Comptes Rendus Mécanique (CRM)

Yancheng Zhang

Qiang Chen

Gildas Guillemot

Charles-André Gandin

Michel Bellet

Numerical modelling of fluid and solid thermomechanics in additive manufacturing by powder-bed fusion: Continuum and level set formulation applied to track- and part-scale simulations

Yuzhu Han

C. R. Mecanique 346 (2018) 1153–1158Contents lists available at ScienceDirect
Comptes Rendus Mecanique
www.sciencedirect.com
A class of fast diffusion p-Laplace equation with arbitrarily 
high initial energy ✩
Yuzhu Han
School of Mathematics, Jilin University, Changchun 130012, PR China
a r t i c l e i n f o a b s t r a c t
Article history:
Received 23 January 2018
Accepted 11 June 2018
Available online 2 August 2018
Keywords:
Fast diffusion
p-Laplace
High initial energy
Global existence
Blow up
In this paper, the authors investigate a class of fast-diffusion p-Laplace equation, which 
was considered by Li, Han and Li (2016) [1], where, among other things, blow-up in finite 
time of solutions was proved for positive but suitably small initial energy. Their results 
will be complemented in this paper in the sense that the existence of finite time blow-up 
solutions for arbitrarily high initial energy will be proved. Moreover, an abstract criterion 
for the existence of global solutions that vanish at infinity will also be provided for high 
initial energy.
© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
1. Introduction
In this paper, we investigate the following fast diffusion p-Laplace equation coupled with the homogeneous Neumann 
boundary condition
⎧⎪⎪⎨
⎪⎪⎩
ut = div(|∇u|p−2∇u) + |u|q−1u − 1||
∫

|u|q−1udx, x ∈ , t > 0
|∇u|p−2 ∂u
∂n
= 0, x ∈ ∂, t > 0
u(x,0) = u0(x), x ∈ 
(1)
where  is a bounded domain in RN (N ≥ 2) with smooth boundary ∂, 2N
N + 2 < p < 2, 2 < q + 1 ≤ p
∗ = Np
N − p , n is the 
unit outward normal on ∂, and the initial datum u0(x) satisfies
0 ≡ u0(x) ∈ L∞() ∩ W 1,p(),
∫

u0(x)dx = 0 (2)
It is easily seen from the structure of the nonlinearity and the homogeneous Neumann boundary condition that the integral 
of the solution u(x, t) to problem (1) with respect to x is conserved, that is 
∫

u(x, t) dx = ∫

u0(x)dx = 0 as long as u(x, t)
exists.
✩ The author is supported by NSFC (11401252) and by Science and Technology Development Project of Jilin Province (20160520103JH).
E-mail address: yzhan@jlu.edu.cn.https://doi.org/10.1016/j.crme.2018.06.013
1631-0721/© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
1154 Y. Han / C. R. Mecanique 346 (2018) 1153–1158In recent years, parabolic equations with nonlocal reaction term like |u|q−1u − 1||
∫

|u|q−1u dx have been investigated 
extensively because of their wide applications in many applied sciences. Interested readers may refer to [2–7,1,8–12] and 
the references therein for the motivation to study problems like (1) and for some results on the existence of local and (or) 
global solutions as well as solutions that blow up in finite time. In particular, the authors in [1] studied finite-time blow-up 
and extinction of solutions to problem (1). To state the blow-up results in [1] a little more precisely, we first recall some 
notations and definitions of functionals in that paper. Denote by ‖u‖r the Lr() norm of a Lebesgue function u ∈ Lr() for 
1 ≤ r ≤ ∞ and let W 1,p∗ () be the subspace of W 1,p(), whose element u satisfies 
∫

u dx = 0. W 1,p∗ () will be equipped 
with the norm ‖u‖
W 1,p∗ ()
= (∫

|∇u|pdx)1/p , which, by recalling Poincaré’s inequality, is equivalent to the classical one 
equipped with W 1,p(). Let B > 0 be the optimal constant of the embedding inequality
‖u‖q+1 ≤ B‖∇u‖p, u ∈ W 1,p∗ () (3)
where 1 < q ≤ (Np − N + p)/(N − p), and set
α1 = B−
q+1
q−p+1 , J1 =
( 1
p
− 1
q + 1
)
B−
p (q+1)
q−p+1 > 0 (4)
Moreover, define the energy functional
J (t)  J (u(x, t)) =
∫

