Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 66, pp. 1–13.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.66

CAUCHY PROBLEM FOR THE LANE-EMDEN HEAT FLOW

WITH SIGN-CHANGING INITIAL DATA

RUI HUANG, SHANMING JI, YANSHENG MA

Abstract. This article concerns the blow-up phenomenon of sign-changing

solutions to the Lane-Emden heat flow. We construct sign-changing weak
sub-solutions and localization of the positive and negative parts of the sign-

changing solutions. We also extend the results to a nonlinear and finite diffu-
sion equations.

1. Introduction

We consider the initial-boundary value problem for n-dimensional evolutionary
Lane-Emden heat flow

∂u

∂t
= ∆u+ |u|p−1u, x ∈ Ω, t > 0,

u(x, t) = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

(1.1)

where p > 1, Ω ⊂ Rn is a bounded domain with smooth boundary, u0 ∈ L∞(Ω) is
non-negative or sign-changing. The closely related stationary problem of (1.1) is

−∆u = |u|p−1u, x ∈ Ω,

u(x) = 0, x ∈ ∂Ω, u(x) > 0, x ∈ Ω.
(1.2)

For a convex domain or a star-shaped domain Ω, it was known by Pohozaev [7]
that the positive solution of (1.2) exists if and only if 1 < p < 2∗, where

2∗ :=

{
+∞, n = 1, 2,
n+2
n−2 , n ≥ 3.

Especially for the radially symmetric case such that Ω = BR, the stationary problem
(1.2) is called the Lane-Emden problem arising from the study of stellar interiors
[1, 6]. If the domain Ω is not star-shaped, an annulus B2R\BR for example, positive
solutions may still exist for super-critical case p ≥ 2∗, see [2, 10] and the references
therein. For periodic problems with periodic coefficients related to (1.1), we refer
the readers to Esteban [3, 4], Quittner [8], Yin and Jin [12].

2020 Mathematics Subject Classification. 35K57, 35B40, 35B44, 35K20.

Key words and phrases. Lane-Emden heat flow; sign-changing solution; blow-up.
©2024. This work is licensed under a CC BY 4.0 license.
Submitted August 22, 2024. Published October 29, 2024.

1



2 R. HUANG, S. JI, Y. MA EJDE-2024/66

Hereafter, we always assume that p > 1 and the stationary problem (1.2) admits
a positive solution, denoted by uΩ as it depends on Ω. Therefore, if Ω is star-shaped,
we assume that 1 < p < 2∗.

It is worth noticing that, from Quittner [9, Theorem 17.8], there is no globally
existent solution “above” the equilibrium uΩ, namely the solutions with u0(x) ≥
uΩ(x) and u0(x) ̸≡ uΩ(x). This paper aims to show that the blow-up phenomenon
is also valid for sign-changing solutions. For any sign-changing function f(x), x ∈ Ω,
we denote

f+(x) := max{f(x), 0}, f−(x) := max{−f(x), 0}, (1.3)

such that f(x) = f+(x)− f−(x) and 0 ≤ f±(x) ≤ |f(x)|. Further, we define

Ω+
f := {x ∈ Ω; f(x) > 0}, Ω−

f := {x ∈ Ω; f(x) < 0},
Γf := {x ∈ Ω; f(x) = 0}.

(1.4)

Clearly, Ω+
f , Ω

−
f , and Γf are disjoint subsets of Ω such that Ω = Ω+

f ∪ Ω−
f ∪ Γf ,

and supp f± = Ω±
f .

It may happen that Γf has interior points and both Ω+
f and Ω−

f are non-empty.

If we take such kind of functions as initial data, meaning that u0(x) ≡ 0 in some
open subset G ⊂ Ω and u±

0 ̸≡ 0, according to the smoothing effect of the parabolic
operator we know that Ω±

u(x,t) are non-empty but Γu(x,t) has no interior points for

small time interval t ∈ (0, δ). Therefore, the interface between Ω+
u(x,t) and Ω−

u(x,t)

is not continuous as t → 0+, which shows one of the main difficulties arising in the
study of asymptotic behaviors of sign-changing solutions to parabolic problems.

Another example illustrating the complexity of large time behaviors of sign-
changing solutions is a special case with geometric symmetry: If Ω is symmetric
with respect to a hyperplane Π (taking Π := {x ∈ Rn;x1 = 0} for simplicity),
i.e., x = (x1, x2, · · · , xn) ∈ Ω if and only if x̂ := (−x1, x2, · · · , xn) ∈ Ω, and the
initial data u0(x) is antisymmetric with respect to the same hyperplane Π, i.e.,
u0(x̂) = −u0(x), then the solution u(x, t) is also antisymmetric with respect to Π.
Further if u0(x) ≥, ̸≡ uΩ+(x) for x ∈ Ω+, where Ω± := {x ∈ Ω;±x1 > 0}, then
the solution u(x, t) blows up to positive infinity on Ω+ and blows up to negative
infinity on Ω− at the same time.

