International Journal of Analysis and Applications
ISSN 2291-8639
Volume 5, Number 2 (2014), 147-153
http://www.etamaths.com

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR A

QUASILINEAR PARABOLIC EQUATION WITH ABSORPTION

AND NONLINEAR BOUNDARY CONDITION

IFTIKHAR AHMED∗, CHUNLAI MU AND PAN ZHENG

Abstract. This paper deals with the evolution r-Laplacian equation with
absorption and nonlinear boundary condition. By using differential inequality

techniques, global existence and blow-up criteria of nonnegative solutions are

determined. Moreover, upper bound of the blow-up time for the blow-up
solution is obtained.

1. Introduction

In this paper, we investigate the global existence and finite time blow-up of
nonnegative solutions for the following initial-boundary value problem


ut = div(|∇u|r−2∇u) −f(u), (x,t) ∈ Ω × (0, t∗),

|∇u|r−2
∂u

∂n
= g(u), (x,t) ∈ ∂Ω × (0, t∗),

u (x, 0) = u0 (x) > 0, x ∈ Ω,

(1.1)

where r ≥ 2, ∂u
∂n

is the outward normal derivative of u on the boundary ∂Ω assumed

sufficiently smooth, Ω is a bounded star-shaped region in RN (N ≥ 2) and t∗ is
the blow-up time if blow-up occurs, or else t∗ = ∞. It is well known that the
functions f and g may greatly affect the behavior of the solution u(x,t) with the
development of time. From the physical standpoint, −f is the cold source function,
g is the heat-conduction function transmitting into interior of Ω from the boundary
of Ω.

The global existence and blow-up for nonlinear parabolic equations have been
extensively investigated by many authors in the last decades (see [1–6] and the
references therein). In recent years, many authors have also studied bounds for
the blow-up time in nonlinear parabolic problems by using differential inequality
techniques (see [7–12]). In particular, Payne et al. [13] considered the following
semilinear heat equation with nonlinear boundary condition


ut = ∆u−f(u), (x,t) ∈ Ω × (0, t∗),
∂u

∂n
= g(u), (x,t) ∈ ∂Ω × (0, t∗),

u (x, 0) = u0 (x) , x ∈ Ω,

(1.2)

2010 Mathematics Subject Classification. 35K55, 35K65.
Key words and phrases. Global existence; Blow-up; Quasilinear parabolic equation; Nonlinear

boundary condition.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

147



148 AHMED, MU AND ZHENG

and established sufficient conditions on the nonlinearities to guarantee that the
solution u(x,t) exists for all time t > 0 or blows up in finite time t∗. Moreover, an
upper bound for t∗ was derived. Under more restrictive conditions, a lower bound
for t∗ was also obtained.

Moreover, in [14], Payne et al. also studied the following initial-boundary prob-
lem 


ut = ∇(|∇u|2p∇u), (x,t) ∈ Ω × (0, t∗),

|∇u|2p
∂u

∂n
= f(u), (x,t) ∈ ∂Ω × (0, t∗),

u (x, 0) = u0 (x) , x ∈ Ω,

(1.3)

and obtained upper and lower bounds for the blow-up time under some conditions
when blow-up does occur at some finite time.

In the present work, by using differential inequality techniques, we give some
sufficient conditions on the functions f and g for the global existence and blow-up
of nonnegative solutions to problem (1.1). Our main results are stated as follows.

Theorem 1.1. (Conditions for global existence). Let u(x,t) be the solution
of problem (1.1) and assume that the non-negative functions f and g satisfy the
following conditions

(1.4) f(ξ) ≥ k1ξp, ξ ≥ 0,

(1.5) g(ξ) ≤ k2ξq, ξ ≥ 0,
for some non-negative constants k1 and k2. Moreover suppose that the positive
constants p and q satisfy the following conditions

(1.6) p > q > r − 1 and rq < (r − 1)(p + 1).
Then the non-negative solution u(x,t) of problem (1.1) exists globally for all time
t > 0.

Theorem 1.2.(Conditions for blow-up in finite time). Let u(x,t) be the solution
of problem (1.1) and assume that the non-negative functions f and g satisfy the
following conditions

(1.7) ξf(ξ) ≤ rF(ξ), ξ ≥ 0,

(1.8) ξg(ξ) ≥ rG(ξ), ξ ≥ 0,
with

(1.9) F(ξ) =

∫ ξ
0

f(η)dη, G(ξ) =

∫ ξ
0

g(η)dη.

Moreover suppose that Ψ(0) > 0, where

(1.10) Ψ(t) = r

∫
∂Ω

G(u)ds−
∫

Ω

|∇u|rdx−r
∫

Ω

F(u)dx.

Then the solution u(x,t) of problem (1.1) blows up at time t∗ < T with

(1.11) T =
Φ(0)

(r − 2)Ψ(0)
, for r > 2,



QUASILINEAR PARABOLIC EQUATION 149

where Φ(t) =
∫

Ω
u2dx. If r = 2, we have T = ∞.

