- Email: [email protected]
INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR NONLINEAR PARABOLIC SYSTEMS WITH MEASURABLE COEFFICIENTS
2003,23B(1) :67-73
..4at~cta57cientia
1~IfmJl1~m INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR NONLINEAR PARABOLIC SYSTEMS WITH MEASURABLE COEFFICIENTS 1 Wen Guochun (
)
1Jj:) l!l;f~
School of Mathematical Sciences, Peking University, Beijing 100871, China
Xu Zuoliang ( -H-ii' tit ) Information School, Renmin University of China, Beijing 100872, China
Abstract
This paper deals with some initial-oblique derivative boundary value problems
for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved. Key words Nonlinear parabolic systems with measurable coefficients, initial-oblique derivative problems, multiply connected domains 2000 MR Subject Classification
35K45, 35K55, 35K60
1 Formulation of Initial-Oblique Derivative Problems for Nonlinear Parabolic Systems
n be a bounded multiply connected domain in R N and the boundary an E C 2 . Denote Q = n x I,I = 0 < t :::; T, 0 < T < 00, aQ = 51 U 52 is the parabolic boundary, where 51 = n x {t = O} is the bottom and 52 = an x I is the lateral boundary. We consider the Let
nonlinear parabolic system of
seco~d
order equations
Fdx, t, u, Dxu, D;u) - Ukt = 0 in Q, k = 1,"', m, where U = (u1,"',u m),D xu can be reduced to the form N
'LJ " aij(k)UkXiXj i,j=l in which (k) _
aij 1 Received
=
(uxJ,D~u
m
N
h=l
i=l
(k)UhXi + c(k)] + "'['" LJ LJ bhi h uh
11 0
= (UXiXJ.
April 16, revised November 4, 2002.
Under certain conditions, system (1.1)
- Ukt -_
(k)_
Fkrrkij(x,t,u,p,Tr)dT,b hi -
(1.1 )
11 0
f (k)·in Q,k -_ 1, ... ,m,
FkTphi(x,t,u,Tp,O)dT,
(1.2)
68
ACTA MATHEMATICA SCIENTIA
Vo1.23 Ser.B
1 1
C~k) =
FkTUh (x, t, TU, 0, O)dT, jCk) = -Fk(X, t, 0, 0, 0), r = D;u,
p = Dxu, rhij = UhXiX;,Phi = UhXi' i,j = 1"", N, h, k = 1"", m. Suppose that (1.1) (or (1.2)) satisfies Condition C, i.e. for arbitrary systems of functions: uj(x,t) = (u{(x,t)"",u!n(x,t))(j = 1,2),Fk(x,t,u,Dxu,D;u (k = 1,"',m) satisfy the conditions
k= 1,"',m
(1.3)
where u~ (x, t) E C~:~/2(n)nwi,1 (fl)(j = 1,2, k = 1"", rn), Wi,l(Q) = W22,a(Q)nW~,1 (Q), = u 1 - u 2 and
f3 < 1, u
-Ck) -_ C h
1 1
- - -
ii;7), b~), c~k), jCk)
1
L
I
j=l
sup]
Ia- ijCk) 1 :::;
1
2
2)],u = u 2 + T(U 1 - u 2), h, k = 1"", m,
~j
N
1
2
:::;
L
i,j=l
N
< qo1 L
iii7)~i~j
N
Q
2
are measurable in Q and satisfy the conditions
N
qa
2
a Fkuh(x,t,u,p,r)dT,r - Dx[u +T(U -u )],
j5 = D x[u2 + T(U - u
and
-_
°<
j=l
I
~j 2 , 0 < qa < 1,
(1.4)
1
N
L (a;7) (x, t) )2]/ inf[L a;7) (x, tW :::; q1 < N ~ 1/2'
i,j=l
(1.5)
Q ;=1
k0, l-bCk)1 I-Ck)1 :::; k0, L p[jCk) , Q] :::; k 1,2,J-1,''', .. N "h khi:::; ka'C -l,"',m, h
where qa, q1, k a, k 1,p(> N
+ 2)
(1.6)
are non-negative constantsl!l. Moreover, for almost every point
(x,t) E Q and D;u, a;7)(x,t,u, Dxu,D;u),b~~)(x,t, u,DxU),c~k\x,t,u) are continuous in u E R, Dxu ERN. If the conditions (1.4), (1.5) hold, and the condition (1.6) is replaced by
lii;7)
1 :::;
ka in Q,Lp[b~~),Q]:::; qkh:::; ka, supc~k) < .
