Global in Space Regularity Results for the Heat Equation with Robin-Neumann Type Boundary Conditions in Time-varying Domains Текст научной статьи по специальности «Математика»

УДК 517.9

Global in Space Regularity Results for the Heat Equation with Robin-Neumann Type Boundary Conditions in Time-varying Domains

Tahir Boudjeriou*

Laboratory of Applied Mathematics, Department of Mathematics Faculty of Exact Sciences, University of Bejaia, Bejaia, 6000

Algeria

Arezki Kheloufit

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

Algeria

Received 27.04.2018, received in revised form 18.01.2019, accepted 06.03.2019 This article deals with the heat equation

dt u - d2u = f in D, D = {(t,x) € R2 : a<t <Ь,ф (t) <x <

with the function ф satisfying some conditions and the problem is supplemented with boundary conditions of Robin-Neumann type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f € L2(D) there exists a unique solution u such that u, dtu, dju € L2 (D) ,j = 1, 2. The proof is based on the domain decomposition method. This work complements the results obtained in [10].

Keywords: heat equation, Unbounded non-cylindrical domains, Robin condition, Neumann condition,

anisotropic Sobolev spaces.

DOI: 10.17516/1997-1397-2019-12-3-355-370.

1. Introduction and preliminaries

This work is devoted to the analysis of the following one-dimensional second order parabolic problem

dtu - dXu = f € L2(il^),

dxu + ¡3u\ri =0, (1)

dxu\r2 = 0,

where L2(Q^) stands for the space of square-integrable functions on with the measure dtdx. The coefficient ¡3 is a real number satisfying the following non-degeneracy assumption

¡3< 0. (2)

Here, (see, Fig. 1) is an open set of R2 defined by

^to := {(t, x) € R2 : a <t <b, ^ (t) < x < ,

* [email protected]

1 [email protected], [email protected] © Siberian Federal University. All rights reserved

where a, b are real numbers such that < a < 0 <b < while ^ is a Lipschitz continuous real-valued function on (a, b), and such that

-1

, (t) = ■ (t) on (a 0},

^ (t)'=1 (t) on [0,b) .

The function (respectively, y2) is positive and decreasing (respectively, increasing) on (a, 0] (respectively, on [0, b)) and verifies the hypothesis (0) = 2 (0) =0. A natural assumption between coefficient 3 and the function of parametrization of the domain which guarantees the uniqueness of the solution of Problem (1) is

-")

- ¡J > 0 almost everywhere t G]a, 0[. (3)

The lateral boundaries r1 and r2 of are defined by

ri = {(t, ^i (t)) G R2 ' a<t< 0} , r = {(t, ^2 (t)) G R2 '0 <t<b} .

Fig. 1. The unbounded domain Notice that the section of in the t direction defined by

Ix '= [^-1(x),^-1(x)}

for x in ]0, is such that the sections In,n G N* become bounded when n becomes large, i. e.,

Vn G N*, 1 (n) - ^-1(n)| < b - a. (4)

The most interesting point of the parabolic problem studied here is the unboundedness of with respect to the space variable x which prevents one using the methods in [16, 17] and [21]. It's the characteristic (4) of the x-sections of which helps us to overcome this difficulty. Also, These specific Robin-Neumann type boundary conditions

dx u + ¡3ulVi = dxv\V2 =0

are important for the originality of this work. Indeed, to our knowledges, results concerning parabolic equations on unbounded (with respect to the space variable x) time-varying domains, subject to such kind of boundary conditions, have not appeared in the literature to date. So, let us consider the anisotropic Sobolev space

H12 (Qto) '= {u G H1'2 (QTO) ' dxu + ¡3ulri = dxulr2 = 0}

with

H1'2 (Qc) := {u G L2 (Q^) : dtu, dju G L2 (Q^) ,j =—, 2} . The space H1'2 (Qc) is equipped with the natural norm, that is

2

\\U\\ni,2{Qxi) = ^M^n^) + \\dtU\\l2{Qxi) + ^ И

Then, the main result of this paper is the following theorem:

Theorem 1.1. Under the conditions (2) and (3), Problem (1) admits a (unique) solution u G H1'2 (Qc) ■

It is not difficult to prove the uniqueness of the solution. Indeed, let us consider u G H^'2 (Qc) a solution of the problem (1) with a null right-hand side term. So,

dtu — d2u = 0 in Qc.

In addition u fulfils the boundary conditions

dxu + f3u\Vi = dxu\V2 =0.