This paper is organized as follows. In Section 2, we establish the conditions
on the functions f and g, which guarantee that u(x,t) exists globally, and prove
Theorem 1.1. In Section 3, we obtain the blow-up condition of the solution and
derive an upper bound estimate for the blow-up time t∗.

2. Conditions for global existence

In this section, we establish the sufficient conditions on the functions f and
g, which guarantee that u(x,t) exists globally, and prove Theorem 1.1. To do this,
we need the following Lemma.

Lemma 2.1. Let Ω be a bounded star-shaped domain in RN , N ≥ 2. Then for
any non-negative C1 function u and γ > 0, we have

(2.1)

∫
∂Ω

uγds ≤
N

ρ0

∫
Ω

uγdx +
γd

ρ0

∫
Ω

uγ−1|∇u|dx,

where

(2.2) ρ0 = min
x∈∂Ω

(x ·n) and d = max
x∈∂Ω

|x|.

Proof. As Ω is a bounded star-shaped domain, it is easy to see that ρ0 > 0.
Integrating the identity

(2.3) div(uγx) = Nuγ + γuγ−1(x ·∇u)

over Ω, it follows from the divergence theorem that

(2.4)

∫
∂Ω

uγ(x ·n)ds = N
∫
·Ω
uγdx + γ

∫
Ω

uγ−1(x ·∇u)dx.

By the definition of ρ0 and d, we obtain

(2.5) ρ0

∫
∂Ω

uγds ≤
∫
∂Ω

uγ(x ·n)ds ≤ N
∫
·Ω
uγdx + γd

∫
Ω

uγ−1|∇u|dx,

which implies the desired conclusion.
Proof of Theorem 1.1. Setting

(2.6) Φ(t) =

∫
Ω

u2dx,

then it follows from (1.1), (1.4) and (1.5) that

Φ′(t) = 2

∫
Ω

uutdx

= 2

∫
Ω

u[div(|∇u|r−2∇u) −f(u)]dx

= 2

∫
∂Ω

u|∇u|r−2
∂u

∂n
ds− 2

∫
Ω

|∇u|rdx− 2
∫

Ω

uf(u)dx

= 2

∫
∂Ω

ug(u)ds− 2
∫

Ω

|∇u|rdx− 2
∫

Ω

uf(u)dx

≤ 2k2
∫
∂Ω

uq+1ds− 2
∫

Ω

|∇u|rdx− 2k1
∫

Ω

up+1dx.

(2.7)



150 AHMED, MU AND ZHENG

By Lemma 2.1, we have

(2.8)

∫
∂Ω

uq+1ds ≤
N

ρ0

∫
Ω

uq+1dx +
(q + 1)d

ρ0

∫
Ω

uq|∇u|dx,

where ρ0 and d are given by (2.2). Combining (2.7) with (2.8), we obtain
(2.9)

Φ′(t) ≤
2k2N

ρ0

∫
Ω

uq+1dx+
2k2(q + 1)d

ρ0

∫
Ω

uq|∇u|dx−2
∫

Ω

|∇u|rdx−2k1
∫

Ω

up+1dx.

By using Young’s inequality with ε > 0, we derive

(2.10)

∫
Ω

uq|∇u|dx ≤
1

rε

∫
Ω

|∇u|rdx +
r − 1
r

ε
1
r−1

∫
Ω

u
rq
r−1 dx,

where ε =
k2(q+1)d
rρ0

> 0. It follows from (2.9) and (2.10) that

(2.11)

Φ′(t) ≤
2k2N

ρ0

∫
Ω

uq+1dx + 2(r − 1)
(
k2(q + 1)d

rρ0

) r
r−1
∫

Ω

u
rq
r−1 dx− 2k1

∫
Ω

up+1dx.

By Hölder’s inequality , we have

(2.12)

∫
Ω

u
rq
r−1 dx ≤

(∫
Ω

uq+1dx

)α (∫
Ω

up+1dx

)1−α
,

where α =
(r−1)(p+1)−rq

(r−1)(p−q) ∈ (0, 1), due to (1.6). By using the fundamental inequality

(2.13) ar11 a
r2
2 ≤ r1a1 + r2a2, a1,a2 > 0,r1,r2 ≥ 0 and r1 + r2 = 1,

it follows from (2.12) that∫
Ω

u
rq
r−1 dx ≤

(
κ
α−1
α

∫
Ω

uq+1dx

)α (
κ

∫
Ω

up+1dx

)1−α
≤ ακ

α−1
α

∫
Ω

uq+1dx + (1 −α)κ
∫

Ω

up+1dx,

(2.14)

where

(2.15) 0 < κ <
k1

(r − 1)(1 −α)

(
k2(q + 1)d

rρ0

) −r
r−1

.

