Zhan Journal of Inequalities and Applications (2018) 2018:7 https://doi.org/10.1186/s13660-017-1596-4

RESEARCH

Open Access

The uniqueness of a nonlinear diffusion equation related to the p-Laplacian Huashui Zhan* *

Correspondence: [email protected] School of Applied Mathematics, Xiamen University of Technology, Xiamen, 361024, P.R. China

Abstract Consider a nonlinear diffusion equation related to the p-Laplacian. Different from the usual evolutionary p-Laplacian equation, the equation is degenerate on the boundary due to the fact that the diffusion coefficient is dependent on the distance function. Not only the existence of the weak solution is established, but also the uniqueness of the weak solution is proved. MSC: 35L65; 35L85; 35R35 Keywords: p-Laplacian; diffusion coefficient; boundary value condition; uniqueness

1 Introduction and the main results Recently, we noticed that Benedikt et al. [1] had studied the equation   ut = div |∇u|p–2 ∇u + q(x)uγ ,

(x, t) ∈ QT  × (0, T),

(1.1)

and shown that the uniqueness of the solutions of equation (1.1) is not true. Here,  is an open bounded domain with a smooth boundary, 0 < γ < 1, p > 1, q(x) ∈ C 1 (), q(x) ≥ 0 and there exists at least a point x0 ∈ , q(x0 ) > 0. This comes more or less as a surprise. In general, we may think that the source time q(x)uγ only affects the existence of the weak solutions. At the same time, in [2], we have considered the following equation:   ut = div ρ α |∇u|p–2 ∇u + f (u, x, t),

(x, t) ∈ QT ,

(1.2)

and we have shown that the uniqueness of the weak solution is true when f (u, ·, ·) is a Lipschitz function, here α > 0, ρ(x) = dist(x, ∂) is the distance function from the boundary. Certainly, since 0 < γ < 1 in equation (1.1), f (u, x, t) = q(x)uγ is not a Lipschitz function about the variable u. Consequently, the results in [1] and [2] are compatible. If α = 0, there are a great deal of papers devoted to equations (1.2), many of them are important and interesting. But it is impossible to list all these papers, and we only list a few of them [3–7] here. In this paper, we assume that q(x) ∈ C 1 (). We will consider a nonlinear convectiondiffusion equation related to the p-Laplacian, N    ∂bi (u) ut = div ρ α |∇u|p–2 ∇u + + q(x)|u|γ –1 u, ∂x i i=1

(x, t) ∈ QT ,

(1.3)

© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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where 0 < γ < 1. The initial value condition u(x, t) = u0 (x),

x ∈ ,

(1.4)

is always necessary. Different from the usual evolutionary p-Laplacian equation or equation (1.1), an obvious feature of equations (1.2), (1.3) lies in that the diffusion coefficient ρ α depends on the distance to the boundary. By this feature, instead of the usual boundary value condition u(x, t) = 0,

(x, t) ∈ ∂ × (0, T),

(1.5)

only a partial boundary condition, u(x, t) = 0,

(x, t) ∈ p × (0, T),

(1.6)

should be imposed generally, where p ⊆ ∂ is a relatively open subset in ∂. One can refer to our previous work [2, 8]. Since equation (1.3) is a nonlinear equation, it is difficult to depict p as the linear degenerate parabolic equation by the Fichera function. The main aim of this paper is to prove the uniqueness of the solutions without any boundary value condition. In the first place, since we had known the interesting result of [1] (i.e. the nonuniqueness of the weak solution of equation (1.1)), we should clarify why the uniqueness of the weak solutions of equation (1.3) can be obtained. Let us introduce some basic functional spaces. For every fixed t ∈ [0, T), we define the Banach space p    Vt () = u(x, t) : u(x, t) ∈ L2 () ∩ W01,1 (), ∇u(x, t) ∈ L1 () , uVt () = u2, + ∇up, , and we denote by Vt () its dual. Also, we denote the Banach space ⎧ ⎨W(Q ) = {u : [0, T] → V ()|u ∈ L2 (Q ), |∇u|p ∈ L1 (Q ), u = 0 on  = ∂}, T t T T ⎩uW(Q ) = ∇up,Q + u2,Q , T