Q
00,
Lp[c~k), Q] < qkh :::; ka, Lp[jCk) , Q] :::; i., i,j = 1,"', N, h, k = 1,"', m, in which qkh(k, h = 1"", m) are non-negative constants, then the corresponding conditions will be called Condition C'. It can be seen that Condition C c Condition C'. The so-called initial-oblique derivative boundary value problem (Problem 0) is to find a solution u(x, t) = (Ul.' ... ,urn) of system (1.2), such that Uk 1," . , m) satisfying the initial-boundary conditions:
= Uk (x, t)
E C)3:13/2 (Q)nW2' (Q) (k =
UdX,O)=gk(X), xEfl,k=l,"',m,
(1. 7)
k l Uk
= aUk all + bUk =
Tdx, t), (x, t) E 52, i.e.
lO
-
-21
No.1
Wen & Xu: INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR PARABOLIC SYSTEMS
69
(1.8)
where dj (x, t), g(x) = (gdx), ... ,gm(x)), 7(X, t) = 2
-
C",[gk(X), 0] :::;
(71 (x,
t), ... ,7m (x, t)) satisfy the conditions
1,1 _ »:C",,"'/2h(x, t), 52] < k2,k -
1,"', m,
C~',~/2[b(x,t),52]:::; ko, C~',~/2[d7(x,t),52]:::; ko,j = 1,"',N, N
COS(lI,
n) :::: do
> 0, i.e,
L dj
COS(lI,
Xj) :::: do
> 0, k = 1, ... ,m,
(1.9)
j=l
where n is the unit outward normal vector on 52, a(O < a < 1), ko, ka, do are non-negative constants. In particular, if Problem 0 with the conditions II = n, b = 0 on 52, then Problem 0 is the initial-Neumann boundary value problem, which will be called Problem N.
2 A Priori Estimates of Solutions for Initial-Oblique Derivative Problem for (1.2) First of all, we prove the uniqueness theorem of solutions of Problem 0 for (1.2).
Theorem 2.1
If system (1.2) satisfies Condition C and a\J) = aij(k = 1,"', m,i,j =
1, ... , N), or (1.2) satisfies Condition C' and qhk (k -I h, k, h = 1, ... ,m) are small enough, then the solution of Problem 0 is unique. Proof If system (1.2) satisfies Condition C and a1~) = aij(i,j = 1,"', N, k = 1,"', m), and let u1(X,t),u2(x,t) be two solutions of Problem 0 for (1.2), it is easily seen that U u 1 - u 2 = (U1, ... ,u m) is a solution of the initial-boundary value problem m
N
' L" aijUkxiXj
i,j=1
+
N
" L ' [L' " b-(k) hi UhXi
h=1
. Q, +C_(k)] h Uh -Ukt -- 0 m
k -- 1 ,"',m,
(2.1)
i=1
Uk(X,O) lkudx, t) = 0 i.e ..
= 0,
0;: +
x E 0, k
= 1,"', m,
(2.2)
bUk = 0, (x, t) E 52, k = 1,"', m,
(2.3)
b~~), c~k) are as stated in (1.3). Introduce a transformation of function U = v exp(Bt) , B is an appropriately large number, such that B > sUPQ c~, k = 1,' .. ,m, thus the initial-boundary where
aij,
value problem (2.1)-(2.3) is reduced to inQ,k=l,"',m,
vdx, 0) = 0 on 0, k = 1, ... ,m, .
t.e.
8Vk Oll
+ bVk = 0
on 52, k
(2.4)
(2.5)
= 1,"', m.
(2.6)
Noting that B - sUPQ c~ > 0, (x, t) E Q, k = 1,"', m, by the maximum principle of solutions for (2.4), we can derive that v(x, t) = u(x, t) = 0, i.e. u 1(x, t) = u 2(x, t), (x, t) E Q. In fact, let
70
ACTA MATHEMATICA SCIENTIA
Vol.23 Ser.B
us multiply Vk to each equation of system (2.4). Thus an equation and boundary condition for
v2 =
2:;:1 V~
can be obtained, which is
m
N
LL
Taking into account
k=liJ=1
m
aijVkx;Vkxj 2: qo
N
LL
k=li=1
1
Vkx;
2
1
,
m
L
h,k=1
c~k)VkVh I:::; mk6v2/2,
if the maximum of v 2 is taken in an inner point Po = (to,xo) E Q, and 1v(Po) l¥o 0, then in a neighborhood of Po, the right hand side of (2.7)2: [B - mNk5l4qo - mk5]v2. We choose the constant B such that B > mNk5/4qo - mk5' By using the maximum principle, we see that v 2 cannot take a positive maximum in Q. If v 2 takes the positive maximum at a point Po E 52, 2 then this contradicts (2.6), because from (2.6) it follows that ~ aavv + bv2 = on 52. Hence Vk(t, x) = 0, k = 1,"', m, i.e. v(t, x) = u(t, x) = 0, thus u 1(t, x) - u 2(t, x) = in
°
Q.
°
If (1.2) satisfies Condition C' and qkh(k ¥o h, k, h = 1,"', m) are small enough, we can prove the result in this theorem by using another method (see [2],[3],[4]). Secondly, we shall give the estimates of solutions u(x, t) of Problem 0 in the spaces 10
-
-21
C13 :/3/2(Q) and W 2' (Q).
