Ona second order linearparabolic equation withvariable coefficients in a non-regular domain of R3 Текст научной статьи по специальности «Математика»

УДК 517.9

On a Second Order Linear Parabolic Equation with Variable Coefficients in a Non-Regular Domain of R3

Ferroudj Boulkouane*

Faculté des science de la nature et de la vie Université de Bejaia, 6000 Bejaia Algerie

Arezki Kheloufim

Faculty of Technology, Lab. of Applied Mathematics Bejaia University, 6000 Bejaia Algeria

Received 11.10.2017, received in revised form 22.01.2018, accepted 06.03.2018 This paper is devoted to the study of the following variable-coefficient parabolic equation in non-divergence form 2 2

dtu - ai(t,xi,x2)diiu + ^ bi(t,xi,x2 )diU + c(t,xi ,X2)u = f (t,xi,x2), i=1 i=l

subject to Cauchy-Dirichlet boundary conditions. The problem is set in a non-regular domain of the form Q = {(t,xi) e R2 : 0 <t<T,pi (t) <xi <ф2 (t)} x ]0,b[,

where фк, k = 1, 2 are "smooth" functions. One of the main issues of this work is that the domain can possibly be non-regular, for instance, the singular case where ф1 coincides with ф2 for t = 0 is allowed. The analysis is performed in the framework of anisotropic Sobolev spaces by using the domain decomposition method. This work is an extension of the constant-coefficients case studied in [15].

Keywords: parabolic equations, non-regular domains, variable coefficients, anisotropic Sobolev spaces. DOI: 10.17516/1997-1397-2018-11-4-416-429.

1. Introduction and main results

This work is devoted to the study of the following two-space dimensional non-divergence parabolic equation

( dtu + Lu = f e L2 (Q),

\ 1 uuqxet =0,

where

2 2 L = - ai(t,xi,X2)dii + bi(t,xi,x2)di + c(t,xi,x2),

i=i i=i

d d2

with di = ——, dii = ——2 ,i = 1,2. L2 (Q) stands for the space of square-integrable functions on

dxi dxf

Q with the measure dtdxidx2, dQ is the boundary of Q, is the part of the boundary of Q where t = T and the coefficients ai, bi, i = 1,2 and c satisfy non-degeneracy-assumptions (to be made more precise later). Here Q (see, Fig. 1) is the three-dimensional non-cylindrical domain

Q = {(t,xi) e R2 : 0 <t<T,^i (t) <xi <^2 (t)} x ]0,b[,

* ferroudjboulkouane@gmail.com © Siberian Federal University. All rights reserved

where T and b are positive numbers, and y2 are two Lipschitz continuous real-valued functions on [0, T] satisfying

y (t) := y (t) - yi (t) > 0, Vt G ]0, T] and y (0) = 0.

Fig. 1. The non-regular domain Q

Besides being interesting in itself, Problem (1.1) governs, for instance, the concentration of the biological oxygen demand in water in the case of a river with variable width and constant depht, see for example, similar problems in [1] and [31]. Also, the particular form of the operator L helps us to prove the "energy" type estimate of Proposition 2.1 which is essential in proving the existence of solutions to Problem (1.1).

The difficulty related to this kind of problems (in addition to the presence of variable coefficients) comes from this singular situation for evolution problems, i.e., is allowed to coincide with ^>2 for t = 0, which prevents the domain Q to be transformed into a regular domain without the appearance of some degenerate terms in the parabolic equation, see for example Sadallah [30]. On the other hand, we cannot recast such problems in semigroups setting like in [6] and [27]. Indeed, since the initial condition is defined on a measure zero set, then the semigroup generating the solution cannot be defined.

It is well known that there are two main approaches for the study of boundary value problems in such non-smooth domains. We can work directly in the non-regular domains and we obtain singular solutions (see, for example [3,16,18] and [20]), or we impose conditions on the non-regular domains (and on the coefficients) to obtain regular solutions (see, for example [2,17] and [30]). It is the second approach that we follow in this work. So, let us consider the anisotropic Sobolev space

U0

(Q) = {u GU1'2 (Q): u\aQ^T =o} ,

with

where

U1'2 (Q) = {u : dtu, dau G L (Q), \«\ < 2}, a = (i1, i2) G N2, \a\ = i1 + i2, dau = d^d¡2u

The space H1'2 (Q) is equipped with the natural norm, that is

1/2

llull

Hi.2(Q)

lldtull

L2(Q)