Using Green formula, we have

f (dtu — dju) u dt dx = i ( — lu^ vt — u dxu vx J da + i (dxu)2 dtdx,

where vt, vx are the components of the unit outward normal vector at the boundary of Qc. We shall rewrite the boundary integral making use of the boundary conditions. On the parts of the boundary of Qc where x = ^ (t), i = —, 2, we have

v = —— =, Vt = , ф (t = and dxu (t,vi (t))+ I3u (t,vi (t))= dxu (t,w (t))=0.

+ (t) + (t) Accordingly, the corresponding boundary integral is

2

Then, we obtain

fX^ — в) u2 (t, Vi(t)) dt + [ Щг u2(t, Mt))dt.

f (dtu - dXu) udtdx = f [«M - ¡^v2 (t,<1(t)) dt + / <^t)u2(t,<2(t))dt + JQ^ Ja \ 2 J JO 2

+ (dxu)2 dtdx.

J Q^

Consequently using the fact that u is the solution yields

/ (dxu)2 dtdx = 0,

because

а {Щт — в) u2 (t, Vi(t)) dt + f Щ1 u2(t, Mt))dt Z 0

2

thanks to the hypothesis (3) and the fact that <^2 is an increasing function on [0, b). This implies that dxu = 0 and consequently d^u = 0. Then, the hypothesis dtu - d^u = 0 gives dtu = 0. Thus, u is constant. The boundary conditions and the fact that 3 = 0 imply that u = 0.

We can find in [10] solvability results for Problem (1) with Dirichlet-Neumann type boundary conditions, corresponding here to the case where 3 = to. In the case of bounded non-cylindrical domains Q; ,l > 0, studies related to Problem (1) can be found in [7, 11] and [8] both in one-dimensional and bi-dimensional cases. It is possible to consider similar questions with some other operators (see, for example, [4] for a 2m-th order operator in bounded non-rectangular domains). Whereas second-order parabolic equations in bounded non-cylindrical domains are well studied (see for instance [1, 6, 9, 12, 14, 15, 18, 19, 20, 23] and the references therein), the literature concerning unbounded non-cylindrical domains does not seem to be very rich. The regularity of the heat equation solution in a non-smooth and unbounded domain (in the t direction) is obtained in [21] and [22] by using two different approaches. In [13], uniqueness classes of solutions of nondivergent second order parabolic equations were obtained. The heat equation in unbounded non-cylindrical domains with respect to the space variable x were considered in [5] and [2]. In Guesmia [5], the analysis is done in the framework of evolution function spaces. However, in Aref'ev and Bagirov [2], properties of solutions of the heat equation with Cauchy-Dirichlet boundary conditions were obtained in the more regular anisotropic Sobolev-Slobodetskii spaces (more precisely, those of functions with t-and-x derivatives are in weighted L2-spaces). The class of domains used in [2] corresponds here to

\ -ayf-i on [a, 0} , ( ) ' [ SVt on [0, b]

for any positive constants a and S.

This paper is organized as follows. The two next sections are devoted to the proof of Theorem 1.1. Indeed, in Section 2, we study an auxiliary problem related to Problem (1) in a bounded domain. Then, in Section 3, prove the energy type estimate

Wum\\u1'2{nm) ^ C \\f WL2,

where C is a constant independent of m and for each m G N*, um G H1'2 (Qm) is the solution (obtained in the Section 2) in truncated bounded domain Qm approximating The previous estimate will allow us to pass to the limit and complete the proof of Theorem 1.1.

2. An auxiliary problem in a bounded domain

In this section, we replace the unbounded domain by the bounded domain Qc, c> 0 (see, Fig. 2) defined by

Qc = {(t, x) G '0 < x < c} and we consider the boundary value problem

dtuc - d2xv,c = fc G L2(Qc),

(5)

uc = 0

c ' J- 0,c

dxuc + 3uclr1c = 0, dxuclV2c = 0,

where fc = f lfic, ^ = {(t,c): d1 <t<d2}, r^ = {(t,^(t)) G R2 ' ¿1 <t < 0} and r2,c = {(t, ^2 (t)) G R2 ' 0 <t<d2} with ¿1 = ^-1(c), ¿2 = ^(c).

(h 0 do

Fig. 2. The bounded domain Qc

2.1. Problem (5) in a reference domain

Here, we replace Qc by

Qcn) = i(t,x) G Qc ' ¿1 + 1 <t<d2 - 1 },

nn

where n is a large enough positive integer such that d1 + 1 < 0 and d2 - 1 > 0 (see, Fig. 3).

n n

Thus, y>1 ( ¿1 +— j < c and y>2 ( ¿2--j < c.

