Existence for Nonlinear Evolution Equations and Application to Degenerate Parabolic Equation

Abstract

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

1. Introduction

Let be a real reflexive Banach space, and let , be maximal monotone operators (possibly multivalued) from to its dual . In this paper, we consider the abstract evolution equation: where and are given. Inspired by some practical applications (see [1–5]), and in our work are assumed to be subdifferentials of proper, convex, and lower semicontinuous functions on .

During the past decades, the problem has been investigated in many papers, such as [1, 2, 6–16]. For the case that and is a subdifferential operator, the existence and uniqueness in the Hilbert space framework (i.e., ) were established in [11, 13, 15, 16], and the unique solvability in the setting was given by Akagi and Ôtani [2]. Assuming that is continuous, is continuous and elliptic in some sense, Alt and Luckhaus proved the existence in [8], and Otto established the -contraction and uniqueness in [17]. For the case that is Lipschitz continuous and is coercive, the existence theory was given in [14]. In fact, if is Lipschitz continuous and invertible, the problem (1) can be rewritten as Under the condition that is a bi-Lipschitz subdifferential operator, this problem was investigated and solved in [11]. Then the result was extended to a more general case in [12], where is a maximal monotone operator.

More generally, both and are possibly nonlinear, and such equations are said to be doubly nonlinear. On the assumption that one of is a subdifferential operator and the other is strongly monotone, the existence was established in [10]. In addition, many practical applications (see [1–5]) suggested that both and are subdifferential operators. For the case that and are subdifferentials of functions on a Hilbert space, the existence was given in [6]. Supposing that is a subdifferential operator in a Hilbert space and is a subdifferential operator in a real reflexive Banach space , respectively, Barbu [9] 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 ) is time dependent. In this paper, we aim to extend the first existence theory in [14] to the case that 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 [8] 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 be a real reflexive Banach space and let be the set of all proper, convex, and lower semicontinuous functions , where proper means that .

For any , its subdifferential of at , denoted by , is given by where . Then, we define the subdifferential operator with the domain .

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

Our basic assumptions are as follows.(A), where , , and is continuous.(B), where with , satisfying that(B1) is coercive; that is, (B2)there exists a nondecreasing function , such that (F).(V)There exists such that .

Remark 1. (1) Condition (A) implies that the operator is single valued.
(2) is equivalent to .
(3) (B1) implies the coerciveness of ; that is,

2.2. Existence Theorem

Theorem 2. Assume (A), (B), (F), and (V) are all satisfied. Then, there exists at least one solution triplet such that is a solution of (1); that is, , , and Moreover, strongly in as .

3. Preliminaries

The proofs related to this section can be found in [3, 18–23].

3.1. Lower Semicontinuous Functions

Lemma 3. Let . Then is bounded from below by an affine function; that is, there exist and such that

Let be a function from to , then its conjugate function , originally developed by Fenchel, is defined as

Lemma 4. Let X be a real reflexive Banach space. (1)For any , is convex and lower semicontinuous.(2)For any , is proper.

Remark 5. Let with . Then, for any . In fact, from the definition of ,

3.2. Maximal Monotone Operators

Let be a real reflexive Banach space. An operator is called monotone, if where . In addition, is called maximal if it has no proper monotone extension in ; that is, for any and any , only if .

Lemma 6. Let be a real reflexive Banach space, and let be maximal monotone. Let and be such that , , and . Then .

Lemma 7. Let be a reflexive Banach space. Let , be maximal monotone operators such that . Then is maximal monotone from to .

Lemma 8. Let , be maximal monotone operators such that (a) is regular; that is, for all and all , we have (b) and ;(c) is maximal monotone.
Then, .

3.3. Subdifferentials

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

Lemma 9. Let be a reflexive Banach space and . Then, (1) is maximal monotone;(2).

Lemma 10. Let be a reflexive Banach space and , where . Then the following conditions are equivalent:

Let and , and let be a function on such that(1)there are two constants and , for all and all , (2) for each and the function is measurable for each . Then, we can define a function on as follows: which is proper, convex, and lower semicontinuous and on . For any , we call in the sense of if and , where and . Then, we have the following conclusion (Proposition 1.1 of [24]).

Lemma 11. Assume that for each and each with , there exists a function such that , , is right-continuous at , and Let be a function in such that and let be a function in . Then, in the sense of , if and only if for a.e. .

Remark 12. Assume and . Let and . Then, in the sense of , if and only if for a.e. .

