JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 567241 10.1155/2014/567241 567241 Research Article Existence for Nonlinear Evolution Equations and Application to Degenerate Parabolic Equation Su Ning http://orcid.org/0000-0003-3682-3853 Zhang Li Primicerio Mario Department of Mathematical Sciences Tsinghua University Beijing 100084 China tsinghua.edu.cn 2014 2442014 2014 06 12 2013 05 03 2014 24 4 2014 2014 Copyright © 2014 Ning Su and Li Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider an abstract Cauchy problem for a doubly nonlinear evolution equation of the form d / d t 𝒜 u + u f t in V , t 0 ,   T , where V is a real reflexive Banach space, 𝒜 and are maximal monotone operators (possibly multivalued) from V to its dual V . In view of some practical applications, we assume that 𝒜 and are subdifferentials. By using the back difference approximation, existence is established, and our proof relies on the continuity of 𝒜 and the coerciveness of . As an application, we give the existence for a nonlinear degenerate parabolic equation.

1. Introduction

Let V be a real reflexive Banach space, and let 𝒜 , be maximal monotone operators (possibly multivalued) from V to its dual V . In this paper, we consider the abstract evolution equation: (1) d d t 𝒜 ( u ( t ) ) + ( u ( t ) ) f ( t ) t ( 0 , T ] , 𝒜 ( u ( 0 ) ) v 0 , where f : [ 0 , T ] V and v 0 V are given. Inspired by some practical applications (see ), 𝒜 and in our work are assumed to be subdifferentials of proper, convex, and lower semicontinuous functions on V .

During the past decades, the problem has been investigated in many papers, such as [1, 2, 616]. For the case that 𝒜 = I and is a subdifferential operator, the existence and uniqueness in the Hilbert space framework (i.e., V = H ) were established in [11, 13, 15, 16], and the unique solvability in the V - V setting was given by Akagi and Ôtani . Assuming that 𝒜 is continuous, is continuous and elliptic in some sense, Alt and Luckhaus proved the existence in , and Otto established the L 1 -contraction and uniqueness in . For the case that 𝒜 is Lipschitz continuous and is coercive, the existence theory was given in . In fact, if 𝒜 is Lipschitz continuous and invertible, the problem (1) can be rewritten as (2) d u d t + ( v ( t ) ) f ( t ) , v ( t ) 𝒜 - 1 ( u ( t ) ) , t ( 0 , T ] . Under the condition that 𝒜 - 1 is a bi-Lipschitz subdifferential operator, this problem was investigated and solved in . Then the result was extended to a more general case in , where 𝒜 - 1 is a maximal monotone operator.

More generally, both 𝒜 and are possibly nonlinear, and such equations are said to be doubly nonlinear. On the assumption that one of 𝒜 , is a subdifferential operator and the other is strongly monotone, the existence was established in . In addition, many practical applications (see ) suggested that both 𝒜 and are subdifferential operators. For the case that 𝒜 and are subdifferentials of functions on a Hilbert space, the existence was given in . Supposing that 𝒜 is a subdifferential operator in a Hilbert space H and is a subdifferential operator in a real reflexive Banach space V , respectively, Barbu  and Akagi [1, 7] obtained the existence with some appropriate assumptions imposed on .

In some papers (such as [6, 11, 12]), the problems investigated are time dependent; that is, (possibly together with 𝒜 ) is time dependent. In this paper, we aim to extend the first existence theory in  to the case that 𝒜 is only continuous but not Lipschitz continuous, and is coercive with some other appropriate conditions. Our basic assumptions and the existence theory are stated in Section 2, and the preliminaries are introduced in Section 3. In Section 4, we use the backward difference quotient to approximate the time derivative as in  and solve the problem by means of convex analysis and uniform estimation, in which we make good use of the properties of subdifferentials and maximal monotone operator. In Section 5, as an application of the abstract existence theorem, we give the existence for a nonlinear degenerate parabolic equation.

