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Lestimates of solutions for the quasilinear parabolic equation with nonlinear gradient term and L1 data

Caisheng Chen*, Fei Yang and Zunfu Yang

Author Affiliations

College of Science, Hohai University, Nanjing 210098, P. R. China

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Boundary Value Problems 2012, 2012:19  doi:10.1186/1687-2770-2012-19


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/19


Received:10 August 2011
Accepted:15 February 2012
Published:15 February 2012

© 2012 Chen et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ℝN, m > 0 and g(u) satisfies |g(u)| ≤ K1|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 0 1 , m + 2 ) for all u0 L1(Ω). Furthermore, if 2q m + 2, we derive the Lestimate for ∇u(t). The asymptotic behavior of global weak solution u(t) for small initial data u0 L2(Ω) also be established if p + α > max{m + 2, q + β}.

2000 Mathematics Subject Classification: 35K20; 35K59; 35K65.

Keywords:
quasilinear parabolic equation; Lestimates; asymptotic behavior of solution

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 ℝN with smooth boundary Ω and m > 0, α, β > 1, 0 ≤ p < q, 1 ≤ q < m + 2.

Recently, Andreu et al. in [1] considered the following quasilinear parabolic problem

u t - Δ 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 u0 L1(Ω). By the so-called stability theorem with the initial data, they proved that there exists a generalized solution u(t) ∈ C([0, T], L1) for (1.2), in which u(t) satisfies A k ( u ) L 2 ( [ 0 , T ] , W 0 1 , 2 ) and

Ω J k ( u ( t ) - ϕ ( t ) ) d x + 0 t Ω ( u A k ( u - ϕ ) + u u β - 2 u q A k ( u - ϕ ) ) d x d s = 0 t Ω ( u u α - 2 u p A k ( u - ϕ ) - A k ( u - ϕ ) ϕ s ) d x d s + Ω J k ( u 0 - ϕ ( 0 ) ) d x (1.3)

for ∀t ∈ [0, T] and ϕ L 2 ( [ 0 , T ] , W 0 1 , 2 ) L ( Q T ) , where QT = Ω × (0, T], and for any k > 0,

A k ( u ) = - k u - k , u - k u k , k u k . (1.4)

Jk(u) is the primitive of Ak(u) such that Jk(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 Lestimate 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 Carathéodory function satisfying the following structure condition

a ( x , t , s , ξ ) ξ θ ξ m , f o r ( x , t , s , ξ ) Ω × + × 1 × N (1.6)

with θ > 0 and u0 Lq(Ω), q ≥ 1. By the integral inequalities method, Porzio derived the Ldecay estimate of the form

u ( t ) L ( Ω ) C u 0 L q ( Ω ) α 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 u0 L1(Ω) and give the Lestimates 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 Lestimates, we use an improved Morser's technique as in [8-10]. Since the equation 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 Lestimates 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 Lestimates of u(t). Also the proof of the main results will be given in Section 3. The Lestimates of ∇u(t) are considered in Section 4. The asymptotic behavior of solution for the small initial data u0(x) is investigated in Section 5.

2 Preliminaries and main results

Let Ω be a bounded domain in ℝN with smooth boundary Ω and ∥·∥r, ∥·∥1,r denote the Sobolev space Lr(Ω) and W1,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).

(H1) the parameters α, β > 1, 0 ≤ p < q < m + 2 < N, p + α < q + β and q(α - 1) ≥ p(β - 1),

(H2) the function g(u) ∈ C1 and ∃K1 ≥ 0 and 0 ≤ ν < max{q + β - 2, m], such that

g ( u ) K 1 u 1 + ν , u 1 ,

(H3) the initial data u0 L1(Ω),

(H4) 2q ≤ 2 + m, α, β < 2 + m(1 + 1/N)/2,

(H5) the mean curvature of H(x) of Ω at x is non-positive with respect to the outward normal.

