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

  • Li Wei1,

    Affiliated with

    • Ravi P Agarwal2, 3Email author and

      Affiliated with

      • Patricia JY Wong4

        Affiliated with

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq1_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ1_HTML.gif
        (1.1)
        whose existence results in L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq2_HTML.gif (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ2_HTML.gif
        (1.2)
        was tackled in [57] and later generalized to one that contains a perturbation term | u | p 2 u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq3_HTML.gif [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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ3_HTML.gif
        (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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ4_HTML.gif
        (1.4)
        several generalizations have been investigated. These include [1114]
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ5_HTML.gif
        (1.5)
        http://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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ7_HTML.gif
        (1.7)

        where 0 C ( x ) L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq4_HTML.gif, 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ8_HTML.gif
        (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ9_HTML.gif
        (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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ10_HTML.gif
        (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 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq5_HTML.gif, where 2 p < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq6_HTML.gif, 1 q < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq7_HTML.gif if p N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq8_HTML.gif, and 1 q 2 N p N p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq9_HTML.gif if p < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq10_HTML.gif.

        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ11_HTML.gif
        (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq11_HTML.gif. We use ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq12_HTML.gif to denote the generalized duality pairing between X and X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq11_HTML.gif. 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq13_HTML.gif 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 ( , + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq14_HTML.gif, Φ is not identically +∞ such that Φ ( ( 1 λ ) x + λ y ) ( 1 λ ) Φ ( x ) + λ Φ ( y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq15_HTML.gif, whenever x , y X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq16_HTML.gif and 0 λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq17_HTML.gif.

        The function Φ : X ( , + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq18_HTML.gif is said to be lower-semicontinuous on X [17] if lim inf y x Φ ( y ) Φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq19_HTML.gif for any x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq20_HTML.gif.

        Given a proper convex function Φ on X and a point x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq20_HTML.gif, we denote by Φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq21_HTML.gif the set of all x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq22_HTML.gif such that Φ ( x ) Φ ( y ) + ( x y , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq23_HTML.gif for every y X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq24_HTML.gif. Such elements x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq25_HTML.gif are called subgradients of Φ at x, and Φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq26_HTML.gif is called the subdifferential of Φ at x [17].

        A mapping T : D ( T ) = X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq27_HTML.gif is said to be demi-continuous on X if w - lim n T x n = T x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq28_HTML.gif for any sequence { x n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq29_HTML.gif strongly convergent to x in X. A mapping T : D ( T ) = X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq27_HTML.gif is said to be hemi-continuous on X if w - lim t 0 T ( x + t y ) = T x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq30_HTML.gif for any x , y X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq31_HTML.gif [17].

        With each multi-valued mapping A : X 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq32_HTML.gif, we associate the subset A 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq33_HTML.gif as follows [17]:
        A 0 x = { y A x : y = | A x | } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equa_HTML.gif

        where | A x | : = inf { z : z A x } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq34_HTML.gif. If X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq11_HTML.gif is strictly convex, then D ( A ) = D ( A 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq35_HTML.gif and A 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq33_HTML.gif is single-valued, which in this case is called the minimal section of A.

        A multi-valued mapping B : X 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq36_HTML.gif is said to be monotone [18] if its graph G ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq37_HTML.gif is a monotone subset of X × X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq38_HTML.gif in the sense that ( u 1 u 2 , w 1 w 2 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq39_HTML.gif for any [ u i , w i ] G ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq40_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq41_HTML.gif. The monotone operator B is said to be maximal monotone if G ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq37_HTML.gif is not properly contained in any other monotone subsets of X × X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq38_HTML.gif.

        Definition 2.1 [18]

        Let C be a closed convex subset of X, and let A : C 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq42_HTML.gifbe a multi-valued mapping. Then A is said to be a pseudo-monotone operator provided that
        1. (i)

          for each x C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq43_HTML.gif, the image Ax is a nonempty closed and convex subset of X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq11_HTML.gif;

           
        2. (ii)
          if { x n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq29_HTML.gif is a sequence in C converging weakly to x C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq44_HTML.gif and if f n A x n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq45_HTML.gif is such that lim sup n ( x n x , f n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq46_HTML.gif, then to each element y C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq47_HTML.gif, there corresponds an f ( y ) A x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq48_HTML.gif with the property that
          ( x y , f ( y ) ) lim inf n ( x n x , f n ) ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equb_HTML.gif
           
        3. (iii)

          for each finite-dimensional subspace F of X, the operator A is continuous from C F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq49_HTML.gif to X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq11_HTML.gif in the weak topology.

