Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 51, pp. 1–18.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE OF GLOBAL SOLUTIONS AND BLOW-UP FOR

p-LAPLACIAN PARABOLIC EQUATIONS WITH LOGARITHMIC

NONLINEARITY ON METRIC GRAPHS

RU WANG, XIAOJUN CHANG

Abstract. In this article, we study the initial-boundary value problem for a p-

Laplacian parabolic equation with logarithmic nonlinearity on compact metric

graphs. Firstly, we apply the Galerkin approximation technique to obtain the
existence of a unique local solution. Secondly, by using the potential well

theory with the Nehari manifold, we establish the existence of global solutions

that decay to zero at infinity for all p > 1, and solutions that blow up at finite
time when p > 2 and at infinity when 1 < p ≤ 2. Furthermore, we obtain

decay estimates of the global solutions and lower bound on the blow-up rate.

1. Introduction

This article is devoted to studying the existence global solutions and blow-up
for the p-Laplacian parabolic equations with logarithmic nonlinearity on compact
metric graphs. A metric graph G = (E, V ) is a connected metric space and consists
of a set of edges E and a set of vertices V , where some of the endpoints of the
edges are glued together at some vertices. Any edge e ∈ E is identified either with
a closed bounded interval Ie = [0, `e] of length `e > 0, or with a closed half-line
Ie = [0,∞). A metric graph is compact if and only if it has a finite number of edges
and vertices, and there are no half-lines.

In recent decades, motivated from various applications in physics, chemistry,
biology, and engineering, the study of partial differential equations on metric graphs
has attracted much attention. In particular, the results for parabolic equations
on metric graphs (also called networks or one-dimensional ramified spaces in this
case) first appeared in 1980s, see for example [36, 39, 43, 46]. Since then, there
have been many studies on this subject, see [7, 11, 13, 22, 23, 26, 38] and the
references therein. Around the same period, there were a lot of research conducted
for nonlinear Schrödinger equations on metric graphs (also called quantum graphs),
the interested reader is referred to [2, 3, 6, 8, 11, 25, 27, 28, 40]. For other evolution
equations on metric graphs, we refer the reader to [4, 5, 9, 16, 31].

2020 Mathematics Subject Classification. 35K92, 35B44, 35B40, 35R02.
Key words and phrases. Metric graphs; p-Laplace operator; logarithmic nonlinearity;

global solution; blow-up.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted April 10, 2022. Published July 18, 2022.

1



2 R. WANG, X. J. CHANG EJDE-2022/51

In this article, we are concerned with the p-Laplacian parabolic problem on a
compact metric graph G = (E, V ),

(ue)t − (|u′e|p−2u′e)′ + |ue|p−2ue = |ue|p−2ue log |ue| for each e ∈ E, t > 0,∑
e�v
|due
dxe

(v, t)|p−2 due
dxe

(v, t) = 0 at each v ∈ V, t > 0,

uei(v, t) = uej (v, t) if ei � v, ej � v, t > 0,

ue(x, 0) = (u0)e(x) for each e ∈ E,

(1.1)

where p > 1, ue := ue(x, t) is the restriction of u : G × [0,+∞) → R on the edge
e for each t ≥ 0 (see Sect. 2.1 for details), e � v means that the sum above is
extended to all edges e incident at v, and due

dxe
(v, t) stands for u′e(0, t) if xe = 0

at v and −u′e(`e, t) if xe = `e at v. The initial value (u0)e is the restriction of
u0 ∈ W 1,p(G) on e, where W 1,p(G) is a Banach space that will be introduced in
Sect. 2.1. Here the second equation is known as the Kirchhoff boundary condition
(see [17, 18, 21]), and the third equation is the so called continuity condition at
each vertex v. Our goal in the present paper is to establish a sufficient criterion for
the global existence and blow-up of solutions to (1.1).

The existence of global solutions, and the finite time blow-up of parabolic prob-
lems in Euclidean space have been studied extensively by many scholars. Among
them there are a great many studies devoted to the problems with power non-
linearity (see [1, 24, 32, 35, 45, 48, 50] and the references therein), based on the
potential well method introduced by Payne and Sattinger in [41] (see also [44]). In
recent years, the logarithmic parabolic equations received considerable interest, see
[10, 12, 14, 15, 29, 30]. In fact, the logarithmic nonlinearity has wide applications
in physics and other applied science, and it describes different blow-up mechanism
from the case of power nonlinearity. In [29], the following p-Laplacian parabolic
equation with logarithmic nonlinearity was investigated:

ut −∆pu = |u|p−2u log |u| in Ω× (0, T ),

u(x, t) = 0 on ∂Ω× (0, T ),

u(x, 0) = u0(x) on Ω,

where p > 2, ∆pu := div(|∇u|p−2∇u), Ω ⊂ RN (N ≥ 2) is a bounded domain with
smooth boundary, while the case p = 2 was studied in [14], and the case 1 < p < 2
was studied in [12]. In these studies, by using the logarithmic Sobolev inequality
(see [19, 20, 33]) and the potential well method, it was shown that the blow-up of
solutions occurs at finite time when p > 2 and at infinity when 1 < p ≤ 2.

Motivated by above-mentioned results, it is natural to ask whether similar results
hold for parabolic problems on metric graphs. Recently, in [13], Cazazu et al.
studied the existence and asymptotic behavior of solutions for large times of a
parabolic equation with linear diffusion and a power nonlinear term on a star-
shaped graph. As far as we know, there are no results yet on Cauchy problems of
nonlinear parabolic equations derived by the quasi-linear operator (|u′|p−2u′)′ on
metric graphs.

To deal with the logarithmic nonlinear term in (1.1), we note that, because of
the presence of Kirchhoff boundary condition, the logarithmic Sobolev inequality
involving Lp estimates on real line (see [42]) is not applicable, and hence we can



EJDE-2022/51 p-LAPLACIAN WITH LOGARITHMIC NONLINEARITY 3

not follow directly the strategy as in [12, 14, 29], where the logarithmic Sobolev
inequality is very useful.

