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Simultaneously, considering the viscous effect of material, damping of medium, geometrical nonlinearity, physical nonlinearity, we set up a more general equation of beam subjected to axial force and external load. We prove the existence and uniqueness of global solutions under non-linear boundary conditions which the model is added one damping mechanism at l end. What is more, we also prove the exponential decay property of the energy of above mentioned system.

The problem is based on the equation

which was proposed by Woinowsky-krieger [

was done by Ball [_{t} is effective near the boundary. In the same direction, Kouemon Patchen [_{t}) was effective in Ω. To [

which the model is clamped at x = 0 and is supported x = l. He proved the existence and decay rates of the solutions. A rather general kirchhoff-type beam equation

was set up by Ball [

with nonlinear boundary conditions

and initial conditions

and (4)

In this paper, our analysis is based on the Sobolev spaces

,

espectively equipped with the norm and. We assume that f, g:R → R are continuously differentiable functions such that

and (5)

where and

and (6)

for some ρ > 0.

Assume that the functions are non-negative functions and respectively satisfy

and (7)

and (8)

Now we come to the following conclusions of the existence and uniqueness of global solutions.

Theorem 1. Assume that the assumptions of (5)-(8) and hold. Then for any satisfying the compatibility condition

There exists a function u satisfying (1)-(4) such that

Proof. Let us solve the variational problem associated with (1)-(4), which is given by: find such that

for all. Let be a complete orthogonal system of W. For each, let us put

.

We search for a function

where is a unknown function such that for any, and it satisfies the approximating equation

with the initial conditions

and (12)

Thus (11) and (12) are equivalent to the Cauchy problem of ODES in the variable t, which is known to have a local solution u^{m}(t) in an interval [0, t_{m}) (t_{m} < T) for any given T > 0.

Estimate 1. By integration of (11) over [0, t] (t < t_{m}) with, we see that

where and.

Considering that

,

and the initial conditions, we get

.

Using Gronwall inequality, we have. Then there exists a constant M_{1} depending only on T such that

for any and for all.

In this paper, C is a constant independent of m, t and denotes different value in different mathematical expression.

Estimate 2. Integrating by parts (11) with and t = 0, and considering the compatibility condition (3) we get

Thus there exists a positive constant M_{2} such that

Estimate 3. Let us fix t, ξ > 0 such that ξ < T – t. Taking the difference of (11) with t = t + ξ and t = t, and replacing ω by, we get

where

.

Let us estimate. Since

we have

Noting that ΔM_{1} = M(z(t + ξ) – M(z(t)) and ΔM_{2} = N(z_{t}(t + ξ)) – N(z_{t}(t)), then integrating by parts we have

Since, by the Mean value theorem, from estimates 1 and (16) we have

where η_{1} is between and.

By the Mean value theorem, we also have

Considering that M(z(t + ξ) ≤ C and N(z(t + ξ) ≤ C, we conclude that there exists constants k_{1} > 0 and k_{2} > 0 such that

A argument for f yields

where k_{3} > 0 is a constant. Putting

and taking into account of (17)-(18) and the assumptions of g, we deduce from (15) that

where k_{4} = max{k_{1} + k_{3}, k_{2}}. Therefore

Dividing the above inequality by ξ^{2} and letting ξ → 0 gives

.

From estimate 2 we find a constant M_{3} > 0 such that

.

With the estimates 1 - 3 we can use Lions-Aubin Lemma to get the necessary compactness in order to pass (11) to the limit. Then it is a matter of routine to conclude the existence of the global solution in [0, T].

Theorem 2. The solution u(t) of theorem 1 is unique.

Proof. Let u, v be two solutions of (1)-(4) with the same initial data. Then writing p = u – v, putting ω = p_{t} in (10) and using mean value theorem, chauchy-schwarz inequality and Gronwall inequality, we may get p = 0. Thus u = v.

In order to establish our decay result, we define the energy of the system by

where. We have Theorem 3. Let u(t) be the solution given by theorem 1 as q(x, t) = 0 and g = 0. And assume that N(s)s ≥ 0 and f(s)s ≥ 0. Then there exist constants λ_{2}, λ_{4} > 0 and λ_{3} < 0 such that.

To prove Theorem 3, we firstly introduce two lemmas.

Let us define .

Then we have the following lemmas.

Lemma 1. Let E_{ε}(t) = μE(t) + εψ(t). Then there exists a constant k_{5} > 0 such that

.

Proof. By, and there exists k_{5} > 0 such that

where.

Lemma 2. There exist constants λ_{0} > 0 and λ_{1} such that

.

Proof. Taking the inner product of (1) with u_{t} and considering that N(s)s ≥ 0, we have

.

Taking the inner product of (1) with u, we have

Thus

.

Set .

Since, we have. Therefore

.

From the Mean value theorem, there exists a constant λ_{1} such that

On writing, we have

The proof of theorem 3. From lemma 1, we have

From Lemma 2, we have

Therefore

.

By Gronwall inequality and combing (21), we have

.

Hence, for sufficiently small ε > 0

.

On writing

and we have .

The proof of theorem 3 is now completed.