Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
AN
EVOLUTION
EQUATION
WITH
COMPETING
p
-LAPLACIAN
OPERATORS
∗
Dumitru
Motreanu
†
Dedicated
to
Professor
Constantin
Fetecau
on
the
occasion
of
his
80th
anniversary
DOI 10.56082/annalsarscimath.2026.4.289
Abstract
The
paper
investigates
an
initial-boundary
value
problem
involv-
ing
the
competing
(
p,
q
)-Laplacian
−
∆
p
+∆
q
.
Under
some
conditions,
which
are
different
from
the
usual
ellipticity
and
monotonicity
con-
ditions,
the
existence
of
a
generalized
solution
is
proven
through
a
special
Galerkin
approach.
An
example
illustrates
the
applicability
of
our
result.
Keywords:
initial-boundary
value
problem,
time
depending
competing
operators,
p
-Laplacian,
generalized
solution.
MSC:
35K92,
47J35,
35K20.
1
Introduction
and
main
result
Our
aim
is
to
study
the
following
initial-boundary
value
problem
u
t
−
∆
p
u
+
∆
q
u
=
f
(
x,
t,
u
)
in
Ω
×
(0
,
T
)
u
=
0
on
∂
Ω
×
[0
,
T
]
u
(
x,
0)
=
u
0
(
x
)
on
Ω
,
(
P
)
∗
Accepted
for
publication
on
October
07,
2026
†
motreanu@univ-perp.fr
,
D´
epartement
de
Math´
ematiques,
Universit´
e
de
Perpignan
Via
Domitia,
Perpignan,
66860,
France
289
An
evolution
equation
with
competing
p
-Laplacian
operators
290
on
a
bounded
domain
Ω
in
R
N
with
Lipschitz
boundary
∂
Ω
and
with
fixed
real
numbers
T
>
0,
p
≥
2,
q
∈
(1
,
p
),
and
a
function
u
0
∈
L
2
(Ω).
The
unknown
in
problem
(
P
)
is
a
function
u
(
x,
t
)
that
will
be
subsequently
viewed
as
u
(
t
)(
x
),
with
(
x,
t
)
∈
Ω
×
(0
,
T
).
The
notation
u
t
(or
u
(
t
))
stands
for
the
time
derivative,
i.e.,
the
partial
derivative
of
u
with
respect
to
the
time
variable
t
.
The
main
difficulty
in
dealing
with
problem
(
P
)
comes
from
the
operator
−
∆
p
+
∆
q
that
we
call
the
competing
(
p,
q
)-Laplacian.
This
operator
is
not
monotone
and
no
ellipticity
condition
can
be
matched.
We
note
that
if
u
∈
W
1
,p
0
(Ω)
is
a
nonzero
function
and
λ
is
a
positive
number,
the
expression
−
∆
p
(
λu
)
+
∆
q
(
λu
)
,
λu
W
1
,p
0
(Ω)
=
λ
p
−
∆
p
u,
u
W
1
,p
0
(Ω)
−
λ
q
−
∆
q
u,
u
W
1
,q
0
(Ω)
is
positive
for
λ
sufficiently
large
and
negative
for
λ
small
enough.
Until
now,
problems
involving
competing
operators
like
−
∆
p
+∆
q
have
been
examined
only
for
stationary
problems.
Their
study
was
initiated
in
[
12
]
and
after
that
developed
in
[
6
,
8
,
11
,
13
,
15
,
17
,
20
]
(refer
also
to
the
recent
book
[
10
]).
Here,
we
consider
for
the
first
time
evolution
problems
with
competing
operators.
Due
to
the
competing
character,
the
monotonicity
method
of
Browder
and
Minty
as
well
as
the
theory
of
semigroups
are
not
applicable
(see,
e.g.,
[
9
]).
Another
major
difficulty
in
treating
problem
(
P
)
is
the
presence
of
the
operator
∆
q
which
is
not
pseudomonotone.
Indeed,
if
∆
q
were
pseudomono-
tone
on
W
1
,q
0
(Ω),
then
for
each
sequence
{
u
n
}
in
W
1
,q
0
(Ω)
with
u
n
u
in
W
1
,q
0
(Ω)
and
lim
sup
n
→∞
∆
q
u
n
,
u
n
−
u
W
1
,q
0
(Ω)
≤
0
it
would
hold
lim
n
→∞
∆
q
u
n
,
u
n
W
1
,q
0
(Ω)
=
∆
q
u,
u
W
1
,q
0
(Ω)
or
equivalently
u
n
→
u
in
W
1
,q
0
(Ω)
because
the
space
W
1
,q
0
(Ω)
is
uniformly
convex.
Never-
theless,
we
always
have
lim
sup
n
→∞
∆
q
u
n
,
u
n
−
u
W
1
,q
0
(Ω)
=
lim
sup
n
→∞
∆
q
u
n
−
∆
q
u,
u
n
−
u
W
1
,q
0
(Ω)
≤
0
due
to
the
monotonicity
of
−
∆
q
.
Therefore
the
pseudomonotonicity
of
∆
q
would
lead
to
the
contradiction
that
the
weak
convergence
and
strong
con-
vergence
in
W
1
,q
0
(Ω)
are
equivalent.