In this article, by combining the potential well method and the Nehari manifold,
together with some modified estimates without the use of logarithmic Sobolev in-
equality, we obtain the existence of bounded global solutions that decay to zero at
infinity for all p > 1. Under suitable conditions, we shall show there exist finite
time blow-up solutions when p > 2, and global solutions that blow up at infinity
when 1 < p ≤ 2. In addition, we also obtain the decay estimates of the bounded
global solutions and lower bound for the blow-up rate.

To study the dynamic properties of the solutions to problem (1.1), we define the
energy functional Φ : W 1,p(G)→ R by

Φ(u) :=
1

p

∫
G
|u′|pdx+

1

p

∫
G
|u|pdx+

1

p2

∫
G
|u|pdx− 1

p

∫
G
|u|p log |u|dx.

We define

f(t) :=

{
|t|p−2t log |t|, if t 6= 0,

0, if t = 0.

Since p > 1, f : R → R is continuous. By elementary computations, for any
q ∈ (p,∞), there exists Cq > 0 such that

|f(t)| ≤ Cq(1 + |t|q−1).

Identifying the logarithmic nonlinear term by f(u), using Lemma 2.1 and standard
arguments as in [47, Lemma 2.16], we have Φ ∈ C1(W 1,p(G),R).

The Nehari functional I : W 1,p(G)→ R is defined by

I(u) := 〈Φ′(u), u〉 =

∫
G
|u′|pdx+

∫
G
|u|pdx−

∫
G
|u|p log |u| dx,

where 〈·, ·〉 denotes the dual pairing between W 1,p(G) and its dual space W−1,p
′
(G).

Clearly,

Φ(u) =
1

p
I(u) +

1

p2
‖u‖pp. (1.2)

We define the potential well W and its outside, respectively, by

W := {u ∈ X0 : Φ(u) < d, I(u) > 0},
Z := {u ∈ X0 : Φ(u) < d, I(u) < 0},

where d is the depth of the potential well W and X0 := W 1,p(G) \ {0}.
We define the Nehari manifold by

N := {u ∈ X0 : I(u) = 0}.
Set

d = inf
u∈N

Φ(u). (1.3)

To state our main results, we introduce the following definition.

Definition 1.1. A function u := u(x, t) is called a weak solution of problem (1.1)
on G × (0, T∗), if u ∈ L∞(0, T∗;W

1,p(G)) with ut ∈ L2(0, T∗;L
2(G)) satisfies (1.1)

in the distribution sense, i.e.,∫
G
utvdx+

∫
G
|u′|p−2u′v′dx+

∫
G
|u|p−2uvdx =

∫
G
|u|p−2uv log |u|dx,

for all v ∈W 1,p(G), a.e. t ∈ (0, T∗), where u(x, 0) = u0(x) ∈ X0.



4 R. WANG, X. J. CHANG EJDE-2022/51

Our first result is about the existence of a local solution.

Theorem 1.2. Let G be a compact metric graph and u0 ∈ X0. Then there exists a
positive constant T∗ such that problem (1.1) has a unique weak solution u := u(x, t)
on G × (0, T∗) in the sense of Definition 1.1. Furthermore, u satisfies the energy
inequality ∫ t

0

‖us(·, s)‖22ds+ Φ(u(x, t)) ≤ Φ(u0), a.e. t ∈ [0, T∗]. (1.4)

The existence of a global solution for problem (1.1) can be stated as follows.

Theorem 1.3. Let G be a compact metric graph and u0 ∈ W. Then problem (1.1)
admits a global weak solution u := u(x, t) satisfying the energy estimate∫ t

0

‖us(·, s)‖22ds+ Φ(u(x, t)) ≤ Φ(u0), a.e. t ≥ 0. (1.5)

Moreover, there exists a constant R ≤ d such that

(i) if p > 2 and Φ(u0) < R, then

‖u(·, t)‖2 ≤ ‖u0‖2
( p

2(1 + ζ(p− 2)‖u0‖p−22 t)

) 1
p−2

, ∀t ≥ 0,

where ζ := |G|
2−p
2

(
1
2 − C( 1

2 )(p2Φ(u0))γ−1
)
> 0 with γ > 1 given in Sect.

2.2;
(ii) if 1 < p ≤ 2 and Φ(u0) < R, then

‖u(·, t)‖2 ≤ ‖u0‖2e
1
2−

1
Cα t, ∀t ≥ 0,

where α := 1
2 − C( 1

2 )(p2Φ(u0))γ−1 > 0.

The next theorem is about the blow-up results of problem (1.1).

Theorem 1.4. Let G be a compact metric graph and u0 ∈ Z. Assume that
u := u(x, t) is the local weak solution of problem (1.1) corresponding to this initial
function and satisfies the energy inequality∫ t

0

‖us(·, s)‖22ds+ Φ(u(x, t)) ≤ Φ(u0), ∀t ∈ [0, Tmax). (1.6)

Then the following statements hold:

(i) if p > 2 and Φ(u0) ≤ 0, then u blows up at finite time, i.e.,

lim
t→T−max

‖u(·, t)‖22 = +∞,

where Tmax := 2p
p−2 |G|

p−2
2 ‖u0‖2−p2 ;

(ii) if 1 < p ≤ 2 and Φ(u0) ≤ 0, then u blows up at infinity, i.e.,

lim
t→+∞

‖u(·, t)‖22 = +∞.

Furthermore, if ‖u0‖22 ≤ −p2Φ(u0), the lower bound for the blow-up rate
can be estimated by

‖u(·, t)‖22 ≥ ‖u0‖22.



