Initial-irregular oblique derivative problems for nonlinear parabolic complex equations of second order with measurable coefficients

Nonlinear Analysis 39 (2000) 937 – 953

www.elsevier.nl/locate/na

Initial-irregular oblique derivative problems for nonlinear parabolic complex equations of second order with measurable coecients ( Guochun Wen, Benteng Zou ∗ Department of Mathematics, Peking University, Beijing 100871, People’s Republic of China Received 21 January 1998; accepted 28 January 1998

Keywords: Initial-irregular oblique derivative problems; Nonlinear and nondivergence parabolic equations; Measurable coecients

In this paper, initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic complex equations of second order in multiply connected domains are discussed, where coecients of equations are measurable. Firstly, the uniqueness of solutions for the above problems is veri ed, and then a priori estimates of solutions for the problems are given. Finally, by using the above estimates and the Leray–Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results in this paper are generalizations of corresponding theorems in [1, 5–7]. 1. Formulation of initial-boundary value problems Let D be an P (N + 1)-connected bounded domain in the z = x + iy plane C with the N boundary = j=0 j ∈ C2 (0¡¡1). Without loss of generality, we may consider PN that D is a circular domain in |z|¡1 with the boundary = j=0 j ; where j = {|z − zj |} = j ; j = 0; 1; : : : ; N; 0 = N +1 = {|z| = 1} and z = 0 ∈ D: Denote G = D × I; in which I = {0¡t ≤ T }; T is a positive constant, and @G = @G1 ∪ @G2 is the parabolic (

The work was supported by the National Natural Science Foundation of China.

∗

Corresponding author. Tel.: 00861 062755937; fax: 00861 062751801. E-mail address: [email protected] (B. Zou)

0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 2 5 8 - 2

938

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

boundary of G, where @G1 ; @G2 are the bottom {z ∈ D; t = 0} and the lateral boundary {z ∈ ; t ∈ I } of the domain G; respectively. We consider the nonlinear nondivergence parabolic equation of second order (x; y; t; u; ux ; uy ; uxx ; uxy ; uyy ) − Hut = 0 in G;

(1.1)

where  is a real-valued function of x; y; t ( ∈ G ); u; ux ; uy ; uxx ; uxy ; uyy (∈ R ) and H (0¡H ¡∞) is a constant. Under certain conditions, Eq. (1.1) can be reduced to the complex form A0 uzz − Re[Quzz + A1 uz ] − A2 u − Hut = A3 ;

(1.2)

where z = x + iy;  = F(z; t; u; uz ; uzz ; uzz ); and Z 1 Fuzz (z; t; u; uz ; uzz ; uzz ) d = A0 (z; t; u; uz ; uzz ; uzz ); A0 = 0

Z

Q = −2

1 0

Z A1 = −2

1 0

Z A2 = −

1 0

Fuzz (z; t; u; uz ; uzz ; uzz ) d = Q(z; t; u; uz ; uzz ; uzz ); Fuz (z; t; u; uz ; 0; 0) d = A2 (z; t; u; uz );

Fu (z; t; u; 0; 0; 0) d = A2 (z; t; u);

A3 = −F(z; t; 0; 0; 0; 0) = A3 (z; t): In particular, for the linear parabolic equation of second order a11 uxx + 2a12 uxy + a22 uyy + b1 ux + b2 uy + cu − Hut = d in G; its complex form is as follows: A0 uzz − Re[Quzz + A1 uz ] − A2 u − Hut = A3

in G;

(1.3)

where A0 = A0 (z; t) = 2(a11 + a22 );

Q = Q(z; t) = 2(−a11 + a22 − 2a12 i);

A1 = A1 (z; t) = −(b1 + b2 i);

A2 = A2 (z; t) = −c; A3 = A3 (z; t) = d:

Suppose that Eq. (1.2) satis es Condition C, namely 1. A0 (z; t; u; uz ; uzz ; uzz ); Q(z; t; u; uz ; uzz ; uzz ); A1 (z; t; u; uz ); A2 (z; t; u); A3 (z; t) are measurable for any continuous function u(z; t) ∈ C 1; 0 (G) and measurable functions uzz ; uzz ∈ L2 (G) and satisfy the conditions 0¡ ≤ A0 ≤ −1 ;

sup(A20 + |Q|2 )= inf A20 ≤ q¡ 43 ; G

G

Lp [Aj ; G] ≤ k0 ; j = 1; 2; Lp [A3 ; G] ≤ k1 ; p¿4; |Aj | ≤ k0 in G = {(z; t) ∈ G; dist((z; t); @G∗ )¡}; j = 1; 2;

(1.4) (1.5)

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

939

in which @G∗ = @G1 ∪ {@G2 ∩ (cos(; n)¿0)}; n is the outer normal vector at every point on @G2 ;  is a suciently small positive number. 2. The above functions with respect to u ∈ R; uz ∈ C are continuous for almost every point (z; t) ∈ G and uzz ∈ C; uzz ∈ R: 3. For almost every point (z; t) ∈ G and u ∈ R; uz ∈ C; U j ∈ C; V j ∈ R ( j = 1; 2); there is F(z; t; u; uz ; U 1 ; V 1 ) − F(z; t; u; uz ; U 2 ; V 2 ) ˜ 1 − U 2 )]; = A˜0 (V 1 − V 2 ) − Re[Q(U  ≤ A˜0 ≤ −1 ;