2. Basic Assumptions and Existence Theorem 2.1. Basic Assumptions

To state our assumptions clearly, we introduce some notations.

Let X be a real reflexive Banach space and let ( X ) be the set of all proper, convex, and lower semicontinuous functions ψ : X ( - , + ] , where proper means that ψ + .

For any ψ ( X ) , its subdifferential of ψ at u , denoted by X ψ ( u ) , is given by (3) X ψ ( u ) = { v X : v , x - u ψ ( x ) - ψ ( u ) , x D ( ψ ) } , where D ( ψ ) = { x X : ψ ( x ) < + } . Then, we define the subdifferential operator X ψ : X 2 X ; u X ψ ( u ) with the domain D ( X ψ ) = { x D ( ψ ) : X ψ ( x ) Ø } .

Let V be a real reflexive Banach space, and let H be a real Hilbert space, where V is densely and compactly embedded in H . Denote the injection by i : V H .

Our basic assumptions are as follows.

𝒜 = V ( Φ A i ) , where Φ A ( H ) , Φ A ( 0 ) 0 , and H Φ A is continuous.

= V Φ B , where Φ B ( V ) with D ( V Φ B ) Ø , satisfying that

Φ B is coercive; that is, (4) lim u V + Φ B ( u ) u V = + ;

there exists a nondecreasing function l ( · ) : , such that (5) v V l ( Φ B ( u ) ) , u D ( V Φ B ) , v V Φ B ( u ) .

f W 1,2 ( 0 , T ; V ) .

There exists u 0 int ( D ( 𝒜 ) ) D ( ) such that v 0 = 𝒜 ( u 0 ) .

Remark 1.

(1) Condition (A) implies that the operator A is single valued.

(2) Φ A ( 0 ) 0 is equivalent to Φ A ( 0 ) < + .

(3) (B1) implies the coerciveness of ; that is, (6) lim u V + v ( u ) v , u u V = + .

2.2. Existence Theorem Theorem 2.

Assume (A), (B), (F), and (V) are all satisfied. Then, there exists at least one solution triplet (7) u L ( 0 , T ; V ) , v L ( 0 , T ; V ) W 1,2 ( 0 , T ; V ) , w L ( 0 , T ; V ) , such that u is a solution of (1); that is, v ( t ) = 𝒜 ( u ( t ) ) , w ( t ) ( u ( t ) ) , and (8) d d t v ( t ) + w ( t ) = f ( t ) . Moreover, v ( t ) v 0 strongly in V as t 0 .

3. Preliminaries

The proofs related to this section can be found in [3, 1823].

3.1. Lower Semicontinuous Functions Lemma 3.

Let ψ ( X ) . Then ψ is bounded from below by an affine function; that is, there exist x 0 * X and β such that (9) ψ ( x ) x 0 * , x + β , x X .

Let ψ be a function from X to ( - , + ] , then its conjugate function ψ * , originally developed by Fenchel, is defined as (10) ψ * ( v ) : = sup { v , u - ψ ( u ) : u X } , v X .

Lemma 4.

Let X be a real reflexive Banach space.

For any ψ : X ( - , + ] , ψ * is convex and lower semicontinuous.

For any ψ ( X ) , ψ * is proper.

Remark 5.

Let ψ ( X ) with ψ ( 0 ) 0 . Then, ψ * ( v ) 0 for any v D ( ψ * ) . In fact, from the definition of ψ * , (11) ψ * ( v ) = sup { v , u - ψ ( u ) : u X } - ψ ( 0 ) 0 .

