EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 1, No. 3, 2008, (33-39)
ISSN 1307-5543 – www.ejpam.com

Blow-up for nonlinear heat equations with absorptions

Hailiang Zhang1,∗, Xiulan Guo2

1 Department of Mathematics, Zhejiang Ocean University, Zhoushan 316004, CHINA
2 College of Science, Henan University of Technology, Zhengzhou 450001, CHINA

Abstract. This paper deals with the blow-up of positive solution of the nonlinear heat equation ut =
∇(a(u)∇u)− f (u) subject to nonlinear boundary condition ∂ u

∂ n
= b(u). Under suitable assumptions on

nonlinear functions a, f , b and initial data u0(x), we obtain the blow-up rate and the blow-up set of
the solutions of the problem by the Nirenberg maximum principle.

AMS subject classifications: 3333

Key words: Nonlinear heat equation, Nonlinear boundary condition, Blow-up, Absorption, Maximum
principle.

1. Introduction

In the last few decades, blow-up phenomena for the nonlinear parabolic equations with
heat sources have been studied by many authors, and the reader is referred to [3-5, 9, 11,
14] and the references therein. For the parabolic equations with no sources, some necessary
conditions for the global existence and blow-up of the solutions are given in [6, 8, 12-13].
Here we are interested in the blow-up phenomena of the solutions of the parabolic problems
with absorptions at interior of the domains.

In this paper, we investigate the following initial-boundary value problem:

ut = div(a(u)∇u)− f (u) in Ω× (0, T ) (1)

∂ u

∂ n
= b(u) on ∂Ω× (0, T ) (2)

u(x , 0) = u0(x)> 0 in Ω (3)

where Ω is a bounded domain in RN with C2 boundary, ∂ u
∂ n

denotes the outward normal

derivative; u0(x) ∈ C(Ω) ∩ C2(Ω) is a positive function, satisfying compatibility condition;
a(·), f (·), b(·) are smooth positive functions.

Problem (1)-(3) has been formulated from physical models arising in various fields of ap-
plied sciences. For example it can be interpreted as a heat conduction problem with nonlinear

∗Corresponding author. Email addresses: hlzhang@zjou.edu.cn (H. Zhang)
guoxiulanlm@yahoo.cn (X. Guo)

http://www.ejpam.com 33 c© 2008 EJPAM All rights reserved.



H. Zhang, X. Guo / Eur. J. Pure Appl. Math, 1 (2008), (33-39) 34

diffusivity, absorption at interior of the domain and a nonlinear radiation law on the boundary
of the material body.

The local existence and uniqueness of positive classical solution of the problem (1)-(3)
were established by Amann [1], finite extinction time is studied by Leung and Zhang [7].
Here we are interested in the blow-up phenomenon of the solution of the problem (1)-(3).We
say that the solution u blows up if there exists a 0< T <+∞ such that lim

t→T
‖ u ‖L∞(Ω)=+∞.

In this paper, we study the blow-up rate and blow-up set by the maximum principles. Our
results generalize and deepen ones from corresponding work in [2-6, 8-12].

2. Blow-up rate and blow-up set

Theorem 1. Let u(x , t) be a solution of the problem (1)-(3). Assume that:
(1)
∫ +∞

0
a(s)
b(s)ds <+∞ and a′(s), ( b′(s)

a(s) )
′, ( b(s)

f (s))
′ ≥ 0 for s > 0,

(2) div(a(u0)∇u0)≥ f (u0).
Then u(x , t) must blow up in finite time T and there exists a constant δ > 0 such that

sup
x∈Ω

u(x , t)≤ H(δ(T − t)) for 0< t < T,

where H = G−1 and G(s) =
∫ +∞

s
a(µ)
b(µ)dµ.

Proof. We will use the ideas of [4] or [14].

Step 1: Growth estimate.
Introduce an auxiliary function

J(x , t) = a(u)ut −δb(u) (δ > 0) (4)

then we have
∇J = a

′
ut∇u+ a∇ut −δb

′
∇u

4J = a4ut + 2a
′
∇u∇ut + a

′′
ut | ∇u |2 +a

′
ut4u−δb

′′
| ∇u |2 −δb

′
4u

and

Jt = a
′
u2

t + a24ut + 2aa
′
∇u∇ut + aa

′′
ut | ∇u |2 +aa

′
ut4u− (a f ′+δb′)ut .

