P-BIHARMONIC PSEUDO-PARABOLIC EQUATION WITH
LOGARITHMIC NON LINEARITY
Sushmitha Jayachandran
Research Scholar, Department of Mathematics,Central University of Kerala, Kerala - 671 320, India.
E-mail:sushmithaakhilesh@gmail.com
ORCID:
Gnanavel Soundararajan
Assistant Professor, Department of Mathematics,Central University of Kerala, Kerala - 671 320, India.
E-mail:gnanavel.math.bu@gmail.com
ORCID:
Reception: 20/09/2022 Acceptance: 05/10/2022 Publication: 29/12/2022
Suggested citation:
Sushmitha Jayachandran and Gnanavel Soundararajan (2022). p-Biharmonic Pseudo-Parabolic Equation with Logarithmic
Non linearity. 3C TIC. Cuadernos de desarrollo aplicados a las TIC,11 (2), 108-122. https://doi.org/10.17993/3ctic.2022.112.108-
122
https://doi.org/10.17993/3ctic.2022.112.108-122
ABSTRACT
This paper deals with the existence of solutions of a p-biharmonic pseudo parabolic partial differential
equation with logarithmic nonlinearity in a bounded domain. We prove the global existence of the weak
solutions using the Faedo-Galerkin method and applying the concavity approach, that the solutions blow
up at a finite time. Further, we provide a maximal limit for the blow-up time.
KEYWORDS
p-Biharmonic, pseudo-parabolic, global existence, blow up
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
108
P-BIHARMONIC PSEUDO-PARABOLIC EQUATION WITH
LOGARITHMIC NON LINEARITY
Sushmitha Jayachandran
Research Scholar, Department of Mathematics,Central University of Kerala, Kerala - 671 320, India.
E-mail:sushmithaakhilesh@gmail.com
ORCID:
Gnanavel Soundararajan
Assistant Professor, Department of Mathematics,Central University of Kerala, Kerala - 671 320, India.
E-mail:gnanavel.math.bu@gmail.com
ORCID:
Reception: 20/09/2022 Acceptance: 05/10/2022 Publication: 29/12/2022
Suggested citation:
Sushmitha Jayachandran and Gnanavel Soundararajan (2022). p-Biharmonic Pseudo-Parabolic Equation with Logarithmic
Non linearity. 3C TIC. Cuadernos de desarrollo aplicados a las TIC,11 (2), 108-122. https://doi.org/10.17993/3ctic.2022.112.108-
122
https://doi.org/10.17993/3ctic.2022.112.108-122
ABSTRACT
This paper deals with the existence of solutions of a p-biharmonic pseudo parabolic partial differential
equation with logarithmic nonlinearity in a bounded domain. We prove the global existence of the weak
solutions using the Faedo-Galerkin method and applying the concavity approach, that the solutions blow
up at a finite time. Further, we provide a maximal limit for the blow-up time.
KEYWORDS
p-Biharmonic, pseudo-parabolic, global existence, blow up
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
109
1 INTRODUCTION
Here we examine the following problem for a p-biharmonic pseudo-parabolic equation with logarithmic
nonlinearity.
ut−∆ut+ ∆(|∆u|p−2∆u)−div(|∇u|q−2∇u)=−div(|∇u|q−2∇ulog |∇u|)if (x, t)∈Ω×(0,T),
u=∂u
∂ν =0 if (x, t)∈∂Ω×(0,T),
u(x, 0) = u0(x)if x∈Ω.
(1)
where Ω
⊂RN
(
N≥
1) represents a bounded domain whose boundary
∂
Ωis smooth enough,
T∈
(0
,∞
),
ν
indicates the normal vector on
∂
Ωpointing outward,
u0∈W2,p
0
(Ω)
\{
0
}
and the condition
2<p<q<p(1 + 2
N+2 )holds for pand q.
Pseudo-parabolic equations address several significant physical processes, like the evolution of two
components of intergalactic material, the leakage of homogeneous fluids through a rock surface, the
biomathematical modeling of a bacterial film, some thin film problems, the straight transmission
of nonlinear, dispersive, long waves, the heat transfer containing two temperatures, a grouping of
populations, etc. Shawalter and Ting [18], [22] first examined the pseudo parabolic equations in 1969.
After their precursory results, there are many papers studied the nonlinear pseudo-parabolic equations,
like semilinear pseudo-parabolic equations, quasilinear pseudo-parabolic equations, and even singular
and degenerate pseudo-parabolic equations (see [1], [28], [4], [6], [15], [24], [25]). A pseudo-parabolic
equation with p-Laplacian ∆
pu
=
div
(
|∇u|p−2∇u
)and logarithmic nonlinearity were studied by Nahn,
and Truong [13] in 2017. Considering the equation,
ut−∆ut−∆pu=|u|p−2ulog |u|
and by using the potential well method proposed by Sattinger [17] and a logarithmic Sobolev inequa-
lity, they proved the existence or nonexistence of global weak solutions. Additionally, they provided
requirements for both the large time decay of weak global solutions and the finite time blow-up of weak
solutions. Later, many authors [26], [27], [23] considered pseudo-parabolic equations with logarithmic
nonlinearity and established results for local and global existence, uniqueness, decay estimate and
asymptotic behaviour of solutions, blow-up results. Logarithmic nonlinearities in parabolic and pseudo-
parabolic equations were studied by Lakshmipriya et.al [11], [10] and other researchers [29], [9], [5] and
they proved the existence of weak solutions and their blow up in finite time. Lower bound of Blow-up
time to a fourth order parabolic equation modelling epitaxial thin film growth
Recently, higher-order equations have gained much importance in studies. Lower bound of Blow-up
time to a fourth order parabolic equation modelling epitaxial thin film growth studied by Liu et.al [3].
The p-biharmonic equation
ut+ ∆(|∆u|p−2∆u)+λ|u|p−2u=0
were studied by Liu and Guo [14], and by using the discrete-time method and uniform estimates, they
established the existence and uniqueness of weak solutions. Hao and Zhou [7] obtained results for blow
up, extinction and non-extinction of solutions for the equation
ut+ ∆(|∆u|p−2∆u)=|u|q−1
|Ω|Ω|u|dx.
Wang and Liu [8] studied the p-biharmonic parabolic equation with logarithmic nonlinearity,
ut+ ∆(|∆u|p−2∆u)=|u|q−1ulog |u|
for 2
<p<q<p
(1 +
4
n
)and proved the global existence, blow up, extinction and no extinction of
solutions. Then Liu and Li [2] studied,
ut+ ∆(|∆u|p−2∆u)=λ|u|q−1ulog |u|.
Based on the difference and variation methods, they showed the existence of weak solutions and observed
large-time behaviour and the transmission of solution perturbations for
λ>
0
,p > q > p
2
+1
,p > n
2
.
https://doi.org/10.17993/3ctic.2022.112.108-122
Comert and Piskin [?] studied a p-biharmonic pseudo-parabolic equation with logarithmic nonlinearity
and used the potential well method and logarithmic Sobolev inequality obtained the existence of the
unique global weak solution. In addition, they also exhibited polynomial decay of solutions. Motivated
by these works, we have formulated our problem (1) for a p-biharmonic pseudo-parabolic equation
with logarithmic nonlinearity and studied their existence and non-existence. The problem (1) for the
case
p
=2is already investigated and proved the existence, uniqueness and blow up of solutions
(see [19], [20], [21]).
The rest of this paper is arranged to the two sections below. The preliminary notations, definitions,
and results we need to support our main findings are described in Section 2. Section 3 contains the
major findings of this paper explained in five theorems.
2 PRELIMINARIES
In this section, we provide some fundamental ideas and facts that are necessary for us to explain our
findings. In this article, we follow the notations listed below throughout.
∥.∥r
denotes the
Lr
(Ω) norm
for 1
≤r≤∞
,
∥.∥H1
0
denotes the norm in
H1
0
(Ω),(
., .
)
1
denotes the
H1
0
(Ω)-inner product.=,
r′
denotes
the Holder conjugate exponent of r>1(that is, r′=r
r−1).
We define the energy functional Jand the Nehari functional Ias follows:
I,J :W2,p
0(Ω) →Rby
J(u)=1
p∥∆u∥p
p+q+1
q2∥∇u∥q
q−1
qΩ|∇u|qlog |∇u|dx (2)
I(u)=∥∆u∥p
p+∥∇u∥q
q−Ω|∇u|qlog |∇u|dx (3)
Then we have,
J(u)=1
qI(u)+1
p−1
q∥∆u∥p
p+1
q2∥∇u∥q
q(4)
We introduce the Nehari manifold as
N={u∈W2,p
0(Ω)\{0}:I(u)=0}
also define the potential well as
W={u∈W2,p
0(Ω)\{0}:J(u)< d, I(u)>0}
where d= infu∈N J(u)is referred to as the depth of the potential well.
Definition 1. A function
u
=
u
(
x, t
)is considered to be a weak solution of problem (1) if
u∈
L∞(0,T;W2,p
0(Ω)),ut∈L2(0,T;H1
0(Ω)) and validates
(ut,ϕ)+(∇ut,∇ϕ)+(|∆u|p−2∆u, ∆ϕ)+(|∇u|q−2∇u, ∇ϕ)=(|∇u|q−2∇ulog |∇u|,∇ϕ)(5)
for all
ϕ∈W2,p
0
(Ω) and a.e 0
≤t≤T
along with
u
(
x,
0) =
u0
(
x
)in
W2,p
0
(Ω)
\{
0
}
. Furthermore, it also
agrees the energy inequality
t
0∥uτ∥2
H1
0dτ +J(u)≤J(u0),0<t≤T. (6)
Lemma 1. [12] Let ρbe a positive number. Then we have the following inequalities:
xplog x≤(eρ)−1for all x≥1
and
|xplog x|≤(ep)−1for all 0<x<1.
