báo cáo hóa học:" L^{infty} estimates of solutions for the quasilinear parabolic equation with nonlinear gradient term and L^1 data" ppt

This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted
PDF and full text (HTML) versions will be made available soon.
L^{\infty} estimates of solutions for the quasilinear parabolic equation with
nonlinear gradient term and L^1 data
Boundary Value Problems 2012, 2012:19 doi:10.1186/1687-2770-2012-19
Caisheng Chen ()
Fei Yang ()
Zunfu Yang ()
ISSN 1687-2770
Article type Research
Submission date 10 August 2011
Acceptance date 15 February 2012
Publication date 15 February 2012
Article URL />This peer-reviewed article was published immediately upon acceptance. It can be downloaded,
printed and distributed freely for any purposes (see copyright notice below).
For information about publishing your research in Boundary Value Problems go to
/>For information about other SpringerOpen publications go to

Boundary Value Problems
© 2012 Chen et al. ; licensee Springer.
This is an open access article distributed under the terms of the Creative Commons Attribution License ( />which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
L
∞
estimates of solutions for the quasilinear parabolic
equation with nonlinear gradient term and L
1
data
Caisheng Chen
1
, Fei Yang
2

and Zunfu Yang
3
College of Science, Hohai University, Nanjing 210098, P. R. China
∗
Corresponding author:
Email addresses:
FY:
ZY:
Abstract
In this article, we study the quasilinear parabolic problem

u
t
− div(|∇u|
m
∇u) + u|u|
β−2
|∇u|
q
= u|u|
α−2
|∇u|
p
+ g(u), x ∈ Ω, t > 0,
u(x, 0) = u
0
(x), x ∈ Ω; u(x, t) = 0, x ∈ ∂Ω, t ≥ 0,
(0.1)
where Ω is a bounded domain in R
N

, m > 0 and g(u) satisfies |g(u)| ≤ K
1
|u|
1+ν
with 0 ≤ ν < m. By the Moser’s technique, we prove that if α, β > 1, 0 ≤ p <
q, 1 ≤ q < m + 2, p + α < q + β, there exists a weak solution u(t) ∈ L
∞
([0, ∞), L
1
) ∩
L
∞
loc
((0, ∞), W
1,m+2
0
) for all u
0
∈ L
1
(Ω). Furthermore, if 2q ≤ m + 2, we derive the
L
∞
estimate for ∇u(t). The asymptotic behavior of global weak solution u(t) for
small initial data u
0
∈ L
2
(Ω) also be established if p + α > max{m + 2, q + β}.
Keywords: quasilinear parabolic equation; L

∞
estimates; asymptotic behavior of solution.
2000 Mathematics Subject Classification: 35K20; 35K59; 35K65.
1 Introduction
In this article, we are concerned with the initial boundary value problem of the quasilinear
parabolic equation with nonlinear gradient term

u
t
− div(|∇u|
m
∇u) + u|u|
β−2
|∇u|
q
= u|u|
α−2
|∇u|
p
+ g(u), x ∈ Ω, t > 0,
u(x, 0) = u
0
(x), x ∈ Ω, u(x, t) = 0, x ∈ ∂Ω, t ≥ 0,
(1.1)
where Ω is a bounded domain in R
N
with smooth boundary ∂Ω and m > 0, α , β > 1, 0 ≤ p <
q, 1 ≤ q < m + 2.
1
Recently, Andreu et al. in [1] considered the following quasilinear parabolic problem


u
t
− ∆u + u|u|
β−2
|∇u|
q
= u|u|
α−2
|∇u|
p
, x ∈ Ω, t > 0,
u(x, 0) = u
0
(x), x ∈ Ω, u(x, t) = 0, x ∈ ∂Ω, t ≥ 0,
(1.2)
where α, β > 1, 0 ≤ p < q ≤ 2, p + α < q + β and u
0
∈ L
1
(Ω). By the so-called stability theorem
with the initial data, they proved that there exists a generalized solution u(t) ∈ C([0, T ], L
1
) for
(1.2), in which u(t) satisfies A
k
(u) ∈ L
2
([0, T ], W
1,2

0
) and

Ω
J
k
(u(t) − φ(t))dx +
t

0

Ω
(∇u · ∇A
k
(u − φ) + u|u|
β−2
|∇u|
q
A
k
(u − φ))dxds
=
t

0

Ω
(u|u|
α−2
|∇u|

p
A
k
(u − φ) − A
k
(u − φ)φ
s
)dxds +

Ω
J
k
(u
0
− φ(0))dx
(1.3)
for ∀t ∈ [0, T ] and ∀φ ∈ L
2
([0, T ], W
1,2
0
) ∩ L
∞
(Q
T
), where Q
T
= Ω × (0, T ], and for any k > 0,
A
k

(u) =



−k u ≤ −k,
u − k ≤ u ≤ k,
k u ≥ k.
(1.4)
J
k
(u) is the primitive of A
k
(u) such that J
k
(0) = 0. The problem similar to (1.2) has also been
extensively considered, see [2–6] and the references therein. It is an interesting problem to prove
the existence of global solution u(t) of (1.2) or (1.1) and to derive the L
∞
estimate for u(t) and
∇u(t).
Porzio in [7] also investigated the solution of Leray-Lions type problem



u
t
= div(a(x, t, u, ∇u)), (x, t) ∈ Ω × (0, +∞),
u(x, 0) = u
0
(x), x ∈ Ω,

u(x, t) = 0, (x, t) ∈ ∂Ω × (0, +∞),
(1.5)
where a(x, t, s, ξ) is a Carath´eodory function satisfying the following structure condition
a(x, t, s, ξ)ξ ≥ θ|ξ|
m
, for ∀(x, t, s, ξ) ∈ Ω × R
+
× R
1
× R
N
(1.6)
with θ > 0 and u
0
∈ L
q
(Ω), q ≥ 1. By the integral inequalities method, Porzio derived the L
∞
decay estimate of the form
u(t)
L
∞
(Ω)
≤ Cu
0