Theorem 2.1. For a large enough positive integer n, the problem

' dtu(n) - dlu{n = f(n) G l2 (n£n)),

(n)

u c

t = di + n

(n)

= u c

3xu(n) + 3u(n)

(6)

-.(l.n)

(n)

r(2,n) r c

admits a (unique) solution u(n) G H1'2 (n(n)) . Here, f(n) = f \(

r(1n) = {(t,^1 (t)) G R2 :di + n <t< 0 j , r(2'n) = j(t, ^ (t)) G R2 : 0 <t<d2 - ^J •

Proof. The uniqueness of the solution is easy to check. Let us prove its existence. The change of variable

(t,x) ^ (t,y)

f x - 4(n) (t)\

V'c - 4(n) (t)J '

where

4(n) (t)

:=

^i (t) on [di + n, 0 , ^2 (t) on [0, d2 - n] ,

transforms n(n) into the rectangle R(n) = ]d1 + n, d2 - n [x]0,1[ • Putting u(n) (t, x) = v(n) (t, y) and f(n) (t, x) = g(n) (t, y), then Problem (6) becomes

dtv(n) (t, y) + a (t, y) dyv(n) (t, y) - d2v(n) (t, y) = g(n) (t, y) in R(n),

(n

)lt

(n

)| 1 =0, ly=1 '

b2 (t)

dy v(n) + ^b(t)v(n)|r(n,di) =0,

dyv(n)|

(7)

yv' '|p(n,d2)

where b(t) := c - 4(n)(t), a(t,y) := (y 1)ff (t), and

c - 4(n)(t)

r(n'dl) = j(t, 0) G R2 ' ¿1 + n <t< 0J , r(n'd2) = j(t, 0) G R2 '0 <t<d2 - 1} .

The aforementioned change of variable conserves the spaces L2 and H1'2 because - 1 and

1 1 b2 (t)

a (t, y) are bounded functions when t G^ + n, ¿2 - n [. In other words

f(n) G L2(Q(n)) ^ g(n) G L2(R(n)), u(n) G H1'2(Q(n)) ^ v(n) G H1'2(R(n)). We need the following lemma:

Lemma 2.1. For a large enough positive integer n, the following operator is compact:

B : H1'2 ( R(n) ) ^ L2 ( R(n) ) , v(n) ^ Bv(n) = a (t, y) dy

,(n)

Here, for a fixed t in ] d1 + n, 0[

( v(n) l = v(n) l = 0

Hi'2 (R(nA = v(n G H1'2 (R(n)) : t=dl +n , 'y=1 ' 7 v ; ' v ' * (n) ■ (n)| dyv(n)L) =0

dyv(n) + 3b (t) v(n)|

r(n , di) = dyv • |r(n, d2)

I

0

x=

0

d

0

x

Proof. has the "horn property" of Besov [3], so

dy : H\'2 (R(n)) ^ H2 '1 (.R(n^ , v(n) ^ dyv(n) is continuous. Since R(n) is bounded, the canonical injection is compact from H2'

1 (rM)

into

L2 (R(n)), see for instance [3]. Here

H 2 '1 (R(n)) = L2 (d1 + 1 ,d2 - H1 ]0, lA n H1 (d1 + 1 ,d2 - L2 ]0, lA , \ J \ n n J \ n n J

see [17] for the complete definitions of the Hr's Hilbertian Sobolev spaces. Then, dy is a compact operator from HYj2 (R(n)) into L2 (R(n)) . Since a(.,.) is a bounded function for t €]d1 + n,d2 -n [, the operator B = ady is also compact from H12 (R(n)) into L2 (R(n)) . □

So, thanks to Lemma 2.1, to complete the proof of Theorem 2.1, it is sufficient to show that the operator

dt -^Wd2 : H12 (R(n)) ^ L2 (R(n)) (c - 'C )2

is an isomorphism. A simple change of variable t = h (s) with h' (s)= (c - 'C^)2 (t), transforms the problem

dtv(n) (t, y) ---d2v(n) (t, y) = g(n) (t, y) € L2 (R(n)) ,

(c - 'C ')2 (t)

v(n)\. , , = v(n)\ _ = 0,

t=di + n v ly=i

dy v(n) + pb(t)v(n)\r(ri,dl) =0,

d v(n) \ = 0 dyv |r(r,d2) = 0,

into the following

' dsw(n) (s, y) - d2w(n) (s, y) = Z(n) (s, y), w(n)\s=h-i(d , 1) = w(n)\y=1 =0,

(8)

dyw(n) + ¡3b(h(s))w(n) l^) = 0,

dy w(n)\,r,d2) =0, r h

with Z(n) (s,y) = gn (t\y, w(n) (s,y) = v(n) (t,y) and h (s)

T(n'dl) 0) € R2 : h-1(di + n) <s< 0} , T(n'd2) 0) € R2 :0 < s < h-1(d2 - .