Theorem 2.2 Suppose that (1.2) satisfies the same conditions in Theorem 2.1. Then any solution u(x, t) of Problem 0 satisfies the estimates
Ru =
Ilullcl,O
(3.(3/2
<
IluIIW;Ol(Q)
L Ilukllc1,o k=1
(3.(3/2
(Q):::; M 1 = M 1(q,po, ex, k,
Q),
(2.8)
m
L IlukIIW;Ol(Q) :::; M 2 = M 2(q,po, ex, k, Q),
(2.9) k=1 (3 :::; ex),Po(2 :::; Po < p),q = (qo,ql),k = (ko,kl,k2),Ml,M2 are non-negative
Su =
where (3(0 constants.
m
(Q) =
=
Proof If Condition C' holds and qhk (h =j:. k, h, k
= 1,"', m)
are small enough, we can
use the results in [3] to prove the estimates in (2.8) ,(2.9). It is clear that system (1.2) can be rewritten as N
N
i,j=l
i=l
'L...J " a (k)Ukx;xj + 'L...J " b(k) ki Ukx; ij
h (k) --
+ C (k) k Uk
~ L...J [~b(k) L...J hi Uhx; + C(k)] h Uh
h=l,h#k
- Ukt -_ f(k)
+ h(k) ,
. Q, k -- 1,"', m. in
(2.10)
i=l
Noting Condition C' and qhk, h =j:. k, h, k = 1,"', m are small enough, there exists a small positive number e, such that
(2.11)
No.1
Wen & Xu: INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR PARABOLIC SYSTEMS
71
By using the results in [3], the solution U = [udx, t),"', um(x, t)] of Problem 0 for system (1.2) satisfies the estimates m
Ru = C~:~/2[U, Q] =
m
l: C~:~/2[Uk' Q] ~ Msl:[k + Ilh(k)/IL2(Q)], l
k=1
k=1
m
Su = Ilull w;.l(Q) =
l: Ilukllw;·l(Q) ~ M
k=1 where k' = max(k l,k 2),Mj = Mj(q,po,a,k,Q),j Ru =
m
+ Ilh(k)IIL2(Q)], k=1 = 3,4. From (2.11), it follows that 4l:W
m
m
k=1
k=1
l: RUk ~ Ms[mk' + cRu], Su = l: SUk ~ M
4[mk'
Provided that e is a sufficiently small positive number, such that cMs Ru
<
1
M~3 mk' =
- c
MI,Su
< 1, then we have
~ M 4[mk' +cRu] ~ M 4mk' + M IM4c =
If system (1.2) satisfies Condition C and a~~) also derive the estimates (2.8) and (2.9).
3
+ cRu].
The Solvability of Problem
M2.
= aij(k = 1"", m, i,j = 1"", N),
we can
0 for the Parabolic System (1.2)
We first consider a special form of system (1.2), namely N
2 ) A _ kl(x, t ,u, D xU, D xu,g kl -uUk_ A '"' kl UUk-Ukt-g L...JaijUkxiXj
-~ [t, b~~Uhxi in which ~Uk
= L:~I 82uk18x;
i,j=1
+ c~lUh] + t"
in Q, k = 1, ... ,m,
(3.1)
and the coefficients in QI, in R N x I}\QI,
where Ql = {(x,t) E Qldist((x,t),8Q) ~ 111},1 is a positive integer and Oii = l,oij = O(i ::f. j, i, j = 1"", N). By using the method as stated in Section 2, we can obtain the following result. T'heorern 3.1 Under the same conditions in Theorem 2.1, if u(x, t) = [UI (x, t), ... ,um(x, t)] is any solution of Problem 0 for system (3.1), then u(x, t) can be expressed in the form U(x, t)
= U(x, t) + V(x, t) = U(x, t) + vo(x, t) + v(x, t),
v(x,t)=Hp=
r G(x,t, (,T)p((,T)da(dT,
lQo
G = { [A(t - T)]-N/2 exp[lx - (12/4(t - T)], t > T, 0, t
~ T,
except t -
T
= [z -
(I =
0,
(3.2)
72
ACTA MATHEMATICA SCIENTIA
Vol.23 Ser.B
where p(x, t) = ~u - Ut and V(x, t) is a solution of Problem Do for (3.1) in Qo = no x 1(n o = {Ixl < R}) with the initial-boundary condition V(x, t) = a on 8Qo, in which R is a large number, such that no :J 0, and U(x, t) is a solution of Problem 6 for LU = ~U - U, Q with the initial-boundary conditions (3.10), (3.11) below, which satisfy the estimates
= a in
where ,8(0 < ,8 ~ a), M j = Mj(q,po, a, k, Qt}(j = 5,6) are non-negative constants, q = (qo, qd, k = (ko, k 1 , k2 ) . Theorem 3.2 If system (1.2) satisfies the same conditions as in Theorem 2.1, then Problem 0 for (3.1) has a solution u(x, t). Proof In order to prove the existence of solutions of Problem 0 for the nonlinear system (3.1) by using the Larey-Schauder theorem, we introduce the system with the parameter h E [0,1]: (3.4) ~Uk - Ukt = hll(x, t, U, Dxu, D~u) in Q, k = 1,"', m. '21
10
-
C13 :13 / 2 (Qo) n W;,l(QO)(O < ,8 ~ a), the elements of which are systems of real functions V(x,t) satisfyDenote by B M a bounded open set in the Banach space B = W 2 ' (Qo)
ing the inequalities
(3.5) . in which W;,l(QO) = W;,o(Qo)
n Wg,l(QO), M 6
is a non-negative constant as stated in (3.3).