3.2. Maximal Monotone Operators

Let X be a real reflexive Banach space. An operator 𝒯 : X 2 X is called monotone, if (12) v 1 - v 2 , u 1 - u 2 0 , u j D ( 𝒯 ) , v j 𝒯 ( u j ) , j = 1,2 , where D ( 𝒯 ) = { x X : 𝒯 ( x ) Ø } . In addition, 𝒯 is called maximal if it has no proper monotone extension in X ; that is, for any x X and any y X , (13) v - y , u - x 0 , u D ( 𝒯 ) , v 𝒯 ( u ) only if y 𝒯 ( x ) .

Lemma 6.

Let X be a real reflexive Banach space, and let 𝒯 : X 2 X be maximal monotone. Let x i X and y i 𝒯 ( x i ) be such that x i x , y i y , and limsup y i , x i y , x . Then y 𝒯 ( x ) .

Lemma 7.

Let X be a reflexive Banach space. Let 𝒯 1 , 𝒯 2 : X 2 X be maximal monotone operators such that int ( D ( 𝒯 1 ) ) D ( 𝒯 2 ) Ø . Then 𝒯 1 + 𝒯 2 is maximal monotone from X to 2 X .

Lemma 8.

Let 𝒯 1 , 𝒯 2 : X 2 X be maximal monotone operators such that

𝒯 2 is regular; that is, for all x D ( 𝒯 2 ) and all y R ( 𝒯 2 ) , we have (14) sup { v - y , x - u : u X , v 𝒯 2 ( u ) } < + ;

D ( 𝒯 1 ) D ( 𝒯 2 ) Ø and R ( 𝒯 2 ) = X ;

𝒯 1 + 𝒯 2 is maximal monotone.

Then, R ( 𝒯 1 + 𝒯 2 ) = X .

3.3. Subdifferentials

For any ψ ( X ) , its subdifferential X ψ , defined as (3), has the following properties.

Lemma 9.

Let X be a reflexive Banach space and ψ ( X ) . Then,

X ψ : X 2 X is maximal monotone;

v X ψ ( u ) u X ψ * ( v ) ψ ( u ) + ψ * ( v ) = v , u .

Lemma 10.

Let X be a reflexive Banach space and A = X ψ , where ψ ( X ) . Then the following conditions are equivalent: (15) lim x V + , x D ( ψ ) ψ ( x ) x V = + ; R ( A ) = X , A - 1 i s b o u n d e d .

Let T > 0 and 1 < p < + , and let ψ be a function on [ 0 , T ] × X such that

there are two constants α and β , for all x V and all t [ 0 , T ] , (16) ψ ( t , x ) + α x X + β 0 ;

ψ ( t , · ) ( X ) for each t [ 0 , T ] and the function t ψ ( t , v ( t ) ) is measurable for each v L p ( 0 , T ; X ) .

Then, we can define a function ψ on L p ( 0 , T ; X ) as follows: (17) ψ ( u ) = { 0 T ψ ( t , u ( t ) ) d t , if ψ ( · , u ( · ) ) L 1 ( 0 , T ) , + , otherwise , which is proper, convex, and lower semicontinuous and ψ > - on L p ( 0 , T ; X ) . For any u L p ( 0 , T ; X ) , we call v ψ ( u ) in the sense of L p ( 0 , T ; X ) if v L p ( 0 , T ; X ) and v Y ψ ( u ) , where ( 1 / p ) + ( 1 / p ) = 1 and Y = L p ( 0 , T ; X ) . Then, we have the following conclusion (Proposition 1.1 of ).

Lemma 11.

Assume that for each t [ 0 , T ] and each z X with ψ ( t , z ) < , there exists a function v L p ( 0 , T ; X ) such that v ( t ) = z , ψ ( · , v ( · ) ) L 1 ( 0 , T ) , v is right-continuous at t , and (18) limsup s t ψ ( s , v ( s ) ) ψ ( t , z ) . Let u be a function in L p ( 0 , T ; X ) such that ψ ( · , u ( · ) ) L 1 ( 0 , T ) and let f be a function in L p ( 0 , T ; X ) . Then, f ψ ( u ) in the sense of L p ( 0 , T ; X ) , if and only if f ( t ) X ψ ( u ( t ) ) for a.e. t [ 0 , T ] .