+

E lldaull

L2 (Q)

a <2

2

2

In this paper we prove that Problem (1.1) admits a unique solution u in H1'2 (Q), under the following additional assumptions on the smooth differentiable coefficients c, ai, bi, i = 1, 2 and on the functions of parametrization yk, k = 1, 2,

y'k (t) y (t) ^ 0 as t ^ 0, k =1,2, (1.2)

( ai > 0 (parabolicity condition) (__ 3)

\ ai, bi, c, dtai, diai e L~ (Q) ,i = 1, 2, ( .3)

with \ai\ ^ c0, \Vai\ ^ c1, \bi\ ^ c2, |c| ^ c3, aiaj ^ a0 > 0 (i; j = 1, 2), b2 ^ b0 > 0, c2 ^ d0 > 0, where c0,c1,c2,c3,a0, b0 and d0 are positive constants. Our main result is

Theorem 1.1. We assume that y1 and y2 fulfil the condition (1.2), and the coefficients ai, bi, i = 1, 2, and c fulfil the condition (1.3), then the operator

22 L = dt ai(t,xi,x2)du + ^ bi(t,xi,x2)di + c(t,xi ,x2)

i=i i=i

is an isomorphism from H1'2 (Q) into L2 (Q).

The case a1 = a2 = 1, b1 = b2 = c = 0, corresponding to the heat operator has been studied in [15] and [17] both in bi-dimensional and multidimensional cases.

Whereas parabolic equations with variables coefficients in cylindrical domains are well studied, the literature concerning such problems in non-cylindrical domains does not seem to be very rich, see [24] for the case of smooth coefficients and [28] for the case of discontinuous coefficients. Concerning parabolic equations in time-varying domains we can find in Fichera [9] and Oleinik [29] solvability results for non-divergence parabolic equations. For the divergence form case, see [5,14] and [25]. In the case of Holder spaces functional framework, we can find in Baderko [4] results for non-cylindrical domains of the same kind but which cannot include our domain. In [10], we can find Wiener type criterion in the framework of continuous spaces which cannot include our L2-case.

Our work is motivated by the interest of researchers for many mathematical questions related to non-regular domains. During the last decades and since many applied problems lead directly to boundary-value problems in "bad" domains, numerous authors studied partial differential equations in non-smooth domains. Among these we can cite [7,8,11,12,19,21,22,32] and the references therein.

The organization of this paper is as follows. In Section 2, we divide the proof of Theorem 1.1 into three steps:

a) We prove well-posedness results for Problem (1.1) when Q is replaced by the truncated domain { }

Qa = {(t,xi) e R2 : a < t < T; yi (t) < xi < y2 (t)} x ]0,b[,

with a > 0, (Theorem 2.1).

b) We approximate Q by a sequence (Qn), n e N*, of such truncated regular domains and we establish a uniform estimate (see Proposition 2.1) of the type

\\un\\H1,2{Qn ) < K \\f \\L2(Q) ,

where un is the solution of Problem (1.1) in Qn and K is a constant independent of n.

c) We build a solution u of Problem (1.1), by considering un the 0-extension to Q of the solutions un (un, n e N* exists by Theorem 2.1), and showing (in virtue of Proposition 2.1) that u^k ^ u, weakly in L2 (Q), for a suitable increasing sequence of integers (nk)k-^1.

Note that this work may be extended at least in the following directions:

1. The function f on the right-hand side of the equation of Problem (1.1), may be taken in Lp (Q), p € ]1, . The domain decomposition method used here does not seem to be appropriate for the space Lp (Q) when p = 2. An idea for this extension can be found in [13] or in [23].

2. The bi-dimensional case in x, can be naturally extended to an upper dimension in x, such as, for example, the following problem

N N

i a,i(t, xi, ...,xn)diiu

i=i i=i

in the domain

dtu - E ai(t, xi,..., xn )duu + ^ bi(t,xi, ...,xn )diu + c(t,x1, ...,xn )u = f (t,xi, ...,xn ),

^(t,xu..,xN) € Rn+1 : 0 <t<T, 0 ^ xl + ... + x2N < ^(t^ , N > 2. These questions will be developed in forthcoming works.

2. Proof of Theorem 1.1

We divide the proof of Theorem 1.1 into three steps.

2.1. Step 1: case of a truncated domain Qa which can be transformed into a parallelepiped

In this subsection, we replace Q by

Qa = {(t,xi) € R2 : a < t < T; (t) < xi < pi (t)} x ]0,6[,

with a > 0, (see, Fig. 2). Thus, we have p (a) > 0.