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
110
1 INTRODUCTION
Here we examine the following problem for a p-biharmonic pseudo-parabolic equation with logarithmic
nonlinearity.
ut−∆ut+ ∆(|∆u|p−2∆u)−div(|∇u|q−2∇u)=−div(|∇u|q−2∇ulog |∇u|)if (x, t)∈Ω×(0,T),
u=∂u
∂ν =0 if (x, t)∈∂Ω×(0,T),
u(x, 0) = u0(x)if x∈Ω.
(1)
where Ω
⊂RN
(
N≥
1) represents a bounded domain whose boundary
∂
Ωis smooth enough,
T∈
(0
,∞
),
ν
indicates the normal vector on
∂
Ωpointing outward,
u0∈W2,p
0
(Ω)
\{
0
}
and the condition
2<p<q<p(1 + 2
N+2 )holds for pand q.
Pseudo-parabolic equations address several significant physical processes, like the evolution of two
components of intergalactic material, the leakage of homogeneous fluids through a rock surface, the
biomathematical modeling of a bacterial film, some thin film problems, the straight transmission
of nonlinear, dispersive, long waves, the heat transfer containing two temperatures, a grouping of
populations, etc. Shawalter and Ting [18], [22] first examined the pseudo parabolic equations in 1969.
After their precursory results, there are many papers studied the nonlinear pseudo-parabolic equations,
like semilinear pseudo-parabolic equations, quasilinear pseudo-parabolic equations, and even singular
and degenerate pseudo-parabolic equations (see [1], [28], [4], [6], [15], [24], [25]). A pseudo-parabolic
equation with p-Laplacian ∆
pu
=
div
(
|∇u|p−2∇u
)and logarithmic nonlinearity were studied by Nahn,
and Truong [13] in 2017. Considering the equation,
ut−∆ut−∆pu=|u|p−2ulog |u|
and by using the potential well method proposed by Sattinger [17] and a logarithmic Sobolev inequa-
lity, they proved the existence or nonexistence of global weak solutions. Additionally, they provided
requirements for both the large time decay of weak global solutions and the finite time blow-up of weak
solutions. Later, many authors [26], [27], [23] considered pseudo-parabolic equations with logarithmic
nonlinearity and established results for local and global existence, uniqueness, decay estimate and
asymptotic behaviour of solutions, blow-up results. Logarithmic nonlinearities in parabolic and pseudo-
parabolic equations were studied by Lakshmipriya et.al [11], [10] and other researchers [29], [9], [5] and
they proved the existence of weak solutions and their blow up in finite time. Lower bound of Blow-up
time to a fourth order parabolic equation modelling epitaxial thin film growth
Recently, higher-order equations have gained much importance in studies. Lower bound of Blow-up
time to a fourth order parabolic equation modelling epitaxial thin film growth studied by Liu et.al [3].
The p-biharmonic equation
ut+ ∆(|∆u|p−2∆u)+λ|u|p−2u=0
were studied by Liu and Guo [14], and by using the discrete-time method and uniform estimates, they
established the existence and uniqueness of weak solutions. Hao and Zhou [7] obtained results for blow
up, extinction and non-extinction of solutions for the equation
ut+ ∆(|∆u|p−2∆u)=|u|q−1
|Ω|Ω|u|dx.
Wang and Liu [8] studied the p-biharmonic parabolic equation with logarithmic nonlinearity,
ut+ ∆(|∆u|p−2∆u)=|u|q−1ulog |u|
for 2
<p<q<p
(1 +
4
n
)and proved the global existence, blow up, extinction and no extinction of
solutions. Then Liu and Li [2] studied,
ut+ ∆(|∆u|p−2∆u)=λ|u|q−1ulog |u|.
Based on the difference and variation methods, they showed the existence of weak solutions and observed
large-time behaviour and the transmission of solution perturbations for
λ>
0
,p > q > p
2
+1
,p > n
2
.
https://doi.org/10.17993/3ctic.2022.112.108-122
Comert and Piskin [?] studied a p-biharmonic pseudo-parabolic equation with logarithmic nonlinearity
and used the potential well method and logarithmic Sobolev inequality obtained the existence of the
unique global weak solution. In addition, they also exhibited polynomial decay of solutions. Motivated
by these works, we have formulated our problem (1) for a p-biharmonic pseudo-parabolic equation
with logarithmic nonlinearity and studied their existence and non-existence. The problem (1) for the
case
p
=2is already investigated and proved the existence, uniqueness and blow up of solutions
(see [19], [20], [21]).
The rest of this paper is arranged to the two sections below. The preliminary notations, definitions,
and results we need to support our main findings are described in Section 2. Section 3 contains the
major findings of this paper explained in five theorems.
2 PRELIMINARIES
In this section, we provide some fundamental ideas and facts that are necessary for us to explain our
findings. In this article, we follow the notations listed below throughout.
∥.∥r
denotes the
Lr
(Ω) norm
for 1
≤r≤∞
,
∥.∥H1
0
denotes the norm in
H1
0
(Ω),(
., .
)
1
denotes the
H1
0
(Ω)-inner product.=,
r′
denotes
the Holder conjugate exponent of r>1(that is, r′=r
r−1).
We define the energy functional Jand the Nehari functional Ias follows:
I,J :W2,p
0(Ω) →Rby
J(u)=1
p∥∆u∥p
p+q+1
q2∥∇u∥q
q−1
qΩ|∇u|qlog |∇u|dx (2)
I(u)=∥∆u∥p
p+∥∇u∥q
q−Ω|∇u|qlog |∇u|dx (3)
Then we have,
J(u)=1
qI(u)+1
p−1
q∥∆u∥p
p+1
q2∥∇u∥q
q(4)
We introduce the Nehari manifold as
N={u∈W2,p
0(Ω)\{0}:I(u)=0}
also define the potential well as
W={u∈W2,p
0(Ω)\{0}:J(u)< d, I(u)>0}
where d= infu∈N J(u)is referred to as the depth of the potential well.
Definition 1. A function
u
=
u
(
x, t
)is considered to be a weak solution of problem (1) if
u∈
L∞(0,T;W2,p
0(Ω)),ut∈L2(0,T;H1
0(Ω)) and validates
(ut,ϕ)+(∇ut,∇ϕ)+(|∆u|p−2∆u, ∆ϕ)+(|∇u|q−2∇u, ∇ϕ)=(|∇u|q−2∇ulog |∇u|,∇ϕ)(5)
for all
ϕ∈W2,p
0
(Ω) and a.e 0
≤t≤T
along with
u
(
x,
0) =
u0
(
x
)in
W2,p
0
(Ω)
\{
0
}
. Furthermore, it also
agrees the energy inequality
t
0∥uτ∥2
H1
0dτ +J(u)≤J(u0),0<t≤T. (6)
Lemma 1. [12] Let ρbe a positive number. Then we have the following inequalities:
xplog x≤(eρ)−1for all x≥1
and
|xplog x|≤(ep)−1for all 0<x<1.
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
111
The following lemma is similar to one in [8], [16]. However, we explain the proof with some changes
due to the occurrence of the non-linear logarithmic term
−div
(
|∇u|q−2∇ulog |∇u|
)and the q-Laplacian
div(|∇u|q−2∇u).
Lemma 2. For any u∈W2,p
0(Ω)\{0}, we have the following:
(i) limγ→0+J(γu)=0and limγ→∞ J(γu)=−∞;
(ii) d
dγ J(γu)= 1
γI(γu)for γ>0;
(iii) there exists a unique
γ∗
=
γ∗
(
u
)
>
0such that
d
dγ J
(
γu
)
|γ=γ∗
=0. Also
J
(
γu
)is increasing on
0<γ≤γ∗, decreasing on γ∗≤γ<∞and takes the maximum at γ=γ∗;
(iv) I(γu)>0for 0<γ<γ
∗,I(γu)<0for γ∗<γ<∞and I(γ∗u)=0.
Proof.
(i) Applying the definition of Jwe have
J(γu)=γp
p∥∆u∥p
p+γq(q+ 1)
q2∥∇u∥q
q−γqlog γ
q∥∇u∥q
q−γq
qΩ|∇u|qlog |∇u|dx
so it is evident that limγ→0+J(γu)=0and limγ→∞ J(γu)=−∞ since 2<p<q.
(ii) Direct computation yields,
d
dγ J(γu)=γp−1∥∆u∥p
p+γq−1∥∇u∥q
q−γq−1Ω|∇u|qlog |γ∇u|dx =1
γI(γu)
(iii) We have,
d
dγ J(γu)=γq−1γp−q∥∆u∥p
p+∥∇u∥q
q−log γ∥∇u∥q
q−Ω|∇u|qlog |∇u|dx
Now define,
g(γ)=γp−q∥∆u∥p
p+∥∇u∥q
q−log γ∥∇u∥q
q−Ω|∇u|qlog |∇u|dx
Then we can observe that gis decreasing since
g′(γ)=(p−q)γp−q−1∥∆u∥p
p−1
γ∥∇u∥q
q<0
Also, limγ→0+g(γ)=∞and limγ→∞ g(γ)=−∞.
Hence, a unique γ∗with g(γ∗)=0is guaranteed.
Also, g(γ)>0for 0<γ<γ
∗and g(γ)<0for γ∗<γ<∞.
Now, since
d
dγ J
(
γu
)=
γq−1g
(
γ
)we obtain
d
dγ J
(
γu
)
|γ=γ∗
=0and also
J
(
γu
)is increasing on
0<γ≤γ∗, decreasing on γ∗≤γ<∞and takes the maximum at γ=γ∗.
(iv) (iv) is obvious since I(γu)=γd
dγ J(γu).
