The first initial-boundary value problem for parabolic equations in a cone with edges Текст научной статьи по специальности «Математика»

YflK 531(075)

BecTHHK Cnöry. Cep. 1. T. 2(60). 2015. Bun. 3

THE FIRST INITIAL-BOUNDARY VALUE PROBLEM FOR PARABOLIC EQUATIONS IN A CONE WITH EDGES

Vu Trong Luong1, Do Van Loi2

1 Department of Mathematics, Taybac University, Sonla city, Sonla, Vietnam

2 Department of Mathematics, Hongduc University, Thanhhoa, Vietnam

The initial-boundary value problem with Dirichlet boundary condition for higher order parabolic equations in a cone with edges is considered. We prove the well-posedness by the similar arguments as in [19]. Moreover, the regularity of the solution are also proven. Refs 25.

Keywords: well-posedness, initial-boundary value problems, regularity, cone with edges.

1. Introduction. Let K = {x G R3 : x/|x| = w £ 0} be open polyhedral domain in R3 with vertex at the origin. Suppose that the boundary dK consists of the vertex x = 0, the edges (half-lines) M1, • • • , Md, and smooth (of class ) faces ri, • • • , rd. This means that Q = K n S2 is a domain of polygonal type on the unit sphere S2 with sides Yk = rk n S2. Let T, 0 < T < to, and j = rj x (0, T), j = 1, • • • , d; KT = K x (0, T).

Let the partial differential operator given by

L(x,t; D) =

E

Dp (apq (x,t)DX ),

where apq are bounded functions with complex values from C°°(1Ct), apq = ( —1 )p+qaqp and aqp denotes the conjugate of aqp.

We also suppose that operator L is strong elliptic uniformly with respect to t e [0, T), that is, there exists a constant c > 0 such that

E

epeq > ciei2m, v(x,t) gkt

(i.i)

| p| ,| q|

forall vector £ e Rn.

Consider the initial-boundary value problem

ut + (-l)mL(x,t; Dx)u = f in KT,

3k-1t

dv k-1

= 0, k = 1,

j = 1, ••• ,d,

u|t=o = 0 in K.

(1.2)

(1.3)

(1.4)

Here function f (x,t) is given on , v denotes the exterior normal to rjT, j = l, • • • ,d.

Elliptic boundary value problems in polyhearal domains have been studied by Maz'ya and Rossman in the monograph [22]. Along with elliptic boundary value problems, mathematicians have paid considerable attention to initial-boundary value problems for parabolic equations in domains with conical points or with edges. In [7, 8] Maz'ya and Kozlov considered the heat equation in domains with conical points in which the asymptotics of the solutions near conical points was studied. For domains with edges, Solonnikov [24, 25] and Nazarov [23] estimated the Green function and proved the existence of solutions of the Dirichlet and Neumann problems for the heat equation in weighted

o

a

m

m

r

Sobolev spaces. In [13, 14] Kozlov and Rossman have been studied the asymptotics of the solutions of the Dirichlet problem for the heat equation near an edge. In [9-11] Kozlov has been dealt with for general second order parabolic equations with time-independent coefficients in domains with conical points, where the asymptotics of solutions and a description of the sigularities of the Green function near the conical points were obtained. In the case of time-dependent coefficients, let us mention some works related to this case. In [1-4] in which the unique existence of weak solutions in Wp-Sobolev spaces was established. In [6] one investigated results on the existence, uniqueness and regularity of generalized solution of equation (1.2) with initial and general boundary conditions in conical domains. Recently, in [19] and [20] we considered the Cauchy—Dirichlet problem for nonstationary equations of second order in domains with edges.

In contrast to the above papers, in this work, we consider higher order parabolic equations with time-dependent coefficients in a cone with edges. By modifying the method suggested in [19] to obtain the well-posedness of problem (1.2)-(1.4). Furthemore, we prove the regularity of the solution in weighted Sobolev spaces with using the help of regularity results for elliptic boundary value problems in [22].

