Existence for Nonlinear Evolution Equations and Application to Degenerate Parabolic Equation

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


Introduction
Let  be a real reflexive Banach space, and let A, B be maximal monotone operators (possibly multivalued) from  to its dual   .In this paper, we consider the abstract evolution equation:
During the past decades, the problem has been investigated in many papers, such as [1,2,[6][7][8][9][10][11][12][13][14][15][16].For the case that A =  and B 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 A is continuous, B is continuous and elliptic in some sense, Alt and Luckhaus proved the existence in [8], and Otto established the  1 -contraction and uniqueness in [17].For the case that A is Lipschitz continuous and B is coercive, the existence theory was given in [14].In fact, if A is Lipschitz continuous and invertible, the problem (1) can be rewritten as   + B (V ()) ∋  () , V () ∈ A −1 ( ()) ,  ∈ (0, ] . ( Under the condition that A −1 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 A −1 is a maximal monotone operator.More generally, both A and B are possibly nonlinear, and such equations are said to be doubly nonlinear.On the assumption that one of A, B is a subdifferential operator and the other is strongly monotone, the existence was established in [10].In addition, many practical applications (see [1][2][3][4][5]) suggested that both A and B are subdifferential operators.For the case that A and B are subdifferentials of functions on a Hilbert space, the existence was given in [6].Supposing that A is a subdifferential operator in a Hilbert space  and B 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 B.
In some papers (such as [6,11,12]), the problems investigated are time dependent; that is, B (possibly together with A) is time dependent.In this paper, we aim to extend the first existence theory in [14] to the case that A is only continuous 2 Journal of Applied Mathematics but not Lipschitz continuous, and B 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.

Basic Assumptions and Existence Theorem
2.1.Basic Assumptions.To state our assumptions clearly, we introduce some notations.
Our basic assumptions are as follows.

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

Lower Semicontinuous Functions
Lemma 3. Let  ∈ F().Then  is bounded from below by an affine function; that is, there exist  * 0 ∈   and  ∈ R 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.

Maximal Monotone Operators.
Let  be a real reflexive Banach space.An operator T :  → 2   is called monotone, if where (T) = { ∈  : T() ̸ = Ø}.In addition, T is called maximal if it has no proper monotone extension in ; that is, for any  ∈  and any  ∈   , only if  ∈ T().
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 V() ∈   (()).Then, for each ℎ > 0, where  −ℎ denotes the backward difference operator, The following chain rule of integral form was proved in [25].
Let Λ :  →  be a linear continuous operator, and assume that  ∈ F() is continuous at some point of (Λ) (the range of Λ).Then where   is the dual operator of Λ. Proof.Since int((A)) ⋂ (B) ̸ = Ø, we get the maximal monotonicity of A + B from Lemmas 9 and 7.
To prove (A + B) =   , we only need to verify that A and B satisfy Lemma 8. Applying (4) and Lemma 10, we can deduce that (B) =   , and then the proof is completed if we could show that B =   Φ  is regular.
In fact, for any real reflexive Banach space  and any  ∈ F(),    is regular.

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.
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).
In virtue of (30), for any  ∈  2 (0, ; ), we have For the second term, applying Lemma 13, we have lim inf since V ℎ () → V() strongly in   and Φ *  is lower semicontinuous.Moreover, from Lemma 14, we can easily get Therefore, lim sup On the other hand, applying (47) on , we have Consequently, we obtain (50).
As the end of the proof, since V ℎ → V in (0, ;   ) and as  → 0, we have V() → V 0 strongly in   as  → 0.

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.

Existence of Solutions.
Applying the existence theorem of abstract form, we could claim the solvability of the problem (59), by some appropriate assumptions.
Assume that (H1)-(H3) hold; then Φ  and Φ  satisfy Theorem 2. Therefore, for any V 0 and  satisfying Theorem 2, the abstract problem (63) is solvable, which implies the existence for the solution of (59) as follows.
Remark 16.Since   Φ  is continuous and  is densely and compactly embedded in , A =   (Φ  ∘ ) =   ∘   Φ  ∘  is compact, where  is the injection from  to .Assume (A), (B), and (V) are satisfied.Then A+B is maximal monotone from  to 2   and (A + B) =   .