We
are
thus
prevented
to
use
the
sur-
jectivity
theorem
for
pseudomonotone
operators
relative
to
time-derivative
(see
[
3
,
Theorem
2.152])
as
well
as
the
corresponding
sub-supersolution
ap-
proach
(refer
to
[
2
,
4
,
5
,
14
]).
The
lower-order
term
in
the
equation
of
(
P
)
is
described
by
a
Carath´
eo-
dory
function
f
:
Ω
×
(0
,
T
)
×
R
→
R
(i.e.,
f
(
·
,
·
,
s
)
is
measurable
for
all
D.
Motreanu
291
s
∈
R
and
f
(
x,
t,
·
)
is
continuous
for
a.e.
(
x,
t
)
∈
Ω
×
(0
,
T
)).
Due
to
the
lack
of
needed
convergence
of
the
gradient
∇
u
of
the
solution
(see
[
1
]),
we
don’t
consider
the
gradient
dependence
in
the
lower-order
term.
We
require
the
following
growth
condition:
(H)
There
exist
a
function
σ
∈
L
p
p
−
1
(Ω
×
(0
,
T
))
and
a
constant
a
∈
(0
,
λ
1
,p
)
such
that
|
f
(
x,
t,
s
)
|
≤
σ
(
x,
t
)
+
a
|
s
|
p
−
1
for
a.e.
(
x,
t
)
∈
Ω
×
(0
,
T
)
and
all
s
∈
R
,
where
λ
1
,p
denotes
the
first
eigenvalue
of
−
∆
p
on
W
1
,p
0
(Ω),
that
is
λ
1
,p
=
inf
u
∈
W
1
,p
0
(Ω)
\{
0
}
∇
u
p
L
p
(Ω)
u
p
L
p
(Ω)
.
(1)
In
order
to
overcome
the
lack
of
ellipticity
and
monotonicity
in
problem
(
P
)
we
weaken
the
notion
of
solution
as
follows.
Definition
1.
By
a
generalized
solution
to
problem
(
P
)
we
mean
a
function
u
∈
L
p
(0
,
T
;
W
1
,p
0
(Ω))
for
which
u
(0)
=
u
0
such
that
there
exists
the
gen-
eralized
derivative
u
∈
L
p
p
−
1
(0
,
T
;
W
−
1
,
p
p
−
1
(Ω))
and
there
exists
a
sequence
{
u
n
}
in
L
p
(0
,
T
;
W
1
,p
0
(Ω))
with
the
properties:
(i)
u
n
u
in
L
p
(0
,
T
;
W
1
,p
0
(Ω))
and
u
n
(0)
u
0
and
u
n
(
T
)
u
(
T
)
in
L
2
(Ω)
;
(ii)
there
exists
the
generalized
derivative
u
n
∈
L
p
p
−
1
(0
,
T
;
W
−
1
,
p
p
−
1
(Ω))
for
every
n
and
u
n
u
in
L
p
p
−
1
(0
,
T
;
W
−
1
,
p
p
−
1
(Ω))
as
n
→
∞
;
(iii)
−
∆
p
u
n
+
∆
q
u
n
f
(
·
,
·
,
u
)
−
u
in
L
p
p
−
1
(0
,
T
;
W
−
1
,
p
p
−
1
(Ω))
;
(iv)
lim
n
→∞
u
n
−
∆
p
u
n
+
∆
q
u
n
,
u
n
−
u
L
p
(0
,T
;
W
1
,p
0
(Ω))
=
0
.
We
state
our
existence
result.
Theorem
1.
Under
hypothesis
(H),
problem
(
P
)
possesses
a
generalized
solution
in
the
sense
of
Definition
1
.
The
proof
of
Theorem
1
proceeds
along
a
special
Galerkin-type
approach
without
any
use
of
monotonicity,
ellipticity
and
coerciveness,
relying
on
a
An
evolution
equation
with
competing
p
-Laplacian
operators
292
finite
dimensional
differential
system
associated
with
problem
(
P
)
that
is
re-
solved
through
the
Carath´
eodory
theorem
for
ordinary
differential
equations.
In
Section
5
we
illustrate
the
applicability
of
Theorem
1
by
an
example.
The
rest
of
the
paper
consists
of
the
following
material.
Section
2
com-
prises
preliminary
results.
Section
3
focuses
on
resolving
the
finite
dimen-
sional
differential
system
associated
with
problem
(
P
).
Section
4
sets
forth
the
analysis
of
the
sequence
required
in
Definition
1
.
Section
5
contains
the
proof
of
Theorem
1
as
well
as
an
example.
2
Preliminary
results
For
any
normed
space
E
we
denote
by
·
E
the
norm
of
E
,
by
E
∗
its
dual
and
by
·
,
·
X
the
duality
pairing
between
E
and
E
∗
.
The
weak
convergence
is
denoted
by
and
the
convergence
in
norm
by
→
.
Throughout
the
paper,
the
space
W
1
,p
0
(Ω)
is
endowed
with
the
norm
u
W
1
,p
0
=
(
Ω
|∇
u
|
p
dx
)
1
p
.
The
gradient
of
u
with
respect
to
x
∈
Ω
is
denoted
∇
u
,
while
|∇
u
(
x,
t
))
|
represents
the
Euclidean
norm
of
the
vector
∇
u
(
x,
t
)
∈
R
N
.