           

        Lemma 2.1 [19]

        Let Ω be a bounded conical domain in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq50_HTML.gif. If m p > N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq51_HTML.gif, then W m , p ( Ω ) C B ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq52_HTML.gif; if 0 < m p N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq53_HTML.gif and q = N p N m p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq54_HTML.gif, then W m , p ( Ω ) L q ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq55_HTML.gif; if m p = N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq56_HTML.gif and p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq57_HTML.gif, then for 1 q < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq58_HTML.gif, W m , p ( Ω ) L q ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq55_HTML.gif.

        Lemma 2.2 [18]

        If B : X 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq59_HTML.gif is an everywhere defined, monotone, and hemi-continuous operator, then B is maximal monotone. If B : X 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq59_HTML.gif is a maximal monotone operator such that D ( B ) = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq60_HTML.gif, then B is pseudo-monotone.

        Lemma 2.3 [18]

        If X is a Banach space and Φ : X ( , + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq61_HTML.gif is a proper convex and lower-semicontinuous function, then Φ is maximal monotone from X to X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq11_HTML.gif.

        Lemma 2.4 [18]

        If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq62_HTML.gif and B 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq63_HTML.gif are two maximal monotone operators in X such that int D ( B 1 ) D ( B 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq64_HTML.gif, then B 1 + B 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq65_HTML.gif is maximal monotone.

        Lemma 2.5 [20]

        Let X and its dual X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq11_HTML.gif be strictly convex Banach spaces. Suppose S : D ( S ) X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq66_HTML.gif is a closed linear operator and S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq67_HTML.gif is the conjugate operator of S. If ( u , S u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq68_HTML.gif u D ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq69_HTML.gif and ( v , S v ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq70_HTML.gif v D ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq71_HTML.gif, then S is a maximal monotone operator possessing a dense domain.

        Lemma 2.6 [18]

        Any hemi-continuous mapping T : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq72_HTML.gif is demi-continuous on Int D ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq73_HTML.gif.

        Theorem 2.1 [16]

        Let X be a real reflexive Banach space with X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq11_HTML.gif being its dual space. Let C be a nonempty closed convex subset of X. Assume that
        1. (i)

          the mapping A : C 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq74_HTML.gif is a maximal monotone operator;

           
        2. (ii)

          the mapping B : C X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq75_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq76_HTML.gif, i.e., there exists an element u 0 C D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq77_HTML.gif and a number r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq78_HTML.gif such that ( u u 0 , B u ) > ( u u 0 , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq79_HTML.gif for all u C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq80_HTML.gif with u > r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq81_HTML.gif.

           

        Then the equation b A u + B u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq82_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equc_HTML.gif
        Let V = L p ( 0 , T ; W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq83_HTML.gif and V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq84_HTML.gif be the dual space of V. The duality pairing between V and V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq84_HTML.gif will be denoted by , V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq85_HTML.gif. The norm in V will be denoted by V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq86_HTML.gif, which is defined by
        u V = ( 0 T u ( t ) W 1 , p ( Ω ) p d t ) 1 p , u ( x , t ) V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equd_HTML.gif
        Let W = L q ( 0 , T ; W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq87_HTML.gif and W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq88_HTML.gif be the dual space of W. The norm in W will be denoted by W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq89_HTML.gif, which is defined by
        v W = ( 0 T v ( t ) W 1 , p ( Ω ) q d t ) 1 q , v ( x , t ) W . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Eque_HTML.gif

        In the integro-differential equation (1.11), Ω is a bounded conical domain of a Euclidean space R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq50_HTML.gif where N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq90_HTML.gif, Γ is the boundary of Ω with Γ C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq91_HTML.gif [5], ϑ denotes the exterior normal derivative to Γ. Here, | | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq92_HTML.gif and , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq93_HTML.gif denote the Euclidean norm and the inner-product in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq50_HTML.gif, respectively. Also, 0 C ( x , t ) L p ( 0 , T ; W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq94_HTML.gif, f ( x , t ) V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq95_HTML.gif is a given function, T and a are positive constants, and ε is a nonnegative constant. Moreover, β x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq96_HTML.gif is the subdifferential of φ x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq97_HTML.gif, where φ x = φ ( x , ) : R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq98_HTML.gif for x Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq99_HTML.gif, and φ : Γ × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq100_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq99_HTML.gif, φ x = φ ( x , ) : R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq101_HTML.gif is a proper, convex, and lower-semicontinuous function and φ x ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq102_HTML.gif.

        Assumption 3 0 β x ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq103_HTML.gif and for each t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq104_HTML.gif, the function x Γ ( I + λ β x ) 1 ( t ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq105_HTML.gif is measurable for λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq106_HTML.gif.