˜ 2 )= inf A˜02 ≤ q¡ 4 ; sup(A˜20 + |Q| 3 G

G

(1.6) (1.7)

in Eqs. (1.4), (1.5) and (1.7),  (0¡¡1); q (≥1); k0 ; k1 ; p(¿4) are nonnegative constants. Condition C with the conditions |Aj | ≤ k0 (j = 1; 2) in G is called Condition C∗ . The solution u(z; t) of complex Eq. (1.2) with Condition C is indicated by a function u(z; t) ∈ C 1; 0 (G) ∩ W˜ 22; 1 (G ∗ ); i.e. u; uz ∈ C 0; 0 (G); uzz ; uzz ; ut ∈ L2 (G ∗ ), and u(z; t) satis es Eq. (1.2) for almost every point (z; t) ∈ G; in which W˜ 22; 1 (G ∗ ) = W22; 0 (G ∗ ) ∩ W20; 1 (G ∗ ); C 0; 0 (G) = C(G) and G ∗ is any closed subset in G. In this case, the solution of Eq. (1.2) is called a generalized solution in G. The so-called initial-irregular oblique derivative problem for complex equation (1:2) is to nd a continuous solution u(z; t) ∈ C 1; 0 (G) of Eq. (1.2) satisfying the initialboundary conditions u(z; 0) = g(z)

on D;

@u + (z; t)u = (z; t); @

(1.8)  z ) + u =  on @G2 ; i:e: 2Re(u

(1.9)

where  is a given vector at every point (z; t) ∈ @G2 parallel to the plane t = 0;  = cos(; x)−i cos(; y); and @g=@+g =  on × {t = 0}: Suppose that g(z); (z; t); (z; t), (z; t) satisfy the conditions: C2 [g; D] ≤ k2 ;

(z; t) cos(; n) ≥ 0

on @G2 ;

1 1 C;1; =2 [; @G2 ] ≤ k0 ;  = {; }; C;1; =2 [; @G2 ] ≤ k2 ;

(1.10) (1.11)

in which n is the outer normal at every point (z; t) ∈ @G2 ; and ( 12 ¡¡1); k0 ; k2 are nonnegative constants. The boundary @G2 can be divided into two parts, namely E + ⊂ {(z; t) ∈ @G2 ; cos(; n) ≥ 0;  ≥ 0} and E − ⊂ {(z; t) ∈ @G2 ; cos(; n) ≤ 0;  ≤ 0}, and E + ∩ E − = ; E + ∪ E − = @G2 ; E + ∩ E − = E 0 : For any component Lt of @G2 ∩ {t = t0 = constant; 0 ≤ t0 ≤ T }; there are three cases: (1) Lt ⊂ E + : (2) Lt ⊂ E − : (3) − + − There exists at least a point on each component of L+ t = E ∩ Lt and Lt = E ∩ Lt ; 0 t t 0t 0t 0 such that cos(; n) 6= 0; and E ∩ Lt = {a1 ; : : : ; am ; a1 ; : : : ; am0 }; 0¡m; m ¡∞; such that − every component of L+ t ; Lt includes its initial point and does not include its terminal − + 0t t point; and aj ∈ Lt ; aj ∈ Lt ; when the direction of  at ajt ; aj0t is equal to the direction

940

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

+ 0t t 0t of Lt ; and ajt ∈ L− t ; aj ∈ Lt ; when the direction of  at aj ; aj is opposite to the direction of Lt ; and cos(; n) changes the sign once on the two components with the end point ajt or aj0t . We may assume that

u(ajt ) = bj (t) on Ij0 ; j = 1; : : : ; m; S herein Ij0 = t|(z; t)=ajt ; bj (t) is a given continuous function satisfying C 1 [bj (t); Ij0 ] ≤ k2 ¡∞;

j = 1; : : : ; m:

(1.12)

(1.13)

In addition, if cos(; n) = 0;  = 0 on a component L0t of @G2 × {t = t0 }; then Eq. (1.9) should be replaced by an appropriate condition u(z; t) = r(z; t) in @G 0

(1.14)

with the condition 1 C;2; =2 [r(z; t); @G 0 ] ≤ k2 ; S where @G 0 = L0t : The above initial-boundary value problem is called Problem P. Problem P with the condition @G 0 = @G2 is called the rst boundary value problem (Problem D), Problem P with the condition cos(; n) = 1;  = 0 on @G2 is called the initial-Neumann boundary value problem (Problem N), and Problem P with the condition cos(; n)¿0 on @G2 is called the initial-regular oblique derivative problem (Problem O).

2. Uniqueness of solutions In order to discuss the uniqueness of solutions of Problem P for Eq. (1.2), we add the condition: 4. For any u j ∈ R; uzj ∈ C ( j = 1; 2); U ∈ C; V ∈ R; there is F(z; t; u1 ; uz1 ; U; V ) − F(z; t; u2 ; uz2 ; U; V ) = Re[A˜1 (u1 − u2 )z ] + A˜2 (u1 − u2 ) on G;

(2.1)

where A˜j ( j = 1; 2) satisfy Lp [A˜j ; G]¡∞ in G: Conditions (1)–(4) will be called Condition C0 . If Eq. (1.2) is linear, then Conditions C0 and C are same. Theorem 2.1. Under Condition C0 , Problem P for Eq. (1.2) has at most one solution u(z; t). Proof. We rst prove the theorem under Conditions C0 and C∗ . Let u j (z; t) (j = 1; 2) be two solutions of Problem P for Eq. (1.2), it is easy to see that u = u(z; t) = u1 (z; t)−

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

941

u2 (z; t) is a solution of the initial-boundary value problem ˜ zz + A˜1 uz ] − A˜2 u − Hut = 0 A˜0 uzz − Re[Qu u(z; 0) = 0

in D;

@u + u = 0 @ where A˜0 =

Z

1 0

Q˜ = −2

(2.2) (2.3)

on @G2 ;

(2.4)

2 Fs (z; t; v; p; q; s) d; s = uz2z + (u1 − u2 )zz ; q = uzz + (u1 − u2 )zz ;

Z

A˜1 = −2

in G;

1 0

Z

Fq (z; t; v; p; q; s) d; p = uz2 + (u1 − u2 )z ; v = u2 + (u1 − u2 );

1 0

Fp (z; t; v; p; q; s) d; A˜2 = −

Z

1 0

Fv (z; t; v; p; q; s) d:

Introduce a transformation U = ue−Bt ; where B is an undetermined real constant, complex Eq. (2.1) can be reduced to the form ˜ zz + A1 Uz ] − (A˜2 + HB)U = HUt A˜0 Uzz − Re[QU