Remark 2.1 The assumptions (H1) and (H3) 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 0 1 , m + 2 )

and u u β - 2 u q , u u α - 2 u p L loc 1 ( [ 0 , ) × Ω ) , and for any ϕ = ϕ ( x , t ) C 1 ( [ 0 , ) , C 0 1 ( Ω ) ) the equality

0 T Ω - u ϕ t + u m u ϕ + u u β - 2 u q ϕ d x d t = Ω ( u 0 ( x ) ϕ ( x , 0 ) - u ( x , T ) ϕ ( x , T ) ) d x + 0 T Ω ( u u α - 2 u p + g ( u ) ) ϕ d x d t (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.

Theorem 2.4 Assume (H1)-(H3). 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 0 1 , m + 2 ) , u t L loc 2 ( ( 0 , ) , L 2 ) (2.2)

and the estimates

u ( t ) C 0 t - λ , 0 < t T . (2.3)

Furthermore, if (H4) is satisfied, the solution u(t) has the following estimates

0 T s 1 + r u t ( s ) 2 2 d s 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 C0 = C0(T, ∥u01).

Theorem 2.5 Assume (H1)-(H5). Then the solution u(t) of (1.1) has the following Lgradient estimate

u ( t ) C 0 t - σ , 0 < t T , (2.6)

with σ = (2 + 2λ + N)(mN + 2m + 4)-1 and C0 = C0(T, ∥u01).

Remark 2.6 The estimates (2.3) and (2.6) give the behavior of ∥u(t)∥and ∥∇u(t)∥as

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, ∃d0 > 0, such that u0 L2(Ω) with ∥u02 < d0, the initial boundary value problem

u t - div ( u m u ) + γ u 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 0 1 , m + 2 , which satisfies

u ( t ) 2 C ( 1 + t ) - 1 / m , t 0 . (2.8)

where C = C(∥u02).

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(q2 + ) with μ = (- )/(q - p) > q + β.

Then, ∃d0 > 0, such that u0 L2 with ∥u02 < d0, the initial boundary value problem

u t - div ( u m u ) + γ u 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 0 1 , m + 2 which satisfies

u ( t ) 2 C ( 1 + t ) - 1 / ( q + β - 2 ) , t 0 . (2.10)

where C = C(∥u02).

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|βu W1,p(Ω), we have

u q C 0 1 / ( β + 1 ) u r 1 - θ u β u 1 , p θ / ( β + 1 )

with θ = (1 + β)(r-1 - q-1)/(N-1 - p-1 + (1 + β)r-1), where the constant C0 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 ) + A t λ θ - 1 y 1 + θ ( t ) B t - k y ( t ) + C t - δ , 0 < t T

with A, θ > 0, λθ ≥ 1, B, C ≥ 0, k ≤ 1. Then, we have

y ( t ) A - 1 / θ ( 2 λ + 2 B T 1 - k ) 1 / θ t - λ + 2 C ( λ + B T 1 - k ) - 1 t 1 - δ , 0 < t T .

3 Lestimate 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 approximate equations and we may assume for our estimates that the solutions under consideration are sufficiently smooth.

Let u 0 , i C 0 2 ( Ω ) and u0,i u0 in L1(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 ui(t)(see Chapter 6 in [12]). We will derive estimates for ui(t). For the simplicity of notation, we write u instead of ui and uk for |u|k-1u where k > 0. Also, let C, Cj be generic constants independent of k, i, n changeable from line to line.

Lemma 3.1 Let (H1)-(H3) hold. Suppose that u(t) is the solution of (3.1), then u(t) ∈ L([0, ∞), L1).

Proof Let n = 1, 2, ..., and

f n ( s ) = 1 , 1 n s n s ( 2 - n s ) , 0 s 1 n - n s ( 2 + n s ) , - 1 n s 0 - 1 , s < - 1 n .

It is obvious that fn(s) is odd and continuously differentiable in ℝ1. Furthermore, f n ( s ) 1 , f n ( s ) 0 and fn(s) → sign(s) uniformly in ℝ1.