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

        Lemma 3.1 Define the function Φ : V R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq107_HTML.gif by
        Φ ( u ) = 0 T Γ φ x ( u | Γ ( x , t ) ) d Γ ( x ) d t , u V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equf_HTML.gif

        Then Φ is a proper, convex, and lower-semicontinuous mapping on V. Therefore, Φ : V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq108_HTML.gif, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq109_HTML.gif, the function x Γ β x 0 ( s ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq110_HTML.gif is measurable, where β x 0 ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq111_HTML.gif denotes the minimal section of β x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq96_HTML.gif. Since for all s 1 , s 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq112_HTML.gif 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 } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equg_HTML.gif

        it implies that for u V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq113_HTML.gif, the function φ x ( u | Γ ( x , t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq114_HTML.gif is measurable on Γ. Then from the property of φ x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq97_HTML.gif, we know that Φ is proper and convex on V.

        To see that Φ is lower-semicontinuous on V, let u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq115_HTML.gif in V. We may assume that there exists a subsequence of u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq116_HTML.gif, for simplicity, we still denote it by u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq116_HTML.gif, such that u n | Γ ( x , t ) u | Γ ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq117_HTML.gif for x Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq118_HTML.gif and t ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq119_HTML.gif a.e. This yields
        φ x ( u | Γ ( x , t ) ) lim inf n φ x ( u n | Γ ( x , t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equh_HTML.gif
        for all x Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq118_HTML.gif and each t ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq119_HTML.gif a.e. since φ x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq97_HTML.gif is lower-semicontinuous for each x Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq118_HTML.gif. It then follows from Fatou’s lemma that for each t ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq119_HTML.gif,
        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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equi_HTML.gif

        So, Φ ( u ) lim inf n Φ ( u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq120_HTML.gif whenever u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq121_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq122_HTML.gif by
        S u = u t + a t Ω u d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equj_HTML.gif

        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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq123_HTML.gif, integrating by parts gives
        http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq124_HTML.gif, where D ( S ) = { w V : w t V , w ( x , 0 ) = w ( x , T ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq125_HTML.gif.

        For u ( x , t ) D ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq126_HTML.gif, 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equl_HTML.gif
        which implies that
        0 T Ω u t u ( x , t ) d x d t = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equm_HTML.gif
        Similarly, for u ( x , t ) D ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq127_HTML.gif,
        http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equo_HTML.gif
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equp_HTML.gif

        In the same manner, we have w , S w V = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq128_HTML.gif for w D ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq129_HTML.gif. 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq130_HTML.gif is maximal monotone.

        Lemma 3.4 [14]

        Define the mapping B p , q : W 1 , p ( Ω ) ( W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq131_HTML.gif 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equq_HTML.gif

        Then B p , q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq132_HTML.gif is maximal monotone.

        Lemma 3.5 [14]

        Let X 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq133_HTML.gif denote the closed subspace of all constant functions in W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq134_HTML.gif. Let X be the quotient space W 1 , p ( Ω ) X 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq135_HTML.gif. For u ¯ W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq136_HTML.gif, define the mapping P : W 1 , p ( Ω ) X 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq137_HTML.gif by
        P u ¯ = 1 meas ( Ω ) Ω u ¯ d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equr_HTML.gif
        Then, there is a constant C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq138_HTML.gif such that for every u ¯ W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq136_HTML.gif,
        u ¯ P u ¯ L p ( Ω ) C u ¯ ( L p ( Ω ) ) N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equs_HTML.gif

        Here meas ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq139_HTML.gif denotes the measure of Ω.

        Definition 3.1 Define A : V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq140_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equt_HTML.gif

        Lemma 3.6 The mapping A : V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq141_HTML.gif 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq142_HTML.gif when p > N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq143_HTML.gif, and W 1 , p ( Ω ) L q ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq144_HTML.gif when p = N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq145_HTML.gif. If p < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq10_HTML.gif, then W 1 , p ( Ω ) L N p N p ( Ω ) L p ( Ω ) L q ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq146_HTML.gif since 1 < q p < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq147_HTML.gif. Thus, for all w ¯ W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq148_HTML.gif, w ¯ L q ( Ω ) k w ¯ W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq149_HTML.gif, where k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq150_HTML.gif is a constant. Therefore, for u , v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq151_HTML.gif, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equu_HTML.gif
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equv_HTML.gif

        Moreover, since 1 < q p < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq152_HTML.gif, then L p ( 0 , T ; W 1 , p ( Ω ) ) L q ( 0 , T ; W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq153_HTML.gif, which implies that u W u V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq154_HTML.gif and v W v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq155_HTML.gif for u , v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq156_HTML.gif.