Remark 12.

Assume T > 0 and ψ ( X ) . Let u L 2 ( 0 , T ; X ) and v L 2 ( 0 , T ; X ) . Then, v ψ ( u ) in the sense of L 2 ( 0 , T ; X ) , if and only if v ( t ) X ψ ( u ( t ) ) for a.e. t [ 0 , T ] .

Next, we introduce some chain rules of subdifferentials in different forms.

By the definition of ψ * and Lemma 9, we can easily verify the following chain rule in the form of difference quotient.

Lemma 13.

Let v ( t ) X ψ ( u ( t ) ) . Then, for each h > 0 , (19) - h ψ ( u ( t ) ) v ( t ) , - h u ( t ) , - h ψ * ( v ( t ) ) - h v ( t ) , u ( t ) , where - h denotes the backward difference operator, (20) - h u ( t ) = u ( t ) - u ( t - h ) h .

The following chain rule of integral form was proved in .

Lemma 14.

Assume p [ 1 , + ) , and ( 1 / p ) + ( 1 / q ) = 1 . Let X be a real reflexive Banach space and ψ ( X ) . Let u W 1 , p ( 0 , T ; X ) and w L q ( 0 , T ; X ) be such that w ( t ) X ψ ( u ( t ) ) almost everywhere in ( 0 , T ) . Then t ψ ( u ( t ) ) is absolutely continuous, and for all v L q ( 0 , T ; X ) with v ( t ) X ψ ( u ( t ) ) almost everywhere in ( 0 , T ) , (21) ψ ( u ( t ) ) - ψ ( u ( s ) ) = s t v , d u d t 0 s t T .

Lemma 15.

Let X be a real reflexive Banach space and let Y be a Hilbert space with X densely and compactly embedded in Y . Let Λ : X Y be a linear continuous operator, and assume that ψ ( Y ) is continuous at some point of R ( Λ ) (the range of Λ ). Then (22) X ( ψ Λ ) = Λ Y ψ Λ : X 2 X , where L is the dual operator of Λ .

Remark 16.

Since H Φ A is continuous and V is densely and compactly embedded in H , 𝒜 = V ( Φ A i ) = i H Φ A i is compact, where i is the injection from V to H .

Lemma 17.

Assume (A), (B), and (V) are satisfied. Then 𝒜 + is maximal monotone from V to 2 V and R ( 𝒜 + ) = V .

Proof.

Since int ( D ( 𝒜 ) ) D ( ) Ø , we get the maximal monotonicity of 𝒜 + from Lemmas 9 and 7.

To prove R ( 𝒜 + ) = V , we only need to verify that 𝒜 and satisfy Lemma 8. Applying (4) and Lemma 10, we can deduce that R ( ) = V , and then the proof is completed if we could show that = V Φ B is regular.

In fact, for any real reflexive Banach space X and any ψ ( X ) , X ψ is regular.

Take x D ( X ψ ) and y R ( X ψ ) . Since R ( X ψ ) D ( ψ * ) , (23) v - y , x - u ψ ( x ) - ψ ( u ) - y , x - u ψ ( x ) + ψ * ( y ) - y , x holds for any u D ( ψ ) and for any v X ψ ( u ) . Since the right-hand side of (23) is a constant independent of v and u , (24) sup { v - y , x - u : u D ( X ψ ) , v X ψ ( u ) } < + ; that is, X ψ is regular.

4. Proof of Theorem <xref ref-type="statement" rid="thm2.1">2</xref>

In this section, to prove Theorem 2, we use the backward difference to approximate the time derivative. Since we can establish the solvability of the resulting approximate equations from Lemma 17, then, combining convex analysis and uniform estimation, we verify the existence.

4.1. Approximate Problems and Approximate Solutions

Let N be a positive integer, and h = T / N .