Multiplying the equation in (3.1) by fn(u) and integrating on Ω, we get

Ω f n ( u ) u t d x + Ω u m + 2 f n ( u ) d x + Ω u u β - 2 f n ( u ) u q d x Ω u u α - 2 f n ( u ) u p d x + Ω u u β - 2 f n ( u ) u q d x (3.2)

and the application of the Young inequality gives

Ω u u α - 2 f n ( u ) u p d x 1 4 Ω u u β - 2 f n ( u ) u q d x + C 1 Ω u μ - 1 d x , (3.3)

where μ = (- )(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 ) = 0 u ( s s β - 2 f n ( s ) ) 1 / q d s , u 1 .

It is easy to verify that Fn(u) is odd in ℝ1. Then, we obtain from the Sobolev inequality that

1 4 Ω u u β - 2 f n ( u ) u q d x = 1 4 Ω F n ( u ) q d x λ 0 Ω F n ( u ) q d x = λ 0 Ω n F n ( u ) q d x + λ 0 Ω n c F n ( u ) q d x (3.4)

with some λ0 > 0 and

Ω n = { x Ω | u ( x , t ) n - 1 } , Ω n c = Ω \ Ω n , n = 1 , 2 , .

We note that |Fn(u)|q n-(q+β-1) in Ω n c and

Ω n c F n ( u ) q d x n - ( q + β - 1 ) Ω .

On the other hand, we have |u(x, t)| ≥ n-1 in Ωn and

F n ( u ) n - 1 u ( s s β - 2 f n ( s ) ) 1 / q d s q q + β - 1 u q + β - 1 q - n - q + β - 1 q in Ω n .

This implies that there exists λ1 > 0, such that

λ 0 Ω n F n ( u ) q d x λ 1 Ω n u q + β - 1 d x - λ 1 Ω n - ( q + β - 1 ) (3.5)

Then it follows from (3.4)-(3.5) that

1 4 Ω u u β - 2 f n ( u ) u q d x λ 1 Ω u q + β - 1 d x - C 2 n - ( q + β - 1 ) (3.6)

with some C2 > 0.

Similarly, we have from the assumption (H2) and the Young inequality that

Ω g ( u ) f n ( u ) d x K 1 Ω u 1 + ν f n ( u ) d x K 1 Ω u 1 + ν d x λ 1 2 Ω n u q + β - 1 d x + C 2 ( 1 + n - 1 - ν ) . (3.7)

Furthermore, the assumption μ < q + β implies that

C 1 Ω n u μ - 1 d x λ 1 2 Ω n u q + β - 1 d x + C 2 . (3.8)

Then (3.2)-(3.3) and (3.6)-(3.8) give that

Ω f n ( u ) u t d x + 1 2 Ω u u β - 2 f n ( u ) u q d x C 3 1 + n - 1 - ν + n - ( q + β - 1 ) . (3.9)

Letting n → ∞ in (3.9) yields

d d t u ( t ) 1 + 1 2 Ω u β - 1 u q d x C 3 . (3.10)

Note that

Ω u β - 1 u q d x = q q + β - 1 q Ω u 1 + β - 1 q q d x 2 λ 2 u 1 q + β - 1

with some λ2 > 0. Then (3.10) becomes

d d t u ( t ) 1 + λ 2 u ( t ) 1 q + β - 1 C 3 . (3.11)

This gives that u(t) ∈ L([0, ∞), L1) if u0 L1.

Remark 3.2 The differential inequality (3.10) implies that the solution ui(t) of (3.1) satisfies

0 T Ω u i β - 1 u i q d x d t C 0 for i = 1 , 2 , . (3.12)

withC0 = C0(T, ∥u01).

Lemma 3.3 Assume (H1)-(H4). 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, C0 = C0(T, ∥u01).

