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 [
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 um(t) in an interval [0, tm) (tm < T) for any given T > 0.
Estimate 1. By integration of (11) over [0, t] (t < tm) with, we see that
where and.
Considering that
,
and the initial conditions, we get
.
Using Gronwall inequality, we have. Then there exists a constant M1 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 M2 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 ΔM1 = M(z(t + ξ) – M(z(t)) and ΔM2 = N(zt(t + ξ)) – N(zt(t)), then integrating by parts we have
Since, by the Mean value theorem, from estimates 1 and (16) we have
where η1 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 k1 > 0 and k2 > 0 such that
A argument for f yields
where k3 > 0 is a constant. Putting
and taking into account of (17)-(18) and the assumptions of g, we deduce from (15) that
where k4 = max{k1 + k3, k2}. Therefore
Dividing the above inequality by ξ2 and letting ξ → 0 gives
.
From estimate 2 we find a constant M3 > 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 = pt 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 λ2, λ4 > 0 and λ3 < 0 such that.
To prove Theorem 3, we firstly introduce two lemmas.
Let us define .
Then we have the following lemmas.
Lemma 1. Let Eε(t) = μE(t) + εψ(t). Then there exists a constant k5 > 0 such that
.
Proof. By, and there exists k5 > 0 such that
where.
Lemma 2. There exist constants λ0 > 0 and λ1 such that
.
Proof. Taking the inner product of (1) with ut 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 λ1 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.