(2.5)

and U (z; t) satis es the initial-boundary conditions U (z; t) = 0

on D;

@U + (z; t)U (z; t) = 0 @ U (ajt ) = 0 U (z; t) = 0

(2.6) on @G2 :

on Ij0 ; j = 1; : : : ; m: on @G 0 :

(2.7) (2.8) (2.9)

We choose the constant B such that HB+inf G A˜2 ¿0; if the positive maximum of U (z; t) takes a point P0 ∈ G; then by using the maximum principle of solutions for Smparabolic Eq. (2.5), it is seen that the maximum point P0 ∈ @G2 \{I 0 ∪ @G 0 }, here I 0 = j=1 Ij0 . As stated before, for any component Lt of @G2 ∪ {t = t0 = constant; 0 ≤ t0 ≤ T }, there are three cases: 1. Lt ⊂ E + . 2. Lt ⊂ E − . 3. There exists at least a point on each component − + − of L+ l = E ∪ Lt and Lt = E ∩ Lt ; such that cos(; n) 6= 0. If P0 ∈ Lt of cases 1 and 2, then we have @U + (z; t)U (z; t)¿0 or @

¡0 at P0 ;

(2.10)

− respectively, this contradicts Eq. (2.7). If P0 belongs to any component of L+ t or Lt 0 0 of case 3, it is easy to see that P0 ∈= {I ∪ @G }; on the basis of the same reason as before, it can be derived that cos(; n) = 0;  = 0 at P0 , otherwise by means of the maximum principle, there are Eqs. (2.7) and (2.10). Denote by L˜t the longest curve

942

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

− ˜ of L+ t or Lt including the point P0 so that cos(; n) = 0;  = 0 on Lt . It is clear that ˜ ˜ aj (1 ≤ j ≤ m) is not an end point of Lt : If Lt does not include aj0t (1 ≤ j ≤ m0 ); then there exists a point P 0 ∈ Lt \L˜t , such that at P 0 , there are three cases: (1) with cos(; n)¿0; @U =@n¿0; cos(; s)¿0(¡0); @U=@s ≥ 0 (≤0); (P 0 ) ≥ 0; the following can be derived:

@U @U @U + U = cos(; n) + cos(; s) + U ¿0 @ @n @s

at P 0 ;

(2.11)

where s is a tangent vector on Lt at P 0 . (2) with cos(; n)¡0; @U=@n¿0; cos(; s)¿0 (¡0); @U=@s ≤ 0 (≥0); (P 0 ) ≤ 0, the following can be derived: @U @U @U + U = cos(; n) + cos(; s) + U ¡0 @ @n @s

at P 0 :

(2.12)

(3) with cos(; n) = 0;  6= 0; at P 0 , the following can be derived: @U + U 6= 0 @

at P 0 :

(2.13)

These inequalities are contradictory with Eq. (2.7). If L˜t includes a point a0tj (1 ≤ j ≤ m0 ), then in the neighborhood of aj0 , there are two possible cases: (1) with the direction of  at aj0t and the direction of Lt are same, there exists a point P˜ belonging to a curve 0t (⊂L− t ) with the initial aj , such that cos(; n) ≤ 0; ¡0; so @U + U ¡0 @

at P˜

(2.14)

˜ thus we have or cos(; n)¡0; @U=@n¿0; cos(; s)¿0; @U=@s ≤ 0;  = 0; at P, Eq. (2.14) too. They contradict Eq. (2.7). (2) With the direction of  at aj0t and the direction of Lt are opposite, there exists a point P˜ belonging to a curve (⊂ L+ t ) with the initial point aj0t , such that cos(; n) ≥ 0; ¿0; so @U=@ + U ¿0

at P˜

(2.15)

˜ thus we have Eq. (2.15). They contradict Eq. (2.7). Hence, or cos(; n)¿0;  = 0; at P, we derive that U (z; t) ≤ 0, i.e. u1 (z; t) − u2 (z; t) ≤ 0 in G: Similarly, we can prove that U (z; t) cannot take the negative minimum on G, hence U (z; t) ≥ 0, i.e. u1 (z; t) − u2 (z; t) ≥ 0 in G: Thus, u1 (z; t) = u2 (z; t) in G. Next, we prove the theorem under Condition C0 . Similar to the earlier proof, if 1 u (z; t); u2 (z; t) are two solutions of Problem P for Eq. (1.2), then u = u1 − u2 is a solution of the initial-boundary value problem (2:2)– (2:4). As the right-hand sides in Eqs. (2.2)–(2.4) are equal to zero, which are less than a small positive number . According to the estimates in Theorem 3.3 below, we have 0 [u; G] ≤ M (k1 + k2 ) = 2M; C ;1; =2

(2.16)

in which M = M14 is a constant as stated in (3.25) and k1 + k2 = 2: Noting that the positive constant  is arbitrary, let  → 0, it is derived that C 1; 0 [u; G] = 0, hence u = u1 − u2 = 0, i.e. u1 (z; t) = u2 (z; t) in G.

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

943

3. A priori estimates of solutions Theorem 3.1. Suppose that complex Eq. (1.2) satis es Condition C and u(z; t) ∈ W˜ 22; 1 (G) is any solution of Problem P satisfying C 1; 0 [u; G] = C 0; 0 [u; G] + C 0; 0 [uz ; G] ≤ M1 ;

(3.1)

where M1 is a nonnegative constant. Then the solution u(z; t) of Problem P for Eq. (1.2) satis es the estimate 0 C ;1; =2 [u; Gm ] ≤ M2 = M2 (; q; ; k; p; Gm ; M1 );

(3.2)

where Gm ={(z; t)∈G; dist(z; @D) ≥ 1=m; t ∈I }; m is a positive integer, and (0¡ ≤ ) and M2 are nonnegative constants only dependent on ; q; ; k = (k0 ; k1 ; k2 ); p; Gm ; M1 : 0 holds, because the set Proof. It suces to prove Eq. (3.2) only when Cylr = Cyl−r; 0; 2r 1=2 1

