Zhan Boundary Value Problems (2016) 2016:178 DOI 10.1186/s13661-016-0684-6

RESEARCH

Open Access

The stability of the solutions of an equation related to the p-Laplacian with degeneracy on the boundary Huashui Zhan* *

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

Abstract The equation related to the p-Laplacian ut = div(ρ α |∇u|p–2 ∇u) +

N  ∂ bi (u) i=1

∂ xi

,

(x, t) ∈  × (0, T),

is considered, where ρ (x) = dist(x, ∂) is the distance function from the boundary. If α < p – 1, the weak solution belongs to Hγ for some γ > 1, the Dirichlet boundary condition can be imposed as usual, the stability of the solutions is proved. If α ≥ p – 1, the weak solution lacks the regularity to define the trace on the boundary. It is surprising that we can still prove the stability of the solutions without any boundary condition. In other words, when α ≥ p – 1, the phenomenon that the solutions of the equation may be free from any limitations of the boundary condition is revealed. MSC: 35L65; 35L85; 35R35 Keywords: stability; boundary degeneracy; the p-Laplacian

1 Introduction and the main results Consider an equation related to the p-Laplacian, N    ∂bi (u) , ut = div ρ α |∇u|p– ∇u + ∂xi i=

(x, t) ∈ QT ,

(.)

with the initial value u(x, ) = u (x),

x ∈ .

(.)

Here  is a bounded domain in RN with appropriately smooth boundary, ρ(x) = dist(x, ∂), p > , α > , QT =  × (, T). Can we impose the homogeneous boundary condition u(x, t) = ,

(x, t) ∈ ∂ × (, T),

(.)

© 2016 Zhan. 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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as usual? Recently, Ji˘rí et al. [] studied the equation   ut = div |∇u|p– ∇u + q(x)uγ ,

(x, t) ∈ QT ,

(.)

with  < γ < , and they showed that the uniqueness of the solutions of equation (.) is not true. The author [] studied the equation   ut = div ρ α |∇u|p– ∇u + f (u, x, t)

(.)

and showed that the stability of the solutions can be obtained without any boundary value condition. Comparing (.) with (.), one can see that the degeneracy of the coefficient ρ α can eliminate the effect from the source term f (u, x, t). Thus, it is naturally to ask whether the coefficient ρ α can eliminate the effect from the convection term ∂b∂xi (u) in equation (.). i Yin and Wang [] studied the following equation:   ∂u – div a(x)|∇u|p– ∇u – bi (x)Di u + c(x, t)u = f (x, t), ∂t

(.)

where Di = ∂x∂ i , a ∈ C(), and a(x) >  in . Yin and Wang classified the boundary into three parts: the nondegenerate boundary, the weakly degenerate boundary, and the strongly degenerate boundary, by means of a reasonable integral description. The boundary value condition should be supplemented definitely on the nondegenerate boundary and the weakly degenerate boundary although the equation is degenerate on this portion of the degenerate boundary. On the strongly degenerate boundary, they formulated a new approach to prescribe the boundary value condition rather than define the Fichera function as treating the linear case. Moreover, they formulated the boundary value condition on this strongly degenerate boundary in a much weaker sense since the regularity of the solution is much weaker near this boundary. Stated succinctly, instead of the usual boundary value condition (.), only the partial boundary condition u(x, t) = ,

(x, t) ∈ p × (, T),

(.)

is imposed in [], where p ⊆ ∂. In our paper, we will study how the degeneracy of ρ α affects the well-posedness of the solutions of equation (.). Definition . A function u(x, t) is said to be a weak solution of equation (.) with the initial value (.), if u ∈ L∞ (QT ),

ρ α |∇u|p ∈ L (QT ),

(.)

and, for any function ϕ ∈ C∞ (QT ), 

 –uϕt + ρ |∇u| α

QT

p–

∇u · ∇ϕ +

N  i=

 bi (u) · ϕxi dx dt = .

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The initial value is satisfied in the sense of that    lim u(x, t) – u (x) dx = . t→ 

(.)

