Abstract In this article, we investigate the general qusi-one-dimensional nozzle flows governed by non-isentropic compressible Euler system. First, the steady states of the subsonic and supersonic flows are analyzed. Then, the existence, stability, and uniqueness of the subsonic temporal periodic solutions around the steady states are proved by constructing a new iterative format technically. Besides, further regularity and stability of the obtained temporal periodic solutions are obtained, too. The main difficulty in the proof is coming from derivative loss, which is caused by the diagonalization. Observing that the entropy is conserved along the second characteristic curve, we overcome this difficulty by transforming the derivative of entropy with respect to x x into a derivative along the direction of first or third characteristic. The results demonstrate that dissipative boundary feedback control can stabilize the non-isentropic compressible Euler equations in qusi-one-dimensional nozzles.

In this paper, we consider a wave equation with a nonlinear weakly dissipative boundary feedback localized on a part of the boundary. We establish an explicit and general decay rate result, using some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term.

We investigate a nonlinear viscoelastic equation with interior time-varying delay and nonlinear dissipative boundary feedback. Under suitable assumptions on the relaxation function and time-varying delay effect together with nonlinear dissipative boundary feedback, we prove the global existence of weak solutions and asymptotic behavior of the energy by using the Faedo-Galerkin method and the perturbed energy method, respectively. This result improves earlier ones in the literature, such as Kirane and Said-Houari (2011) and Ammari et al. (2010). Moreover, we give an positive answer to the open problem given by Kirane and Said-Houari (2011).

We study a wave equation in one dimensional space with nonlinear dissipative boundary feedback at both ends. We prove existence and uniqueness of solution, strong and uniform exponential decay of energy under some conditions in the nonlinear feedback. Decay rate estimates of the energy are given under weak growth assumptions on the feedback functions.

We prove that the Schrodinger equation defined on a bounded open domain of \( \mathbb{R}^{n} \) and subject to a certain attractive, nonlinear, dissipative boundary feedback is (semigroup) well-posed on L2(Ω) for any n = 1, 2, 3, ..., and, moreover, stable on L2(Ω) for n = 2, 3, with sharp (optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in a given L2(Ω)-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically rely—at the outset—on a far general result of interest in its own right: an energy estimate at the L2(Ω)-level for a fully general Schrodinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/micro-local machinery [L-T-Z.2, Section 10], to shift down the more natural H1(Ω)-level energy estimate to the L2(Ω)-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear (interior and) boundary dissipation.

The wave equation with variable coefficients with a nonlinear dissipative boundary feedback is studied. By the Riemannian geometry method and the multiplier technique, it is shown that the closed loop system decays exponentially or asymptotically, and hence the relation between the decay rate of the system energy and the nonlinearity behavior of the feedback function is established.

Asymptotic behavior of Timoshenko beam with dissipative boundary feedback

We study the uniform stabilization of the wave equation by means of a nonlinear dissipative boundary feedback. We consider a Neumann condition on the whole boundary, and the observation is the boundary displacement and velocity. We obtain, in a nonlinear framework, estimates of the decay, for any displacement. We establish a similar result for the one-dimensional wave equation with a variable coefficient.

The uniform stabilization problem is studied for the Euler–Bernoulli equation (though the methods apply also to the corresponding nonconstant coefficient case) defined in a smooth, bounded domain $\Omega $ of $R^n $, with suitable dissipative boundary feedback operators. These either are active in both the Dirichlet and Neumann boundary conditions, or are active in only the Dirichlet and inactive in the Neumann boundary condition. The uniform stabilization results presented are fully consistent with recently established exact controllability and optimal regularity theories for the solutions, which in fact motivate the choices of functional spaces in the first place. In particular, these uniform stabilization results require no geometrical conditions on $\Omega $ in the case of active Dirichlet/ Neumann feedback operators, and require some geometrical conditions on $\Omega $ in the case of an active feedback operator only in the Dirichlet boundary condition, as is the case of recent exact controllability theories [I. Lasiecka and R. Triggiani, SIAM J. Control Optim., 27 (1989), pp. 330–373]. Moreover, the forms of the dissipative feedback controls are natural consequences of (i) the type of boundary conditions selected; (ii) the choice that the control in the lowest boundary condition be $L_2 $ in time and space.