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

whose existence results in L p ( Ω ) (for various ranges of p) can be found in 1234; 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

was tackled in 567 and later generalized to one that contains a perturbation term | u | p − 2 u 89

{ − 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 .

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 ) ,

several generalizations have been investigated. These include 11121314

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 ,

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 .

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

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 .

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 .

Our discussion is based on some results on the ranges for maximal monotone operators and pseudo-monotone operators in 161718. 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).

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 A0 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

(i) for each x ∈ C , the image Ax is a nonempty closed and convex subset of X∗;

(ii) if {xn} 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 ) ;

(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 < + ∞ , Wm,p(Ω)↪Lq(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→2X∗ 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

(i) the mapping A : C → 2 X ∗ is a maximal monotone operator;

(ii) the mapping B : C → X ∗ is pseudo-monotone, bounded, and demi-continuous;

(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.

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 RN 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 RN, 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 514.

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 βx. 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 φx, 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 un, 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 φx 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

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 ) ,

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¯∈W1,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

which implies that A is everywhere defined and bounded.

If 1 < p < 2 , then for u,v∈V, we have

which also implies that A is everywhere defined and bounded.

Since Bp,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 Bp,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 → + ∞ ,

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,

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 .

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 .

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

If 1<p<2, then

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

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

If p≥2, then

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 ) .

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 ) .

Moreover,

Let w = u ± ψ , where ψ ∈ C 0 ∞ ( Ω × ( 0 , T ) ) . Then we have

From the properties of a generalized function, we get

Noting (3.10) again, by using Green’s formula, we have

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. □

The authors declare that they have no competing interests.

All authors approve the final manuscript.