On Mixed Problems for Parabolic Equations in General Holder
Spaces
MARTÍN LOPEZ MORALES
Departamento de Computación
Tecnológico de Monterrey. Campus Ciudad de México
Calle del Puente 222. Ejidos de Huipulco. Tlalpan. 14380.México D.F.
MEXICO
Abstract: - We consider the second and the third boundary value problems for linear parabolic
equations in a cylindrical domain .We establish new a priori estimates in general Hölder norms
for the solutions to these problems, under the assumption that the coefficients satisfy the general
Holder condition with respect to the space variables only. In this connection, however, we also
obtain an estimate of the modulus of continuity with respect to the time of the second
derivatives with respect to x of the corresponding solutions.
On the basis of new a priori estimates for the solutions to these problems, we establish the
corresponding solvability theorems in general Hölder spaces.
We apply these results to obtain the solvability to these problems for nonlinear parabolic
equations
Key-Words: - estimates, solvability, equations, parabolicity, solution, problem
1. Introduction
,      . .
We establish new a priori estimates in
general anisotropic Hölder norms for the
solutions of the problems, (1), (2), (3) and
(1), (2), (3´) under the assumption of the
general anisotropic Holder continuity of the
coefficients with respect to space variables
only. In this connection, however, we also
obtain an estimate of the modulus of
continuity with respect to the time of the
We consider the linear parabolic equation
n
n
u t   a ij ( t , x )u x x   b i ( t , x )u x i 
i , j1
i j i 1
 c ( t , x )u  f ( t , x )
in
a
Q
T
(1)
cylindrical
 
 0, T 
with
domain
the
initial
condition
u
t 0
leading derivatives
i
 ( x )
n
 h (t, x)
i 1 i
S T of Q T
u xi
n
 h (t, x)u
i  1
i
ST
  ( t , x ),
(3)
 b(t, x)u
x
 ψ (t, x).
ST
i
(3´)
Here
x  (x1 ,..., xn) is a point of ,
is a bounded domain in n-dimensional
Euclidean space
 
E , t  0, T , S
n
  (
 
1,...,
 0, T    ,
T
j
Note that in the works [ 1 ] - [ 11] and in
many others the a priori estimates are
obtained under the fulfilment
Hölder
condition with respect to the totality of
variables ( t , x ) on the coefficients of
Equation (1).On the basis of new a priori
estimates for the solutions of the problems
(1), (2), (3) and (1), (2), (3´) we establish
the corresponding theorems on the
solvability for these problems in general
anisotropic Hölder spaces.
We assume that the coefficients of Equation
(1) satisfy the uniform parabolicity
condition: for any
nonzero vector
(2)
and one of the following conditions on the
lateral surface
ux x (but not ut ).
n
a
where   is the boundary of
i 1
1
 )E
n
n
and
2
ij
(t , x) QT
(t , x) i  j    ,  const .  0.
(4)
We apply the results in the linear theory to
establish the local solvability with respect to
the time t, in general Hölder anisotropic
spaces, to the boundary valued problem for
the nonlinear parabolic equation
functions
A(t , x, u, p, r ), ( x), (t , x), (t , x)
than in the works [ 8 ] , [ 9 ] and [ 15 ].
Some close results have been established in
[14] [16] and [21]-[23].
The results of this work must find
applications in the problems of
heat
conduction, diffusion, Mechanics of Fluids
and many others.
(5)
u A(t, x, u, u , u )
in Q with the initial condition (2) and one
t
x
xx
T
.2.Notations
of the boundary conditions (3) or (3´), where
u x (u ,..., u ),uxx  (..., u ,...),
x1
xn
xi x j
1 i , j n.
We
shall say that
the function
u(t , x) defined in the cylinder Q T satisfies
In the present work, like in the works the
equation (5) is linearized directly. No
conditions are imposed here on the nature of
the growth of the non linearity of the function
A
,
which
is
defined
for
(t, x)  QT andany u,u x ,u xx . The main
assumption concerning to the function
A(t , x, u, p, r ) ,where
p(p ..., p )
1,
n
the general Hölder condition in
u ( t , x ) u ( t , y )  C x  y
(t , x), (t , y )  Q
properties:
Ia.
l     0,1 if l  0 
 l  l ln  0  if l  0 
(l ), l  0
or
l  
  0 then
l  ln   for l  0  and
 l    l  l ln  0 for l  0. l 0 
where l 0 is sufficiently small number (we
suppose that the derivative β l  exists and it
Ic. If
denote the
In all the work we suppose that in the
equation
(1),
the
functions
f  f 1 f 2 , a ij , b i , c and f i satisfy the
general Hölder condition in
or l  
Ib.
( t , x, u , p , r )
and (, ) here and below
usual scalar product in E n
.
βl  is defined and continuous
in 0  l   . Moreover it has the following
where
ri j
T
( x y )
The function
1  i , jn, is the parabolicity condition:for
any non zero vector (1 ,...,  n )En
and
any
(t, x)Q T ,u, p, r
(6)
( A r (t, x, u, p, r), ) 0,
A r (...) isthe matrix A
Q T of
exponent ( l ), l  0 with respect to the
space variables, if there exists a constant
C > 0 such that
r (..., r ij ,...),
and
and Propositions
is a continuous function in
R 0  0. l 0    1l 0 , .
Q T of exponent
Note that the condition Ic introduces a new
set of functions ( the functions u(t , x) that
satisfy the general Hölder condition with
respect to space variables only ).In this new
set of functions we will obtain the
corresponding existence and uniqueness
theorems for the solutions to the problems
(1),(2),(3); and (5) , (2) , (3) and (5) , (3) ,(3’)
with respect to the space
f 2 satisfies the general
Hölder condition in Q T of exponent
(l ), l  0 with respect to the space
variables only and
variables only.
All the coefficients and the independent
terms are continuous in the cylinder Q T .
In the case of the problems (5) , (2) ,
(3) and (5) , (3) , (3`) we require less
smoothness conditions from the
We denote by  i  1,2, the set of functions
 ( l ) for which
i
1 (l )
 l
2
l  ( t ) 1
  ( l )  t
dt  ;
0
2 ( l )
 l