1

0 1=2 1 Gm can be covered by nite domains in the form Cyltz0;; t2r+r ; t ¡t¡t 1 + 1=2 = {|z−z |¡2r r}; and make the transformation  = z − z 0 ;  = t − t 1 − r; where r is small enough such 1 1 that 2r 1=2 ¡1=m; thus we map the domain Cylzt 0;; t2r+r 1=2 onto Cylr : This requirement can be realized. Making a second-order continuously dierentiable function ( 1 in Cylr ; g(z; t) = 0 ≤ g(z; t) ≤ 1 in Cyl2 r \Cylr ; (3.3) 0 in {C × I }\Cyl2 r ;

introducing ( E(z; t) =

t −1 e−|z| 0

2

=4t

if t¿0; if t ≤ 0

except t = |z| = 0

(3.4)

and applying the inequality [1, 5] Z Z Z |Lu|2 E(−z; −t)(−t) dx dy dt Cylr

Z Z Z

≤ M3

2

Cylr

−

(Lu) E(−z; −t)(−t)

(3.5) dx dy dt

to the function v(z; t) = g(z; t)[u(z; t) − u(0; 0) − zuz (0; 0)];

(3.6)

where (0¡ ¡ 0 ) is a suciently small constant and Lu = uzz − ut =4; A˜j = Aj =;

˜ zz + A˜1 uz ] − A˜2 u − ut =4; Lu = A˜0 uzz − Re[Qu

j = 0; 1; 2; 3;

Q˜ = Q=;  = 4 inf A0 =3 G

944

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

and M3 = M3 (; q; ; k; G), the inequality Z Z Z |u(z; t) − u(0; 0) − zuz (0; 0)|2 E(−z; −t)(−t)−2− dx dy dt Cylr

Z Z Z ≤

Cyl2 r

|v(z; t)|2 E(−z; −t)(−t)−2− dx dy dt

(Z Z Z ≤ M4 Z Z Z +

Cyl2 r

Cyl2 r

|Lu|2 E(−z; −t)(−t)− dx dy dt

|u(z; t) − u(0; 0) − zuz (0; 0)|2 E(−t)− dx dy dt

(Z Z Z +

Cyl2 r

"Z Z Z ≤ M5

) 2

Cyl2 r

−

|uz (z; t) − uz (0; 0)| E(−t)

dx dy dt #

(Lu)2 E(−t)− dx dy dt + M6

(3.7)

is derived, in which Condition C is used. We choose positive number small enough, such that Z Z Z Z Z Z (Lu)2 E(−t)− dx dy dt = |A˜3 |2 E(−t)− dx dy dt Cyl2 r

Cyl2 r

"Z Z Z ≤

Cyl2 r

#2=p |A˜3 |p dx dy dt

"Z Z Z ×

#1=q q

Cyl2 r

− q

E (−t)

dx dy dt

≤ k1 M7 ; (3.8)

where q = p=(p − 2); Mj = Mj (; q; k; p; ; Gd ; M1 ) ( j = 4; : : : ; 7) are nonnegative constants. Furthermore, by using L2 [uzz ; G] ≤ L2 [Lu; G]; HL2 [ut ; G] ≤ L2 [Lu; G]; L2 [uzz ; G] ≤ L2 [uzz ; G] ≤ L2 [Lu; G]

(3.9)

[1], estimate (3:2) is obtained. Theorem 3.2. Suppose that Condition C holds and the solution u(z; t) of Problem P for Eq. (1.2) satis es estimate (3:1). Then the solution u(z; t) satis es the estimate 0 C ;1; =2 [u; G] ≤ M8 = M8 = M8 (; q; ; k; p; G; M1 );

where k = (k0 ; k1 ; k2 ); (0¡ ≤ ); M8 are nonnegative constants.

(3.10)

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

945

Proof. By using Theorem 3.1, we can prove that u(z; t) satis es the estimate 0 (u; G ) ≤ M9 = M9 (; q; ; k; p; ; G); C ;1; =2

(3.11)

where G = {(z; t) ∈ G|dist(z; @D) ≥ ¿0; t ∈ I };  is a small positive number. Next, we estimate u(z; t) on G\G : We are free to choose a point z∗ ∈ ; there is no harm in assuming that z∗ ∈ 0 ; otherwise when z∗ ∈ j (1 ≤ j ≤ N ); through a transformation  = j =(z − zj ); the circumference j (1 ≤ j ≤ N ) is reduced to the unit circumference. Moreover, we can assume that | cos(; n)|¡ 12 at z∗ , because if | cos(; n)| ≥ 12 at z∗ , a priori estimates of the solution u(z; t) in the neighborhood of z∗ can be obtained as stated in [7]. For a small positive number d, denote ˜ = ∩ {|z − z∗ | ≤ d}; @G˜ 2 = ˜ × I ; and nd a curve ˆ , such that ˆ ∩ ˜ = ; ˆ ∪ ˜ ∈ C2 is a closed curve, which is a boundary of a simply connected domain D∗ such that D∗ ∩ D 6= : ˜ t); (z; Moreover, we appropriately choose the functions (z; ˜ t) on @Gˆ 2 = ˆ × I and g˜ on D∗ \D; so that ( ( ( ;  − u; (z; t) ∈ @G˜ 2 ; u; z ∈ D; ∗ ∗ ∗  = g =  = ∗ ˜ g; ˜ z ∈ D \D; ; ; ˜ (z; t) ∈ @Gˆ 2 ; satisfy the condition 0 0 (∗ ; @G ∗ ) ≤ 2k0 ; C;1; =2 (∗ ; @G ∗ ) ≤ 2(k0 + k1 ); C2 (g∗ ; D∗ ) ≤ 2k2 ; C;1; =2

(3.12)

and K = (1=2)
ˆ ∪ ˜ arg∗ (z; t) = 0; t ∈ I : Afterwards, a solution v(z; t) of the following initial-boundary value problem can be found vzz − Hvt = 0 in G ∗ = D∗ × I;

(3.13)

v(z; 0) = g∗ on D∗ ;

(3.14)