We choose any function V(x, t) E B M and substitute it into the appropriate positions in the right hand side of (3.4), and then we make an integral ii(x, t) = Hp as follows:
ii(x,t) = Hp, p(x,t) = ~V -~.
(3.6)
Next denote by iiO(x, t) a solution of the initial-boundary value problem in Qo, i.e.
(3.7)
iio(x, t) = -ii(x, t) on 8Qo. It is clear that V(x, t) = ii(x, t)
+ iiO(x, t)
(3.8)
is a solution of of the corresponding Problem Do in
Qo, namely ~V - ~ = p(x, t) in Qo, V(x, t) =
a
on 8Qo.
Moreover on the basis of the result in [2], we can find a solution U(x, t) of the following initialboundary problem in Q: ~U
- ir; = a
(3.9)
in Q,
U(x, 0) = g(x) - V(x, 0) on 0, 8U 8v
-
+ b(x, t)U = r(x, t) -
8V 8v
•
+ b(x, t)V
(3.10) on 8 2 .
(3.11)
Now we discuss the equation ~V -
vt =
hll(x, t, U, Dxu, D~U + D~V), k = 1,,,,, m,
a ~ h ~ 1,
(3.12)
No.1
Wen & Xu: INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR PARABOLIC SYSTEMS
73
where u = fj + V. By Condition C', applying the principle of contracting mapping, we can find a unique solution V(x, t) of Problem Do for system (3.12) in Qo satisfying the initial-boundary condition (3.13) V(x, t) = 0 on 8Qo. Denote u(x, t) = U(x, t) + V(x, t), where the relation between U and V is as stated in Theorem 3.1, and by V = S(V,h),u = Sl(V,h)(O ~ h ~ 1) the mappings from V onto V and u respectively. Furthermore, if V(x, t) is a solution of Problem Do in Qo for the system ~V -
vt = hll(x,t,u,Dxu,D;(U + V)),O ~ h ~ 1,
(3.14)
with the initial-boundary condition V(x, t) = 0 on 8Qo, where u = Sl (V, h), then from Theorem 3.1, the solution V(x, t) of Problem Do for (3.14) satisfies the second estimate in (3.3), consequently the solution V(x, t) E B M. Set B o = B M X [0,1], we can verify that the mapping V = S(V, h) satisfies the three conditions of the Leray-Schauder theorem: 1. For every h E [0,1], V = S(V, h) continuously maps the Banach space B into itself, and is completely continuous on B M . Besides, for every function V(x,t) E B M , S(V,h) is uniformly continuous with respect to h E [0,1]. 2. For h = 0, from (3.5) and (3.12), it is clear that V = S(V,O) E B M . 3. From Theorems 2.2 and (3.5), we see that the functional equation V = S(V, h) (0 ~ h ~ 1) does not have a solution V(x, t) on the boundary 8B M = BM\BM. Hence by the Leray-Schauder theorem, we know that Problem Do for system (3.12) with h = 1 has a solution V(z, t) E BM, and then Problem 0 of system (3.4) with h = 1, i.e. (3.1) has a solution u(x, t) = 51 (V, h) = U(x, t) + V(x, t) = U(x, t) + vO(x,t) + v(x, t) E B. Theorem 3.3 Under the same conditions in Theorem 2.1, Problem 0 for system (1.2) has a solution. Proof By Theorems 2.2 and 3.2, Problem 0 for system (3.1) possesses a solution ul(x, t), and the solution ul(x, t) of Problem 0 for (3.1) satisfies the estimates (2.8) and (2.9), where l = 1,2, . ". Thus, we can choose a subsequence {u 1k (x, tn, such that {u 1k (x, tn, {U~ki (x, tn(i = 1,"', N) in Q uniformly converge to uO(x,t), U~i (x, t)(i = 1,"', N) respectively. Obviously, uO(x,t) satisfies the initial-boundary conditions of Problem O. On the basis of principle of compactness of solutions for system. (3.1), we can see that uO(x,t) is a solution of Problem 0 for (1.2). We mention that the initial-irregular oblique derivative problem for system (1.2) remains to be continuously considered. References 1 Alkhutov Yu A, Mamedov I T. The first boundary value problem for nondivergence second order parabolic equations