1l
the Banach space of functions u(t , x ) that

( l )
1
For the functions ( l )   (( l )  
introduce the functions
1
 (l )  B (l ) 
2
B (l )  B B
(l )
1
ln
ln l
 ( l )
2
are
) we
j
The functions B  ( l ) are functions of the type
 ( l ), j  1, 2. See examples of functions of the
type  ( l ) in [12].
For the functions u(t , x ) defined and
Holder continuous ( in the general sense ) in
the cylinder Q T of exponent (l ) , l  0
with respect to space variables, we introduce
the following norms :
t 
 
t 
 
2
n


d( P, R)  t    (x i y i ) 
i 1


0   t
u
where
QT
0, 0
 sup u( P ) ,
PQ
t
u 0,0 sup u(t , x) ;
Q
x
H
t
u 0, l   u 0,0,  H
t
 l 
H l  (u ) sup
x, y 
t
(u),
x y
space variables in the cylinder
the norms
n
t
i 1
  x y
u

n
0 , ( l )
t
i , j 1
u
,



u
,m=0,1,2.
m , ( l ) sup
m, ( l )
0   t
We will denote by

d( P , R )
(d( P , R )
QT
0, ( l )
u
QT
0,0
,
(14)
H
QT
(l )
(u ),
n
Q
i 1
(15)
QT
0 , ( l )
Q
Q
,
(16)
n
u 2,T( l )  u 1,T( l )   u xi x j
i , j 1
QT
0 , ( l )
,
(17)
We denote by C m , ( l ) (Q ), m  0,1,2 the
t
Banach space of functions u( t , x ) that are
continuous in Q t  0, t    together with
all derivatives respect to x up to the order
m, m0,1,2 and have finite norms (15) (17), respectively.
It is possible to consider all the preceding
definitions in the layer    0,   En (see
(10)
,
P , R Q
u 1, ( l )  u 0,T( l )   u xi

t
0 , ( l )
u ( P ) u ( R )