Note that this change of variable preserves the spaces L2 and H1'2. It follows from Lions and Magenes [17], for instance, that there exists a unique w(n) € H1'2 solution of the problem (8). In other words, the operator

Li := dt--:-— d

2y

(c - 4n))

is an isomorphism from H Y'2 (R(n)) into L2 (R(n)) . On the other hand, the operator ady is compact (see Lemma 2.1). Consequently, L1 + ady is a Fredholm operator from HY'2 (R(n)) into L2 (Rny) . Thus the invertibility of L1 + ady follows from its injectivity. This implies that

Problem (6) admits a unique solution u(n) G H1'2 ^Q(n) j . We obtain the function u(n) by setting u(n) (t, x) = v(n) (t, y) = w(n) (h-1 (t), y) . This ends the proof of Theorem 2.1. □

We shall need the following result in order to justify the calculus of the next section.

Lemma 2.2. The space

Vm x ]0,; u(n)

t=di + n y=1

{u(n) G H2 (

di + -, 0

x ]0,1[ ; u

»

0, dyu(n) + ßb (t)'

,(n)

r(n,di)

is dense in the space

|u(n) GH1^

di + 1, 0

x ]0,1[

. i[>

(n)

t=di + n

= U(n)

y=i

= 0, dyu(n) + ßb (t) u(n)

r(n,di)

i) = »} 1. = 0}-

Proof. It is a consequence of [17, Vol. 1, Theorem 2.1].

Remark 2.1. We can replace in Lemma 2.2, ]d1 + n, 0[ x ]0, 1[ by Q(n) change of variable defined above.

□

t<0

with the help of the

2.2. Problem (5) in the non-rectangular bounded domain Qc

Now, we return to the non-rectangular bounded domain Qc. For a large enough positive

integer n such that d1 +— < 0 < d2--, we set fCn) = f Lm and denote by u(n) G H1'2 (0(n))

n n Ua V /

the solution of Problem (6) in Such a solution exists by Theorem 2.1.

An energy type estimate

First, let us denote

Qi = ^n) , Q2 = ^(n)

t<0

t>0

and fi = f \Q, ,i = 1, 2.

Then, consider the following problems:

dtui - d2xui = fi in Qi,

ui\

t=d1 + n

= ui\x=c = 0,

dxui + ßui\r(n,di) = 0,

r c

dtv - d2v = f2 in Q2,

v\r = v \x=c = 0,

dxv\r(„,d2) = 0,

where

(9)

(10)

r Cn'dl) = j(t,^i (t)) G R2 : di + n <t< 0 j , r(n'd2) = j(t, ^2 (t)) G R2 : 0 < t < d2 - ^J and

r = {(0, x) G R2 : x G ]0,c[} .

By a similar argument like that used in Subsection 2.1, Problems (9) and (10) admit (unique) solutions ui G H1'2 (Qi) and v G H1'2 (Q2) -

The following Lemmas will be needed in order to establish the uniform estimate of Proposition 2.1.

Lemma 2.3. The solutions ui and v of Problems (9) and ( 10) verify the following estimates: Wflf = W^UlWLïQ) + \\dïul\\lHQl) + \\dxul\\2L2 (r) + In, (11)

WhW = \\dtv\\h(Q2) + \\dïv\\L2{Q2) + Wdxv\\t2(n , (12)

12 + \\dXv\\X , 11« „,llx

32) + I ldxvll L2(Q2)

where

C 0

In = -P(ui(0,^i(0)))2 — <pi(t)(dxu(t,vi(t)))2 dt

di + n

r= {(0, x) £ R2 : x £ ]0,c[} , V = ^ (d2 - -,x^j £ R2 : x £ ^d2 - -^c Proof. Let us denote the inner product in L2 (Qi) by (.,.), then we have

WfiWliQ) = (dtui - d2xui, dtui - dXui) =

= \\dfUi\\2L2(Ql) + \\d2ui\\2L2{Ql) - 2{dtu1,d2u1) Calculating the last term of the previous relation, we obtain

