A plate equation with a memory term and a time delay term in the internal feedback is investigated. Under suitable assumptions, we establish the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations and some energy estimates. Moreover, by using energy perturbation method, we prove a general decay result of the energy provided that the weight of the delay is less than the weight of the damping.

In this paper, we are concerned with the following plate equation with a memory term and a time delay term in the internal feedback:

Equation (

In this paper, we consider the following initial conditions:

Fourth order equations with lower order perturbation are related to models of elastoplastic microstructure flows. For the single plate equation without delay, that is,

In recent years, many mathematical workers studied some systems with time delay effects. Datko et al. [

Equation (

The outline of this paper is as follows. In Section

The notation in this paper will be as follows:

In this section, we give some preparations for our consideration and our main results.

We assume that

(if

For the memory kernel

where

There exists a positive nonincreasing differentiable function

The nonlinear term

where

In order to deal with the delay feedback term, motivated by [

Let

Now we define the weak solutions of (

Next we state the global well-posedness of problem (

Let

If the initial data

If the initial data

then the above weak solution has higher regularity

In both cases, we have that the solution

We define the energy of problem (

Finally, we give the energy decay of problem (

Let

In this section, we will prove the global existence and the uniqueness of the solution of problem (

Let

We define for

Given initial data

By using standard ordinary differential equations theory, the problem (

Now multiplying the first approximate equation of (

Multiplying the second approximate equation of (

Consider

Consider

From (

Firstly we prove the continuous dependence and uniqueness for stronger solutions of problem (

Let

We can prove the continuous dependence and uniqueness for weak solutions by using density arguments (see, e.g., Cavalcanti et al. [

This ends the proof of Theorem

In this section, we will establish the decay property of the solution for problem (

We first consider stronger solutions. Define the modified energy by

It follows from (

Under the assumptions in Theorem

For the same argument as (

Now we define the following functional:

Under the assumptions in Theorem

By taking a derivative of (

In order to handle the term

Under the assumptions in Theorem

Differentiating (

We define the Lyapunov functional

First, we claim that there exist two positive constants

Next, combining (

This proves the general decay for regular solutions. We can extend the result to weak solutions by using a standard density argument; one can refer to Cavalcanti et al. [

There are some open problems concerning our present work, and here we give some of them.

It is obvious that the weak damping term

We only obtain the general decay for

It is interesting to study that the weight of the delay is bigger than the weight of the damping; that is,

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to thank the referees for their helpful comments. This work was supported by the Fundamental Research Funds for the Central Universities with Contract no. JBK150128.