(u ) 
QT
Q T , we define
t
u 2, ( l )  u 1,( l )   u xi x j
(l )
T
PR
u(t , x)u(t , y )
u 1, ( l )  u 0,( i )   u xi
T
(8)
xy
.
(9)
For the functions u(t , x ) that have continuous
derivatives with respect to x up to the order
m(m 1,2) inclusively in the cylinder Q T
and satisfy the general Holder condition of
exponent (l ) , l  0 with respect to the
t
(13)
QT we consider the usual norms
cylinder
(7)
t
1/ 2
For the functions u( t , x )
that have
continuous derivatives with respect to x up to
the order l , ( l=0,1,2 ) inclusively in the
u 0,0 sup u 0,0 ; u 0,( l ) sup u 0,( l ) ,
t
,
We define the parabolic distance between
each
two
points
P(t , x),R ( , y ),0t T
by
the
magnitude
,
0   t
Q T 0 , T
in
0  t  T together with all derivatives respect
to x up to the order ,(0,1,2) inclusively
and have a finite norm (12).
,
 (1)
continuous
(11)

20 ) or in any domain contained in   .
With respect to the coefficients of
equation (1) we assume that
T
the
a i j (t, x), b(t, x), c(t, x)C 0,( l ) (Q T ), and
(12)
n
n
T
T
T
 a i j 0, ( l )   b i 0, ( l )  c 0, ( l )  B,
i , j1
i 1
t 
Cm,( l ) (QT ), m0,1,2
(18)
3
moreover
the
a i j (t , x ), i , j1,...,n
coefficients
uniformly
are
continuous with respect to t on
right- hand side is considered in the cylinder
0,t  Dl .
t ,S
 m , (tl )  , we will say that
If
t 
 
S T: for any
  0 there exists   ( ) such that for all
(t 1, x), (t 2 , x)  S T we have
a ij (t 1 , x)  a ij (t, x)  
Moreover we will consider in S t the norm

(19)
for t 1  t 2  ( )
Definition
C m ,( l ) (S T ).
We will say that  A 2, ( l )
if for every point
(t 1 ,x)(t 2 ,x)
x 0   there exists an
n-dimensional ball B with centre
sup
x q  h( x 1 ,..., x q 1 , x q 1 ,..., x n ),


( l )
,
for 0tT,
and
 0 0 , such
that the part of the boundary   lying in the
n
b 1 T,B ( l )   h k 1 T,B ( l ) D.
k 1
(22)
For equation ( 5 ) we consider in addition to
the parabolicity condition ( 6 ) that there
exists a domain
ball
K x 00  x : x  x 0   0 ,x 0 
the
t 
1,B
Heine-Borel lemma) a number
of
(
.
 1 t,B ( l )  , h k 1 t,B ( l ) , k 1,..., n;
and the function h has second derivatives that
are bounded and Hölder continuous of
exponent (l ), l  0 .
In what follows, we assume that
   A 2 ,  ( l ) , then there exists (by the
one
t 1 t 2

( t  t )
1
2
)
2
(21)
With respect to the boundary " data " in the
problems ( 1 ) , ( 2 ) , ( 3 ) and (1), ( 2) , (3´)
we assume that
q(1  q  n in the form
by
1 B
0 t  t  t
1
2
( t 1, x ),( t 2 , x )  S T
x 0 such
that B    can be represented for some
represented
t 
t 
 

1,B 
1,B 
  l 
  l 
is
equations


k  1,..., N
H M  (t,x)Q T ; u  M, p  M, r  M
in which the function ( t , x, u, p, r ),
where the point
together with its derivatives with respect to
u , p i , ri j i , j  1,..., n , up to the second
x q  h k ( x'q ), x'q  ( x 1 ,..., x q 1 , x q 1 ,..., x n ),
'
xq
belongs to the
order inclusively is continuous and satisfies
the Lipschitz condition with respect to
u, p, r and a general Hölder condition of
domain
Dk E n 1 and h k
2 ,( l )
const ., .
exponent ( l ),l  0 with respect to x and
For the functions
(t,x) defined in S T 0,T  
C M .Moreover
(t, x,0,0,0)C 0T,( l ) (Q T )
with the constant
we introduce the following norms
T 
(t , x,0,0,0) 0,( l ) C