Fig. 2. The truncated domain Qa

We can find a change of variable ^ mapping Qa into the parallelepiped

Pa = ]a, T[ x ]0,1[ x ]0,5[,

which leaves the variable t unchanged. ^ is defined as follows:

■0 : Qa -> Pa,

, . . . , f x1 — p1 (t) \ (t,xi,x2) I—> 0 (t,xi,xi) = (t,yi,yi) = t,-—-,x2) .

V p(t) J

The mapping ^ transforms the parabolic equation in the domain Qa into a variable-coefficient parabolic equation in the parallelepiped Pa. Indeed, the equation

dtu ai(t, xi, X2)dnu + ^ bi(t,xi,x2)diu + c(t,xi,x2)u = f (t,xi,x2) i=i i=i

in Qa is equivalent to the following

2 _____ 2 _____ _____ dtv ai (t,Vi,V2)diiV + ^2 bi (t,yi,V2)diV + c (t,yi,y2)v = g(t,yi,y2)

i=i i=i

in Pa, where ai (t,yi,y2), bi (t,yi,y2) and c (t,yi,y2) are defined by

\ ai (t,f(t) y1 + fi (t) ,y2) T^^ \ u , \

ai (t, yi,y2) = -^-, a2 (t, y^ y2) = a2 (t, f (t) yi + fi (t) ,y2 ) ,

V2 (t)

, N bi (t,f(t) yi + fi (t) ,y2) h , m ' ,,,,

b1 {t, yi,y2) = -77\- I1 - f (t) yi - fi (t) ,

f (t)

b2 (t, yi,y2) = b2 (t, f (t) yi + fi (t) ,y2) , c (t, y^ y2 ) = c (t,f (t) yi + fi (t) ,y2) ,

and

g (t, yi,y2) = f (t, xi, x2) , v (t, yi,y2) = u (t, xi, x2) .

Since the functions ai, bi, i = 1, 2, c and f are bounded, and using the fact that the mapping ^ is tri-Lipschitz, then, it is easy to check the following

Lemma 2.1. u e H1'2 (Qa) if and only if v e H1'2 (Pa).

The boundary conditions on V which correspond to the boundary conditions on u are the following

V\epa^rT =0,

where rT is the part of the boundary of Pa where t = T. In the sequel, the variables (t,yi,y2) will be denoted again by (t, xi, x2).

Theorem 2.1. The operator

22 L = dt -^2 ai (t , xi , x2 )dii +J2 bi (t

, xi , x2 )di + c (t , xi , x2 )

i=i i=i is an isomorphism from H^'2 (Pa) into L2 (Pa), with

0 v-1 a)

H'2 (Pa) = {v G H1'2 (Pa) : v\dPa^ = 0} .

Proof. Since the differentiable coefficients ai (t, x1, x2 ),bi (t , x1, x2 ), i = 1, 2 and c (t,

x 1, x2

) are

bounded in Pa, the optimal regularity is given by Ladyzhenskaya-Solonnikov-Ural'tseva [24]. □ We shall need the following result in order to justify all the calculus of the next subsection. Lemma 2.2. The space

is dense in the space

{v G H4 (Pa): v\dppa =0} {v GH1'2 (Pa): v\dppa .

Here, dpPa is the parabolic boundary of Pa and H4 stands for the usual Sobolev space defined, for instance, in Lions-Magenes [26].

The proof of the above lemma may be found in [15].

Remark 2.1. In Lemma 2.2, we can replace Pa by Qa with the help of the change of variable 0 defined above.

2.2. Step 2: uniform estimate

We denote un € H1'2 (Qn),n € N*, the solution of Problem (1.1) corresponding to a second member fn = f\Qn € L2 (Qn) in

{(t,:

Qn = \ (t,xl) G R : n<t< T,V1 (t) <x1 <v2 (t) j X ]0,b[. Proposition 2.1. There exists a constant K1 independent of n such that

\\Un\\W.2(Qn) ^ K1 \\fn\\L2(Qn) ^ K1 \\f \\L2(Q) ■

In order to prove Proposition 2.1, we need some preliminary results.

Lemma 2.3. Let ]a, ¡[ C R. There exists a constant K2 (independent of a and ¡)such that

So)

L2(]a,ß[)

< K (ß - af2-j)

,(2)

L2(]a,ß[)

0,1,

for every w G H2 (ja, ß[) O H0 (ja, ß[), where w(j), j = 1, 2, denotes the derivative of order j of w on ja, ß[ and w(0) = w.

Lemma 2.4. For every e > 0, chosen such that p (t) ^ e, there exists a constant C\ independent of n such that for i = 1, 2

dj un

L2(Qn)

< Cie2(2-j) \\diiun\\l2{Qn) , j =0, 1,

where dlun = diun and d0un = un.

