Logarithmic wave equation involving variable-exponent
nonlinearities:Well-posedness and blow-up
ABITA RAHMOUNE
Department Of Technical sciences
Université Amar Telidji, Laghouat
Laboratory of Pure and Applied Mathematics
ALGERIA
.
Abstract: In this paper, we focus on a class of existence, uniqueness, and explosion in a nite time of
solving a logarithmic wave equation model with nonlinearities with variable exponents and nonlinear
source terms under homogeneous Dirichlet boundary conditions.
utt −∆u+|ut|m(.)−2ut=|u|p(.)−2uln |u|
We applied the Faedo-Galerkin method in combination with the Banach xed point theorem to determine
the existence and uniqueness of a local solution in time. Various inequality techniques were used under
appropriate conditions to obtain the blow-up of a solution. This type of equation is related to uid
dynamics, electrorheological uids, quantum mechanics theory, nuclear physics, optics, and geophysics.
Key-Words: Wave equations; Logarithmic nonlinearity; variable exponents spaces; Existence; Finite
time blow-up.
Received: October 3, 2022. Revised: November 7, 2022. Accepted: November 19, 2022. Published: December 12, 2022.
1 Introduction
In recent years, many authors have paid attention
to the study of nonlocal logarithmic dierential
equations. This is partly due to the wide use of
this species to model various phenomena such as
uid dynamics, electrorheological uids, nuclear
physics, optics, geophysics, quantum mechanics
theory. In this work we treat the following semi-
linear wave equation with logarithmic nonlinear
source term under homogeneous Dirichlet bound-
ary condition
utt −∆u+|ut|m(.)−2ut=|u|p(.)−2uln |u|,
in
Ω×(0,T)
u(x, t) = 0,
on
∂Ω×(0, T )
u(x, 0) = u0(x), ut(x, 0) = u1(x),
in
Ω,
(1.1)
In (1.1),
Ω
be a bounded domain in
Rn(n≥
1)
with a smooth boundary
∂Ω,
for all
m(.),
p(.) : Ω→R
measurable functions satisfying
2≤q1≤q(x)≤q2≤2n
n−2, n ≥3,
2≤q1≤q(x)≤q2<∞, n ≤2,
(1.2)
with
q1:= ess inf
x∈Ωq(x), q2:= ess sup
x∈Ω
q(x)
and the logHölder continuity condition:
|q(x)−q(y)| ≤ −A
log |x−y|,
for a.e.
x, y ∈Ω,
with
0<|x−y|< δ, A > 0, δ < 1
(1.3)
In case
m
,
p
are constants, local, global exis-
tence and long-time behavior have been consid-
ered by many authors. For example, the log-
arithmic nonlinearity term
|u|p−2uln(|u|)
in the
absence of the damping term
|ut|m−2ut
causes
an innite time blow -up of solutions with nega-
tive initial energy [4, 1, 11, 12], in contrast to the
power source term
|u|p−2u
, which causes a nite
time blow-up of solutions [5, 6], it is known that
the damping term
|ut|m−2ut
for any initial data
[7, 8, 13] ensures global existence. We also refer
to [9, 10] and its references for logarithmic nonlin-
earity problems. These semilinear wave equations
arise when studying various problems and can be
used as models for viscoelastic liquids, processes
of ltration through a porous medium and liq-
uids with temperature-dependent viscosity, ltra-
tion theory, etc. (see [36, 35]). We also refer to
[14, 15] and its references for other issues in this
direction.
In recent years, some partial dierential equa-
tions with logarithmic nonlinearity term have at-
tracted much attention due to their wide applica-
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tion in physics and other applied sciences, such
as heat conduction with two temperature sys-
tems [17], seepage of homogeneous uids through
a ssured rock [16], unidirectional propagation of
nonlinear, dispersive, long waves [17, 18], uid
ow in ssured porous media [19], two-phase ow
in porous media with dynamic capillary pressure
[20, 21] and the aggregation of populations [22].
Pseudo-parabolic equations can also be viewed as
Sobolev-type or Sobolev-Galpern-type equations,
see [23, 24] and many articles have been devoted
to the study of well-posedness and qualitative
properties of solutions to these partial dierential
equations with constant exponents. It is impor-
tant to point out that the calculation of blow-up
time and rate on nonlinear evolution equations is
an important topic (see [25, 26]), and such evalu-
ations be able conclusively characterize the blow-
up phenomenon. The terminology variable ex-
ponents comes from the fact that
m(.)
and
p(.)
are functions and not real numbers. This term
|ut|m(.)−2ut− |u|p(.)−2uln |u|
is then a generaliza-
tion of
|ut|m−2ut− |u|p−2u
, which corresponds to
m(.), p (.)>1
and
ln |u|
. In fact, (1.1) can be
cast as an extension of the variable case of the
second-order viscoelastic wave equation with vari-
able growth conditions
utt −∆u+|ut|m(.)−2ut=|u|p(.)−2u,
in
Ω×(0, T )
(1.4)
what one gets when
|ut|m(.)−2ut− |u|p(.)−2uln |u|
considered. Equation (1.4) is a well-known model
for electrorheological uids [32] that occurs in the
treatment of uid dynamics. On the other hand,
results for the viscoelastic wave equation with log-
arithmic damping and variable growth conditions
are limited and rare, and the literature on these
equations is much less extensive, see [37, 39, 38].