[ 1
p
|∇u(x, t)|p − 1
q + 1 |u(x, t)|
q+1]dx (5)
Then the main blow-up result in [1] is the following.
Theorem 1.1. (Blow-up with positive initial energy) Assume that max{1, 2NN+2 } < p < 2, 1 < q ≤ (Np − N + p)/(N − p) and that 
the initial datum u0(x) is chosen to satisfy J (u0) < J1 and ‖∇u0‖p > α1 , where J1 and α1 are given in (4). Then the weak solution 
u(x, t) to problem (1) blows up in finite time.
In this paper, we will extend the blow-up result obtained in [1] to the case where the initial energy J (u0) is bigger 
than J1. We will show, for any M > J1, that there exists a u0 such that J (u0) > M , and that the solution to problem (1)
with u0 as initial datum blows up in finite time. The organization of the remaining of this paper is as follows. In Section 2, 
we give some notations, definitions, and lemmas concerning the basic properties of the related functionals and sets. The 
main results will be stated and proved in Sections 3.
Remark 1.1. In a recent paper [8], the authors investigated problem (1) for p > 2 and obtained the blow-up results when 
the initial datum u0 satisfies 
p (q + 1)
q + 1 − p ||
q−1
2 J (u0) < ‖u0‖q+12 , by applying Levine’s concavity arguments (see [13]). The 
condition that p > 2 plays an essential role in their proof and it seems that the proof cannot be generalized to the case 
p < 2 trivially. It is not difficult to check that the methods used in this paper can also be applied to the case p > 2, and 
our assumption on the initial data is a little weaker than that in [8]. Moreover, we also give an abstract criterion for the 
existence of global solutions that tend to 0 in W 1,p∗ () as t tends to ∞ (vanish at infinity) in this case.
2. Preliminaries
Since 1 < p < 2, the equation in (1) is singular at the points where ∇u = 0, and therefore classical solutions may not 
exist in general. We first give the definition of weak solutions to problem (1).
Definition 2.1. We say that a function u ∈ L∞( × (0, T )) ∩ Lp(0, T ; W 1,p()) with ut ∈ L2( × (0, T )) is a weak solution to 
problem (1) if
t∫
0
∫