The above lemmas are useful to prove the main results in the following section.
https://doi.org/10.17993/3ctic.2022.112.108-122
3 MAIN RESULTS
In this section, we prove the existence of weak local solutions to the problem (1). Further, we show
that the weak solution exists globally using the potential well method when the initial energy of the
system is subcritical and critical. We show that the solution becomes unbounded in finite time and
specifies an upper limit for the blow-up time.
Theorem 1. (The Local existence)
Let
u0∈W2,p
0
(Ω)
\{
0
}
and 2
<p<q<p
(1 +
2
N+2
). Then a
T>
0and a unique weak solution
u
(
t
)of
problem(1) agreeing the energy inequality
t
0∥uτ∥2
H1
0dτ +J(u(t)) ≤J(u0),0≤t≤T(7)
and u(0) = u0exists.
Proof.Existence
Let {wi}i∈Nbe an orthonormal basis for W2,p
0(Ω). We use the approximation,
uk(x, t)=
k
i=1
ak,i(t)wi(x),k=1,2,...
where ak,i(t):[0,T]→Raccepts the below ODE.
(ukt,w
i)+(∇ukt,∇wi)+(|∆uk|p−2∆uk,∆wi)+(|∇uk|q−2∇uk,∇wi)
=(|∇uk|q−2∇uklog |∇uk|,∇wi)(8)
i=1,2,...,k and
uk(x, 0) =
k
i=1
ak,i(0)wi(x)→u0(x)in W2,p
0(Ω)\{0}
By Peano’s theorem, the above ODE has a solution
ak,i
and we can find a
Tk>
0with
ak,i ∈C1
([0
,T
k
]),
which implies uk∈C1([0,T
k]; W2,p
0(Ω)).
Now by multiplying (8) by
ak,i
(
t
), summing it for
i
=1
,
2
,...,k
and integrating with respect to
t
from
0to twe obtain,
1
2∥uk∥2
H1
0+t
0
(∥∆uk∥p
p+∥∇uk∥q
q)dt =1
2∥uk(0)∥2
H1
0+t
0Ω|∇uk|qlog |∇uk|dxdt
That is,
ψk(t)=ψk(0) + t
0Ω|∇uk|qlog |∇uk|dxdt (9)
where
ψk(t)=1
2∥uk∥2
H1
0+t
0
(∥∆uk∥p
p+∥∇uk∥q
q)dt (10)
We obtain the following by employing lemma(1), Gagliardo-Nirenberg interpolation inequality and
Young’s inequality.
Ω|∇uk|qlog |∇uk|dx ≤{x∈Ω:|∇uk|≥1}|∇uk|qlog |∇uk|dx
≤(eρ)−1∥∇uk∥q+ρ
q+ρ
≤(eρ)−1Cq+ρ
1∥∆uk∥θ(q+ρ)
p∥uk∥(1−θ)(q+ρ)
2
≤ϵ∥∆uk∥p
p+C(ϵ)∥uk∥
p(1−θ)(q+ρ)
p−θ(q+ρ)
2(11)
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
112
The following lemma is similar to one in [8], [16]. However, we explain the proof with some changes
due to the occurrence of the non-linear logarithmic term
−div
(
|∇u|q−2∇ulog |∇u|
)and the q-Laplacian
div(|∇u|q−2∇u).
Lemma 2. For any u∈W2,p
0(Ω)\{0}, we have the following:
(i) limγ→0+J(γu)=0and limγ→∞ J(γu)=−∞;
(ii) d
dγ J(γu)= 1
γI(γu)for γ>0;
(iii) there exists a unique
γ∗
=
γ∗
(
u
)
>
0such that
d
dγ J
(
γu
)
|γ=γ∗
=0. Also
J
(
γu
)is increasing on
0<γ≤γ∗, decreasing on γ∗≤γ<∞and takes the maximum at γ=γ∗;
(iv) I(γu)>0for 0<γ<γ
∗,I(γu)<0for γ∗<γ<∞and I(γ∗u)=0.
Proof.
(i) Applying the definition of Jwe have
J(γu)=γp
p∥∆u∥p
p+γq(q+ 1)
q2∥∇u∥q
q−γqlog γ
q∥∇u∥q
q−γq
qΩ|∇u|qlog |∇u|dx
so it is evident that limγ→0+J(γu)=0and limγ→∞ J(γu)=−∞ since 2<p<q.
(ii) Direct computation yields,
d
dγ J(γu)=γp−1∥∆u∥p
p+γq−1∥∇u∥q
q−γq−1Ω|∇u|qlog |γ∇u|dx =1
γI(γu)
(iii) We have,
d
dγ J(γu)=γq−1γp−q∥∆u∥p
p+∥∇u∥q
q−log γ∥∇u∥q
q−Ω|∇u|qlog |∇u|dx
Now define,
g(γ)=γp−q∥∆u∥p
p+∥∇u∥q
q−log γ∥∇u∥q
q−Ω|∇u|qlog |∇u|dx
Then we can observe that gis decreasing since
g′(γ)=(p−q)γp−q−1∥∆u∥p
p−1
γ∥∇u∥q
q<0
Also, limγ→0+g(γ)=∞and limγ→∞ g(γ)=−∞.
Hence, a unique γ∗with g(γ∗)=0is guaranteed.
Also, g(γ)>0for 0<γ<γ
∗and g(γ)<0for γ∗<γ<∞.
Now, since
d
dγ J
(
γu
)=
γq−1g
(
γ
)we obtain
d
dγ J
(
γu
)
|γ=γ∗
=0and also
J
(
γu
)is increasing on
0<γ≤γ∗, decreasing on γ∗≤γ<∞and takes the maximum at γ=γ∗.
(iv) (iv) is obvious since I(γu)=γd
dγ J(γu).
The above lemmas are useful to prove the main results in the following section.
https://doi.org/10.17993/3ctic.2022.112.108-122
3 MAIN RESULTS
In this section, we prove the existence of weak local solutions to the problem (1). Further, we show
that the weak solution exists globally using the potential well method when the initial energy of the
system is subcritical and critical. We show that the solution becomes unbounded in finite time and
specifies an upper limit for the blow-up time.
Theorem 1. (The Local existence)
Let
u0∈W2,p
0
(Ω)
\{
0
}
and 2
<p<q<p
(1 +
2
N+2
). Then a
T>
0and a unique weak solution
u
(
t
)of
problem(1) agreeing the energy inequality
t
0∥uτ∥2
H1
0dτ +J(u(t)) ≤J(u0),0≤t≤T(7)
and u(0) = u0exists.
Proof.Existence
Let {wi}i∈Nbe an orthonormal basis for W2,p
0(Ω). We use the approximation,
uk(x, t)=
k
i=1
ak,i(t)wi(x),k=1,2,...
where ak,i(t):[0,T]→Raccepts the below ODE.
(ukt,w
i)+(∇ukt,∇wi)+(|∆uk|p−2∆uk,∆wi)+(|∇uk|q−2∇uk,∇wi)
=(|∇uk|q−2∇uklog |∇uk|,∇wi)(8)
i=1,2,...,k and
uk(x, 0) =
k
i=1
ak,i(0)wi(x)→u0(x)in W2,p
0(Ω)\{0}
By Peano’s theorem, the above ODE has a solution
ak,i
and we can find a
Tk>
0with
ak,i ∈C1
([0
,T
k
]),
which implies uk∈C1([0,T
k]; W2,p
0(Ω)).
Now by multiplying (8) by
ak,i
(
t
), summing it for
i
=1
,
2
,...,k
and integrating with respect to
t
from
0to twe obtain,
1
2∥uk∥2
H1
0+t
0
(∥∆uk∥p
p+∥∇uk∥q
q)dt =1
2∥uk(0)∥2
H1
0+t
0Ω|∇uk|qlog |∇uk|dxdt
That is,
ψk(t)=ψk(0) + t
0Ω|∇uk|qlog |∇uk|dxdt (9)
where
ψk(t)=1
2∥uk∥2
H1
0+t
0
(∥∆uk∥p
p+∥∇uk∥q
q)dt (10)
We obtain the following by employing lemma(1), Gagliardo-Nirenberg interpolation inequality and
Young’s inequality.
Ω|∇uk|qlog |∇uk|dx ≤{x∈Ω:|∇uk|≥1}|∇uk|qlog |∇uk|dx
≤(eρ)−1∥∇uk∥q+ρ
q+ρ
≤(eρ)−1Cq+ρ
1∥∆uk∥θ(q+ρ)
p∥uk∥(1−θ)(q+ρ)
2
≤ϵ∥∆uk∥p
p+C(ϵ)∥uk∥
p(1−θ)(q+ρ)
p−θ(q+ρ)
2(11)
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
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113
where θ=1
n+1
2−1
q+ρ2
n+1
2−1
p−1,ϵ∈(0,1),
C(ϵ)=pϵ
θ(q+ρ)θ(q+ρ)
θ(q+ρ)−pp−θ(q+ρ)
p(eρ)−1Cq+ρ
1p
p−θ(q+ρ)and,
ρis chosen so that 2<q+ρ<p(1 + 2
n+2 ).
Let β=p(1−θ)(q+ρ)
2(p−θ(q+ρ)) =np+(p−n)(q+ρ)
p(4+n)−(n+2)(q+ρ). Then β>1and
Ω|∇uk|qlog |∇uk|dx ≤ϵ∥∆uk∥p
p+C(ϵ)∥uk∥2β
2(12)
Then (9) implies that,
ψk(t)≤ψk(0) + ϵt
0∥∆uk∥p
pdt +C(ϵ)t
0∥uk∥2β
2dt
≤C2+ϵψk(t)+C(ϵ)2βt
01
2∥uk∥2
H1
0β
+s
0
(∥∆uk∥p
p+∥∇uk∥q
q)dsβdt
≤C2+ϵψk(t)+C3t
0
ψk(t)βdt
Hence we get,
ψk(t)≤C4+C5t
0
ψk(t)βdt
Then the Gronwall-Bellman-Bihari type integral inequality gives a Tsuch that 0<T < C1−β
4
C5(1−β)and
ψk(t)≤CTfor all t∈[0,T].(13)
Hence the solution of (8) exists in [0,T]for all k.
