Asymptotic stability for third-order non-homogeneous differential-operator equations

In this article, global asymptotic stability of solutions of non-homogeneous differential-operator equations of the third order is studied. It is proved that every solution of the equations decays exponentially under the Routh–Hurwitz criterion for the third order equations.

For each equation, they give a suitable Lyapunov function, which is a powerful tool to study the qualitative aspects of the solutions of these equations. They have showed that if the stability conditions fail, they can obtain instability solutions for suitable domains. It has also be noted that the two equations are heat and wave type forT , respectively. In [4], the homogeneous differential-operator equation of a third order is investigated. It is shown under some conditions, compatible with the Routh-Hurwitz criterion, that the solution of the equation is asymptotically stable, and that every solution of the equation tends to zero with an exponential rate. For other studies on stability in the third-order problems, one can see [8][9][10][11][12].
In this article, in contrast to the works on homogenous equations mentioned in [2][3][4], a non-homogeneous differential-operator equation of the third order has been investigated. It has been shown that every solution to the equation is asymptotically stable, provided that a condition on the function on the right-hand side is established.

Preliminaries
Let H be a real Hilbert space with inner product (·, ·) and norm ||·||. We consider in H the following third-order non-homogeneous equation: where A, B, and C are linear (not necessarily bounded), positively defined, and self-adjoint operators. The unique solvability of the problem can be established by the Faedo-Galerkin method [11]. The domains of these operators D(A), D(B), and D(C) are dense linear subspace of H . The symbol " " stands for differentiation with respect to t. Also ||h(t)|| → 0 as t → ∞. To show that every solution of (2.1) tends to zero as t → ∞, we need the following lemma.
Then, there exists a positive function φ(t), which is some measure of the solution, satisfying Proof Assume that u = u(t) is an arbitrary solution of Eq. (2.1). Taking the inner product in H of (2.1) with u + εu , where ε is a positive number specified below, and using the standard inequality to the right-hand side, we obtain where ε 1 and ε 2 are positive numbers which are specified below. It follows that d dt then the last inequality becomes Using the Cauchy inequality, together with conditions (2.3) and (2.4), we get where ε 3 and ε 4 are constants to be chosen below. Hence, due to (2.8) and (2.9), we find the following estimate for φ (u(t)): From condition (2.2), it is clear that there is a positive number α < α, such that We can also choose ε 3 and ε 4 , such that and we see that For some specific works on the stability for the third-order problem, one can see [4] and [12].

Theorem 2.2 Suppose that all conditions of lemma are satisfied. Assume also that D(B) = D(C) ⊆ D(A)
, and there exist positive numbers β, γ 2 , and α 1 , such that and Moreover, every solution of the Cauchy problem for Eq. (2.1) tends to zero as t → ∞.

Applications
Application 3.1 One of the models describing acoustic waves in a non-uniform compressible relaxing medium whose density in the unperturbed state depends only on x is given by the following: For more detailed physical interpretation, existence, and uniqueness of the solutions, we refer to [10]. It can be stated that (3.1) differs from the equation for acoustic waves in uniform medium only by the fact that the coefficients are not constant. Taking into account one-dimensional case of (3.1) in x, introducing x = x/ [C ∞ (0)T i ] and t = t/T i as dimensionless variables, where T i is the characteristic period of internal interactions, and setting γ ( 2 and u = p/ p(0)C 2 ∞ (0), we lead to the following non-homogeneous initial-boundary value problem: 2) can be rewritten in the form of (2.1) as follows: u ttt + Au tt + Bu t + Cu = f (x, t). (3.4) Thus, Theorem 2.2 allows us the possibility to confirm that, under the required conditions on the function f (x, t), every solution of (3.4) satisfying the conditions in (3.3) tends to zero with an exponential rate.
Application 3.2 Any physical process including the photon interactions in metal films and photon scattering in dielectric media occurs in a finite time. These microscopic interactions are viewed as impeding sources causing a delayed response on the macroscopic scale. In [9], the heat flux is eliminated using the Taylor series expansion with respect to t by considering the one-dimensional equation obtained in [9], which shows that the heat flux and the temperature gradient occur in a sequence of time, and by employing the conservation of energy, the following equation is derived