Definition . Let p > . The function u(x, t) is said to be the weak solution of equation (.) with the initial value (.) and the usual boundary condition (.), if u satisfies Definition ., and the usual boundary condition (.) is satisfied in the sense of trace. The existence of the solutions of equation (.) with the initial value (.) can be obtained similar to [, ]. The main aim of the paper is to study the stability of the solutions. Theorem . Let p > , α < p – , bi (s) be a Lipschitz function. If u, v are two solutions of equation (.) with the same homogeneous value condition (.) and with the initial values u (x), v (x), respectively, then 

  u(x, t) – v(x, t) dx ≤

 |u – v |,



∀t ∈ [, T).

(.)



Theorem . Let p ≥ , α ≥ p – , bi (s) be bounded when s is bounded. If u, v are two solutions of equation (.) with the initial values u (x), v (x), respectively, then 

  u(x, t) – v(x, t) dx ≤





  u (x) – v (x) dx.

(.)



Theorem . Let p > , α ≥ p – , bi (s) satisfy α+(p–)   bi (s ) – bi (s ) ≤ c|s – s | p .

(.)

If u, v are two solutions of equation (.) with the initial values u (x), v (x), respectively, such that   u(x, t) ≤ cρ(x),

  v(x, t) ≤ cρ(x),

(.)

then the stability (.) is true. Theorem . Let p > , α ≥ p – , bi (s) be a C  function which satisfies   b (s) ≤ c|s|r . i

(.)

Assume the following: (i) When  < p < , α ≤  + p(r – ), r ≥ . (ii) When p = , α ≤ pr. . (iii) When p > , r ≥  , α < (r+)p–  Let u, v be two solutions of equation (.) with the initial values u (x), v (x), respectively, such that (.) is true. Then the stability (.) is true. Since the solution u lacks the regularity when α ≥ p – , we cannot define the trace of u on the boundary. How to construct a suitable test function to get the stability (.) is a full

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challenge. By Theorem ., the remaining important problem is that if p > , α ≥ p – , the same conclusion is true or not. Theorems . and . partially solve the problem. Certainly, the conditions of (.)-(.) in Theorem ., and the conditions (i)-(iii) in Theorem ., may be not the best.

2 The case of α < p – 1 Lemma . If α < p – , let u be the solution of equation (.) with the initial value (.), then u ∈ H γ () for some γ >  the trace of u on the boundary ∂ can be defined in the traditional way. If bi ≡ , the lemma had been proved in Theorem . in []. For the general case, the lemma had been proved in [] recently. By the lemma, the initial-boundary value problem (.)-(.)-(.) is reasonable. Proof of Theorem . Let u and v be two weak solutions with the different initial values u(x, ), v(x, ), respectively. From the definition of the weak solution, we have ρ α |∇u|p , ρ α |∇v|p ∈ L (Q), and for all ϕ ∈ C∞ (QT ), 

∂(u – v) dx dt = – ϕ ∂t QT



–

  ρ α |∇u|p– ∇u – |∇v|p– ∇v · ∇ϕ dx dt QT

N   i=

QT



 bi (u) – bi (v) · ϕxi dx dt.

(.)

For any given positive integer n, let gn (s) be an odd function, and when s > , , s > n , gn (s) =   –n s , s ≤ n . n s e

(.)

By the usual boundary condition (.), u(x, t) = v(x, t) = ,

(x, t) ∈ ∂ × (, T).

By a process of limit, we can choose gn (u – v) as the test function in (.), then  gn (u – v) QT



+ QT

+

  ρ α |∇u|p– ∇u – |∇v|p– ∇v · ∇(u – v)gn dx dt

N   i=

Thus 

 QT



QT

 bi (u) – bi (v) · (u – v)xi gn dx dt = .

d ∂(u – v) dx = u – v  , ∂t dt   ρ α |∇u|p– ∇u – |∇v|p– ∇v · ∇(u – v)gn dx dt ≥ ,

gn (u – v) 

∂(u – v) dx dt ∂t

(.)

Zhan Boundary Value Problems (2016) 2016:178

lim

n→∞

N   QT

i=

Page 5 of 13

  bi (u) – bi (v) gn (u – v)(u – v)xi dx dt = .

(.)