Proof. Replacing in Lemma 2.3 w by un and ]a,ft[ by ]v1 (t), v2 (t)[, for a fixed t, we obtain

f V2(t) , ,2 2(2 ,) f V2(t) 2

/ (djun) dx1 < K2 (v (t))2{2-3] (diiUn) dx1 <

Jv 1 (t) v ' -'vi(t)

f V2(t)

< K2e2(2-j) (diiUn)2 dx1,

■'Vl(t)

with i = 1,2 and j = 0,1. Integrating in the previous inequality with respect to t, then with respect to x2, we get the desired result with C1 = K2. □

Proof of Proposition 2.1. Let us denote the inner product in L2 (Qn) by (.,.), then we have

\fn\\L2(Qn) = dtun -J2 ai(t,x1,x2)duun + Ë bi(t,xi,x2)diun + c(t,xi,x2)u i=l i=l 22

= \\dtun\\2L2(Qn) + E \aidiiun\2L2{Qn) + E \\bidiun\\2L2{Qn) + \\Cun\\2L2{Qn) -i=l i=l 2 2 2

-2 E {dtun, aidiiun) + 2 E {dtun, bidiun) + 2(dtun, cun) - 2 E (aidiu, bidiun)-i=l i=l i=l 2 2 2

-2 E (aidiiun, b2d2un) - 2 E (aidüun, cun) + 2 E (bidiun, curi) + i=l i=l i=l +2(alldnun, a22d22un) - 2(bldluri, b2d2un).

L2(Qn)

2

2

j

2

2

1) Estimation of -2{dtun, aidiiun), i = 1, 2 : We have

1 2

dtUnduUn = di (dtUndiUn) — ^dt (du)

Then

-2(dtUn, aiduUn)

—2 aidtUndiiUndt dx1dx2 =

J Qn

—2di (dtUndiUn) + dt (du)

dt dx1 dx2

' 8Q„

(diUn) vt — 2dtUndiUnVi

d,a+

+

2di0,ii (dtUndiUn) — dtau (diUn)2

dt dx1 dx2,

where vt,vi, i = 1,2 are the components of the unit outward normal vector at dQn. We shall rewrite the boundary integral making use of the boundary conditions. On the parts of the

boundary of Qn where t = —, x2 =0 and x2 = b we have un = 0 and consequently diun = 0.

n

The corresponding boundary integral vanishes. On the part of the boundary where t = T, we have vi = 0 and vt = 1. Accordingly the corresponding boundary integral

,-b ,V2{T) 2

/ / ai(T,x1 ,x2)(diUn) dx1dx2 JO Jf 1(T)

is nonnegative, since a,i(T, x1,x2) > 0. On the part of the boundary where x\ = (t), k = 1,2, we have

v1

( — 1)"

vt = ( —1)fc+1 * (t) and V2 =0.

+ ) (t)

Consequently, the corresponding boundary integral is

\/l + (v'k? (t)

,-b T

In,i = y]( — 1)k+i+l / ai(t,ipk (t) ,x2)y'k (t) [diUn (t,ipk (t) ,x2)]2 dt dx2. k=1 Jo h

Furthermore,

dtai (diUn) dt dx1dx2

< c1 \\diUn\\2L2{Qn) ,

and for every e > 0

/ diai (dtundiun) dt dxidx2 J Qn

Then for i = 1,2 we have

^ cW \dtUn\ \diUn\dt dx1dx2 ^ J Qn

+ C1

+ 2e

< c1 2 \\dtUn\\2L2(Qn) + 2e \\diUn\\2L2(Qn) .

— 2(dtUn,diUn) > — \In,1,i\ — \In,2,i\— c1 \\diUn\\2L2(Qn) —c1e \\dtUn fL2(Qn) — — \\diUn\\2L2(Qn) (2.1)

where

i'b fT

In,k'i = ( — 1)k+l+J J^ ai(t,<fik (t) ,x2)<f'k (t) [diUn (t,<fik (t) ,x2)]2 dt dx2, k =1, 2.