        If p 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq157_HTML.gif, then for u , v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq158_HTML.gif, we have
        http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq159_HTML.gif, then for u , v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq158_HTML.gif, we have
        http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq132_HTML.gif is monotone, we can easily see that for u , v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq151_HTML.gif,
        u v , A u A v V = 0 T ( u v , B p , q u B p , q v ) d t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equy_HTML.gif

        which implies that A is monotone.

        To show that A is hemi-continuous, it suffices to show that for any u , v , w V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq160_HTML.gif and k [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq161_HTML.gif, w , A ( u + k v ) A u V 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq162_HTML.gif, as k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq163_HTML.gif. Noting the fact that B p , q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq132_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equz_HTML.gif

        Hence, A is hemi-continuous.

        This completes the proof. □

        Lemma 3.7 The mapping A : V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq141_HTML.gif satisfies that for u D ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq164_HTML.gif,
        u u 0 , A u V u V + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ12_HTML.gif
        (3.1)

        as u V + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq165_HTML.gif in V.

        Proof First, we shall show that for u V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq166_HTML.gif,
        u V + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equaa_HTML.gif
        is equivalent to
        u 1 meas ( Ω ) Ω u d x V + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equab_HTML.gif
        In fact, from Lemma 3.5, we know that for u V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq113_HTML.gif,
        u 1 meas ( Ω ) Ω u d x L p ( Ω ) C u ( L p ( Ω ) ) N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equac_HTML.gif
        where C is a positive constant. Thus,
        http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ13_HTML.gif
        (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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equae_HTML.gif
        which implies that
        u W 1 , p ( Ω ) u 1 meas ( Ω ) Ω u d x W 1 , p ( Ω ) + const . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equaf_HTML.gif
        Hence,
        u V u 1 meas ( Ω ) Ω u d x V + const . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ14_HTML.gif
        (3.3)

        In view of (3.2) and (3.3), we have shown that for u V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq166_HTML.gif, u V + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq167_HTML.gif is equivalent to u 1 meas ( Ω ) Ω u d x V + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq168_HTML.gif.

        Next, we shall show that A satisfies (3.1). In fact, we have
        http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq159_HTML.gif, then
        http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equag_HTML.gif
        Also,
        0 T Ω C ( x , t ) p 2 d x d t C ( x , t ) V p < + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equah_HTML.gif
        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 + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equai_HTML.gif

        as u V + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq167_HTML.gif.

        Moreover, we have
        http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq169_HTML.gif.

        If p 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq157_HTML.gif, then
        http://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 + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equaj_HTML.gif
        as u V + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq167_HTML.gif. Moreover, if 0 T Ω | u | q d x d t < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq170_HTML.gif, 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equak_HTML.gif
        as u V + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq167_HTML.gif; while if 0 T Ω | u | q d x d t + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq171_HTML.gif,
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equal_HTML.gif

        Hence, the right side of (3.7) tends to +∞ as u V + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq167_HTML.gif, which implies that A satisfies (3.1).

        This completes the proof. □

        Lemma 3.8 If w ( x , t ) Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq172_HTML.gif, then w ( x , t ) = w ˜ ( x , t ) β x ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq173_HTML.gif a.e. on Γ × ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq174_HTML.gif.

        Proof If w ( x , t ) Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq172_HTML.gif, 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equam_HTML.gif

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq175_HTML.gif.

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq176_HTML.gif such that
        0 = S u + A u + Φ ( u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ19_HTML.gif
        (3.8)
        Then we have for all w V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq177_HTML.gif,
        u w , S u V + u w , A u V + u w , Φ ( u ) V = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equan_HTML.gif
        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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equao_HTML.gif
        From the definition of S, we have
        u ( x , 0 ) = u ( x , T ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equ20_HTML.gif
        (3.9)
        Moreover,
        http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq178_HTML.gif, where ψ C 0 ( Ω × ( 0 , T ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq179_HTML.gif. Then we have
        http://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
        http://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
        http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equar_HTML.gif

        Thus, ϑ , ( C ( x , t ) + | u | 2 ) p 2 2 u Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq180_HTML.gif.

        In view of Lemma 3.8, we have ϑ , ( C ( x , t ) + | u | 2 ) p 2 2 u β x ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq181_HTML.gif a.e. on Γ × ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq174_HTML.gif. 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq182_HTML.gif and v ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq183_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_Equas_HTML.gif

        since S is monotone. But A + Φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq184_HTML.gif is monotone too, so u v , S u S v V = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq185_HTML.gif, which implies that u ( x , t ) = v ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-131/MediaObjects/13661_2012_Article_232_IEq186_HTML.gif.

        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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