Open Access

Study on integro-differential equation with generalized p-Laplacian operator

Boundary Value Problems20122012:131

DOI: 10.1186/1687-2770-2012-131

Received: 13 June 2012

Accepted: 24 October 2012

Published: 13 November 2012

Abstract

We tackle the existence and uniqueness of the solution for a kind of integro-differential equations involving the generalized p-Laplacian operator with mixed boundary conditions. This is achieved by using some results on the ranges for maximal monotone operators and pseudo-monotone operators. The method used in this paper extends and complements some previous work.

MSC: 47H05, 47H09.

Keywords

maximal monotone operator pseudo-monotone operator generalized p-Laplacian operator integro-differential equation mixed boundary conditions

1 Introduction

Nonlinear boundary value problems (BVPs) involving the p-Laplacian operator Δ p arise from a variety of physical phenomena such as non-Newtonian fluids, reaction-diffusion problems, petroleum extraction, flow through porous media, etc. Thus, the study of such problems and their generalizations have attracted numerous attention in recent years. Some of the BVPs studied in the literature include the following:
{ Δ p u + g ( x , u ( x ) ) = f ( x ) , a.e. in  Ω , u n = 0 , a.e. on  Γ
(1.1)
whose existence results in L p ( Ω ) (for various ranges of p) can be found in [14]; a related BVP
{ Δ p u + g ( x , u ( x ) ) = f ( x ) , a.e. in  Ω , ϑ , | u | p 2 u β x ( u ( x ) ) , a.e. on  Γ
(1.2)
was tackled in [57] and later generalized to one that contains a perturbation term | u | p 2 u [8, 9]
{ Δ p u + | u | p 2 u + g ( x , u ( x ) ) = f ( x ) , a.e. in  Ω , ϑ , | u | p 2 u β x ( u ( x ) ) , a.e. on  Γ .
(1.3)
Motivated by Tolksdorf’s work [10] where the following Dirichlet BVP has been discussed:
{ div [ ( C ( x ) + | u | 2 ) p 2 2 u ] = f ( x ) , a.e. in  K ( 1 , S ) , u = g , a.e. in  Σ ( 1 , S ) ,
(1.4)
several generalizations have been investigated. These include [1114]
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ5_HTML.gif
(1.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ6_HTML.gif
(1.6)
and
{ div [ ( C ( x ) + | u | 2 ) p 2 2 u ] + ε | u | q 2 u + g ( x , u ( x ) ) = f ( x ) , a.e. in  Ω , ϑ , ( C ( x ) + | u | 2 ) p 2 2 u β x ( u ( x ) ) , a.e. on  Γ ,
(1.7)

where 0 C ( x ) L p ( Ω ) , ε is a nonnegative constant and ϑ denotes the exterior normal derivative of Γ.

Inspired by all this research, recently we have studied the following nonlinear parabolic equation with mixed boundary conditions [15]:
{ u t div [ ( C ( x , t ) + | u | 2 ) p 2 2 u ] + ε | u | p 2 u = f ( x , t ) , ( x , t ) Ω × ( 0 , T ) , ϑ , ( C ( x , t ) + | u | 2 ) p 2 2 u β ( u ) h ( x , t ) , ( x , t ) Γ × ( 0 , T ) , u ( x , 0 ) = u ( x , T ) , a.e.  x Ω .
(1.8)
We tackle the existence of solutions for (1.8) via the study of existence of solutions for two BVPs: (i) the elliptic equation with Dirichlet boundary conditions
{ div [ ( C ( x ) + | u | 2 ) p 2 2 u ] + ε | u | q 2 u = f ( x ) , a.e. in  Ω , γ u = w , a.e. on  Γ
(1.9)
and (ii) the elliptic equation with Neumann boundary conditions
{ div [ ( C ( x ) + | u | 2 ) p 2 2 u ] + ε | u | q 2 u = f ( x ) , a.e. in  Ω , ϑ , ( C ( x ) + | u | 2 ) p 2 2 u β ( u ) h ( x ) , a.e. in  Γ .
(1.10)

By setting up the relations between the auxiliary equations (1.9) and (1.10) and by employing some results on ranges for maximal monotone operators, we showed that (1.8) has a unique solution in L p ( 0 , T ; W 1 , p ( Ω ) ) , where 2 p < + , 1 q < + if p N , and 1 q 2 N p N p if p < N .