α
L
q
(Ω)

t
−λ
, t > 0 (1.7)
with C = C(N, q, m, θ), α = mq(N(m − 2) + mq)
−1
, λ = N(N(m − 2) + mq)
−1
.
In this article, we will consider the global existence of solution u(t) of (1.1) with u
0
∈ L
1
(Ω)
and give the L
∞
estimates for u(t) under the similar condition in [1]. More specially, we will
study the behavior of solution u(t) as t → 0
+
. Obviously, if m = 0 and g ≡ 0, problem (1.1)
is reduced to (1.2). We remark that the methods used in our article are different from that of
[1]. In L
∞
estimates, we use an improved Morser’s technique as in [8–10]. Since the equation
2
in (1.1) contains the nonlinear gradient term u|u|
α−2
|∇u|
p
and u|u|
β−2

|∇u|
q
, it is difficult to
derive L
∞
estimates for u(t) and ∇u(t).
This article is organized as follows. In Section 2, we state the main results and present some
Lemmas which will be used later. In Section 3, we use these Lemmas to derive L
∞
estimates of
u(t). Also the proof of the main results will be given in Section 3. The L
∞
estimates of ∇u (t)
are considered in Section 4. The asymptotic behavior of solution for the small initial data u
0
(x)
is investigated in Section 5.
2 Preliminaries and main results
Let Ω be a bounded domain in R
N
with smooth boundary ∂Ω and  · 
r
,  · 
1,r
denote the
Sobolev space L
r
(Ω) and W
1,r
(Ω) norms, respectively, 1 ≤ r ≤ ∞. We often drop the letter Ω

in these notations.
Let us state our precise assumptions on the parameters p, q, α, β and the function g(u).
(H
1
) the parameters α, β > 1, 0 ≤ p < q < m + 2 < N, p +α < q +β and q(α − 1) ≥ p(β −1),
(H
2
) the function g(u) ∈ C
1
and ∃K
1
≥ 0 and 0 ≤ ν < max{q + β − 2, m}, such that
|g(u)| ≤ K
1
|u|
1+ν
, ∀u ∈ R
1
,
(H
3
) the initial data u
0
∈ L
1
(Ω),
(H
4
) 2q ≤ 2 + m, α, β < 2 + m(1 + 1/N)/2,
(H

5
) the mean curvature of H(x) of ∂Ω at x is non-positive with respect to the outward
normal.
Remark 2.1 The assumptions (H
1
) and (H
3
) are similar to as in [1].
Definition 2.2 A measurable function u(t) = u(x, t) on Ω × [0, ∞) is said to be a global
weak solution of the problem (1.1) if u(t) is in the class
C([0, ∞), L
1
) ∩ L
∞
loc
((0, ∞), W
1,m+2
0
)
and u|u|
β−2
|∇u|
q
, u|u|
α−2
|∇u|
p
∈ L
1
loc

([0, ∞)×Ω), and for any φ = φ(x, t) ∈ C
1
([0, ∞), C
1
0
(Ω)),
the equality
T

0

Ω

−uφ
t
+ |∇u|
m
∇u∇φ + u|u|
β−2
|∇u|
q
φ

dxdt
=

Ω
(u
0
(x)φ(x, 0) − u(x, T )φ(x, T ))dx +

T

0

Ω
(u|u|
α−2
|∇u|
p
+ g(u))φdxdt
(2.1)
is valid for any T > 0.
Remark 2.3 In [1], the concept of generalized solution for (1.2) was introduced. A similar
concept can be found in [7, 11]. By the definition, we know that weak solution is the generalized
solution. Conversely, a generalized solution is not necessarily weak solution.
Our main results read as follows.
3
Theorem 2.4 Assume (H
1
)–(H
3
). Then the problem (1.1) admits a global weak solution
u(t) which satisfies
u(t) ∈ L
∞
([0, ∞), L
1
) ∩ C([0, ∞), L
1
) ∩ L

∞
loc
((0, ∞), W
1,m+2
0
), u
t
∈ L
2
loc
((0, ∞), L
2
) (2.2)
and the estimates
u(t)
∞
≤ C
0
t
−λ
, 0 < t ≤ T. (2.3)
Furthermore, if (H
4
) is satisfied, the solution u(t) has the following estimates
T

0
s
1+r
u

t
(s)
2
2
ds ≤ C
0
, (2.4)
∇u(t)
m+2
≤ C
0
t
−(1+λ)/(m+2)
, 0 < t ≤ T, (2.5)
with r > λ = N(mN + m + 2)
−1
and C
0
= C
0
(T, u
0

1
).
Theorem 2.5 Assume (H
1
)–(H
5
). Then the solution u(t) of (1.1) has the following L

∞
gradient estimate
∇u(t)
∞
≤ C
0
t
−σ
, 0 < t ≤ T, (2.6)
with σ = (2 + 2λ + N)(mN + 2m + 4)
−1
and C
0
= C
0
(T, u
0

1
).
Remark 2.6 The estimates (2.3) and (2.6) give the behavior of u(t)
∞
and ∇u(t)
∞
as
t → 0
+
.
Theorem 2.7 Assume the parameters α, β > 1, γ ≥ 0, 0 ≤ q < m + 2 < N and p <
m + 2 < p + α, α ≤ (m + 2 − p)(1 + 2N

−1
).
Then, ∃d
0
> 0, such that u
0
∈ L
2
(Ω) with u
0

2
< d
0
, the initial boundary value problem

u
t
− div(|∇u|
m
∇u) + γu|u|
β−2
|∇u|
q
= |u|
α−2
u|∇u|
p
, x ∈ Ω, t > 0,
u(x, 0) = u

0
(x), x ∈ Ω, u(x, t) = 0, x ∈ ∂Ω, t ≥ 0,
(2.7)
admits a solution u(t) ∈ L
∞
([0, ∞), L
2
) ∩ W
1,m+2
0
, which satisfies
u(t)
2
≤ C(1 + t)
−1/m
, t ≥ 0. (2.8)
where C = C(u
0

2
).
Theorem 2.8 Assume the parameters γ > 0, α, β > 1, 1 ≤ p < q < m + 2 < N and
τ = N(µ − q)(q + β) ≤ 2(q
2
+ Nβ) with µ = (qα − pβ)/(q − p) > q + β.
Then, ∃d
0
> 0, such that u
0
∈ L