T

T

and we denote by W (QT ) its dual. According to Antontsev-Shmarev [9], we know

w ∈ W (QT )

⇐⇒

⎧ ⎨w = w + N D w , w ∈ L2 (Q ), w ∈ Lp (Q ), 0 0 T i T i=1 i i ⎩∀φ ∈ W(QT ),

w, φ = (w0 φ + N wi Di φ) dx dt. QT

i

The norm in W (QT ) is defined by   vW (QT ) = sup

v, φ|φ ∈ W(QT ), φW(QT ) ≤ 1 . Basing on these functional spaces, we can give the definition of the weak solution.

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Definition 1.1 A nonnegative function u(x, t) is said to be a weak solution of equation (1.3) with the initial value (1.4), if u ∈ L∞ (QT ),

ut ∈ W (QT ),

ρ α |∇u|p ∈ L1 (QT ),

(1.7)

and, for any function ϕ ∈ L∞ (0, T; W0 ()) ∩ W(QT ), 1.p





ut , ϕ +

ρ |∇u| α

p–2

∇u · ∇ϕ +

QT



N 

 bi (u) · ϕxi dx dt

i=1

q(x)|u|γ –1 uϕ dx dt.

=

(1.8)

QT

The initial value is satisfied in the sense that

  lim u(x, t) – u0 (x) dx = 0. t→0 

(1.9)

The most important of Definition 1.1 lies in ut ∈ W (QT ). Once the weak solution comes with this property, then we have Lemma 3.1 below, and just by this lemma, we can prove the uniqueness. By comparing the analysis in [1], we know the weak solution defined in [1] does not have this property. Second, we introduce the existence result. Theorem 1.2 If p > 1, 0 < γ < 1, bi (s) is a C 1 function, and u0 ∈ L∞ (),

ρ α |∇u0 |p ∈ L1 (),

(1.10)

then equation (1.1) with initial value (1.4) has a weak solution. Last but not least we will prove the following local stability. Theorem 1.3 Let p > 1, γ > 0, bi (s) be a Lipschitz function. If u, v are two solutions of equation (1.3) with the initial values u0 (x), v0 (x), respectively, then there exists a positive , 2, αp} such that constant β ≥ max{ p–α p–1

 2 ρ β u(x, t) – v(x, t) dx ≤ c



2  ρ β u0 (x) – v0 (x) dx.

(1.11)



In particular, for any small enough constant λ > 0,

  u(x, t) – v(x, t)2 dx ≤ cλ–β

λ

  u0 (x) – v0 (x)2 dx.

(1.12)



Here, λ = {x ∈  : dist(x, ∂) > λ}, by the arbitrariness of λ, we have the uniqueness of the solution. This conclusion implies that the degeneracy of the diffusion coefficient can take place of the usual boundary value condition. We would like to suggest that, if ρ α is substituted by a nonnegative diffusion coefficient a(x) ∈ C 1 () with a(x)|x∈ > 0,

a(x)|x∈∂ = 0,

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a similar conclusion to Theorem 1.3 is still true. For some special cases, one can see our recent work [10]. Actually, we had used some ideas of [10] to prove Theorem 1.3. This paper is arranged as follows. In Section 1, we give the basic definition and introduce the main results. In Section 2, we prove the existence of the solution to equation (1.1) with initial value (1.4). In Section 3, we prove Theorem 1.3 and obtain the uniqueness of the solution.

2 The weak solutions dependent on the initial value We consider the weak solution of the initial value problem for equation (1.3) in this section. It is supposed that u0 satisfies u0 ∈ L∞ (),

|∇u0 |p ∈ L1 ().