[
uϕs − |∇u|p−2∇u · ∇ϕ +
(
|u|q−1u − 1||
∫

|u|q−1udx
)
ϕ
]
dx ds
=
∫

u(x, t)ϕ(x, t)dx −
∫

u0(x)ϕ(x,0)dx (6)
holds for all ϕ ∈ C1( × [0, T ]).
Y. Han / C. R. Mecanique 346 (2018) 1153–1158 1155For u ∈ W 1,p∗ (), set
J (u) = 1
p
‖∇u‖pp − 1q + 1‖u‖
q+1
q+1
I(u) = ‖∇u‖pp − ‖u‖q+1q+1
and define the Nehari’s manifold
N = {u ∈ W 1,p∗ ()| I(u) = 0, ‖∇u‖p = 0}
N+ = {u ∈ W 1,p∗ ()| I(u) > 0}
N− = {u ∈ W 1,p∗ ()| I(u) < 0}
Since q +1 ≤ Np/(N − p), the functionals J and I are well defined and continuous on W 1,p∗ (). Next, we define the potential 
well (see [14]) and the set outside the potential well respectively by
W = {u ∈ W 1,p∗ ()| I(u) > 0, J (u) < d} ∪ {0}
V = {u ∈ W 1,p∗ ()| I(u) < 0, J (u) < d}
where
d = inf
0 =u∈W 1,p∗ ()
sup
λ≥0
J (λu) = inf
u∈N J (u)
is the depth of the potential well W. The positivity of d will be given in the next lemma.
Lemma 2.1. The depth d of the potential well W is positive.
Proof. Fix u ∈ N . It follows from (3) and the definition of N that
‖∇u‖pp = ‖u‖q+1q+1 ≤ Bq+1‖∇u‖q+1p
which implies ‖∇u‖p ≥
( 1
Bq+1
) 1
q+1−p
. Noticing that q > 1 > p − 1, we have
J (u) = 1
p
‖∇u‖pp − 1q + 1‖∇u‖
p
p = q + 1 − pp (q + 1) ‖∇u‖
p
p (7)
≥ q + 1 − p
p (q + 1)
( 1
Bq+1
) p
q+1−p = J1
Taking infimum over N , we see that d ≥ J1 > 0. The proof is complete. 
Remark 2.1. Since B is the best embedding constant in (3), it is easy to check that d = J1.
For any s > d, define the (closed) sublevels of J by
J s = {u ∈ W 1,p∗ ()| J (u) ≤ s}
By the definition of J (u), N, J s and d, we see that
Ns  N ∩ J s =
{
u ∈ N
∣∣∣
( 1
p
− 1
q + 1
)
‖∇u‖pp ≤ s
}
= ∅, ∀ s > d (8)
We also define
λs = inf{‖u‖2 | u ∈ Ns}, s = sup{‖u‖2 | u ∈ Ns} (9)
It is clear that λs is nonincreasing with respect to s, and that s is nondecreasing.
Finally, we introduce the following sets:
B = {u0 ∈ W 1,p∗ () | the solution u = u(t) of (1) blows up in finite time}
G0 = {u0 ∈ W 1,p∗ () | the solution u = u(t) of (1) tends to 0 in W 1,p∗ () as t → ∞}
To study the long-time behaviors of solutions to problem (1) with high initial energy, we need the following properties 
of the functionals and sets defined above.
1156 Y. Han / C. R. Mecanique 346 (2018) 1153–1158Lemma 2.2.
(i) J (t) defined in (5) satisfies J ′(t) = − ∫