Observing the above two phenomena, we present the following asymptotic be-
havior of sign-changing solutions to the initial-boundary-value problem (1.1) with
conditions on the initial data such that the solution cannot blow up to negative
infinity.

Theorem 1.1. Assume that u0 ∈ L∞(Ω) is sign-changing and satisfies the follow-

ing conditions: there exist disjoint non-empty open subsets Ω̂±
u0

⊂ Ω and relatively

closed subset Γ̂u0
⊂ Ω such that

Ω−
u0

⊂ Ω̂−
u0
, Ω̂+

u0
⊂ Ω+

u0
, Γ̂u0

= ∂Ω̂+
u0

∩ ∂Ω̂−
u0

∩Ω, Ω = Ω̂+
u0

∪ Ω̂−
u0

∪ Γ̂u0
, (1.5)

we further assume that the boundary of Ω̂±
u0

is piecewise smooth and the unit normal

vector at x ∈ Γ̂u0
is denoted by ν pointing in the direction from Ω̂+

u0
to Ω̂−

u0
, the

stationary Lane-Emden problem (1.2) on Ω̂±
u0

admits positive solutions uΩ̂±
u0

(zero

extended to Ω). Moreover,

u−
0 (x) ≤, ̸≡ uΩ̂−

u0
(x), x ∈ Ω̂−

u0
, u+

0 (x) ≥, ̸≡ uΩ̂+
u0
(x), x ∈ Ω̂+

u0
, (1.6)



EJDE-2024/66 LANE-EMDEN HEAT FLOW WITH SIGN-CHANGING INITIAL DATA 3

and ( ∂

∂ν
uΩ̂+

u0

)−
≤

( ∂

∂ν
uΩ̂−

u0

)+

, x ∈ Γ̂u0 , (1.7)

where
(

∂f
∂ν

)±
denotes the one-side directional derivative of f(x) with respect to ν.

Under the above conditions, we have that

(i) the solution u(x, t) ≥ uΩ̂+
u0
(x)−uΩ̂−

u0
(x) for all t ∈ (0, Tmax), where Tmax ∈

(0,+∞) is the maximal existence time of u(x, t);
(ii) the solution u(x, t) blows up to positive infinity in finite time

lim
t→T−

max

sup
x∈Ω

u(x, t) = +∞.

According to the blow-up rate estimates established by Giga-Matsui-Sasayama
[5] in convex domain with 1 < p < 2∗, it holds

∥u(x, t)∥L∞(Ω) ≤ C(Tmax − t)−
1

p−1 , for t ∈ (0, Tmax).

Our result in Theorem 1.1 shows that the Lane-Emden problem does admit a
solution blowing up in finite time with sign-changing initial data.

The main idea of the proof is the localization of positive and negative parts of
the sign-changing solutions based on the comparison with weak sub-solutions.

We present a simple one-dimensional example of initial data satisfying the con-
ditions in Theorem 1.1: let Ω = (a, b), c ∈ [a+b

2 , b), and

u0(x) :=

{
λ(x− a)(x− c), x ∈ (a, c),

−µ(x− c)(x− b), x ∈ (c, b),

where λ > 0 is sufficiently large and µ > 0 is sufficiently small. Then, according to
Theorem 1.1, the solution blows up to positive infinity in finite time.

The rest of this articleis organized as follows. In Section 2, we construct weak
sub-solutions, localize the positive and negative parts of the sign-changing solutions,
and utilize them to provide a proof of the main theorem. In Section 3, we will extend
the method developed in Section 2 to study nonlinear diffusion equations with finite
propagation speed and obtain their sign-changing blow-up solutions.

2. Proof of the main results

The key ingredient is the construction and verification of weak sub-solutions,
which makes it possible to localize the positive and negative parts of the sign-
changing solution.

Lemma 2.1. Under the assumptions in Theorem 1.1, the sign-changing function

u(x) := uΩ̂+
u0
(x)− uΩ̂−

u0
(x)

is a weak sub-solution to the initial-boundary-value problem (1.1), where uΩ̂±
u0
(x)

is the positive solution (with zero extension to Ω) of the stationary Lane-Emden

problem (1.2) on Ω̂±
u0
.