The
dual
of
W
1
,p
0
(Ω)
is
the
space
W
−
1
,
p
p
−
1
(Ω).
Noting
that
p
≥
2,
we
have
the
evolution
triple
W
1
,p
0
(Ω)
⊂
L
2
(Ω)
⊂
W
−
1
,
p
p
−
1
(Ω)
(2)
with
each
embedding
dense
and
compact.
Recall
that
L
2
(Ω)
is
a
Hilbert
space
with
the
scalar
product
(
u,
v
)
L
2
(Ω)
=
Ω
u
(
x
)
v
(
x
)
dx
.
The
space
X
:=
L
p
(0
,
T
;
W
1
,p
0
(Ω))
is
equipped
with
the
norm
u
X
:=
T
0
u
(
t
)
p
W
1
,p
0
dt
1
p
making
it
a
separable
and
uniformly
convex
Banach
space.
The
dual
of
X
is
X
∗
=
L
p
p
−
1
(0
,
T
;
W
−
1
,
p
p
−
1
(Ω)).
There
is
a
dense
and
continuous
embedding
C
([0
,
T
]
,
W
1
,p
0
(Ω))
⊂
X.
(3)
Moreover,
the
set
P
of
polynomials
m
i
=0
t
i
v
i
with
v
i
∈
W
1
,p
0
(Ω)
and
m
≥
0,
is
dense
in
X
,
that
is,
P
=
X.
(4)
In
the
statement
of
problem
(
P
)
there
appears
the
time-dependent
neg-
D.
Motreanu
293
ative
p
-Laplacian
−
∆
p
:
X
→
X
∗
given
by
−
∆
p
u,
v
X
=
T
0
−
∆
p
u
(
t
)
,
v
(
t
)
W
1
,p
0
(Ω)
dt
=
T
0
Ω
|∇
u
(
x,
t
)
|
p
−
2
∇
u
(
x,
t
)
∇
v
(
x,
t
)
dxdt
(5)
for
all
u,
v
∈
X
.
The
operator
−
∆
p
in
(
5
)
is
the
time
realization
on
the
space
X
of
the
ordinary
negative
p
-Laplacian
−
∆
p
:
W
1
,p
0
(Ω)
→
W
−
1
,
p
p
−
1
(Ω).
The
fact
that
the
operator
in
(
5
)
is
well
defined
is
seen
by
T
0
−
∆
p
u
(
t
)
p
p
−
1
W
−
1
,
p
p
−
1
(Ω)
dt
=
T
0
u
(
t
)
p
W
1
,p
0
(Ω)
dt
for
all
u
∈
X.
Likewise,
the
time-dependent
negative
q
-Laplacian
−
∆
q
:
L
q
(0
,
T
;
W
1
,q
0
(Ω))
→
L
q
q
−
1
(0
,
T
;
W
−
1
,
q
q
−
1
(Ω))
is
defined
by
−
∆
q
u,
v
L
q
(0
,T
;
W
1
,q
0
(Ω))
=
T
0
−
∆
q
u
(
t
)
,
v
(
t
)
W
1
,q
0
(Ω)
dt
=
T
0
Ω
|∇
u
(
x,
t
)
|
q
−
2
∇
u
(
x,
t
)
∇
v
(
x,
t
)
dxdt
(6)
for
all
u,
v
∈
L
q
(0
,
T
;
W
1
,p
0
(Ω)).
Since
q
<
p
,
H¨
older’s
inequality
produces
a
constant
C
>
0
independent
of
t
such
that

t
0
Ω
|∇
u
(
x,
s
)
|
q
dxds
1
q
≤
C

t
0
Ω
|∇
u
(
x,
s
)
|
p
dxds
1
p
(7)
for
all
u
∈
X
and
t
∈
[0
,
T
].
Taking
t
=
T
,
we
obtain
the
continuous
embedding
X
⊂
L
q
(0
,
T
;
W
1
,q
0
(Ω)).
Therefore
the
operator
−
∆
p
+∆
q
is
well
defined
on
the
space
X
,
so
the
formulation
of
problem
(
P
)
is
meaningful.
The
following
lemma
provides
important
properties
of
the
operator
−
∆
p
in
(
5
).
Similarly,
we
can
proceed
with
the
operator
−
∆
q
in
(
6
).
Lemma
1.
The
time-dependent
negative
p
-Laplacian
−
∆
p
:
X
→
X
∗
is
well
defined,
demicontinuous,
strictly
monotone,
bounded
(i.e.,
bounded
on
bounded
sets),
and
satisfies
−
∆
p
u
X
∗
=
u
p
−
1
X
for
all
u
∈
X.
(8)
An
evolution
equation
with
competing
p
-Laplacian
operators
294
Proof.
From
(
5
)
and
H¨
older’s
inequality
we
infer
that
|−
∆
p
u,
v
X
|
≤
T
0
Ω
|∇
u
(
x,
t
)
|
p
−
1
|∇
v
(
x,
t
)
|
dxdt
≤

T
0
Ω
|∇
u
(
x,
t
)
|
p
dxdt
p
−
1
p

T
0
Ω
|∇
v
(
x,
t
)
|
p
dxdt
1
p
=
u
p
−
1
X
v
X
for
all
u,
v
∈
X
.