All the mentioned derivatives are bounded in
H M by the constant C .
 t ,S T

m , ( l )
 max (t,x 1 ,...,x q 1 ,h k (x q ),x q 1 ,...,x n ) mt ,( l ) ,
On the lateral surface S T of Q T the
function
 r (t, x, u, p, r ), are
1 k  N
ij
(20)
where the norms
m , ( l )
uniformly continuous with respect to t
(see [18] ).
, m0,1,2 in the
4
x'i D  R n  1 , h
Now we shall consider the equation (1)
with the initial zero condition
u t  0 0
k
s1,..., N.
(23)
and one of the boundary conditions (3)
or ( 3' ).
Denote by
2 , ( l )
const .,
k
Q 0T the cylinder 0, TK x 00 .
The transformations
3. Bounds for solutions to the
mixed problems
Theorem 1.Let
y x 1,..., y i1 x i1,y i x i1,..., y n1x n,
1
y n x i h ( x'i ),s1,..., N
s
T
u(t, x)C 2,B2 (Q T )
(25)
( l )
maps
be a solution of the problem (1), (23), (3) in
the cylinder Q T . Assume that
T
T
K x 00 onto a domain lying in the half
(l ),(l ) 2 , (l )  1 ,
(l )   2 if r  0  or r   ,
0   1   2  1.
T 
1,B
space
0,  ( l )



 (24)



K denotes here and below a constant
depending on
n, (l ), (l ), ,  , , D,   and ( ) .
(t 0 ,x 0 ) be an arbitrary point of
since
S T ,t 00, T,x 0.
Proof. Let
  A
 

2
1

,2   f1  ,2   () 2
v
K
0,  ( l )
2

2,B
( l )
     , 2  
 1,B ( l )

2 , ( l )
 0 0and N such that the part of
the boundary   lying in the ball
numbers

half
ball
(26)
then there exists two
K x 00 x; xx 0  0
some
K y n 0,y 0 y(x 0 ) is contained in
( l )
t 
and
this domain. The mapping (25) is one-to-one,
non degenerated and its transforms the
equation (1) into the equation of the same
type.
With the aid of the results to the Cauchy
problem ( see [ 12 ] ), the properties of the
function ( l ), l  0 , the estimates for the
solutions to the mixed problems for parabolic
equations in the a half space ( see [8] and [9]
the interpolation inequalities of Lemma 2 in
[12] with  small enough we get the
estimates


  t  ,  2  1
t0 , 

 0 
2
 f1
t 0
f 2 0,(l )  


0,( l )
 t 0 ,


u  2
K
  t ,

2,B ( l )
   0 

1,B
( l )


(19) hold . Then the following inequality
holds for 0tT .
f2
y n 0
y0

  and the conditions (4) , (18) ,
 

2 1


 f t t 2
t  K  1 0,(l )
u
 t 
2
2, B
( l )  
 1, B ( l )

onto a set which lies on
y n 0 and, as we may assume, it maps
f 1 C 0,( l ) (Q T ), f 2 C 0,( l ) (Q T ),

( )K
x0
0
is representable by
one of the equations
x h s ( x' ),
i
i


,2  
f2

0,( l )




where
v(t,x)u(t,x)u; u u( ,x)
x' x ,...,x
,x
,...,x n},
i  1
i 1 i  1
s1,...,N}
v
5
 ,2 

 sup v
,k  0,1,2
2,B k
2,B k






2

( l )
( l )
(27)
The Prof. of theorem 2 is similar to the proof
of theorem 1.
Remark 3. Arguing as in the proof of
Remark 2 we can get the estimate
This means that the estimate ( 24 ) holds for
t  ,2 . Similarly the estimate (24)






yields for t  2,3 , t  3,4 , then
by means of a finite number of steps with
length  we run over the interval 0,T .
This complete the proof of theorem 1
.
 

T t 2 2 1
u t
K  f 1
2
 0 , ( l )
2, B
( l )

 
Q
Remark 1. By virtue of Lemma 4 in the
work [16] ( estimates for the solution
to the Cauchy problem parabolic
equations ) , with the aid of theorem 1 in
[16] and using the interpolation
inequalities of Lemma 2 in [12 ] we
conclude that for 0  t  T
 2  1
 t ,
f

t
Q
 1 0,( l ) 0 2
u t 2 K 
2 , B ( l )
   t ,
1,B ( l )

f2
t ,
0, ( l ) 

uu(t,x)(x),(x)C 2,( l ) ().