@v = ∗ ; i:e: 2Re(∗ vz ) = ∗ on @D∗ × I ; (3.15) @∗ in which ∗ = cos(∗ ; x)−i cos(∗ ; y); cos(∗ ; n)∗ ≥ 0; and v(z; t) satis es the estimate C 2; 1 (v; G ∗ ) ≤ M10 = M10 (; q; ; k; p; G):

(3.16)

[4, 6]. Setting that U = u − v; it is clear that U (z; t) satis es the following equation and initial-boundary condition: An0 Uzz − Re[Qn Uzz ] − HUt = A∗ U (z; 0) = 0

in G ∗ ;

on D∗ ;

(3.18)

@U = 0; i:e: 2Re(∗ Uz ) = 0 on @G˜ 2 ; @∗ where A∗ = A − A0 vzz + Re[Qvzz ] + Hvt : Now, we nd a solution the following boundary value problem: [ (z; t)]z = 0

in D∗ ;

Re (z; t) = −arg ∗ (z; t)

(3.17)

on @D∗ :

(3.19) (z; t) in z ∈ D∗ of

(3.20)

946

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

Moreover, we nd a solution of the following boundary value problem: Wz∗ = W z in G ∗ ; Re[W ∗ (z; t)] = 0 on @G ∗ ;

Im[W ∗ (z∗ ; t)] = 0;

where we rst assume that the coecients of Eq. (3.17) on D∗ are in nite-order continuously dierentiable and satisfy the similar Condition C, because we can choose the in nite-order continuously dierentiable functions such that they converge to the coecients of Eq. (3.17), respectively, according to the norm Lp (G ∗ ), but nally which can be cancelled. Denote by G˜ ∗ the symmetrical domain of G ∗ and  = exp[i (z; t)]. Let Z z ∗ V (z; t) = W (z; t) dz + W˜ (z; t) d z − v(z∗ ; t) on Gˆ = G ∗ ∪ G˜ ∗ ; z∗

( W (z; t) =

(z; t)Uz (z; t); −(1= z; t)U1=z (1= z; t);

( W˜ (z; t) =

W ∗ (z; t); (z; t) ∈ G ∗ ;

(3.21)

−W ∗ (1= z; t); (z; t) ∈ G˜ ∗ :

Thus, the complex function V (z; t) satis es the equation in the form A∗0 Vzz − Re[Q∗ Vzz ] − HVt = A˜∗ ;

(3.22)

satisfying the conditions similar to Condition C. According to the method as before, we obtain the estimate 0 C ;1; =2 (V; Gd ) ≤ M11 = M11 (; q; ; k; G);

(3.23)

in which Gd = {[|z − z∗ | ≤ d=2] × I } ∩ G: It implies that 0 C ;1; =2 (u; Gd ) ≤ M12 = M12 (q; ; k; p; G):

(3.24)

Combining (3.11), (3.16) and (3.24), estimate (3:10) is obtained. Theorem 3.3. Suppose that Condition C holds and u(z; t) is any solution u(z; t) of Problem P for Eq. (1.2). Then the solution u(z; t) satis es the estimates 0 C ;1; =2 [u; G] ≤ M13 ;

0 C ;1; =2 [u; G] ≤ M14 (k1 + k2 );

(3.25)

where = (; q; ; k; p; G)¿0, M13 =M13 (; q; ; k; p; G) and M14 = M14 (; q; ; k0 ; p; G) are nonnegative constants. Proof. We rst prove that u(z; t) satis es estimate (3:1), where M1 = M1 (; q; ; k; p; G) is a nonnegative constant. Suppose that Eq. (3.1) is not true, then there exist sequences of functions {Anj } ( j = 0; 1; 2; 3); {Qn }; {gn }; {n }; {n }{n }; {bjn } (j = 1; : : : ; m); {r n }; which satisfy same conditions on Aj ( j = 0; 1; 2; 3); Q; g; ; ; ; bj (j = 1; : : : ; m); r; and {Anj }; {Qn } weakly converge to A0j ( j = 0; 1; 2; 3); Q0 in G, {gn }; {n }; {n }; {n }; {bjn }; {r n } uniformly converge to g; 0 ; 0 ; 0 ; bj0 (j = 1; : : : ; m); r 0 on D, @G2 ; Ij0 (j = 1; : : : ; m) or @G 0 respectively. Moreover the initial-boundary value problem An0 uzz − Re[Qn uzz + An1 uz ] − An2 u − An3 = Hut

in G;

(3.26)

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

u(z; 0) = gn (z)

947

on D;

@u + n u = n @n

(3.27)

on @G2 ;

(3.28)

u(ajt ) = bjn (t) on Ij0 ; j = 1; : : : ; m; u(z; t) = r n (z; t) on @G 0

(3.29)

has the solution un (z; t) (n = 1; 2; : : :); such that Jn = C 1; 0 (un ; G) → ∞ as n → ∞; we may assume that Jn ≥ 1; n = 1; 2; : : : : It is easy to see that U n = un =Jn is a solution of the following initial-boundary value problem: An0 Uznz − Re[Qn Uzzn ] − HUtn = An ; An = Re[An1 Uzn ] + An2 U n + An3 =Jn U n (z; 0) = gn (z)=Jn @U n + n U n = n =Jn @n U n (ajt ) = bjn =Jn

on D;

in G;

(3.30) (3.31)

on @G2 ;

(3.32)

on Ij0 ; j = 1; : : : ; m; U n (z; t) = r n =Jn

on @G 0 :

(3.33)

Noting that An ; gn =Jn ; n =Jn ; bjn =Jn ; r n =Jn satisfy 1 (n =Jn ; @G2 ) ≤ k2 ; Lp (An ; G) ≤ 2k0 + k1 ; C2 (gn =Jn ; D) ≤ k2 ; C;1; =2 2; 1 (r n =Jn ; @G 0 ) ≤ k2 : C1 (bjn =Jn ; Ij0 ) ≤ k2 ; j = 1; : : : ; m; C;=2