with discontinuous coefficients. Math USSR Sbornik, 1988, 59: 471-495 2 Ladyzhenskaya 0 A, Solonnikov V A, Ural'tseva N N. Linear and Quasilinear Equations of Parabolic Type. Trans Math Monographs 23, Providence, RI: Amer Math Soc, 1968 3 Wen G C, Tain M Y. Initial-oblique derivative problems for nonlinear parabolic equations with measurable coefficients. Comm in Nonlinear Sci & Numer Simu, 1998, 3:109-113 4 Wen G C, Zou B T. Initial-oblique derivative problems for linear parabolic equations of second order with measurable coefficients in a higher dimensional domain. Acta Math Appl Sinica, 1999, 22:579-588 (in Chinese)
..4at~cta57cientia
1~IfmJl1~m INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR NONLINEAR PARABOLIC SYSTEMS WITH MEASURABLE COEFFICIENTS 1 Wen Guochun (
)
1Jj:) l!l;f~
School of Mathematical Sciences, Peking University, Beijing 100871, China
Xu Zuoliang ( -H-ii' tit ) Information School, Renmin University of China, Beijing 100872, China
Abstract
This paper deals with some initial-oblique derivative boundary value problems
for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved. Key words Nonlinear parabolic systems with measurable coefficients, initial-oblique derivative problems, multiply connected domains 2000 MR Subject Classification
35K45, 35K55, 35K60
1 Formulation of Initial-Oblique Derivative Problems for Nonlinear Parabolic Systems
n be a bounded multiply connected domain in R N and the boundary an E C 2 . Denote Q = n x I,I = 0 < t :::; T, 0 < T < 00, aQ = 51 U 52 is the parabolic boundary, where 51 = n x {t = O} is the bottom and 52 = an x I is the lateral boundary. We consider the Let
nonlinear parabolic system of
seco~d
order equations
Fdx, t, u, Dxu, D;u) - Ukt = 0 in Q, k = 1,"', m, where U = (u1,"',u m),D xu can be reduced to the form N
'LJ " aij(k)UkXiXj i,j=l in which (k) _
aij 1 Received
=
(uxJ,D~u
m
N
h=l
i=l
(k)UhXi + c(k)] + "'['" LJ LJ bhi h uh
11 0
= (UXiXJ.
April 16, revised November 4, 2002.
Under certain conditions, system (1.1)
- Ukt -_
(k)_
Fkrrkij(x,t,u,p,Tr)dT,b hi -
(1.1 )
11 0
f (k)·in Q,k -_ 1, ... ,m,
FkTphi(x,t,u,Tp,O)dT,
(1.2)
68
ACTA MATHEMATICA SCIENTIA
Vo1.23 Ser.B
1 1
C~k) =
FkTUh (x, t, TU, 0, O)dT, jCk) = -Fk(X, t, 0, 0, 0), r = D;u,
p = Dxu, rhij = UhXiX;,Phi = UhXi' i,j = 1"", N, h, k = 1"", m. Suppose that (1.1) (or (1.2)) satisfies Condition C, i.e. for arbitrary systems of functions: uj(x,t) = (u{(x,t)"",u!n(x,t))(j = 1,2),Fk(x,t,u,Dxu,D;u (k = 1,"',m) satisfy the conditions
k= 1,"',m
(1.3)
where u~ (x, t) E C~:~/2(n)nwi,1 (fl)(j = 1,2, k = 1"", rn), Wi,l(Q) = W22,a(Q)nW~,1 (Q), = u 1 - u 2 and
f3 < 1, u
-Ck) -_ C h
1 1
- - -
ii;7), b~), c~k), jCk)
1
L
I
j=l
sup]
Ia- ijCk) 1 :::;
1
2
2)],u = u 2 + T(U 1 - u 2), h, k = 1"", m,
~j
N
1
2
:::;
L
i,j=l
N
< qo1 L
iii7)~i~j
N
Q
2
are measurable in Q and satisfy the conditions
N
qa
2
a Fkuh(x,t,u,p,r)dT,r - Dx[u +T(U -u )],
j5 = D x[u2 + T(U - u
and
-_
°<
j=l
I
~j 2 , 0 < qa < 1,
(1.4)
1
N
L (a;7) (x, t) )2]/ inf[L a;7) (x, tW :::; q1 < N ~ 1/2'
i,j=l
(1.5)
Q ;=1
k0, l-bCk)1 I-Ck)1 :::; k0, L p[jCk) , Q] :::; k 1,2,J-1,''', .. N "h khi:::; ka'C -l,"',m, h
where qa, q1, k a, k 1,p(> N
+ 2)
(1.6)
are non-negative constantsl!l. Moreover, for almost every point
(x,t) E Q and D;u, a;7)(x,t,u, Dxu,D;u),b~~)(x,t, u,DxU),c~k\x,t,u) are continuous in u E R, Dxu ERN. If the conditions (1.4), (1.5) hold, and the condition (1.6) is replaced by
lii;7)
1 :::;
ka in Q,Lp[b~~),Q]:::; qkh:::; ka, supc~k) < .