2. The well-posedness of the problem. Fistly, we will introduce some Sobolev spaces as usual on K and KT.

1. Hm(K) is a Sobolev space complex functions u(x) defined on K with the norm

\Dpu\2dx)j 2 < +oo.

\P\Sm k

2. Hm(K) denotes the closure of C§°(K) in Hm(K).

3. Hm'h(KT) denotes the Sobolev space complex functions u(x,t) defined on KT with the norm

h 1

MlH^{KT)= (J (12 \Dpu\2 +Y,\uv\2)dxdt)2

KT \p\<m j=1

where p = (p1, ■ ■ ■ ,pn); m, k are nonnegative integers.

4. The space Hm'k(KT) is the closure in Hm'k(KT) of the set consiting of all functions u e C°°(1Ct), which vanish near 8K,t = U^=i

Let us denote by

/m

12 (-l)lplaPq(x,t)DluD^dx

k \p\ ?\q\ =0

the time-dependent bilinear form. Then, we have the following Green's formula:

(L(x, t; D)u, v)l2(k) = B(u, v; t), which is valid for all u,v e C0f(K) and a.e. t e [0,T].

Definition 2.1. A function u e iim'1(KT) is called a generalized solution of problem (1.2)-(14), if and only if u(x, 0) =0, Vx e K and the equality

(ut, v)l2(k) + (-1)mB(u, v; t) = (f, v)i2(K), a.e. t e [0, T], (2.1)

holds for all v e Fm(K).

From the assumptions above, we also have the Garding's inequality, i.e., there exist constants mo > 0, Ao > 0 such that

( —1)mB(u, u; t) > ^oyuyHm(K) - AoHuH^c) (2.2)

holds for all u e Fm(K) and a.e. t e [0,T].

We note that the constant A0 can be chosen with 0, since by a substitution v = e-Aotu the operator L can be transformed to L = L + A0, with the time-dependent bilinear form associated with L is B(.,.; t) satisfying (2.2) with the constant A0 = 0. Hence, throughout the present paper we also suppose that B(.,.; t) satisfying the following inequality:

( —1)mB(u,u; t) > Mo||u||Hm(K) (2.3)

for all u e Fm(K) and a.e. t e [0, T].

By Galerkin's approximating method and arguments similar as in [19], we have the following theorem.

Theorem 2.1. Let f e L2(KT), and suppose that the coefficients of the operator L satisfy

sup{|apq|, |apqt| : (x,t) e KT} < m, m = const.

Then problem (1.2)-(1.4) has unique generalized solution u in the space iim,1(KT) and the following estimate holds:

||u||Hm.i(KT) < C||f ||2(KT), (2.4)

here C is a constant independent of u and f. This solution depends continuously on f.

The results above shows the unique solvability of problem (1.2)-(1.4). Furthemore, the next observation shows that the generalized solution dependens continuously on the right-hand side f of (1.2).

Now we will prove the continuous dependence on the coefficients of the operator L. Let S > 0, we denote by

Ls = Ls(x,t; D) := £ Dp(apg(x,t)Dq),

o<|p|, | q| <m

the operator depends on S, the coefficients

apq are bounded functions with complex values from CTO(KT), apq = ( — 1)|p|+|q|a,qp, denotes the transposed conjugate matrix of a£p. Set

Bs(u,v;t)= j ("1)'i'1aspq{x,t)DqxuDpxvdx,

° |p|,|q|=o

assume also that the Garding's inequality

( —1)mB<5(u, u; t) > Mo|u|Hm(K), Mo > 0, (2.5)

holds for all u e Hm(K) and a.e. t e [0,T]. Let us be the generalized solution of problem (1.2)-(1.4) with replacing the operator L by Ls. Then we have the following theorem. Theorem 2.2. Let u be the generalized solution of problem (1)-(3). Suppose that

sup{|apq (x,t) — aSpq (x, t) | :0 < |p|, |q| < m, (x,t) e KT }< 0(5) ^ 0 as 5 ^ 0.

Then functions us conveges to u as 5 ^ 0.