Hence,
Jt − a4J + f

′
J = a

′
u2

t +δ(ab
′′
− a

′
b
′
) | ∇u |2 +δ(b′ f − b f ′). (5)

Under the assumptions of theorem 1, we obtain

Jt − a4J + f ′J ≥ 0. (6)

Since u0(x)> 0 and div(a(u0)∇u0)≥ f (u0), by the Nirenberg maximum principle, u≥ 0 and
ut ≥ c in Ω× (ε0, T ), where c, ε0 are some positive constants. It following that

J(x ,ε0) = a(u(x ,ε0))ut(x ,ε0)−δb(u(x ,ε0))≥ 0 (7)



H. Zhang, X. Guo / Eur. J. Pure Appl. Math, 1 (2008), (33-39) 35

provided δ is small enough. Note that

∂ u

∂ n
= b(u) on ∂Ω× (0, T ),

we have
∂ ut

∂ n
= b′(u)ut on ∂Ω× (0, T ).

Thus
∂ J

∂ n
= a
∂ ut

∂ n
+ (a′ut −δb′)

∂ u

∂ n
= (ab)′ut −δbb′. (8)

Using (4) and (8) we have

∂ J

∂ n
−
(ab)′

a
J =
δa′b2

a
≥ 0 on ∂Ω× (ε0, T ). (9)

Thus, by the maximum principle for parabolic problems, J(x , t)≥ J(x ,ε0)≥ 0 in Ω× (ε0, T ),
i.e.,

ut ≥ δ
b(u)
a(u)

in Ω× (ε0, T ). (10)

Step 2: Blow-up rate.
Set

G(s) =

∫ ∞

s

a(s)
b(s)

ds

then

−(G(u))t =
a(u)
b(u)

ut .

By the growth estimate (10) we have

−(G(u))t ≥ δ in Ω× (ε0, T ).

By integration from t to T

G(u(x , t))− G(u(x , T ))≥ δ(T − t) ε0 < t < T

therefore also
G(u(x , t))≥ δ(T − t). (11)

Since
∫ +∞

0
a(s)
b(s)ds <+∞, by (11), u(x , t) must blow up in finite time T and

sup
x∈Ω

u(x , t)≤ G−1 (δ(T − t)) for 0< t < T.

The proof of Theorem 1 is completed.

With analogy to Theorem 1, one can also obtain bounds for the global solutions.



H. Zhang, X. Guo / Eur. J. Pure Appl. Math, 1 (2008), (33-39) 36

Corollary 1. Let u(x , t) be a solution of the problem (1)-(3). If
∫ +∞

0
a(s)
b(s)ds = +∞, a′(s) ≤ 0,

and ( b′(s)
a(s) )

′ ≤ 0 for s > 0, then u(x , t) exists globally and

sup
x∈Ω

u(x , t)≤ G−1((t + G(M))) for t > 0,

where M =max
Ω

u0.

Proof. Let w(x , t) be a smooth positive solution of the following problem:

wt = div(a(w)∇w) in Ω× (0, T ), (12)

∂ w

∂ n
= b(w) on ∂Ω× (0, T ), (13)

u(x , 0) = M =max
Ω

u0 in Ω. (14)

With analogy to the proof of Theorem 1, by the maximum principle we have L = a(w)wt −
b(w)≤ 0, i.e.,

wt ≤
b(w)
a(w)

in Ω× (0, T ). (15)

For each fixed x ∈ Ω, we get by integration (15)

∫ w(x ,t)

M

a(s)
b(s)

ds ≤ t. (16)

It follows from assumptions that w(x , t) must be a global solution. With inequality (16), one
gets

G(w(x , t))− G(M) =

∫ w(x ,t)

C

dβ(s)
f (s)
≤ t

and
w(x , t)≤ G−1(t + G(M)).

By the comparison principle we know that w(x , t) is an upper solution of (1)-(3). Thus

u(x , t)≤ w(x , t) in Ω× (0, T ).

The proof of Corollary 1 is complete.

We shall prove in the following theorem that the blowup will occur only at the boundary
of the domain.

Theorem 2. Suppose that the assumptions of Theorem 1 hold, and there exists a positive
constant C0 such that s( b

a
)′(H(s))≤ C0 for s > 0. Then for any Ω′ ⊂⊂ Ω and ε0 > 0,

sup
x∈Ω′, t∈[ε0,T )

u(x , t)<+∞.