In this paper, we shall employ the technique used in (1.8), viz. using the results on ranges for nonlinear operators, to study the existence and uniqueness of the solution to a nonlinear integro-differential equation with the generalized p-Laplacian operator. We note that most of the existing methods in the literature used to investigate such problems are based on the finite element method, hence our technique is new in tackling integro-differential equations. We shall consider the following nonlinear integro-differential problem with mixed boundary conditions:
{ u t div [ ( C ( x , t ) + | u | 2 ) p 2 2 u ] + ε | u | q 2 u + a t Ω u d x = f ( x , t ) , ( x , t ) Ω × ( 0 , T ) , ϑ , ( C ( x , t ) + | u | 2 ) p 2 2 u β x ( u ) , ( x , t ) Γ × ( 0 , T ) , u ( x , 0 ) = u ( x , T ) , x Ω .
(1.11)

Our discussion is based on some results on the ranges for maximal monotone operators and pseudo-monotone operators in [1618]. Some new methods of constructing appropriate mappings to achieve our goal are employed. Moreover, we weaken the restrictions on p and q. The paper is outlined as follows. In Section 2 we shall state the definitions and results needed, and in Section 3 we shall establish the existence and uniqueness of the solution to (1.11).

2 Preliminaries

Let X be a real Banach space with a strictly convex dual space X . We use ( , ) to denote the generalized duality pairing between X and X . For a subset C of X, we use IntC to denote the interior of C. We also use ‘→’ and ‘w-lim’ to denote strong and weak convergences, respectively.

Let X and Y be Banach spaces. We use X Y to denote that X is embedded continuously in Y.

The function Φ is called a proper convex function on X [17] if Φ is defined from X to ( , + ] , Φ is not identically +∞ such that Φ ( ( 1 λ ) x + λ y ) ( 1 λ ) Φ ( x ) + λ Φ ( y ) , whenever x , y X and 0 λ 1 .

The function Φ : X ( , + ] is said to be lower-semicontinuous on X [17] if lim inf y x Φ ( y ) Φ ( x ) for any x X .

Given a proper convex function Φ on X and a point x X , we denote by Φ ( x ) the set of all x X such that Φ ( x ) Φ ( y ) + ( x y , x ) for every y X . Such elements x are called subgradients of Φ at x, and Φ ( x ) is called the subdifferential of Φ at x [17].

A mapping T : D ( T ) = X X is said to be demi-continuous on X if w - lim n T x n = T x for any sequence { x n } strongly convergent to x in X. A mapping T : D ( T ) = X X is said to be hemi-continuous on X if w - lim t 0 T ( x + t y ) = T x for any x , y X [17].

With each multi-valued mapping A : X 2 X , we associate the subset A 0 as follows [17]:
A 0 x = { y A x : y = | A x | } ,

where | A x | : = inf { z : z A x } . If X is strictly convex, then D ( A ) = D ( A 0 ) and A 0 is single-valued, which in this case is called the minimal section of A.

A multi-valued mapping B : X 2 X is said to be monotone [18] if its graph G ( B ) is a monotone subset of X × X in the sense that ( u 1 u 2 , w 1 w 2 ) 0 for any [ u i , w i ] G ( B ) , i = 1 , 2 . The monotone operator B is said to be maximal monotone if G ( B ) is not properly contained in any other monotone subsets of X × X .

Definition 2.1 [18]

Let C be a closed convex subset of X, and let A : C 2 X be a multi-valued mapping. Then A is said to be a pseudo-monotone operator provided that
  1. (i)

    for each x C , the image Ax is a nonempty closed and convex subset of X ;

     
  2. (ii)
    if { x n } is a sequence in C converging weakly to x C and if f n A x n is such that lim sup n ( x n x , f n ) 0 , then to each element y C , there corresponds an f ( y ) A x with the property that
    ( x y , f ( y ) ) lim inf n ( x n x , f n ) ;
     
  3. (iii)

    for each finite-dimensional subspace F of X, the operator A is continuous from C F to X in the weak topology.