Let uε,0 ∈ C0∞ () and ρεα |∇uε,0 |p ∈ L1 () be uniformly bounded, and uε,0 converges to u0 1,p in W0 (). Here ρε = ρ ∗ δε + ε, ε > 0, δε is the mollifier as usual. By the results of [11, Section 8], we have the following important lemma. Lemma 2.1 If uε ∈ L∞ (0, T; L2 ()) ∩ W(QT ), uεt W (QT ) ≤ c, ∇(|uε |q–1 uε )p,QT ≤ c, then there is a subsequence of {uε } which is relatively compact in Ls (QT ) with s ∈ (1, ∞). Here q ≥ 1. We now consider the following regularized problem: N      p–2 ∂bi (uε ) uεt – div ρεα |∇uε |2 + ε 2 ∇uε – ∂xi i=1

= q(x)|uε |γ –1 uε ,

(x, t) ∈ QT ,

(x, t) ∈ ∂ × (0, T),

uε (x, t) = 0,

x ∈ ,

uε (x, 0) = uε,0 (x),

(2.1) (2.2) (2.3)

since 0 < γ < 1, it is well known that the above problem has an unique classical solution [12, 13]. By the maximum principle, there is a constant c only dependent on u0 L∞ () but independent on ε, such that uε L∞ (QT ) ≤ c.

(2.4)

Multiplying (2.1) by uε and integrating it over QT , we get 1 2



u2ε dx +  N



QT

  p–2 ρεα |∇uε |2 + ε 2 |∇uε |2 dx dt

∂bi (uε ) dx dt ∂xi QT i=1

1 u20 dx + q(x)|uε |γ –1 uε dx dt. = 2  QT +

uε

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By the fact N

 i=1

 ∂bi (uε ) uε dx dt = – ∂xi QT i=1 N

=–



∂uε bi (uε ) dx dt ∂xi

QT

N  

i=1

∂ ∂xi

uε

bi (s) ds dx = 0, 0

then 1 2



u2ε dx +

QT



  p–2 ρεα |∇uε |2 + ε 2 |∇uε |2 dx dt ≤ c.

(2.5)

It is also easy to show that



ρ α |∇uε |p dx dt ≤ c

QT

QT

ρεα |∇uε |p dx dt ≤ c.

(2.6)

Now, for any v ∈ W(QT ), vW (QT ) = 1,

 p–2  ρεα |∇uε |2 + ε 2 ∇uε · ∇v dx dt

uεt , v = – QT

–

QT

∂v bi (uε ) dx dt + ∂xi

q(x)|uε |γ –1 uε v dx dt,

(2.7)

QT

by Young inequality, we can show that 

   uεt , v ≤ c QT

ρεα |∇uε |p dx dt +



  |v|p + |∇v|p dx dt + 1 ≤ c,

QT

then uεt W (QT ) ≤ c.

(2.8)

Now, let ϕ ∈ C01 (), 0 ≤ ϕ ≤ 1 such that ϕ|2λ = 1,

ϕ|\λ = 0.

Then        (ϕuε )t , v  =  ϕuεt , v ≤  uεt , v, we have     ϕ(x)u 

QT

≤ uεt W (QT ) ≤ c,  T   ∇(ϕuε )p dx dt ≤ c(λ) 1 + εt W (QT )

0

λ

(2.9)  |∇uε |p dx dt ≤ c(λ),

(2.10)

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and so   ∇(ϕuε ) ≤ c. p,Q

(2.11)

T

By Lemma 2.1, ϕuε is relatively compact in Ls (QT ) with s ∈ (1, ∞). Then ϕuε → ϕu a.e. in QT . In particular, due to the arbitrariness of λ, uε → u a.e. in QT . Hence, by (2.4), (2.7), there exists a function u and an n-dimensional vector function ζ = (ζ1 , . . . , ζn ) satisfying p

|ζ | ∈ L p–1 (QT ),

u ∈ L∞ (QT ), and uε ∗ u,

in L∞ (QT ),

ρεα |∇uε |p–2 uεxi  ζi

uε → u,

a.e. in QT .

p

in L p–1 (QT ).

In order to prove that u satisfies equation (1.3), we notice that, for any function ϕ ∈ C0∞ (QT ), 



–uε ϕt + ρεα

QT



|∇uε | + ε 2

 p–2 2

∇uε · ∇ϕ +

N 

 bi (uε ) · ϕxi dx dt

i=1

q(x)|uε |γ –1 uε ϕ dx dt,

=

(2.12)

QT

and uε → u is almost everywhere convergent, so bi (uε ) → bi (u), |uε |γ –1 uε → |u|γ –1 u. Then 

QT



 N  ∂u ϕ + ς · ∇ϕ + bi (u) · ϕxi dx dt ∂t i=1 q(x)|u|γ –1 uϕ dx dt.