The interest in the mathematical analysis of
partial dierential equations in recent years has
been driven by inhomogeneous dierential opera-
tors with variable exponents (see eg [29, 28, 27]).
The study of these systems is based on the use
of Lebesgue and Sobolev spaces with variable ex-
ponents. Note that the problems of dierential
equations with non-standard
p(x)
growth are an
unfamiliar and interesting topic. These are non-
linear theory of elasticity, electrorheological u-
ids, etc. These uids retain the motivating prop-
erty that their viscosity depends on the electric
eld in the uid. For general accounts of the
underlying physics see [31] and for the mathe-
matical visions see [30]. A number of papers on
problems in so-called rheological and electrorhe-
ological uids that indicate spaces with variable
exponents have recently been published by Dien-
ing and Ruzicka [32, 33]. The results of this work
were summarized in the books [32, 33]. Numerous
mathematical models in uid mechanics, elastic-
ity theory (recently in image processing), see eg
[34], etc. have been shown which are obviously re-
lated to the non-standard local growth problem.
In this article we consider (1.1) and establish a
local existence result. We also show that the so-
lution explodes in nite time
T
for suitable initial
dates.
2 Preliminaries
Let
p: Ω →[1,∞]
be a measurable function.
Lp(.)(Ω)
denotes the set of the real measurable
functions
u
on
Ω
such that
Ω
|λu (x)|p(x)dx < ∞
for some
λ > 0
.
The variable-exponent space
Lp(.)(Ω)
equipped
with the Luxemburgtype norm
∥u∥p(.)
=inf λ > 0
,
Ω
u(x)
λ
p(x)
dx≤1
,
is a Banach space. Throughout the paper, we use
∥.∥q
to indicate the
Lq
-norm for
1≤q≤+∞
.
H1
0(Ω)
is the closure of
C∞
0(Ω)
with respect to
the following norm:
∥u∥H1
0(Ω) =∥∇u∥2
2+∥u∥2
21
2.
It is known that for the elements of
H1
0(Ω)
the
Poincaré inequality holds,
∥u∥2≤C∗∥∇u∥2,
for all
u∈H1
0(Ω).
and an equivalent norm of
H1
0(Ω)
can be dened
by
∥u∥H1
0(Ω) =∥∇u∥2=Ω
|∇u(x)|2dx1
2
.
Lemma 2.1 [28, 29]. If
p: Ω →[1,∞)
is a mea-
surable function and
2≤p1≤p(x)≤p2<2n
n−2, n ≥3.
(2.1)
Then, the embedding
H1
0(Ω) →Lp(.)(Ω)
is
continuous and compact.
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3 Existence of weak solutions
In this section we present the local existence and
uniqueness of solutions for the system (1.1). Our
proof method is based on Banach's xed point
theorem.
Theorem 3.1 Let
m(.)
, and
p(.)
satises (1.2),
(1.3), and in addition
p(.)
satisfy
2< p1≤p(x)≤p2<2n−1
n−2, n ≥3.
(3.1)
Then, for any given
(u0, u1)∈H1
0(Ω) ×L2(Ω)
it exists
T > 0
and a unique solution
u
of the
problem (1.1) on
(0, T )
such that
u∈C(0, T ), H1
0(Ω)∩C1(0, T ), L2(Ω)
(3.2)
∩Lm(.)(Ω ×(0, T ))
,
utt ∈L2(0, T ), H−1(Ω)
.
To prove the main theorem we need the lo-
cal existence and uniqueness of the solution of a
related problem. Then, given
v
, consider the fol-
lowing initial boundary value problem:
utt −∆u+|ut|m(.)−2ut=v(x, t),
in
Ω×(0, T ),
u(x, t) = 0,
on
∂Ω×(0, T ),
u(x, 0) = u0(x), ut(x, 0) = u1(x),
in
Ω,
(3.3)
where the exponent
m(.)
is a given measurable
function satisfying (1.2) and (1.3). We now have
the following existence result of the local solution
of the problem (3.3) for
v∈L2(Ω ×(0, T ))
, and
suitable initial value
(u0, u1)∈H1
0(Ω) ×L2(Ω)
,
which we created using the Galerkin method as
in [2], or in [3, Theorem 3.1, Chapter 1].
Lemma 3.2 Suppose that
m(.)
satises (1.2), and
(1.3). Then, for all
(u0, u1)∈H1
0(Ω) ×L2(Ω)
and
v∈L2(Ω ×(0, T ))
, there is a unique local
solution
u
of the problem (3.3),
u∈L∞(0, T ), H1
0(Ω),
ut∈L∞(0, T ), L2(Ω)∩Lm(.)(Ω ×(0, T ))
utt ∈L2(0, T ), H−1(Ω).
(3.4)
proof.
1. Uniqueness: If the problem (3.3) has two so-
lutions
u
and
v
. Then,
w=u−v
must verify
wtt −∆w+ut|ut|m(.)−2−vt|vt|m(.)−2= 0,
in
Ω×(0, T ),
w(x, t) = 0,
on
∂Ω×(0, T ),
w(x, 0) = wt(x, 0) = 0,
in
Ω.
Formally, multiplying by
ut
and integrate
over
Ω×(0, t)
, gives
Ωw2
t+|∇w|2
+2 t
0Ωut|ut|m(x)−2−vt|vt|m(x)−2(ut
−vt)dxds= 0.
