ) almost all t on Qt which is the

intersection of the cylinder QT with hyperplane t = const. Taking into

account the condition of the theorem and applying the theorem on smoothness of the solution of the Dirichlet's problem for the second order elliptic equation in domain with non smooth boundary [2], we get that u(t, x) e C 5 (Qt),

0 < 5 < 1 . Using the suggested methods in [4] we prove the desired results. The analogous results for linear equation has been obtained in [3].

2 2

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REFERENCES

[1] I.V. Suveyka, Differential Equations I. XIX, 2, 1983.

[2] E. Mierzemen, Zur Regular tat verallgemeinerter hosing Ven guesilinearen elliptischen

Differetialgleichungen ZwWEI ter Ordiny in Gebieten mit Eckem. AZ eitschrift fur Analis

und there Anwendungen, Bd 1(4), 1982, pp.69-71.

[3] V.A. Kondratiev, Boundary Problems for Elliptic equations in domain with conical or angle

domains. Notes MMO, pp. 205-292.

[4] I.I. Melnikov, Singularities Solutions of mixed Problems for second order hyperbolic

Equations in Domain with piecewise smooth Boundary. Uspehi. Math. Nauk. V37, 1, 1982,pp.149-150.