(dtui,32ui) = / dtuid2uidtdx = jQi

/ dxdtui.dxuidtdx + dtui.dxui vxda. Iqi jSQi

So,

—2 (dtui,dxui} = dt (dxui)2 dtdx — 2 dtui.dxu\vxda

JQi JdQi

x

'Qi

JxL" ' ^2

da

= / (dxui) vt - 23tui.3xuivx

JdQi L J

where vt, vx are the components of the unit outward normal vector at dQi. We shall rewrite the boundary integral making use of the boundary conditions. On the parts of the boundary of Qi

where t = di +— and x = c, we have ui =0 and consequently dxui = 0. The corresponding n

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

/ (dxui)2 dx J0

is nonnegative. On the part of the boundary where x = (t), we have

vx = , 1 , vt = tfi (t and dxui (t,wi (t)) + ¡3ui (t,wi (t))=0.

y/i + Wi)2 (t) + (t)

Consequently, the corresponding boundary integral is the following:

0 0 In =i wi (t) [dxui (t,wi (t))]2 dt + 2 dtui(t,wi(t))dxui(t,wi(t)) dt.

Jdi+n Jdi+n

By putting h(t) := ui(t,^i(t)), t £ [di + n, 0], we obtain

dtu(t,<pi(t))dxu(t,<pi(t)) = h'(t)dxu(t,<pi(t)) — <f'i(t) (dxu(t, <fii(t)))2 .

So, by using the boundary conditions, we get

2 / dtu1(t,^1(t))dxu1(t,^1(t)) dt =

r° r0 ' 2

2 h'(t)dxu(t,n(t)) dt - 2 ^1(t)(dxu(t,^1(t)))2 dt

Jdi + n Jd1 + n

f0 f0 ' 2 -23/ h'(t)h(t) dt - 2 ^1(t)(3xu(t,^1(t)))2 dt =

Jd1 +1 Jd1 +1

l n l n

0 0 -3 (h(t)2)' dt - 2/ ^1(t)(dxu(t,^1(t)))2 dt =

•/di + n Jdi + n

f 0

-3(h(0))2 - 2 <p[(t) (dxu(t, <p1 (t)))2 dt.

di + n

Finally,

-2{dtuu d2u) = -3(u1(0, ^(0)))2 - J i ^(t) (dxu(t, V1(t))Y ¿t + \\dxu1 Wh2(r)

dl + 11

and formula (11) follows. By using a similar argument, we can prove formula (12). □

Let us now, consider the following problem

dtw - d2xw = 0 in Q2,

wlr = u1lr , (13)

wlx=c = dxwlr(„,d2) = 0,

r c

where u1 is the solution of Problem (9). Thanks to [17, Theorem 4.3, Vol.2], Problems (13) admits a unique solution w G H1'2 (Q2). Note that we can approach u1 lr (which is in H 1(r)) by regular functions (for example, by functions in H2(r)), then it is easy to prove that

Lemma 2.4. The solution w of Problem (13) verifies

R u1 11 L2 (r) = WdtWW\2{Q2) + ||d2w|L2 (Q2) + RH^r') •

(14)

Now, we set

u(n) i u1in Ql,

uc = 1

u2 in Q2,

where u2 = v + w. Note that uCn) G H1'2 is then the solution of Problem (6) obtained in

Theorem 2.1.

Proposition 2.1. There exists a constant C > 0 independent of n such that

dt un

(nCn))

L2 n.

+

dx uCn)

(nCn))

L2 n.

< C WfcWL2(nc) •

Proof. Summing up the estimates (11), (12) and (14), we then obtain

2

L2( nCn)) =

(n)

L2(Qi) + 11 f2 WL2(Q2) ^

> Wdtu1WL2(Qi) + WdtvWL2(Q2) + WdtWfL2 (Q2) +

+ 1 dx,u11 L2(Qi) + ||dxv|L2(Q2) + ||dxw||L2(Q2) •

0

2

2

(n)

Consequently,

2

L2( n<">)

But

2 1 2 2 1 2 ^ > ||dt UiWl^Q!) + 2 WdtU2\\L2(Q2) + Hd2ul|L2(Ql) + 2 lldXu2 ||L2(Q2) >

2

> 2

Otuin)

(o<n))

l2 a

+

diun

2

l2 acc

f(n)

l2 a

(a(n))

< Wfcl\L2(nc

then,

dtu(n

l2 a

+

dlu(n

L2(a(n))

< 2

/(n) 2 2

c" L2(a(n)) ^ 2 ||fc||L2(Qc) •

l2 a

This ends the proof of Proposition 2.1.