Now multiplying (8) by a′
k,i(t)and summing for i=1,2,...,k we get,
(ukt,u
kt)+(∇ukt,∇ukt)+(|∆uk|p−2∆uk,∆ukt)+(|∇uk|q−2∇uk,∇ukt)
=(|∇uk|q−2∇uklog |∇uk|,∇ukt)
integrating with respect to t,
t
0∥ukt∥2
H1
0dt +J(uk(t)) = J(uk(0)) for all t∈[0,T].(14)
As contrast to that, a constant C6>0satisfying
J(uk(0)) ≤C6for all k. (15)
exists since
uk
(0)
→u0
and by the continuity of
J
. Then from (12),(13),(14) and (15) we can see that
C6≥t
0∥ukt∥2
H1
0dt +1
p∥∆uk∥p
p+q+1
q2∥∇uk∥q
q−1
qΩ|∇uk|qlog |∇uk|dx
≥t
0∥ukt∥2
H1
0dt +1
p−ϵ
q∥∆uk∥p
p+q+1
q2∥∇uk∥q
q−C(ϵ)
q∥uk∥2β
H1
0
≥t
0∥ukt∥2
H1
0dt +1
p−ϵ
q∥∆uk∥p
p+q+1
q2∥∇uk∥q
q−C(ϵ)
q2βCβ
T
Let ˜
C=C6+C(ϵ)2β
qCβ
T. Then we gain that
t
0∥ukt∥2
2dt ≤˜
C
t
0∥∇ukt∥2
2dt ≤˜
C
∥∆uk∥p
p<˜
C1
p−ϵ
q−1
∥∇uk∥q
q<˜
Cq2
q+1
https://doi.org/10.17993/3ctic.2022.112.108-122
Thus we have
{uk}k∈N
is bounded in
L∞
(0
,T
;
W2,p
0
(Ω)) and
{ukt}k∈N
is bounded in
L2
(0
,T
;
H1
0
(Ω)).
Hence there exists a subsequence, however indicated by {uk}k∈Nwhich agrees,
uk→uweakly* in L∞(0,T;W2,p
0(Ω))
ukt →utweakly in L2(0,T;H1
0(Ω))
uk→uweakly* in L∞(0,T;W1,q
0(Ω))
since
ukt →utweakly in L2(0,T;L2(Ω))
by Aubin-Lions lemma we get,
uk→ustrongly in C(0,T;L2(Ω))
Therefore,
|∆uk|p−2∆uk→ξ1weakly* in L∞(0,T;W−2,p′
0(Ω))
and,
|∇uk|q−2∇uk→ξ2weakly* in L∞(0,T;W−1,q′
0(Ω))
where
W−2,p′
0
(Ω) is the dual space of
W2,p
0
(Ω) and
W−1,q′
0
(Ω) is the dual space of
W1,q
0
(Ω). Now from
the theory of monotone operators, it concludes,
ξ1=|∆u|p−2∆uand ξ2=|∇u|q−2∇u.
Now let Φ(u)=|u|q−2ulog |u|.Wehave
∇uk→∇uweakly* in L∞(0,T;L2(Ω))
∇ukt →∇utweakly in L2(0,T;L2(Ω))
Therefore,
∇uk→∇ustrongly in C(0,T;L2(Ω))
and
Φ(∇uk)→Φ(∇u)a.e in Ω×(0,T)
We again use Lemma(1) and Gagliardo-Nirenberg interpolation inequality to emerge the below.
Ω
(Φ(∇uk))q′dx ≤{x∈Ω:|∇uk|≤1}|∇uk|q−1|log |∇uk||q′
dx
+{x∈Ω:|∇uk|≥1}|∇uk|q−1|log |∇uk||q′
dx
≤(e(q−1))−q′|Ω|+(eµ)−q′∥∇uk∥r
r
≤(e(q−1))−q′|Ω|+(eµ)−q′Cr
7∥∆uk∥rα
p∥uk∥r(1−α)
2
<C
8
where r=(q−1+µ)q′,q′=q
q−1and α=1
n+1
2−1
r2
n+1
2−1
p−1. Hence,
Φ(∇uk)→Φ(∇u)weakly* in L∞(0,T;Lq′(Ω))
Now for a fixed iin (8) letting ktends to ∞we get,
(ut,w
i)+(∇ut,∇wi)+(|∆u|p−2∆u, ∆wi)+(|∇u|q−2∇u, ∇wi)=(|∇u|q−2∇ulog |∇u|,∇wi)
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
114
where θ=1
n+1
2−1
q+ρ2
n+1
2−1
p−1,ϵ∈(0,1),
C(ϵ)=pϵ
θ(q+ρ)θ(q+ρ)
θ(q+ρ)−pp−θ(q+ρ)
p(eρ)−1Cq+ρ
1p
p−θ(q+ρ)and,
ρis chosen so that 2<q+ρ<p(1 + 2
n+2 ).
Let β=p(1−θ)(q+ρ)
2(p−θ(q+ρ)) =np+(p−n)(q+ρ)
p(4+n)−(n+2)(q+ρ). Then β>1and
Ω|∇uk|qlog |∇uk|dx ≤ϵ∥∆uk∥p
p+C(ϵ)∥uk∥2β
2(12)
Then (9) implies that,
ψk(t)≤ψk(0) + ϵt
0∥∆uk∥p
pdt +C(ϵ)t
0∥uk∥2β
2dt
≤C2+ϵψk(t)+C(ϵ)2βt
01
2∥uk∥2
H1
0β
+s
0
(∥∆uk∥p
p+∥∇uk∥q
q)dsβdt
≤C2+ϵψk(t)+C3t
0
ψk(t)βdt
Hence we get,
ψk(t)≤C4+C5t
0
ψk(t)βdt
Then the Gronwall-Bellman-Bihari type integral inequality gives a Tsuch that 0<T < C1−β
4
C5(1−β)and
ψk(t)≤CTfor all t∈[0,T].(13)
Hence the solution of (8) exists in [0,T]for all k.
Now multiplying (8) by a′
k,i(t)and summing for i=1,2,...,k we get,
(ukt,u
kt)+(∇ukt,∇ukt)+(|∆uk|p−2∆uk,∆ukt)+(|∇uk|q−2∇uk,∇ukt)
=(|∇uk|q−2∇uklog |∇uk|,∇ukt)
integrating with respect to t,
t
0∥ukt∥2
H1
0dt +J(uk(t)) = J(uk(0)) for all t∈[0,T].(14)
As contrast to that, a constant C6>0satisfying
J(uk(0)) ≤C6for all k. (15)
exists since
uk
(0)
→u0
and by the continuity of
J
. Then from (12),(13),(14) and (15) we can see that
C6≥t
0∥ukt∥2
H1
0dt +1
p∥∆uk∥p
p+q+1
q2∥∇uk∥q
q−1
qΩ|∇uk|qlog |∇uk|dx
≥t
0∥ukt∥2
H1
0dt +1
p−ϵ
q∥∆uk∥p
p+q+1
q2∥∇uk∥q
q−C(ϵ)
q∥uk∥2β
H1
0
≥t
0∥ukt∥2
H1
0dt +1
p−ϵ
q∥∆uk∥p
p+q+1
q2∥∇uk∥q
q−C(ϵ)
q2βCβ
T
Let ˜
C=C6+C(ϵ)2β
qCβ
T. Then we gain that
t
0∥ukt∥2
2dt ≤˜
C
t
0∥∇ukt∥2
2dt ≤˜
C
∥∆uk∥p
p<˜
C1
p−ϵ
q−1
∥∇uk∥q
q<˜
Cq2
q+1
https://doi.org/10.17993/3ctic.2022.112.108-122
Thus we have
{uk}k∈N
is bounded in
L∞
(0
,T
;
W2,p
0
(Ω)) and
{ukt}k∈N
is bounded in
L2
(0
,T
;
H1
0
(Ω)).
Hence there exists a subsequence, however indicated by {uk}k∈Nwhich agrees,
uk→uweakly* in L∞(0,T;W2,p
0(Ω))
ukt →utweakly in L2(0,T;H1
0(Ω))
uk→uweakly* in L∞(0,T;W1,q
0(Ω))
since
ukt →utweakly in L2(0,T;L2(Ω))
by Aubin-Lions lemma we get,
uk→ustrongly in C(0,T;L2(Ω))
Therefore,
|∆uk|p−2∆uk→ξ1weakly* in L∞(0,T;W−2,p′
0(Ω))
and,
|∇uk|q−2∇uk→ξ2weakly* in L∞(0,T;W−1,q′
0(Ω))
where
W−2,p′
0
(Ω) is the dual space of
W2,p
0
(Ω) and
W−1,q′
0
(Ω) is the dual space of
W1,q
0
(Ω). Now from
the theory of monotone operators, it concludes,
ξ1=|∆u|p−2∆uand ξ2=|∇u|q−2∇u.
Now let Φ(u)=|u|q−2ulog |u|.Wehave
∇uk→∇uweakly* in L∞(0,T;L2(Ω))
∇ukt →∇utweakly in L2(0,T;L2(Ω))
Therefore,
∇uk→∇ustrongly in C(0,T;L2(Ω))
and
Φ(∇uk)→Φ(∇u)a.e in Ω×(0,T)
We again use Lemma(1) and Gagliardo-Nirenberg interpolation inequality to emerge the below.
