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Comptes Rendus
Mécanique
Jun Zhou and Xiongrui Wang
Blow-up time of solutions to a class of pseudo-parabolic equations
Volume 351 (2023), p. 219-226
Published online: 25 May 2023
https://doi.org/10.5802/crmeca.189
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
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e-ISSN : 1873-7234
Comptes Rendus
Mécanique
2023, Vol. 351, p. 219-226
https://doi.org/10.5802/crmeca.189
Spontaneous Articles / Articles spontanés
Blow-up time of solutions to a class of
pseudo-parabolic equations
Jun Zhou∗, a and Xiongrui Wangb
a School of Mathematics and Statistics, Southwest University, Chongqing, 400715,
P. R. China
b Department of Mathematics, Yibin University, Yibin, Sichuan, 644000, P. R. China
E-mails: jzhou@swu.edu.cn, wxr888x@163.com
Abstract. In this paper, we study the Dirichlet problem for a semilinear pseudo-parabolic equation. By using
the energy estimates and ordinary differential inequalities, we studied the upper and lower bounds of blow-
up time of the solutions. The results of this paper extend and complete the results on this model.
Keywords. Pseudo-parabolic equation, Blow-up, Blow-up time, Potential well method, Global existence.
Mathematical subject classification (2010). 35K70, 35A01.
Funding. This work is supported by Natural Science Foundation of Chongqing (CSTB2022NSCQ-MSX1674).
Manuscript received 25 October 2022, revised and accepted 7 April 2023.
1. Introduction
The pseudo-parabolic equation
ut −a∆ut −∆u = f (u) (1)
with a > 0 can be used to describe many interesting physical and biological phenomena [1–6],
for example, the non-stationary process in semiconductors in the presence of sources, where
a∆ut −ut stands for the free electron density rate, ∆u stands for the linear dissipation of free
charge current and the nonlinear term stands for the source of free electron current (see [3, 5]).
Showalter and Ting [7] investigated the initial boundary value problem of (1) with f (u) = 0,
and the global existence, uniqueness and regularity of solutions were studied. When f (u) is a
polynomial, i.e., f (u) = up−1 or f (u) = |u|p−2u, the initial and boundary value problem of (1)
was studied in [8–13], and the existence, asymptotic behavior of the global solutions and global
nonexistence of solutions were studied. When f (u) is a logarithmic function, i.e., f (u) = u ln |u|,
the initial boundary value problem of (1) was studied in [14], where global existence, infinite-
time blow-up of solutions, and behavior of vacuum isolation of solutions were studied. When
f (u) is a nonlocal function, i.e., f (u) = |u|p−2u − 1|Ω|
∫
Ω |u|p−2ud x, where Ω ⊂ RN is a bounded
domain, problem (1) with homogeneous Neumann boundary and initial value was studied [15],
∗Corresponding author.
ISSN (electronic) : 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/
220 Jun Zhou and Xiongrui Wang
and the existence, uniqueness and asymptotic behavior of the global solution and the blow-up
phenomena of solution were studied. For the study of global existence and finite time blow-up of
solutions for pseudo-parabolic equation with potential terms, we refer the readers to [16, 17].
In this paper, we consider the following initial boundary-value problem (IBVP) for a class of
pseudo-parabolic equation proposed by Zhu el al. in [18]:
ut −∆ut −∆u +u = |u|p−2u, x ∈Ω, t > 0,
u(x,0) = u0(x), x ∈Ω,
u(x, t ) = 0, x ∈ ∂Ω, t > 0,
(2)
where p ∈ (2,2∗) and either Ω= RN (N ≥ 3) or Ω⊂RN (N ≥ 3) is a bounded domain with smooth
boundary ∂Ω, u0 ∈ H 10 (Ω). Here
2∗ = 2N
N −2 .
When Ω=RN , the boundary condition that u(x, t ) = 0 for x ∈ ∂Ω and t > 0 is ineffective.
Problem (2) was studied by Zhu el al. in [18] via the potential well method (see, for example,
[19–21]), where the existence, uniqueness, polynomial decay of the global solutions, and blow-up
of solutions were studied. Zhou [22] improve the results of [18] by showing the global solutions
decay exponentially. The main purpose of this paper is to complete the blow-up result given in [18]
by estimating the upper and lower bounds of the blow-up time. To introduce the previous results
and give the main results of the present paper, let’s firstly recall some notations used in [18].