Here, (.) is established by   c g (s) ≤ , n s

 |s| ≤ , n

according to the definition of gn (s), and by the following facts:    

QT ∩{|u–v|< n }

   bi (u) – bi (v) gn (u – v)xi dx dt 

  = 



QT ∩{|u–v|< n }

   bi (u) – bi (v) gn (u – v)(u – v)xi dx dt 

   bi (u) – bi (v)    (u – v)x  dx dt i   u–v QT ∩{|u–v|< n }      – α bi (u) – bi (v)  α ρ p (u – v)x  dx dt  p ρ =c i   u–v QT ∩{|u–v|< n } 

≤c

 ≤c

 p

  p– p  – α bi (u) – bi (v)  p– ρ p  dx dt    u–v QT ∩{|u–v|< n }

 ·

Q∩{|u–v|< n }

 α  ρ ∇(u – v)p dx dt

 p .

(.)

Since α < p – , and bi (u) is a Lipschitz function,  p

   – α bi (u) – bi (v)  p– – α ρ p  dx dt ≤ c ρ p– dx dt ≤ c.   u–v QT ∩{|u–v|< n } QT



(.)

In (.), let n → ∞. If {x ∈  : |u – v| = } is a set with  measure, then  lim

n→∞

QT ∩{|u–v|< n }



α

ρ p– dx dt =

α

QT ∩{|u–v|=}

ρ p– dx dt = .

(.)

If the set {x ∈  : |u – v| = } has a positive measure, then  lim

n→∞

QT ∩{|u–v|< n }

 p ρ α ∇(u – v) dx dt =

 QT ∩{|u–v|=}

 p ρ α ∇(u – v) dx dt = . (.)

Therefore, in both cases, (.) goes to  as n → ∞. Now, let n → ∞ in (.). Then d u – v  ≤ . dt It implies that 

  u(x, t) – v(x, t)dx ≤



Theorem . is proved.

 |u – v |dx,

∀t ∈ [, T).





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Remark . If bi (s) is a C  (R), the conclusion of Theorem . had been proved by the author and Yuan in [] recently, the method used here is very similar as that in []. Since [] is written in Chinese, we give the details here.

3 The stability of the case α ≥ p – 1 When α ≥ p – , let u be a weak solution of equation (.) with the initial value (.). Generally, we cannot define the trace of u on the boundary, how to prove the uniqueness of the solutions seems very difficult. Theorem . solves the problem when p ≥ , in the following, we give its proof. Proof of Theorem . Denote λ = {x ∈  : dist(x, ∂) > λ}. Let α

α ξλ = dist(x,  \ λ ) p = dλp .

(.)

From the definition of the weak solution, we have  ϕ QT

∂(u – v) dx dt = – ∂t



–

  ρ α |∇u|p– ∇u – |∇v|p– ∇v ∇ϕ dx dt QT

N   i=

QT



 bi (u) – bi (v) ϕxi dx dt,

(.)

for any ϕ ∈ C∞ (QT ). For any fixed τ , s ∈ [, T], we may choose χ[τ ,s] (uε – vε )ξλ as a test function in the above equality, where χ[τ ,s] is the characteristic function on [τ , s], uε and vε are the mollified functions of the solutions u and v, respectively. Thus, letting Qτ s =  × [τ , s],  (uε – vε )ξλ Qτ s



=–

∂(u – v) dx dt ∂t

  

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

Qτ s

–

N   i=

Qτ s



 

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

(.)

For any give λ > , denoting QTλ = λ × (, T), we know that ∇u ∈ Lp (QTλ ), ∇v ∈ Lp (QTλ ). Thus according to the definition of the mollified functions uε and vε , we have uε ∈ L∞ (QT ),

vε ∈ L∞ (QT ),

∇uε p,λ ≤ ∇u p,λ ,

∇vε p,λ ≤ ∇v p,λ .

By the Young inequality,     |∇u|p– ∇u – |∇v|p– ∇v ∇(uε – vε )  p  p –    |∇u|p– ∇u – |∇v|p– ∇v p– + ∇(uε – vε ) p p     ≤ c |∇u|p + |∇v|p + c |∇uε |p + |∇vε |p . ≤

(.) (.)

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Since on λ , ρ α ≥ λα , we have  α   ρ |∇u|p– ∇u – |∇v|p– ∇v ∇[(uε – vε )     ≤ c(λ) |∇u|p + |∇v|p + c |∇uε |p + |∇vε |p   ≤ c(λ) |∇u|p + |∇v|p , by the Lebesgue control convergence theorem, 



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

lim

ε→

Qτ s



  

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

= 

Qτ s

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

= Qτ s



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

+

(.)