Ω
(Φ(∇uk))q′dx ≤{x∈Ω:|∇uk|≤1}|∇uk|q−1|log |∇uk||q′
dx
+{x∈Ω:|∇uk|≥1}|∇uk|q−1|log |∇uk||q′
dx
≤(e(q−1))−q′|Ω|+(eµ)−q′∥∇uk∥r
r
≤(e(q−1))−q′|Ω|+(eµ)−q′Cr
7∥∆uk∥rα
p∥uk∥r(1−α)
2
<C
8
where r=(q−1+µ)q′,q′=q
q−1and α=1
n+1
2−1
r2
n+1
2−1
p−1. Hence,
Φ(∇uk)→Φ(∇u)weakly* in L∞(0,T;Lq′(Ω))
Now for a fixed iin (8) letting ktends to ∞we get,
(ut,w
i)+(∇ut,∇wi)+(|∆u|p−2∆u, ∆wi)+(|∇u|q−2∇u, ∇wi)=(|∇u|q−2∇ulog |∇u|,∇wi)
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115
for all i=1,2,...,k. Then for all ϕ∈W2,p
0(Ω) and for a.e. t∈[0,T],
(ut,ϕ)+(∇ut,∇ϕ)+(|∆u|p−2∆u, ∆ϕ)+(|∇u|q−2∇u, ∇ϕ)=(|∇u|q−2∇ulog |∇u|,∇ϕ)
and u(x, 0) = u0(x)in W02,p
2(Ω)\{0}.
Uniqueness
Let uand ˜ube two weak solutions of problem (1). For any ϕ∈H2
0(Ω), it is noted that,
(ut,ϕ)+(∇ut,∇ϕ)+(|∆u|p−2∆u, ∆ϕ)+(|∇u|q−2∇u, ∇ϕ)=(|∇u|q−2∇ulog |∇u|,∇ϕ)
(˜ut,ϕ)+(∇˜ut,∇ϕ)+(|∆˜u|p−2∆˜u, ∆ϕ)+(|∇˜u|q−2∇˜u, ∇ϕ)=(|∇˜u|q−2∇˜ulog |∇˜u|,∇ϕ)
On subtraction of one equation from the other and taking ϕ=u−˜u, the above yields that
(ϕt,ϕ)+(∇ϕt,∇ϕ)+Ω
(|∆u|p−2∆u−|∆˜u|p−2∆˜u)(∆u−∆˜u)dx
+Ω
(|∇u|q−2∇u− |∇˜u|q−2∇˜u)(∇u−∇˜u)dx
=Ω
(|∇u|q−2∇ulog |∇u|−|∇˜u|q−2∇˜ulog |∇˜u|)(∇u−∇˜u)dx
Then by the monotonicity of q-Laplacian
div
(
|∇u|q−2∇u
)and the p-Biharmonic operator ∆(
|
∆
u|p−2
∆
u
)
and by the Lipschitz continuity of |x|q−2xlog |x|we get,
(ϕt,ϕ)1≤LΩ
(∇u−∇˜u)2dx
where L>0is the Lipschitz constant. Thus we obtain,
(ϕt,ϕ)1≤L∥∇ϕ∥2
2≤L∥ϕ∥2
H1
0
By the integration from 0to twith respect to twe obtain that,
∥ϕ∥2
H1
0−∥ϕ(0)∥2
H1
0≤Lt
0∥ϕ∥2
H1
0dt.
Since ϕ(0) = u(0) −˜u(0) = 0, apply Gronwall’s inequality to gain,
∥ϕ∥2
H1
0=0
Therefore, ϕ=0a.e. in Ω×(0,T). That is, u=˜ua.e. in Ω×(0,T).
Energy inequality
Let χ∈C[0,T]be a non-negative function. Then (14) implies
T
0
χ(t)t
0∥ukt∥2
H1
0dsdt +T
0
J(uk(t))χ(t)dt =T
0
J(uk(0))χ(t)dt
Since, we have the lower semi-continuity
T
0J
(
uk
(
t
))
χ
(
t
)
dt
with respect to the weak topology of
L2(0,T;W2,p
0(Ω)).T
0
J(u(t))χ(t)dt ≤lim inf
k→∞ T
0
J(uk(t))χ(t)dt
also T
0J(uk(0))χ(t)dt →T
0J(u0)χ(t)dt as k→∞. Thus we get,
T
0
χ(t)t
0∥ut∥2
H1
0dsdt +T
0
J(u(t))χ(t)dt ≤T
0
J(u0)χ(t)dt
Since χ(t)is arbitrary,
t
0∥uτ∥2
H1
0dτ +J(u(t)) ≤J(u0)for 0≤t≤T.
Hence the proof is complete.
Next theorem address the case of the initial energy of the system is sub-critical, i.e,
J
(
u0
)
<d
. We
will demonstrate the existence of weak global solutions.
https://doi.org/10.17993/3ctic.2022.112.108-122
Theorem 2. (Global Existence for J(u0)<d)
A unique global weak solution usatisfying the energy estimate,
t
0∥uτ∥2
H1
0dτ +J(u(t)) ≤J(u0)for 0≤t<∞(16)
exists for problem(1) if the conditions
J
(
u0
)
<d
and
I
(
u0
)
>
0holds for the initial value
u0∈
W2,p
0(Ω)\{0}.
Proof. Define
{wi}i∈N
and
{uk}k∈N
as in the proof of Theorem(1). Multiplying (8) by
a′
k,i
(
t
)and
summing over iand integrating with respect to tfrom 0to twe identify,
t
0∥ukt∥2
H1
0dt +J(uk(t)) = J(uk(0)) for all t∈[0,T
max)(17)
where Tmax is the maximum time for solution uk(x, t)to exist.
We have J(uk(0)) →J(u0)as k→∞and J(u0)<d. Therefore,
t
0∥ukt∥2
H1
0dt +J(uk(t)) < d, t ∈[0,T
max)(18)
Since
I
(
u0
)
>
0we have
I
(
uk
(0))
>
0for sufficiently large
k
. We claim that
I
(
uk
)
>
0for sufficiently
large
k
. Otherwise we can locate a
t0
such that
I
(
uk
(
t0
)) = 0,
uk
(
t0
)
=0. Then
uk
(
t0
)
∈N
and
J(uk(t0)) ≥d, which is a contradiction to (18).
Therefore I(uk)>0for appropriately large k.
Then we get,
J(uk)=1
qI(uk)+1
p−1
q∥∆uk∥p
p+1
q2∥∇uk∥q
q>0
Therefore, t
0∥ukt∥2
H1
0dt < d
also 1
p−1
q∥∆uk∥p
p+1
q2∥∇uk∥q
q<J(uk)<d
Let K0= min{1
p−1
q,1
q2}and K=d+d
K0then
∥∆uk∥p
p+∥∇uk∥q
q< d/K0
and t
0∥ukt∥2
H1
0dt +∥∆uk∥p
p+∥∇uk∥q
q<K (19)
where
K>
0. Hence we take
Tmax
=
∞
. Now it is noticeable that problem (1) has a weak global
solution by applying identical ideas used to prove the Theorem(1), and the solution
u
also agrees with
the energy inequality t
0∥uτ∥2
H1
0dτ +J(u(t)) ≤J(u0),0≤t<∞.
We will explain the global existence of weak solutions in the following theorem for the critical initial
energy. That is when J(u0)=d.
Theorem 3. (Global existence for J(u0)=d)
Observe the conditions
J
(
u0
)=
d
and
I
(
u0
)
>
0holds for the initial value
u0∈W2,p
0
(Ω)
\{
0
}
.
Subsequently problem(1) possesses a unique global weak solution
u∈L∞
(0
,T
;
W2,p
0
(Ω)) with
ut∈
L2(0,T;L2(Ω)) for 0≤t≤Tand it also accepts the energy estimate (16).
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
116
for all i=1,2,...,k. Then for all ϕ∈W2,p
0(Ω) and for a.e. t∈[0,T],
(ut,ϕ)+(∇ut,∇ϕ)+(|∆u|p−2∆u, ∆ϕ)+(|∇u|q−2∇u, ∇ϕ)=(|∇u|q−2∇ulog |∇u|,∇ϕ)
and u(x, 0) = u0(x)in W02,p
2(Ω)\{0}.
Uniqueness
Let uand ˜ube two weak solutions of problem (1). For any ϕ∈H2
0(Ω), it is noted that,
(ut,ϕ)+(∇ut,∇ϕ)+(|∆u|p−2∆u, ∆ϕ)+(|∇u|q−2∇u, ∇ϕ)=(|∇u|q−2∇ulog |∇u|,∇ϕ)
(˜ut,ϕ)+(∇˜ut,∇ϕ)+(|∆˜u|p−2∆˜u, ∆ϕ)+(|∇˜u|q−2∇˜u, ∇ϕ)=(|∇˜u|q−2∇˜ulog |∇˜u|,∇ϕ)
On subtraction of one equation from the other and taking ϕ=u−˜u, the above yields that
(ϕt,ϕ)+(∇ϕt,∇ϕ)+Ω
(|∆u|p−2∆u−|∆˜u|p−2∆˜u)(∆u−∆˜u)dx
+Ω
(|∇u|q−2∇u− |∇˜u|q−2∇˜u)(∇u−∇˜u)dx
=Ω
(|∇u|q−2∇ulog |∇u|−|∇˜u|q−2∇˜ulog |∇˜u|)(∇u−∇˜u)dx
Then by the monotonicity of q-Laplacian
div
(
|∇u|q−2∇u
)and the p-Biharmonic operator ∆(
|
∆
u|p−2
∆
u
)
and by the Lipschitz continuity of |x|q−2xlog |x|we get,
(ϕt,ϕ)1≤LΩ
(∇u−∇˜u)2dx
where L>0is the Lipschitz constant. Thus we obtain,
(ϕt,ϕ)1≤L∥∇ϕ∥2
2≤L∥ϕ∥2
H1
0
By the integration from 0to twith respect to twe obtain that,
∥ϕ∥2
H1
0−∥ϕ(0)∥2
H1
0≤Lt
0∥ϕ∥2
H1
0dt.