     

Lemma 2.1 [19]

Let Ω be a bounded conical domain in R N . If m p > N , then W m , p ( Ω ) C B ( Ω ) ; if 0 < m p N and q = N p N m p , then W m , p ( Ω ) L q ( Ω ) ; if m p = N and p > 1 , then for 1 q < + , W m , p ( Ω ) L q ( Ω ) .

Lemma 2.2 [18]

If B : X 2 X is an everywhere defined, monotone, and hemi-continuous operator, then B is maximal monotone. If B : X 2 X is a maximal monotone operator such that D ( B ) = X , then B is pseudo-monotone.

Lemma 2.3 [18]

If X is a Banach space and Φ : X ( , + ] is a proper convex and lower-semicontinuous function, then Φ is maximal monotone from X to X .

Lemma 2.4 [18]

If B 1 and B 2 are two maximal monotone operators in X such that int D ( B 1 ) D ( B 2 ) , then B 1 + B 2 is maximal monotone.

Lemma 2.5 [20]

Let X and its dual X be strictly convex Banach spaces. Suppose S : D ( S ) X X is a closed linear operator and S is the conjugate operator of S. If ( u , S u ) 0 u D ( S ) and ( v , S v ) 0 v D ( S ) , then S is a maximal monotone operator possessing a dense domain.

Lemma 2.6 [18]

Any hemi-continuous mapping T : X X is demi-continuous on Int D ( T ) .

Theorem 2.1 [16]

Let X be a real reflexive Banach space with X being its dual space. Let C be a nonempty closed convex subset of X. Assume that
  1. (i)

    the mapping A : C 2 X is a maximal monotone operator;

     
  2. (ii)

    the mapping B : C X is pseudo-monotone, bounded, and demi-continuous;

     
  3. (iii)

    if the subset C is unbounded, then the operator B is A-coercive with respect to the fixed element b X , i.e., there exists an element u 0 C D ( A ) and a number r > 0 such that ( u u 0 , B u ) > ( u u 0 , b ) for all u C with u > r .

     

Then the equation b A u + B u has a solution.

3 Existence and uniqueness of the solution to (1.11)

We begin by stating some notations and assumptions used in this paper. Throughout, we shall assume that
1 < q p < + , 1 p + 1 p = 1 and 1 q + 1 q = 1 .
Let V = L p ( 0 , T ; W 1 , p ( Ω ) ) and V be the dual space of V. The duality pairing between V and V will be denoted by , V . The norm in V will be denoted by V , which is defined by
u V = ( 0 T u ( t ) W 1 , p ( Ω ) p d t ) 1 p , u ( x , t ) V .
Let W = L q ( 0 , T ; W 1 , p ( Ω ) ) and W be the dual space of W. The norm in W will be denoted by W , which is defined by
v W = ( 0 T v ( t ) W 1 , p ( Ω ) q d t ) 1 q , v ( x , t ) W .

In the integro-differential equation (1.11), Ω is a bounded conical domain of a Euclidean space R N where N 1 , Γ is the boundary of Ω with Γ C 1 [5], ϑ denotes the exterior normal derivative to Γ. Here, | | and , denote the Euclidean norm and the inner-product in R N , respectively. Also, 0 C ( x , t ) L p ( 0 , T ; W 1 , p ( Ω ) ) , f ( x , t ) V is a given function, T and a are positive constants, and ε is a nonnegative constant. Moreover, β x is the subdifferential of φ x , where φ x = φ ( x , ) : R R for x Γ , and φ : Γ × R R is a given function.

To tackle (1.11), we need the following assumptions which can be found in [5, 14].

Assumption 1 Green’s formula is available.

Assumption 2 For each x Γ , φ x = φ ( x , ) : R R is a proper, convex, and lower-semicontinuous function and φ x ( 0 ) = 0 .

Assumption 3 0 β x ( 0 ) and for each t R , the function x Γ ( I + λ β x ) 1 ( t ) R is measurable for λ > 0 .

We shall present a series of lemmas before we prove the main result.

Lemma 3.1 Define the function Φ : V R by
Φ ( u ) = 0 T Γ φ x ( u | Γ ( x , t ) ) d Γ ( x ) d t , u V .

Then Φ is a proper, convex, and lower-semicontinuous mapping on V. Therefore, Φ : V V , the subdifferential of Φ, is maximal monotone.