Qτ s

The first term on the right hand side of (.), 

  ρ α ξλ |∇u|p– ∇u – |∇v|p– ∇v ∇(u – v) dx dt ≥ .

(.)

Qτ s

The last term on the right hand side of (.), since, when x ∈ λ , dλ (x) > , obeys    



(u – v)ρ |∇u| α

p–

∇u – |∇v|

p–

Qτ s



   ∇v ∇ξλ dx dt 

  |u – v|ρ α |∇u|p– + |∇v|p– |∇ξλ | dx dt

≤ Qτ s

 s  ≤c τ

  ρ α |∇u|p + |∇v|p dx dt

 p–

 s  p · τ

λ



 s    ≤c ρ α |∇u|p + |∇v|p dx dt τ

·

 p

λ

p– p



 s  τ

p( αp –) ρ α dλ |∇dλ |p |u – v|p dx dt

 p

λ

 s  ≤c τ

ρ α |∇ξλ |p |u – v|p dx dt

p( αp –) ρ α dλ |u – v|p dx dt

 p .

(.)

λ

Here, we have used the fact that |∇dλ | =  is true almost everywhere. Now, since p ≥  and α ≥ p –  imply that α – p ≥ , we have   lim 

λ→

Qτ s

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

 s  ≤ lim c λ→

τ

λ

α

ρ (ρ – λ)

p( αp –)

|u – v| dx dt p

 p

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 s   p α+p( αp –) p =c ρ |u – v| dx dt τ



 s   p ≤c ρ α–p |u – v|p dx dt τ



 s   p  ≤c |u – v| dx dt . τ

(.)



Using the Hölder inequality



 |∇uε | dx ≤

|∇uε | dx p

λ

 p

λ

 ≤

|∇u| dx p

 p

≤ c(λ),

λ

|∇uε | ∈ L (λ ). Then by the definition of the mollified function uε , one has |∇uε | ≤ |∇u| . By the Lebesgue control convergence theorem, we have 



lim

ε→

i

Qτ s





= 

Qτ s

Since α ≥ p – , ρ–

–δ p

 

bi (u) – bi (v) (u – v)ξλ x dx dt i

Qτ s

=



 

bi (u) – bi (v) (uε – vε )ξλ x dx dt

 bi (u) – bi (v) (u – v)ξλxi dx dt + –≥

α p

– . p

 Qs

 bi (u) – bi (v) (u – v)xi ξλ dx dt.

For any  > δ > , by p ≥ ,

–δ p

(.)

<  and

dx ≤ c,



due to  being appropriately smooth. By these two facts, we have  lim

λ→

Qτ s

 bi (u) – bi (v) (u – v)ξλxi dx dt

≤ c lim

 s

λ→ τ

≤ c lim

λ

 s

λ→ τ



α –  p bi (u) – bi (v) (u – v)dλ |dλxi | dx α – p

|u – v| dλ

dx





 s   |u – v| ρ – p dx ≤c τ



–δ     –δ  –δ

 s  s – –δ  –δ p –δ |u – v| dx ρ dx ≤c

τ



–δ

 s   –δ ≤c |u – v| dx

τ



τ



(.)

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and  lim

λ→

Qτ s

 bi (u) – bi (v) (u – v)xi ξλ dx dt



= lim

λ→

≤

Qτ s

 s 

α

 p bi (u) – bi (v) (u – v)xi dλ dx dt



 bi (u) – bi (v) p dx dt

   s  p



τ

τ

  ρ α |∇u|p + |∇v|p dx dt

 p



 s   

p

p    bi (u) – bi (v) dx dt ≤c τ



 s   

 s   

p p ≤c |u – v| dx dt ≤c |u – v| dx dt . τ



τ

(.)



By (.)-(.), after letting ε → , we let λ →  in (.). Then 



 u(x, s) – v(x, s) dx –







 u(x, τ ) – v(x, τ ) dx



 s  q     ≤c u(x, t) – v(x, t) dx dt , 

(.)



where q < . Let κ(s) = ( κ(s) – κ(τ ) ≤c s–τ



s τ

 [u(x, s) – v(x, s)]



dx. Then

κ(t) dt)q . s–τ

By the L’Hospital rule, we have κ(s) κ (τ ) ≤ c lim  s s→τ ( κ(t) dt)–q τ

 q s κ (s) = c lim κ(t) dt = . s→τ κ(s) τ

(.)