Since ϕ(0) = u(0) −˜u(0) = 0, apply Gronwall’s inequality to gain,
∥ϕ∥2
H1
0=0
Therefore, ϕ=0a.e. in Ω×(0,T). That is, u=˜ua.e. in Ω×(0,T).
Energy inequality
Let χ∈C[0,T]be a non-negative function. Then (14) implies
T
0
χ(t)t
0∥ukt∥2
H1
0dsdt +T
0
J(uk(t))χ(t)dt =T
0
J(uk(0))χ(t)dt
Since, we have the lower semi-continuity
T
0J
(
uk
(
t
))
χ
(
t
)
dt
with respect to the weak topology of
L2(0,T;W2,p
0(Ω)).T
0
J(u(t))χ(t)dt ≤lim inf
k→∞ T
0
J(uk(t))χ(t)dt
also T
0J(uk(0))χ(t)dt →T
0J(u0)χ(t)dt as k→∞. Thus we get,
T
0
χ(t)t
0∥ut∥2
H1
0dsdt +T
0
J(u(t))χ(t)dt ≤T
0
J(u0)χ(t)dt
Since χ(t)is arbitrary,
t
0∥uτ∥2
H1
0dτ +J(u(t)) ≤J(u0)for 0≤t≤T.
Hence the proof is complete.
Next theorem address the case of the initial energy of the system is sub-critical, i.e,
J
(
u0
)
<d
. We
will demonstrate the existence of weak global solutions.
https://doi.org/10.17993/3ctic.2022.112.108-122
Theorem 2. (Global Existence for J(u0)<d)
A unique global weak solution usatisfying the energy estimate,
t
0∥uτ∥2
H1
0dτ +J(u(t)) ≤J(u0)for 0≤t<∞(16)
exists for problem(1) if the conditions
J
(
u0
)
<d
and
I
(
u0
)
>
0holds for the initial value
u0∈
W2,p
0(Ω)\{0}.
Proof. Define
{wi}i∈N
and
{uk}k∈N
as in the proof of Theorem(1). Multiplying (8) by
a′
k,i
(
t
)and
summing over iand integrating with respect to tfrom 0to twe identify,
t
0∥ukt∥2
H1
0dt +J(uk(t)) = J(uk(0)) for all t∈[0,T
max)(17)
where Tmax is the maximum time for solution uk(x, t)to exist.
We have J(uk(0)) →J(u0)as k→∞and J(u0)<d. Therefore,
t
0∥ukt∥2
H1
0dt +J(uk(t)) < d, t ∈[0,T
max)(18)
Since
I
(
u0
)
>
0we have
I
(
uk
(0))
>
0for sufficiently large
k
. We claim that
I
(
uk
)
>
0for sufficiently
large
k
. Otherwise we can locate a
t0
such that
I
(
uk
(
t0
)) = 0,
uk
(
t0
)
=0. Then
uk
(
t0
)
∈N
and
J(uk(t0)) ≥d, which is a contradiction to (18).
Therefore I(uk)>0for appropriately large k.
Then we get,
J(uk)=1
qI(uk)+1
p−1
q∥∆uk∥p
p+1
q2∥∇uk∥q
q>0
Therefore, t
0∥ukt∥2
H1
0dt < d
also 1
p−1
q∥∆uk∥p
p+1
q2∥∇uk∥q
q<J(uk)<d
Let K0= min{1
p−1
q,1
q2}and K=d+d
K0then
∥∆uk∥p
p+∥∇uk∥q
q< d/K0
and t
0∥ukt∥2
H1
0dt +∥∆uk∥p
p+∥∇uk∥q
q<K (19)
where
K>
0. Hence we take
Tmax
=
∞
. Now it is noticeable that problem (1) has a weak global
solution by applying identical ideas used to prove the Theorem(1), and the solution
u
also agrees with
the energy inequality t
0∥uτ∥2
H1
0dτ +J(u(t)) ≤J(u0),0≤t<∞.
We will explain the global existence of weak solutions in the following theorem for the critical initial
energy. That is when J(u0)=d.
Theorem 3. (Global existence for J(u0)=d)
Observe the conditions
J
(
u0
)=
d
and
I
(
u0
)
>
0holds for the initial value
u0∈W2,p
0
(Ω)
\{
0
}
.
Subsequently problem(1) possesses a unique global weak solution
u∈L∞
(0
,T
;
W2,p
0
(Ω)) with
ut∈
L2(0,T;L2(Ω)) for 0≤t≤Tand it also accepts the energy estimate (16).
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
117
Proof. Let
ηj
=1
−1
j,j
=1
,
2
,...
then
ηj→
1when
j→∞
. Take into account the below problem:
ut−∆ut+ ∆(|∆u|p−2∆u)−div(|∇u|q−2∇u)=−div(|∇u|q−2∇ulog |∇u|)if (x, t)∈Ω×(0,T),
u=∂u
∂ν =0 if (x, t)∈∂Ω×(0,T),
u(x, 0) = ηju0(x)=uj
0if x∈Ω.
(20)
Since I(u0)>0, lemma (2)(iv) gives a γ∗>1with I(γ∗u0)=0.
Again from lemma (2)(iii) and (iv) we gain I(ηju0)>0and J(ηju0)<J(u0)since ηj<1<γ
∗.
Thus we have J(uj
0)<dand I(uj
0)>0.
Then by Theorem(2), for each
j
problem (20) has a global weak solution
uj∈L∞
(0
,T
;
Wp−2
0
(Ω)) with
uj
t∈L2(0,T;L2(Ω)) which satisfies the energy inequality,
t
0∥uj
τ∥2
H1
0dτ +J(uj(t)) ≤J(uj
0)for 0≤t<∞.
Thus we have t
0∥uj
τ∥2
H1
0dτ +J(uj)<d for 0≤t<∞.
Now by applying ideas similar to the one used to prove Theorem(1), we obtain a subsequence of
{uj}j∈N
converging to a function
u
, which is a weak solution of problem (1). It also fulfils the energy inequality
(16). The solution’s uniqueness can also be proved as in Theorem(1).
Hence the proof is over.
The following theorem gives the blow-up of solutions for the subcritical initial energy and an upper
bound for blow up time.
Theorem 4. (Blow up for J(u0)<d)
Let
u0∈H2
0
(Ω)
\{
0
}
,
J
(
u0
)
<d
and
I
(
u0
)
<
0. Then the weak solution
u
of problem (1) blows up in a
finite time
T∗
in the notion,
limt→T−
∗∥u∥2
H1
0
=
∞
. Furthermore, the upper bound of blow-up time
T∗
is
given by
T∗≤4(q−1)∥u0∥2
H1
0
q(q−2)2(d−J(u0)).
Proof.First we prove
J
(
u
(
t
))
<d
and
I
(
u
(
t
))
<
0for
t∈
[0
,T
], where
T
indicates the maximum time
for which u(x, t)exists.
We have J(u(t)) <J(u0)<dby (6).
If we can choose a
t0∈
(0
,T
)with
I
(
u
(
t0
))=0or
J
(
u
(
t0
)) =
d
, since
J
(
u
(
t0
))
<d
, we must have
I(u(t0))=0.
Which implies u(t0)∈Nand thus d≤J(u(t0)), a contradiction.
Hence, J(u(t)) <dand I(u(t)) <0for t∈[0,T]. Now define
P(t)=t
0∥u∥2
H1
0dt
Then,
P′(t)=∥u∥2
H1
0
and
P′′(t)=2(u, ut)1=−2I(u)>0
Hence for t>0,P′(t)≥P′(0) = ∥u0∥2
H1
0>0.
Now fix t1>0. Then for t1≤t<∞,
P(t)≥P(t1)≥t1∥u0∥2
H1
0>0
By Holder’s inequality, we have,
1
4(P′(t)−P′(0))2≤t
0∥u∥2
H1
0dt t
0∥ut∥2
H1
0dt (21)
https://doi.org/10.17993/3ctic.2022.112.108-122
Since I(u(t)) <0, Lemma 2 (iv), gives a γ∗with 0<γ
∗<1and I(γ∗u)=0. Therefore,
d≤1
p−1
q(γ∗)p∥∆u∥p
p+1
q2(γ∗)q∥∇u∥q
q
≤1
p−1
q∥∆u∥p
p+1
q2∥∇u∥q
q(22)
Now by using (4),(6) and (22) we see that,
P′′(t)≥2q(d−J(u0))+2qt
0∥ut∥2
H1
0dt (23)
Then from (21) and (23) it follows that
P′′(t)P(t)−q
2(P′(t)−P′(0))2≥P(t)2q(d−J(u0)) >0for t∈[t1,∞)(24)
Now choose ˜
T>0large enough to introduce,
Q(t)=P(t)+(˜
T−t)∥u0∥2
H1
0for t∈[t1,˜
T]
Then Q(t)≥P(t)>0for t∈[t1,˜
T],Q′(t)=P′(t)−P′(0) >0and Q′′(t)=P′′(t)>0.
Hence from (24) we observe,
Q(t)Q′′(t)−q
2(Q′(t))2≥P(t)2q(d−J(u0)) + P′′(t)( ˜
T−t)∥u0∥2
H1
0>0(25)
Now define
R(t)=Q(t)−q−2
2
Then,
R′(t)=−q−2
2Q(t)−q
2Q′(t)
and
R′′(t)=q−2
2Q(t)−q+2
2q
2(Q′(t))2−Q(t)Q′′(t)<0
Hence
R
(
t
)is a concave function in [
t1,˜
T
]for any sufficiently large
˜
T >t
1
. Also since
R
(
t1
)
>
0
and
R′′
(
t1
)
<
0, there appears a finite time
T∗>t
1>
0having
limt→T−
∗R
(
t
)=0. That yields
limt→T−
∗Q(t)=+∞, which in turn gives limt→T−
∗P(t)=+∞. Hence we get
lim
t→T−
∗∥u∥2
H1
0=+∞.