By using the inequality
|a|m(x)−2a− | bm(x)−2b.(a−b)≥0
(3.5)
for all
a,b∈Rn
and a.e
x∈Ω
, we get
Ωw2
t+|∇w|2= 0
which means that
w= 0
, since
w= 0
on
∂Ω
.
Therefore, the uniqueness follows.
2. Existence. Let
(vj)∞
j=1
be an orthonormal
basis of
H1
0(Ω)
, with
−∆vj=λjvj
in
Ω, vj= 0,
on
∂Ω,
let determine the nite-dimensional subspace
Vk=span {v1, . . . , vk}
, without loss of gener-
ality we may take
∥vj∥2= 1
. We will con-
struct a convergent sequence
uk(x, t),
uk(x, t) =
k
j=1
akj (t)vj,
where
uk(x, t)
satisfy the system of linear dif-
ferential equations
Ωuk
tt(x, t)vj(x)dx+Ω∇uk(x, t)∇vj(x)dx
+Ωuk
t(x, t)m(x)−2uk
t(x, t)vj(x)dx=Ωv(t)vj(x)dx
uk(x, 0) = uk
0, uk
t(x, 0) = uk
1∀j= 1.2. . . . . . k,
(3.6)
where
uk
0=
k
i=1
(u0, vi)v→u0
in
H1
0(Ω),
uk
1=
k
i=1
(u1, vi)vi→u1
in
L2(Ω).
Note that (3.6) is a system of ordinary dif-
ferential equations for
akj (t).
The local exis-
tence of solutions of the system (3.6) is guar-
anteed by the Picard-Lindelof Theorem on
functional analysis concepts, which is known
to have a local solution in an interval
[0, Tk)
with
0< Tk≤Tmax <+∞
. The extension of
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the solution to the entire interval
[0,+∞)
is
a consequence of the following estimates.
Multiplying (3.6) by
a′
kj (t)
and sum over
j
to
nd
1
2
d
dtΩuk
t(x, t)2dx+∇uk(x, t)2dx
+Ωuk
t(x, t)m(x)dx=Ωv(x, t)uk
t(x, t)dx
A simple integration on
(0, t)
yields
1
2Ωuk
t(x, t)2dx+∇uk(x, t)2dx
+t
0Ωuk
t(x, s)m(x)dxds
=1
2Ωuk
12+∇uk
02dx
+t
0Ωv(x, s)uk
t(x, s)dxds
≤1
2Ωu2
1+|∇u0|2dx
+εt
0Ωuk
t2dxds+cεT
0Ωv2dxds
≤Cε+εsup(0,tk)Ωuk
t(x, t)2dx,
∀t∈[0, tk)
(3.7)
Hence
1
2sup(0,tk)Ωuk
t(x, t)2dx
+1
2sup(0,tk)Ω∇uk(x, t)2dx
+tk
0Ωuk
t(x, s)m(x)dxds≤Cε
+εsup(0,tk)Ωuk
t(x, t)2dx
Taking
ε=1
4
, we arrive at
sup(0,tk)Ωuk
t(x, t)2dx
+sup(0,tk)Ω∇uk(x, t)2dx
+tk
0Ωuk
t(x, s)m(x)dxds≤C
Therefore, the solution can be prolonged to
[0, T )
and, besides, we have
uk
is a bounded sequence
in
L∞(0, T ), H1
0(Ω),
uk
t
is a bounded sequence in
L∞(0, T ), L2(Ω)∩Lm(.)(Ω ×(0, T )),
uk
tm(.)−2uk
t
is a bounded sequence
in
L
m(.)
m(.)−1(Ω ×(0, T )).
From DunfordPettis theorem, we can ex-
tract from
uk
a subsequence still denoted
by
(uk)
such that
uk→u
weakly
∗
in
L∞(0, T ), H1
0(Ω),
(3.8)
uk
t→ut
weakly
∗
in
L∞(0, T ), L2(Ω)
and weakly in
Lm(.)(Ω ×(0, T )),
(3.9)
uk
t
m(.)−2uk
t→ψ
weakly
in
L
m(.)
m(.)−1(Ω ×(0, T )).
(3.10)
Limits (3.8)(3.10) allow us to pass to the
limit in the approximate equation so that we
can deduce that
u∈C[0, T ], L2(Ω)
, and therefore
u(x, 0)
has a sense.
Now we show that
u∈C[0, T ], L2(Ω)
is
a solution to the system (3.3). First we
try to prove that
ψ=|ut|m(.)−2ut,
for all
v∈L∞(0, T ), L2(Ω),
in (3.6), integrate
over
(0, t)
, and make
k→ ∞
in the results,
we can derive for a.e
t∈[0, T ]
that
1
2
d
dtΩ
utϕ+Ω
(∇u.∇ϕ+ψϕ)dx=Ω
vϕdx, ∀ϕ∈H1
0(Ω).
(3.11)
For simplicity let
A(ϕ) = |ϕ|m(x)−2ϕ
and de-
ne (see [2, Proposition 2.5. ]),
Xk=t
0ΩAuk
t−A(ϕ)uk
t−ϕdt≥0,
∀ϕ∈Lm(.)(0, T ); H1
0(Ω)
So if we using (3.7) we get
Xk=t
0Ωvuk
tdxds+12Ωuk
12+∇uk
02dxds
−1
2Ωuk
t(x, t)2dx
−1
2Ω∇uk(x, t)2dx−t
0ΩAuk
tϕdxds
−t
0ΩA(ϕ)uk
t−ϕdxds
Taking
k→ ∞
we get
0≤lim sup
k
Xk≤t
0Ω
vutdxds
+1
2Ωu2
1+|∇u0|2dxds−1
2Ω
|ut(t)|2dx
−1
2Ω
|∇u(x, t)|2dx−t
0Ω
ψϕdxds
−t
0Ω
A(ϕ) (ut−ϕ)dxds.