Proof. According to condition (1.6),

u(x) = uΩ̂+
u0
(x)− uΩ̂−

u0
(x) ≤ u+

0 (x)− u−
0 (x) = u0(x)



4 R. HUANG, S. JI, Y. MA EJDE-2024/66

for x ∈ Ω, and u(x) = 0 for x ∈ ∂Ω. We only need to show the differential inequality∫
Ω

∇u(x) · ∇φ(x)dx ≤
∫
Ω

up(x)φ(x)dx, (2.1)

for any 0 ≤ φ(x) ∈ C∞
0 (Ω). We employ the partition of unity such that

φ(x) ≡ χ+(x)φ(x) + χ−(x)φ(x) + χ0(x)φ(x) =: φ+(x) + φ−(x) + φ0(x), (2.2)

where suppχ± ⊂ Ω̂±
u0
, 0 ≤ χ+(x), χ−(x), χ0(x) ≤ 1 are smooth functions. For the

differential inequality (2.1) supported in Ω̂±
u0
, we have∫

Ω

∇u(x) · ∇φ±(x)dx =

∫
Ω̂±

u0

−∆u(x) · φ±(x)dx

=

∫
Ω̂±

u0

up(x) · φ±(x)dx =

∫
Ω

up(x) · φ±(x)dx.

(2.3)

If we take χ±(x) sufficiently large such that Ω̂±
u0
\ suppχ± is sufficiently “narrow”,

then suppχ0 is sufficiently close to Γ̂u0 , meaning that there exists ε > 0 (sufficiently
small) such that

suppχ0 ⊂ Γ̂ε
u0

:= {x ∈ Ω : dist(x, Γ̂u0
) < ε}.

Therefore, near any point x ∈ Γ̂u0
, noticing that uΩ̂±

u0
(x) = 0 for x ∈ Γ̂u0

, the gradi-

ent ∇u(x) in the differential inequality (2.1) can be approximated by ( ∂
∂νuΩ̂+

u0
)− ·ν

for the negative side of Γ̂u0
and approximated by ( ∂

∂νuΩ̂−
u0
)+ · ν for positive side of

Γ̂u0 , where ν is the unit normal vector at x ∈ Γ̂u0 pointing in the direction from

Ω̂+
u0

to Ω̂−
u0
. That is,

∇u(x) =
( ∂

∂ν
uΩ̂+

u0

)−
· ν · χΩ̂+

u0
(x) +

( ∂

∂ν
uΩ̂−

u0

)+

· ν · χΩ̂−
u0
(x) + o(1), x ∈ Γ̂ε

u0
,

where χΩ̂±
u0
(x) is the characteristic function of the set Ω̂±

u0
, and o(1) is an infinites-

imal as ε → 0+. Then we have∫
Ω

∇u(x) · ∇φ0(x)dx

=

∫
Γ̂ε
u0

∇u(x) · ∇φ0(x)dx

=

∫
Γ̂ε
u0

∩Ω̂+
u0

∇u(x) · ∇φ0(x)dx+

∫
Γ̂ε
u0

∩Ω̂−
u0

∇u(x) · ∇φ0(x)dx

=

∫
Γ̂ε
u0

∩Ω̂+
u0

( ∂

∂ν
uΩ̂+

u0

)−
· ∂

∂ν
φ0(x)dx

+

∫
Γ̂ε
u0

∩Ω̂−
u0

( ∂

∂ν
uΩ̂−

u0

)+

· ∂

∂ν
φ0(x)dx+ o(1)

=

∫
Γ̂u0

( ∂

∂ν
uΩ̂+

u0

)−
· φ0(x)dx−

∫
Γ̂u0

( ∂

∂ν
uΩ̂−

u0

)+

· φ0(x)dx+ o(1).

(2.4)

On the other hand,∫
Ω

up(x)φ0(x)dx =

∫
Γ̂ε
u0

up(x)φ0(x)dx = O(ε), as ε → 0+, (2.5)



EJDE-2024/66 LANE-EMDEN HEAT FLOW WITH SIGN-CHANGING INITIAL DATA 5

since the measure meas(Γ̂ε
u0
) = O(ε). Combining the above estimates (2.3), (2.4),

(2.5), and noticing that ( ∂

∂ν
uΩ̂+

u0

)−
≤

( ∂

∂ν
uΩ̂−

u0

)+

,

for x ∈ Γ̂u0
according to the condition (1.7), letting ε tends to zero, we see that

the differential inequality (2.1) is valid and thus u(x) is a weak sub-solution to
problem (1.1). Similar computation of second order generalized derivatives ∆u(x)
of piecewise continuous function u(x) can be found in [11]. □

Compared with the above weak sub-solution, we can localize the positive part
u+(x, t) and the negative part u−(x, t) of the sign-changing solution u(x, t) to the
subsets Ω±

u(x,t) respectively (see notations (1.3) and (1.4) for the meaning of sym-

bols u±(x, t) and Ω±
u(x,t)). Comparison principle of the heat equation implies that

u(x, t) ≥ u(x) for t ∈ (0, Tmax), which means that Ω̂+
u0

⊂ Ω+
u(x,t) and Ω−

u(x,t) ⊂ Ω̂−
u0
.