It
turns
out
that
the
operator
−
∆
p
from
X
into
X
∗
is
well
defined
and
bounded.
Taking
v
=
u
shows
that
(
8
)
holds
true.
Since
−
∆
p
in
(
5
)
is
the
time
realization
on
the
space
X
of
the
negative
p
-Laplacian
−
∆
p
:
W
1
,p
0
(Ω)
→
W
−
1
,p
(Ω),
the
operator
−
∆
p
:
X
→
X
∗
is
demicontinuous
(see
[
3
,
Theorem
2.153]).
The
strict
monotonicity
of
−
∆
p
in
(
5
)
is
valid
because
the
mapping
ξ
→
|
ξ
|
p
−
2
ξ
is
strictly
monotone
on
R
N
.
The
proof
is
complete.
Recall
that
the
Nemytskii
operator
N
f
associated
with
the
Carath´
eodory
function
f
:
Ω
×
(0
,
T
)
×
R
→
R
is
defined
as
N
f
(
u
)
:=
f
(
·
,
·
,
u
(
·
,
·
)).
Lemma
2.
Assume
condition
(H)
holds.
Then
the
mapping
N
f
:
X
→
X
∗
is
well
defined,
bounded
and
continuous.
Proof.
The
growth
condition
in
(H)
guarantees
that
f
(
·
,
·
,
u
(
·
,
·
))
∈
L
p
p
−
1
(Ω
×
(0
,
T
))
for
all
u
∈
X.
(9)
The
continuity
and
density
of
the
embedding
X
⊂
L
p
(Ω
×
(0
,
T
))
implies
that
L
p
p
−
1
(Ω
×
(0
,
T
))
⊂
X
∗
continuously.
According
to
(
9
),
through
Kras-
noselskii’s
theorem
(see
[
7
]),
the
mapping
N
f
is
bounded
and
continuous
from
L
p
(Ω
×
(0
,
T
))
to
L
p
p
−
1
(Ω
×
(0
,
T
)).
Then
the
restriction
N
f
:
X
→
X
∗
is
bounded
and
continuous,
which
completes
the
proof.
Next
we
briefly
discuss
the
generalized
derivative
u
associated
with
the
evolution
triple
(
2
)
for
a
function
u
∈
X
.
More
details
can
be
found
in
[
3
,
16
,
18
,
19
].
The
function
u
∈
X
∗
is
called
the
generalized
derivative
of
u
∈
X
if
there
holds
T
0
ϕ
(
t
)
u
(
t
)
dt
=
−
T
0
ϕ
(
t
)
u
(
t
)
dt
for
all
ϕ
∈
C
∞
c
(0
,
T
)
.
D.
Motreanu
295
A
necessary
and
sufficient
condition
is
that
T
0
(
u
(
t
)
,
v
)
L
2
(Ω)
ϕ
(
t
)
dt
=
−
T
0
u
(
t
)
,
v
W
1
,p
0
(Ω)
ϕ
(
t
)
dt
(10)
for
all
ϕ
∈
C
∞
c
(0
,
T
)
and
v
∈
W
1
,p
0
(Ω).
Definition
1
shows
that
each
generalized
solution
to
problem
(
P
)
belongs
to
W
:=
{
w
∈
X
:
w
∈
X
∗
}
,
(11)
which
is
a
Banach
space
endowed
with
the
norm
w
W
=
w
X
+
w
X
∗
.
There
is
the
continuous
embedding
W
⊂
C
([0
,
T
]
,
L
2
(Ω))
(12)
(see
[
18
,
Proposition
23.23]).
Due
to
inclusion
(
12
),
the
initial
condition
u
(0)
=
u
0
∈
L
2
(Ω)
in
problem
(
P
)
makes
sense.
Thanks
to
the
Lions-Aubin
theorem,
we
have
the
compact
embedding
W
⊂
L
p
(Ω
×
(0
,
T
))
(13)
(see
[
16
,
p.
106]).
Lemma
3.
Assume
condition
(H).
Then
the
mapping
N
f
:
W
→
X
∗
is
completely
continuous
(i.e.,
u
n
u
in
W
implies
N
f
(
u
n
)
→
N
f
(
u
)
in
X
∗
).
Proof.
As
noted
in
the
proof
of
Lemma
2
,
the
mapping
N
f
is
bounded
and
continuous
from
L
p
(Ω
×
(0
,
T
))
to
X
∗
.
In
view
of
the
compact
embedding
(
13
),
the
restriction
of
N
f
to
W
is
completely
continuous,
so
achieving
the
result.
3
Finite
dimensional
approximation
for
problem
(
P
)
Since
the
space
W
1
,p
0
(Ω)
is
separable,
there
is
a
countable
family
{
w
k
}
k
≥
1
of
linearly
independent
vectors
in
W
1
,p
0
(Ω)
such
that
if
E
n
is
the
linear
span
of
{
w
1
,
.
.
.
,
w
n
}
,
we
have
∞
n
=1
E
n
=
W
1
,p
0
(Ω)
.