Q

a

Lemma 2.5. There exists a positive constant K4 independent of n such that \In,k,i\ < K^e \\diiUnW\2(Qn), k = 1, 2,

\In,k,2\ < K4e \\d22UnfL2 (Qn) + c0e \\di2UnfL2(Qn) , k = 1 2 I. i> d2Un

where di2un

dxidx2

Proof. We convert the boundary integral In,iti into a surface integral by setting

2 p2 (t) — xi 2 X1=V2(t)

[diUn (t, pi (t), x2)]2 =--2—y-—^ [diUn (t, xi,x2)]2

p ( t)

Xl=Ifl(t)

f dl{ P2 (t\. xl [dlun]^ dxl =

K i(t) i p(t) J

rv2(t) p2 (t) _ xl rv2(t) 1 2

- 2 -—-dlun.dllun dxl + —^[dlun] dxl.

ki(t) p (t) jfi(t) p (t)

Then, we have

In,i,i

fb fT

J yi al (t,pl (t), x2)p'l (t)[dlun (t,pl (t) ,x2)]2 dt dx2 =

[ al(t, pl (t), x2) Pl( ,) (dlun)2 dt dxldx2 + JQn P (t)

f p (t) x

+ 2 al(t,pl (t) ,x2)-t--p'l (t)(dlun)(dnun) dt dxldx2.

JQ„ P (t)

'Qn

Thanks to Lemma 2.4, we can write

Therefore

r2(t) 2 2 f^2(t) 2

/ [du] dxl < K2P (t) / [duun] dxl.

Jvi (t) -'vi(t)

f ^2(t) 2 \p' \ f V2(t) 2

[du] — dxl < K2 \p'l\p [duun] d

Jvi(t) P Jvi(t)

consequently

\In,l,l\ < K2 c0 IpII p (dnun)2 dt dxldx2 + 2 c0 \pll\dlunl\dllun\ dt dxldx2,

J Qn J Qn

since

P2 (t) - xl

P (t)

< 1. Using the inequality

2 \p'ldlun\ \dnun\ < e (duun)2 + 1 (pl) (du)2

e

for all e > 0, we obtain

I 2 cq I 2 2

\In,l,l\ < K2 I [cq \pl\P + cQe](dllun) dt dxldx2 +-- (pl) (du) dt dxldx2.

Qn e Qn

Lemma 2.4 yields

— I (fi)2 (diun) dt dxidx2 ^ K2 — I (fi)2 f2 (diiu,n)2 dt dxidx2. e JQn e JQn

Thus,

1 1 2 2 2 1 2

\In ' i 'i\ < K2 i co \fi\\f\ + -(f'i) \f\ (dii un) dt dxidx2 + coe (Biu) dt dxidx2 < JQn L e J JQn

^ (2K2 + 1) coe (diiun)2 dt dxidx2,

Qn

since \f[f\ < e. Finally, taking K4 = (2K2 + 1) c0, we obtain

\!n,i,i\ < K4e \\9iiunfL2(Qn) .

The inequalities

\In,2'i\ < K4e \\9iiunfL2(Qn) ,

and

\In,k'2\ < K4e \\d22un\\2L2(Qn) + coe \\di2un\\2L2^Qn) , k = 1, 2 can be proved by a similar method. This ends the proof of Lemma 2.5. 2) Estimation of 2(aidiiun,a2d22un) : We have

diiuTi.d22uri = di (diun.d22un) - 82 (diun.di2un) + (di2un)2 .

Then

2(aidiiun, a2&22un) = 2 aia2diiun.d22un dtdxidx2 =

Qn

= 2 a\a2

JQn

di (diuri.d22uri) — d2 (d\un.di2un) + (d^Un)2

dt dxidx2 =

= 2 aia2 [diun.d22Unvi — du.di2'unv2] da+ JdQn

+ 2 aia2 (di2un)2 dtdxidx2-

JQn

— 2 di (aia2). (diun.d22un) dt dxidx2 +

Qn

+ 2 d2 (aia2) (diun.di2un) dtdxidx2,

Qn

where vt ,vi, i = 1,2 are the components of the unit outward normal vector at dQn. We shall rewrite the boundary integral making use of the boundary conditions. On the parts of the

boundary of Qn where t = —, x2 =0 and x2 = b we have un =0 and consequently diun = 0.

n

The corresponding boundary integral vanishes. On the part of the boundary where t = T, we have vi = v2 = 0. Accordingly the corresponding boundary integral vanishes. On the part of the boundary where xi = fk (t), k = 1,2, we have v2 =0, un = 0 and consequently d22un = 0. The corresponding boundary integral vanishes. So,

2 aia2 [diun.d22unVi - diun.di2unV2] da = 0.