In addition, by using the method in the proof of Theorem 3.4 below, it can be derived that kU n kW˜ 2; 1 (G) = L2 [|Uzzn | + |Uznz |; G] + L2 [HUtn ; G] ≤ M15 ; 2

(3.34)

where M15 = M15 (; q; ; k; p; G) is a nonnegative constant. Hence, we can choose a subsequence {U nk } of {U n }; such that {U nk }; {Uznk } uniformly converge to U 0 ; Uz0 on G; and {Uznzk }; {Uzznk }; {Utnk } weakly converge to Uz0z ; Uzz0 ; Ut0 in G, respectively, and U 0 = U 0 (z; t) is a solution of the following initial-boundary value problem: A00 Uz0z − Re[Q0 Uzz0 + A01 Uz0 ] − A02 U 0 = HUt0 U 0 (z; 0) = 0

on D;

@U 0 + U 0 = 0 @ U 0 (ajt ) = 0

in G;

on @G2 ;

on Ij0 ; j = 1; : : : ; m; U 0 (z; t) = 0 on @G 0 :

(3.35) (3.36) (3.37) (3.38)

By using the method in [3] and Theorem 2.1, we can prove U 0 (z; t) = 0 in G. However, from C 1; 0 (U n ; G) = 1, it follows that there exists a point P ∗ = (z ∗ ; t ∗ ); such that |U 0 (P ∗ )| + |Uz0 (P ∗ )|¿0: This contradiction shows that Eq. (3.1) is true. From

948

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

Theorem 3.2, it follows that the rst estimate in Eq. (3.25) holds. Moreover, it is easy to derive the second estimate in Eq. (3.25). Theorem 3.4. Under the same condition in Theorem 3.3, any solution u(z; t) of Problem P for Eq. (1.2) satis es the estimate kukW˜ 2; 1 (G) ≤ M15 ; kukW˜ 2; 1 (G) ≤ M16 (k1 + k2 ); 2

2

(3.39)

in which M15 = M15 (; q; ; k; p; G); M16 = M16 (; q; ; k0 ; p; G): Proof. Similar to Theorem 3.3, it suces to prove the rst estimate in Eq. (3.39) and the estimate kukW˜ 2; 1 (G0 ) ≤ M17 = M17 (; q; ; k; p; ; G); 2



(3.40)

where G0 = {|z − z∗ | ≤ ; 0 ≤ t ≤ T }; herein z∗ is an arbitrary point in D, and  is a suciently small positive number. We construct a continuously dierentiable function g(z) up to second order, such that ( 1; z ∈ G0 ; 0 \G0 : (3.41) 0 ≤ g(z) ≤ 1; z ∈ G2 g(z) = 0 ; 0; z ∈ G\G2 Denote U (z; t) = u(z; t)g(z), it is clear that U (z; t) is a solution of the following initialboundary value problem: A0 Uzz − Re[QUzz ] − HUt = A in G;

(3.42)

U (z; 0) = 0 on D;

(3.43)

U (z; t) = 0 on @G2 ;

(3.44)

where A = g[Re(A1 Uz ) + A2 U + A3 ] + U [A0 gzz − Re(Qgzz )] + 2Re(A0 gz Uz − Qgz Uz ): Let  = 2 inf G A0 and divide Eq. (3.42) by ; we obtain ˜ zz ) − H˜ Ut = A˜ in G; LU = A˜0 Uzz − Re(QU

(3.45)

in which A˜0 = A0 =; Q˜ = Q=; H˜ = H=; A˜ = A=; and Eq. (3.45) can be rewritten as ˜ zz ) = A˜ in G; LU = LU + (A˜0 − 1)Uzz + Re(QU

(3.46)

herein LU = Uzz − H˜ Ut . From Eq. (3.45) it follows ˜ zz ) in G |LU | ≤ |LU | + (A˜0 − 1)Uzz − Re(QV

(3.47)

and then there exists a suciently small positive number 0 , such that ˜ zz )|2 |LU |2 ≤ (1 + 0−1 )|LU |2 + (1 + 0 )|(A˜0 − 1)Uzz − Re(QU ˜ 2 + (1 + 0 ) sup[(A˜0 − 1)2 + |Q| ˜ 2 )(|Uzz |2 + |Uzz |2 )] ≤ (1 + 0−1 )|A| G

(3.48)

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

949

and 2(1 + 0 )¡1; where we assume that ˜ 2 ] = sup[A˜02 + |Q| ˜ 2 − 2A˜0 + 1]¡ 1 :  = sup[(A˜0 − 1)2 + |Q| 2 G

G

(3.49)

˜ 2 ) = supG (|A0 |2 + |Q|2 )=4 inf G |A0 |2 ¡ 1 , it follows that In fact, from supG (|A˜0 |2 + |Q| 2 Eq. (3.49) holds. By Eq. (3.9), we have Z Z Z Z Z Z −1 2 ˜ 2 dz dt + 2(1 + 0 ) |LU | dz dt ≤ (1 + 0 ) |A| G

Z Z Z |LU |2 dz dt; ×

G

G

it implies that Z Z Z G

|LU |2 dz dt ≤

(1 + 0−1 ) 1 − 2(1 + 0 )

Z Z Z G

˜ 2 dz dt: |A|

(3.50)

Noting that U (z; t) = u(z; t) in G0 ; we obtain k|uzz | + |uzz | + H |ut |kL2 (G0 ) ≤ M18 = M18 (; q; ; k; ; p; G): This shows that estimate (3:40) is valid. Finally, we mention that if condition (1:4) holds, then Eq. (3.49) must be true. 4. Existence of solutions First of all, we give an expression of solutions of Problem P for the parabolic equation ( f(z; t) on Gm ; Lu = uzz − ut = fm (z; t); fm (z; t) = (4.1) 0 on {C × I }\Gm ; where Gm = {(z; t) ∈ G; dist((z; t); @G) ≥ 1=m}; m is a positive integer, and Lp [ f(z; t); G m ] ≤ k1 , herein p (¿4); k1 are positive constants. 0 Theorem 4.1. If u(z; t) is any solution of Problem P for Eq. (4.1), and u(z; t) ∈ C ;1; =2 (G) ∩ W˜ 22; 1 (G), then u(z; t) can be expressed in the form

u(z; t) = U (z; t) + V (z; t) = U (z; t) + v0 (z; t) + v(z; t);