2
with u
0

2
< d
0
, the initial boundary value problem

u
t
− div(|∇u|
m
∇u) + γu|u|
β−2
|∇u|
q
= |u|
α−2
u|∇u|
p
, x ∈ Ω, t > 0
u(x, 0) = u
0
(x), x ∈ Ω; u(x, t) = 0, x ∈ ∂Ω, t ≥ 0
(2.9)
admits a solution u(t) ∈ L
∞
([0, ∞), L
2

) ∩ W
1,m+2
0
which satisfies
u(t)
2
≤ C(1 + t)
−1/(q+β−2)
, t ≥ 0. (2.10)
4
where C = C(u
0

2
).
To obtain the above results, we will need the following Lemmas.
Lemma 2.9 (Gagliardo–Nirenberg type inequality) Let β ≥ 0, N > p ≥ 1, q ≥ 1 + β and
1 ≤ r ≤ q ≤ pN(1 + β)/(N − p). Then for |u|
β
u ∈ W
1,p
(Ω), we have
u
q
≤ C
1/(β+1)
0
u
1−θ
r

|u|
β
u
θ/(β+1)
1,p
with θ = (1 + β)(r
−1
− q
−1
)/(N
−1
− p
−1
+ (1 + β )r
−1
), where the constant C
0
depends only on
p, N.
The Proof of Lemma 2.9 can be obtained from the well-known Gagliardo–Nirenberg–Sobolev
inequality and the interpolation inequality and is omitted here.
Lemma 2.10 [10] Let y(t) be a nonnegative differentiable function on (0, T ] satisfying
y

(t) + At
λθ−1
y
1+θ
(t) ≤ Bt
−k

y(t) + Ct
−δ
, 0 < t ≤ T
with A, θ > 0, λθ ≥ 1, B, C ≥ 0, k ≤ 1. Then, we have
y(t) ≤ A
−1/θ
(2λ + 2BT
1−k
)
1/θ
t
−λ
+ 2C(λ + BT
1−k
)
−1
t
1−δ
, 0 < t ≤ T.
3 L
∞
estimate for u(t)
In this section, we derive a priori estimates of the assumed solutions u(t) and give a proof of
Theorem 2.4. The solutions are in fact given as limits of smooth solutions of appropriate approx-
imate equations and we may assume for our estimates that the solutions under consideration
are sufficiently smooth.
Let u
0,i
∈ C
2

0
(Ω) and u
0,i
→ u
0
in L
1
(Ω) as i → ∞. For i = 1, 2, . . . , we consider the
approximate problem of (1.1)



u
t
− div

(|∇u|
2
+ i
−1
)
m
2
∇u

+ u|u|
β−2
|∇u|
q
= u|u|

α−2
|∇u|
p
+ g(u), x ∈ Ω, t > 0,
u(x, 0) = u
0,i
(x), x ∈ Ω, u(x, t) = 0, x ∈ ∂Ω, t ≥ 0.
(3.1)
The problem (3.1) is a standard quasilinear parabolic equation and admits a unique smooth
solution u
i
(t)(see Chapter 6 in [12]). We will derive estimates for u
i
(t). For the simplicity of
notation, we write u instead of u
i
and u
k
for |u|
k−1
u where k > 0. Also, let C, C
j
be generic
constants independent of k, i, n changeable from line to line.
Lemma 3.1 Let (H
1
)–(H
3
) hold. Suppose that u(t) is the solution of (3.1), then u(t) ∈
L

∞
([0, ∞), L
1
).
Proof Let n = 1, 2, . . ., and
f
n
(s) =







1,
1
n
≤ s
ns(2 − ns), 0 ≤ s ≤
1
n
−ns(2 + ns), −
1
n
≤ s ≤ 0
−1, s < −
1
n
.

It is obvious that f
n
(s) is odd and continuously differentiable in R
1
. Furthermore, |f
n
(s)| ≤
1, f

n
(s) ≥ 0 and f
n
(s) → sign(s) uniformly in R
1
.
5
Multiplying the equation in (3.1) by f
n
(u) and integrating on Ω, we get

Ω
f
n
(u)u
t
dx +

Ω
|∇u|
m+2

f

n
(u)dx +

Ω
u|u|
β−2
f
n
(u)|∇u|
q
dx
≤

Ω
u|u|
α−2
f
n
(u)|∇u|
p
dx +

Ω
g(u)f
n
(u)dx
(3.2)
and the application of the Young inequality gives


Ω
u|u|
α−2
f
n
(u)|∇u|
p
dx ≤
1
4

Ω
u|u|
β−2
f
n
(u)|∇u|
q
dx + C
1

Ω
|u|
µ−1
dx, (3.3)
where µ = (qα − pβ)(q − p )
−1
≥ 1, i.e q(α − 1) ≥ p(β − 1).
In order to get the estimate for the third term of left-hand side in (3.2), we denote

F
n
(u) =
u

0
(s|s|
β−2
f
n
(s))
1/q
ds, u ∈ R
1
.
It is easy to verify that F
n
(u) is odd in R
1
. Then, we obtain from the Sobolev inequality
that
1
4

Ω
u|u|
β−2
f
n
(u)|∇u|

q
dx =
1
4

Ω
|∇F
n
(u)|
q
dx
≥λ
0

Ω
|F
n
(u)|
q
dx = λ
0

Ω
n
|F
n
(u)|
q
dx + λ
0


Ω
c
n
|F
n
(u)|
q
dx
(3.4)
with some λ
0
> 0 and
Ω
n
= {x ∈ Ω||u(x, t)| ≥ n
−1
}, Ω
c
n
= Ω\Ω
n
, n = 1, 2, . . . .
We note that |F
n
(u)|
q
≤ n
−(q+β−1)
in Ω

c
n
and

Ω
c
n
|F
n
(u)|
q
dx ≤ n
−(q+β−1)
|Ω|.
On the other hand, we have |u(x, t)| ≥ n
−1
in Ω
n
and
|F
n
(u)| ≥
|u|

n
−1
(s|s|
β−2
f
n

(s))
1/q
ds ≥
q
q + β − 1

|u|
q+β−1
q
− n
−
q+β−1
q

in Ω
n
.
This implies that there exists λ
1
> 0, such that
λ
0

Ω
n
|F
n
(u)|
q
dx ≥ λ

1

Ω
n
|u|
q+β−1
dx − λ
1
|Ω|n
−(q+β−1)
(3.5)
6
Then it follows from (3.4)–(3.5) that
1
4