Throughout this paper, we use ∥ · ∥p as the usual Lp -norm, (·, ·) and ∥ · ∥ as the inner product and
the associated norm on H 10 (Ω) respectively, that is
(u, v) =
∫
Ω
(∇u ·∇v +uv)d x (3)
and
∥u∥ =
√
(u,u). (4)
By [18, Theorem 1.3], when p ∈ (2,2∗) and u0 ∈ H 10 (Ω), IVBP (2) admits a (weak) solution
u ∈C 1 ([0,Tmax) , H 10 (Ω)) , (5)
which satisfies u(0) = u0 ∈ H 10 (Ω) and(
u′(t ), v)+ (u(t ), v)= ∫
Ω
|u(t )|p−2u(t )vd x, ∀ v ∈ H 10 (Ω) (6)
where Tmax is the maximum existence time.
As in [18], the energy functional J (u) can be defined through
J (u):=p −2
2p
∥u∥2 + 1
p
I (u), ∀ u ∈ H 10 (Ω), (7)
where
I (u):=∥u∥2 −∥u∥pp , ∀ u ∈ H 10 (Ω). (8)
By (5), J (u(t )) and I (u(t )) are well-defined and are continuous differentiable with respect to t .
Let
N := {u ∈ H 10 (Ω) \ {0} : I (u) = 0} (9)
denote the Nehari manifold. We also define two sets N± related to N as follows:
N+ :=
{
u ∈ H 10 (Ω) : I (u) > 0
}
,
N− :=
{
u ∈ H 10 (Ω) : I (u) < 0
}
.
(10)
Jun Zhou and Xiongrui Wang 221
The main result on global existence and blow-up of [18] is the following theorem (see [18,
Theorem 1.4]):
Theorem 1. Let u = u(t ) be a solution of IBVP (2), whose maximal existence time is Tmax. Then
Tmax <∞ if and only if there exists a t0 ∈ [0,Tmax) such that u(t0) ∈N−. Moreover, if Tmax <∞, then
limt↑Tmax ∥u(t )∥ =∞.
When blow-up occurs, the blow-up time Tmax cannot usually be computed exactly. It is
therefore of great importance in practice to determine the upper and lower bounds for Tmax.
So we complete Theorem 1 by showing the upper and lower bounds of Tmax:
Theorem 2. Let u = u(t ) be a solution of IBVP (2) with initial value u0 ∈ N−, i.e., I (u0) =
∥u0∥2 −∥u0∥pp < 0. Then
1
2
∞∫
∥u0∥2
dξ
κpξ
p 2 −ξ
≤ Tmax ≤ 4(p −1)∥u0∥
2
(p −2)2 (∥u0∥pp −∥u0∥2) , (11)
where κ is the optimal constant of the embedding H 10 (Ω) ,→ Lp (Ω), i.e.,
κ= sup
u∈H 10 (Ω)∥u∥=1
∥u∥p . (12)
Remark 3.
(1) Since p ∈ (2,2∗), it follows H 10 (Ω) ,→ Lp (Ω);
(2) We show that
1
2
∞∫
∥u0∥2
dξ
κpξ
p 2 −ξ
< 4(p −1)∥u0∥
2
(p −2)2 (∥u0∥pp −∥u0∥2) . (13)
So, (11) makes sense. By using I (u0) < 0, it follows from (8) and (12) that
∥u0∥2 < ∥u0∥pp ≤ κp∥u0∥p ,
which implies
∥u0∥ > κ
−p
p−2 . (14)
Let f (ξ) =σκpξ p 2 −ξ for ξ≥ 0, where
σ= κ−p∥u0∥2−p .
Since p > 2, by (14),
σ< κ−p
(
κ
−p
p−2
)2−p
= 1.
Then, for ξ ∈ [∥u0∥2,∞),
f ′(ξ) = p 2σκpξ p 2−1 −1 = p 2∥u0∥2−pξ
p 2−1 −1
≥ p 2∥u0∥2−p
(∥u0∥2) p 2−1 −1
= p 2−1 > 0.
(15)
So,
f (ξ) ≥ f (∥u0∥2)= 0, ξ ∈ [∥u0∥2,∞).
222 Jun Zhou and Xiongrui Wang
Then, note σ ∈ (0,1), we can estimate the integral to the left of (13) as
1
2
∞∫
∥u0∥2
dξ
κpξ
p 2 −ξ
=1
2
∞∫
∥u0∥2
dξ
(1−σ)κpξ p 2 + f (ξ)
≤ 1
2(1−σ)κp
∞∫
∥u0∥2
dξ
ξ
p 2
= 1
(1−σ)κp (p −2)∥u0∥
2−p
= 1(
1−κ−p∥u0∥2−p
)
κp (p −2)∥u0∥
2−p
= 1
(p −2)(κp∥u0∥p−2 −1) .
(16)
By (12), we can estimate the right of (13) as
4(p −1)∥u0∥2
(p −2)2 (∥u0∥pp −∥u0∥2) ≥ 4(p −1)∥u0∥
2
(p −2)2 (κp∥u0∥p −∥u0∥2)
= 4(p −1)
(p −2)2 (κp∥u0∥p−2 −1) .