Therefore, we define the localized part u±(x, t) of u(x, t) as follows:

u+(x, t) := u(x, t) · χΩ̂+
u0
(x), u−(x, t) := −u(x, t) · χΩ̂−

u0
(x), (2.6)

where χΩ̂±
u0
(x) is the characteristic function of the set Ω̂±

u0
. Then

u(x, t) = u+(x, t)− u−(x, t) = u+(x, t)− u−(x, t),

and u+(x, t) ≥ u+(x, t), u−(x, t) ≥ u−(x, t), u±(x, t) coincides with u±(x, t) in

Ω̂±
u0

∩ Ω±
u(x,t). However, since Ω̂+

u0
⊂ Ω+

u(x,t) and Ω−
u(x,t) ⊂ Ω̂−

u0
, we have

0 = u+(x, t) < u+(x, t) = u(x, t), −u(x, t) = u−(x, t) < u−(x, t) = 0,

for all x ∈ Ω+
u(x,t)\Ω̂

+
u0
.

Lemma 2.2. The localized part u+(x, t) satisfies

∂u+

∂t
= ∆u+ + |u+|p−1u+, x ∈ Ω̂+

u0
, t > 0,

u+(x, t) ≥ 0, x ∈ ∂Ω̂+
u0
, t > 0,

u+(x, 0) = u+
0 (x) ≥, ̸≡ uΩ̂+

u0
(x), x ∈ Ω̂+

u0
,

(2.7)

and the localized part u−(x, t) satisfies

∂u−

∂t
= ∆u− + |u−|p−1u−, x ∈ Ω̂−

u0
, t > 0,

u−(x, t) ≤ 0, x ∈ ∂Ω̂−
u0
, t > 0,

u−(x, 0) = u−
0 (x) ≤, ̸≡ uΩ̂−

u0
(x), x ∈ Ω̂−

u0
.

(2.8)

Proof. The properties of the localized part u±(x, t) follow from the definition (2.6)
and the comparison principle of the heat equation compared with the weak sub-
solution u(x) proved in Lemma 2.1. □

Now that we have localized the solution u(x, t) to disjoint subsets Ω̂±
u0
, we can

analyze the asymptotic behavior of different localized parts u±(x, t) separately,
following the similar line as Quittner and Souplet [9, Theorem 17.8].

Lemma 2.3. The localized part u+(x, t) blows up in finite time, i.e., Tmax < +∞.



6 R. HUANG, S. JI, Y. MA EJDE-2024/66

Proof. We denote v(x) := uΩ̂+
u0
(x) for simplicity in this proof. We define

z(t) :=

∫
Ω̂+

u0

u+(x, t) · uΩ̂+
u0
(x)dx =

∫
Ω̂+

u0

u+(x, t) · v(x)dx.

Noticing that v(x) = 0 and ∂v(x)
∂ν < 0 for x ∈ ∂Ω̂+

u0
, we have

z′(t) =

∫
Ω̂+

u0

∂u+(x, t)

∂t
· v(x)dx

=

∫
Ω̂+

u0

∆u+(x, t) · v(x)dx+

∫
Ω̂+

u0

up
+(x, t) · v(x)dx

= −
∫
Ω̂+

u0

∇u+(x, t) · ∇v(x)dx+

∫
Ω̂+

u0

up
+(x, t) · v(x)dx

=

∫
Ω̂+

u0

u+(x, t) ·∆v(x)dx−
∫
∂Ω̂+

u0

u+(x, t)
∂v(x)

∂ν
dx

+

∫
Ω̂+

u0

up
+(x, t) · v(x)dx

≥ −
∫
Ω̂+

u0

u+(x, t) · vp(x)dx+

∫
Ω̂+

u0

up
+(x, t) · v(x)dx

=

∫
Ω̂+

u0

(
1−

(u+(x, t)

v(x)

)1−p
)
up
+(x, t) · v(x)dx.

(2.9)

Similar to the proof of [9, Lemma 17.9], for each fixed τ ∈ (0, Tmax), there exists a
constant α > 1 such that

u+(x, t) ≥ αv(x), x ∈ Ω̂+
u0
, t ∈ (τ, Tmax),

since u+(x, 0) ≥, ̸≡ v(x). Applying Jensen’s inequality to (2.9), we have

z′(t) ≥ (1− α1−p)

∫
Ω̂+

u0

up
+(x, t) · v(x)dx

≥ (1− α1−p)
(∫

Ω̂+
u0

u+(x, t) · v(x)dx
)p(∫

Ω̂+
u0

v(x)dx
)1−p

= (1− α1−p) ·
(∫

Ω̂+
u0

v(x)dx
)1−p

zp(t), t ∈ (τ, Tmax),

(2.10)

which implies that z(t) blows up in finite time since p > 1 and z(τ) > 0. □

Proof of Theorem 1.1. By the comparison principle and the weak sub-solution proved
in Lemma 2.1, we know that the solution is bounded from below such that u(x, t) ≥
u(x) for all t ∈ (0, Tmax). Lemma 2.3 shows that the localized part u+(x, t)
blows up in finite time, which also implies that u(x, t) blows up in finite time
and Tmax < +∞. □