H. Zhang, X. Guo / Eur. J. Pure Appl. Math, 1 (2008), (33-39) 37

Proof. We will use the ideas of [6]. Let d(x) = dist(x ,∂Ω) and

v(x) = d2(x) for x ∈ Nε(∂Ω)

where Nε(∂Ω) = {x ∈ Ω : d(x) < ε}. Since ∂Ω is C2, the function v(x) is in C2(Nε(∂Ω)) if ε
is small enough. Therefore, there exists a constant C > 0 such that

div(a(v)∇v)−
C0

v
|∇v|2 ≥−C in Nε0(∂Ω)

if ε0 is small enough. We next extend v(x) to a function on Ω such that v ∈ C2(Ω) and
v ≥ c0 > 0 on Ω/Nε0(∂Ω). Then

div(a(v)∇v)−
C0

v
|∇v|2 ≥−C∗ on Ω

for some 1≥ C∗ > 0.
Set w(x , t) = C1H(τ), where τ= δ(v(x) + C∗(T − t)) and C1 > 0. Then

wt − div(a(w)∇w) + f (w)≥ 0 in Ω× (ε0, T ).

By Theorem 1 we have

sup
x∈Ω

u(x , t)≤ H(δ(T − t)) on ∂Ω× (ε0, T ).

Thus
w(x , t) = C1H(δC∗(T − t)))> H(δ(T − t))≥ u(x , t) on ∂Ω× (ε0, T )

if C1 > 1. Take C1 to be large enough so that w(x ,ε0)≥ u(x ,ε0). Then the maximum principle
implies that w(x , t)≥ u(x , t) in Ω× (ε0, T ). Therefore for Ω′ ⊂⊂ Ω

u(x , t)≤ C1H(δ(v(x) + C∗(T − t)))≤ C1H(δv(x))

i.e.,
sup

x∈Ω′, t∈[ε0,T )
u(x , t)<+∞.

The theorem 2 is proved.

In our theorems, if f (u) ≡ 0, a(u) ≡ 1 and b(u) = up (p > 1), then the following conclu-
sion holds:
Corollary 2. Let u(x , t) be a smooth solution of the following problem:







ut =4u in Ω× (0, T )
∂ u
∂ n
= up on ∂Ω× (0, T )

u(x , 0) = u0(x)> 0 in Ω.

If 4u0 ≥ 0, then u(x , t) blows up in finite time and blowup will occur only at the boundary of
the domain.

This is the case of [6].



REFERENCES 38

3. Concluding remarks and applications

Problem (1)-(3) arises in the nonlinear diffusion process, in which div(a(u)∇u) denotes
the nonlinear diffusion effect, f (u) denotes absorption in the interior of the domain, and
∂ u
∂ n

denotes the boundary flux along the outward normal direction to the domain. With this
model, all of the results obtained in the preceding sections are physically meaningful.

Our results show that the strength of the boundary flux plays a key role in the blowup
properties of the problem (1)-(3). If the boundary flux is sufficiently strong, then it will bring
about blowup in a finite time, and the blowup will occur only at the boundary of the domain.
If the boundary flux is not sufficiently strong, it is probable that the solution may never blow
up.

As the application of theorems, now we consider the following porous medium problem

ut =4up − uq in Ω× (0, T ),

∂ u

∂ n
= ur on ∂Ω× (0, T ),

u(x , 0) = u0(x)> 0 in Ω,

where p, q, r > 0.
If r ≤ p ≤ 1, then Corollary 1 hold, every positive solution u(x , t) of the problem exists

globally. If r ≥ max{1, p, q} and 4up
0 ≥ uq

0, by Theorems 1-2 we know that every positive
solution u(x , t) of the problem blows up in a finite time T and the blowup will occur only at
the boundary of the domain. Moreover, there exits a constant C > 0 such that

sup
x∈Ω

u(x , t)≤
C

(T − t)
1

r−p

ε0 < t < T, ε0 > 0.

Acknowledgements

The authors would like to thank Professor C.V.Pao for his helpful discussion. We should
also like to thank the referees for their advice on amendments. This work is supported by
the National Natural Science Foundation of China and the Key Project of Zhejiang Ocean
University.

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