Proof Multiplying the equation in (3.1) by uk-1, k ≥ 2, we have

1 k d d t 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 d x Ω u α + k - 2 u p d x + K 1 Ω u ν + k d x . (3.14)

It follows from the Hölder and Sobolev inequalities that

K 1 Ω u ν + k d x C u k θ 1 u 1 θ 2 u s θ 3 C u k θ 1 u k + m m + 2 m + 2 ( m + 2 ) θ 3 k + m k - 1 2 m + 2 k + 2 m + 2 u k + m m + 2 m + 2 m + 2 + C k σ u k k ,

in which θ1 = kλ(m - ν + (m + 2)N-1), θ2 = νλ(m + 2)N-1, θ3 = νλ(k + m), σ = νλ, s = N(k + m)(N - m - 2)-1.

Note that

Ω u α + k - 2 u p d x 1 4 Ω u β + k - 2 u q d x + C Ω u μ + k - 2 d x

and

1 2 Ω u β + k - 2 u q d x C 1 k - q Ω u q + β + k - 2 q q d x

with some C1 independent of k and μ = (- )(q - p)-1 < q + β.

Without loss of generality, we assume k > 3 - μ. Similarly, we derive

C Ω u μ + k - 2 d x C u k - 2 μ 1 u 1 μ 2 u k * μ 3 C ξ 1 μ 2 u k μ 1 u k * μ 3 C u k μ 1 u q k / q q q μ 3 / q k A k

with ξ1 = supt≥0u(t)∥1 and

μ 1 = λ 0 ( k - 2 ) ( q + β - μ + q N - 1 ) , μ 2 = λ 0 μ q N - 1 , μ 3 = λ 0 μ q 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 C η u q k / q q q + C η - θ / θ u k μ 1 θ (3.15)

with μλ0θ = 1, (1 - μλ0)θ' = 1.

Note that μ1θ' < k. Let η = C 1 2 C k - q . Then it follows from (3.15) that

A k C 1 2 k - q u q k / q q q + C k γ ( u k k + 1 ) (3.16)

with γ = qθ'θ-1 = λ0/(1 - μλ0). Then, (3.14) becomes

1 k d d t 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 C k σ 0 ( u k k + 1 )

or

d d t u k k + C 1 k - m u k + m m + 2 m + 2 m + 2 C k 1 + σ 0 ( u k k + 1 ) (3.17)

with σ0 = max{σ, γ} = max{νλ, γ}.

Now we employ an improved Moser's technique as in [8,9]. Let {kn} be a sequence defined by k1 = 1, kn = Rn-2(R - m - 1) + m(R - 1)-1(n = 2, 3, ...) with R > max{m + 1, m + 4 - μ} such that kn ≥ 3 - μ(n ≥ 2). Obviously, kn → ∞ as n → ∞.

By Lemma 2.9, we have

u ( t ) k n C 0 m + 2 m + k n u ( t ) k n - 1 1 - θ n u m + k n m + 2 m + 2 θ n ( m + 2 ) m + k n (3.18)

with θ n = R N ( 1 - k n - 1 k n - 1 ) ( m + 2 + N ( R - 1 ) ) - 1 .

Then, inserting (3.18) into (3.17) (k = kn), we find that

d d t u ( t ) k n k n + C 1 C 0 - m + 2 θ n k n - m u ( t ) k n - 1 ( 1 - 1 / θ n ) ( m + k n ) u ( t ) k n ( m + k n ) / θ n C k n 1 + σ 0 ( u ( t ) k n k n + 1 ) , 0 < t T , (3.19)

or

d d t u ( t ) k n k n + C 1 C 0 - m + 2 θ n k n - m u ( t ) k n - 1 m - β n u ( t ) k n k n + β n C k n 1 + σ 0 ( u ( t ) k n k n + 1 ) , (3.20)

where β n = ( m + k n ) θ n - 1 - 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 n - 1 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 n - m ( y n ( t ) ) 1 + β n / k n u ( t ) k n - 1 m - β n C k n 1 + σ 0 ( y n ( t ) + 1 ) . (3.21)

We claim that there exist a bounded sequence {ξn} and a convergent sequence {λn}, such that

u ( t ) k n ξ n t - λ n , 0 < t T . (3.22)