To prove Theorem 2, we approximate the time derivative in (1) by - h and approximate f by f h : (25) f h ( t ) = 1 h ( k - 1 ) h k h f ( s ) d s , ( k - 1 ) h < t k h , k = 1,2 , , N . These lead to the approximate problem (26) - h 𝒜 ( u ( t ) ) + ( u ( t ) ) f h ( t ) in V , t ( 0 , T ] , 𝒜 ( u h ( t ) ) = v 0 , t ( - h , 0 ] , and we can solve the solution u h ( t ) inductively for t ( - h , k h ] , k = 0,1 , , N , as follows.

For t ( - h , 0 ] , we set u h ( t ) = u 0 , v h ( t ) = v 0 = 𝒜 ( u 0 ) . Suppose that we have a solution of (26) with T = k h , 0 k < N , which implies that we have a solution triplet (27) u h ( t ) V , v h ( t ) = 𝒜 ( u h ( t ) ) , w h ( t ) ( u h ( t ) ) , t ( - h , k h ] . Consider the problem (26) with T = ( k + 1 ) h , which is equivalent to (28) 𝒜 ( u h ( t ) ) + h ( u h ( t ) ) h f h ( t ) + v h ( t - h ) in V , - h < t ( k + 1 ) h . Since for t ( k h , ( k + 1 ) h ] , h f h ( t ) + v ( t - h ) V , by Lemma 17, this problem has at least one solution u h ( t ) for t ( k h , ( k + 1 ) h ] . Then we can solve u h ( t ) inductively for t ( - h , k h ] , k = 0,1 , , N , and consequently the problem (26) has at least one solution triplet: (29) u h ( t ) V , v h ( t ) = 𝒜 ( u h ( t ) ) , w h ( t ) ( u h ( t ) ) , such that (30) - h v h ( t ) + w h ( t ) = f h ( t ) in V , 0 < t T , (31) v h ( 0 ) = v 0 . Obviously, the triplet ( u h ( t ) , v h ( t ) , w h ( t ) ) is piecewise constant; that is, the triplet is constant in each interval ( ( k - 1 ) h , k h ] , k = 0,1 , , N .

In the following, we aim to obtain some uniform estimates on the approximation solutions (see Section 4.2) and then solve the problem (1) by taking the limit of an appropriate subsequence (see Section 4.3).

4.2. Uniform Estimates Lemma 18.

There exists a constant C > 0 , such that (32) sup 0 t T u h ( t ) V C , (33) sup 0 t T w h ( t ) V C , (34) - h v h L 2 ( 0 , T ; V ) C .

Proof.

Applying (30), for any τ = k h , 1 k N , we have (35) 0 τ - h v h ( t ) , - h u h ( t ) + 0 τ w h ( t ) , - h u h ( t ) = 0 τ f h ( t ) , - h u h ( t ) . In view of maximal monotonicity of subdifferentials, we have (36) 0 τ - h v h ( t ) , - h u h ( t ) 0 . By virtue of Lemma 13, the second term in (35) could be estimated as follows: (37) 0 τ w h ( t ) , - h u h ( t ) 0 τ    - h Φ B ( u h ( t ) ) d t = Φ B ( u h ( τ ) ) - Φ B ( u 0 ) . As for the third term, applying integration by parts, we have (38) 0 τ f h ( t ) , - h u h ( t ) = f h ( τ ) , u h ( τ ) - f h ( h ) , u h ( 0 ) - 0 τ - h h f h ( t ) , u h ( t ) d t . Since f h f in W 1,2 ( 0 , T ; V ) as h 0 and the embedding W 1,2 ( 0 , T ; V ) C ( [ 0 , T ] ; V ) is continuous, there exists a constant C > 0 such that (39) 0 τ f h ( t ) , - h u h ( t ) C + C sup 0 t T u h ( t ) V . Then, from the assumption on initial value (see (V) Section 2.1), it follows that (40) sup 0 t T Φ B ( u h ( t ) ) C + C sup 0 t T u h ( t ) V . From the coercivity of Φ B (see (4)), we get (32) and (33) from (40) and (5). Applying - h v h ( t ) = f h - w h , we get (34).