Theorem 2.2. There exists a constant K > 0 independent of n and c such that

,(n)

Proof. The majoration of

2

H1,2 a

< K

f(n)

2

2

dtucn) 2 ( r^\\ + oxun

L2 (nCrn

2

□

is given by Proposition 2.1. The

majorations of

dx

(n)

l2( a(n))

in Lemma 3.1 and Lemma 3.2.

and

(n)

uc

l2( a(n))

l2( a(n))

can be obtained by similar arguments used

□

Passing to the limit

We are now in position to prove the first main result of this work.

Theorem 2.3. Problem (5) admits a (unique) solution uc belonging to

H1/ (fic) = {u G H1,2 (fic) ; u|r0 c = 0xu + Pu\rlc = 0xu\r2c = o} •

Proof. Choose a sequence ( fi

(n)

nEN«

of the domains defined above. Then, we have fiCn) ^ fic,

as n ^ Consider the solution ui"' G H1'2 of the mixed Robin-Neumann boundary

value problem

dtu" - d2xuCn) = f("' G L2 ,

(n)

uc

t=d1 + i-

(n)

uc

dxun + I3u(n

-.(1 , n)

dxun \t

(2,n)

0.

(n)

Such a solution uc exists by Theorem 2.1. Let us define

r(n) r1 •-

(n)

• =

(t,x) G fic • d1 <t <d1 +--

n

(t, x) G fic • d2--<t < d2

n

• = Ut,x) G fic • t = d2--

c

2

2

2

c

2

c

0

x=c

V

and consider uc the 0-extension of uCn) to and the extension by symmetry with respect to

the vertical segment a to . This extension noted by uCn) is then in 'H1'2(Qc) and verifies in particular

2

(n)

< K WfcW2L2(nc) .

h1 , 2(^c)

The following compactness result is well known: A bounded sequence in a reflexive Banach space (and in particular in a Hilbert space) is weakly convergent. So, for a suitable increasing sequence of integers nk, k = 1,2,..., there exist functions

uc, vc and vc j, j = 1, 2

in L2 (Qc) such that

uink ^ uc weakly in L2 (Qc), k ^ m, dtu>ch) ^ vc weakly in L2 (Qc), k ^ m,

(nk) „, . weakly T2

dx uck) ^ vc,j weakly in L2 (Qc), k ^ m, j = 1, 2.

Then, vc = dtuc, vCi1 = dxuc and vc,2 = d1uc in the sense of distributions in Qc and so in L2 So, we have

dtuc — dxuc = fc in Qc-On the other hand, the solution uc satisfies the boundary conditions

uc\ro, c = dxuc + ¡3uc\ri, c = dxu,c\r2, c =0

since

Vu £ N*, uc\n(n) = uin). This proves the existence of solution to Problem (5). □

3. Back to Problem (1)

For a large enough positive integer m, we define Qm by

= {(t, x) £ : 0 < x < m} . Let um £ H1/ (Qm) the solution of the following problem:

dtum - 32xum = fm £ L2(Qm),

um\r0m = dxum + ¡Sum\rim = 0, (15)

dxum\r2 m 0

where { }

fm = f\Qm , r0,m = {(t,m) : W-1 (m) <t < W-1 (m)} ,

rhm = {(t, wi (t)) £ R2 : W-1 (m) <t< 0} , T2,m = {(t, W2 (t)) £ R2 : 0 <t<v-1 (m)} . Such a solution um exists by Theorem 2.3.

Theorem 3.1. There exists a constant K > 0 independent of m such that

W^W^ . 2(nm) ^ K WfmWL2(nm) .

In order to show the desired inequality, we need the following lemmas: Lemma 3.1. There exists a constant K1 > 0 independent of m such that

\\um\\ь*{Пт) ^ K1 Wfm\\L2{nm) ■

Proof. For a real number Л — 0, we have

j f'^ft* u*^^ e ^dfcdf^c — / dt e ^dt^d^c / д x u*^^ u*^^ e ^dfcdi^c,

I m ^m^ ujvuj^V — i ^t^rn ^m^ ^ t/U'J' I ^x m m

d^2ule-2x2t^ - dx (0xUmUme-2x2t) '' 2 -2\2tiii , \2 f „2 -2Л21.

dtdx +

+ (dxum)2е-2Л dtdx + ЛМ и2ГГке-2Л t dtdx

J Qm ^Qm

Г - P)u2m(t,vi(t))e-2^ dt +

+Л2 u2me-^ t dtdx + / (dxum)2е-2Л t dtdx +

СФ-2 1(m)