Proof The proof of this lemma is analogous to that of Lemma 3.1 in [1]. We give the outline of the proof as follows.

Note that for each s R , the function x Γ β x 0 ( s ) R is measurable, where β x 0 ( s ) denotes the minimal section of β x . Since for all s 1 , s 2 R we have
{ x Γ : φ x ( s 1 ) > s 2 } = n { x Γ : i = 1 n s 1 n β x 0 ( i s 1 n ) > s 2 } ,

it implies that for u V , the function φ x ( u | Γ ( x , t ) ) is measurable on Γ. Then from the property of φ x , we know that Φ is proper and convex on V.

To see that Φ is lower-semicontinuous on V, let u n u in V. We may assume that there exists a subsequence of u n , for simplicity, we still denote it by u n , such that u n | Γ ( x , t ) u | Γ ( x , t ) for x Γ and t ( 0 , T ) a.e. This yields
φ x ( u | Γ ( x , t ) ) lim inf n φ x ( u n | Γ ( x , t ) )
for all x Γ and each t ( 0 , T ) a.e. since φ x is lower-semicontinuous for each x Γ . It then follows from Fatou’s lemma that for each t ( 0 , T ) ,
Γ φ x ( u | Γ ( x , t ) ) d Γ ( x ) Γ lim inf n φ x ( u n | Γ ( x , t ) ) d Γ ( x ) lim inf n Γ φ x ( u n | Γ ( x , t ) ) d Γ ( x ) .

So, Φ ( u ) lim inf n Φ ( u n ) whenever u n u in V. This completes the proof. □

Lemma 3.2 Define S : D ( S ) = { u V : u t V , u ( x , 0 ) = u ( x , T ) } V by
S u = u t + a t Ω u d x .

Then S is a linear maximal monotone operator possessing a dense domain in V.

Proof It is obvious that S is closed and linear.

For u ( x , t ) , w ( x , t ) D ( S ) , integrating by parts gives
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equk_HTML.gif

Then S w = w t a t Ω w d x , where D ( S ) = { w V : w t V , w ( x , 0 ) = w ( x , T ) } .

For u ( x , t ) D ( S ) , we find
0 T Ω u t u ( x , t ) d x d t = Ω | u ( x , T ) | 2 d x Ω | u ( x , 0 ) | 2 d x 0 T Ω u t u ( x , t ) d x d t = 0 T Ω u t u ( x , t ) d x d t ,
which implies that
0 T Ω u t u ( x , t ) d x d t = 0 .
Similarly, for u ( x , t ) D ( S ) ,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equn_HTML.gif
which implies that
a 0 T Ω u ( x , t ) ( t Ω u d x ) d x d t = 0 .
Thus,
u , S u V = 0 T Ω u t u ( x , t ) d x d t + a 0 T Ω u ( x , t ) ( t Ω u d x ) d x d t = 0 .

In the same manner, we have w , S w V = 0 for w D ( S ) . Therefore, noting Lemma 2.5 the result follows. □

In view of Lemmas 2.3 and 2.4, we have the following result.

Lemma 3.3 S + Φ : V V is maximal monotone.

Lemma 3.4 [14]

Define the mapping B p , q : W 1 , p ( Ω ) ( W 1 , p ( Ω ) ) as follows:
( v ¯ , B p , q u ¯ ) = Ω ( C ( x , t ) + | u ¯ | 2 ) p 2 2 u ¯ , v ¯ d x + ε Ω | u ¯ | q 2 u ¯ v ¯ d x , u ¯ , v ¯ W 1 , p ( Ω ) .

Then B p , q is maximal monotone.

Lemma 3.5 [14]

Let X 0 denote the closed subspace of all constant functions in W 1 , p ( Ω ) . Let X be the quotient space W 1 , p ( Ω ) X 0 . For u ¯ W 1 , p ( Ω ) , define the mapping P : W 1 , p ( Ω ) X 0 by
P u ¯ = 1 meas ( Ω ) Ω u ¯ d x .
Then, there is a constant C > 0 such that for every u ¯ W 1 , p ( Ω ) ,
u ¯ P u ¯ L p ( Ω ) C u ¯ ( L p ( Ω ) ) N .

Here meas ( Ω ) denotes the measure of Ω.