JdQn

Furthermore,

2 aia2 (di2Un f dt dxidx2 > 2ao Wd^UrW^Q^ ,

Qn

and for every e > 0

-2 i di (aa). (diUn.d22Un) dt dxidx2 > -ßie Wd^u^^Q^ - — \\diun\\2L2

■J Qn e

(Qn)

+2 i d2 (a1a2) [d1un.d12un) dt dx1dx2 > -fa \\di2un\\\2(Qn) - — \\diun\\\2(QQn),

■'Qn e

with ¡3\ is a positive constant. Then, we have

2 2 2j3~i 2

2(a1d11un,a2d22un) > (2ao -fa) \\d12 un\\L2(Qn)-ftie \\d22un W^q^--— \\diun\\L2(Qn) . (2.2)

It is easy to establish the following estimates.

Lemma 2.6. Set c4 = c0c2, c5 = c0c3 and c6 = c2c3. Then, for every e > 0 we have

2{dtUn, bidiUn) > -ec2 WdtUnWL2(Qn) - 7 WdiUnW2L2(Qn) ,i = 1, 2,

2(dtUn, cun) > -ec3 WdtUnWL2(Qn) c3 II ||2 7 WUn WL2(Qn) ,

2{aidnUn, bkdkUn) > -c4e Wdii UnW L2(Qn) - ~ Wdk UnW2L2(Qn , i = 1, 2

-2{aidiiUn, cun) > -c5e Wdii UnWL2(Qn) c5 11 ||2 7 WUn WL2(Qn) , i = 1, 2,

2{bidiUn, cun) > -c6e WunWL2(Qn) - 7 WdiUnW2L2(Qn) ,

2{bidiUn, b2d2Un) > -boe d UnW2L2(Qn) - ^W d2 un W W 2L2(Qn)

Now, summing up the estimates (2.1), (2.2) and making use of Lemma 2.5 and Lemma 2.6 then we obtain

WL2(Qn) > (1 - ae) WdtUn\2L2(Qn) + d0 Wun\2L2(Qn) - Wun\2L2(Qn) +

+ bo( Wfan W2L2{Qn) + Wd2UnW2L2{Qn)) - a ^ + ^ ( Wd1UnWL2(Qn) + W d2 UnfL2(Qn)) +

+ (ao - ae)^2 WdiiUnW2L2(Qn) + (2a0 - ßie - c0 e) Wduu^L

(Qn)

where a is a positive constant independent of n. Thanks to Lemma 2.4, it follows that for i = 1,2

(e + 1)

1 i " ^ "2 , f , , 1 \ 2 no ii2

-a[e + - ) WdiUnWL2{Qn) > -a[e + ~)Cie WdiiUnW

(•+1)

\L2(Qn)

and

-a(ye + ^ WunWL2 (Qn) > -a(e + ^ C1e4 Wdiiunf

L2(Qn) ■

Therefore,

\\fn\\2L2{Qn) > (1 — ae) \\dtun\\2L2{Qn) + d0 \\un\\2L2( — aie + - I Cie4( \\dn un\\2L2{Qn) + \\d22un\\2L2{Qn)) + bo( \\diun\\2L2{Qn) + \d

un\\L2(Qn) )

— +1) Cie2 (Wd2iun\\iL2{Qn) + \\d2un\\2L2(Qn) ) + 2

+ (a0 — ae)£ \\diiun\\2L2(Qn) + (2a0 — Pie — c0e) \\di2un\\2L2(Qn) , (2-3) i=i

which implies

\\fn\\2L2{Qn) > (1 — ae) \\dtun\\2L2 (Qn) + d0 \\un\\2L2{Qn) + bo{ \\diun\\L2 (Qn) + \\d2 un\\2L2{Qn)) + + (ao — ae — aCi (e2 + e) — aCi (e5 + e3)) ( \\diiun\\2L2{Qn) + \\d22un\\2L2{Qn)) +

+ (2ao — Pie — coe) \\di2un\\2L2(Qn) . Then, it is sufficient to choose e verifying

(1 — ae) > 0, (2a0 — ¡3ie — c0e) > 0 and (a0 — ae — aCi(e2 + e) — aCi(e5 + e3)) > 0 to get a constant K0 > 0 independent of n such that

\\fn\\L2(Qn) > K0 \\un\\U1,2(Qn ) ,

and since

\\fn\\L2(Qn) < f \l2(Q) , there exists a constant Ki > 0, independent of n satisfying

\\un\\u1.2(Qn) < Ki \\fn\\L2(Qn) < Ki \\f \\l2(Q) . This completes the proof of Proposition 2.1. □

2.3. Step 3: passage to the limit

Choose a sequence Qn n = 1,2,... of reference domains (see the above subsection) such that Qn Ç Q. Then we have Qn ^ Q, as n ^ <x. Consider the solution un G H1'2 (Qn) of the Cauchy-Dirichlet problem

22 du — Y^ ai(t,xi,x2)duun +J2 bi(t,xi,x2)du + c(t, xi, x2)un = f in Qn,

i=i i=i

un\dQn-^T =

Such a solution un exists by Theorem 2.1. Let un the 0-extension of un to Q. In virtue of Proposition 2.1, we know that there exists a constant C such that

\\urn\\H1'2(Qn ) C \\f \\L2(Q) .