(4.2)

where V (z; t) = v0 (z; t) + v(z; t) is a solution of Eq. (4.1) in G0 = D0 × I = {|z|¡1} × I with the homogeneous Dirichlet boundary condition V (z; t) = 0 on @G0 = @G01 ∪ @G02 ; G01 = D0 × {t = 0}; G02 = {|z| = 1} × I;

(4.3)

950

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

(Problem D0 ), and v(z; t) = H˜ fm = ( E(z; t; ; ) =

Z tZ Z D0

0

E(z; t; ; )fm (; ) d d;

(t − )−1 e−|z−| 0

2

=4(t−)

(4.4)

if t¿; if t ≤  except (z; t) = (; );

and U (z; t) is a solution of Problem P for LU = 0 in G with the initial-boundary conditions U (z; 0) = g(z) − V (z; 0) on D;

(4.5)

@V @U + (z; t)U =  − + (z; t)V on @G2 ; @ @

(4.6)

U (atj ) = bj (t) − V (atj )

on Ij0 ; j = 1; : : : ; m; U (z; t) = r(z; t) − V (z; t)

on @G 0 (4.7)

which satis es the estimates 0 0 C ;1; =2 [U; G]+ kU kW˜ 2; 1 (G) ≤ M19 ; C ;1; =2 [V; G]+ kV kW˜ 2; 1 (G) ≤ M20 ; 2

(4.8)

2

where (0¡ ≤ ); Mj = Mj (; k1 ; p; Gm )( j = 19; 20) are nonnegative constants. Proof. It is clear that the solution u(z; t) can be expressed as Eq. (4.2). On the basis of Theorems 3.3 and 3.4, it is easy to see that V (z; t) satis es the second estimate in Eq. (4.8), and then we know that U (z; t) satis es the rst estimate of Eq. (4.8). Next, we consider the special complex Eq. (1.2), namely uzz − ut = fm (z; t; u; uz ; uzz ; uzz ); fm = (1 − A0m =H )uzz + {Re[Qm uzz + A1m uz ] +A2m u + A3m }=H in G; where  = 4 inf G A0 =3, and the coecients ( ( ( A0 =; Q=; Aj = on Gm ; Qm = Ajm = A0m = H=; 0 on {C × I }\Gm ; 0;

(4.9)

j = 1; 2; 3:

Theorem 4.2. If Eq. (1.2) satis es Condition C, then Problem P for Eq. (4.9) has a solution u(z; t). Proof. In order to prove the existence of solutions of Problem P for Eq. (4.9) by using the Larey–Schauder theorem, we introduce the equation with the parameter h ∈ [0; 1]: uzz − ut = hfm (z; t; u; uz ; uzz ; uzz ) in G:

(4.10)

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

951

2; 1

0 Denote by BM a bounded open set in the Banach space B = Wˆ 2 (G) = C ;1; =2 (G) ∩ W˜ 22; 1 (G) (0¡ ≤ ), the elements of which are real functions u(z; t) satisfying the inequalities 0 [u; G]+ kukW˜ 2; 1 (G) ¡M21 = M13 + M15 + 1; kukWˆ 2; 1 (G) = C ;1; =2 2

2

(4.11)

in which W˜ 22; 1 (G) = W22; 0 (G) ∩ W20; 1 (G); M13 ; M15 are nonnegative constants similar to those in Eqs. (3.25) and (3.39). We choose any function u(z; ˜ t) ∈ BM and substitute into the appropriate position in the right-hand side of Eq. (4.10), afterwards we make an integral v(z; ˜ t) = H˜  as follows: Z tZ Z ˜ E(z; t; ; )(; ) d d; (4.12) v(z; ˜ t) = H  = D0

0

where (z; t) = u˜ zz − u˜ t , and E(z; t; ; ) is as stated in Eq. (4.4). Next, we nd a solution v˜0 (z; t) of the initial-boundary value problem on G0 : v˜0zz − v˜0t = 0

on G0 ;

˜ t) v˜0 (z; t) = −v(z;

(4.13)

on @G0

(4.14)

and denote V˜ (z; t) = v(z; ˜ t) + v˜0 (z; t). Moreover, on the basis of the result in [6], we can nd a solution U˜ (z; t) of the initial-boundary value problem on G: U˜zz − U˜t = 0

on G;

(4.15)

U˜ (z; 0) = g(z) − V˜ (z; 0)

on D;

(4.16)

@V˜ @U˜ + (z; t)U˜ =  − + (z; t)V˜ @ @ U˜ (ajt ) = bj (t) − V˜ (ajt )

on @G2 ;

(4.17)

on Ij0 ; j = 1; : : : ; m;

U˜ (z; t) = r(z; t) − V˜ (z; t)

on @G 0 :

(4.18)

Now, we discuss the equation ˜ u˜ z ; U˜ zz + Vzz ; U˜ zz + Vzz ); Vzz − Vt = hfm (z; t; u;

0 ≤ h ≤ 1:

(4.19)

By Condition C, the principle of contracting mapping and the results in Section 3, Problem D0 for Eq. (4.19) in G0 has a unique solution V (z; t) with the initial-boundary condition V (z; t) = 0

on @G0 :

(4.20)

Setting u(z; t) = U (z; t) + V (z; t), denote by u = S(u; ˜ h) (0 ≤ h ≤ 1) the mapping from u˜ onto u, where the relation between U and V is the same as that between U˜ and V˜ . Furthermore, if u(z; t) is a solution of Problem P in G for the equation uzz − ut = hfm (z; t; u; uz ; uzz ; uzz );

0≤h≤1

(4.21)