(3.12)
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If we put
ϕ=ut
in (3.11) and integrate over
(0, T )
, we get
t
0Ωvutdxds=1
2Ω|ut(x, t)|2dxds
−1
2Ωu2
1dxds+1
2Ω|∇u(x, t)|2dx
−1
2Ω|∇u0|2dx+t
0Ωψutdxds.
(3.13)
Combine (3.12) and (3.13) gives
0≤lim sup
k
Xk≤t
0Ω
ψutdxds
−t
0Ω
ψϕdxds−t
0Ω
A(ϕ) (ut−ϕ)dxds.
That is
t
0Ω
(ψ−A(ϕ)) (ut−ϕ)dxds≥0,
∀ϕ∈Lm(.)(0, T ); H1
0(Ω).
Consequently
t
0Ω
(ψ−A(ϕ)) (ut−ϕ)dxds≥0,
∀ϕ∈Lm(.)(Ω ×(0, T )),
by density of
H1
0(Ω)
in
Lm(.)(Ω)
.
Now, let
ϕ=λw +ut, w ∈Lm(.)(Ω ×(0, T )).
Hence, we know
−λt
0Ω
(ψ−A(λw +ut)) wdxds≥0,
∀w∈Lm(.)(Ω ×(0, T )),
for
λ > 0,
we have
t
0Ω
(ψ−A(λw +ut)) wdxds≤0,
∀w∈Lm(.)(Ω ×(0, T )).
If we take
λ→0
and using the hemi-
continuity of
A
, we get
t
0Ω
(ψ−A(ut)) wdxds≤0,
∀w∈Lm(.)(Ω ×(0, T ))
(3.14)
Similarly we nd for
λ < 0
t
0Ω
(ψ−A(ut)) wdxds≥0,
∀w∈Lm(.)(Ω ×(0, T ))
(3.15)
From (3.14) and (3.15), for
k→+∞
we get
ψ=A(ut)
and
uk
t
m(.)−2uk
t→ |ut|m(.)−2ut
weakly in
L
m(.)
m(.)−1(Ω ×(0, T )).
Therefore, from the above result and (3.8)
(3.10), we deduce that there is
u∈
C[0, T ], L2(Ω)
that satises the following
equation
utt −∆u+|ut|m(.)−2ut−v, ϕ= 0
for all
ϕ∈H1
0(Ω)
and the initial conditions
u(0) = u0, ut(0) = u1,
which completes the existence proof in
Lemma (3.2).
The following lemma crucial for the proof of
our main result
Lemma 3.3 For a.e
x∈Ω
and
p(.)
that satisfy
(3.1), the function
F(s) = |s|p(x)−2s(ln |s|)
is dif-
ferentiable and
|F′(s)| ≤ (p2−1) |s|p(x)−2|ln |s||
+|s|p(x)−2
≤2(p2−1)
e((p1−2)−k1)|s|k1+2(p2−1)
e(k2−(p2−2)) |s|k2
+|s|p1−2+|s|p2−2, s = 0,
(3.16)
where
p1−2≤p2−2< k2≤2
n−2,
for
n≥3,
0< p1−2≤p2−2< k2,
for
n= 1,2,
(3.17)
and
0< k1< p1−2≤p2−2≤2
n−2,
for
n≥3,
0< k1< p1−2≤p2−2,
for
n= 1,2.
(3.18)
proof. Obviously we have for
k= 0
since
ln ζ≤
1
ek ζk
for every
ζ≥1
and
ln ζ≥ − 1
ek ζ−k
,
ζ < 1
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then for every
k > 0
|F′(s)|=(p(x)−1) |s|p(x)−2(ln |s|) + |s|p(x)−2
≤p2−1
ek |s|p1+k−2+|s|p2+k−2
+p2−1
ek |s|p1−k−2+|s|p2−k−2
+|s|p1−2+|s|p2−2
≤2p2−1
ek |s|p2+k−2+ 2p2−1
ek |s|p1−k−2
+|s|p1−2+|s|p2−2
=2(p2−1)
e((p1−2)−k1)|s|k1+2(p2−1)
e(k2−(p2−2)) |s|k2
+|s|p1−2+|s|p2−2,
with
k1,
and
k2
are in (3.17)-(3.18).
Proof of Theorem (3.1).
1. Existence. Let
v∈L∞(0, T ), H1
0(Ω)\{0}
.
Then
|v|p(.)−2vln |v|
2
2
≤
Ω
|v|2p1−2(ln |v|)2dx
+
Ω
|v|2p2−2(ln |v|)2dx
=
{x∈Ω:|v(t)|<1}
|v|2p1−2(ln |v|)2dx
+
{x∈Ω:|v(t)|<1}
|v|2p2−2(ln |v|)2dx
+
{x∈Ω:|v(t)|≥1}
|v|2p1−2(ln |v|)2dx
+
{x∈Ω:|v(t)|≥1}
|v|2p2−2(ln |v|)2dx.