(14)
An
evolution
equation
with
competing
p
-Laplacian
operators
296
Hence
E
n
is
a
vector
subspace
of
W
1
,p
0
(Ω)
of
finite
dimension
n
with
the
basis
{
w
1
,
.
.
.
,
w
n
}
that
we
endow
with
the
scalar
product
of
L
2
(Ω).
Given
the
initial
condition
u
0
∈
L
2
(Ω),
we
fix
a
sequence
{
u
0
,n
}
n
≥
1
with
u
0
,n
∈
E
n
for
each
n
satisfying
u
0
,n
→
u
0
in
L
2
(Ω).
This
is
possible
according
to
(
14
)
(note
p
≥
2).
We
can
uniquely
express
u
0
,n
=
1
≤
k
≤
n
c
k,n
w
k
,
with
c
k,n
∈
R
,
1
≤
k
≤
n
.
On
the
interval
[0
,
T
]
we
consider
the
Cauchy
problem
n
k
=1
θ
k
(
t
)(
w
k
,
w
i
)
L
2
(Ω)
+
(
−
∆
p
+
∆
q
−
N
f
)(
n
k
=1
θ
k
(
t
)
w
k
)
,
w
i
W
1
,p
0
(Ω)
=
0
,
θ
i
(0)
=
c
i,n
,
1
≤
i
≤
n,
(
S
P
)
with
the
unknowns
(
θ
1
(
t
)
,
.
.
.
,
θ
n
(
t
))
∈
R
n
.
Proposition
1.
Assume
that
condition
(H)
holds.
Then
the
system
of
or-
dinary
differential
equations
(
S
P
)
admits
an
absolutely
continuous
solution
(
θ
1
(
t
)
,
.
.
.
,
θ
n
(
t
))
which
is
defined
on
the
entire
interval
[0
,
T
]
.
Proof.
The
Gram
determinant
det(
w
k
,
w
i
)
L
2
(Ω)
is
nonzero
because
the
vec-
tors
{
w
1
,
.
.
.
,
w
n
}
are
linearly
independent,
so
the
matrix
((
w
k
,
w
i
)
L
2
)
1
≤
k,i
≤
n
is
invertible.
Hypothesis
(H)
permits
to
apply
Carath´
eodory’s
theorem
(see
[
19
,
p.
1044])
to
the
ordinary
differential
system
(
S
P
)
noting
that
the
term
involving
N
f
can
be
locally
bounded
by
an
integrable
function
in
t
due
to
the
imposed
growth
condition.
Consequently,
there
exists
an
ab-
solutely
continuous
solution
(
θ
1
(
t
)
,
·
·
·
,
θ
n
(
t
))
of
system
(
S
P
)
on
a
maximal
subinterval
[0
,
T
n
)
of
[0
,
T
].
We
are
going
to
show
that
T
n
=
T
.
Set
u
n
(
t
)
=
n
k
=1
θ
k
(
t
)
w
k
.
(15)
Then
system
(
S
P
)
takes
the
form
n
k
=1
θ
k
(
t
)(
w
k
,
w
i
)
L
2
(Ω)
+
(
−
∆
p
+
∆
q
−
N
f
)(
u
n
(
t
))
,
w
i
W
1
,p
0
(Ω)
=
0
θ
i
(0)
=
c
i,n
,
1
≤
i
≤
n.
(16)
It
follows
that
(
u
n
(
t
)
,
u
n
(
t
))
L
2
(Ω)
+
(
−
∆
p
+
∆
q
−
N
f
)(
u
n
(
t
))
,
u
n
(
t
)
W
1
,p
0
(Ω)
=
0
(17)
D.
Motreanu
297
for
almost
all
t
∈
[0
,
T
n
).
Integration
over
the
interval
[0
,
t
]
results
in
1
2
u
n
(
t
)
2
L
2
(Ω)
+
t
0
(
−
∆
p
+∆
q
−
N
f
)(
u
n
(
s
))
,
u
n
(
s
)
W
1
,p
0
(Ω)
ds
=
1
2
u
0
,n
2
L
2
(Ω)
for
all
t
∈
[0
,
T
n
).
By
hypothesis
(H)
and
H¨
older’s
inequality,
in
conjunction
with
(
1
),
it
turns
out
1
2
u
n
(
t
)
2
L
2
(Ω)
+
t
0
u
n
(
s
)
p
W
1
,p
0
(Ω)
ds
−
t
0
u
n
(
s
)
q
W
1
,q
0
(Ω)
ds
≤
1
2
u
0
,n
2
L
2
(Ω)
+
t
0
Ω
σ
(
x
)
u
n
(
x,
s
)
dxds
+
a
t
0
Ω
|
u
n
(
x,
s
)
|
p
dxds
≤
1
2
u
0
,n
2
L
2
(Ω)
+
σ
L
p
p
−
1
(Ω
×
(0
,T
))
λ
−
1
p
1
,p

t
0
u
n
(
s
)
p
W
1
,p
0
(Ω)
ds
1
p
+
aλ
−
1
1
,p
t
0
u
n
(
s
)
p
W
1
,p
0
(Ω)
ds.