Ω
u|u|
β−2
f
n
(u)|∇u|
q
dx ≥ λ
1

Ω
|u|
q+β−1
dx − C

2
n
−(q+β−1)
(3.6)
with some C
2
> 0.
Similarly, we have from the assumption (H
2
) and the Young inequality that

Ω
|g(u)f
n
(u)|dx ≤K
1

Ω
|u|
1+ν
|f
n
(u)|dx
≤K
1

Ω
|u|
1+ν
dx ≤

λ
1
2

Ω
n
|u|
q+β−1
dx + C
2
(1 + n
−1−ν
).
(3.7)
Furthermore, the assumption µ < q + β implies that
C
1

Ω
n
|u|
µ−1
dx ≤
λ
1
2

Ω
n
|u|

q+β−1
dx + C
2
. (3.8)
Then (3.2)–(3.3) and (3.6)–(3.8) give that

Ω
f
n
(u)u
t
dx +
1
2

Ω
u|u|
β−2
f
n
(u)|∇u|
q
dx ≤ C
3

1 + n
−1−ν
+ n
−(q+β−1)


. (3.9)
Letting n → ∞ in (3.9) yields
d
dt
u(t)
1
+
1
2

Ω
|u|
β−1
|∇u|
q
dx ≤ C
3
. (3.10)
Note that

Ω
|u|
β−1
|∇u|
q
dx =

q
q + β − 1


q

Ω
|∇u
1+
β−1
q
|
q
dx ≥ 2λ
2
u
q+β−1
1
with some λ
2
> 0. Then (3.10) becomes
d
dt
u(t)
1
+ λ
2
u(t)
q+β−1
1
≤ C
3
. (3.11)
This gives that u(t) ∈ L

∞
([0, ∞), L
1
) if u
0
∈ L
1
. ✷.
Remark 3.2 The differential inequality (3.10) implies that the solution u
i
(t) of (3.1)
satisfies
T

0

Ω
|u
i
|
β−1
|∇u
i
|
q
dxdt ≤ C
0
for i = 1, 2, . . . . (3.12)
with C
0

= C
0
(T, u
0

1
).
7
Lemma 3.3 Assume (H
1
)–(H
4
). Then, for any T > 0, the solution u(t) of (3.1) also
satisfies the following estimates:
u(t)
∞
≤ C
0
t
−λ
, 0 < t ≤ T, (3.13)
where λ = N(mN + m + 2)
−1
, C
0
= C
0
(T, u
0


1
).
Proof Multiplying the equation in (3.1) by u
k−1
, k ≥ 2, we have
1
k
d
dt
u(t)
k
k
+ (k − 1)

m + 2
k + 2

m+2
∇u
k+m
m+2

m+2
m+2
+

Ω
|u|
β+k−2
|∇u|

q
dx
≤

Ω
|u|
α+k−2
|∇u|
p
dx + K
1

Ω
|u|
ν+k
dx.
(3.14)
It follows from the H¨older and Sobolev inequalities that
K
1

Ω
|u|
ν+k
dx ≤ Cu
θ
1
k
u
θ

2
1
u
θ
3
s
≤ Cu
θ
1
k
∇u
k+m
m+2

(m+2)θ
3
k+m
m+2
≤
k − 1
2

m + 2
k + 2

m+2
∇u
k+m
m+2


m+2
m+2
+ Ck
σ
u
k
k
,
in which θ
1
= kλ(m − ν + (m + 2)N
−1
), θ
2
= νλ(m + 2)N
−1
, θ
3
= νλ(k + m), σ = νλ, s =
N(k + m)(N − m − 2)
−1
.
Note that

Ω
|u|
α+k−2
|∇u|
p
dx ≤

1
4

Ω
|u|
β+k−2
|∇u|
q
dx + C

Ω
|u|
µ+k−2
dx
and
1
2

Ω
|u|
β+k−2
|∇u|
q
dx ≥ C
1
k
−q

Ω
|∇u

q+β+k−2
q
|
q
dx
with some C
1
independent of k and µ = (qα − pβ)(q − p)
−1
< q + β.
Without loss of generality, we assume k > 3 − µ. Similarly, we derive
C

Ω
|u|
µ+k−2
dx ≤ Cu
µ
1
k−2
u
µ
2
1
u
µ
3
k
∗
≤ Cξ

µ
2
1
u
µ
1
k
u
µ
3
k
∗
≤ Cu
µ
1
k
∇u
q
k
/q

qµ
3
/q
k
q
≡ A
k
with ξ
1

= sup
t≥0
u(t)
1
and
µ
1
= λ
0
(k − 2)(q + β − µ + qN
−1
), µ
2
= λ
0
µqN
−1
, µ
3
= λ
0
µq
k
,
λ
0
= (q + β + q/N)
−1
, k
∗

= q
k
N(N − q)
−1
, q
k
= q + β + k − 2.
Then, for any η > 0,
A
k
≤ Cη∇u
q
k
/q

q
q
+ Cη
−θ

/θ
u
µ
1
θ

k
(3.15)
8
with µλ

0
θ = 1, (1 − µλ
0
)θ

= 1.
Note that µ
1
θ

< k. Let η =
C
1
2C
k
−q
. Then it follows from (3.15) that
A
k
≤
C
1
2
k
−q
∇u
q
k
/q


q
q
+ Ck
γ
(u
k
k
+ 1)
(3.16)
with γ = qθ

θ
−1
= qµλ
0
/(1 − µλ
0
). Then, (3.14) becomes
1
k
d
dt
u
k
k
+
k − 1
2

m + 2

k + 2

m+2
∇u
k+m
m+2

m+2
m+2
+
C
1
2
k
−q
∇u
q
k
/q

q
q
≤ Ck
σ
0
(u
k
k
+ 1)
or

d
dt
u
k
k
+ C
1
k
−m
∇u
k+m
m+2

m+2
m+2
≤ Ck
1+σ
0
(u
k
k
+ 1) (3.17)
with σ
0
= max{σ, γ} = max{νλ, γ}.
Now we employ an improved Moser’s technique as in [8, 9]. Let {k
n
} be a sequence defined
by k
1