Thus, by the arbitrary of τ , we have 

  u(x, s) – v(x, s) dx ≤



The proof is complete.

 |u – v | dx.

(.)





4 Proof of Theorems 1.5 and 1.6 Instead of the condition p ≥ , if we only assume that p > , then we also can obtain the uniqueness of the solutions in some cases. Proof of Theorems . and . Denote λ = {x ∈  : dist(x, ∂) > λ} as before, let ξλ ∈ C∞ (λ ) such that ξλ =  on λ ,  ≤ ξλ ≤ , and c |∇ξλ | ≤ . λ

(.)

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From the definition of the weak solution, we have  ϕ QT

∂(u – v) dx dt = – ∂t



–

  ρ α |∇u|p– ∇u – |∇v|p– ∇v ∇ϕ dx dt QT

N   i=



QT

 bi (u) – bi (v) ϕxi dx dt,

(.)

for any ϕ ∈ C∞ (QT ). For any fixed τ , s ∈ [, T], we may choose χ[τ ,s] (uε – vε )ξλ as a test function in the above equality, where χ[τ ,s] is the characteristic function on [τ , s], uε and vε are the mollified functions of the solutions u and v, respectively. Thus, letting Qτ s =  × [τ , s],  (uε – vε )ξλ Qτ s



∂(u – v) dx dt ∂t

  

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

=– Qτ s

–

N  



Qτ s

i=



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

(.)

i

For any give λ > , denoting QTλ = λ × (, T), we know that ∇u ∈ Lp (QTλ ), ∇v ∈ Lp (QTλ ). Thus according to the definition of the mollified functions uε and vε , similarly, we have (.)-(.). The first term on the right hand side of (.), 

  ρ α ξλ |∇u|p– ∇u – |∇v|p– ∇v ∇(u – v) dx dt ≥ .

(.)

Qτ s

The last term on the right hand side of (.), since α ≥ p – ,    

Qτ s

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



  |u – v|ρ α |∇u|p– + |∇v|p– |∇ξλ | dx dt

≤ Qτ s

 s  ≤c τ

·

λ \λ

 s  λ \λ

τ

 s  ≤c τ

≤ cλ

λ \λ

(α+–p) p

τ

λ \λ

p

ρ α |∇ξλ |p dx dt

p



 p– p

 p

  ρ α |∇u|p + |∇v|p dx dt

 p–

 s  p · τ

 s  τ

 s  ≤c



ρ |∇u| + |∇v| dx dt α

λ \λ

  ρ α |∇u|p + |∇v|p dx dt

  ρ α |∇u|p + |∇v|p dx dt

 p– p

λ \λ

λα–p dx dt

 p

 p– p

→ ,

as λ → .

(.)

Zhan Boundary Value Problems (2016) 2016:178

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Also by the Lebesgue control convergence theorem, 



lim

ε→

i

Qτ s





= 

 

bi (u) – bi (v) (uε – vε )ξλ x dx dt

i

Qτ s



=

 

bi (u) – bi (v) (u – v)ξλ x dx dt

Qτ s

 bi (u) – bi (v) (u – v)ξλxi dx dt +

 Qs

 bi (u) – bi (v) (u – v)xi ξλ dx dt.

(.)

By that |u(x, t)| ≤ cρ(x), |v(x, t)| ≤ cρ(x), 



lim

λ→

 s

 bi (u) – bi (v) (u – v)ξλxi dx dt ≤ c lim

λ→ τ

Qτ s

λ \λ

dx = .

(.)

By the Hölder inequality  lim

λ→

Qτ s

 bi (u) – bi (v) (u – v)xi ξλ dx dt



= lim

λ→

≤

Qτ s

 s  τ



 α α ρ – p ξλ bi (u) – bi (v) ρ p (u – v)xi dx dt

ρ

 p

bi (u) – bi (v) dx dt

– αp 

   s  p

τ



  ρ |∇u|p + |∇v|p dx dt α

 p



 s   

p

p  –α  ρ p bi (u) – bi (v) dx dt ≤c . τ

(.)



(I) If p > ,  α+(p–)   α bi (u) – bi (v) ≤ c|u – v| p |u – v| p = c|u – v| p ,

by (.), 



lim

λ→

Qτ s

 s   

p   bi (u) – bi (v) (u – v)xi ξλ dx dt ≤ c |u – v| dx dt . 