Choosing
σ
such that
2≤2 (p1−1) ≤2 (p2−1) < σ ≤2n
n−2,
for
n≥3,
2≤2 (p1−1) ≤2 (p2−1) < σ
,
for
n= 1,2,
and by
ln ζ≤1
es ζs
for any
ζ≥1, s > 0,
we
have
{x∈Ω:|v(t)|<1}
|v|2p1−2(ln |v|)2dx
+
{x∈Ω:|v(t)|≥1}
|v|2p1−2(ln |v|)2dx
≤|Ω|
e2+1
e22
σ+2−2p12
Ω
|v|σdx
≤|Ω|
e2+1
e2Cσ
s2
σ+2−2p12∥∇v∥σ
2<∞,
(3.19)
similarly
{x∈Ω:|v(t)|<1}
|v|2p2−2(ln |v|)2dx
+
{x∈Ω:|v(t)|≥1}
|v|2p2−2(ln |v|)2dx
≤|Ω|
e2+1
e2Cσ
s2
σ+2−2p22∥∇v∥σ
2<∞,
(3.20)
where
Cs
is the optimal constant of Sobolev
embedding
H1
0(Ω) →Lσ(Ω)
. So, in this case.
|v|p(.)−2vln |v| ∈ L∞(0, T ), L2(Ω)
⊂L2(Ω ×(0, T ))
Thus for every
v∈L∞(0, T ), H1
0(Ω)\{0}
,
there is a unique
u
such that
u∈L∞(0, T ), H1
0(Ω),
ut∈L∞(0, T ), L2(Ω)∩Lm(.)(Ω ×(0, T )),
(3.21)
solve the nonlinear problem
utt −∆u+|ut|m(.)−2ut=|v|p(.)−2vln |v|,
in
Ω×(0, T )
u(x, t) = 0,
on
∂Ω×(0, T )
u(x, 0) = u0(x), ut(x, 0) = u1(x),
in
Ω.
(3.22)
Let
R0
be a positive real number such that
R0=2|u1|2+|∇u0|2,
for a suciently small time
T > 0
we dene
the space
BT(R0)
by
BT(R0) =
v(t)∈L∞(0, T ), H1
0(Ω),
vt(t)∈L∞(0, T ), L2(Ω)),
|v′(t)|2+|∇v(t)|2≤R2
0
on
[0, T ],
v(0) = v0, v′(0) = u1.
We introduce the metric
d
on the space
BT(R0)
d(u, v) = sup
0≤t≤T|ut(t)−vt(t)|2+|∇u(t)− ∇v(t)|2
for
u, v ∈BT(R0).
Obviously the space
BT(R0)
is the complete
metric space. Let
v∈BT(R0)
. Then
|∇v(t)| ≤ R0
,
|v′(t)| ≤ R0
for all
t∈[0, T ]
.
Dene the mapping
Φ
Φ (v) = u,
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where
u
satises (3.21) and (3.22). Then we
have
Φ (v) = u∈BT(R0)
for
v∈BT(R0),
(3.23)
Φ : BT(R0)→BT(R0)
is a contractive mapping
.
(3.24)
For showing (3.23), multiply (3.22) by
ut
1
2
d
dtΩ
u2
tdx+Ω
|∇u|2dx+Ω
|ut|m(x)dx
=Ω
|v|p(x)−2v(ln |v|)utdx
(3.25)
From Young's inequality, (3.19) and (3.20) for
all
ε > 0
the following estimates hold:
Ωvp(x)−2v(ln |v|)utdx
≤Ωu2
tdx+1
4Ω|v|2p(x)−2(ln |v|)2dx
≤Ωu2
tdx+1
4Ω|v|2p2−2(ln |v|)2dx
+Ω|v|2p1−2(ln |v|)2dx
≤Ωu2
tdx
+1
42|Ω|
e2+1
e2Cσ
s2
σ+2−2p12∥∇v∥σ
2
+1
e2Cσ
s2
σ+2−2p22∥∇v∥σ
2.
So (3.25) becomes
d
dt∥ut∥2
2+∥∇u∥2
2
≤1
e2|Ω|+2
e2Cσ
s2
σ+ 2 −2p22
Rσ
0+∥ut∥2
2.
Thus, we have
ψv(u) (t)≤ψv(u) (0)
+t
01
e2|Ω|+2
e2Cσ
s2
σ+2−2p22Rσ
0+ψv(u) (t)ds
≤1
2R2
0+β0t
0(1 + ψv(u) (t)) ds,
where
β0=max 1
e2|Ω|+2
e2Cσ
s2
σ+2−2p22Rσ
0,1
and
ψv(u) (t) = ∥ut∥2
2+∥∇u∥2
2
.
By the Gronwall inequality and simple calcu-
lations we have
∥ut∥2
2+∥∇u∥2
2≤1
2R2
0+β0T0eβ0T0< R2
0,0≤t≤T0,
for suciently small
0< T0≤T
. Thus (3.23)
is fullled.
Next we show (3.24). Let
w=u1−u2
,
where
u1= Φ (v1), u2= Φ (v2)
with
v1,
v2∈BT(R0)
. Then we have
(wtt, v)−(∆w, v)
+|u1t(t)|m(x)−1u1t(t)− |u2t(t)|m(x)−1u2t(t), v
=|v1|p(x)−2v1ln |v1|−|v2|p(x)−2v2ln |v2|, v,
in
L20, T1;H−1(Ω).
(3.26)
Now, set
βv(w) (t) = |wt(t)|2+|∇w(t)|2.