EJDE-2024/66 LANE-EMDEN HEAT FLOW WITH SIGN-CHANGING INITIAL DATA 7

In this section, we extend our results to the Lane-Emden heat flow with general
nonlinear terms of the form

∂u

∂t
= ∆u+ f(u), x ∈ Ω, t > 0,

u(x, t) = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

(2.11)

where Ω ⊂ Rn is a bounded domain with smooth boundary, u0 ∈ L∞(Ω) is non-
negative or sign-changing, and f is a C1-function with a superlinear growth.

Theorem 2.4. Assume that f is a convex C1-function with f(0) = 0, f ′ is non-
constant near 0. Then, under the assumptions in Theorem 1.1, we have

(i) the solution u(x, t) ≥ uΩ̂+
u0
(x)−uΩ̂−

u0
(x) for all t ∈ (0, Tmax), where Tmax ∈

(0,+∞) is the maximal existence time of u(x, t);
(ii) the solution u(x, t) blows up to positive infinity in finite time

lim
t→T−

max

sup
x∈Ω

u(x, t) = +∞.

Proof. The proof of this theorem is similar to that of Theorem 1.1. For simplicity,
we only present the key and distinct parts. We firstly note that the conditions on
the function f in Theorem 2.4 ensure that equation (2.11) has a positive (classical)
equilibrium v0 (see [9, Theorem 17.10]), which will help us construct a sub-solution
similar to that in Lemma 2.1. Then, by the comparison principle compared with the
sub-solution, we can localize the sign-changing solution u(x, t) to disjoint subsets

Ω̂±
u0
, getting the positive part u+(x, t) and the negative part u−(x, t). Finally,

similar to Lemma 2.3, we prove that the positive part u+(x, t) blows up in finite
time. The key inequality is as follows.

z′(t) =

∫
Ω̂+

u0

∂u+(x, t)

∂t
· v(x)dx

=

∫
Ω̂+

u0

∆u+(x, t) · v(x)dx+

∫
Ω̂+

u0

f(u+(x, t)) · v(x)dx

= −
∫
Ω̂+

u0

∇u+(x, t) · ∇v(x)dx+

∫
Ω̂+

u0

f(u+(x, t)) · v(x)dx

=

∫
Ω̂+

u0

u+(x, t) ·∆v(x)dx−
∫
∂Ω̂+

u0

u+(x, t)
∂v(x)

∂ν
dx

+

∫
Ω̂+

u0

f(u+(x, t)) · v(x)dx

≥ −
∫
Ω̂+

u0

u+(x, t) · f(v(x))dx+

∫
Ω̂+

u0

f(u+(x, t)) · v(x)dx

=

∫
Ω̂+

u0

(
1− f(v(x))u+(x, t)

f(u+(x, t))v(x)

)
f(u+(x, t))v(x)dx.

(2.12)

Based on the convexity of function f and inequality ([9, Lemma 17.9])

u+(x, t) ≥ αv(x), x ∈ Ω̂+
u0
, t ∈ (τ, Tmax),



8 R. HUANG, S. JI, Y. MA EJDE-2024/66

we can obtain

z′(t) ≥
(
1− 1

α

) ∫
Ω̂+

u0

f(u+(x, t)) · v(x)dx

=
(
1− 1

α

) 1∫
Ω̂+

u0
v(x)dx

f(z(t)), t ∈ (τ, Tmax),
(2.13)

with α > 1. This completes the proof. □

3. Extension

In the previous section, we considered the blow-up phenomenon of sign-changing
solutions to the linear diffusion equations. In this section, we will focus on the case
of nonlinear diffusion. As an example, we consider the following porous media
equation with finite diffusion speed,

∂u

∂t
= ∆um + |u|p−1u, x ∈ Ω, t > 0,

u(x, t) = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

(3.1)

where p > 1, m > 1, Ω ⊂ Rn is a bounded domain with smooth boundary,
u0 ∈ L∞(Ω) is non-negative or sign-changing. Porous medium equations serve
as fundamental models for describing fluid flow, heat transfer, and mass diffusion
processes within porous materials, such as soil, rocks, and biological tissues. The
classical heat equation exhibits infinite propagation speed, which may not align
with real-world scenarios. In contrast, porous medium equations with m > 1 offer
a more realistic representation as they exhibit finite propagation speed. These equa-
tions are widely studied due to their relevance in fields like hydrology, petroleum
engineering, environmental science, and biomedicine.