Taking
into
account
(
7
),
this
amounts
to
saying
that
1
2
u
n
(
t
)
2
L
2
(Ω)
+
1
−
aλ
−
1
1
,p
t
0
u
n
(
s
)
p
W
1
,p
0
(Ω)
ds
≤
1
2
u
0
,n
2
L
2
(Ω)
+
C
1

t
0
u
n
(
s
)
p
W
1
,p
0
(Ω)
ds
1
p
+
C
2

t
0
u
n
(
s
)
p
W
1
,p
0
(Ω)
ds
q
p
,
(18)
with
positive
constants
C
1
and
C
2
independent
of
t
.
Using
that
1
−
aλ
−
1
1
,p
>
0
and
p
>
q
>
1,
it
is
seen
from
(
18
)
that
the
set
{
u
n
(
t
)
:
t
∈
[0
,
T
n
)
}
is
uniformly
bounded
in
L
2
(Ω).
Hence
the
solution
(
θ
1
(
t
)
,
·
·
·
,
θ
n
(
t
))
of
system
(
16
)
is
uniformly
bounded
on
[0
,
T
n
),
which
renders
T
n
=
T
.
This
completes
the
proof.
Corollary
1.
The
function
u
n
:
[0
,
T
]
→
W
1
,p
0
(Ω)
in
(
15
)
fulfills
u
n
∈
W
,
where
W
is
the
space
described
in
(
11
)
.
Proof.
Proposition
1
combined
with
(
3
)
entails
that
u
n
∈
X
.
From
(
15
)
and
Proposition
1
we
infer
that
the
derivative
u
n
=
n
k
=1
θ
k
w
k
,
defined
almost
everywhere
on
[0
,
T
],
is
an
element
of
X
∗
.
Altogether,
we
have
u
n
∈
W
.
An
evolution
equation
with
competing
p
-Laplacian
operators
298
4
The
sequence
of
approximate
solutions
Now,
we
focus
on
the
sequence
{
u
n
}
n
≥
1
⊂
W
introduced
in
(
15
).
The
next
proposition
relies
on
key
uniform
estimates
for
the
Galerkin
approximations
u
n
and
their
time
derivatives
u
n
.
Proposition
2.
Assume
that
hypothesis
(H)
holds.
Then
there
exists
u
∈
W
such
that
(i)
u
n
u
in
X
;
(ii)
u
n
u
in
X
∗
,
so
u
n
u
in
W
and
in
C
([0
,
T
]
,
L
2
(Ω))
;
(iii)
−
∆
p
u
n
+
∆
q
u
n
N
f
(
u
)
−
u
in
X
∗
;
(iv)
u
(0)
=
u
0
and
u
n
(0)
u
0
and
u
n
(
T
)
u
(
T
)
in
L
2
(Ω)
.
Proof.
From
(
18
)
we
infer
for
t
=
T
the
estimate
1
2
u
n
(
T
)
2
L
2
(Ω)
+
1
−
aλ
−
1
1
,p
u
n
p
X
≤
1
2
u
0
,n
2
L
2
(Ω)
+
C
1
u
n
X
+
C
2
u
n
q
X
,
with
positive
constants
C
1
and
C
2
.
Since
1
−
aλ
−
1
1
,p
>
0
and
p
>
q
>
1,
it
follows
that
the
sequence
{
u
n
}
n
≥
1
is
bounded
in
X
and
the
sequence
{
u
n
(
T
)
}
n
≥
1
is
bounded
in
L
2
(Ω).
Thanks
to
the
reflexivity
of
X
and
of
L
2
(Ω)
we
can
pass
to
a
subsequence
for
which
there
exist
u
∈
X
and
h
∈
L
2
(Ω)
such
that
u
n
u
in
X
and
u
n
(
T
)
h
in
L
2
(Ω)
.
(19)
The
first
weak
limit
in
(
19
)
proves
item
(i).
By
virtue
of
Lemmas
1
and
2
,
the
sequences
{−
∆
p
u
n
}
n
≥
1
,
{−
∆
q
u
n
}
n
≥
1
and
{
N
f
(
u
n
)
}
n
≥
1
are
bounded
in
X
∗
.
Due
to
the
reflexivity
of
X
∗
,
up
to
subsequences
there
hold
−
∆
p
u
n
ξ,
−
∆
q
u
n
η,
N
f
(
u
n
)
ζ
in
X
∗
,
(20)
with
some
ξ
∈
X
∗
,
η
∈
X
∗
and
ζ
∈
X
∗
.
From
(
16
)
we
get
for
almost
all
t
∈
|
0
,
T
]
and
all
ϕ
∈
C
1
([0
,
T
])
and
v
∈
E
n
that
(
u
n
(
t
)
,
ϕ
(
t
)
v
)
L
2
(Ω)
+
ϕ
(
t
)
(
−
∆
p
+
∆
q
−
N
f
)(
u
n
(
t
))
,
v
W
1
,p
0
(Ω)
=
0
.
(21)
D.
Motreanu
299
Let
v
∈
∞
n
=1
E
n
.
Fix
an
m
with
v
∈
E
m
and
thus
(
21
)
is
valid
for
all
n
≥
m
.