EJDE-2022/51 p-LAPLACIAN WITH LOGARITHMIC NONLINEARITY 5

In the sequel, for the sake of simplicity, we denote u(x, t) by u(t). This paper
is organized as follows. In Sect. 2, we first present some notation, definitions and
lemmas that will be used throughout the paper, and then obtain the existence of
ground state solutions for a stationary problem associated with (1.1). In Sect. 3,
we obtain the existence and uniqueness of local weak solutions of problem (1.1). In
Sect. 4, we establish the existence of global solutions of problem (1.1). Moreover,
we give the decay estimates. In Sect. 5, under some appropriate conditions, we
prove that, problem (1.1) admits a finite time blow-up weak solution when p > 2,
and a global weak solution that blows up at infinity when 1 < p ≤ 2, respectively.
The lower bound for the blow-up rate is also given.

2. Preliminaries

2.1. Sobolev Space. A function u on a metric graph G is a map u : G → R,
and it can be interpreted as a family of functions {ue}e∈E , where ue : Ie → R
is the restriction of u to the interval Ie such that u|e = ue. Similarly, a function
u : G × [0,+∞) can be seen as the family {ue(x, t)}e∈E with ue : Ie× [0,+∞)→ R
is the restriction of u to the interval Ie for each t ≥ 0.

If u is defined on G (G × [0,+∞) respectively), we denote by u′ (u′(x, t) respec-
tively) the functions with restriction to every Ie given by u′e (u′e(x, t) respectively),
which is the derivative with respect to x. Throughout this paper, we will also
denote u′(x, t) by u′(t) for simplicity if there is no confusion.

Let 1 ≤ p ≤ ∞. Endowing each edge with Lebesgue measure, one can define
the space Lp(G) in a natural way, i.e., we say that u ∈ Lp(G) if ue ∈ Lp(Ie) for all
e ∈ E and

‖u‖pp := ‖u‖pLp(G) :=
∑
e∈E
‖ue‖pLp(Ie) <∞.

The Sobolev space W 1,p(G) is defined as the space of continuous functions u on G
such that ue ∈W 1,p(Ie) for all e ∈ E and

‖u‖p1,p := ‖u‖pW 1,p(G) :=
∑
e∈E

(
‖ue‖pLp(Ie) + ‖u′e‖

p
Lp(Ie)

)
<∞.

By [18], we know that W 1,p(G) is a Banach space for 1 ≤ p ≤ ∞. It is reflexive
for 1 < p <∞ and separable for 1 ≤ p <∞. Throughout the paper, we will denote
by C a positive constant that may vary from place to place. The following two
lemmas will be needed later.

Lemma 2.1 ([18]). Let G be a compact metric graph and 1 < p < ∞. Then the
injection W 1,p(G) ⊂ Lq(G) is compact for all 1 ≤ q ≤ ∞.

Lemma 2.2 ([29]). Let η be a positive number. Then

log s ≤ e−1

η
sη, ∀s ∈ [1,+∞).

By Lemma 2.1 and the interpolation inequality, we obtain the following Gagliardo-
Nirenberg type inequality involving Lp norm on compact metric graphs.

Lemma 2.3. Let 1 < p < +∞ and r > 0. Then, for any u ∈W 1,p(G), there exists
C > 0 such that

‖u‖p+r ≤ C‖u‖θ1,p‖u‖(1−θ)p ,



6 R. WANG, X. J. CHANG EJDE-2022/51

where θ := p+kr
k(p+r) ∈ (0, 1) with k ≥ 2 a positive integer, and C > 0 depends only

on G, p, r, and k.

2.2. Potential well. In this subsection, we consider the minimization problem
(1.3). Let u ∈ X0. We consider the fibering map j : R+ → R given by

j(λ) = Φ(λu), ∀λ > 0.

Clearly,

j(λ) =
λp

p
‖u‖p1,p +

λp

p2
‖u‖pp −

λp log λ

p
‖u‖pp −

λp

p

∫
G
|u|p log |u|dx.

Moreover, we have the following properties.

Lemma 2.4. Let u ∈ X0. Then we have

(1) limλ→0+ j(λ) = 0 and limλ→+∞ j(λ) = −∞;
(2) there is a unique λ∗ := λ∗(u) > 0 such that j′(λ∗) = 0;
(3) j(λ) is increasing on (0, λ∗), decreasing on (λ∗,+∞) and attains the max-

imum at λ∗;
(4) I(λu) > 0 for 0 < λ < λ∗, I(λu) < 0 for λ∗ < λ < +∞ and I(λ∗u) = 0.

Proof. Fix u ∈ X0. (1) is obvious. Differentiating j(λ), we obtain

d

dλ
j(λ) = λp−1

(
‖u‖p1,p − log λ‖u‖pp −

∫
G
|u|p log |u|dx

)
= λp−1

(
I(u)− log λ‖u‖pp

)
.

Therefore, taking λ∗ := λ∗(u) := exp
( I(u)
‖u‖pp

)
, we obtain that (2) and (3) hold. To

show (4), it suffices to check that

I(λu) = λp
(
‖u‖p1,p − log λ‖u‖pp −

∫
G
|u|p log |u|dx

)
= λj′(λ). �

For the Nehari functional I, we have the following result.

Lemma 2.5. Let u ∈ X0. Then there exists a constant l := l(p) > 0 such that

(1) if 0 < ‖u‖p < l, then I(u) > 0;
(2) if I(u) < 0, then ‖u‖p > l;
(3) if I(u) = 0, then ‖u‖p ≥ l.