To obtain an upper limit for blow-up time we define,
S(t)=P(t)+(T∗−t)∥u0∥2
H1
0+σ(t+φ)2for t∈[0,T
∗]
where the constants σ, φ > 0will be given later.
Then,
S′(t)=∥u∥2
H1
0−∥u0∥2
H1
0+2σ(t+φ)>2σ(t+φ)>0(26)
also by (23) we get,
S′′(t)≥2q(d−J(u0))+2qt
0∥ut∥2
H1
0dt +2σ(27)
By Schwartz’s inequality, we have,
t
0
d
dt∥u∥2
H1
0dt ≤2t
0∥u∥2
H1
0dt t
0∥ut∥2
H1
0dt (28)
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
118
Proof. Let
ηj
=1
−1
j,j
=1
,
2
,...
then
ηj→
1when
j→∞
. Take into account the below problem:
ut−∆ut+ ∆(|∆u|p−2∆u)−div(|∇u|q−2∇u)=−div(|∇u|q−2∇ulog |∇u|)if (x, t)∈Ω×(0,T),
u=∂u
∂ν =0 if (x, t)∈∂Ω×(0,T),
u(x, 0) = ηju0(x)=uj
0if x∈Ω.
(20)
Since I(u0)>0, lemma (2)(iv) gives a γ∗>1with I(γ∗u0)=0.
Again from lemma (2)(iii) and (iv) we gain I(ηju0)>0and J(ηju0)<J(u0)since ηj<1<γ
∗.
Thus we have J(uj
0)<dand I(uj
0)>0.
Then by Theorem(2), for each
j
problem (20) has a global weak solution
uj∈L∞
(0
,T
;
Wp−2
0
(Ω)) with
uj
t∈L2(0,T;L2(Ω)) which satisfies the energy inequality,
t
0∥uj
τ∥2
H1
0dτ +J(uj(t)) ≤J(uj
0)for 0≤t<∞.
Thus we have t
0∥uj
τ∥2
H1
0dτ +J(uj)<d for 0≤t<∞.
Now by applying ideas similar to the one used to prove Theorem(1), we obtain a subsequence of
{uj}j∈N
converging to a function
u
, which is a weak solution of problem (1). It also fulfils the energy inequality
(16). The solution’s uniqueness can also be proved as in Theorem(1).
Hence the proof is over.
The following theorem gives the blow-up of solutions for the subcritical initial energy and an upper
bound for blow up time.
Theorem 4. (Blow up for J(u0)<d)
Let
u0∈H2
0
(Ω)
\{
0
}
,
J
(
u0
)
<d
and
I
(
u0
)
<
0. Then the weak solution
u
of problem (1) blows up in a
finite time
T∗
in the notion,
limt→T−
∗∥u∥2
H1
0
=
∞
. Furthermore, the upper bound of blow-up time
T∗
is
given by
T∗≤4(q−1)∥u0∥2
H1
0
q(q−2)2(d−J(u0)).
Proof.First we prove
J
(
u
(
t
))
<d
and
I
(
u
(
t
))
<
0for
t∈
[0
,T
], where
T
indicates the maximum time
for which u(x, t)exists.
We have J(u(t)) <J(u0)<dby (6).
If we can choose a
t0∈
(0
,T
)with
I
(
u
(
t0
))=0or
J
(
u
(
t0
)) =
d
, since
J
(
u
(
t0
))
<d
, we must have
I(u(t0))=0.
Which implies u(t0)∈Nand thus d≤J(u(t0)), a contradiction.
Hence, J(u(t)) <dand I(u(t)) <0for t∈[0,T]. Now define
P(t)=t
0∥u∥2
H1
0dt
Then,
P′(t)=∥u∥2
H1
0
and
P′′(t)=2(u, ut)1=−2I(u)>0
Hence for t>0,P′(t)≥P′(0) = ∥u0∥2
H1
0>0.
Now fix t1>0. Then for t1≤t<∞,
P(t)≥P(t1)≥t1∥u0∥2
H1
0>0
By Holder’s inequality, we have,
1
4(P′(t)−P′(0))2≤t
0∥u∥2
H1
0dt t
0∥ut∥2
H1
0dt (21)
https://doi.org/10.17993/3ctic.2022.112.108-122
Since I(u(t)) <0, Lemma 2 (iv), gives a γ∗with 0<γ
∗<1and I(γ∗u)=0. Therefore,
d≤1
p−1
q(γ∗)p∥∆u∥p
p+1
q2(γ∗)q∥∇u∥q
q
≤1
p−1
q∥∆u∥p
p+1
q2∥∇u∥q
q(22)
Now by using (4),(6) and (22) we see that,
P′′(t)≥2q(d−J(u0))+2qt
0∥ut∥2
H1
0dt (23)
Then from (21) and (23) it follows that
P′′(t)P(t)−q
2(P′(t)−P′(0))2≥P(t)2q(d−J(u0)) >0for t∈[t1,∞)(24)
Now choose ˜
T>0large enough to introduce,
Q(t)=P(t)+(˜
T−t)∥u0∥2
H1
0for t∈[t1,˜
T]
Then Q(t)≥P(t)>0for t∈[t1,˜
T],Q′(t)=P′(t)−P′(0) >0and Q′′(t)=P′′(t)>0.
Hence from (24) we observe,
Q(t)Q′′(t)−q
2(Q′(t))2≥P(t)2q(d−J(u0)) + P′′(t)( ˜
T−t)∥u0∥2
H1
0>0(25)
Now define
R(t)=Q(t)−q−2
2
Then,
R′(t)=−q−2
2Q(t)−q
2Q′(t)
and
R′′(t)=q−2
2Q(t)−q+2
2q
2(Q′(t))2−Q(t)Q′′(t)<0
Hence
R
(
t
)is a concave function in [
t1,˜
T
]for any sufficiently large
˜
T >t
1
. Also since
R
(
t1
)
>
0
and
R′′
(
t1
)
<
0, there appears a finite time
T∗>t
1>
0having
limt→T−
∗R
(
t
)=0. That yields
limt→T−
∗Q(t)=+∞, which in turn gives limt→T−
∗P(t)=+∞. Hence we get
lim
t→T−
∗∥u∥2
H1
0=+∞.
To obtain an upper limit for blow-up time we define,
S(t)=P(t)+(T∗−t)∥u0∥2
H1
0+σ(t+φ)2for t∈[0,T
∗]
where the constants σ, φ > 0will be given later.
Then,
S′(t)=∥u∥2
H1
0−∥u0∥2
H1
0+2σ(t+φ)>2σ(t+φ)>0(26)
also by (23) we get,
S′′(t)≥2q(d−J(u0))+2qt
0∥ut∥2
H1
0dt +2σ(27)
By Schwartz’s inequality, we have,
t
0
d
dt∥u∥2
H1
0dt ≤2t
0∥u∥2
H1
0dt t
0∥ut∥2
H1
0dt (28)
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
119
Therefore,
(S′(t))2=4
1
2t
0
d
dt∥u∥2
H1
0dt +σ(t+φ)2
≤4t
0
d
dt∥u∥2
H1
0dt +σ(t+φ)2t
0
d
dt∥ut∥2
H1
0dt +σ
=4
S(t)−(T∗−t)∥u0∥2
H1
0t
0
d
dt∥ut∥2
H1
0dt +σ
≤4S(t)t
0
d
dt∥ut∥2
H1
0dt +σ(29)
Now by applying (27) and (29) we can see that,
S(t)S′′(t)−q
2(S′(t))2≥S(t)(2q(d−J(u0)) −2σ(q−1))
If σ∈0,q(d−J(u0))
q−1, then
S(t)S′′(t)−q
2(S′(t))2>0.
Also we have
S
(0) =
T∗∥u0∥2
H1
0
+
σφ2>
0and
S′
(0) = 2
σφ >
0. Then by Levine’s Concavity approach,
we obtain the upper bound for blow-up as,
T∗≤S(0)
(q
2−1)S′(0) =
T∗∥u0∥2
H1
0
(q−2)σφ +φ
q−2
Therefore,
T∗≤σφ2
(q−2)σφ −∥u0∥2
H1
0
thus we must have
φ∈(q−1)∥u0∥2
H1
0
q(q−2)(d−J(u0)),∞
Let υ=σφ ∈0,q(d−J(u0))φ
q−1, then T∗≤φυ
(q−2)υ−∥u0∥H1
0
.
Now let h(φ, υ)= φυ
(q−2)υ−∥u0∥H1
0
. Since his monotonically decreasing concerning υ, we have
inf
{(φ,υ)}h(φ, υ) = inf
{φ}hφ, q(d−J(u0))φ
q−1
= inf
{b}k(φ)
where,
k(φ)=hφ, q(d−J(u0))φ
q−1=φ2q(d−J(u0))
q(q−2)(d−J(u0))φ−(q−1)∥u0∥H1
0
now since k(φ)takes the minimum at φ∗=2(q−1)∥u0∥H1
0
q(q−2)(d−J(u0)) we can conclude that,
T∗≤k(φ∗)=
4(q−1)∥u0∥2
H1
0
q(q−2)2(d−J(u0)).□
The following theorem show that the weak solution of the system blow-up when the initial energy of
the system is critical.