= 1, k
n
= R
n−2
(R − m − 1) + m(R − 1)
−1
(n = 2, 3, . . .) with R > max{m + 1, m + 4 − µ}
such that k
n
≥ 3 − µ(n ≥ 2). Obviously, k
n
→ ∞ as n → ∞.
By Lemma 2.9, we have
u(t)
k
n
≤ C
m+2
m+k
n
0
u(t)
1−θ
n
k
n−1
∇u
m+k
n
m+2


θ
n
(m+2)
m+k
n
m+2
(3.18)
with θ
n
= RN (1 − k
n−1
k
−1
n
)(m + 2 + N(R − 1))
−1
.
Then, inserting (3.18) into (3.17) (k = k
n
), we find that
d
dt
u(t)
k
n
k
n
+ C
1

C
−
m+2
θ
n
0
k
−m
n
u(t)
(1−1/θ
n
)(m+k
n
)
k
n−1
u(t)
(m+k
n
)/θ
n
k
n
≤ Ck
1+σ
0
n
(u(t)
k

n
k
n
+ 1), 0 < t ≤ T,
(3.19)
or
d
dt
u(t)
k
n
k
n
+ C
1
C
−
m+2
θ
n
0
k
−m
n
u(t)
m−β
n
k
n−1
u(t)

k
n
+β
n
k
n
≤ Ck
1+σ
0
n
(u(t)
k
n
k
n
+ 1), (3.20)
where β
n
= (m + k
n
)θ
−1
n
− k
n
, n = 2, 3, . . It is easy to see that
θ
n
→ θ
0

=
N(R − 1)
m + 2 + N(R − 1)
, β
n
k
−1
n
→
m + 2
N(R − 1)
, as n → ∞.
Denote
y
n
(t) = u(t)
k
n
k
n
, 0 < t ≤ T.
Then (3.20) can be rewritten as follows
y

n
(t) + C
1
C
−
m+2

θ
n
k
−m
n
(y
n
(t))
1+β
n
/k
n
u(t)
m−β
n
k
n−1
≤ Ck
1+σ
0
n
(y
n
(t) + 1). (3.21)
We claim that there exist a bounded sequence {ξ
n
} and a convergent sequence {λ
n
}, such
that

u(t)
k
n
≤ ξ
n
t
−λ
n
, 0 < t ≤ T. (3.22)
9
Indeed, by Lemma 3.1, the estimate (3.22) holds for n = 1 if we take λ
1
= 0, ξ
1
=
sup
t≥0
u(t)
1
. If (3.22) is true for n − 1, then we have from (3.21) and (3.22) that
y

n
(t) + C
1
C
−
m+2
θ
n

k
−m
n
(ξ
n−1
)
m−β
n
t
Λ
n
τ
n
−1
y
1+τ
n
n
(t) ≤ Ck
1+σ
0
n
(y
n
(t) + 1), 0 ≤ t ≤ T, (3.23)
where
τ
n
=
β

n
k
n
, Λ
n
= k
n
λ
n
, λ
n
=
1 + λ
n−1
(β
n
− m)
β
n
.
Applying Lemma 2.10 to (3.23), we have
y
n
(t) ≤

C
1
C
−
m+2

θ
n
k
−m
n
ξ
m−β
n
n−1

−1/τ
n
(2k
n
λ
n
+ 2CT k
1+σ
0
n
)
1/τ
n
t
−k
n
λ
n
. (3.24)
This implies that for t ∈ (0, T),

u(t)
k
n
≤

C
1
C
−
m+2
θ
n
k
−m
n
ξ
m−β
n
n−1

−1/β
n
(2k
n
λ
n
+ 2CT k
1+σ
0
n

)
1/β
n
t
−λ
n
≤ ξ
n
t
−λ
n
, (3.25)
where
ξ
n
= ξ
n−1

C
1
C
−
m+2
θ
n
k
−m
n

−1/β

n
(2k
n
λ
n
+ 4CT k
1+σ
0
n
)
1/β
n
, (3.26)
in which the fact k
n
∼ β
n
as n → ∞ has been used.
It is not difficult to show that {ξ
n
} is bounded. Furthermore, by Lemma 4 in [9], we have
1 + λ
n−1
(β
n
− m)
β
n
→ λ =
N

m + 2 + mN
, as n → ∞.
Letting n → ∞ in (3.22) implies that (3.13) and we finish the Proof of Lemma 3.3. ✷.
Lemma 3.4. Let (H
1
)–(H
4
) hold. Then, the solution u(t) of (3.1) has the following
estimates
T

0
s
1+r
u
t
(s)
2
2
ds ≤ C
0
(3.27)
and
∇u(t)
m+2
≤ C
0
t
−(1+λ)/(m+2)
, 0 < t ≤ T, (3.28)

with r > λ = N(mN + m + 2)
−1
, C
0
= C
0
(T, u
0

1
).
Proof We first choose r > λ and η(t) ∈ C[0, ∞) ∩ C
1
(0, ∞) such that η(t) = t
r
when
t ∈ [0, 1]; η(t) = 2, when t ≥ 2 and η(t), η

(t) ≥ 0 in [0, ∞). Multiplying the equation in (3.1) by
η(t)u, we have
1
2
η(t)u(t)
2
2
+
t

0
η(s)∇u(s)

m+2
m+2
ds +
t

0

Ω
|u|
β
|∇u|
q
η(s)dxds
≤
1
2
t

0
η

(s)u(s)
2
2
ds +
t

0

Ω

|u|
α
|∇u|
p
η(s)dxds + K
1
t

0

Ω
|u|
2+ν
η(s)dxds.
(3.29)
10
Note that
t

0

Ω
|u|
α
|∇u|
p
η(s)dxds ≤
1
2
t


0

Ω
|u|
β
|∇u|
q
η(s)dxds + C
t

0

Ω
|u|
µ
η(s)dxds.
Hence, we have
1
2
η(t)u(t)
2
2
+
t

0
η(s)∇u(s)
m+2
m+2

ds +
1
2
t

0

Ω
|u|
β
|∇u|
q
η(s)dxds
≤
1
2
t

0
η

(s)u(s)
2
2
ds + C
t

0

Ω

|u|
µ
η(s)dxds + K
1
t

0

Ω
|u|
2+ν
η(s)dxds.
(3.30)
By Lemma 3.1 and the estimate (3.13), we get
t

0
η

(s)u(s)
2
2
ds ≤ C
t

0
s
r−1
u(t)
1

u(t)
∞
ds ≤ Ct
r−λ
, 0 ≤ t < T.
(3.31)
Since µ < q + β, we have from Sobolev inequality that
C
t