Multiplying (3.26) by
wt
and using (3.5) we
get
1
2
d
dt|wt(t)|2+|∇w(t)|2
≤|v1|p(x)−2v1ln |v1|−|v2|p(x)−2v2ln |v2|, wt.
Now we estimate
I=Ω
|F(v1(s)) −F(v2(s))| |wt|dx
=ΩF′(ξ)∥v∥wtdx,
where
v=v1−v2
and
ξ=av1+(1−a)v2,0≤a≤1.
By Holders, Youngs inequalities and Lemma
(3.3) we have
I2≤Ωw2
tdxΩ|F′(ξ)|2|v|2dx
≤4Ωw2
tdx2(p2−1)
e((p1−2)−k1)2
Ω|αv1+ (1 −α)v2|2k1|v|2dx
+2(p2−1)
e(k2−(p2−2)) 2Ω|αv1+ (1 −α)v2|2k2|v|2dx
+ 4 Ω|αv1+ (1 −α)v2|2(p1−2)|v|2dx
+4 Ω|αv1+ (1 −α)v2|2(p2−2)|v|2dx
≤c∗Ωw2
tdxΩ|v|2n
n−2dxn−2
n
Ω|αv1+ (1 −α)v2|k1n2
ndx
+Ω|αv1+ (1 −α)v2|nk22
ndx
+Ω|αv1+ (1 −α)v2|2(p1−2)dx
+Ω|αv1+ (1 −α)v2|2(p2−2)dx,
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If we recall (3.1) and (3.12) we come to
I2≤c∗csΩw2
tdx∥∇v∥2
2∥∇v1∥2k1
2+∥∇v1∥2k2
2
+∥∇v2∥2k1
2+∥∇v2∥2k2
2
+∥∇v1∥2(p1−2)
2+∥∇v1∥2(p2−2)
2
+∥∇v2∥2(p1−2)
2+∥∇v2∥2(p2−2)
2
≤8c∗csR2(k2+p2−2)
0d(v1, v2)βv1(w) (t),
where
c∗=c(e, p1, p2, k1, k2)
and
cs
the
Sobolev embedding
H1
0(Ω) →L2n
n−2(Ω)
.
If we combine, it follows
d
dtβv(w) (t)≤ξd(v1, v2)1
2βv(w) (t)1
2.
Since
βv(w) (0) = 0
, by the Gronwall lemma
d(u1, u2)≤ξ2T
4d(v1, v2)eT.
Choose a
0< T1≤T
small enough to satisfy
ξ2
4T1eT1<1.
Thus, according to Banach's contraction
mapping theorem, there exists a xed point
u= Φ(u)∈BT1(R0)
, which is a locally weak
solution in time to (1.1).
2. Uniqueness. Suppose we have two solutions
u
and
v
and set
w(s) = u1(s)−u2(s), s ∈[0, t]
0, s ∈[t, T ],
then
w∈L20, T ;W1,p(.)
0(Ω),
wt∈L20, T ;H1
0(Ω)
and
w
fullled
1
2Ω
w2
tdx+1
2Ω
|∇w|2dx
≤t
0Ω
(F(u)−F(v)) wtdx
Consequently, the uniqueness results from the
local Lipschitz continuity of
F:R∗→R
and
the embedding
H1
0(Ω) →L2(Ω)
. This com-
pletes the proof of the theorem.
4 Blow-up of weak solutions
Finally, we give the sucient conditions for
m(.)
for inating weak solutions of the problem (1.1)
in nite time if
2< m1≤m(x)≤m2
< p1≤p(x)≤p2<2n−1
n−2, n ≥3,
(4.1)
holds, and
E(0) <0
, where
E(t) = 1
2Ω|ut(x, t)|2+|∇u(x, t)|2dx
−Ω
1
p(x)|u(x, t)|p(x)ln(|u(x, t)|)dx
+Ω
1
p2(x)|u(x, t)|p(x)dx.
(4.2)
For our purpose we need to the following lemma
showing the decrease in energy
E.
Lemma 4.1 The energy associated with the prob-
lem (1.1) given by (4.2) satises the
dE(t)
dt=−Ω
|ut|m(x)dx≤0,
(4.3)
and the inequality
E(t)≤E(0)
holds, where
E(0) = 1
2Ω|u1|2+|∇u0|2dx
−Ω
1
p(x)|u0|p(x)ln(|u0|)dx
+Ω
1
p2(x)|u0|p(x)dx.
(4.4)
Let
H(t) = −E(t)
for
t≥0,
(4.5)
since
E(t)
is absolutely continuous, hence
H′(t)≥
0
and
0< H(0) ≤H(t)≤Ω
1
p(x)|u(x, t)|p(x)ln(|u|)dx.
Lemma 4.2 Let the assumptions (2.1) be fullled
and let
u
be the solution of (1.1). Then,
Ω
|u|p(x)dx≥Ω2
|u|p1dx:= ∥u∥p1
p1,Ω2,
(4.6)
where
Ω2={x∈Ω/|u(x, t)| ≥ 1}.
proof. Let
Ω1={x∈Ω/|u(x, t)|<1},
so, we have
Ω|u|p(x)dx=Ω2|u|p(x)dx+Ω1|u|p(x)dx
≥Ω2|u|p1dx+Ω1|u|p2dx
≥Ω2|u|p1dx:= ∥u∥p1
p1,Ω2.
Thus (4.6).