A closely related stationary problem to (3.1) is

−∆um = |u|p−1u, x ∈ Ω,

u(x) = 0, x ∈ ∂Ω, u(x) > 0, x ∈ Ω.
(3.2)

By the transform w = um, (3.2) can be rewritten as

−∆w = |w|
p
m−1w, x ∈ Ω,

w(x) = 0, x ∈ ∂Ω, w(x) > 0, x ∈ Ω.
(3.3)

It follows from the results in [7] that a positive solution of (3.3) exists if and only
if p

2∗ < m < p, where

2∗ :=

{
+∞, n = 1, 2,
n+2
n−2 , n ≥ 3.

Thus, under the assumption

0 < m < p < +∞, n = 1, 2,

n− 2

n+ 2
< m < p <

n+ 2

n− 2
, n ≥ 3,

(3.4)

equation (3.3) admits positive solution.
We denote w0 = um

0 . The following theorem shows the blow-up phenomenon for
the porous medium equation (3.1).



EJDE-2024/66 LANE-EMDEN HEAT FLOW WITH SIGN-CHANGING INITIAL DATA 9

Theorem 3.1. Assume that u0 ∈ L∞(Ω) is sign-changing and satisfies the follow-

ing conditions: there exist disjoint non-empty open subsets Ω̂±
w0

⊂ Ω and relatively

closed subset Γ̂w0
⊂ Ω such that

Ω−
u0

⊂ Ω̂−
w0

, Ω̂+
w0

⊂ Ω+
u0
, Γ̂w0

= ∂Ω̂+
w0

∩∂Ω̂−
w0

∩Ω, Ω = Ω̂+
w0

∪Ω̂−
w0

∪Γ̂w0
, (3.5)

we further assume that the boundary of Ω̂±
w0

is piecewise smooth and the unit normal

vector at x ∈ Γ̂w0
is denoted by ν pointing in the direction from Ω̂+

w0
to Ω̂−

w0
, the

stationary problem (3.3) on Ω̂±
w0

admits positive solutions wΩ̂±
w0

(zero extended to

Ω). Moreover,

u−
0 (x) ≤, ̸≡ w

1
m

Ω̂−
w0

(x), x ∈ Ω̂−
u0
, u+

0 (x) ≥, ̸≡ w
1
m

Ω̂+
u0

(x), x ∈ Ω̂+
u0
, (3.6)( ∂

∂ν
wΩ̂+

w0

)−
≤

( ∂

∂ν
wΩ̂−

w0

)+

, x ∈ Γ̂w0 , (3.7)

where
(
∂f
∂ν

)±
denotes the one-side directional derivative of f(x) with respect to ν.

Under the above conditions, we have that

(i) the solution u(x, t) ≥ w
1
m

Ω̂+
w0

(x)−w
1
m

Ω̂−
w0

(x) for all t ∈ (0, Tmax), where Tmax ∈
(0,+∞) is the maximal existence time of u(x, t);

(ii) the solution u(x, t) blows up to positive infinity in finite time

lim
t→T−

max

sup
x∈Ω

u(x, t) = +∞.

The main idea of the proof of Theorem 3.1 is similar to Theorem 1.1. We will
initially construct a weak sub-solution to equation (3.1), then localize the positive
and negative parts of the sign-changing solution based on the comparison with this
weak sub-solution. The main difference in the proof lies in considering the weak
solutions for the problem considered in this section. Firstly, we construct the weak
sub-solution.

Lemma 3.2. Under the assumptions in Theorem 3.1, the sign-changing function

ũ(x) := w
1
m

Ω̂+
w0

(x)− w
1
m

Ω̂−
w0

(x)

is a weak sub-solution to the initial-boundary-value problem (3.1), where uΩ̂±
w0
(x)

is the positive solution (with zero extension to Ω) of the stationary problem (3.3)

on Ω̂±
w0

.

Proof. It follows from condition (3.6) that

ũ(x) = w
1
m

Ω̂+
w0

(x)− w
1
m

Ω̂−
w0

(x) ≤ u+
0 (x)− u−

0 (x) = u0(x)

for x ∈ Ω, and ũ(x) = 0 for x ∈ ∂Ω. It suffices to demonstrate the differential
inequality ∫

Ω

∇ũm(x) · ∇φ(x)dx ≤
∫
Ω

ũp(x)φ(x)dx, (3.8)



10 R. HUANG, S. JI, Y. MA EJDE-2024/66

for any 0 ≤ φ(x) ∈ C∞
0 (Ω). Using a similar partition of unity φ+(x), φ−(x), φ0(x)

in (2.2), we have∫
Ω

∇ũm(x) · ∇φ±(x)dx =

∫
Ω̂±

w0

−∆ũm(x) · φ±(x)dx

=

∫
Ω̂±

w0

ũm(x) · φ±(x)dx =

∫
Ω

ũp(x) · φ±(x)dx.