Proof. By setting Us = us — u, we get from (2.1) that

(Uf,v) + (-l)mB6(U6,v;t)= J2 (-1 )M+m(apq-a6pq)DquD^dx (2.6)

0< \ p\, \ q\ <m

holds for all v e Hm(K). Let {i^k(x)}f=1, as in Theorem 2.1, set US'N(x,t) = Sfc=1 Ck'N(t)wk(x), with {C^N}N=1 are the solution of the system of the following ordinary differential equations

(UpN,Uk) + ( — l)mB(US'N,Ukl t) =

Y ("1)lpl+mKi - aspq)DquDP^dx, t £ [0, T), k = I,N, (2.7)

0< \ p\ , \ q\ <m

with the initial conditions

Ci'N (0)=0,k =!,..., N. (2.8)

Let us multiply (2.7) by CkN (t), sum k = 1,... N, to find

(USt'N, U S'N) + ( — l)mB(U S'N, U S'N; t) =

= J2 (-l)|p|+m(aM - aspq)D"uDPU^dx. (2.9)

0< \ p\, \ q\ <m

Now adding this equality to its complex conjugate, we get

¿(II^HLaq) + (~irmUS'N, Us'»;t) =

= 2Re J2 (-l)lfl+m(aM - aspq)DquDPUs,Ndx. (2.10)

0< \ p\, \ q\ <m

Employing inequality (2.5) and the Cauchy inequality, we obtain from (2.10) the estimate

¿(ll^'Ar|||2(/c))+(2M0-e)||t/5'Ar||lim(,C) <Ce(S)\\u\\2Hm(K), (2.11)

where C only depens on e. Choosing 0 < e < 2j20, we have

¿(ll^llLaq) <ce(6)\\u\\2Hm(K). (2.12)

Integrating them with respect to t from 0 to t,t e (0, T), we obtain

T

J (l11^11^)) dt - c^MH^kt) < cemn^y

0

By arguments analogous to the proof of Theorem 2.1 of [19], we arrive at

||U5'N ||

Therefore,

||U5 |Hm,1(KT ) < IN^inf S'N |Hm,1(KT ) < C0(S)|f ||L2(KT ). It means that ||U5||Hm.i(KT) ^ 0 as S ^ 0. The theorem is proved.

3. The regularity of the generalized solution. In this section, we discuss the regularity of the generalized solution u of problem (1.2)-(1.4). Firstly, we give a needed auxiliary lemma, which deal with the regularity the solution with respect to time variable. It is proved by repeating almost word for word in the proof of Theorem 3.1 of [6]. Lemma 3.1. Let h e N*, and we assume that

(i) sup {|apqtfc | : i, j = 1,..., n; (x, t) e kt, k < h + 1} < M,

(ii) ftk e L2(Kt), k < h; ftk(x,0)=0, 0 < k < h — 1.

Then the generalized solution u e iim'1(KT) of problem (1.2)-(1.4) has derivatives with respect to t up to order h with utk e iim'1(KT), k = 0,..., h, and

h

||uth 1(KT ) < CY, Hftj ||l2(Kt ), (3.1)

j=o

where C is a constant independent of u and f.

Next, we will show the global regularity of the solution. To do this, we introduce operator pencils generated by the Dirichlet problem for elliptic equation in cone K. Let Mk be an edge of the cone K, and let rk+, rk- be the faces adjacent to Mk. Then by we denote the dihedron which is bounded by the half-planes Tk± tangent to rk± at Mk. Let r, < be polar coordinates in the plane perpendicular to Mk such that

r°fc± = {x e R3 : r > 0,< = ±0fc/2}. Fix t e [0, T], we define the operator Ak (A, t) as follows: Ak(A, t)U = r2m-ALo(0,t, D)(rAU), where Lo(0,t,D) = £ Dp(apq(0,t)Dq), u(x) = rAU(<), A e C. The operator

|p| = |q|=m

Ak(A, t) realizes a continuous mapping from W22m(Ik) n ) into L2(Ik) for every

A e C, where Ik denotes the interval (—0k/2, 0k/2). A complex number Ao is called an eigenvalue of the pencil Ak(A, t) if there exists a nonzero function U e W22 m (Ik) n Wm (Ik) such that Ak(Ao,t)U = 0. We denote by S+k)(t) and S(k)(t) the greatest positive real numbers such that the strip

m — 1 — SW (t) < ReA < m — 1 + S+k) (t)

is free of eigenvalues of the pencil Ak(A, t). Furthemore, we define

r(k)

inf S±(t)

te[o,T ]

for k = 1,..., d.