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Lemma 4.3 Under the assumptions of Theorem
(3.1), the function
H(t)
presented above yields
the following estimates:
0< H(0) ≤H(t)≤|Ω|
p1e+Bs
(s−p2)ep1
∥∇u∥s
2, t ≥0,
(4.7)
where
s
is chosen suciently small such that
p1≤p2< s ≤2n
n−2,
for
n≥3,
(4.8)
p1≤p2< s < ∞
for
n= 1,2,
and
Bs
is a positive constant of embedding
H1
0(Ω)
in
Ls(Ω)
such that
∥u∥s≤Bs∥∇u∥2,∀u∈H1
0(Ω).
(4.9)
proof. By Lemma (4.1),
H(t)
is nondecreasing in
t
. Thus
H(t)≥H(0) = −E(0) >0, t ≥0.
(4.10)
Combining (4.2), (4.3), (4.5) and using the fact
that
ln ζ≤1
eσ ζσ
for any
σ > 0
we have
0< H (t)<1
p1Ω|u(x, t)|p(x)ln(|u(x, t)|)dx
=1
p1{x∈Ω:|u(x)|<1}|u(x, t)|p(x)−1(|u(x, t)|
(ln(|u(x, t)|))) dx
+1
p1{x∈Ω:|u(x)|≥1}|u(x, t)|p(x)ln(|u(x, t)|)dx
≤|Ω|
p1e+1
σep1
{x∈Ω:|u(x)|≥1}
|u|p2+σdx
≤|Ω|
p1e+1
σep1∥u∥p2+σ
p2+σ
≤|Ω|
p1e+Bs
(s−p2)ep1∥∇u∥s
2,
(4.11)
and (4.7) follows.
Theorem 4.4 Suppose the conditions of Theorem
(3.1) are satised. Moreover, let (4.1) hold as well
as
E(0) <0
. Then the solution of problem (1.1)
given by Theorem (3.1) blows up in nite time.
proof. for each
t
in
[0, T )
let dene
L(t) := H1−α(t) + εΩ
u(x, t)ut(x, t)dx,
(4.12)
with
ε > 0
is small enough to be chosen later and
α
such that
0< α ≤min p1−2
2p1
,p1−m2
p1(m2−1),
2 (p1−m1)
s(m1−1) p1
,2 (p1−m1)
s(m2−1) p1.
(4.13)
A straightforward derivation of (4.12) using Eq.
(1.1), we obtain
L′(t) = (1 −α)H−α(t)H′(t)
+εΩu2
t− |∇u|2
+εΩ
|u|p(x)(ln |u|)−εΩ
|ut|m(x)−2uut
(4.14)
On the right-hand side of (4.14) by adding and
subtracting
ε(1−η)p1H(t)
with
0< η < p1−2
p1,
we
obtain
L′(t) = (1 −α)H−α(t)H′(t) + ε(1 −η)p1H(t)
+ηΩ|u|p(x)(ln |u|)dx
+ε(1−η)p1
2+ 1∥ut∥2
2+ε(1−η)p1
2−1∥∇u∥2
2
−εΩuut|ut|m(x)−2dx,
(4.15)
Due to the fact that (4.6), taking into account
1
p2
2Ω
|u(x, t)|p(x)dx < 1
p1Ω
|u|p(x)(ln |u|)dx,
(4.15) result in
L′(t)≥(1 −α)H−α(t)H′(t)−εΩ|ut|m(x)−2uutdx
+εβ H(t) + ∥ut∥2
2+∥∇u∥2
2+Ω|u(x, t)|p(x)dx
≥(1 −α)H−α(t)H′(t)−εΩ|ut|m(x)−2uutdx
+εβ H(t) + ∥ut∥2
2+∥∇u∥2
2+∥u∥p1
p1,Ω2,
(4.16)
where
β=min (1 −η)p1,p1
p2
2
η, (1 −η)p1
2+ 1
,(1 −η)p1
2−1>0.
Now, using Young's inequality, we estimate the
last term in (4.14) in the manner shown below
Ω
|ut|m(x)−1|u|dx≤1
m1Ω
ζm(x)|u|m(x)dx
+m2−1
m2Ω
ζ−m(x)
m(x)−1|ut|m(x)dx, ∀ζ > 0.
(4.17)
Consequently, by taking
δ
such that
ζ−m(x)
m(x)−1=kH−α(t), k > 0,
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By putting it in (4.17) with
k
large enough to be
determined later, we obtain
Ω
|ut|m(x)−1|u|dx≤
1
m1Ω
k1−m(x)|u|m(x)Hα(m(x)−1)(t)dx
+(m2−1) k
m2
H−α(t)H′(t).
(4.18)
The result of joining (4.16) with (4.18)
L′(t)≥(1 −α)−εm2−1
m2kH−α(t)H′(t)
+εβ H(t) + ∥ut∥2
2+∥∇u∥2
2+∥u(t)∥p1
p1
−εk1−m1
m1Hα(m2−1)(t)Ω|u|m(x)dx.
(4.19)
Applying lemma (4.3) we have
Hα(m2−1)(t)Ω|u(t)|m(x)dx
≤C2α(m2−1)−1|Ω|
p1eα(m2−1)
+2α(m2−1)−11
(s−p2)ep1∥∇u∥sα(m2−1)
2
∥u∥m1
p1,Ω2+∥u∥m2
p1,Ω2
≤2α(m2−1)−1C|Ω|
p1eα(m2−1)
×∥u∥p1
p1,Ω2m1
p1+∥u∥p1
p1,Ω2m2
p1
+2α(m2−1)−1C1
(s−p2)ep1∥∇u∥sα(m2−1)
2
×∥u∥m1
p1,Ω2+∥u∥m2
p1,Ω2.