u2t dx, and therefore is nonincreasing in t.
(ii) 0 is away from both N and N− , i.e. dist(0, N) > 0, dist(0, N−) > 0.
(iii) For any s > d, the set J s ∩ N+ is bounded in W 1,p∗ ().
Proof. (i) The proof of (i) is accomplished by taking ut as a test function for smooth solutions. By approximation, we can 
see that it also holds for weak solutions.
(ii) For any u ∈ N , by the definition of d and (7) we see that
d ≤ J (u) = 1
p
‖∇u‖pp − 1q + 1‖∇u‖
p
p = q + 1 − pp (q + 1) ‖∇u‖
p
p
which implies that dist(0, N) = inf
u∈N ‖∇u‖p ≥
( p d (q + 1)
q + 1 − p
)1/p
.
For any u ∈ N− , we have ‖∇u‖p = 0. Then it follows that
‖∇u‖pp < ‖u‖q+1q+1 ≤ Bq+1‖∇u‖q+1p
which yields
‖∇u‖p ≥
( 1
Bq+1
) 1
q+1−p
Therefore, dist(0, N−) = inf
u∈N−
‖∇u‖p > 0.
(ii) For any u ∈ J s ∩ N+ , J (u) ≤ s and I(u) > 0. Therefore, we have
s ≥ J (u) = ( 1
p
− 1
q + 1 )‖∇u‖
p
p + 1q + 1 I(u) >
q + 1 − p
p (q + 1) ‖∇u‖
p
p
and
‖∇u‖pp ≤ p s (q + 1)q + 1 − p
The proof is complete. 
Lemma 2.3. For any s > d, λs and s defined in (9) satisfy
0 < λs ≤ s < +∞ (10)
Proof. To show the positivity of λs we need the following Gagliardo–Nirenberg inequality for u ∈ W 1,p∗ () (see [15]
page 241),
‖u‖q+1q+1 ≤ C‖∇u‖α(q+1)p ‖u‖(1−α)(q+1)2 , ∀ u ∈ W 1,p∗ () (11)
where α is determined by (
1
2
+ 1
N
− 1
p
)α = 1
2
− 1
q + 1 , and C is a positive constant depending only on N, p, and q. Since 
2N
N + 2 < p < 2 and 2 < q + 1 ≤
Np
N − p , it is easy to check that α ∈ (0, 1). Therefore, for any s > d and u ∈ Ns , we obtain 
from (11) that
‖∇u‖pp = ‖u‖q+1q+1 ≤ C‖∇u‖α(q+1)p ‖u‖(1−α)(q+1)2 (12)
which then yields that
‖∇u‖p−α(q+1)p ≤ C‖u‖(1−α)(q+1)2 (13)
By Lemma 2.2 (ii) and (8), we see that the left-hand side of (13) is bounded away from 0 no matter what the sign of 
p − α(q + 1) is. This proves λs > 0 by the definition of λs . On the other hand, the fact that s < ∞ just follows from (8)
and the Sobolev embedding inequality ‖u‖2 ≤ C∗‖∇u‖p since 2N
N + 2 < p is equivalent to 2 < p
∗ = Np
N − p . The proof is 
complete. 
Y. Han / C. R. Mecanique 346 (2018) 1153–1158 11573. Main results
With the help of Lemmas 2.2 and 2.3 and inspired by [16,17], we can give an abstract criterion for the existence of 
global solutions that tend to 0 as t tends to ∞ or finite-time blow-up solutions in terms of λs and s for supercritical 
initial energy, i.e. J (u0) > d.
Theorem 3.1. Assume that J (u0) > d, then the following statements hold:
(i) if u0 ∈ N+ and ‖u0‖2 ≤ λ J (u0) , then u0 ∈ G0;
(ii) if u0 ∈ N− and ‖u0‖2 ≥  J (u0) , then u0 ∈ B.
Proof. Denote by T (u0) or T the maximal existence time of the solutions to problem (1) with initial datum u0. When the 
solution is global, i.e. T = ∞, we denote by
ω(u0) =
⋂
t≥0
{u(s) : s ≥ t}W
1,p∗ ()
the ω-limit set of u0.
(i) Assume that u0 ∈ N+ with ‖u0‖2 ≤ λ J (u0) . We first claim that u(t) ∈ N+ for all t ∈ [0, T ). If not, there would exist 
a t0 ∈ (0, T ) such that u(t) ∈ N+ for t ∈ [0, t0) and u(t0) ∈ N . Recalling Lemma 2.2, (i) we have J (u(t0)) ≤ J (u0), which 
implies that u(t0) ∈ J J (u0) . Therefore, u(t0) ∈ N J (u0) . According to the definition of λ J (u0) , we have
‖u(t0)‖2 ≥ λ J (u0) (14)
On the other hand, it follows by choosing uχ[t1,t2] as a test function in (6) and then using Lebesgue’s differentiation theorem 
that
d
dt
‖u(t)‖22 = −2 I(u) (15)
Noticing that I(u(t)) > 0 for t ∈ [0, t0), we then have
‖u(t0)‖2 < ‖u0‖2 ≤ λ J (u0)
which contradicts the definition of λ J (u0) and the claim is proved. Lemma 2.2 (iii) shows that the orbit {u(t)} remains 
bounded in W 1,p∗ () for t ∈ [0, T ) so that T = ∞. Let ω be an arbitrary element in ω(u0), then by (15) and Lemma 2.2 (iii), 
we have
‖ω‖2 < λ J (u0), J (ω) ≤ J (u0)
which, recalling the definition of λ J (u0) again, implies ω(u0) ∩ N = ∅. Therefore, ω(u0) = {0}, i.e. u0 ∈ G0.
(ii) Assume that u0 ∈ N− with ‖u0‖2 ≥  J (u0) . By applying a similar argument as above, we see that u(t) ∈ N− for all 
t ∈ [0, T ). Now suppose, on the contrary, that T = ∞, then for every ω ∈ ω(u0), it follows from (15) and Lemma 2.2 (i) that
‖ω‖2 >  J (u0), J (ω) ≤ J (u0)
By the definition of  J (u0) again, we then infer that ω(u0) ∩ N = ∅. Therefore, it must hold that ω(u0) = {0}, which is 
contradictive with the fact that dist(0, N−) > 0. Hence, T < ∞ as claimed, and the proof of this theorem is complete. 
Remark 3.1. Since λ J (u0) > 0, Theorem 3.1 (i) is nontrivial. Moreover, Theorem 3.1 (ii) implies that there exists a u0 such 
that J (u0) is arbitrarily large, while the corresponding solution u(x, t) to problem (1) with u0 as initial datum blows up in 
finite time. To illustrate this, we need the following proposition.
Proposition 3.1. Let u0 ∈ W 1,p∗ () with J (u0) > d. If p (q + 1)q + 1 − p ||
q−1
2 J (u0) ≤ ‖u0‖q+12 , then u0 ∈ N− ∩B.
Proof. Since 
p (q + 1)
q + 1 − p ||
q−1
2 J (u0) ≤ ‖u0‖q+12 , we obtain, by making use of Hölder’s inequality, that
p (q + 1)
q + 1 − p ||
q−1
2 J (u0) ≤ ‖u0‖q+12 < ‖u0‖q+1q+1||
q−1
2 (16)
The last inequality in (16) is strict since u0 is not a constant. Recalling the definition of J (u0), I(u0) and noticing (16) we 
have
1158 Y. Han / C. R. Mecanique 346 (2018) 1153–1158J (u0) = 1
p
‖∇u0‖pp − 1q + 1‖u0‖
q+1
q+1 = (
1
p
− 1
q + 1 )‖u0‖
q+1
q+1 +
1
p
I(u0)
= q + 1 − p
p (q + 1) ‖u0‖
q+1
q+1 +
1
2
I(u0) > J (u0) + 1
p
I(u0)
which implies that I(u0) < 0, i.e. u0 ∈ N− .
To show that u0 ∈ B, it remains to prove ‖u0‖2 ≥  J (u0) by Theorem 3.1 (ii). For this, ∀ u ∈ N J (u0) , we have
‖u‖q+12 ≤ ||
q−1
2 ‖u‖q+1q+1 = ||
q−1
2 ‖∇u‖pp
= || q−12 p (q + 1)
q + 1 − p (
1
p
− 1
q + 1 )‖∇u‖
p
p
≤ || q−12 p (q + 1)
q + 1 − p J (u0)
The last inequality holds because of (8). Taking supremum over N J (u0) on the left-hand side of the above inequality and 
recalling (16), we obtain