Definition 3.1 Define A : V V as follows:
v , A u V = 0 T ( v , B p , q u ) d t 0 T Ω f ( x , t ) v ( x , t ) d x d t , u , v V .

Lemma 3.6 The mapping A : V V is everywhere defined, bounded, monotone, and hemi-continuous. Therefore, Lemma  2.2 implies that it is also pseudo-monotone.

Proof From Lemma 2.1, we know that W 1 , p ( Ω ) C B ( Ω ) when p > N , and W 1 , p ( Ω ) L q ( Ω ) when p = N . If p < N , then W 1 , p ( Ω ) L N p N p ( Ω ) L p ( Ω ) L q ( Ω ) since 1 < q p < + . Thus, for all w ¯ W 1 , p ( Ω ) , w ¯ L q ( Ω ) k w ¯ W 1 , p ( Ω ) , where k > 0 is a constant. Therefore, for u , v V , we have
0 T u L q ( Ω ) q d t const 0 T u W 1 , p ( Ω ) q d t = const u W q
and
0 T v L q ( Ω ) q d t const 0 T v W 1 , p ( Ω ) q d t = const v W q .

Moreover, since 1 < q p < + , then L p ( 0 , T ; W 1 , p ( Ω ) ) L q ( 0 , T ; W 1 , p ( Ω ) ) , which implies that u W u V and v W v V for u , v V .

If p 2 , then for u , v V , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equw_HTML.gif

which implies that A is everywhere defined and bounded.

If 1 < p < 2 , then for u , v V , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equx_HTML.gif

which also implies that A is everywhere defined and bounded.

Since B p , q is monotone, we can easily see that for u , v V ,
u v , A u A v V = 0 T ( u v , B p , q u B p , q v ) d t 0 ,

which implies that A is monotone.

To show that A is hemi-continuous, it suffices to show that for any u , v , w V and k [ 0 , 1 ] , w , A ( u + k v ) A u V 0 , as k 0 . Noting the fact that B p , q is hemi-continuous and using the Lebesgue’s dominated convergence theorem, we have
0 lim k 0 | w , A ( u + k v ) A u V | 0 T lim k 0 | ( w , B p , q ( u + k v ) B p , q u ) | d t = 0 .

Hence, A is hemi-continuous.

This completes the proof. □

Lemma 3.7 The mapping A : V V satisfies that for u D ( S ) ,
u u 0 , A u V u V + ,
(3.1)

as u V + in V.

Proof First, we shall show that for u V ,
u V +
is equivalent to
u 1 meas ( Ω ) Ω u d x V + .
In fact, from Lemma 3.5, we know that for u V ,
u 1 meas ( Ω ) Ω u d x L p ( Ω ) C u ( L p ( Ω ) ) N ,
where C is a positive constant. Thus,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equad_HTML.gif
which implies that
u 1 meas ( Ω ) Ω u d x V [ ( C p + 1 ) 0 T u ( L p ( Ω ) ) N p d t ] 1 p ( C p + 1 ) 1 p u V .
(3.2)
On the other hand, we have
u 1 meas ( Ω ) Ω u d x W 1 , p ( Ω ) u W 1 , p ( Ω ) 1 meas ( Ω ) Ω u d x W 1 , p ( Ω ) ,
which implies that
u W 1 , p ( Ω ) u 1 meas ( Ω ) Ω u d x W 1 , p ( Ω ) + const .
Hence,
u V u 1 meas ( Ω ) Ω u d x V + const .
(3.3)

In view of (3.2) and (3.3), we have shown that for u V , u V + is equivalent to u 1 meas ( Ω ) Ω u d x V + .

Next, we shall show that A satisfies (3.1). In fact, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ15_HTML.gif
(3.4)
If 1 < p < 2 , then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ16_HTML.gif
(3.5)
From (3.2) and (3.3), we know that
0 T Ω | u | p d x d t 1 C p + 1 u 1 meas ( Ω ) Ω u d x V p 1 C p + 1 u V p + const .
Also,
0 T Ω C ( x , t ) p 2 d x d t C ( x , t ) V p < + .
It follows from (3.5) that
0 T Ω ( C ( x , t ) + | u | 2 ) p 2 2 u , u d x d t u V + ε 0 T Ω | u | q d x d t u V + ,

as u V + .

