We consider an abstract Cauchy problem for a doubly nonlinear evolution equation of the form

Let

During the past decades, the problem has been investigated in many papers, such as [

More generally, both

In some papers (such as [

To state our assumptions clearly, we introduce some notations.

Let

For any

Let

Our basic assumptions are as follows.

there exists a nondecreasing function

There exists

(1) Condition (A) implies that the operator

(2)

(3) (B1) implies the coerciveness of

Assume (A), (B), (F), and (V) are all satisfied. Then, there exists at least one solution triplet

The proofs related to this section can be found in [

Let

Let

Let X be a real reflexive Banach space.

For any

For any

Let

Let

Let

Let

Let

Then,

For any

Let

Let

Let

there are two constants

Assume that for each

Assume

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

By the definition of

Let

The following chain rule of integral form was proved in [

Assume

Let

Since

Assume (A), (B), and (V) are satisfied. Then

Since

To prove

In fact, for any real reflexive Banach space

Take

In this section, to prove Theorem

Let

To prove Theorem

For

In the following, we aim to obtain some uniform estimates on the approximation solutions (see Section

There exists a constant

Applying (

There exist

From (

Since

In virtue of

To complete the proof, we need to show that

In view of (

Applying (

As the end of the proof, since

The abstract existence can be applied to many models in fluid mechanics (see [

Let

For the case

Let

Applying the existence theorem of abstract form, we could claim the solvability of the problem (

Assume (H1)–(H3). Let

More generally, instead of the boundary condition on

there exists a constant

there exists a nondecreasing function

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

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