Combining (2.11) with (2.14), we obtain

(2.16) Φ′(t) ≤ K1
∫

Ω

uq+1dx−K2
∫

Ω

up+1dx,

where

(2.17) K1 =
2k2N

ρ0
+ 2(r − 1)ακ

α−1
α

(
k2(q + 1)d

rρ0

) r
r−1

> 0,

and

(2.18) K2 = 2k1 − 2(r − 1)(1 −α)κ
(
k2(q + 1)d

rρ0

) r
r−1

> 0,

due to (2.15). According to Hölder’s inequality, we derive

(2.19)

∫
Ω

uq+1dx ≤
(∫

Ω

up+1dx

)q+1
p+1

|Ω|
p−q
p+1 ,



QUASILINEAR PARABOLIC EQUATION 151

where |Ω| =
∫

Ω
dx is the N-volume of Ω. It follows from (2.16) and (2.19) that

(2.20) Φ′(t) ≤
(∫

Ω

up+1dx

)q+1
p+1

[
K1|Ω|

p−q
p+1 −K2

(∫
Ω

up+1dx

)p−q
p+1

]
.

By Hölder’s inequality again, we have

(2.21) Φ(t) =

∫
Ω

u2dx ≤
(∫

Ω

up+1dx

) 2
p+1

|Ω|
p−1
p+1 .

Therefore, we deduce from (2.20) and (2.21) that

(2.22) Φ′(t) ≤
(∫

Ω

up+1dx

)q+1
p+1 [

K1|Ω|
p−q
p+1 −K2|Ω|

(1−p)(p−q)
2(p+1) Φ

p−q
2

]
.

Hence, we infer from (2.22) that Φ(t) is decreasing in each time interval on which
we have

(2.23) Φ(t) >

(
K1
K2

) 2
p−q

|Ω|,

so that Φ(t) remains bounded for all time under the conditions stated in Theorem
1.1, which completes the proof. �

3. Conditions for blow-up in finite time

In this section, we obtain the blow-up condition of the solution and derive an
upper bound estimate for the blow-up time t∗.
Proof of Theorem 1.2. Using Green formula and the assumptions stated in
Theorem 1.2, we have

Φ′(t) = 2

∫
Ω

uutdx

= 2

∫
Ω

u[div(|∇u|r−2∇u) −f(u)]dx

= 2

∫
∂Ω

u|∇u|r−2
∂u

∂n
ds− 2

∫
Ω

|∇u|rdx− 2
∫

Ω

uf(u)dx

= 2

∫
∂Ω

ug(u)ds− 2
∫

Ω

|∇u|rdx− 2
∫

Ω

uf(u)dx

≥ 2r
∫
∂Ω

G(u)ds− 2
∫

Ω

|∇u|rdx− 2r
∫

Ω

F(u)dx

≥ 2Ψ(t).

(3.1)

Differentiating (1.10), we obtain

Ψ′(t) = r

∫
∂Ω

utg(u)ds−
∫

Ω

(|∇u|r)tdx−r
∫

Ω

utf(u)dx

= r

∫
Ω

utdiv(|∇u|r−2∇u)dx−r
∫

Ω

utf(u)dx

= r

∫
Ω

(ut)
2dx ≥ 0.

(3.2)



152 AHMED, MU AND ZHENG

As Ψ(0) > 0, then Ψ(t) > 0 for all t ∈ (0, t∗). By using Hölder’s inequality , we
derive

(3.3) (Φ′(t))2 = 4

(∫
Ω

uutdx

)2
≤ 4

∫
Ω

u2dx

∫
Ω

(ut)
2dx =

4

r
Φ(t)Ψ′(t).

It follows from (3.1) and (3.3) that

(3.4) Φ(t)Ψ′(t) ≥
r

4
(Φ′(t))2 ≥

r

2
Φ′(t)Ψ(t),

that is

(3.5) (Φ−
r
2 Ψ)′(t) ≥ 0.

Integrating from 0 to t, we have

(3.6) Φ−
r
2 (t)Ψ(t) ≥ Φ−

r
2 (0)Ψ(0) =: K > 0.

Therefore, we deduce from (3.1) that

(3.7) Φ′(t) ≥ 2Ψ ≥ 2KΦ
r
2 (t).

If r > 2, it follows from integrating over (0, t) that

(3.8) Φ(t) ≥
[
Φ

2−r
2 (0) −K(r − 2)t

]− 2
r−2

,

which implies Φ(t) → +∞ as t → T = Φ
2−r
2 (0)

K(r−2) =
Φ(0)

(r−2)Ψ(0) . Hence, for r > 2, we

have

(3.9) t∗ ≤
Φ(0)

(r − 2)Ψ(0)
.

If r = 2, we infer from (3.7) that

(3.10) Φ(t) ≥ Φ(0)e2Kt, for all t > 0,

which implies t∗ = ∞, this completes the proof. �

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QUASILINEAR PARABOLIC EQUATION 153

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College of Mathematics and Statistics, Chongqing University, Chongqing 401331,

China

∗Corresponding author