Exponential Decay Rate of the Perturbed Energy of the Wave Equation with Zero Order Term

In the case of V. Komornick and E. Zuazua in [3] have shown that = 0, q E decays exponentially. When 0 q  , there is some difficulty to obtain this result since we have a lower order term with respect to the energy in some multiplier estimate (see (3.5) in [3]). The purpose of this paper is to overcome this kind of difficulty, where we prove an useful estimate then by the compactness uniqueness argument we absorb the lower order term. Finally, we employ some result of I. Lasiecka and D. Tataru in [4]. In all this paper, is a generic positive constant independent of the initial data and it may change from line to line. C


Introduction
Here is a bounded domain of with smooth boundary and  is a partition of and are two positive bounded functions, that is there exists four positive constants  such that for all and for all :


For all > 0,  we define the perturbed energy of the system (1) by for all Here is the usual energy defined by In the case of V. Komornick and E. Zuazua in [3] have shown that = 0, q E  decays exponentially.When 0 q  , there is some difficulty to obtain this result since we have a lower order term with respect to the energy in some multiplier estimate (see (3.5) in [3]).The purpose of this paper is to overcome this kind of difficulty, where we prove an useful estimate then by the compactness uniqueness argument we absorb the lower order term.Finally, we employ some result of I. Lasiecka and D. Tataru in [4].
In all this paper, is a generic positive constant independent of the initial data and it may change from line to line.

Exponential decay rate of E 
Using the multiplier method we can show that the energy of the system (1) is a decreasing function.That is, for all we have The proof of the main theorem involves two lemmas.Lemma 1 For all two positive constants and S such that , where 0 is some sufficiently large positive constant, we have for Here where  is the constant verifying On the other hand, we have (see (3.5) in [3]) Integrate over   , S T use (4) with  sufficiently small and by the decreasing of , we find With  sufficiently small we find the desired result.
To absorb the lower order term Proof.First, we have from (3) On the other hand, by ( 2) Then, with  sufficiently small, we find Now, we come back to the proof of (5).
It is sufficient to prove (see [1]] that, for some large enough, we have 0 We argue by contradiction.There exists a sequence of solutions   where represents the energy of . If we apply (7) with and 0 , we obtain by (8), ( 9) and (4) that Proof.If we insert (5) in ( 3) we find If we choose sufficiently large we find such that 0 for all 0 This imply that If we apply (10) repeatedly on the intervals , , we get The last system have the solution then Then by (11) We can treat exactly in the same way the situation of the second order hyperbolic equation with variable coefficients, linear zero order term and polynomial growth of the nonlinear feedback near the origin.In this case, we use the Riemann geometric approach to handle the case of the variable coefficients principal part (see [2]] to show, directly, that we have an exponential or polynomial decay rate of the perturbed energy functional defined for all > 0  by

Thus
where  depends on the behavior of nonlinear damping at the origin.

References
, s E mT

bounded and therefore there exists aTheorem 3
If we multiply the first equation by , integrate over and use the first Green's formula we find We give, now, the proof of the main result.For any initial data   0 1 , y y D  , the energy perturbed E  is exponentially stable.
the solution of the system