q+1
J (u0)
≤ || q−12 2 (q + 1)
q − 1 J (u0) ≤ ‖u0‖
q+1
2
i.e. ‖u0‖2 ≥  J (u0) . Therefore, u0 ∈ N− ∩B, and this completes the proof. 
Theorem 3.2. For any M > d, there exists a uM ∈ N− ∩B such that J (uM) ≥ M.
Proof. Similar treatments have been used in [16,17]. We repeat the proof here for the convenience of the readers. For any 
M > d, let 1 and 2 be two arbitrary disjoint open subdomains of , and assume that v ∈ W 1,p∗ (1) is an arbitrary 
nontrivial function. Since q > p − 1, we can choose α > 0 large enough that J (αv) ≤ 0 and ‖αv‖q+12 > ||
q−1
2
p (q + 1)
q + 1 − p M .
Fix α and choose a function w ∈ W 1,p∗ (2) such that J (w) + J (αv) = M . Extend v and w to be 0 in  \ 1 and  \ 2, 
respectively, and set uM = αv + w . Then J (uM) = J (αv) + J (w) = M , and
‖uM‖q+12 ≥ ‖αv‖q+12 > ||
q−1
2
p (q + 1)
q + 1 − p J (uM)
By Proposition 3.1, it is seen that uM ∈ N− ∩B. The proof is complete. 
Acknowledgement
The author would like to express his sincere gratitude to Professor Wenjie Gao for his enthusiastic guidance and constant 
encouragement.
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A class of fast diffusion <i>p</i>-Laplace equation with arbitrarily high initial energy