Q
00,
Lp[c~k), Q] < qkh :::; ka, Lp[jCk) , Q] :::; i., i,j = 1,"', N, h, k = 1,"', m, in which qkh(k, h = 1"", m) are non-negative constants, then the corresponding conditions will be called Condition C'. It can be seen that Condition C c Condition C'. The so-called initial-oblique derivative boundary value problem (Problem 0) is to find a solution u(x, t) = (Ul.' ... ,urn) of system (1.2), such that Uk 1," . , m) satisfying the initial-boundary conditions:
= Uk (x, t)
E C)3:13/2 (Q)nW2' (Q) (k =
UdX,O)=gk(X), xEfl,k=l,"',m,
(1. 7)
k l Uk
= aUk all + bUk =
Tdx, t), (x, t) E 52, i.e.
lO
-
-21
No.1
Wen & Xu: INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR PARABOLIC SYSTEMS
69
(1.8)
where dj (x, t), g(x) = (gdx), ... ,gm(x)), 7(X, t) = 2
-
C",[gk(X), 0] :::;
(71 (x,
t), ... ,7m (x, t)) satisfy the conditions
1,1 _ »:C",,"'/2h(x, t), 52] < k2,k -
1,"', m,
C~',~/2[b(x,t),52]:::; ko, C~',~/2[d7(x,t),52]:::; ko,j = 1,"',N, N
COS(lI,
n) :::: do
> 0, i.e,
L dj
COS(lI,
Xj) :::: do
> 0, k = 1, ... ,m,
(1.9)
j=l
where n is the unit outward normal vector on 52, a(O < a < 1), ko, ka, do are non-negative constants. In particular, if Problem 0 with the conditions II = n, b = 0 on 52, then Problem 0 is the initial-Neumann boundary value problem, which will be called Problem N.
2 A Priori Estimates of Solutions for Initial-Oblique Derivative Problem for (1.2) First of all, we prove the uniqueness theorem of solutions of Problem 0 for (1.2).
Theorem 2.1
If system (1.2) satisfies Condition C and a\J) = aij(k = 1,"', m,i,j =
1, ... , N), or (1.2) satisfies Condition C' and qhk (k -I h, k, h = 1, ... ,m) are small enough, then the solution of Problem 0 is unique. Proof If system (1.2) satisfies Condition C and a1~) = aij(i,j = 1,"', N, k = 1,"', m), and let u1(X,t),u2(x,t) be two solutions of Problem 0 for (1.2), it is easily seen that U u 1 - u 2 = (U1, ... ,u m) is a solution of the initial-boundary value problem m
N
' L" aijUkxiXj
i,j=1
+
N
" L ' [L' " b-(k) hi UhXi
h=1
. Q, +C_(k)] h Uh -Ukt -- 0 m
k -- 1 ,"',m,
(2.1)
i=1
Uk(X,O) lkudx, t) = 0 i.e ..
= 0,
0;: +
x E 0, k
= 1,"', m,
(2.2)
bUk = 0, (x, t) E 52, k = 1,"', m,
(2.3)
b~~), c~k) are as stated in (1.3). Introduce a transformation of function U = v exp(Bt) , B is an appropriately large number, such that B > sUPQ c~, k = 1,' .. ,m, thus the initial-boundary where
aij,
value problem (2.1)-(2.3) is reduced to inQ,k=l,"',m,
vdx, 0) = 0 on 0, k = 1, ... ,m, .
t.e.
8Vk Oll
+ bVk = 0
on 52, k
(2.4)
(2.5)
= 1,"', m.
(2.6)
Noting that B - sUPQ c~ > 0, (x, t) E Q, k = 1,"', m, by the maximum principle of solutions for (2.4), we can derive that v(x, t) = u(x, t) = 0, i.e. u 1(x, t) = u 2(x, t), (x, t) E Q. In fact, let
70
ACTA MATHEMATICA SCIENTIA
Vol.23 Ser.B
us multiply Vk to each equation of system (2.4). Thus an equation and boundary condition for
v2 =
2:;:1 V~
can be obtained, which is
m
N
LL
Taking into account
k=liJ=1
m
aijVkx;Vkxj 2: qo
N
LL
k=li=1
1
Vkx;
2
1
,
m
L
h,k=1
c~k)VkVh I:::; mk6v2/2,
if the maximum of v 2 is taken in an inner point Po = (to,xo) E Q, and 1v(Po) l¥o 0, then in a neighborhood of Po, the right hand side of (2.7)2: [B - mNk5l4qo - mk5]v2. We choose the constant B such that B > mNk5/4qo - mk5' By using the maximum principle, we see that v 2 cannot take a positive maximum in Q. If v 2 takes the positive maximum at a point Po E 52, 2 then this contradicts (2.6), because from (2.6) it follows that ~ aavv + bv2 = on 52. Hence Vk(t, x) = 0, k = 1,"', m, i.e. v(t, x) = u(t, x) = 0, thus u 1(t, x) - u 2(t, x) = in
°
Q.
°
If (1.2) satisfies Condition C' and qkh(k ¥o h, k, h = 1,"', m) are small enough, we can prove the result in this theorem by using another method (see [2],[3],[4]). Secondly, we shall give the estimates of solutions u(x, t) of Problem 0 in the spaces 10
-
-21
C13 :/3/2(Q) and W 2' (Q).