This means that uT, dtun, daun for 1 < \a\ < 2 are bounded functions in L2 (Q). So, for a suitable increasing sequence of integers nk, k = 1,2,..., there exist functions

u, v and va, 1 ^ \a\ ^ 2

in L2 (Q) such that

unk ^ u

weakly in L2 (Q), k oo,

dtunk ^ v weakly in L2 (Q), k ^ <x, daunk ^ va weakly in L2 (Q), k ^ m,

1 < |a| < 2. Clearly,

v = dtu, va = dau, 1 ^ |a| ^ 2 in the sense of distributions in Q, then in L2 (Q). So, u e H1'2 (Q) and

22

(a' = 1

dtu - ai(t, xi,x2)duu + ^ bi(t,xi, X2)diU + c(t,xi,X2)u = f in Q.

On the other hand, the solution u satisfies the boundary conditions u\ôqxst = 0, since

yn G N*, u\q = un.

This proves the existence of a solution to Problem (1.1). Notice that we have the estimate

\\u\\Hi*{Q) < K Wf L2 (Q) ,

which implies the uniqueness of the solution.

Remark 2.2. If (0) < y2 (0) and (T) = y2 (T), then the result given in Theorem 1.1 holds true under the assumption

y'k (t) y (t) ^ 0 as t ^ T, k = 1, 2

instead of hypothesis (1.2).

The authors want to thank the anonymous referee for a careful reading of the manuscript and for his/her helpful suggestions.

References

[1] B.-E.Ainseba, J.-P. Kernevez, R.Luce, Application des sentinelles a l'identification des pollutions dans une riviere, Mathematical modelling and Numerical analyses, 3(1994), no. 28, 297-312.

[2] Yu.A.Alkhutov, Lp-Estimates of solutions of the Dirichlet problem for the heat equation in a ball, J. Math. Sc., 142(2007), no. 3, 2021-2032.

[3] V.N.Aref'ev, L.A.Bagirov, Asymptotic behavior of solutions to the Dirichlet problem for parabolic equations in domains with singularities, Mathematical Notes, 5(1996), no. 1, 10-17.

[4] E.A.Baderko, M.F.Cherepova, The first boundary value problem for parabolic systems in plane domains with nonsmooth lateral boundaries, Doklady Mathematics, 90(2014), no. 2, 573-575.

[5] R.M.Brown, W.Hu, G.M.Lieberman, Weak solutions of parabolic equations in non-cylindrical domains, Proc. Amer. Math. Soc., 125(1997), 1785-1792.

[6] P.Cannarsa, G.Da Prato, J.-P.Zolèsio, Evolution equations in non-cylindrical domains, Atti Accad. Naz-lincei cl. Sci. Fis Mat. Natur., 88(1990) no. 8, 73-77.

[7] M.Dauge, Elliptic Boundary Value Problems on Corner Domains, Springer, Berlin, 1988.

[8] S.P.Degtyarev, The solvability of the first initial-boundary problem for parabolic and degenerate parabolic equations in domains with a conical point, Sbornik Math., 201(2010), no. 7, 999-1028.

[9] G.Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Acc. Naz. Lincei Mem. Ser. 5, (1956), no. 8, 1-30.

10] N.Garofalo, E.Lanconelli, Wiener's criterion for parabolic equation with variable coefficients and its consequences, Trans. Amer. Math. Soc., 308(1988), 811-836.

11] P.Grisvard, Elliptic Problems in Non-smooth Domains, Monographs and Studies in Mathematics, 24 Pitman, Boston, MA, 1985.

12] P.Grisvard, Singularities in Boundary Value Problems, RMA, 22 Masson, Paris, 1992.

13] A.F.Guliyev, S.H.Ismayilova, Mixed boundary-value problem for linear second-order nondivergent parabolic equations with discontinuous coefficients, Ukrainian Mathematical Journal, 66(2015), no. 11, 1615-1638.

14] Y.Jiongmin, Weak solutions of second order parabolic equations in noncylindrical domains, J. Partial Differential Equations, 2(1989), no. 2, 76-86.

15] A.Kheloufi, R.Labbas, B.K.Sadallah, On the resolution of a parabolic equation in a nonreg-ular domain of R3, Differ. Equat. Appl., 2(2010), no. 2, 251-263.