(4.20)
We are to analyze the terms on the right-hand
side of (4.20). By using Young's inequality, we
have
∥∇u∥sα(m2−1)
2∥u∥m1
p1,Ω2≤m1
p1∥u(t)∥p1
p1,Ω2
+Cp1−m1
p1∥∇u∥
sα(m2−1)p1
p1−m1
2
=m1
p1∥u(t)∥p1
p1,Ω2
+Cp1−m1
p1∥∇u∥2
2sα(m2−1)p1
2(p1−m1),
similarly
∥∇u∥sα(m2−1)
2∥u(t)∥m2
p1,Ω2≤m2
p1
∥u(t)∥p1
p1,Ω2
+Cp1−m2
p1∥∇u∥2
2sα(m2−1)p1
2(p1−m2).
Using the following well-known algebraic inequal-
ity:
zτ≤z+1 ≤1 + 1
d(z+d),∀z≥0,0< τ ≤1, d ≥0,
(4.21)
with
z=∥u(t)∥p1
p1,Ω2, a = 1 + 1
H(0) , d =H(0)
and
τ=m1
p1τ=m2
p1
, respectively, then the condi-
tion (4.1) implies that
0< τ ≤1
and therefore
∥u(t)∥p1
p1,Ω2m1
p1+∥u(t)∥p1
p1,Ω2m2
p1
≤2a∥u(t)∥p1
p1,Ω2+H(0)≤2a∥u(t)∥p1
p1,Ω2+H(t),
similarly, with
z=∥∇u∥2
2, b = 1+ 1
H(0) , d =H(0)
and
τ=sα(m2−1)p1
2(p1−m1)
, then the condition (4.13) im-
plies that
0< τ ≤1
and therefore
∥∇u∥2
2sα(m2−1)p1
2(p1−m1)
≤b∥∇u∥2
2+H(0)≤b∥∇u∥2
2+H(t),
also, with
z=∥∇u∥2
2, h = 1 + 1
H(0) , d =H(0)
and
τ=sα(m2−1)p1
2(p1−m2),
∥∇u∥2
2sα(m2−1)p1
2(p1−m2)≤h∥∇u∥2
2+H(t),
therefore, (4.20) leads to
Hα(m2−1)(t)Ω
|u(t)|m(x)dx
≤C∥u(t)∥p1
p1,Ω2+H(t) + ∥∇u∥2
2,∀t∈[0, T ].
(4.22)
where
C
to indicate a generic positive constant
depending on
(Ω, e, h, p1,2, m1,2)
only. Combining
(4.19) and (4.22) yields
L′(t)≥(1 −α)−εm2−1
m2kH−α(t)H′(t)
+εβ−k1−m1
m1C
×H(t) + ∥ut∥2
2+∥∇u∥2
2+∥u(t)∥p1
p1,Ω2.
(4.23)
At this point we pick
γ=β−k1−m1
m1C > 0
, (it is
the case when
k > βm1
C1
1−m1
).
Once
k
is xed we pick
ε > 0
sucient small
so that
(1 −α)−εm2−1
m2k≥0
and
L(0) = H1−α(0) + εΩ
u0(x)u1(x)dx > 0.
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Hence (4.23) takes the form
L′(t)≥γH(t) + ∥ut∥2
2+∥∇u∥2
2+∥u(t)∥p1
p1,Ω2.
(4.24)
Therefore, we have
L(t)≥L(0) >0
, for all
t≥0
On the other hand from (4.12),
L1
1−α(t)≤21/(1−α)H(t) + Ω
uut(x, t)dx
1
1−α,
(4.25)
By applying Holder's inequality we see that
Ω
uut(x, t)dx≤C∥u∥p1∥ut∥2≤2C∥u∥p1,Ω2∥ut∥2.
Again, algebraic inequality (4.21), with
z=
∥u∥p1
p1,Ω2, h = 1 + 1
H(0) , d =H(0)
and
0< τ =
2
(1−2α)p1≤1
(see (4.13)), gives
∥u∥p1
p1,Ω22
(1−2α)p1≤C∥u∥p1
p1,Ω2+H(t),
Thus, Young's inequality gives
Ωuut(x, t)dx1/(1−α)
≤C∥u∥
2(1−α)
1−2α
p1,Ω2+∥ut∥2(1−α)
21/(1−α)
≤C∥u∥p1
p1,Ω22
(1−2α)p1+∥ut∥2
2
≤C∥u∥p1
p1,Ω2+H(t) + ∥ut∥2
2,
for all
t≥0,
joining it with (4.24) and (4.25) yields
L′(t)≥δL 1
1−α(t),
for all
t≥0,
(4.26)
where
δ
is a positive constant depending on
(ε, γ, C)
. With a simple integration of (4.26) over
(0, t)
we infer that
Lα
1−α(t)≥1
Lα
1−α(0) −α
1−αδt.
(4.27)
Consequently,
L(t)
blows up in a nite time
T
T≤1−α
δαL α
1−α(0).
Data Availability
No data is used in the manuscript.
Disclosure statement
No potential conict of interest was reported by
the author
Acknowledgments
The author expressed their thanks to the anony-
mous referees/editors for their comments and
valuable suggestions.
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