(3.9)

With the help of the approximation

∇w(x) =
( ∂

∂ν
wΩ̂+

w0

)−
· ν · χΩ̂+

w0
(x) +

( ∂

∂ν
wΩ̂−

w0

)+

· ν · χΩ̂−
w0
(x) + o(1), x ∈ Γ̂ε

w0
,

we obtain∫
Ω

∇ũm(x) · ∇φ0(x)dx

=

∫
Γ̂ε
w0

∇ũm(x) · ∇φ0(x)dx

=

∫
Γ̂ε
w0

∩Ω̂+
w0

∇w(x) · ∇φ0(x)dx+

∫
Γ̂ε
w0

∩Ω̂−
w0

∇w(x) · ∇φ0(x)dx

=

∫
Γ̂ε
w0

∩Ω̂+
w0

( ∂

∂ν
wΩ̂+

w0

)−
· ∂

∂ν
φ0(x)dx

+

∫
Γ̂ε
w0

∩Ω̂−
w0

( ∂

∂ν
wΩ̂−

w0

)+

· ∂

∂ν
φ0(x)dx+ o(1)

=

∫
Γ̂w0

[( ∂

∂ν
wΩ̂+

w0

)−
−

( ∂

∂ν
wΩ̂−

w0

)+
]
· φ0(x)dx+ o(1).

(3.10)

Because meas(Γ̂ε
w0

) = O(ε), we have∫
Ω

ũp(x)φ0(x)dx =

∫
Γ̂ε
w0

ũp(x)φ0(x)dx = O(ε), as ε → 0+, (3.11)

Thus it follows from (3.9), (3.10), (3.11) and the condition (3.7) that the differential
inequality (3.8) is valid which implies ũ(s) is a weak sub-solution to problem (3.1).

□

Next, we localize the positive part u+(x, t) and the negative part u−(x, t) of the
sign-changing solution u(x, t) to the subsets Ω±

u(x,t) respectively.

We define the localized part u±(x, t) of u(x, t) as follows:

u+(x, t) := u(x, t) · χΩ̂+
w0
(x), u−(x, t) := −u(x, t) · χΩ̂−

w0
(x), (3.12)

where χΩ̂±
w0
(x) is the characteristic function of the set Ω̂±

w0
. The comparison prin-

ciple of equation (3.1) implies that u(x, t) ≥ ũ(x) for t ∈ (0, Tmax), which means

that Ω̂+
w0

⊂ Ω+
u(x,t) and Ω−

u(x,t) ⊂ Ω̂−
w0

. Thus we have

u(x, t) = u+(x, t)− u−(x, t) = u+(x, t)− u−(x, t),

and u+(x, t) ≥ u+(x, t), u
−(x, t) ≥ u−(x, t). u±(x, t) coincides with u±(x, t) in

Ω̂±
u0

∩ Ω±
u(x,t). However, for all x ∈ Ω+

u(x,t)\Ω̂
+
u0
, we have

0 = u+(x, t) < u+(x, t) = u(x, t), −u(x, t) = u−(x, t) < u−(x, t) = 0.



EJDE-2024/66 LANE-EMDEN HEAT FLOW WITH SIGN-CHANGING INITIAL DATA 11

From the definition (3.12) and the comparison principle for the equation (3.1)
compared with the weak sub-solution ũ(x) proved in Lemma 3.2, we derive the
following lemma.

Lemma 3.3. The localized part u+(x, t) satisfies

∂u+

∂t
= ∆um

+ + |u+|p−1u+, x ∈ Ω̂+
w0

, t > 0,

u+(x, t) ≥ 0, x ∈ ∂Ω̂+
w0

, t > 0,

u+(x, 0) = u+
0 (x) ≥, ̸≡ w

1
m

Ω̂+
w0

(x), x ∈ Ω̂+
w0

,

(3.13)

in the distribution sense and the localized part u−(x, t) satisfies

∂u−

∂t
= ∆u− + |u−|p−1u−, x ∈ Ω̂−

w0
, t > 0,

u−(x, t) ≤ 0, x ∈ ∂Ω̂−
w0

, t > 0,

u−(x, 0) = u−
0 (x) ≤, ̸≡ w

1
m

Ω̂−
w0

(x), x ∈ Ω̂−
w0

,

(3.14)

in the distribution sense.

Following the approach in the proof of [9, Lemma 17.9], we derive the subsequent
result for (3.1), which will be used in the next lemma.

Lemma 3.4. Assume that u
(1)
0 , u

(2)
0 ∈ L∞(Ω) and u

(1)
0 ≥, ̸≡ u

(2)
0 . Let u(1)(x, t) and

u(2)(x, t) be the corresponding solutions of (3.1), then for any fixed τ ∈ (0, Tmax),
there exists a constant α > 1 such that

u(1)(x, t) ≥ αu(2)(x, t), t ∈ (τ, Tmax).