Indeed, by Lemma 3.1, the estimate (3.22) holds for n = 1 if we take λ1 = 0, ξ1 = supt≥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 n - m ( ξ n - 1 ) m - β n t Λ n τ n - 1 y n 1 + τ n ( t ) C k n 1 + σ 0 ( 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 n - m ξ n - 1 m - β n - 1 / τ n ( 2 k n λ n + 2 C T k n 1 + σ 0 ) 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 n - m ξ n - 1 m - β n - 1 / β n ( 2 k n λ n + 2 C T k n 1 + σ 0 ) 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 n - m - 1 / β n ( 2 k n λ n + 4 C T k n 1 + σ 0 ) 1 / β n , (3.26)

in which the fact kn ~ βn as n → ∞ has been used.

It is not difficult to show that {ξn} is bounded. Furthermore, by Lemma 4 in [9], we have

1 + λ n - 1 ( β n - m ) β n λ = N m + 2 + m N , as n .

Letting n → ∞ in (3.22) implies that (3.13) and we finish the Proof of Lemma 3.3.

Lemma 3.4. Let (H1)-(H4) hold. Then, the solution u(t) of (3.1) has the following estimates

0 T s 1 + r u t ( s ) 2 2 d s 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, C0 = C0(T, ∥u01).

Proof We first choose r > λ and η(t) ∈ C[0, ∞) ⋂ C1(0, ∞) such that η(t) = tr 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 + 0 t η ( s ) u ( s ) m + 2 m + 2 d s + 0 t Ω u β u q η ( s ) d x d s 1 2 0 t η ( s ) u ( s ) 2 2 d s + 0 t Ω u α u p η ( s ) d x d s + K 1 0 t Ω u 2 + ν η ( s ) d x d s . (3.29)

Note that

0 t Ω u α u p η ( s ) d x d s 1 2 0 t Ω u β u q η ( s ) d x d s + C 0 t Ω u μ η ( s ) d x d s .

Hence, we have

1 2 η ( t ) u ( t ) 2 2 + 0 t η ( s ) u ( s ) m + 2 m + 2 d s + 1 2 0 t Ω u β u q η ( s ) d x d s 1 2 0 t η ( s ) u ( s ) 2 2 d s + C 0 t Ω u μ η ( s ) d x d s + K 1 0 t Ω u 2 + ν η ( s ) d x d s . (3.30)

By Lemma 3.1 and the estimate (3.13), we get

0 t η ( s ) u ( s ) 2 2 d s C 0 t s r - 1 u ( t ) 1 u ( t ) d s C t r - λ , 0 t < T . (3.31)

Since μ < q + β, we have from Sobolev inequality that

C 0 t Ω u μ η ( s ) d x d s 1 4 0 t Ω u β u q η ( s ) d x d s + C 0 t η ( s ) d s . (3.32)

Similarly, we have from 2 + ν < q + β that

K 1 0 t Ω u 2 + ν η ( s ) d x d s 1 4 0 t Ω u β u q η ( s ) d x d s + C 0 t η ( s ) d s . (3.33)

Therefore, it follows from (3.30)-(3.33) that

0 t Ω u m + 2 η ( s ) d x d s C t r - λ , 0 t T . (3.34)

Next, let G ( u ) = 0 u g ( s ) d s , u 1 , ρ ( t ) = 0 t η ( s ) d s , t ( 0 , ) . Furthermore, multiplying the equation in (3.1) by ρ(t)ut yields

ρ ( t ) u t ( t ) 2 2 + 1 m + 2 d d t Ω ρ ( t ) ( u 2 + i - 1 ) m + 2 2 d x + ρ ( t ) Ω G ( u ) d x ρ ( t ) m + 2 Ω ( u 2 + i - 1 ) m + 2 2 d x + d d t Ω ρ ( t ) G ( u ) d x + Ω ρ ( t ) u β - 1 u t u q d x + Ω ρ ( t ) u α - 1 u t u p d x . (3.35)