=

(2.13)

QT

Now, if we can prove that



ρ |∇u| α

QT

p–2

ζ · ∇ϕ1 dx dt

∇u · ∇ϕ1 dx dt =

(2.14)

QT

for any function ϕ1 ∈ C0∞ (QT ), then u satisfies equation (1.3). In what follows, we will use a similar method to that in [14] to prove (2.14). Let 0 ≤ ψ ∈ C0∞ (QT ) and ψ = 1 in supp ϕ1 . Let v ∈ L∞ (QT ), ρ α |∇v|p ∈ L1 (QT ). It is well known that

QT

  ψρεα |∇uε |p–2 ∇uε – |∇v|p–2 ∇v · (∇uε – ∇v) dx dt ≥ 0.

(2.15)

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By choosing ϕ = ψuε in (2.12), we can obtain

QT

  p–2 ψρεα |∇uε |2 + ε 2 |∇uε |2 dx dt

1 2

=



–

QT

ψt u2ε dx dt –

N





QT

i=1

QT

  p–2 ρεα uε |∇uε |2 + ε 2 ∇uε · ∇ψ dx dt q(x)|uε |γ –1 uε ϕ dx dt.

bi (uε )(uεxi ψ + uε ψxi ) dx dt +

(2.16)

QT

Noticing that, when p ≥ 2,  

|∇uε |2 + ε |∇uε |2 + ε

 p–2 2  p–2 2

|∇uε |2 ≥ |∇uε |p ,   |∇uε | ≤ |∇uε |p–1 + 1 ,

and, when 1 < p < 2,  

|∇uε |2 + ε |∇uε |2 + ε

 p–2 2

 p p |∇uε |2 ≥ |∇uε |2 + ε 2 – ε 2 ,

 p–2 2

  p–1 |∇uε | ≤ |∇uε |2 + ε 2 ,

then in both cases, by (2.15), we have 1 2



ψt u2ε dx dt –

QT

–

N

 QT

i=1 p

QT

QT

QT

q(x)|uε |γ –1 uε ϕ dx dt

bi (uε )(uεxi ψ + uε ψxi ) dx dt +

+ ε 2 c() – –

QT

  p–2 ρεα uε |∇uε |2 + ε 2 ∇uε · ∇ψ dx dt

ρεα ψ|∇uε |p–2 ∇uε ∇v dx dt

ρεα ψ|∇v|p–2 ∇(uε – v) dx dt ≥ 0.

(2.17)

Thus 1 2



ψt u2ε dx dt –

QT

–

N

 i=1 p

QT

bi (uε )(uεxi ψ + uε ψxi ) dx dt +

+ ε 2 c() –

QT

  p–2 ρεα uε |∇uε |2 + ε 2 ∇uε · ∇ψ dx dt

QT

q(x)|uε |γ –1 uε ϕ dx dt QT

ρεα ψ|∇uε |p–2 ∇uε ∇v dx dt

ψρ α |∇v|p–2 ∇v · (∇uε – ∇v) dx dt

–

QT

+ QT

  ψ ρ α – ρεα |∇v|p–2 ∇v · (∇uε – ∇v) dx dt ≥ 0.

(2.18)

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Notice 

  

QT

    ψ ρ α – ρεα |∇v|p–2 ∇v · (∇uε – ∇v) dx dt  |ψ(ρ α – ρεα )| ρα

≤ sup (x,t)∈QT

|ψ(ρ α – ρεα )| ρα

≤ sup (x,t)∈QT

ρ α |∇v|p–1 |∇uε – ∇v| dx dt QT





ρ α |∇v|p dx dt +

ρ α |∇v|p–1 |∇uε | dx dt

QT

(2.19)

QT

and by the Hölder inequality

ρ α |∇v|p–1 |∇uε | dx dt QT



 m s ρ |∇v|p–1 dx dt

≤

1/s 

·

QT

where m =



p ρ n |∇uε | dx dt

1/p ,

QT

α(p–1) , p

n = αp , s =



p . p–1

Due to ρ α |∇u|p , ρ α |∇v|p ∈ L1 (QT ), we have

ρ |∇v| dxdt +

ρ α |∇v|p–1 |∇uε | dx dt ≤ c.