Proof. By Lemma 2.2, we obtain∫
G
|u|p log |u|dx =

∫
G1
|u|p log |u|dx+

∫
G2
|u|p log |u|dx

≤ 1

ηe

∫
G2
|u|p+ηdx

≤ 1

ηe
‖u‖p+ηp+η,

where η ∈ (0, p2 ), G1 := {x ∈ G : |u(x)| ≤ 1}, and G2 := {x ∈ G : |u(x)| > 1}. Then,
by Lemma 2.3 it follows that, for some C > 0,∫

G
|u|p log |u|dx ≤ C

ηe
‖u‖θ(p+η)1,p ‖u‖(1−θ)(p+η)p ,



EJDE-2022/51 p-LAPLACIAN WITH LOGARITHMIC NONLINEARITY 7

where θ ∈ (0, 1) is given in Lemma 2.3 with k = 2. Note that θ(p+η) < p, applying
the Young inequality with ε, for any ε > 0, there exists C(ε) > 0 such that∫

G
|u|p log |u|dx ≤ ε‖u‖p1,p + C(ε)‖u‖pγp , (2.1)

where

γ :=
(1− θ)(p+ η)

p− θ(p+ η)
> 1.

Thus, by (2.1) and the definition of I, it follows that

I(u) ≥ (1− ε)‖u‖p1,p − C(ε)(‖u‖pp)γ

= (1− ε)‖u′‖pp + ‖u‖pp[(1− ε)− C(ε)‖u‖p(γ−1)p ].

Taking ε = 1/2 yields

I(u) ≥ ‖u‖pp[
1

2
− C(

1

2
)‖u‖p(γ−1)p ]. (2.2)

Set

l :=
( 1

2C( 1
2 )

) 1
p(γ−1)

.

Then, in view of (2.2), it is easily seen that (1)–(3) hold. �

From Lemma 2.4, it is obvious that N is not empty. Moreover, since

Φ(u) =
1

p2
‖u‖pp, ∀u ∈ N ,

by (2.1), we have the following lemma.

Lemma 2.6. Φ is coercive on N .

Next, we prove that the infimum in (1.3) can be attained at some u0 ∈ N . Then,
by standard arguments as in [47], we know that u0 is a critical point of Φ and thus
it is a ground state solution of the stationary problem associated with (1.1).

Lemma 2.7. The following statements hold:

(1) d = infu∈X0 supλ>0 Φ(λu) ≥ R := lp

p2 ;

(2) there exists u0 ∈ N such that Φ(u0) = d.

Proof. For (1), by Lemma 2.4, for any u ∈ X0, there exists a unique λ∗ > 0 such
that λ∗u ∈ N . Specially, λ∗ = 1 if u ∈ N . Furthermore,

d ≤ Φ(λ∗u) = inf
u∈X0

sup
λ>0

Φ(λu) ≤ inf
u∈N

sup
λ>0

Φ(λu) = inf
u∈N

Φ(u) = d. (2.3)

By Lemma 2.4, we have I(λ∗u) = 0, which implies that ‖λ∗u‖p ≥ l. Then, by (2.3),
we obtain

d = Φ(λ∗u) =
1

p2
‖λ∗u‖pp ≥ R.

To show (2), let {un}∞n=1 ⊂ N be a minimizing sequence for Φ such that

lim
n→∞

Φ(un) = d.



8 R. WANG, X. J. CHANG EJDE-2022/51

By Lemma 2.4, {un}∞n=1 is bounded in W 1,p(G). Using Lemma 2.1, up to a subse-
quence if necessary, there exists u0 ∈W 1,p(G) such that

un ⇀ u0 in W 1,p(G),

un → u0 in Lq(G) (1 ≤ q ≤ ∞),

un → u0 a.e. in G.
(2.4)

Then, by Lemma 2.2, (2.4) and similar arguments as in [47, Lemma A.1], we can
apply the Lebesgue dominated convergence theorem to get

lim
n→∞

∫
G
|un|p log |un|dx =

∫
G
|u0|p log |u0|dx.

Hence, we deduce that

Φ(u0) =
1

p
‖u0‖p1,p −

1

p

∫
G
|u0|p log |u0|dx+

1

p2
‖u0‖pp

≤ lim inf
n→∞

(1

p
‖un‖p1,p −

1

p

∫
G
|un|p log |un|dx+

1

p2
‖un‖pp

)
= lim inf

n→∞
Φ(un) = d.

It suffices to show that I(u0) = 0. By similar arguments as above, we have

I(u0) = ‖u0‖p1,p −
∫
G
|u0|p log |u0|dx

≤ lim inf
n→∞

(
‖un‖p1,p −

∫
G
|un| log |un|dx

)
= lim inf

n→∞
I(un) = 0.

We assume by contradiction that I(u0) < 0. By Lemma 2.4, there exists a unique

λ∗ := exp( I(u0)
‖u0‖pp ) < 1 such that I(λ∗u0) = 0. Then, using (2.3),

d ≤ 1

p2
‖λ∗u‖pp ≤

(λ∗)p

p2
lim inf
n→∞

‖un‖pp = (λ∗)p lim inf
n→∞

Φ(un) = (λ∗)pd < d.

A contradiction. This completes the proof. �

3. Existence and uniqueness of local weak solutions

In this section, we prove Theorem 1.2 by using the Galerkin approximation. The
proof is divided into four steps.