4. Existence and uniqueness
theorems
Theorem 3 . Suppose that all conditions of
the Theorem 1 are true.
Assume,
furthermore, that the following consistency
condition holds
 ( 0, x )    0

uu(t,x)(x), (x)C 2,( l ) ().
u(t ,x)C
T
2,B
2
( l )
cylinder
T 
Q
T
.
(Q T ) be the
C
Assume
l  0  or

T 
1,B
that
(l )  1 , (l )   2 if
l   , 0  1   2  1 ,
 ( 0, x )    0
Then
( l )
K
is
a
constant
f2
0,  ( l )
a
unique
solution
(Q ) to the problem (1), (2),
T
(3') in Q T with a continuous derivative
u t in Q .
T
We can get the proof to these theorems on the
basis of our new a priori estimates established
in this work and with the aid of the method of
continuity in a parameter (see [5] and [20] ).
We proceed now to formulate the local
existence theorem for solutions to the nonlinear problems for the equation (5).Here we
consider that the function
With respect to the initial and boundary
functions in the problems (5), (2), (3) and (5),



1, B
(l ) 
T

depending
(32)
exists
2
2, B
( l )
(19) hold . Then the following inequality holds
for 0 t T
T
there
u( t , x )  C
  and conditions (4) , (18) and
 

2 1


T

K f 1
t 2
2
0 , ( l )

2, B
( l )

( Q ) with a continuous derivative
T
Theorem.4 Suppose that all assumptions of
the Theorem 2 hold. Moreover, the next
consistency condition is satisfied
T 
T
u
2
B ( l )
u t in Q .
T
f 1 C 0,( l ) (Q T ), f 2 C 0,( l ) (Q T ),
2
 ( l ),  ( l )   ,
(31)
Then there exists a unique solution to the
problem (1), (23), (3) in the space
solution of the problem (1 ) , ( 23 ) , ( 3` ) in
the
0,  ( l )



1, B
( l ) 

T

(30)
Remark 4. We can reduce the mixed
problem ( 1 ) , ( 2 ) , ( 3´ ) to the mixed
problem ( 1 ),
( 23 ) , (3´ ) by means of the transformation
(28)
Remark 2. We can reduce the mixed
problem ( 1 ) , ( 2 ) , ( 3 ) to the mixed
problem ( 1 ),
( 23 ) , ( 3 ) by means of the transformation
Theorem 2 Let
f2
T
(29)
on
n, , , B,  , D, T,  and( )
6
(2), (3') we assume that


1, ( l )   ,

t 
1,B
5. REFERENCES
, 
t 
1,B
( l )

[1] R. B. Barrar, "Some estimates for
solutions of parabolic equations", J. Math.
Anal. App. (1961) 373 - 397.
( l )
(33)
and
(34)
[2] C. Ciliberto, "Formule di maggiorazine e
teoremi di esistenza per le soluzione della
equazioni paraboliche in due variabili",
Ricerche Mat. 3 (1945) 40 - 75.
Theorem 5 Suppose that all assumptions
with respect to the function  ( t , x, u, p, r )
hold and the following consistency condition
is satisfied
[3] L.I "Kaminin , V. N. Masliennikova,
"Boundary estimates of solution of third
boundary value problem for a parabolic
T 
n
b 1,B
  hi
 ( l ) i 1
[T]
 E  .
1, B
( l )
(0, x)
(35)
0.

Then there exists t 0 , to determined by the
equation", Dok. Akad. NAUK SSS R.T. 153 3
(1963) 526 - 529. English translation: Soviet
Mathematics - Doklady, 3, No. 3 (1963) 1711
- 1714.
above assumptions such that the problem (5),
(23),(3) has in the cylinder
with
a
u( t , x )  C
2, B 2
( l )
Q t 0 0, t 0 
[4] A. Friedman. Partial Differential
Equations of Parabolic Type. Prince - Hall,
Englewood Cliffs, New Jersey.( 1964).
unique
solution
( Q t ) with a continuous
0
[5] A. Solonnikov, "On boundary value
problems for linear parabolic systems of
differential equations of general forms",
Trudy Mat. Inst. Steklov 83 (1965) Proc.
Steklov Inst. Math. 83 (1965).[6].
derivative u t in Q t .
0
The proof of the theorem 5 is similar to the
proof of the theorem 3, in the work [14 ]
and to the Prof. of theorem 3 in the work
[18].
[6]I.. I Kaminin, " On smooth of thermal
potentials", I, II, III, Differentsialnie
Uravnienia, 1, No. 6 (1965) 799 - 839; 2, No.
5 647 - 687; No. 10 1333 - 1357; No.11, 1484
- 1501 (1966).
Remark.5
We can reduce the mixed
problem (5), (2), (3) to the problem with zero
initial condition (5), (23), (3), by the means
of the transformation
[7] A. Friedman, "Boundary estimates for
second order parabolic equations and their
applications", J. Math. and Mech. 7, 5 (1968)
771 – 792
.
[8] Ivanovich M.D.. On the nature of
continuity of solutions of linear parabolic
equations of the second order . Vesnik
Moskov Univ.Ser.I, Mat. Mech. 21 , 4 (1966)
, 31-41. Moscow University Mathematics
Bulletin N. 4 (1966) 31-41.
û  u ( t , x ) ( x ),( x )C 2, ( l ) (  ).
Theorem 6
Suppose that all assumptions
with respect to the function  ( t , x, u, p, r )
hold and the following consistency condition
is satisfied (0, x )