0

Ω
|u|
µ
η(s)dxds ≤
1
4
t

0

Ω
|u|
β
|∇u|
q
η(s)dxds + C
t


0
η(s)ds. (3.32)
Similarly, we have from 2 + ν < q + β that
K
1
t

0

Ω
|u|
2+ν
η(s)dxds ≤
1
4
t

0

Ω
|u|
β
|∇u|
q
η(s)dxds + C
t

0
η(s)ds. (3.33)
Therefore, it follows from (3.30)–(3.33) that

t

0

Ω
|∇u|
m+2
η(s)dxds ≤ Ct
r−λ
, 0 ≤ t ≤ T.
(3.34)
Next, let G(u) =

u
0
g(s)ds, u ∈ R
1
, ρ(t) =

t
0
η(s)ds, t ∈ (0, ∞). Furthermore, multiplying
the equation in (3.1) by ρ(t)u
t
yields
ρ(t)u
t
(t)
2
2

+
1
m + 2
d
dt

Ω
ρ(t)(|∇u|
2
+ i
−1
)
m+2
2
dx + ρ

(t)

Ω
G(u)dx
≤
ρ

(t)
m + 2

Ω
(|∇u|
2
+ i

−1
)
m+2
2
dx +
d
dt

Ω
ρ(t)G(u)dx
+

Ω
ρ(t)|u|
β−1
|u
t
||∇u|
q
dx +

Ω
ρ(t)|u|
α−1
|u
t
||∇u|
p
dx.
(3.35)

11
By the assumption p < q and the Cauchy inequality, we deduce

Ω
|u|
β−1
|u
t
||∇u|
q
dx ≤
1
4
u
t
(t)
2
2
+ C

Ω
|u|
2(β−1)
|∇u|
2q
dx (3.36)
and

Ω
|u|

α−1
|u
t
||∇u|
p
dx ≤
1
4
u
t
(t)
2
2
+ C

Ω
|u|
2(α−1)
|∇u|
2p
dx
≤
1
4
u
t
(t)
2
2
+ C


Ω
|u|
2(β−1)
|∇u|
2q
dx + C

Ω
|u|
2(µ−1)
dx
(3.37)
and

Ω
|G(u)|dx ≤ C
1

Ω
|u|
2+ν
dx ≤ Ch
2+ν
(t) (3.38)
with h(t) = u(t)
∞
.
Now, it follows from (H
4

) and (3.35)–(3.38) that
1
2
t

0
ρ(s)u
t
(s)
2
2
ds +
1
m + 2
ρ(t)∇u(t)
m+2
m+2
≤
1
2
t

0
η(s)∇u(s)
m+2
m+2
ds + Cρ(t)h
2+ν
(t)
+ C

t

0
ρ(s)h
2(β−1)
(s)(1 + ∇u(s)
m+2
m+2
)ds + C
t

0
(ρ(s)h
2(µ−1)
(s) + η(s)h
2+ν
(s))ds
≤ C(t
r−λ
+ t
r+2−2(β−1)λ
+ t
r+2−2(µ−1)λ
+ t
r+1−(2+ν)λ
)
+ C
t

0

ρ(s)h
2(β−1)
(s)∇u(s)
m+2
m+2
ds,
(3.39)
or
1
2
t

0
ρ(s)u
t
(s)
2
2
ds +
ρ(t)
m + 2
∇u(t)
m+2
m+2
≤ C
0
t
r−λ
+ C
0

t

0
ρ(s)h
2(β−1)
(s)∇u(s)
m+2
m+2
ds
(3.40)
where C
0
= C
0
(T, u
0

1
) and the fact 2 + λ ≥ 2(µ − 1)λ has been used.
Since the function h
2(β−1)
(t) ∈ L
1
([0, T ]), the application of the Gronwall inequality to (3.40)
gives
t

0
ρ(s)u
t

(s)
2
2
ds + ρ(t)∇u
m+2
m+2
≤ C
0
t
r−λ
, 0 < t ≤ T.
(3.41)
12
Hence,
∇u
m+2
≤ C
0
t
−(1+λ)/(m+2)
, 0 < t ≤ T.
(3.42)
and the Proof of Lemma 3.4 is completed. ✷
Proof of Theorem 2.4 Noticing that the estimate constant C
0
in (3.12)–(3.13) and (3.27)–
(3.28) is independent of i, we have from the standard compact argument as in [1,13,14] that there
exists a subsequence (still denoted by u
i
) and a function u ∈ L

s
([0, T ], W
1,s
0
(Ω)), (1 ≤ s ≤ m+2)
satisfying
u
i
 u weakly in L
s
([0, T ], W
1,s
0
(Ω)),
u
i
→ u in L
s
(Q
T
) and a.e. in Q
T
,
|u
i
|
β−1
|∇u
i
|

q
→ |u|
β−1
|∇u|
q
in L
1
(Q
T
),
|u
i
|
α−1
|∇u
i
|
p
→ |u|
α−1
|∇u|
p
in L
1
(Q
T
),
u
i
→ u in C([0, T ]; L

1
(Ω)),
∂u
i
∂t
→
∂u
∂t
weakly in L
2
loc
(0, T ; L
2
).
(3.43)
Since A
i
(u
i
) = −div((|∇u
i
|
2
+ i
−1
)
m
2
∇u
i

) is bounded in (W
1,m+2
0
)
∗
= W
−1,
m+2
m+1
0
, we see
further that
A
i
(u
i
) → χ weakly
∗
in L
∞
loc
(0, T ; (W
1,m+2
0
)
∗
) (3.44)
for some χ ∈ L
∞
loc