Step 1. Approximate problem. Let {Xm}m∈N be a Galerkin scheme of the
separable Banach space W 1,p(G), i.e.,

Xm := span{ϕ1, ϕ2, . . . , ϕm},

where {ϕj}∞j=1 is an orthonormal basis in L2(G). Take u0 ∈ X0. Then there exists
u0m ∈ Xm such that

um(0) := u0m → u0 strong in W 1,p(G) as m→∞. (3.1)

We find the approximate solution um(t) := um(x, t) of problem (1.1) in the form

um(x, t) :=

m∑
j=1

gjm(t)ϕj(x),



EJDE-2022/51 p-LAPLACIAN WITH LOGARITHMIC NONLINEARITY 9

where the coefficients gjm(t) satisfy∫
G
umt(t)ϕidx+

∫
G
|u′m(t)|p−2u′m(t)ϕ′idx+

∫
G
|um(t)|p−2um(t)ϕidx

=

∫
G
|um(t)|p−2um(t)ϕi log |um(t)|dx

(3.2)

with the initial conditions

um(0) =

m∑
j=1

gjm(0)ϕj(x) = u0m. (3.3)

Equivalently, it suffices to consider the initial value problem

g′im(t) = Fi(g(t)), i = 1, 2, . . . ,m, t ∈ [0, t0],

gim(0) = aim, i = 1, 2, . . . ,m,
(3.4)

where

Fi(g(t)) :=−
∫
G
|u′m(t)|p−2u′m(t)ϕ′idx−

∫
G
|um(t)|p−2um(t)ϕidx

+

∫
G
|um(t)|p−2um(t)ϕi log |um(t)|dx.

By the Peano’s theorem in standard ODEs theory, there is t0,m > 0 depending on
|ajm| such that problem (3.4) admits a local solution gjm ∈ C1([0, t0,m]).

Step 2. A priori estimates. Multiplying the ith equation in (3.2) by gim(t) and
summing over i, we obtain

1

2

d

dt
‖um(·, t)‖22 + ‖um(·, t)‖p1,p =

∫
G
|um(t)|p log |um(t)|dx. (3.5)

On the other hand, by Lemma 2.3 and similar arguments as in Sect. 2, for some
γ̂ > 1, we have∫
G
|um(t)|p log |um(t)|dx ≤ ε‖um(·, t)‖p1,p + Cε‖um(·, t)‖2γ̂2 , ∀t ∈ [0, t0,m]. (3.6)

Combining (3.5) with (3.6), we obtain

1

2

d

dt
‖um(·, t)‖22 + (1− ε)‖um(·, t)‖p1,p ≤ Cε‖um(·, t)‖2γ̂2 .

Taking ε = 1/2, there exists C̃1 > 0 such that

d

dt
‖um(·, t)‖22 + ‖um(·, t)‖p1,p ≤ 2C̃1‖um(·, t)‖2γ̂2 ,

which implies that

‖um(·, t)‖22 ≤
( 1

C̃1−γ̂
2 − 2C̃1(γ̂ − 1)t

) 1
γ̂−1

for all 0 < t < T0 :=
C̃1−γ̂

2

2C̃1(γ̂−1)
, where C̃2 := supm∈N∗ ‖u0m‖22. Hence,

‖um(·, t)‖22 ≤ 2
1

γ̂−1 C̃2, ∀t ≤ min{t0,m,
T0
2
}.

Then, for some C > 0,

‖um(·, t0,m)‖22 ≤ 2
1

γ̂−1 (C̃2 + C).



10 R. WANG, X. J. CHANG EJDE-2022/51

Therefore, by extending the solution to the interval [0, T∗] with T∗ := T0

2 , we obtain

‖um(·, t)‖22 ≤ 2
1

γ̂−1 C̃2, ∀t ∈ [0, T∗],

which together with (3.6) implies that, for any ε ∈ (0, 1),∫
G
|um(t)|p log |um(t)|dx ≤ ε‖um(·, t)‖p1,p + 2

γ
γ−1CεC̃

γ̂
2 , ∀t ∈ [0, T∗]. (3.7)

Multiplying (3.2) by g′im(t), summing over i, and integrating over (0, t) we obtain∫ t

0

‖ums(·, s)‖22ds+ Φ(um(t)) = Φ(um(0)). (3.8)

Using (3.1) and the continuity of Φ, there exists C > 0 such that

Φ(um(0)) ≤ C, ∀m.

From (3.7), we have

Φ(um(t)) ≥ 1− ε
p
‖um(·, t)‖p1,p +

1

p2
‖um(·, t)‖pp −

1

p
2

γ
γ−1CεC̃

γ̂
2 .

Then, using (3.8), it follows that, for some C > 0,

‖um‖L∞(0,T∗;W 1,p(G)) ≤ C, (3.9)

‖umt‖L2(0,T∗;L2(G)) ≤ C. (3.10)

Step 3. Passage to the limit. By the prior estimates (3.9) and (3.10), there
exists u ∈ L∞(0, T∗;W

1,p(G)) such that, up to a subsequence,

um → u weakly∗ in L∞(0, T∗;W
1,p(G)), (3.11)

umt → ut weakly∗ in L2(0, T∗;L
2(G)), (3.12)

|u′m|p−2u′m → χ weakly∗ in L∞(0, T∗;W
−1,p′(G)). (3.13)

Since {um}∞m=1 ⊂ L∞(0, T∗;W
1,p(G)) and {umt}∞m=1 ⊂ L2(0, T∗;L

2(G)), by the
Aubin-Lions compactness theorem (see [34]) we obtain, up to a subsequence,

um → u in C([0, T∗];L
r(G)), ∀r ∈ [2,∞).

By (3.1) it follows that, for a.e. (x, t) ∈ G × (0, T∗),

|um(t)|p−2um(t) log |um(t)| → |u(t)|p−2u(t) log |u(t)|. (3.14)

By Lemma 2.2, ∫
G

∣∣|um(t)|p−2um(t) log |um(t)|
∣∣p′dx

=

∫
G1

∣∣|um(t)|p−2um(t) log |um(t)|
∣∣p′dx

+

∫
G2

∣∣|um(t)|p−2um(t) log |um(t)|
∣∣p′dx

≤ C[|G|+
∫
G2
|um(t)|qdx]

≤ C[|G|+ ‖um(·, t)‖q1,p] ≤ CT∗ ,

(3.15)



EJDE-2022/51 p-LAPLACIAN WITH LOGARITHMIC NONLINEARITY 11

where q = (p − 1 + η)p′ > p, p′ = p
p−1 , and G1 := {x ∈ G : |um(t)| ≤ 1},

G2 := {x ∈ G : |um(t)| > 1}. Then, by (3.14)-(3.15) and [34, Lemma 1.3], we obtain

|um|p−2um log |um| → |u|p−2u log |u| weakly∗ in L∞(0, T∗;L
p′(G)). (3.16)

Similarly,

|um|p−2um → |u|p−2u weakly∗ in L∞(0, T∗;L
p′(G)). (3.17)

Using (3.11)-(3.12) and [49, Lemma 3.1.7], we have

um(0)→ u(0) weakly∗ in L2(G).