Ang Liu

Guanghui Tian

Qingzhao Zhang

Wenli Lin

Jingcai Jiang

Shear relaxation characteristics of rock joints under stepwise loadings

Gilles A. Francfort

Recovering convexity in non-associated plasticity

Elias Cueto

Piotr Breitkopf

Francisco Chinesta

C. R. Mecanique 346 (2018) 483Contents lists available at ScienceDirect
Comptes Rendus Mecanique
www.sciencedirect.com
Model reduction, data-based and advanced discretization in computational mechanics
Foreword
The present volume groups a choice of contributions issued from the biannual encounters of the Franco–Spanish scientific 
community of computational methods in applied sciences and engineering, held in Jaca (Spain). During this third workshop 
(after the two first, held in Jaca in 2013 and Biarritz in 2015), we took the opportunity to pay tribute to our colleague and 
friend Pierre Villon, on the occasion of his retirement and subsequent appointment as emeritus professor at the “Université 
de technologie de Compiègne” (France).
The range of topics covers a vivid scope of recent research subjects, on the frontier between applied mathematics and 
computational mechanics. The contributions range from the fundamentals of solid mechanics, and PDE revisited, through 
the physical interpretation of numerical models, mathematical foundations of reduced-order modeling up to methodological 
developments in time and parametric domains.
The issue includes eleven contributions from both sides of the Pyrenees. Model order reduction methods play a fun-
damental role in many of them. For instance, proper generalized decompositions (PGDs) are the main topic of the works 
by A. Leon and coworkers, who present a PGD version based upon wavelets, or of the contribution by P. Díez et al., who 
describe an algebraic version of PGDs.
More fundamental issues of model order reduction are covered in the contributions by A. Falcó et al., who present a 
geometrical interpretation of reduction of dynamical systems, or the work by C. Paillet et al., who analyze the possibility 
of employing PGD for problems with a large number of parameters. A. Madra et al. present a work on diffuse manifold 
interpolation, while A. Cosimo et al. present a work on hyper-reduction methods.
Application of model order reduction to industrial processes is the main topic of the work by N. Bur et al., while, on 
a more general setting, A. Boujelben and A. Ibrahimbegovic present a paper on finite element formulation for large strains 
in nonlinear dynamics, E. Nadal and coworkers discuss the physical sense of fractional derivatives, while M.H. Malik et al. 
propose a new formulation of parametric dynamics within the PGD framework. Finally, J. Yang et al. present a work on the 
application of Taylor series to the derivation of fundamental solutions to PDEs.
Elias Cueto a, Piotr Breitkopf b, Francisco Chinesta c
aI3A, University of Zaragoza, Maria de Luna s/n, 50018 Zaragoza, Spain
bLaboratoire Roberval, UMR 7337 UTC–CNRS, Université de technologie de Compiègne, Centre de recherches de Royallieu,
CS 60319, 60203 Compiègne cedex, France
cPIMM Laboratory & ESI GROUP Chair, ENSAM ParisTech, 151, bd de l’Hôpital, 75013 Paris, France
E-mail address: Francisco .CHINESTA @ensam .eu (F. Chinesta)
3 February 2017
Available online 27 April 2018https://doi.org/10.1016/j.crme.2018.04.012
1631-0721/© 2018 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).


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