4.3. Completion of Proof Lemma 19.

There exist u L ( 0 , T ; V ) , v ( t ) L ( 0 , T ; V ) W 1,2 ( 0 , T ; V ) , and w ( t ) L ( 0 , T ; V ) such that for almost all t [ 0 , T ] , v ( t ) = 𝒜 ( u ( t ) ) , w ( t ) ( u ( t ) ) , and (41) d d t v ( t ) + w ( t ) = f ( t ) i n V . Moreover, v ( t ) v 0 strongly in V as t 0 .

Proof.

From (32) and (33), there exist u L ( 0 , T ; V ) and w L ( 0 , T ; V ) such that (42) u h ( t ) u weakly in L 2 ( 0 , T ; V ) , (43) w h ( t ) w weakly in L 2 ( 0 , T ; V ) .

Since 𝒜 is compact from V to V (see Remark 16) and u h is uniformly bounded in L ( 0 , T ; V ) , v h ( t ) is precompact in V for each t [ 0 , T ] . Then combining (34), there exist a subsequence of v h (still denoted by v h ) and v L ( 0 , T ; V ) such that (44) v h ( t ) v strongly    in    C ( 0 , T ; V ) . In addition, from (34), there exists χ L ( 0 , T ; V ) such that (45) - h v h ( t ) χ weakly in L 2 ( 0 , T ; V ) , and we can easily testify that χ = d v / d t combining (44).

In virtue of (30), for any x L 2 ( 0 , T ; V ) , we have (46) 0 T - h v h ( t ) , x d t + 0 T w h ( t ) , x d t = 0 T f h ( t ) , x d t . Applying (43) and (45) and letting h 0 , it follows that (47) 0 T d d t v ( t ) , x d t + 0 T w ( t ) , x d t = 0 T f ( t ) , x d t , which implies that for almost all t ( 0 , T ] , (48) d d t v ( t ) + w ( t ) = f ( t )    in    V .

To complete the proof, we need to show that v = 𝒜 ( u ) , w ( u ) for almost all t [ 0 , T ] . Since 𝒜 , are maximal monotone operators, then, combining Lemmas 6 and 11, it suffices to prove that (49) limsup h 0 0 T v h ( t ) , u h ( t ) d t 0 T v ( t ) , u ( t ) d t , (50) limsup h 0 0 T w h ( t ) , u h ( t ) d t 0 T w ( t ) , u ( t ) d t .

In view of (42) and (44), we have (51) lim h 0 0 T v h ( t ) , u h ( t ) d t = 0 T v ( t ) , u ( t ) d t , which implies (49).

Applying (30), we have (52) 0 T w h ( t ) , u h ( t ) d t = 0 T f h ( t ) , u h ( t ) d t - 0 T - h v h ( t ) , u h ( t ) d t . Since f h f strongly in L 2 ( 0 , T ; V ) and u h u weakly in L 2 ( 0 , T ; V ) , we have (53) lim h 0 0 T f h ( t ) , u h ( t ) d t = 0 T f ( t ) , u ( t ) d t . For the second term, applying Lemma 13, we have (54) liminf h 0 0 T - h v h ( t ) , u h ( t ) d t liminf h 0 0 T - h Φ A * ( v h ( t ) ) d t Φ A * ( v ( T ) ) - Φ A * ( v 0 ) , since v h ( T ) v ( T ) strongly in V and Φ A * is lower semicontinuous. Moreover, from Lemma 14, we can easily get (55) Φ A * ( v ( T ) ) - Φ A * ( v 0 ) = 0 T d d t v ( t ) , u ( t ) d t . Therefore, (56) limsup h 0 0 T w h ( t ) , u h ( t ) d t 0 T f ( t ) - d d t v ( t ) , u ( t ) d t . On the other hand, applying (47) on u , we have (57) 0 T f ( t ) - d d t v ( t ) , u ( t ) d t = 0 T w ( t ) , u ( t ) d t . Consequently, we obtain (50).