Integration
by
parts
over
|
0
,
T
]
in
(
21
)
entails
−
T
0
ϕ
(
t
)(
u
n
(
t
)
,
v
)
L
2
(Ω)
dt
+
ϕ
(
T
)(
u
n
(
T
)
,
v
)
L
2
(Ω)
−
ϕ
(0)(
u
0
,n
,
v
)
L
2
(Ω)
+
T
0
ϕ
(
t
)
(
−
∆
p
+
∆
q
−
N
f
)(
u
n
(
t
))
,
v
W
1
,p
0
(Ω)
dt
=
0
,
∀
n
≥
m.
(22)
Letting
n
→
∞
,
by
means
of
(
19
)
and
(
20
)
we
find
−
T
0
ϕ
(
t
)(
u
(
t
)
,
v
)
L
2
(Ω)
dt
+
ϕ
(
T
)(
h,
v
)
L
2
(Ω)
−
ϕ
(0)(
u
0
,
v
)
L
2
(Ω)
+
T
0
ϕ
(
t
)
ξ
−
η
−
ζ,
v
W
1
,p
0
(Ω)
dt
=
0
.
(23)
Thanks
to
the
density
in
(
14
),
we
have
that
(
23
)
holds
for
all
v
∈
W
1
,p
0
(Ω).
In
particular,
taking
ϕ
∈
C
∞
c
(0
,
T
),
we
arrive
at
T
0
ϕ
(
t
)(
u
(
t
)
,
v
)
L
2
(Ω)
dt
=
−
T
0
ϕ
(
t
)
(
−
ξ
+
η
+
ζ,
v
W
1
,p
0
(Ω)
dt
for
all
ϕ
∈
C
∞
c
(0
,
T
)
and
v
∈
W
1
,p
0
(Ω).
According
to
(
10
),
there
exists
the
generalized
derivative
u
∈
X
∗
of
u
with
u
=
−
ξ
+
η
+
ζ.
(24)
From
(
11
)
we
conclude
that
u
∈
W
.
We
pass
now
to
uniformly
estimate
the
time
derivatives
u
n
.
Return
to
(
22
)
that
can
be
written
in
the
form
T
0
(
u
n
(
t
)
,
v
)
L
2
(Ω)
ϕ
(
t
)
dt
+
T
0
ϕ
(
t
)
(
−
∆
p
+
∆
q
−
N
f
)(
u
n
(
t
))
,
v
W
1
,p
0
(Ω)
dt
=
0
whenever
ϕ
∈
C
1
([0
,
T
]),
v
∈
E
m
and
n
≥
m
.
In
the
limit
as
n
→
∞
we
obtain
lim
n
→∞

T
0
(
u
n
(
t
)
,
ϕ
(
t
)
v
)
L
2
(Ω)
dt
+
T
0
(
−
∆
p
+
∆
q
−
N
f
)(
u
n
(
t
))
,
ϕ
(
t
)
v
W
1
,p
0
(Ω)
dt
=
0
An
evolution
equation
with
competing
p
-Laplacian
operators
300
for
all
ϕ
∈
C
1
([0
,
T
])
and
v
∈
∞
n
=1
E
n
.
Using
the
density
properties
de-
scribed
in
(
4
)
and
(
14
),
we
get
lim
n
→∞

T
0
(
u
n
(
t
)
,
w
)
L
2
(Ω)
dt
+
T
0
(
−
∆
p
+
∆
q
−
N
f
)(
u
n
(
t
))
,
w
W
1
,p
0
(Ω)
dt
=
0
(25)
for
all
w
∈
X
.
On
the
basis
of
(
20
)
and
(
24
),
this
yields
u
n
u
in
X
∗
.
In
view
of
(
11
)
and
assertion
(i),
we
infer
that
u
n
u
in
W
.
The
continuous
embedding
(
12
)
implies
u
n
u
in
C
([0
,
T
]
,
L
2
(Ω)),
thereby
assertion
(ii)
follows.
The
fact
that
u
n
u
in
W
allows
us
to
apply
Lemma
3
that
provides
the
strong
convergence
N
f
(
u
n
)
→
N
f
(
u
)
in
X
∗
.
Combining
with
(
20
)
and
(
24
)
leads
to
item
(iii)
in
the
statement
of
proposition.
Insert
(
24
)
(with
ζ
=
N
f
(
u
))
into
(
23
),
which
gives
−
T
0
ϕ
(
t
)(
u
(
t
)
,
v
)
L
2
(Ω)
dt
+
ϕ
(
T
)(
h,
v
)
L
2
(Ω)
−
ϕ
(0)(
u
0
,
v
)
L
2
(Ω)
−
T
0
ϕ
(
t
)
u
(
t
))
,
v
W
1
,p
0
(Ω)
dt
=
0
for
all
ϕ
∈
C
1
([0
,
T
])
and
v
∈
W
1
,p
0
(Ω).
The
integration
by
parts
provides
ϕ
(
T
)(
h,
v
)
L
2
(Ω)
−
ϕ
(0)(
u
0
,
v
)
L
2
(Ω)
=
ϕ
(
T
)(
u
(
T
)
,
v
)
L
2
(Ω)
−
ϕ
(0)(
u
(0)
,
v
)
L
2
(Ω)
.
Since
ϕ
∈
C
1
([0
,
T
])
and
v
∈
W
1
,p
0
(Ω)
are
arbitrary,
and
W
1
,p
0
(Ω)
is
dense
in
L
2
(Ω),
we
deduce
that
u
(0)
=
u
0
and
u
(
T
)
=
h
.