952

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

then from Theorems 3.3 and 3.4, the solution u(z; t) of Problem P for Eq. (4.21) satis es Eqs. (3.25) and (3.39), consequently, u(z; t) ∈ BM . Let B0 = BM × [0; 1]. In the following, we shall verify that the mapping u = S(u; ˜ h) satis es the three conditions of Leray–Schauder’s theorem: 1. For every h ∈ [0; 1]; u = S(u; ˜ h) continuously maps the Banach space B into itself, ˜ t) ∈ BM ; S(u; ˜ h) and is completely continuous on BM . Besides, for every function u(z; is uniformly continuous with respect to h ∈ [0; 1]. In fact, we arbitrarily choose u˜ n (z; t) ∈ BM ; n = 1; 2; : : : ; it is clear that from {u˜ n (z; t)} there exists a subsequence {u˜ nk (z; t)}, such that {u˜ nk (z; t)}; {u˜ nk z (z; t)} and corresponding functions {U˜nk (z; t)}; {U˜nk z (z; t)} uniformly converge to u˜ 0 (z; t); u˜ 0z (z; t); U˜0 (z; t); U˜ 0z (z; t) in G respectively. One can nd a solution V0 (z; t) of Problem D0 for the equation V0zz − V0t = hfm (z; t; u˜ 0 ; u˜ 0z ; U˜ 0zz + V0zz ; U˜ 0zz + V0zz );

0 ≤ h ≤ 1 in G0 :

(4.22)

From unk = S(u˜ nk ; h) and u0 = S(u˜ 0 ; h), we have (Vnk − V0 )zz − (Vnk − V0 )t = h[fm (z; t; u˜ nk ; u˜ nk z ; U˜ nk zz + Vnk zz ; U˜ nk zz + Vnk zz ) − fm (z; t; u˜ nk ; u˜ nk z ; U˜ nk zz + V0zz ; U˜ nk zz + V0zz ) + Cnk (z; t)];

0 ≤ h ≤ 1;

(4.23)

where Cnk = fm (z; t; u˜ nk ; u˜ nk z ; U˜ nk zz + V0zz ; U˜ nk zz + V0zz ) − fm (z; t; u˜ 0 ; u˜ 0 ; U˜ 0zz + V0zz ; U˜ 0zz + V0zz );

(z; t) ∈ G0 :

(4.24)

We can verify that L2 [Cnk ; G0 ] → 0

as k → ∞:

(4.25)

Moreover, the following can be derived: kVnk − V0 kW˜ 2; 1 (G) ≤ L2 [Cnk ; G0 ]=[1 − q0 ]; 2

(4.26)

where q0 = 2(1 + 0 )¡1. This shows that kVnk − V0 kW˜ 2; 1 (G) → 0 as k → ∞. Moreover, 2 from Theorems 3.3 and 3.4, we can verify that from {Vnk (z; t) − V0 (z; t)}, there exists a subsequence, for convenience we denote the subsequence again by {Vnk (z; t) − V0 (z; t)}, 0 [Vnk − V0 ; G0 ] → 0 as k → ∞. This shows the complete continuity of such that C ;1; =2 u = S(u; ˜ h) (0 ≤ h ≤ 1) on BM . By using a similar method, we can prove that u = S(u; ˜ h) (0 ≤ h ≤ 1) continuously maps BM into B, and u = S(u; ˜ h) is uniformly continuous with respect to h ∈ [0; 1] for u˜ ∈ BM . 2. For h = 0, from Eqs. (4.11) and (4.21), it is clear that u = (u; ˜ 0) = U (z; t) ∈ BM . 3. From Theorems 3.3, 3.4 and Eq. (4.11), we see that u = S(u; ˜ h) (0 ≤ h ≤ 1) does not have a solution u(z; t) on the boundary @BM = BM \BM .

G. Wen, B. Zou / Nonlinear Analysis 39 (2000) 937 – 953

953

Hence, by the Leray–Schauder theorem, we know that Problem P for Eq. (4.21) with h = 1, namely Eq. (4.9) has a solution u(z; t) = U (z; t) + V (z; t) = U (z; t) + v0 (z; t) + v(z; t) ∈ BM . Theorem 4.3. Under Condition C, Problem P for Eq. (1.2) has a solution. Proof. By Theorems 3.3, 3.4 and 4.2, Problem P for Eq. (4.9) possesses a solution um (z; t), and the solution um (z; t) satis es estimates (3:25) and (3:39), where m = 1; 2; : : : . Thus, we can choose a subsequence {umk (z; t)}, such that {umk (z; t)}; {umk z (z; t)} on G uniformly converge to u0 (z; t); u0z (z; t); respectively. Obviously, u0 (z; t) satis es the initial-boundary conditions of Problem P. On the basis of the principle of compactness of solutions for Eq. (4.9), it can be seen that u0 (z; t) is a solution of Problem P for Eq. (1.2). Acknowledgements The authors would like to thank Professor H. Begehr for reading the manuscript and proposing some bene cial suggestions. References [1] Y.A. Alkhutov, I.T. Mamedov, The rst boundary value problem for nondivergence second order parabolic equations with discontinuous coecients, Math. USSR Sbornik 59 (1988) 471– 495. [2] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968. [3] A.I. Nazarov, Holder estimates for solutions of problems with an oblique derivative for parabolic equations of nondivergence structure, J. Soviet Math. 64 (1993) 1247 –1252. [4] G. Wen, H. Begehr, Boundary Value Problems for Elliptic Equations and Systems, Longman Scienti c and Technical, New York, 1990. [5] G. Wen, Two boundary value problems for second order nonlinear parabolic equations with measurable coecients, J. Yantai Univ. (Natur. Sci. and Engin.) 1 (1993) 1– 8 (Chinese). [6] G. Wen, Initial and general nonlinear oblique derivative problems for full nonlinear parabolic complex equations, Complex Analysis and its Applications, Longman Scienti c Technical, New York, 1994, pp. 334 – 343. [7] G. Wen, Initial-mixed boundary value problems for nonlinear parabolic complex equations of second order with measurable coecients, Acta Sci. Natur. Univ. Pekin. 31 (1995) 511– 519.