As the end of the proof, since v h v in C ( 0 , T ; V ) and (58) v h ( t ) - v 0 V 0 t - h v h ( τ ) V d τ C t 0 , as t 0 , we have v ( t ) v 0 strongly in V as t 0 .

5. Application to an Initial Boundary Value Problem

The abstract existence can be applied to many models in fluid mechanics (see [26, 27]). We shall illustrate the application of Theorem 2 to establish the existence of a solution to a nonlinear parabolic initial-boundary-value problem with nonlinear degenerate terms under the time derivative. This problem includes a nonlinear dynamic boundary condition.

Let Ω be a bounded domain in 3 with smooth boundary Γ = Ω and Γ has the partition Γ = Γ 1 Γ 2 . Consider (59) t a ( u ) - i = 1 3 x i ( | u | p - 2 u x i ) = f ( x , t ) , ( x , t ) Ω × ( 0 , T ] , t c ( u ) + | u | p - 2 u · n + g ( x ) | u | r - 2 u = 0 , ( x , t ) Γ 1 × ( 0 , T ] , u = 0 , ( x , t ) Γ 2 × ( 0 , T ] , a ( u ) = a 0 ( x ) , ( x , t ) Ω × { 0 } , c ( u ) = c 0 ( x ) , ( x , t ) Γ 1 × { 0 } , where 2 p < + , 1 r p , a ( · ) and c ( · ) are continuous and nondecreasing and g ( x ) L ( Ω ) with g ( x ) g 0 > 0 .

Remark 20.

For the case m e a s ( Γ 1 ) > 0 , the problem (59) cannot be covered by .

5.1. Formulation of Abstract Form

Let H = L 2 ( Ω ) L 2 ( Γ 1 ) and V = { u W 1 , p ( Ω ) : u = 0          o n          Γ 2 } , equipped with the norms u V = u L p ( Ω ) + u L p ( Γ 1 ) and ( u 1 , u 2 ) H = u 1 L 2 ( Ω ) + u 2 L 2 ( Γ 1 ) , respectively. Then V is embedded in H densely and compactly, and denote the injection by i . Assume that a ( · ) , c ( · ) : are nondecreasing and continuous, and ϕ a , ϕ c : ( - , + ] satisfy (60) ϕ a ( x ) = 0 x a ( s ) d s , ϕ c ( x ) = 0 x c ( s ) d s . Define (61) Φ a ( u ) = Ω ϕ a ( u ) d x , u L 2 ( Ω ) , Φ c ( u ) = Γ 1 ϕ c ( u ) d s , u L 2 ( Γ 1 ) , Φ A ( u ) = Φ a ( u 1 ) + Φ c ( u 2 ) , u = ( u 1 , u 2 ) H , Φ B ( u ) = 1 p Ω | u | p d x + 1 r Γ 1 g ( x ) | u | r d s , u V . Let 𝒜 = V ( Φ A i ) and = V Φ B ; then v V , (62) 𝒜 ( u ) , v = Ω a ( u ) v d x + Γ 1 c ( u ) v d s , ( u ) , v = Ω | u | p - 2 u v d x + Γ 1 g ( x ) | u | r - 2 u v d s , F ( t ) , v = Ω f ( x , t ) v d x . Assuming a 0 L 2 ( Ω ) and v 0 L 2 ( Γ 1 ) , the problem (59) can be rewritten as (63) ( CP ) { d d t 𝒜 ( u ) + ( u ) F ( t ) t ( 0 , T ) , 𝒜 ( u ) v 0 t = 0 , where v 0 = ( a 0 , c 0 ) H V .