Theorem 2.2 Suppose that (1.2) satisfies the same conditions in Theorem 2.1. Then any solution u(x, t) of Problem 0 satisfies the estimates
Ru =
Ilullcl,O
(3.(3/2
<
IluIIW;Ol(Q)
L Ilukllc1,o k=1
(3.(3/2
(Q):::; M 1 = M 1(q,po, ex, k,
Q),
(2.8)
m
L IlukIIW;Ol(Q) :::; M 2 = M 2(q,po, ex, k, Q),
(2.9) k=1 (3 :::; ex),Po(2 :::; Po < p),q = (qo,ql),k = (ko,kl,k2),Ml,M2 are non-negative
Su =
where (3(0 constants.
m
(Q) =
=
Proof If Condition C' holds and qhk (h =j:. k, h, k
= 1,"', m)
are small enough, we can
use the results in [3] to prove the estimates in (2.8) ,(2.9). It is clear that system (1.2) can be rewritten as N
N
i,j=l
i=l
'L...J " a (k)Ukx;xj + 'L...J " b(k) ki Ukx; ij
h (k) --
+ C (k) k Uk
~ L...J [~b(k) L...J hi Uhx; + C(k)] h Uh
h=l,h#k
- Ukt -_ f(k)
+ h(k) ,
. Q, k -- 1,"', m. in
(2.10)
i=l
Noting Condition C' and qhk, h =j:. k, h, k = 1,"', m are small enough, there exists a small positive number e, such that
(2.11)
No.1
Wen & Xu: INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR PARABOLIC SYSTEMS
71
By using the results in [3], the solution U = [udx, t),"', um(x, t)] of Problem 0 for system (1.2) satisfies the estimates m
Ru = C~:~/2[U, Q] =
m
l: C~:~/2[Uk' Q] ~ Msl:[k + Ilh(k)/IL2(Q)], l
k=1
k=1
m
Su = Ilull w;.l(Q) =
l: Ilukllw;·l(Q) ~ M
k=1 where k' = max(k l,k 2),Mj = Mj(q,po,a,k,Q),j Ru =
m
+ Ilh(k)IIL2(Q)], k=1 = 3,4. From (2.11), it follows that 4l:W
m
m
k=1
k=1
l: RUk ~ Ms[mk' + cRu], Su = l: SUk ~ M
4[mk'
Provided that e is a sufficiently small positive number, such that cMs Ru
<
1
M~3 mk' =
- c
MI,Su
< 1, then we have
~ M 4[mk' +cRu] ~ M 4mk' + M IM4c =
If system (1.2) satisfies Condition C and a~~) also derive the estimates (2.8) and (2.9).
3
+ cRu].
The Solvability of Problem
M2.
= aij(k = 1"", m, i,j = 1"", N),
we can
0 for the Parabolic System (1.2)
We first consider a special form of system (1.2), namely N
2 ) A _ kl(x, t ,u, D xU, D xu,g kl -uUk_ A '"' kl UUk-Ukt-g L...JaijUkxiXj
-~ [t, b~~Uhxi in which ~Uk
= L:~I 82uk18x;
i,j=1
+ c~lUh] + t"
in Q, k = 1, ... ,m,
(3.1)
and the coefficients in QI, in R N x I}\QI,
where Ql = {(x,t) E Qldist((x,t),8Q) ~ 111},1 is a positive integer and Oii = l,oij = O(i ::f. j, i, j = 1"", N). By using the method as stated in Section 2, we can obtain the following result. T'heorern 3.1 Under the same conditions in Theorem 2.1, if u(x, t) = [UI (x, t), ... ,um(x, t)] is any solution of Problem 0 for system (3.1), then u(x, t) can be expressed in the form U(x, t)
= U(x, t) + V(x, t) = U(x, t) + vo(x, t) + v(x, t),
v(x,t)=Hp=
r G(x,t, (,T)p((,T)da(dT,
lQo
G = { [A(t - T)]-N/2 exp[lx - (12/4(t - T)], t > T, 0, t
~ T,
except t -
T
= [z -
(I =
0,
(3.2)
72
ACTA MATHEMATICA SCIENTIA
Vol.23 Ser.B
where p(x, t) = ~u - Ut and V(x, t) is a solution of Problem Do for (3.1) in Qo = no x 1(n o = {Ixl < R}) with the initial-boundary condition V(x, t) = a on 8Qo, in which R is a large number, such that no :J 0, and U(x, t) is a solution of Problem 6 for LU = ~U - U, Q with the initial-boundary conditions (3.10), (3.11) below, which satisfy the estimates
= a in
where ,8(0 < ,8 ~ a), M j = Mj(q,po, a, k, Qt}(j = 5,6) are non-negative constants, q = (qo, qd, k = (ko, k 1 , k2 ) . Theorem 3.2 If system (1.2) satisfies the same conditions as in Theorem 2.1, then Problem 0 for (3.1) has a solution u(x, t). Proof In order to prove the existence of solutions of Problem 0 for the nonlinear system (3.1) by using the Larey-Schauder theorem, we introduce the system with the parameter h E [0,1]: (3.4) ~Uk - Ukt = hll(x, t, U, Dxu, D~u) in Q, k = 1,"', m. '21
10
-
C13 :13 / 2 (Qo) n W;,l(QO)(O < ,8 ~ a), the elements of which are systems of real functions V(x,t) satisfyDenote by B M a bounded open set in the Banach space B = W 2 ' (Qo)
ing the inequalities
(3.5) . in which W;,l(QO) = W;,o(Qo)
n Wg,l(QO), M 6
is a non-negative constant as stated in (3.3).