However, by (3.1) it follows that um(0) → u0 in L2(G). Hence, u(0) = u0, which
implies that u satisfies the initial condition.

In what follows, we take m → ∞ in (3.2). For φ ∈ L2(0, T∗) and ϕi ∈ Xm, we
have ∫ T∗

0

∫
G
umt(t)ϕidxφ(t)dt+

∫ T∗

0

∫
G
|u′m(t)|p−2u′m(t)ϕ′idxφ(t)dt

+

∫ T∗

0

∫
G
|um(t)|p−2um(t)ϕidxφ(t)dt

=

∫ T∗

0

∫
G
|um(t)|p−2um(t)ϕi log |um(t)|dxφ(t)dt.

(3.18)

Letting m → ∞ in (3.18) and using (3.12), (3.13), (3.16) and (3.17), we deduce
that ∫ T∗

0

∫
G
ut(t)ϕidxφ(t)dt+

∫ T∗

0

∫
G
χ(t)ϕ′idxφ(t)dt

+

∫ T∗

0

∫
G
|u(t)|p−2u(t)ϕidxφ(t)dt

=

∫ T∗

0

∫
G
|u(t)|p−2u(t)ϕi log |u(t)|dxφ(t)dt,

which implies that∫
G
ut(t)ϕidx+

∫
G
χ(t)ϕ′idx+

∫
G
|u(t)|p−2u(t)ϕidx

=

∫
G
|u(t)|p−2u(t)ϕi log |u(t)|dx, a.e. in (0, T∗).

By the density of Xm in W 1,p(G), it follows that∫
G
ut(t)ωdx+

∫
G
χ(t)ω′dx+

∫
G
|u(t)|p−2u(t)ωdx

=

∫
G
|u(t)|p−2u(t)ω log |u(t)|dx, a.e. in (0, T∗), ∀ω ∈W 1,p(G).

(3.19)

It suffices to show that χ = |u′|p−2u′ in the weak sense, which can be proven to be
true by using standard arguments from the theory of monotone operators. Hence,
u is the desired solution of problem (1.1).

Step 4. Uniqueness. For p ≥ 2, the uniqueness is derived from the locally
Lipschitz continuity of the nonlinearity f(u) = |u|p−2u log |u|. For 1 < p < 2, f is
just Hölder continuous but not locally Lipschitz continuous, and we need different



12 R. WANG, X. J. CHANG EJDE-2022/51

arguments. Assume by contradiction that (1.1) admits two weak solution u1(t) and
u2(t). Set w(t) = u1(t)− u2(t). Then w(t) satisfies the equation

(we)t(t)− ((p− 1)|w̃′e(t)|p−2w′e(t))′ + (p− 1)|w̃e(t)|p−2we(t)

= ((p− 1) log |w̃e(t)|+ 1)|w̃e(t)|p−2we(t), ∀t > 0,
(3.20)

on each edge e ∈ E, and the initial-boundary value conditions∑
e�v
|dw̃e

dxe
(v, t)|p−2 dwe

dxe
(v, t) = 0 at each v ∈ V, t > 0,

wei(v, t) = wej (v, t) if ei � v, ej � v, t > 0,

we(x, 0) = 0 for each e ∈ E,

(3.21)

where w̃(t) = τu1(t) + (1− τ)u2(t) with τ ∈ [0, 1].
Multiplying (3.20) by we(t), integrating on Ie and summing over all e ∈ E, by

(3.21) we have

1

2

d

dt

∫
G
|w(t)|2dx+

∫
G

(p− 1)|w̃′(t)|p−2|w′(t)|2dx+

∫
G

(p− 1)|w̃(t)|p−2|w(t)|2dx

=

∫
G

((p− 1) log |w̃(t)|+ 1)|w̃(t)|p−2|w(t)|2dx.

Integrating on (0, t) with t ∈ (0, T ), by 1 < p < 2 it follows that

1

2

∫
G
|w(t)|2dx ≤

∫ t

0

∫
G

((p− 1) log |w̃(s)|+ 1)|w̃(s)|p−2|w(s)|2 dx ds

≤ C
∫ t

0

∫
G
|w(s)|2 dx ds

for some constant C > 0 independent of u1(t) and u2(t). By the Gronwall’s in-
equality we obtain

∫
G |w(s)|2dx = 0 for a.e. s ∈ (0, t). Hence w(t) = 0 a.e. in

G × (0, T ).
Finally, we prove (1.4). Let ψ ∈ C([0, T∗]) such that ψ ≥ 0. By (3.8) we obtain∫ T∗

0

ψ(t)

∫ t

0

‖ums(·, s)‖22 ds dt+

∫ T∗

0

Φ(um(t))ψ(t)dt

=

∫ T∗

0

Φ(um(0))ψ(t)dt.

(3.22)

Using Lemma 2.1 and (3.1), we obtain∫ T∗

0

Φ(um(0))ψ(t)dt→
∫ T∗

0

Φ(u0)ψ(t)dt as m→∞.

From this and (3.22), it follows that∫ T∗

0

ψ(t)

∫ t

0

‖us(·, s)‖22 ds dt+

∫ T∗

0

Φ(u(t))ψ(t)dt ≤
∫ T∗

0

Φ(u0)ψ(t)dt.