±

We introduce spherical coordinates p = |x|, w = x/|x| in K and define U(A, t)U = p2m-AL°(0, t, D)(pAU), where u(x) = pAU(w). The operator U(A, t) realizes a continuous mapping

w22m(n) n W2m(n) ^ l2(Q).

An eigenvalue of U(A, t) is a complex number A° such that U(A°, t)U = 0 for some nonzero function U G W22m(Q) n W2m(n).

Let l be a nonnegative integer, fl G R, 5 = (¿1, • • • , 5d) G Rd. Furthermore, let S = {0} U M1 U • • • U Md be the set of the singular boundary points. Then Vg s(K) is defined as the closure of the set Cq°(1C \ S) with respect to the norm

IMk^) = (/ E P2i"-l+lal) n P)2(,fc~,+HW«i2<fe)i < +00, (3.2)

where p = |x| is the distance of the point x from the origin 0, while rk denotes the distance of the point x from the edge Mk. The closure of the set Cg°(K) with respect to the norm (3.2) is denoted by Vgj(5(K).

Obviously, from (3.2) we have the following imbedding

Vgj(5(K) c V£!m_I(K) c • • • c (K).

We consider the Dirichlet problem for elliptic equations

Lu = F on K,

(3.3)

= 0, j = !,•••, d. '

d'"u

r,-

For the following lemma on the regularity of the solutions to elliptic boundary value problems in domains of polyhedral type, we refer to Corollary 4.1.10 and Theorem 4.1.11 of [22].

Lemma 3.2. Let u G Vg s(K) be a solution of the problem (3.3), where

F G V^2m(K) n V/,^2m(K), l > m, l' > m.

3 3

Suppose that the closed strip between the lines ReA = I — ¡3— — and ReA = I' — ¡3' — - is free

of eigenvalues of the pencil U and that the components of 5 and 5' satisfy the inequalities

—5+k) < 5k - l + m < 5W, —5+k) < 5k - l' + m < 5(k). Then u G Vg' (K) and

INI^,(K) < CHFy^-^-cK),

where C is a constant independent of u and F.

From Lemma 3.1.3 and Lemma 3.1.6 in [22], for 5k G [—m, m], k = 1, 2,...d, we have following imbeddings

Hm(K) c V°m°(K) c V°a(K), V°m°(K) c - ,(K),

and

V0,s(K) c V0-m(K),

where V0~m(.K) is the dual space of V^^K).

We denote by Hm^(KT) the weighted Sobolev space of functions u defined in KT

with the norm

d

W»-y ( £ n

' ^ M<™ k=1 H j=1

Theorem 3.3. Let l, h be nonnegative integers, l ^ 2m, and ft e R, 5 = (51,..., 5d) e Rd, ft, 5k e [—m, m], k = 1, 2, ■ ■ ■ ,d. Assume that the following conditions are satisfied

(i) ftk e l2(kt) n V^s2m(kt), k = 0,1, ■■■ ,h +1,

(ii) ftk (x, 0) = 0, k = 0,1, ■■■ ,h — 1.

Additionally, suppose that the closed strip between the lines ReA = m — 3/2 and ReA = l — ft — 3/2 does not contain eigenvalues of the operator pencils u(A,t),t e [0,T], and

—5+k) <5k — l + m<5(k), k =1,...,d.

Let u e Hm,1(KT) be the generalized solution of problem (1.2)-(1.4). Then utk e v^kt), k = 0,1, ■■■ ,h, and

h h h+1

^2\\utk\\vi,o (kt ) < Wftk Wvl.-2™ (KT) + 53 Wftk\\h2 (Kt ), (3.4)

Vß,S ) ~ *-'---- " V ß

k=0 k=0 k=0

where C is a constant independent of u and f.