By the assumption p < q and the Cauchy inequality, we deduce

Ω u β - 1 u t u q d x 1 4 u t ( t ) 2 2 + C Ω u 2 ( β - 1 ) u 2 q d x (3.36)

and

Ω u α - 1 u t u p d x 1 4 u t ( t ) 2 2 + C Ω u 2 ( α - 1 ) u 2 p d x 1 4 u t ( t ) 2 2 + C Ω u 2 ( β - 1 ) u 2 q d x + C Ω u 2 ( μ - 1 ) d x (3.37)

and

Ω G ( u ) d x C 1 Ω u 2 + ν d x C h 2 + ν ( t ) (3.38)

with h(t) = ∥u(t)∥.

Now, it follows from (H4) and (3.35)-(3.38) that

1 2 0 t ρ ( s ) u t ( s ) 2 2 d s + 1 m + 2 ρ ( t ) u ( t ) m + 2 m + 2 1 2 0 t η ( s ) u ( s ) m + 2 m + 2 d s + C ρ ( t ) h 2 + ν ( t ) (3.39)

or

1 2 0 t ρ ( s ) u t ( s ) 2 2 d s + ρ ( t ) m + 2 u ( t ) m + 2 m + 2 C 0 t r - λ + C 0 0 t ρ ( s ) h 2 ( β - 1 ) ( s ) u ( s ) m + 2 m + 2 d s (3.40)

where C0 = C0(T, ∥u01) and the fact 2 + λ ≥ 2(μ - 1)λ has been used.

Since the function h2(β-1)(t) ∈ L1([0, T]), the application of the Gronwall inequality to (3.40) gives

0 t ρ ( s ) u t ( s ) 2 2 d s + ρ ( t ) u m + 2 m + 2 C 0 t r - λ , 0 < t T . (3.41)

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 C0 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 ui) and a function u L s ( [ 0 , T ] , W 0 1 , s ( Ω ) ) , ( 1 s m + 2 ) satisfying

u i u weakly in L s ( [ 0 , T ] , W 0 1 , s ( 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 loc 2 ( 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 0 1 , m + 2 ) * = W 0 - 1 , m + 2 m + 1 , we see further that

A i ( u i ) χ weakly * in L l o c ( 0 , T ; ( W 0 1 , m + 2 ) * ) (3.44)

for some χ L loc ( 0 , T ] , ( W 0 1 , m + 2 ) * ) . As the Proof of Theorem 1 in [9], we have χ = A(u) = -div((∥∇umu).

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 Lestimate 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 ∥∇ufor the smooth solution u(t) of (3.1). As above, let C, Cj be the generic constants independent of k and i. Denote

D 2 u 2 = i , j = 1 N u i j 2 , u i j = 2 u x i x j .

Multiplying (3.1) by -div(|∇u|k-2u), k m + 2 and integrating by parts, we have

1 k d d t u ( t ) k k + Ω u k + m - 2 D 2 u 2 d x + k - 2 4 Ω u k + m - 4 ( u 2 ) 2 d x - ( N - 1 ) Ω H ( x ) u k + m d S = Ω u u β - 2 u q div ( u k - 2 u ) d x - Ω u u p u α - 2 div ( u k - 2 u ) d x + Ω g ( u ) div ( u k - 2 u ) d x I + I I + I I I . (4.1)

Since

div ( u k - 2 u ) = u k - 2 Δ 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 d x = ( k - 1 ) Ω u k + m - 2 2 D 2 u u k + 2 q - m - 2 2 u β - 1 d x 1 4 Ω u k + m - 2 D 2 u 2 d x + C 0 k 2 Ω u k + 2 q - m - 2 u 2 ( β - 1 ) d x . (4.4)

Similarly, we obtain the following estimates

I I 1 4 Ω u m + k - 2 D 2 u 2 d x + C 0 k 2 Ω u k + 2 p - m - 2 u 2 ( α - 1 ) d x (4.5)

and

I I I = Ω g ( u ) div ( u k - 2 u ) d x = - Ω g ( u ) u k d x K 1 Ω y ν u k d x C h ν ( t ) u ( t ) k k , (4.6)

where h(t) = ∥u(t)∥Ct.