Since ψ is arbitrary, we obtain∫ t

0

‖us(·, s)‖22ds+ Φ(u(t)) ≤ Φ(u0), a.e. t ∈ [0, T∗].



EJDE-2022/51 p-LAPLACIAN WITH LOGARITHMIC NONLINEARITY 13

4. Existence of global weak solutions

In this section, we prove Theorem 1.3. When u0 ∈ W, we prove that problem
(1.1) admits a global weak solution for all p > 1. Furthermore, we show that the
norm ‖u(·, t)‖2 decays polynomially when p > 2, and decays exponentially when
1 < p ≤ 2. We first recall the following result.

Lemma 4.1 ([37]). Let h : R+ → R+ be a non-increasing function, σ, ρ are
constants such that ρ > 0, σ ≥ 0 and∫ +∞

t

h1+σ(s)ds ≤ 1

%
fσ(0)h(t), ∀t ≥ 0.

Then

(1) if σ = 0, then h(t) ≤ h(0)e1−%t, for all t ≥ 0;

(2) if σ > 0, then h(t) ≤ h(0)
(

1+σ
1+%σt

)1/σ
, for all t ≥ 0.

Proof of Theorem 1.3. Assume that u0 ∈ W. Clearly, Φ(u0) > 0. Denote by Tm
the maximal time of existence of um(t). Then, by (3.8), we have∫ t

0

‖ums(·, s)‖22ds+ Φ(um(t)) = Φ(u0m), 0 ≤ t < Tm. (4.1)

Since Φ(u0) < d, by (3.1) and the continuity of Φ it follows that∫ t

0

‖ums(·, s)‖22ds+ Φ(um(t)) < d, 0 ≤ t < Tm (4.2)

for sufficiently large m.
In the following, we shall prove that Tm = +∞ and, for m large enough,

um(t) ∈ W, ∀t ≥ 0. (4.3)

In fact, if not, there exists t∗ ∈ [0, Tm) such that um(t∗) ∈ ∂W, which implies that

Φ(um(t∗)) = d or I(um(t∗)) = 0. (4.4)

By (4.2) we obtain I(um(t∗)) = 0. Then by the definition of d, it follows that
Φ(um(t∗)) ≥ d, which is contrary to (4.2). Hence (4.3) holds. Thus

‖um(·, t)‖pp < dp2 and

∫ t

0

‖ums(·, s)‖22ds < d (4.5)

hold for sufficiently large m and t ∈ [0, Tm). Moreover, by (2.1), for any ε ∈ (0, 1),
we have

‖um(·, t)‖p1,p = pΦ(um(t)) +

∫
G
|um(t)|p log |um(t)|dx− 1

p
‖um(·, t)‖pp

≤ pd+ ε‖um(·, t)‖p1,p + C(ε)(dp2)γ ,

which implies that for some Cd > 0,

‖um(·, t)‖p1,p ≤ Cd, ∀t ∈ [0, Tm). (4.6)

Thus Tm = +∞ for all m. Now, using (4.5), (4.6), and (3.1), by similar arguments
as in previous section, it follows that problem (1.1) admits a global solution u :=
u(t).



14 R. WANG, X. J. CHANG EJDE-2022/51

In what follows, we consider the decay of ‖u(·, t)‖2. Let R be as in Lemma 2.7
and assume Φ(u0) < R. By um(t) ∈ W and (4.1) we obtain (4.5) holds. Then,
using (2.1) and similar arguments as above, we obtain

I(um(t)) ≥ (
1

2
− C(

1

2
)(p2Φ(u0m))γ−1)‖um(·, t)‖pp. (4.7)

On the other hand, for each T > 0, multiplying both sides of (3.2) by gim(t),
summing over i ∈ {1, 2, . . . ,m} and integrating over (t, T ) yields∫ T

t

I(um(s))ds = −
∫ T

t

∫
G
usm(s)um(s) dx ds ≤ 1

2
‖um(·, t)‖22, ∀t ∈ [0, T ]. (4.8)

Case 1: p > 2. By (4.7) and the Hölder’s inequality, we have

I(um(t)) ≥ |G|
2−p
2 (

1

2
− C(

1

2
)(p2Φ(u0m))γ−1)‖um(·, t)‖p2 = ζm‖um(·, t)‖p2, (4.9)

where ζm := |G|
2−p
2 ( 1

2 − C( 1
2 )(p2Φ(u0m))γ−1) > 0. By (4.8) and (4.9) it follows

that ∫ T

t

‖um(·, s)‖p2ds ≤
1

2ζm
‖um(·, t)‖22, ∀t ∈ [0, T ]. (4.10)

On the other hand, by (4.5) and (4.6), we have {um}∞m=1 ⊂ L∞(0, T ;W 1,p(G))
and {umt}∞m=1 ⊂ L2(0, T ;L2(G)). By (3.1) and the continuity of Φ, we may assume
that ζm → ζ asm→∞. Lettingm→∞ in (4.10), by the Aubin-Lions compactness
theorem it follows that∫ T

t

‖u(·, s)‖p2ds ≤
1

2ζ
‖u(·, t)‖22, ∀t ∈ [0, T ].

Letting T →∞, by Lemma 4.1(2), we obtain

‖u(·, t)‖2 ≤ ‖u0‖2
( p

2(1 + ζ(p− 2)‖u0‖p−22 t)

) 1
p−2

, ∀t ≥ 0.

Case 2: 1 < p ≤ 2. By Lemma 2.1 and (4.6), we have∫
G
|um(t)|2dx ≤ sup

G
|um(t)|2−p

∫
G
|um(t)|pdx ≤ C‖um(·, t)‖pp.