Theorem 5. (Blow up for J(u0)=d)
Let
u0∈W2,p
0
(Ω)
\{
0
},J
(
u0
)=
d
and
I
(
u0
)
<
0, then the weak solution
u
(
t
)of problem (1) blows up in
the sense, there appears a T∗<∞such that limt→T−
∗∥u∥2
H1
0=∞.
https://doi.org/10.17993/3ctic.2022.112.108-122
Proof.Since
J
(
u0
)=
d>
0and
J
(
u
)is continuous with respect to
t
, there appears a
t0
with
J
(
u
(
x, t
))
>
0for 0
<t≤t0
. Also, it is easy to see
I
(
u
(
t
))
<
0for every
t
. Therefore from the energy
inequality, t0
0∥uτ∥2
H1
0dτ +J(u(t0)) <J(u0)=d, it follows that J(u(t0)) <d.
Now choose t=t0as initial time, we have J(u(t0)) <dand I(u(t0)) <0. Now define
P(t)=t
t0∥u∥2
H1
0for t>t
0
and the rest of proof resembles the proof of Theorem (4). □
ACKNOWLEDGMENT
The first author acknowledges the Council of Scientific and Industrial Research(CSIR), Govt. of India,
for supporting by Junior Research Fellowship(JRF).
REFERENCES
[1]
Brill, H. (1977), A semilinear Sobolev evolution equation in a Banach space. J. Differential
Equations, 24, 412-425.
[2]
Changchun Liu, and Pingping Li,(2019). A parabolic p-biharmonic equation with logarithmic
non linearity, U.P.B. Sci. Bull., Series A, 81.
[3]
Changchun Liu, Yitong Ma, and Hui Tang,(2020). Lower bound of Blow-up time to a fourth
order parabolic equation modelling epitaxial thin film growth, Applied Mathematics Letters.
[4]
David, C., and Jet, W.(1979). Asymptotic behaviour of the fundamental solution to the equation
of heat conduction in two temperatures, J. Math. Anal. Appl., 69, 411-418.
[5]
Fugeng Zeng, Qigang Deng, an Dongxiu Wang,(2022). Global Existence and Blow
-
Up for
the Pseudo
-
parabolic p(x)
-
Laplacian Equation with Logarithmic Nonlinearity, Journal of Nonlinear
Mathematical Physics, 29, 41-57.
[6]
Gopala Rao, V.R., and Ting, T. W.,(1972). Solutions of pseudo-heat equations in the whole
space, Arch. Ration. Mech. Anal., 49.
[7]
Hao, A. J., and Zhou, J.,(2017). Blow up, extinction and non extinction for a non local
p-biharmonic parabolic equation, Appl. Math. Lett., 64, 198-204.
[8]
Jiaojiao Wang, and Changchun Liu,(2019). p-Biharmonic parabolic equations with logarithmic
nonlinearity, Electronic J. of Differential Equations, 08, 1-18.
[9]
Lakshmipriya Narayanan, and Gnanavel Soundararajan,(2022). Nonexistence of global
solutions of a viscoelastic p(x)-Laplacian equation with logarithmic nonlinearity, AIP Conference
proceedings 2451.
[10]
Lakshmipriya Narayanan, and Gnanavel Soundararajan, Existence of solutions of a
viscoelastic p(x)-Laplacian equation with logarithmic, nonlinearity Discontinuity, Nonlinearity, and
Complexity, Accepted.
[11]
Lakshmipriya, N., Gnanavel, S., Balachandran, K., and Yong-Ki Ma,(2022) Existence
and blow-up of weak solutions of a pseudo-parabolic equation with logarithmic nonlinearity, Boundary
Value Problems.
[12]
Le, C. N., and Le, X. T.,(2017). Global solution and blow up for a class of p-Laplacian evolution
equations with logarithmic non linearity. Acta. Appl. Math., 151, 149-169.
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
120
Therefore,
(S′(t))2=4
1
2t
0
d
dt∥u∥2
H1
0dt +σ(t+φ)2
≤4t
0
d
dt∥u∥2
H1
0dt +σ(t+φ)2t
0
d
dt∥ut∥2
H1
0dt +σ
=4
S(t)−(T∗−t)∥u0∥2
H1
0t
0
d
dt∥ut∥2
H1
0dt +σ
≤4S(t)t
0
d
dt∥ut∥2
H1
0dt +σ(29)
Now by applying (27) and (29) we can see that,
S(t)S′′(t)−q
2(S′(t))2≥S(t)(2q(d−J(u0)) −2σ(q−1))
If σ∈0,q(d−J(u0))
q−1, then
S(t)S′′(t)−q
2(S′(t))2>0.
Also we have
S
(0) =
T∗∥u0∥2
H1
0
+
σφ2>
0and
S′
(0) = 2
σφ >
0. Then by Levine’s Concavity approach,
we obtain the upper bound for blow-up as,
T∗≤S(0)
(q
2−1)S′(0) =
T∗∥u0∥2
H1
0
(q−2)σφ +φ
q−2
Therefore,
T∗≤σφ2
(q−2)σφ −∥u0∥2
H1
0
thus we must have
φ∈(q−1)∥u0∥2
H1
0
q(q−2)(d−J(u0)),∞
Let υ=σφ ∈0,q(d−J(u0))φ
q−1, then T∗≤φυ
(q−2)υ−∥u0∥H1
0
.
Now let h(φ, υ)= φυ
(q−2)υ−∥u0∥H1
0
. Since his monotonically decreasing concerning υ, we have
inf
{(φ,υ)}h(φ, υ) = inf
{φ}hφ, q(d−J(u0))φ
q−1
= inf
{b}k(φ)
where,
k(φ)=hφ, q(d−J(u0))φ
q−1=φ2q(d−J(u0))
q(q−2)(d−J(u0))φ−(q−1)∥u0∥H1
0
now since k(φ)takes the minimum at φ∗=2(q−1)∥u0∥H1
0
q(q−2)(d−J(u0)) we can conclude that,
T∗≤k(φ∗)=
4(q−1)∥u0∥2
H1
0
q(q−2)2(d−J(u0)).□
The following theorem show that the weak solution of the system blow-up when the initial energy of
the system is critical.
Theorem 5. (Blow up for J(u0)=d)
Let
u0∈W2,p
0
(Ω)
\{
0
},J
(
u0
)=
d
and
I
(
u0
)
<
0, then the weak solution
u
(
t
)of problem (1) blows up in
the sense, there appears a T∗<∞such that limt→T−
∗∥u∥2
H1
0=∞.
https://doi.org/10.17993/3ctic.2022.112.108-122
Proof.Since
J
(
u0
)=
d>
0and
J
(
u
)is continuous with respect to
t
, there appears a
t0
with
J
(
u
(
x, t
))
>
0for 0
<t≤t0
. Also, it is easy to see
I
(
u
(
t
))
<
0for every
t
. Therefore from the energy
inequality, t0
0∥uτ∥2
H1
0dτ +J(u(t0)) <J(u0)=d, it follows that J(u(t0)) <d.
Now choose t=t0as initial time, we have J(u(t0)) <dand I(u(t0)) <0. Now define
P(t)=t
t0∥u∥2
H1
0for t>t
0
and the rest of proof resembles the proof of Theorem (4). □
ACKNOWLEDGMENT
The first author acknowledges the Council of Scientific and Industrial Research(CSIR), Govt. of India,
for supporting by Junior Research Fellowship(JRF).
REFERENCES
[1]
Brill, H. (1977), A semilinear Sobolev evolution equation in a Banach space. J. Differential
Equations, 24, 412-425.
[2]
Changchun Liu, and Pingping Li,(2019). A parabolic p-biharmonic equation with logarithmic
non linearity, U.P.B. Sci. Bull., Series A, 81.
[3]
Changchun Liu, Yitong Ma, and Hui Tang,(2020). Lower bound of Blow-up time to a fourth
order parabolic equation modelling epitaxial thin film growth, Applied Mathematics Letters.
[4]
David, C., and Jet, W.(1979). Asymptotic behaviour of the fundamental solution to the equation
of heat conduction in two temperatures, J. Math. Anal. Appl., 69, 411-418.
[5]
Fugeng Zeng, Qigang Deng, an Dongxiu Wang,(2022). Global Existence and Blow
-
Up for
the Pseudo
-
parabolic p(x)
-
Laplacian Equation with Logarithmic Nonlinearity, Journal of Nonlinear
Mathematical Physics, 29, 41-57.
[6]
Gopala Rao, V.R., and Ting, T. W.,(1972). Solutions of pseudo-heat equations in the whole
space, Arch. Ration. Mech. Anal., 49.
[7]
Hao, A. J., and Zhou, J.,(2017). Blow up, extinction and non extinction for a non local
p-biharmonic parabolic equation, Appl. Math. Lett., 64, 198-204.
[8]
Jiaojiao Wang, and Changchun Liu,(2019). p-Biharmonic parabolic equations with logarithmic
nonlinearity, Electronic J. of Differential Equations, 08, 1-18.
[9]
Lakshmipriya Narayanan, and Gnanavel Soundararajan,(2022). Nonexistence of global
solutions of a viscoelastic p(x)-Laplacian equation with logarithmic nonlinearity, AIP Conference
proceedings 2451.
[10]
Lakshmipriya Narayanan, and Gnanavel Soundararajan, Existence of solutions of a
viscoelastic p(x)-Laplacian equation with logarithmic, nonlinearity Discontinuity, Nonlinearity, and
Complexity, Accepted.
[11]
Lakshmipriya, N., Gnanavel, S., Balachandran, K., and Yong-Ki Ma,(2022) Existence
and blow-up of weak solutions of a pseudo-parabolic equation with logarithmic nonlinearity, Boundary
Value Problems.
[12]
Le, C. N., and Le, X. T.,(2017). Global solution and blow up for a class of p-Laplacian evolution
equations with logarithmic non linearity. Acta. Appl. Math., 151, 149-169.
https://doi.org/10.17993/3ctic.2022.112.108-122
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
121
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3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
122