(17)
Since 4(p −1)/(p −2) > 4, (13) follows from (16) and (17).
(3) It obvious that
∞∫
∥u0∥2
dξ
κpξ
p 2 −ξ
≥
∞∫
∥u0∥2
dξ
κpξ
p 2
= 2
κp (p −2)∥u0∥
2−p .
Then it follows from (11) that
1
κp (p −2)∥u0∥
2−p ≤ Tmax ≤ 4(p −1)∥u0∥
2
(p −2)2 (∥u0∥pp −∥u0∥2) .
2. Proof of Theorem 2
In this section, we give the proof of Theorem 2 by using the following lemma:
Lemma 4 ( [23]). Suppose that 0 < T ≤ +∞ and suppose a nonnegative function F (t ) ∈ C 2[0,T )
satisfies
F ′′(t )F (t )− (1+γ)(F ′(t ))2 ≥ 0
for some constant γ> 0. If F (0) > 0, F ′(0) > 0, then
T ≤ F (0)
γF ′(0)
<∞
and F (t ) →+∞ as t ↑ T .
Proof of Theorem 2. Let u = u(t ) be a solution of problem (2) with initial value u0 ∈ N−. By
Theorem 1, the maximal existence Tmax <∞. So we only need to prove the inequality (11).
(1) Upper bound estimate. In the part, we will give the upper bound estimate, i.e., show that
Tmax ≤ 4(p −1)∥u0∥
2
(p −2)2 (∥u0∥pp −∥u0∥2) , (18)
Jun Zhou and Xiongrui Wang 223
For t ∈ [0,Tmax), let
ξ(t ) :=
 t∫
0
∥u(s)∥2d s

1
2
,
η(t ) :=
 t∫
0
∥u′(s)∥2d s

1
2
.
The following facts can be found in [18, Lemma 3.2 and the proof of Theorem 5.1]:
(R1) J (u(t ))+η2(t ) = J (u0), t ∈ [0,Tmax);
(R2)
d
d t ∥u(t )∥2 =−2I (u(t )), t ∈ [0,Tmax);
(R3) u(t ) ∈N−, i.e., I (u(t )) < 0, t ∈ [0,Tmax).
By (R1)-(R3), it is easy to see
(R4) ∥u(t )∥2 is strictly increasing with respect to t ;
(R5) the function φ(t ) := (p −2)∥u(t )∥2 −2p(J (u(t ))) satisfies
φ(t ) ≥ (p −2)∥u0∥2 −2p
(
J (u0)−η2(t )
)
= 2(∥u0∥pp −∥u0∥2)+2pη2(t )︸ ︷︷ ︸
by (7) and (8)
, t ∈ [0,Tmax) .
Consider the following functional
F (t ) := ξ2(t )+ (Tmax − t )∥u0∥2 +β(t +α)2, t ∈ [0,Tmax) , (19)
where α and β are two positive constants to be determined later. Then by (R4), we have
F ′(t ) = ∥u(t )∥2 −∥u0∥2 +2β(t +α)
≥ 2β(t +α) > 0, t ∈ [0,Tmax) ,
(20)
which implies
F (t ) ≥ F (0) = Tmax∥u0∥2 +βα2 > 0, t ∈ [0,Tmax) (21)
and (by (R2), (7) and (R5))
F ′′(t ) =−2I ((t ))+2β
=φ(t )+2β
≥ 2(∥u0∥pp −∥u0∥2)+2pη2(t )+2β, t ∈ [0,Tmax) . (22)
By (3), the Cauchy–Schwarz inequality and Hölder’s inequality, we have
1
2
t∫
0
d
d s
∥u(s)∥2d s =
t∫
0
(u(s),u′(s))d s
≤
t∫
0
∥u(s)∥∥u′(s)∥d s
≤ ξ(t )η(t ), t ∈ [0,Tmax),
224 Jun Zhou and Xiongrui Wang
which, together with (19), implies(
F (t )− (Tmax − t )∥u0∥22
)(
η2(t )+β)= (ξ2(t )+β(t +α)2)(η2(t )+β)
= ξ2(t )η2(t )+βξ2(t )+β(t +α)2η2(t )+β2(t +α)2
≥ ξ2(t )η2(t )+2ξ(t )η(t )β(t +α)+β2(t +α)2
≥ (ξ(t )η(t )+β(t +α))2
≥
1
2
t∫
0
d
d s
∥u(s)∥2d s +β(t +α)
2 , t ∈ [0,Tmax) .
Then it follows from (20) and the above inequality that
(
F ′(t )
)2 = 4
1
2
t∫
0
d
d s
∥u(s)∥2d s +β(t +α)
2
≤ 4F (t )(η2(t )+β) , t ∈ [0,Tmax) .