p

α

QT

QT

Let ε → 0 in (2.19). It converges to 0. Once more, we notice that 

|∇uε |2 + ε

 p–2 2

∇uε

lim

ε→0

QT

p–2 ε 2



p–2 ε 2

= |∇uε |p–2 ∇uε +

1

1

|∇uε |2 + εs

 p–4 2

ds∇uε ,

0

|∇uε |2 + εs

 p–4 2

ds∇uε ∇ψuε dx dt = 0.

0

Let ε → 0 in (2.18), we have 1 2





uζ · ∇ψ dx dt –

ψt u2 dx dt – QT

QT



ψ ζ · ∇v dx dt –

–

N

 QT

i=1



bi (u)(uxi ψ + uψxi ) dx dt

ψρ α |∇v|p–2 ∇v · (∇u – ∇v) dx dt

QT

QT

q(x)|u|γ –1 uϕ dx dt ≥ 0.

+ QT

Let ϕ = ψu in (2.13), we get

ψ ζ · ∇u dx dt –

QT

+

N

 i=1

1 2





uζ · ∇ψ dx dt

u2 ψt dx dt + QT

QT

QT

bi (u)(uxi ψ + uψxi ) dx dt +

q(x)|u|γ –1 uψu dx dt = 0. QT

(2.20)

Zhan Journal of Inequalities and Applications (2018) 2018:7

Thus

  ψ ζ – ρ α |∇v|p–2 ∇v · (∇u – ∇v) dx dt ≥ 0.

Page 9 of 14

(2.21)

QT

Let v = u – λϕ1 , λ > 0, ϕ1 ∈ C0∞ (QT ) is given in (2.14), then

 p–2   ψ ζ – ρ α ∇(u – λϕ1 ) ∇(u – λϕ1 ) · ∇ϕ1 dx dt ≥ 0.

QT

If λ → 0, then

  ψ ζ – ρ α |∇u|p–2 ∇u · ∇ϕ1 dx dt ≥ 0.

QT

Moreover, if λ < 0, similarly we can get

  ψ ζ – ρ α |∇u|p–2 ∇u · ∇ϕ1 ≤ 0.

QT

Thus

  ψ ζ – ρ α |∇u|p–2 ∇u · ∇ϕ1 dx dt = 0.

QT

Noticing that ψ = 1 on supp ϕ1 , (2.14) holds. At same time, we are able to prove (1.9) as in [15], thus we have Theorem 1.2.

3 The uniqueness without the boundary value condition Lemma 3.1 Let u ∈ W(QT ), ut ∈ W (QT ). Then ∀ a.e. t1 , t2 ∈ (0, T),

t2

t1

1 uut dx dt = 2 



  2  2 u (x, t2 ) – u (x, t1 ) dx .

(3.1)



This is Corollary 2.1 of [9]. Proof of Theorem 1.3 Let u, v be two solutions of equation (1.3) with the initial values u0 (x), v0 (x), respectively. Denote λ = {x ∈  : dist(x, ∂) > λ}, let the constant β ≥ , 2, αp} and max{ p–α p–1  β β ξλ = dist(x,  \ λ ) = dλ .

(3.2)

We may choose χ[τ ,s] (uε –vε )ξλ as a test function, where uε and vε are the mollified function of the solutions u and v, respectively. Then   (u – v)t , χ[τ ,s] (uε – vε )ξλ

∂(u – v) dx dt (uε – vε )ξλ = ∂t Qτ s

    =– ρ α |∇u|p–2 ∇u – |∇v|p–2 ∇v ∇ (uε – vε )ξλ dx dt Qτ s

Zhan Journal of Inequalities and Applications (2018) 2018:7

–

N

 Qτ s

i=1





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  bi (u) – bi (v) (uε – vε )ξλ x dx dt i

  q(x) |u|γ –1 u – |v|γ –1 v (uε – vε )ξλ dx dt,

+

(3.3)

Qτ s

where Qτ s =  × (τ , s). For any give λ > 0, since ∇u ∈ Lp (λ ), ∇v ∈ Lp (λ ), according to the definition of the mollified function uε and vε , we have uε ∈ L∞ (QT ),

vε ∈ L∞ (QT ),

∇uε p,λ ≤ ∇up,λ , uε  u,

vε  v,

uε → u,

vε → v,

a.e. in QT ,

∇vε p,λ ≤ ∇vp,λ ,

in W 1,p (λ ).