(.)



(II) (i) If p < , |b i (ζ )| ≤ |ζ |r , r ≥ , 

p

 –α  ρ p bi (u) – bi (v) dx dt ≤

Qs

 ρ

 α  – p– b (ζ ) p– |u – v| p– – |u – v| dx dt i

ρ

α + – p– p– – |u – v| dx dt

p

Qτ s



p

p(+r)

≤ Qτ s

 ≤c

|u – v| dx dt.

(.)

Qτ s

Here, the last inequality is based on the assumption α ≤ p(r – ) + , which implies that –

p( + r) α + –  ≥ . p– p–

Zhan Boundary Value Problems (2016) 2016:178

Page 12 of 13

Then 



lim

λ→

 s   

p   |u – v| dx dt . bi (u) – bi (v) (u – v)xi ξλ dx dt ≤ c 

Qτ s

(.)



(ii) If p = , |b i (ζ )| ≤ |ζ |r , r ≥ αp ,     bi (u) – bi (v) ≤ b (ζ )|u – v| , i then 



lim

λ→

 s      bi (u) – bi (v) (u – v)xi ξλ dx dt ≤ c |u – v| dx dt . 

Qτ s

(.)



(iii) If p > , by |b i (ζ )| ≤ |ζ |s , s ≥  , 



p

α ρ – p bi (u) – bi (v) dx dt

Qτ s



 α  – p– b (ζ ) p– |u – v| p– i p

≤

ρ Qs





≤

ρ

Qτ s

 ≤

ρ

α – p–

p

dx dt

  p  (p–) b (ζ ) p– p– dx dt i

(sp–α) p–

(p–)

|u – v| dx dt 

(p–)

 p– 

Qτ s

 (p–)  |u – v| dx dt

dx dt

Qτ s

p

 (p–) 

p (p–)

,

Qτ s

then, based on the assumption α <

(s+)p– , 

which implies that

(sp – α) > –, p– we have



p

 –α  ρ p bi (u) – bi (v) dx dt

 

p

 ≤c

Qτ s

|u – v| dx dt 

  ,

(.)

Qτ s

and we have (.) too. By (.)-(.), after letting ε → , we let λ →  in (.). Then 



 u(x, s) – v(x, s) dx –







 u(x, τ ) – v(x, τ ) dx



 s  q     ≤c u(x, t) – v(x, t) dx dt , 

(.)



 where q < . Let κ(s) =  [u(x, s) – v(x, s)] dx. As the proof of Theorem ., we have the conclusions. At the end of the paper, we should point out that the conditions (.)-(.) and (i)-(iii)  ∂bi (u) are used to deal with the convection term N i= ∂xi . In other words, if bi ≡ , then all these conditions are not necessary, the same conclusions had been obtained in []. 

Zhan Boundary Value Problems (2016) 2016:178

Page 13 of 13

Competing interests The author declares that they have no competing interests. Acknowledgements The paper is supported by NSF of China (no. 11371297), supported by NSF of Fujian Province (no. 2015J01592), China. Received: 23 April 2016 Accepted: 19 September 2016 References 1. Ji˘rí, B, Peter, G, Luká˘s, K, Peter, T: Nonuniqueness and multi-bump solutions in parabolic problems with the p-Laplacian. J. Differ. Equ. 260, 991-1009 (2016) 2. Zhan, H: On a parabolic equation related to the p-Laplacian. Bound. Value Probl. 2016, 78 (2016). doi:10.1186/s13661-016-0587-6 3. Yin, J, Wang, C: Evolutionary weighted p-Laplacian with boundary degeneracy. J. Differ. Equ. 237, 421-445 (2007) 4. Zhan, H: The solution of convection-diffusion equation. Chin. Ann. Math., Ser. A 34(2), 235-256 (2013) (in Chinese) 5. Zhan, H: The boundary value condition of an evolutionary p(x)-Laplacian equation. Bound. Value Probl. 2015, 112 (2015). doi:10.1186/s13661-015-0377-6 6. Yin, J, Wang, C: Properties of the boundary flux of a singular diffusion process. Chin. Ann. Math., Ser. B 25(2), 175-182 (2004) 7. Zhan, H, Yuan, H: Diffusion convection equation with boundary degeneracy. J. Jilin Univ. Sci. Ed. 53(3), 353-358 (2015) (in Chinese)