Proof. The first, we prove theorem in the case of l = 2m and for arbitrary h e N. Since V°S(K) c V0-0n(K) and L2(K) c V0~m(K), ft, 5k e —m, m],k =1,...,d, we get from (i) that

ftk e V0-0m(K) n V0 , (K), k = 0,1, ■■■ ,h +1. (3.5)

Using the hypothesis (i), we have ftk e L2(KT), k = 0,1, ■ ■ ■ ,h + 1. Thus, by Lemma 3.1, we obtain utk e Hm(K) c v0m0 (K), k = 0, ■■■ ,h +1. Furthermore, v^K) c V°} s (K) for ft, 5k e [—m, m], k = 1, ■ ■ ■ ,d so

utk e V0}S(K) = V^m(K) n V0S(K). (3.6)

We have from (3.5) and (3.6) that f — ut e V°s(K), a.e. t e [0,T]. Applying Lemma 3.2 for following problem

Lu =( — 1)m(f — ut) in K, (3.7)

dk u

dv k

j = 1, ••• ,d, (3.8)

(in the case of I = m, ß = 0, 5k = 0, I' = 2m, ß' = ß, 5' = 5), we obtain u(t) G V2"}(K), a.e. t G [0,T], and

Wu(t)Wv2m(K) < C\\f - uA\vls(K) < C\\f (K) + CWuWvl,(K).

r

Integrating with respect to t from 0 to T and using Lemma 3.1, we obtain

k2T0 (Kt ) <

||u||V2-,0 ) < 11/Hv»,i(KT ) + y^i^H-CKT )) <

vIs(Kt) + ||f Hl2(Kt) + |/i|L2(KT)) ,

< C| II / lh/0 (KT)

where C is a constant indefendent of / and u. Thus the assertions of the theorem hold for h = 0 (in the case l = 2m). Now, assume inductive they are true for h — 1. Differentiating both sides of (3.7) and (3.8) h times with respect to t, we have

Luth = F := (-1)m /th + uth+i + £ (h)Lth—kwtk =0 in K,

^ t—n '

h —1 h

ith = F := (-l)m( fth + uth+1 + ^ (fcj Lth —kut

k—0

4 =0, j = l,---,d, fe = 0, ■■■,m- 1

r,

dv k

where Lth-k = Dp(apqth-k ). Set F = uth, we get

H,kl—n

LF = F in K, (3.9)

dku dvk

= 0, j = 1, • • • , d, k = 0, • • • , m - 1. (3.10)

From the inductive assumptions, we see that

h —1

h

(fc)Lth-fcutk

k—n

h—1

< ^^ ||utk 1 V|-(KT) <

V» (Kt ) k—n

,

h—1

< |ftk yvo, (Kt ) + E ||f 1 L 2 ( K t ) ) ,

/ o /0,<

k—n k—n

and uth+i G iim(K) c Vg5 (K). This together the hypothesis (i) imply that F G Vg5(K), a.e. t G [0, T]. Therefore F G L2(K). Thus, we can use the same arguments as above to get from (3.9), (3.10) that F = uth G Vg^^T) and

h+1

IIuth II0(Kt) < C(E y/tk IIV°0,i(KT) + E /ifc HL2(KT) k=° k=°

The assertions of theorem hold for the case of l = 2m and h G N.

Next, we proof the theorem by induction on l for any h. Suppose that the theorem is true l — 1, for any h. We have

Lu =( —1)m(/ — ut)=: F,

ku (3.11)

= 0, j = l,---,d, k = 0. ••• ,m-l.

dvk

From hypothesis / G Vg /m (K), and by inductive assertions ut G Vg / (K) c Vg ¿2m(K). According to Lemma 3.2, we obtain u G Vg 5(K). Differentiating both sides of (3.11) with

respect to t, we get

'Lut = ( — 1)mft — utt — Ltu) =: F1,

^rr- =0, j = l,-.-,d, k = 0, • • • ,m - 1. dvk r

By the same arguments as above, we have ut e Vp. s (K). Continuing above process, differentiating both sides of (3.11) i times with respect to t, we get

Luti = Fi

= 0, j = 1, • • • ,d, к = 0, • • • ,то - 1.