(3.4) (3.5)

Let us analyze every term in (3.3). For a start, we deal with the first term on the right hand side of (3.3). Since on λ , p  α  ρ |∇u|p–2 ∇u – |∇v|p–2 ∇v  ∈ L p–1 (λ )

by the weak convergency of (3.5)

  ρ α ξλ |∇u|p–2 ∇u – |∇v|p–2 ∇v ∇(uε – vε ) dx dt

lim

ε→0

Qτ s



  ρ α ξλ |∇u|p–2 ∇u – |∇v|p–2 ∇v ∇(u – v) dx dt.

= Qτ s

By (3.4)-(3.5), using the Lebesgue dominated convergence theorem,

  ρ α |∇u|p–2 ∇u – |∇v|p–2 ∇v (uε – vε )∇ξλ dx dt

lim

ε→0

Qτ s



  ρ α |∇u|p–2 ∇u – |∇v|p–2 ∇v (u – v)∇ξλ dx dt.

= Qτ s

So

    ρ α |∇u|p–2 ∇u – |∇v|p–2 ∇v ∇ (uε – vε )ξλ dx dt

lim

ε→0

Qτ s



  ρ α ξλ |∇u|p–2 ∇u – |∇v|p–2 ∇v ∇(u – v) dx dt

= Qτ s



+

  ρ α |∇u|p–2 ∇u – |∇v|p–2 ∇v (u – v)∇ξλ dx dt.

(3.6)

Qτ s

We have

Qτ s

  ρ α ξλ |∇u|p–2 ∇u – |∇v|p–2 ∇v ∇(u – v) dx dt ≥ 0

(3.7)

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and  

    α p–2 p–2  lim ρ |∇u| ∇u – |∇v| ∇v (u – v)∇ξλ dx dt  λ→0 Qτ s  

    β α p–2 p–2  (u – v)ρ |∇u| ∇u – |∇v| ∇v ∇ρ dx dt  =

Qτ s

  |u – v|ρ α |∇u|p–1 + |∇v|p–1 |∇ρ β | dx dt

≤ Qτ s

 s  p1  s  p–1 p  p   α p p ρ |∇u| + |∇v| dx dt · ρ α ∇ρ β  |u – v|p dx dt ≤c τ

τ





 p–1  s  s  p1 p   α p p α+p(β–1) p ≤c ρ |∇u| + |∇v| dx dt · ρ |u – v| dx dt τ

τ





 s  p1 ≤c ρ α+p(β–1) |u – v|p dx dt .

(3.8)



τ

Here, we have used the fact that |∇ρ| = 1 is true almost everywhere. Now, by β ≥ have 

  

p–α , p–1

we

    (u – v)ρ α |∇u|p–2 ∇u – |∇v|p–2 ∇v ∇ρ β dx dt 

Qτ s

 s  p1 β p ≤c ρ |u – v| dx dt . τ

(3.9)



If p ≥ 2,  s τ

ρ β |u – v|p dx dt

 p1

 s  p1 ≤c ρ β |u – v|2 dx dt . τ



(3.10)



If 1 < p < 2, by the Hölder inequality  s

ρ |u – v| dx dt p

β

τ

 p1

 s  12 β 2 ≤c ρ |u – v| dx dt . τ



(3.11)



Now we deal the second term on the right hand side of (3.3). By the Lebesgue dominated convergence theorem and the Hölder inequality



lim lim

λ→0 ε→0

Qτ s



i



= lim

λ→0

  bi (u) – bi (v) (uε – vε )ξλ x dx dt

Qτ s

  bi (u) – bi (v) (u – v)ξλ x dx dt



= lim

λ→0

Qτ s

+ Qs





= Qτ s

i



 bi (u) – bi (v) (u – v)ξλxi dx dt

  bi (u) – bi (v) (u – v)xi ξλ dx dt

 bi (u) – bi (v) (u – v)ρxβi dx dt +



Qs



 bi (u) – bi (v) (u – v)xi ρ β dx dt.