+ yo ^ u2m(t,V2(t))e-2^dt >

^ >2 -2Л2Ь II и2

> Ле \um\L2(fim) ■

On the other hand, for all e > 0, we have

fmume 2Л t dtdx < f \\/m\L2(nm) + e \\um\\L2 (Qm) ) e 'Qm Vе /

Therefore,

2

Hence, by choosing e small enough, we obtain the desired inequality. □

Lemma 3.2. There exists a constant K2 > 0 independent of m such that Proof. We have

\\dxurn\\L2(fim) ^ K2 \\fm\L2(nm) ■

/ dx(umdxum)dtdx — / umdxumVxda,

JQm J

where vt,vx are the components of the unit outward normal vector at dQm. On the part of the boundary of Qm where x — m, we have um — 0. The corresponding boundary integral vanishes. On the part of the boundary where x — ф2 (t), we have dxum — 0. Consequently, the corresponding boundary integral vanishes. On the part of the boundary where x — ф1 (t), we have

-1

Vx — , and dxum (t, ф1 (t)) + @um (t,<pi (t)) — 0.

V1 + (ф1)2 (t)

Accordingly, the corresponding boundary integral is

Г o

-в u2m(t,^i(t)) dt■

Jvi 1(m)

Finally,

/ dx(nmdxum)dtdx = —P ^ u2m(t,^i(t)) dt. J Qm J(m)

On the other hand, we have

/ dx (umdXum)dtdx = / umdXum dtdxdy + / (dXum)2dtdx.

JQm Qm Qm

-P _ u2m(t,Vl(t)) dt =1 umdXumdtdx + ¡¡dx^^ (Qm J 1 (m) JQm

Consequently,

\\dxUm\\2L2(Qjm) = — umdlumdtdx + 3 u2m(t,^i(t)) dt <

^ u^dtdx + / (d2xum)2 dtdx =

JQm J^m

2 II 2 \\2

= \\um\\L2(Q.m) + IIdxumIIL2(nm) ■

Lemma 3.1 and Proposition 2.1 which remains valid in ilm give

\dxum\\2L2{nm) ^ Kl \fm\2L2(nm) + 2 \fm\2L2(nm) ^ K2 \\fmWL (nm) ■

□

Theorem 3.1 is a direct consequence of Lemma 3.1, Lemma 3.2 and Proposition 2.1 which remains valid in ilm. We obtain the solution u of Problem (1) by letting m go to infinity in Theorem 3.1. This ends the proof of Theorem 1.1.

Remark. Let us consider the following problem:

dtv - 32xv = f e L2(D),

dxv + av\ri =0, (16)

dxv\r2 = 0,

where

D := {(t, x) e R2 : ll <t < l2; -to < x < $(t)} ,

where ll,l2 are real numbers such that —to < ll < 0 < l2 < +to, while $ is a Lipschitz continuous real-valued function on (ll,l2), and such that

t , Mt) on (lu 0], ^ Mt) on [0,l2).

The function (respectively, ^2 ) is a negative and increasing (respectively, decreasing) on (li, 0] (respectively, on [0, l2)) and verifies the hypothesis ^i(0) = ^2(0) = 0. Here, the coefficient a is a positive real number and r is the part of the boundary of D where x = ^i(t), i = 1, 2.

By using the same arguments like those used in solving Problem (1), we can show that Problem (16) admits a (unique) solution v belonging to H1,2(D), under the assumption

— a < 0 almost everywhere t G]l1, 0[. (17)

References

[1] [2]

[3]

[4]

[5]

[6]

[7]

[8]

[9;

10

11 12

13

14

15

16 17

Yu.A.Alkhutov, Lp-estimates for the solution of the Dirichlet problem for the heat equation in a ball, J. Math. Sci., 142(2007), no. 3, 2021-2032.

V.N.Aref'ev, L.A.Bagirov, Solutions of the heat equation in domains with singularities, Math. Notes, 64(1998), no. 1-2, 139-153.

O.V.Besov, Continuation of functions from Lpl and Wpl, Trudy Mat. Inst. Steklov., 89(1967), 5-17 (in Russian).

S.Cherfaoui, A.Kessab, A.Kheloufi, On 2m-th order parabolic equations with mixed boundary conditions in non-rectangular domains, Sib. Elektron. Mat. Izv., 14(2017), 73-91.

S.Guesmia, Large time and space size behaviour of the heat equation in non-cylindrical domains, Arch. Math., 101(2013), 293-299.