д kuti

dv k

/ i — 1 \ where Fi := (-1)m( U - uti+i - E ©Lti-kщЛ, i < h. Notice that

^ k=0 '

fti g Yl—s2m(K), uti+1 g v'^1(K) с v^2™(K),

and

Lti-kutk e У^2т(К), k < i - 1.

I - 2mt^\ л--i •„„ T о о „„„• ----„. л- /

Therefore, Fi G Vв s m(K). Applying Lemma 3.2 again, we obtain uti G V(KT) and

i i i+1

j2\\utk Wv^o (/gr) < (j2\\ftk Wy^^T) + ^ \\ftk\\l2 (kt )) ■

k=0 ' k=0 k=0

The proof is completed. Remark:

of x such that

Remark: Let r = min rk. Then there exist positive constants C1 ,C2 independent

1<k<d

cip(x) TT rj¥r < r < C2p(x) TT for all XG1C.

-LJ- p(x) -LJ- p(x)

k=1 ' k=1 '

Thus, the norm in Vp(K) := Vpp(K) equivalent following norm

И1да = (£ Jdxy.

\a\<l k

From Theorem 3.3, we get the following theorem.

Theorem 3.4. Let l, h be nonnegative integers, l > 2m, в G R, в G [-m, m]. Assume that the following conditions are satisfied

1) ftk G Vl—2m'0(Kr), к = 0, ■■■ ,h +1,

2) ftk (x, 0) = 0, к < h.

Additionally, suppose that the closed strip between the lines ReA = m — 3/2 and ReA = l — в — 3/2 does not contain eigenvalues of the operator pencils U(A,t),t G [0, T], and

S{k) < ß - l + m< S{k).

Let u G Hm'1(K.T) be the generalized solution of problem (1.2)-(1.4). Then utk G Vg'0(KT), k = 0, • • • , h and

h h h+1

El|utfcllvj'0(KT) - Hftkllvj-2m(KT) + E lftk\\l2(Kt)• (3-12)

k=0 k=0 k=0

In the case m = 1, we have similar results for problem (1.2)-(1.4) in a polyhedral domain, which is proven in [21].

4. An example. To illustrate the Theorem 3.3, in this section we consider as example the case of operator L = A. For the following information concerning the eigenvalues of pencils Ak(A, t) and U(A, t) introduced in the previous section, we refer to [18, Chapter 2]. The eigenvalue of the operator pencil Ak(A) are

Aj = W^k, j = ±1,±2,...,

(see [18, section 2.1.1]). We see that S+k) = sLk) = n/0k are the greatest positive real numbers such that the strip

-n/0k < ReA < n/dk

is free of eigenvalues of the pencils Ak(A).

Let A be the eigenvalues of the Laplace—Beltrami operator — S (with the Dirichlet condition) on the subdomain Q of the unit sphere (Q is defined in the previous section). Then the eigenvalues of the pencils U(A) are given by

A±k =-i ± ^1 + 7/4.

It is well-known that the spectrum —S is a countable set of positive eigenvalues (see [18, section 2.2.1]). Hence, the interval [—1,0] is free of eigenvalues of the pencils U(A). We denote the smallest positive eigenvalue of the U(A) by A+. Then the interval [—1 — A+, A+] does not contain eigenvalues of the pencils U(A). Now, the conditions about the eigenvalues of pencils Ak (A) and U(A) in Theorem 3.3 can be written down simply as follows

— 1 — A+ < m — 3/2, l — $ — 3/2 < A+,

and

|Sk + l — m| < n/Qk, k = 1,. .., d.

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Статья поступила в редакцию 26 марта 2015 г. Сведения об авторах

Vu Trong Luong — Department of Mathematics, Taybac University, Sonla city, Sonla, Vietnam; vutrongluong@gmail.com

Do Van Loi — Department of Mathematics, Hongduc University, Thanhhoa, Vietnam; 37loilinh@gmail.com