5.2. Existence of Solutions

Applying the existence theorem of abstract form, we could claim the solvability of the problem (59), by some appropriate assumptions.

a ( · ) , c ( · ) : are nondecreasing and continuous.

( u 1 , u 2 ) H = L 2 ( Ω ) L 2 ( Γ 1 ) , a ( u 1 ) L 2 ( Ω ) , and c ( u 2 ) L 2 ( Γ 1 ) .

g L ( Γ 1 ) with g g 0 > 0 .

Assume that (H1)–(H3) hold; then Φ A and Φ B satisfy Theorem 2. Therefore, for any v 0 and F satisfying Theorem 2, the abstract problem (63) is solvable, which implies the existence for the solution of (59) as follows.

Theorem 21.

Assume (H1)–(H3). Let 2 p < + , 1 r p , and p = p / ( p - 1 ) . Assume that f W 1,2 ( 0 , T ; W - 1 , p ( Ω ) ) . Assume ( a 0 , c 0 ) L 2 ( Ω ) L 2 ( Γ 1 ) satisfying (64) u 0 { u W 1 , p ( Ω ) : u = 0    o n    Γ 2 } s u c h t h a t a ( u 0 ) = a 0 , c ( u 0 ) = c 0 . Then there exists at least one solution u ( x , t ) of (59); that is, u is a measurable function such that (65) u L ( 0 , T ; W 1 , p ( Ω ) ) , a ( u ) L ( 0 , T ; W - 1 , p ( Ω ) ) W 1,2 ( 0 , T ; W - 1 , p ( Ω ) ) , c ( u ) L ( 0 , T ; L p ( Γ 1 ) ) W 1,2 ( 0 , T ; L p ( Γ 1 ) ) , and for a. e. t ( 0 , T ) , (66) Ω t a ( u ) ϕ d x + Ω | u | p - 2 u · ϕ d x + Γ 1 ( t c ( u ) + g ( x ) | u | p - 2 u ) ϕ d s = Ω f ( x , t ) ϕ d x for any ϕ W 1 , p ( Ω ) . Moreover, a ( u ( t ) ) a 0 strongly in W - 1 , p ( Ω ) and c ( u ( t ) ) c 0 strongly in L p ( Γ 1 ) as t 0 .

Remark 22.

More generally, instead of the boundary condition on Γ 1 in the problem (59), we assume that (see ) (67) t c ( u ) + | u | p - 2 u · n + b ( u ) + g ( x ) | u | r - 2 u 0 , ( x , t ) Γ 1 × ( 0 , T ] , where b ( · ) is multivalued and maximal monotone. Assume that there exists a proper, convex, lower semicontinuous function ϕ b defined on with ϕ b ( 0 ) = 0 such that b ( · ) = ϕ b ( · ) . Define (68) Φ b ( u ) = Γ 1 ϕ b ( u ) d s u V , and make a modification of Φ B as follows: (69) Φ B ( u ) = 1 p Ω | u | p d x + 1 r Γ 1 g ( x ) | u | r d s + Φ b ( u ) u V . Then the problem can be solved by imposing some appropriate assumptions on b . For a simple case, we could suppose that

there exists a constant b 0 , such that y b 0 for any x , and y b ( x ) ;

there exists a nondecreasing function l b ( · ) : such that (70) b ( u ) L p ( Γ 1 ) l b ( Φ b ( u ) ) , u L p ( Γ 1 ) .

Then Φ B satisfy Theorem 2. In addition, (b1) and (b2) can be satisfied by extensive functions. For example, for r 0 , set (71) b ( s ) = { 1 , s > r 0 , [ - 1,1 ] , s = r 0 , - 1 , s < r 0 , and Φ b ( u ) = Γ 1 | u - r 0 | d s .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported partially by NSFC under Grant no. 11271218.

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