Moreover, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ17_HTML.gif
(3.6)

Therefore, it follows from (3.4), (3.5), and (3.6) that A satisfies (3.1) when 1 < p < 2 .

If p 2 , then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ18_HTML.gif
(3.7)
where M is a positive constant. We can easily see that
u 1 | Ω | Ω u d x V p u 0 V u 1 | Ω | Ω u d x V p p u V + ,
as u V + . Moreover, if 0 T Ω | u | q d x d t < + , then
ε ( 0 T Ω | u | q d x d t ) 1 q [ ( 0 T Ω | u | q d x d t ) 1 1 q u 0 V ] u V 0 ,
as u V + ; while if 0 T Ω | u | q d x d t + ,
ε ( 0 T Ω | u | q d x d t ) 1 q [ ( 0 T Ω | u | q d x d t ) 1 1 q u 0 V ] u V > 0 .

Hence, the right side of (3.7) tends to +∞ as u V + , which implies that A satisfies (3.1).

This completes the proof. □

Lemma 3.8 If w ( x , t ) Φ ( u ) , then w ( x , t ) = w ˜ ( x , t ) β x ( u ) a.e. on Γ × ( 0 , T ) .

Proof If w ( x , t ) Φ ( u ) , then from the definition of subdifferential, we have
0 T Γ φ x ( u | Γ ( x , t ) ) d Γ ( x ) d t 0 T Γ φ x ( w | Γ ( x , t ) ) d Γ ( x ) d t + 0 T Γ w ( x , t ) ( u w ) d Γ ( x ) d t ,

which implies that the result is true. □

We are now ready to prove the main result.

Theorem 3.1 The integro-differential equation (1.11) has a unique solution in V for f ( x , t ) V .

Proof First, we shall show the existence of a solution. Noting Lemmas 2.6, 3.6, 3.7 and 3.3, and by using Theorem 2.1, we know that there exists u ( x , t ) D ( S ) V such that
0 = S u + A u + Φ ( u ) .
(3.8)
Then we have for all w V ,
u w , S u V + u w , A u V + u w , Φ ( u ) V = 0 .
The definition of subdifferential implies that
u w , u t V + u w , a t Ω u d x V + u w , A u V + Φ ( u ) Φ ( w ) 0 .
From the definition of S, we have
u ( x , 0 ) = u ( x , T ) .
(3.9)
Moreover,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ21_HTML.gif
(3.10)
Let w = u ± ψ , where ψ C 0 ( Ω × ( 0 , T ) ) . Then we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equap_HTML.gif
From the properties of a generalized function, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ22_HTML.gif
(3.11)
Noting (3.10) again, by using Green’s formula, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equaq_HTML.gif
Then using (3.10), we obtain
Φ ( w ) Φ ( u ) 0 T Γ ϑ , ( C ( x , t ) + | u | 2 ) p 2 2 u ( w u ) | Γ d Γ ( x ) d t .

Thus, ϑ , ( C ( x , t ) + | u | 2 ) p 2 2 u Φ ( u ) .

In view of Lemma 3.8, we have ϑ , ( C ( x , t ) + | u | 2 ) p 2 2 u β x ( u ) a.e. on Γ × ( 0 , T ) . Combining it with (3.8) and (3.11), we know that (1.11) has a solution in V.

Next, we shall prove the uniqueness of the solution. Let u ( x , t ) and v ( x , t ) be two solutions of (1.11). By (3.8), we have
u v , ( A + Φ ) u ( A + Φ ) v V = u v , S u S v V 0

since S is monotone. But A + Φ is monotone too, so u v , S u S v V = 0 , which implies that u ( x , t ) = v ( x , t ) .

The proof is complete. □

Declarations

Acknowledgements

Li Wei is supported by the National Natural Science Foundation of China (No. 11071053), the Natural Science Foundation of Hebei Province (No. A2010001482) and the Project of Science and Research of Hebei Education Department (the second round in 2010).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Hebei University of Economics and Business
(2)
Department of Mathematics, Texas A&M University — Kingsville
(3)
Department of Mathematics, Faculty of Science, King Abdulaziz University
(4)
School of Electrical and Electronic Engineering, Nanyang Technological University

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