(3.12)

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Since β ≥ 2, |ρxi | ≤ |∇ρ| = 1, by the Hölder inequality,

Qτ s

  bi (u) – bi (v) (u – v)ρxβi dx dt

s



= λ

τ

≤

s τ

 bi (u) – bi (v) (u – v)ρ β–1 |ρxi | dx

|u – v|ρ

β–1

 s  12 β 2 dx ≤ c ρ |u – v| dx dt . τ



(3.13)



Since β ≥ αp, we have   α p β– ≥ β, p p–1 by this result, we have 

  

 Qτ s

≤

   bi (u) – bi (v) (u – v)xi ρ β dx dt 

N  s  τ

i=1

×

 s

  ρ α |∇u|p + |∇v|p dx dt

 1 p

 p1



N  s 

p  ρ bi (u) – bi (v) dx dt β

 1 p



τ

i=1

p bi (u) – bi (v) dx dt



τ

≤c

ρ

(β– αp )p 

 s  1 p  ≤c ρ β |u – v|p dx dt .

(3.14)



τ

If p > 2, then 1 < p < 2. By the Hölder inequality,  s

p

ρ |u – v| dx dt β

 1 p



τ

 s  12 β 2 ≤c ρ |u – v| dx dt , τ

(3.15)



If 1 < p ≤ 2, then p ≥ 2,  s

p

ρ |u – v| dx dt β

τ

 1 p

 s  1 p β 2 ≤c ρ |u – v| dx dt . τ



(3.16)



Again, for the third term on the right hand side of (3.3),

  q(x) |u|γ –1 u – |v|γ –1 v (uε – vε )ξλ dx dt

lim lim

λ→0 ε→0

Qτ s



  q(x)uγ – vγ |u – v|ρ β dx dt ≤ c

≤ Qτ s

≤c Qτ s

q(x)ρ β |u – v| dx dt Qτ s



 s  12 ρ β |u – v| dx dt ≤ c ρ β |u – v|2 dx dt . τ



(3.17)

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At last, by Lemma 3.1,

∂(u – v) dx dt ∂t Qτ s √

 ∂ ξλ (u – v) = lim (u – v) ξλ dx dt λ→0 ∂t Qτ s

  1 = lim ξλ (u – v)2 (x, s) – (u – v)2 (x, τ ) dx 2 λ→0   

 2  2 1 β β = ρ u(x, s) – v(x, s) dx – ρ u(x, τ ) – v(x, τ ) dx . 2   (uε – vε )ξλ

lim lim

λ→0 ε→0

(3.18)

Now, after letting ε → 0, let λ → 0 in (3.3). Then, by (3.7)-(3.18),

 2 ρ β u(x, s) – v(x, s) dx –



 2 ρ β u(x, τ ) – v(x, τ ) dx



 s k  2 ≤c ρ β u(x, t) – v(x, t) dx dt , τ

(3.19)



where k < 1. By this inequality, we are able to show that

 2 ρ β u(x, s) – v(x, s) dx ≤



 2 ρ β u(x, τ ) – v(x, τ ) dx.

(3.20)



Thus, by the arbitrariness of τ , we have



2 ρ u(x, s) – v(x, s) dx ≤ β



ρ β |u0 – v0 |2 dx.

(3.21)



By (3.21), we clearly have (1.10). The proof is complete.



4 Conclusion The equations considered in this paper come from many applied fields such as mechanics, biology, etc. The main points of focus of this paper are two aspects. One is that the weak solution defined in this paper satisfies ut ∈ W  (QT ), then the uniqueness can be proved. The other one is to show that the degeneracy of the diffusion coefficient ρ α can take place with the usual boundary value condition. Acknowledgements The paper is supported by Natural Science Foundation of China (no: 11371297), Natural Science Foundation of Fujian province (no: 2015J01592), supported by Science Foundation of Xiamen University of Technology, China. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the final manuscript.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 10 August 2017 Accepted: 15 December 2017

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