A global approach to fully nonlinear parabolic problems
Authors: Kartsatos Athanassios , Skrypnik Igor 
Published: July 2002

We consider the general initial-boundary value problem (1)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential u Over partial-differential t EndFraction minus upper F left-parenthesis x comma t comma u comma script upper D Superscript 1 Baseline u comma script upper D squared u right-parenthesis equals f left-parenthesis x comma t right-parenthesis comma left-parenthesis x comma t right-parenthesis element-of upper Q Subscript upper T Baseline identical-to normal upper Omega times left-parenthesis 0 comma upper T right-parenthesis comma">

∂
u

∂
t

−
F
(
x
,
t
,
u
,

D

1

u
,

D

2

u
)
=
f
(
x
,
t
)
,

(
x
,
t
)
∈

Q

T

≡
Ω
×
(
0
,
T
)
,

\displaystyle {\frac {\partial u}{\partial t}-F(x,t,u,\mathcal {D}^{1}u, \mathcal {D}^{2}u)=f(x,t),\quad (x,t)\in Q_{T}\equiv \Omega \times (0,T),}

(2)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis x comma t comma u comma script upper D Superscript 1 Baseline u right-parenthesis equals g left-parenthesis x comma t right-parenthesis comma left-parenthesis x comma t right-parenthesis element-of upper S Subscript upper T Baseline identical-to partial-differential normal upper Omega times left-parenthesis 0 comma upper T right-parenthesis comma">

G
(
x
,
t
,
u
,

D

1

u
)
=
g
(
x
,
t
)
,

(
x
,
t
)
∈

S

T

≡
∂
Ω
×
(
0
,
T
)
,

\displaystyle {G(x,t,u,\mathcal {D}^{1}u)=g(x,t),\quad (x,t)\in S_{T}\equiv \partial \Omega \times (0,T),}

(3)

u
(
x
,
0
)
=
h
(
x
)
,

x
∈
Ω
,

\displaystyle {u(x,0)=h(x),\quad x\in \Omega ,}

where

Ω
\Omega

is a bounded open set in

R

n

\mathcal {R}^{n}

with sufficiently smooth boundary.  The problem (1)-(3) is first reduced to the analogous problem in the space

W

p

(
4
)
,
0

(

Q

T

)

W^{(4),0}_{p}(Q_{T})

with zero initial condition and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper W Subscript p Superscript left-parenthesis 2 right-parenthesis comma 0 Baseline left-parenthesis upper Q Subscript upper T Baseline right-parenthesis comma g element-of upper W Subscript p Superscript left-parenthesis 3 minus StartFraction 1 Over p EndFraction right-parenthesis comma 0 Baseline left-parenthesis upper S Subscript upper T Baseline right-parenthesis period">

f
∈

W

p

(
2
)
,
0

(

Q

T

)
,
 
g
∈

W

p

(
3
−

1
p

)
,
0

(

S

T

)
.

f\in W^{(2),0}_{p}(Q_{T}),~g \in W^{(3-\frac {1}{p}),0}_{p}(S_{T}).

\]
The resulting problem is then reduced to the problem

A
u
=
0
,

Au=0,

where the operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A colon upper W Subscript p Superscript left-parenthesis 4 right-parenthesis comma 0 Baseline left-parenthesis upper Q Subscript upper T Baseline right-parenthesis right-arrow left-bracket upper W Subscript p Superscript left-parenthesis 4 right-parenthesis comma 0 Baseline left-parenthesis upper Q Subscript upper T Baseline right-parenthesis right-bracket Superscript asterisk">

A
:

W

p

(
4
)
,
0

(

Q

T

)
→

[

W

p

(
4
)
,
0

(

Q

T

)
]

∗

A:W^{(4),0}_{p}(Q_{T})\to \left [W^{(4),0}_{p}(Q_{T})\right ]^{*}

satisfies Condition

(
S

)

+

.

(S)_{+}.

 This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces.  The local and global solvability of the operator equation

A
u
=
0

Au=0

are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.