We consider the following quasilinear parabolic initial boundary value problem (IBVP for short):

where
is a bounded strictly Lipschitz domain with its boundary
and
,
,

and
is a second-order strongly elliptic differential operator with the boundary operator given by

The coefficient matrix
satisfies regularity conditions on
, respectively. The directional derivative
,
is the outer unit-normal vector on
; the function
is defined as
for
;
denotes the trace operator.

We introduce precise assumptions:

where
) are Carathéodory functions; that is,
(resp.,
) is measurable in
(resp., in
for each
and continuous in
for a.e.
(resp.,
. More general, the function
can be a nonlocal function, for example,
or
.

Let
and
be Banach spaces, we introduce some notations as follows:

(i)
,
.
,
.

(ii)
for
,
.

(iii)
all continuous linear operators from
into
, and
.

(iv)
denotes the Nemytskii operator induced by
.

(v)
denotes the set of all locally Lipschitz-continuous functions from
into
.

(vi)
,
, and
, denotes the set of all Carathéodory functions
on
such that
, and there exists a nondecreasing function
with

Particularly,
is independent of
if
.

(vii)

denotes the Sobolev-Slobodeckii space for

and

with the norm

, especially,

; and

(viii)
,
(
is the set of integral numbers), is defined as

where
,
is the dual space of
, and
is the formally adjoint operator.

(x)
if
and
is an interval in
.

(xi)
denotes all maps
possessing the property of *maximal *
* regularity* on
with respect to
, that is, given
, the initial problem

has a unique solution
.

Now we turn to discuss the local existence result. We write

Exactly,
as
, where
denotes the Banach space of all functions being bounded and uniformly continuous in
. So, we will not emphasize it in the following.

*A* (

*weak*)

* solution*
of IBVP (2.1) is defined as a

function

,

, satisfying

where
and
denote the obvious duality pairings on
and
, respectively.

Set

After these preparations we introduce the following hypotheses:

(H1)
and
*.*

,
with
, and
.

(H3)
for some
.

Theorem 2.1.

Let assumptions (H1)-(H3) be satisfied. Then for each
the quasilinear problem (2.1) possesses a unique weak solution
for some
.

Proof.

The Nemytskii operator

is defined as

. The fact

shows the maximal regularity of the operator
. By [1, Theorem 2.1], if, for
,
for some
, then the existence and the uniqueness of a local solution will be proved.

The remain work is to check the Lipschitz-continuity. Set

Then

. So, for

with

we have

From

, we infer that

where

. Note that

, we can choose

such that

On the other hand, the hypotheses guarantee that

Due to

and

, Hölder inequality follows that

The hypothesis of

means that one can find an

such that

Obviously, if

, the above inequality is followed from (2.20) immediately. Hence it follows from (2.19) and (2.22) that

This ends the proof.

We apply the above theorem to the following two examples in next sections. For this, in the remainder we suppose that hypotheses (H1)-(H2) hold and that