0.
Then there exists
t 0 determined by the
above assumptions such that the problem (5),
(26), (3') has in the cylinder
a unique solution
Q t 0 0, t 0 
u( t , x )  C
B2
( l )
[9] Ivanovich M.D.. Estimates of solutions of
general boundary- value problems for
parabolic systems in C l   ( r ) ( Q ) . Soviet
(Q t )
0
Math.Dokl. 9, 4 (1968),786-789.
with a continuous derivative u t inQ t 0 .
[10] Matichuk M.I. , Eidelman S.D. Parabolic
Systems with coefficients satisfying Dini’s
condition.Soviet.Math.Dokl. I , 165 , 3 (1965)
1461-1464
The proof of this theorem is similar to the
proof of the theorem 5.
7
equations. Differential Equations and
Applications. Vol. I,II ( Colombo OH,1988 )
170-178.Ohio Univ. Press Athens, OH,1989.
[11] Sperner E. Jr. Schauder’s existences
therem for  - Dini continuos data. Ark.
Mat.1,19 , 2 (1981) ; 193-216
.
[12] Lopez M. , Che J. On the Cauchy
Problem for Parabolic Equations . Journal of
the Ramanujan Mathematics Society. India.
Vol. 11. NO. 1. (1996) pp 1-36 ( to appear).
[22] López M. On Solvability for Higher
Order
Parabolic
Equations
WSEAS
Transactions on Mathematics. Issue 2,Vol.3,
April 2004.400-406
[23] Dmytruk I. Integrating nonlinear
heat conduction equation with source
term. Proceedings 5th WSEAS Int. Conf. on
Applied Mathematics (MATH 2004).
Miami, Florida, April 21-23, 2004. 484-211
[13] Lopez M. On the Mixed Problem for
Parabolic Equations. Aportaciones
Matematicas ( Mexican Mathematical
Society).Comunicaciones 12, (1993) ; 81-109
[14] Lopez M. , Che J . On the Solvability of
the First Initial Boundary Value Problem for
Parabolic
Equations.
Aportaciones
Matematicas ( Mexican Mathematical
Society).Comunicaciones 14, (1994) ; 123139.
[15]
D. Eidelman. Parabolic Systems.
Noordhoff, Grioningen and North - Holland,
Amsterdam. (1969).
[16] López M. , 0n the solvability to the
Cauchy problem for linear parabolic
equations", Ciencias Matemáticas, Vol. IX, 3
(1988) Universidad de La Habana, 31- 48.
[17] S.N. Kruzhkov, A. Castro, M. López,
Schauder type estimates and existence
theorems for the solution of the Cauchy
problem for linear and non-linear parabolic
equations (II). Ciencias Matemáticas. Vol. I
(1980). Universidad de La Habana .55 - 76.
[18] López M .On Solvability for Parabolic
Equations with one Space Variable WSEAS
Transactions on Mathematics. Issue 3,Vol.3,
July 2004.451-458.
[19]N. Kruzhkov, A. Castro, M. López,
Schauder type estimates and existence
theorems for the solution of the Cauchy
problem for linear and non-linear parabolic
equations, Vol. 1(1968) Universidad de la
Habana 37 - 55.
[20] A. Ladyzhenskaia, V. A. Solonnikov and
N. N. Urateseva, Linear and quasilinear
equations of parabolic type", Trans. Math.
Mon. Vol. 23 American Mathematical Society
(1968).
[21] A. Lunardi. Existence in the small and in
the large in fully nonlinear parabolic
8