(23)
In view of (21), (22) and (23), we have
F (t )F ′′(t )− p 2(F ′(t ))2 ≥ F (t )(2(∥u0∥pp −∥u0∥2)+2pη2(t )+2β−2p (η2(t )+β))
= F (t )(2(∥u0∥pp −∥u0∥2)−2(p −1)β) , t ∈ [0,Tmax) ,
which is nonnegative if we take β small enough such that
0 <β≤ ∥u0∥
p
p −∥u0∥2
p −1 . (24)
Then it follows from Lemma 4 that
Tmax ≤ F (0)( p 2−1)F ′(0)
= 1
p −2
(
α+ ∥u0∥
2
βα
Tmax
)
.
(25)
By taking α large enough such that
α> ∥u0∥
2
(p −2)β , (26)
we get from (25) that
Tmax ≤ βα
2
(p −2)βα−∥u0∥2
. (27)
The above analysis shows that (ρ :=βα)
Tmax ≤ inf
(ρ,α)∈Φ
f
(
ρ,α
)
, (28)
where
Φ :=
{(
ρ,α
)
: ρ > ∥u0∥
2
p −2 , α≥
(p −1)ρ
∥u0∥pp −∥u0∥2
}
,
f
(
ρ,α
)
:= ρα
(p −2)ρ−∥u0∥2
.
Jun Zhou and Xiongrui Wang 225
Since f (ρ, ·) is increasing, we get
Tmax ≤ inf
ρ> ∥u0∥2p−2
f
(
ρ,
(p −1)ρ
∥u0∥pp −∥u0∥2
)
= inf
ρ> ∥u0∥2p−2
(p −1)ρ2(
(p −2)ρ−∥u0∥2
)(∥u0∥pp −∥u0∥2)
= (p −1)ρ
2(
(p −2)ρ−∥u0∥2
)(∥u0∥pp −∥u0∥2)
∣∣∣∣∣
ρ= 2∥u0∥2p−2
= 4(p −1)∥u0∥
2
(p −2)2 (∥u0∥pp −∥u0∥2) .
So, (18) is true.
(2) Lower bound estimate. In the part, we will give the low bound estimate, i.e., show that
Tmax ≥ 1
2
∞∫
∥u0∥2
dξ
κpξ
p 2 −ξ
. (29)
By (R2), (8), and (12), we have
d
d t
∥u(t )∥2 =−2I (u(t ))
=−2∥u(t )∥2 +2∥u(t )∥pp
≤−2∥u(t )∥2 +2κp∥u(t )∥p , t ∈ [0,Tmax).
Since I (u(t )) < 0 (see (R3)), we get −2∥u(t )∥2 + 2κp∥u(t )∥p ≥ −2I (u(t )) > 0, then the above
inequality can be rewritten as
d
d t ∥u(t )∥2
−2∥u(t )∥2 +2κp∥u(t )∥p ≤ 1, t ∈ [0,Tmax). (30)
Since u blow up at Tmax, by Theorem 1,
lim
t↑Tmax
∥u(t )∥2 =∞,
integrating (30) from 0 to Tmax, we get (29). □
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Comptes Rendus
Mécanique
William Pons and Stanley Pons
Addendum: Average Surface Temperature Over the Circular Disk Heat Source/Sink
Embedded in an Insulating Boundary Plane. Analytic Solution
Volume 351 (2023), p. 335
Published online: 7 August 2023
https://doi.org/10.5802/crmeca.217
This article is licensed under the
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http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mécanique sont membres du
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e-ISSN : 1873-7234
Comptes Rendus
Mécanique
2023, Vol. 351, p. 335
https://doi.org/10.5802/crmeca.217
Spontaneous Articles / Articles spontanés
Addendum: Average Surface Temperature
Over the Circular Disk Heat Source/Sink
Embedded in an Insulating Boundary Plane.
Analytic Solution
William Ponsa and Stanley Pons ∗, b, c
a Department of Chemistry, Clemson University, Clemson SC 29634, USA
b Department of Chemistry, University of Utah, Salt Lake City, UT 84112,
USA(Affiliation where work was conceived.)
c Pons(EI), 14, rue Eugene Giraud, 06560 Valbonne, France
E-mails: wpons@clemson.edu (W. Pons), bpons@laposte.net (S. Pons)
Manuscript received 28 June 2023, accepted 6 July 2023.
We find that a result, Equation (20) in “N. Laraqi, International Journal of Thermal Sciences 49
(2010) 529-533”, is equivalent to Equation (10) of “W. Pons, S. Pons, Comptes Rendus. Mécanique,
Volume 351 (2023) p. 17-27”.
While a full independent analysis of the problem in question was accomplished, we apologize
for this omission.
∗Corresponding author.
ISSN (electronic) : 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/


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