Hence
item
(iv)
is
proven.
The
proof
is
complete.
Corollary
2.
Assume
that
condition
(H)
holds.
Then
the
time
derivative
u
of
the
function
u
∈
W
constructed
in
Proposition
2
can
be
estimated
as
follows
u
X
∗
≤
lim
sup
n
→∞
−
∆
p
u
n
X
∗
+
lim
sup
n
→∞
−
∆
q
u
n
X
∗
+
N
f
(
u
)
X
∗
.
Proof.
From
item
(iii)
in
Proposition
2
and
the
weak
lower
semicontinuity
of
the
norm
we
infer
that
u
X
∗
≤
lim
inf
n
→∞
−
∆
p
u
n
−
∆
q
u
n
X
∗
+
N
f
(
u
)
X
∗
≤
lim
sup
n
→∞
−
∆
p
u
n
−
∆
q
u
n
X
∗
+
N
f
(
u
)
X
∗
.
The
stated
conclusion
holds
true.
D.
Motreanu
301
5
The
generalized
solution
Proof
of
Theorem
1
.
Proposition
2
ensures
the
existence
of
an
u
∈
W
for
which
assertions
(i)-(iv)
are
satisfied
with
the
sequence
{
u
n
}
n
≥
1
⊂
W
constructed
in
Proposition
1
.
We
check
that
the
requirements
in
Definition
1
for
a
generalized
solution
to
problem
(
P
)
are
fulfilled.
We
know
from
Proposition
2
that
u
∈
L
p
(0
,
T
;
W
1
,p
0
(Ω))
with
u
∈
L
p
p
−
1
(0
,
T
;
W
−
1
,
p
p
−
1
(Ω)).
Item
(iv)
in
Proposition
2
entails
u
(0)
=
u
0
.
Part
(i)
in
Definition
1
is
the
direct
consequence
of
items
(i)
and
(iv)
of
Proposition
2
.
Parts
(ii)
and
(iii)
in
Definition
1
are
verified
by
the
statements
(ii)
and
(iii)
of
Proposition
2
;
respectively.
It
remains
to
justify
property
(iv)
in
Definition
1
.
By
(
17
)
we
have
that
u
n
−
∆
p
u
n
+
∆
q
u
n
−
N
f
(
u
n
)
,
u
n
X
=
0
(26)
for
all
n
.
On
the
other
hand,
items
(ii)
and
(iii)
of
Proposition
2
provide
u
n
−
∆
p
u
n
+
∆
q
u
n
−
N
f
(
u
n
)
,
u
X
→
0
as
n
→
∞
.
(27)
Then
(
26
)
and
(
27
)
ensure
that
u
n
−
∆
p
u
n
+
∆
q
u
n
−
N
f
(
u
n
)
,
u
n
−
u
X
→
0
as
n
→
∞
.
Since
u
n
u
in
W
,
Lemma
3
implies
that
N
f
(
u
n
)
,
u
n
−
u
X
→
0
as
n
→
∞
.
Therefore
property
(iv)
in
Definition
1
holds
true.
The
proof
of
Theorem
1
is
complete.
Finally,
we
present
an
example
illustrating
the
applicability
of
Theo-
rem
1
.
Example
1.
Consider
the
initial-boundary
value
problem
u
t
−
∆
p
u
+
∆
q
u
=
ρ
(
x,
t
)
+
|
u
|
α
+1
1+
|
u
|
in
Ω
×
(0
,
T
)
u
=
0
on
∂
Ω
×
(0
,
T
)
u
(
x
)
=
u
0
(
x
)
on
∂
Ω
(28)
on
a
bounded
domain
Ω
in
R
N
with
Lipschitz
boundary
∂
Ω
and
for
fixed
real
numbers
T
>
0,
p
≥
2,
q
∈
(1
,
p
),
α
∈
[0
,
p
−
1),
and
given
functions
ρ
∈
L
p
p
−
1
(Ω
×
(0
,
T
))
and
u
0
∈
L
2
(Ω).
An
evolution
equation
with
competing
p
-Laplacian
operators
302
Problem
(
28
)
matches
the
statement
of
(
P
)
by
setting
f
(
x,
t,
s
)
=
ρ
(
x,
t
)
+
|
s
|
α
+1
1
+
|
s
|
whenever
(
x,
t,
s
)
∈
Ω
×
(0
,
T
)
×
R
.
Using
the
assumption
α
∈
[0
,
p
−
1),
we
obtain
through
Young’s
inequality
that
for
each
ε
>
0
there
is
a
constant
c
(
ε
)
>
0
such
that
|
f
(
x,
t,
s
)
|
≤
ρ
(
x,
t
)
+
|
s
|
α
≤
|
ρ
(
x,
t
)
|
+
ε
|
s
|
p
−
1
+
c
(
ε
)
.
Choose
ε
∈
(0
,
λ
1
,p
).
It
results
that
hypothesis
(H)
is
verified
with
σ
(
x,
t
)
=
|
ρ
(
x,
t
)
|
+
c
(
ε
)
and
a
=
ε
,
which
permits
to
apply
Theorem
1
to
problem
(
28
).
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