16] A.Kheloufi, B.-K.Sadallah, On the regularity of the heat equation solution in non-cylindrical domains: two approaches, Appl. Math. Comput., 218(2011), 1623-1633.

17] A.Kheloufi, Parabolic equations with Cauchy-Dirichlet boundary conditions in a non-regular domain of RN+1, Georgian Math. J., 21(2014), no. 2, 199-209.

18] V.A.Kondrat'ev, Boundary problems for parabolic equations in closed regions, Am. Math. Soc. Providence. R I., (1966), 450-504.

19] V.A.Kondrat'ev, O.A.Oleinik, Boundary-value problems for partial differential equations in nonsmooth domains, Usp. Mat. Nauk, 38(1983), no. 2, 1-66 (in Russian).

20] V.A.Kozlov, Coefficients in the asymptotic solutions of the Cauchy boundary-value parabolic problems in domains with a conical point, Siberian Math. J, 29(1988), 222-233.

21] V.A.Kozlov, V.G.Maz'ya, On singularities of a solution to the first boundaryvalue problem for the heat equation in domains with conical points, I, Izv. Vyssh. Uchebn. Zaved., Mat, 2(1987), 38-46 (in Russian).

22] V.A.Kozlov, V.G.Maz'ya, On singularities of a solution to the first boundaryvalue problem for the heat equation in domains with conical points II, Izv. Vyssh. Uchebn. Zaved., Mat., 3(1987), 37-44 (in Russian).

23] R.Labbas, M.Moussaoui, On the resolution of the heat equation with discontinuous coefficients, Semigroup Forum, 60(2000), 187-201.

24] O.A.Ladyzhenskaya, V.A.Solonnikov, N.N.Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, A.M.S., Providence, Rhode Island, 1968.

[25] J.L.Lions, Sur les problemes mixtes pour certains systemes parboliques dans des ouverts non cylindriques, (1957), Ann. Inst. Fourier, 143-182.

[26] J.L.Lions, E.Magenes, Problemes aux Limites Non Homogenes et Applications, 1, 2 Dunod, Paris, 1968.

[27] G.Lumer, R.Schnaubelt, Time-dependent parabolic problems on non-cylindrical domains with inhomogeneous boundary conditions, J. evol. equ., 1(2001), 291-309.

[28] A.Maugeri, D.K.Palagachev, L.G.Softova, Elliptic and parabolic equations with discontinuous coefficients, Vol. 109 of Mathematical Research, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000.

[29] O.A.Oleinik, A problem of Fichera, Doklady Akad. Nauk SSSR, 157(1964), 1297-1301 (in Russian).

[30] B.K.Sadallah, Etude d'un probleme 2m-parabolique dans des domaines plan non rectangulaires, Boll. Un. Mat. Ital., 2-B(1983), no. 5, 51-112.

[31] M.-E.Stoeckel, R.Mose, P.Ackerer, Application of the sentinel method in a groundwater transport model, Transactions on Ecology and the environment, 17(1998), 673-680

[32] M.Taniguchi, Initial boundary value problem for the wave equation in a domain with a corner, Tokyo J. Math, 16(1993), no. 1, 61-98.

О линейном параболическом уравнении второго порядка с переменными коэффициентами в нерегулярной области R3

Ферроди Булкоан

Факультет естественных наук и жизни Университет Беджайа, 6000, Беджайа

Алжир

Арезки Келуфи

Факультет технологии, лаб. прикладной математики Университет Беджайа, 6000, Беджайа

Алжир

Настоящая 'работа посвящена изучению следующего параболического уравнения с переменными коэффициентами в недивергентной форме:

2 2 dtu - ai(t,xi,x2)dnu + ^ bi(t,xi,x2)diu + c(t,xi,x2)u = f (t,xi,x2), i=1 i=1 с учетом граничных условий Коши-Дирихле. Задача задана в нерегулярной области вида

Q = {(t,xi) € R2 : 0 <t<T,pi (t) < xi <ф2 (t)} x ]0,b[,

где фк, k = 1,2 являются гладкими функциями. Одной из основных задач этой работы служит то, что область может быть нерегулярной, например, допускается особый случай, когда ф1 совпадает с ф2 при t = 0. Анализ проводится в рамках анизотропных пространств Соболева с использованием метода декомпозиции областей. Эта работа является обобщением случая постоянных коэффициентов, изучаемого в [15].

Ключевые слова: параболические уравнения, нерегулярные области, переменные коэффициенты, анизотропные пространства Соболева.