Finally, with the localized solution u(x, t) to disjoint subsets Ω̂±
w0

, we establish
the following result about the asymptotic behavior of the localized part u+(x, t),
which will complete the proof the Theorem 3.1.

Lemma 3.5. The localized part u+(x, t) defined in (3.12) blows up in finite time,
i.e., Tmax < +∞.

Proof. Inspired by [9], we define

z(t) :=

∫
Ω̂+

w0

u+(x, t) · uΩ̂+
w0
(x)dx =

∫
Ω̂+

u0

u+(x, t) · v(x)dx,



12 R. HUANG, S. JI, Y. MA EJDE-2024/66

where we denote v(x) := wΩ̂+
w0
(x) for simplicity. Noticing that v(x) = 0 and

∂v(x)
∂ν < 0 for x ∈ ∂Ω̂+

w0
, we have

z′(t) =

∫
∂Ω̂+

w0

∂um
+ (x, t)(x)

∂ν
· v(x)dx−

∫
Ω̂+

w0

∇um
+ (x, t) · ∇v(x)dx

+

∫
Ω̂+

w0

up
+(x, t) · v(x)dx

= −
∫
Ω̂+

w0

∇um
+ (x, t) · ∇v(x)dx+

∫
Ω̂+

w0

up
+(x, t) · v(x)dx

=

∫
Ω̂+

w0

u+(x, t) ·∆v(x)dx−
∫
∂Ω̂+

w0

u+(x, t)
∂v(x)

∂ν
dx

+

∫
Ω̂+

w0

up
+(x, t) · v(x)dx

≥ −
∫
Ω̂+

w0

um
+ (x, t) · v

p
m (x)dx+

∫
Ω̂+

w0

up
+(x, t) · v(x)dx

=

∫
Ω̂+

w0

(
up
+(x, t) · v(x)− um

+ (x, t) · v
p
m

)
dx.

(3.15)

Taking u
(1)
0 = u+(x, 0), u

(2)
0 = v

1
m (x) and applying lemma 3.4, we have that for any

fixed τ ∈ (0, Tmax), there exists a constant α > 1 such that

u+(x, t) ≥ αv
1
m (x), ∀x ∈ Ω̂+

w0
, t ∈ (τ, Tmax),

since u+(x, 0) ≥, ̸≡ v
1
m (x). Thus we have from (3.15) that

z′(t) ≥
∫
Ω̂+

w0

(
1−

(u+(x, t)

v
1
m (x)

)m−p )
up
+(x, t)v(x)dx.

≥ (1− αm−p)

∫
Ω̂+

w0

up
+(x, t)v(x)dx, t ∈ (τ, Tmax).

Employing Jensen’s inequality, we derive

z′(t) ≥ (1− αm−p)
(∫

Ω̂+
w0

u+(x, t) · v(x)dx
)p(∫

Ω̂+
w0

v(x)dx
)1−p

= (1− αm−p)
(∫

Ω̂+
w0

v(x)dx
)1−p

zp(t), t ∈ (τ, Tmax),

(3.16)

which implies that z(t) blows up in finite time since p > 1 and z(τ) > 0. □

Proof of Theorem 3.1. Based on the comparison principle and the weak sub-solution
derived in Lemma 3.2, we know that the solution is bounded from below, ensuring
u(x, t) ≥ u(x) for all t ∈ (0, Tmax). Furthermore, Lemma 3.5 demonstrates that the
localized part u+(x, t) blows up in finite time, which consequently indicates that
the solution u(x, t) to porous medium equation (3.1) blows up in finite time and
Tmax < +∞. □

According to (3.16), when m > 1, the blow-up rate of the porous medium equa-
tion (3.1) decreases relative to the classical heat equation (1.1).



EJDE-2024/66 LANE-EMDEN HEAT FLOW WITH SIGN-CHANGING INITIAL DATA 13

Acknowledgements. R. Huang was supported by the NSFC Grant No. 12371205.
S. Ji was supported by the NSFC Grant Nos. 12271178 and 12171166, by the
Guangzhou Basic and Applied Basic Research Foundation No. 2024A04J2022, and
by the TCL Young Scholar (2024-2027). Y. Ma was supported by the National Key
R&D Program of China under grant 2023YFA1009002 and by the National Natural
Science Foundation of China No. 11801067.

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Rui Huang

School of Mathematical Sciences, South China Normal University, Guangzhou, 510631,
China

Email address: huangrui@m.scnu.edu.cn

Shanming Ji

School of Mathematics, South China University of Technology, Guangzhou, 510641,
China

Email address: jism@scut.edu.cn

Yansheng Ma (corresponding author)
School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024,

China

Email address: mays538@nenu.edu.cn


	1. Introduction
	2. Proof of the main results
	3. Extension
	Acknowledgements

	References