Using (4.7), we obtain

I(um(t)) ≥ C(
1

2
− C(

1

2
)(p2Φ(u0m))γ−1)‖um(·, t)‖22 = Cαm‖um(·, t)‖22, (4.11)

where αm := 1
2 − C( 1

2 )(p2Φ(u0m))γ−1 > 0. Combining (4.8) with (4.11) yields∫ T

t

‖um(·, s)‖22ds ≤
1

Cαm
‖um(·, t)‖22, ∀t ∈ [0, T ]. (4.12)

Assume αm → α as m→∞. Letting m→∞ in (4.12) we obtain∫ T

t

‖u(·, s)‖22ds ≤
1

Cα
‖u(·, t)‖22, ∀t ∈ [0, T ].

Letting T →∞, by Lemma 4.1 (1) it follows that

‖u(·, t)‖2 ≤ ‖u0‖2e
1
2−

1
Cα t, ∀t ≥ 0.

This completes the proof. �



EJDE-2022/51 p-LAPLACIAN WITH LOGARITHMIC NONLINEARITY 15

5. Blow-up

In this section, we prove Theorem 1.4. When u0 ∈ Z, we prove that the local
solution of problem (1.1) blows up at finite time when p > 2, and the global solution
of problem (1.1) blows up at infinity when 1 < p ≤ 2.

Proof Theorem 1.4. Assume that u(t) is the unique local solution obtained in The-
orem 1.2, which admits a maximal existence interval [0, Tmax) for some Tmax > 0.
Clearly, (1.6) holds on [0, Tmax).

We claim that

if u0 ∈ Z, then u(t) ∈ Z for all t ∈ [0, Tmax). (5.1)

Note that, by similar arguments as in Theorem 1.3, we can prove that

um(t) ∈ Z, ∀t ∈ [0, Tmax)

for sufficiently large m. By (3.11), (3.12) and similar arguments as in previous
section, we obtain

u(t) ∈ Z, ∀t ∈ [0, Tmax).

Hence the claim holds.
We define M(t) := ‖u(·, t)‖22. Then from (5.1), we obtain

M ′(t) = 2

∫
G
ut(t)u(t)dx

= 2
(
− ‖u(·, t)‖p1,p +

∫
G
|u(t)|p log |u(t)|dx

)
= −2I(u(t)) > 0, ∀t ∈ [0, Tmax).

(5.2)

Furthermore, using (1.6) and (5.2), we have

M ′(t) ≥ 2p

∫ t

0

‖us(·, s)‖22ds+
2

p
‖u(·, t)‖pp − 2pΦ(u0). (5.3)

Case 1: p > 2. By Φ(u0) ≤ 0 and (5.3) we obtain

M ′(t) ≥ 2

p
‖u(·, t)‖pp ≥

2

p
|G|

2−p
2 ‖u(·, t)‖p2 =

2

p
|G|

2−p
2 M(t)

p
2 ,

which implies that

‖u(·, t)‖22 ≥
( 1

‖u0‖2−p2 − κt

) 2
p−2

,

where κ := p−2
2p |G|

2−p
2 . Hence, letting Tmax := ‖u0‖2−p2 /κ, we obtain

lim
t→T−max

‖u(·, t)‖22 = +∞.

Case 2: 1 < p ≤ 2. By (1.6), (5.2) and Φ(u0) ≤ 0, it follows that

M ′(t) ≥ p
∫ t

0

‖us(·, s)‖22ds. (5.4)

We claim that, for any t0 > 0,∫ t0

0

‖us(·, s)‖22ds > 0.



16 R. WANG, X. J. CHANG EJDE-2022/51

If not, we may assume that there exists t0 > 0 such that
∫ t0
0
‖us(·, s)‖22ds = 0,

which implies that ut(t) = 0 for a.e. (x, t) ∈ G × (0, t0). Then I(u(t)) = 0 for a.e.
t ∈ (0, t0), and thus

Φ(u(t)) =
1

p2
‖u(·, t)‖pp.

Since Φ(u(t)) ≤ Φ(u0) ≤ 0, we obtain ‖u(·, t)‖p = 0 for all t ∈ [0, t0], which
produces a contradiction. Hence, the claim holds.

Fix t0 > 0. Clearly, µ :=
∫ t0
0
‖us(·, s)‖22ds > 0. Integrating (5.4) over (t0, t), we

obtain

M(t) ≥M(t0) + p

∫ t

t0

∫ τ

0

‖us(·, s)‖22dsdτ

≥ ‖u(·, t0)‖22 + pµ(t− t0)

≥ pµ(t− t0).

Hence limt→∞M(t) =∞. Furthermore, by (5.2), we have

M ′(t) = −2I(u(t)) ≥ −2pΦ(u0)− 2

p
‖u(·, t)‖22 = −2pΦ(u0)− 2

p
M(t),

which implies that (
e

2
p tM(t)

)′ ≥ −2pe2t/pΦ(u0).

Integrating over (0, t), by Φ(u0) ≤ 0 and ‖u0‖22 ≤ −p2Φ(u0), we obtain

M(t) ≥ ‖u0‖22.

The proof is complete. �

Acknowledgments. This research was supported by the NSFC (11971095) and
the NSFJL (20190201136JC). This work was done when X. J. Chang visited the
Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, during
the period from 2021 to 2022 under the support of China Scholarship Council
(202006625034). X. J. Chang would like to thank the Laboratoire for their support
and kind hospitality.

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Ru Wang

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024,

Jilin, China
Email address: wangr076@nenu.edu.cn

Xiaojun Chang (corresponding author)
School of Mathematics and Statistics & Center for Mathematics and Interdisciplinary

Sciences, Northeast Normal University, Changchun 130024, Jilin, China

Email address: changxj100@nenu.edu.cn


	1. Introduction
	2. Preliminaries
	2.1. Sobolev Space
	2.2. Potential well

	3. Existence and uniqueness of local weak solutions
	4. Existence of global weak solutions
	5. Blow-up
	Acknowledgments

	References