([0, T ], (W
1,m+2
0
)
∗
). As the Proof of Theorem 1 in [9], we have χ = A(u) =
−div((|∇u|
m
∇u).
Then, the function u is a global weak solution of (1.1). Furthermore, it follows from Lemma
3.4 that u(t) satisfies the estimate (2.4)–(2.5). The Proof of Theorem 2.4 is now completed. ✷
4 L
∞
estimate for ∇u(t)
In this section, we use an argument similar to that in [9, 10, 15] and give the Proof of
Theorem 2.5. Hence, we only consider the estimate of ∇u
∞
for the smooth solution u(t) of
(3.1). As above, let C, C
j
be the generic constants independent of k and i. Denote
|D
2
u|
2
=
N

i,j=1
u

2
ij
, u
ij
=
∂
2
u
∂x
i
∂x
j
.
13
Multiplying (3.1) by −div(|∇u|
k−2
∇u), k ≥ m + 2 and integrating by parts, we have
1
k
d
dt

∇u(t)
k
k

+

Ω
|∇u|

k+m−2
|D
2
u|
2
dx +
k − 2
4

Ω
|∇u|
k+m−4
|∇(|∇u|
2
)|
2
dx
− (N − 1)

∂Ω
H(x)|∇u|
k+m
dS
=

Ω
u|u|
β−2
|∇u|
q

div(|∇u|
k−2
∇u)dx −

Ω
u|∇u|
p
|u|
α−2
div(|∇u|
k−2
∇u)dx
+

Ω
g(u)div(|∇u|
k−2
∇u)dx ≡ I + II + III.
(4.1)
Since
div(|∇u|
k−2
∇u) = |∇u|
k−2
u +
k − 2
2
|∇u|
k−4
∇u∇(|∇u|

2
), (4.2)
we have
|div(|∇u|
k−2
∇u)| ≤ (k − 1)|∇u|
k−2
|D
2
u| (4.3)
and
|I| ≤ (k − 1)

Ω
|u|
β−1
|∇u|
q+k−2
|D
2
u|dx
= (k − 1)

Ω
|∇u|
k+m−2
2
|D
2
u||∇u|

k+2q−m−2
2
|u|
β−1
dx
≤
1
4

Ω
|∇u|
k+m−2
|D
2
u|
2
dx + C
0
k
2

Ω
|∇u|
k+2q−m−2
|u|
2(β−1)
dx.
(4.4)
Similarly, we obtain the following estimates
|II| ≤

1
4

Ω
|∇u|
m+k−2
|D
2
u|
2
dx + C
0
k
2

Ω
|∇u|
k+2p−m−2
|u|
2(α−1)
dx (4.5)
and
III =

Ω
g(u)div(|∇u|
k−2
∇u)dx = −

Ω

g

(u)|∇u|
k
dx
≤K
1

Ω
|u|
ν
|∇u|
k
dx ≤ Ch
ν
(t)∇u(t)
k
k
,
(4.6)
where h(t) = u(t)
∞
≤ Ct
−λ
.
14
Moreover, we assume that 2q ≤ m + 2, 2p ≤ m + 2, then (4.1) becomes
1
k
d

dt

∇u
k
k

+
1
2

Ω
|∇u|
k+m−2
|D
2
u|
2
dx +
k − 2
4

Ω
|∇u|
k+m−4
|∇(|∇u|
2
)|
2
dx
− (N − 1)


∂Ω
H(x)|∇u|
k+m
dS
≤ C
0
k
2

Ω

|∇u|
k+2q−m−2
|u|
2(β−1)
+ |∇u|
k+2p−m−2
|u|
2(α−1)

dx + Ch
ν
(t)∇u(t)
k
k
≤ C
0
k
2

h
1
(t)

1 + ∇u(t)
k
k

,
(4.7)
where h
1
(t) = max{h
2(α−1)
(t), h
2(β−1)
(t), h
ν
(t)}. Since α, β < 2 +
m
2

1 +
1
N

, ν < m + 2 +
m
N
,

we get h
1
(t) ∈ L
1
([0.T ]) for any T > 0.
If H(x) ≤ 0 on ∂Ω and N > 1, then by an argument of elliptic eigenvalue problem in [15],
there exists λ
1
> 0, such that
∇v
2
2
− (N − 1)

∂Ω
v
2
H(x)dS ≥ λ
1
v
2
1,2
, ∀v ∈ W
1,2
(Ω). (4.8)
Hence, by (4.7) and (4.8), we see that there exists C
1
and C
2
such that

d
dt

∇u(t)
k
k

+ C
1
|∇u(t)|
k+m
2

2
1,2
≤ Ck
3
h
1
(t)(1 + ∇u(t)
k
k
). (4.9)
Let k
1
= m+2, R > m+1, k
n
= R
n−2
(R−1−m)+m(R−1)

−1
, θ
n
= RN (1−k
n−1
k
−1
n
)(R(N −
1) + 2)
−1
, n = 2, 3, . . Then, the application of Lemma 2.9 gives
∇u||
k
n
≤ C
2
k
n
+m
∇u
1−θ
n
k
n−1
|∇u|
k
n
+m
2


2θ
n
k
n
+m
1,2
. (4.10)
Inserting this into (4.9)(k = k
n
), we get
d
dt

∇u||
k
n
k
n

+ C
1
C
−2/θ
n
∇u(t)
(k
n
+m)(1−1/θ
n

)
k
n−1
∇u(t)
(k
n
+m)/θ
n
k
n
≤ C
2
k
3
n
h
1
(t)(1 + ∇u(t)
k
n
k
n
).
(4.11)
By (3.28), we take y
1
= max{1, C
0
}, z
1

= (1 + λ)/(m + 2). As the Proof of Lemma 3.3, we
can show that there exist bounded sequences y
n
and z
n
such that
∇u(t)
k
n
≤ y
n
t
−z
n
, 0 < t ≤ T, (4.12)
in which z
n
→ σ = (2 + 2λ + N)(mN + 2m + 4)
−1
. Letting n → ∞ in (4.12), we have the
estimate (2.6). This completes the Proof of Theorem 2.5. ✷.
15
5 Asymptotic behavior of solution
In this section, we will prove that the problem (1.1) admits a global solution if the initial data
u
0
(x) is small under the assumptions of Theorems 2.7 and 2.8. Also, we derive the asymptotic
behavior of solution u(t).
Proof of Theorem 2.7 The existence of solution for (1.1) in small u
0

can be obtained by
a similar argument as the Proof of Theorem 2.4. So, it is sufficient to derive the estimate (2.8).
Multiplying the equation in (2.7) by u and integrating over Ω, we obtain
1
2
d
dt
u(t)
2
2
+ C
1
∇u(t)
m+2
m+2
≤