We choose any function V(x, t) E B M and substitute it into the appropriate positions in the right hand side of (3.4), and then we make an integral ii(x, t) = Hp as follows:
ii(x,t) = Hp, p(x,t) = ~V -~.
(3.6)
Next denote by iiO(x, t) a solution of the initial-boundary value problem in Qo, i.e.
(3.7)
iio(x, t) = -ii(x, t) on 8Qo. It is clear that V(x, t) = ii(x, t)
+ iiO(x, t)
(3.8)
is a solution of of the corresponding Problem Do in
Qo, namely ~V - ~ = p(x, t) in Qo, V(x, t) =
a
on 8Qo.
Moreover on the basis of the result in [2], we can find a solution U(x, t) of the following initialboundary problem in Q: ~U
- ir; = a
(3.9)
in Q,
U(x, 0) = g(x) - V(x, 0) on 0, 8U 8v
-
+ b(x, t)U = r(x, t) -
8V 8v
•
+ b(x, t)V
(3.10) on 8 2 .
(3.11)
Now we discuss the equation ~V -
vt =
hll(x, t, U, Dxu, D~U + D~V), k = 1,,,,, m,
a ~ h ~ 1,
(3.12)
No.1
Wen & Xu: INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR PARABOLIC SYSTEMS
73
where u = fj + V. By Condition C', applying the principle of contracting mapping, we can find a unique solution V(x, t) of Problem Do for system (3.12) in Qo satisfying the initial-boundary condition (3.13) V(x, t) = 0 on 8Qo. Denote u(x, t) = U(x, t) + V(x, t), where the relation between U and V is as stated in Theorem 3.1, and by V = S(V,h),u = Sl(V,h)(O ~ h ~ 1) the mappings from V onto V and u respectively. Furthermore, if V(x, t) is a solution of Problem Do in Qo for the system ~V -
vt = hll(x,t,u,Dxu,D;(U + V)),O ~ h ~ 1,
(3.14)
with the initial-boundary condition V(x, t) = 0 on 8Qo, where u = Sl (V, h), then from Theorem 3.1, the solution V(x, t) of Problem Do for (3.14) satisfies the second estimate in (3.3), consequently the solution V(x, t) E B M. Set B o = B M X [0,1], we can verify that the mapping V = S(V, h) satisfies the three conditions of the Leray-Schauder theorem: 1. For every h E [0,1], V = S(V, h) continuously maps the Banach space B into itself, and is completely continuous on B M . Besides, for every function V(x,t) E B M , S(V,h) is uniformly continuous with respect to h E [0,1]. 2. For h = 0, from (3.5) and (3.12), it is clear that V = S(V,O) E B M . 3. From Theorems 2.2 and (3.5), we see that the functional equation V = S(V, h) (0 ~ h ~ 1) does not have a solution V(x, t) on the boundary 8B M = BM\BM. Hence by the Leray-Schauder theorem, we know that Problem Do for system (3.12) with h = 1 has a solution V(z, t) E BM, and then Problem 0 of system (3.4) with h = 1, i.e. (3.1) has a solution u(x, t) = 51 (V, h) = U(x, t) + V(x, t) = U(x, t) + vO(x,t) + v(x, t) E B. Theorem 3.3 Under the same conditions in Theorem 2.1, Problem 0 for system (1.2) has a solution. Proof By Theorems 2.2 and 3.2, Problem 0 for system (3.1) possesses a solution ul(x, t), and the solution ul(x, t) of Problem 0 for (3.1) satisfies the estimates (2.8) and (2.9), where l = 1,2, . ". Thus, we can choose a subsequence {u 1k (x, tn, such that {u 1k (x, tn, {U~ki (x, tn(i = 1,"', N) in Q uniformly converge to uO(x,t), U~i (x, t)(i = 1,"', N) respectively. Obviously, uO(x,t) satisfies the initial-boundary conditions of Problem O. On the basis of principle of compactness of solutions for system. (3.1), we can see that uO(x,t) is a solution of Problem 0 for (1.2). We mention that the initial-irregular oblique derivative problem for system (1.2) remains to be continuously considered. References 1 Alkhutov Yu A, Mamedov I T. The first boundary value problem for nondivergence second order parabolic equations with discontinuous coefficients. Math USSR Sbornik, 1988, 59: 471-495 2 Ladyzhenskaya 0 A, Solonnikov V A, Ural'tseva N N. Linear and Quasilinear Equations of Parabolic Type. Trans Math Monographs 23, Providence, RI: Amer Math Soc, 1968 3 Wen G C, Tain M Y. Initial-oblique derivative problems for nonlinear parabolic equations with measurable coefficients. Comm in Nonlinear Sci & Numer Simu, 1998, 3:109-113 4 Wen G C, Zou B T. Initial-oblique derivative problems for linear parabolic equations of second order with measurable coefficients in a higher dimensional domain. Acta Math Appl Sinica, 1999, 22:579-588 (in Chinese)