S.Hofmann, J.L.Lewis, The Lp Neumann problem for the heat equation in non-cylindrical domains, J. Funct. Anal., 220(2005), no. 1, 1-54.

A.Kheloufi, B.-K.Sadallah, Parabolic equations with Robin type boundary conditions in a non-rectangular domain, Electron. J. Differential Equations, (2010), no. 25.

A.Kheloufi, Existence and uniqueness results for parabolic equations with Robin type boundary conditions in a non-regular domain of R3, Applied Mathematics and Computation, 220(2013), 756-769.

A.Kheloufi, B.K.Sadallah, Study of the heat equation in a symmetric conical type domain of RN +1, Mathematical Methods in the Applied Sciences, 37(2014), 1807-1818.

A.Kheloufi, B.K.Sadallah, Study of a parabolic equation with mixed Dirichlet-Neumann type boundary conditions in unbounded noncylindrical domains, Journal of Advanced Research in Applied Mathematics, 7(2015), no. 4, 62-77.

A.Kheloufi, On parabolic equations with mixed Dirichlet-Robin type boundary conditions in a non-rectangular domain, Mediterr. J. Math., 13(2016), 1787-1805.

K.Kuliev, L.E.Persson, An extension of Rothe's method to noncylindrical domains, Applications of Mathematics, 52(2007), no. 5, 365-389.

V.V.Kurta, A.E.Shishkov, Uniqueness classes of solutions of boundary problems for nondivergent second order parabolic equations in noncylindrical domains, Ukrainian Mathematical Journal, 42(1990), no. 7 819-825.

R.Labbas, A.Medeghri, B.-K.Sadallah, Sur une equation parabolique dans un domaine non cylindrique, C. R. Math. Acad. Sci. Paris, 335(2002), 1017-1022.

R.Labbas, A. Medeghri, B.-K.Sadallah, An Lp-approach for the study of degenerate parabolic equations, Electron. J. Differential Equations, (2005), no. 36.

O.A.Ladyzhenskaya, V.A.Solonnikov, N.N.Uraltseva, Linear and Quasilinear Equations of Parabolic Type, American Math. Soc. Translation, Providence, R.I., Vol. 23, 1968.

J.L.Lions, E.Magenes, Problemes aux Limites Non Homogenes et Applications, Vols. 1,2, Mathematic Works and Research, Dunod, Paris, 1968.

[18] F.Paronetto, An existence result for evolution equations in non-cylindrical domains, Nonlinear Differ. Equ. Appl., 20(2013), 1723-1740.

[19] B.-K.Sadallah, Etude d'un problème 2m-parabolique dans des domaines plan non rectangulaires, Boll. Un. Mat. Ital. B (6), 2(1983), no. 1, 51-112.

[20] B.K.Sadallah, Existence de la solution de l'equation de la chaleur dans un disque, C. R. Acad. Sci. Paris Ser. I. Math., 327(1998), 813-816.

[21] B.K.Sadallah, Regularity of a parabolic equation solution in a non-smooth and unbounded domain, J. Aust. Math. Soc., 84(2008), no. 2, 265-276.

[22] B.K.Sadallah, A remark on a parabolic problem in a sectorial domain, Applied Mathematics E-Notes, 8(2008), 263-270.

[23] G.Savare, Parabolic problems with mixed variable lateral conditions: an abstract approach, J. Math. Pures Appl. (9), 76(1997), no. 4, 321-351.

Результаты исследования регулярности в пространстве для уравнения теплопроводности с граничными условиями типа Робина-Неймана в изменяющихся во времени областях

Тахир Буджериу

Лаборатория прикладной математики, Факультет точных наук, Университет Беджая, Беджая, 6000

Алжир

Арезки Хелуфи

Лаборатория прикладной математики Технологический факультет, Университет Беджая, Беджая, 6000

Алжир

Эта статья посвящена уравнению теплопроводности

dtu - d2xu = f in D, D = {(t,x) € Rx : a<t <Ъ,ф (t) <x <

с функцией ф, удовлетворяющей некоторым условиям, и задача дополняется граничными условиями типа Робина-Неймана. Мы изучаем проблему глобальной регулярности в подходящем параболическом пространстве Соболева. Докажем, в частности, что для f € Lx(D) существует единственное решение u такое, что u, dtu, dju € Lx (D) , j = 1, 2. Доказательство основано на методе декомпозиции области. Эта 'работа дополняет результаты, полученные в [10].

Ключевые слова: уравнение теплопроводности, неограниченные нецилиндрические области, условие Робина, условие Неймана, анизотропные пространства Соболева.