Ω
|u|
α
|∇u|
p
dx (5.1)
with C
1
=

m+2
4


m+2
.
Since p < m + 2 < p + α, it follows from Lemma 2.9 that

Ω
|u|
α
|∇u|
p
dx ≤ ∇u(t)
p
m+2
u
α
s
≤ C
0
∇u
p
m+2
u
α(1−θ)
r
∇u
αθ
m+2
≤ C
0
∇u(t)
m+2

m+2
u(t)
p
1
r
(5.2)
with
s =
α(m + 2)
m + 2 − p
, θ =

1
r
−
1
s

1
N
+
1
r
−
1
m + 2

−1
, r =
Np

1
m + 2 − p
, p
1
= p + α − m − 2.
The assumption on α shows that r ≤ 2. Then, (5.1) can be rewritten as
1
2
d
dt
u(t)
2
2
+ ∇u(t)
m+2
m+2
(C
1
− C
0
u(t)
p
1
2
) ≤ 0. (5.3)
By the Sobolev embedding theorem,
∇u(t)
m+2
m+2
≥ C

2
u(t)
m+2
m+2
≥ C
2
u(t)
m+2
2
, (5.4)
we obtain from (5.3) and (5.4) that ∃d
0
> 0, λ
0
> 0, such that u
0

2
< d
0
and
φ

(t) + λ
0
φ
1+m/2
(t) ≤ 0, t ≥ 0 (5.5)
with φ(t) = u(t)
2

2
. This implies that
u(t)
2
≤ C(1 + t)
−1/m
, t ≥ 0, (5.6)
where the constant C depends only u
0

2
. This completes the Proof of Theorem 2.7. ✷
Proof of Theorem 2.8 Multiplying the equation in (2.9) by u and integrating over Ω, we
obtain
1
2
d
dt
u(t)
2
2
+ γ

Ω
|u|
β
|∇u|
q
dx ≤


Ω
|u|
α
|∇u|
p
dx. (5.7)
16
Since p < q, q + β < p + α, it follows from the H¨older inequality that

Ω
|u|
α
|∇u|
p
dx ≤ ∇u
1+β/q

p
q
u
µ(1−p/q)
µ
≤ C
1
∇u
1+β/q

p
q
u

µ
1
(1−p/q)
τ
∇u
1+β/q

µ
2
(1−p/q)
q
≤ C
1
∇u
1+β/q

q
q
u
µ
1
(1−p/q)
τ
≤ C
1
∇u
1+β/q

q
q

u
µ
3
2
(5.8)
with µ
2
= q, µ
1
= µ − q, µ
3
= µ
1
(1 − p/q) and τ = N(µ − q)(q + β)(q
2
+ Nβ)
−1
≤ 2.
Then (5.7) becomes
d
dt
u(t)
2
2
+ ∇u
1+β/q

q
q
(C

0
− C
1
u(t)
µ
3
2
) ≤ 0. (5.9)
This implies that ∃d
0
> 0, λ
1
> 0, such that u
0

2
< d
0
and
φ

(t) + λ
1
φ
(q+β)/2
(t) ≤ 0, t ≥ 0 (5.10)
with φ(t) = u(t)
2
2
. This implies that

u(t)
2
≤ C(1 + t)
−1/(q+β−2)
, t ≥ 0. (5.11)
This is the estimate (2.10) and we finish the Proof of Theorem 2.8. ✷
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
CC proposed the topic and the main ideas. The main results in this article were derived by
CC. FY and ZY participated in the discussion of topic. All authors read and approved the final
manuscript.
Acknowledgments
The authors wish to express their gratitude to the referees for useful comments and suggestions.
References
[1] Andreu, F, Segura de le´on, S, Toledo, J: Quasilinear diffusion equations with gradient terms
and L
1
data. Nonlinear Anal. 56, 1175–1209 (2004)
[2] Andreu, F, Maz´on, JM, Simondon, F, Toledo, J: Global existence for a degenerate nonlinear
diffusion problem with nonlinear gradient term and source. Math. Ann. 314, 703–728 (1999)
17
[3] Andreu, F, Maz´on, JM, Segura de le´on, S, Toledo, J: Existence and uniqueness for a de-
generate parabolic equation with L
1
data. Trans. Amer. Math. So c. 351, 285–306 (1999)
[4] Souplet, PH: Finite time blow-up for a nonlinear parabolic equation with a gradient term
and applications. Math. Methods in the Appl. Sci. 19, 1317–1333 (1996)
[5] Quittner, P: On global existence and stationary solutions for two classes of semilinear
parabolic equations. Comment. Math. Univ. Carolinear, 34, 105–124 (1993)

[6] Fila, M: Remarks on blow up for a nonlinear parabolic equation with a gradient term. Proc.
Amer. Math. Soc. 111, 795–801 (1991)
[7] Porzio, MM: On decay estimates. J. Evol. Equ. 9, 561–591 (2009)
[8] Chen, CS, Wang, RY: L
∞
estimates of solution for the evolution m-Laplacian equation with
initial value in L
q
. Nonlinear Anal. 48, 607–616 (2002)
[9] Nakao, M, Chen, CS: Global existence and gradient estimates for the quasilinear parabolic
equations of m-Laplacian type with a nonlinear convection term. J. Diff. Equ. 162, 224–250
(2000)
[10] Ohara, Y: L
∞
estimates of solution of some nonlinear degenerate parabolic equations.
Nonlinear Anal. 18, 413–426 (1992)
[11] Prignet, A: Existence and uniqueness of entropy solutions of parabolic problems with L
1
data. Nonlinear Anal. 28, 1943–1954 (1997)
[12] Ladyzenskaya, OA, Solonnikov, VA, Uraltseva, NN: Linear and Quasilinear Equations of
Parabolic Type. AMS, Providence (1969)
[13] Souplet, PH, Weissler, FB: Self-similar subsolutions and blow-up for nonlinear parabolic
equations. J. Math. Anal. Appl. 212, 60–74 (1997)
[14] Simon, J: Compact sets in the space L
p
(0, T ; B). Ann. Mat. Pura. Appl. 146, 65–96 (1987)
[15] Engler, HK, Luckhaus, S: Gradient estimates for solution of parabolic equations and sys-
tems. J. Math. Anal. Appl. 147, 309–329 (1990)
18