Moreover, we assume that 2q m + 2, 2p m + 2, then (4.1) becomes

1 k d d t u k k + 1 2 Ω u k + m - 2 D 2 u 2 d x + k - 2 4 Ω u k + m - 4 ( u 2 ) 2 d x - ( N - 1 ) Ω H ( x ) u k + m d S C 0 k 2 Ω u k + 2 q - m - 2 u 2 ( β - 1 ) + u k + 2 p - m - 2 u 2 ( α - 1 ) d x + C h ν ( t ) u ( t ) k k C 0 k 2 h 1 ( t ) 1 + u ( t ) k k , (4.7)

where h1(t) = max{h2(α-1)(t), h2(β-1)(t), hν(t)}. Since α , β < 2 + m 2 1 + 1 N , ν < m + 2 + m N , we get h1(t) ∈ L1([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 λ1 > 0, such that

v 2 2 - ( N - 1 ) Ω v 2 H ( x ) d S λ 1 v 1 , 2 2 , v W 1 , 2 ( Ω ) . (4.8)

Hence, by (4.7) and (4.8), we see that there exists C1 and C2 such that

d d t u ( t ) k k + C 1 u ( t ) k + m 2 1 , 2 2 C k 3 h 1 ( t ) ( 1 + u ( t ) k k ) . (4.9)

Let k1 = m + 2, R > m + 1, kn = Rn-2 (R-1-m) + m (R-1)-1, θ n = R N ( 1 - k n - 1 k n - 1 ) ( 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 k n - 1 1 - θ n u k n + m 2 1 , 2 2 θ n k n + m . (4.10)

Inserting this into (4.9)(k = kn), we get

d d t u k n k n + C 1 C - 2 / θ n u ( t ) k n - 1 ( k n + m ) ( 1 - 1 / θ n ) u ( t ) k n ( k n + m ) / θ n C 2 k n 3 h 1 ( t ) ( 1 + u ( t ) k n k n ) . (4.11)

By (3.28), we take y1 = max{1, C0}, z1 = (1 + λ)/(m + 2). As the Proof of Lemma 3.3, we can show that there exist bounded sequences yn and zn such that

u ( t ) k n y n t - z n , 0 < t T , (4.12)

in which zn 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.

5 Asymptotic behavior of solution

In this section, we will prove that the problem (1.1) admits a global solution if the initial data u0(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 u0 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 d t u ( t ) 2 2 + C 1 u ( t ) m + 2 m + 2 Ω u α u p d x (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 d x u ( t ) m + 2 p u s α C 0 u m + 2 p u r α ( 1 - θ ) u m + 2 α θ C 0 u ( t ) m + 2 m + 2 u ( t ) r p 1 (5.2)

with

s = α ( m + 2 ) m + 2 - p , θ = 1 r - 1 s 1 N + 1 r - 1 m + 2 - 1 , r = N p 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 d t u ( t ) 2 2 + u ( t ) m + 2 m + 2 ( C 1 - C 0 u ( t ) 2 p 1 ) 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 ) 2 m + 2 , (5.4)

we obtain from (5.3) and (5.4) that ∃d0 > 0, λ0 > 0, such that ∥u02 < d0 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 ∥u02. 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 d t u ( t ) 2 2 + γ Ω u β u q d x Ω u α u p d x . (5.7)

Since p < q, q + β < p + α, it follows from the Hölder inequality that

Ω u α u p d x u 1 + β / q q p u μ μ ( 1 - p / q ) C 1 u 1 + β / q q p u τ μ 1 ( 1 - p / q ) u 1 + β / q q μ 2 ( 1 - p / q ) C 1 u 1 + β / q q q u τ μ 1 ( 1 - p / q )