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

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

Lemma 13. Let . Then, for each , where denotes the backward difference operator,

The following chain rule of integral form was proved in [25].

Lemma 14. Assume , and . Let X be a real reflexive Banach space and . Let and be such that almost everywhere in . Then is absolutely continuous, and for all with almost everywhere in ,

Lemma 15. Let be a real reflexive Banach space and let be a Hilbert space with densely and compactly embedded in . Let be a linear continuous operator, and assume that is continuous at some point of (the range of ). Then where is the dual operator of .

Remark 16. Since is continuous and is densely and compactly embedded in , is compact, where is the injection from to .

Lemma 17. Assume (A), (B), and (V) are satisfied. Then is maximal monotone from to and .

Proof. Since , we get the maximal monotonicity of from Lemmas 9 and 7.
To prove , we only need to verify that and satisfy Lemma 8. Applying (4) and Lemma 10, we can deduce that , and then the proof is completed if we could show that is regular.
In fact, for any real reflexive Banach space and any , is regular.
Take and . Since , holds for any and for any . Since the right-hand side of (23) is a constant independent of and , that is, is regular.

4. Proof of Theorem 2

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 be a positive integer, and .

To prove Theorem 2, we approximate the time derivative in (1) by and approximate by : These lead to the approximate problem and we can solve the solution inductively for , , as follows.

For , we set , . Suppose that we have a solution of (26) with , , which implies that we have a solution triplet Consider the problem (26) with , which is equivalent to Since for , + , by Lemma 17, this problem has at least one solution for . Then we can solve inductively for , , and consequently the problem (26) has at least one solution triplet: such that Obviously, the triplet () is piecewise constant; that is, the triplet is constant in each interval , .

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 , such that

Proof. Applying (30), for any , , we have In view of maximal monotonicity of subdifferentials, we have By virtue of Lemma 13, the second term in (35) could be estimated as follows: As for the third term, applying integration by parts, we have Since in as and the embedding is continuous, there exists a constant such that Then, from the assumption on initial value (see (V) Section 2.1), it follows that From the coercivity of (see (4)), we get (32) and (33) from (40) and (5). Applying , we get (34).

4.3. Completion of Proof

Lemma 19. There exist , , and such that for almost all , , , and Moreover, strongly in as .

Proof. From (32) and (33), there exist and such that
Since is compact from to (see Remark 16) and is uniformly bounded in , is precompact in for each . Then combining (34), there exist a subsequence of (still denoted by ) and such that In addition, from (34), there exists such that and we can easily testify that combining (44).
In virtue of (30), for any , we have Applying (43) and (45) and letting , it follows that which implies that for almost all ,
To complete the proof, we need to show that , for almost all . Since , are maximal monotone operators, then, combining Lemmas 6 and 11, it suffices to prove that
In view of (42) and (44), we have which implies (49).
Applying (30), we have Since strongly in and weakly in , we have For the second term, applying Lemma 13, we have since strongly in and is lower semicontinuous. Moreover, from Lemma 14, we can easily get Therefore, On the other hand, applying (47) on , we have Consequently, we obtain (50).
As the end of the proof, since in and as , we have strongly in as .

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 be a bounded domain in with smooth boundary and has the partition . Consider where , , and are continuous and nondecreasing and with .

Remark 20. For the case , the problem (59) cannot be covered by [8].

5.1. Formulation of Abstract Form

Let and , equipped with the norms and , respectively. Then is embedded in densely and compactly, and denote the injection by . Assume that are nondecreasing and continuous, and , satisfy Define Let and ; then , Assuming and , the problem (59) can be rewritten as where .

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.(H1) are nondecreasing and continuous.(H2), , and .(H3) with .Assume that (H1)–(H3) hold; then and satisfy Theorem 2. Therefore, for any and 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 , , and . Assume that . Assume satisfying Then there exists at least one solution of (59); that is, is a measurable function such that and for a. e. , for any . Moreover, strongly in and strongly in as .

Remark 22. More generally, instead of the boundary condition on in the problem (59), we assume that (see [27]) where is multivalued and maximal monotone. Assume that there exists a proper, convex, lower semicontinuous function defined on with such that . Define and make a modification of as follows: Then the problem can be solved by imposing some appropriate assumptions on . For a simple case, we could suppose that (b1)there exists a constant , such that for any , and ;(b2)there exists a nondecreasing function such that Then satisfy Theorem 2. In addition, (b1) and (b2) can be satisfied by extensive functions. For example, for , set and .

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.