Uniform Decay Rates Research Articles

General decay of energy for a viscoelastic equation with nonlinear damping

Abstract In this paper, we investigate a nonlinear viscoelastic equation with nonlinear damping. The general uniform decay rate of the energy is established.

On viscoelastic wave equation with nonlinear boundary damping and source term

In this paper, we consider the viscoelastic wave equation withnonlinear boundary damping and source term. This work is devoted to prove the existence of solutions and uniform decay rates of the energy without imposing any restrictive growth assumption on the damping term and weakening the usual assumptions on the relaxation function.

Nonlinear exact controllability and nonlinear stabilization of hyperbolic or hyperbolic-like evolution equations

The present manuscript provides a comprehensive and detailed version of the content of the lectures delivered by the author at the Winter School on Nonlinear Analysis, Instituto de Matematica/UFRJ, August 3-7, 2009, Rio de Janeiro, Brazil. They encompassed two distinct yet related topics: (1) nonlinear global exact controllability, in the space of finite energy, for a second-order hyperbolic equation with Neumannboundary control; and(2) sharp uniform decay rates, at the L2(Ω)-level, of the Schrodinger equation with nonlinear boundary dissipation. ∗Research partially supported by the National Science Foundation under grant NSF-DMS01043-5 and by the Army Research Office under grant ARO-DAAD19-02-1-0179. Mathematics Subject Classification. Primary: 35L05, 35L20. Secondary: 93.

Periodic Solutions of Piecewise Affine Gene Network Models with Non Uniform Decay Rates: The Case of a Negative Feedback Loop

This paper concerns periodic solutions of a class of equations that model gene regulatory networks. Unlike the vast majority of previous studies, it is not assumed that all decay rates are identical. To handle this more general situation, we rely on monotonicity properties of these systems. Under an alternative assumption, it is shown that a classical fixed point theorem for monotone, concave operators can be applied to these systems. The required assumption is expressed in geometrical terms as an alignment condition on so-called focal points. As an application, we show the existence and uniqueness of a stable periodic orbit for negative feedback loop systems in dimension 3 or more, and of a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only.

Large time behavior of anisotropic electromagnetic/elasticity equations in exterior domains

We find uniform rates of decay of the total energy of the coupled system of anisotropic electromagnetic/elasticity model in exterior domains provided mild dissipative effects are present. The decay of the total energy is of polynomial type. The conclusions of this paper improve previous results on the subject.

Uniform decay for a hyperbolic system with differential inclusion and nonlinear memory source term on the boundary

We prove the existence and uniform decay rates of global solutions for a hyperbolic system with a discontinuous and nonlinear multi-valued term and a nonlinear memory source term on the boundary.

On the large-time behavior of anisotropic Maxwell equations

Anisotropic Maxwell equations with electric conductivity are considered. Electromagnetic waves propagate in the exterior of a bounded connected obstacle with Lipschitz boundary. Our main result says that we can obtain uniform rates of decay of the total energy as $t \rightarrow + \infty$. No special requirements on the geometry of the obstacle are required. Previous results of this type were only given in the isotropic case. We use multipliers and properties of an associated evolution coupled system of first order.

On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms

This paper is concerned with the study of the nonlinearly damped system of wave equations with Dirichlét boundary conditions: <p align="center"> $u_{t t}$ $- \Delta u + |u_t|^{m-1}u_t = F_u(u,v) \text{ in }\Omega\times ( 0,\infty )$, <br> $v_{t t}$$ - \Delta v + |v_t|^{r-1}v_t = F_v(u,v) \text{ in }\Omega\times( 0,\infty )$, <p align="left" class="times"> where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n=1,2,3$ with a smooth boundary $\partial\Omega=\Gamma$ and $F$ is a $C^1$ function given by <p align="center"> $ F(u,v)=\alpha|u+v|^{p+1}+ 2\beta |uv|^{\frac{p+1}{2}}. $ <p align="left" class="times"> Under some conditions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain several results on the global existence, uniform decay rates, and blow up of solutions in finite time when the initial energy is nonnegative.

Well-posedness and uniform decay rates for the Klein–Gordon equation with damping term and acoustic boundary conditions

In this paper, we consider the Klein–Gordon equation u″+−αΔu+β|u′|pu′+γ|u|qu=0 with damping term and acoustic boundary conditions. This work is devoted to prove the existence of global solutions and uniform decay rates of solutions of the Klein–Gordon equation with damping term and acoustic boundary conditions.

On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source

This paper is concerned with the study of the nonlinear damped wave equation $${u_{tt} - \Delta u+ h(u_t)= g(u) \quad \quad {\rm in}\,\Omega \times ] 0,\infty [,}$$ where Ω is a bounded domain of $${\mathbb{R}^2}$$ having a smooth boundary ∂Ω = Γ. Assuming that g is a function which admits an exponential growth at the infinity and, in addition, that h is a monotonic continuous increasing function with polynomial growth at the infinity, we prove both: global existence as well as blow up of solutions in finite time, by taking the initial data inside the potential well. Moreover, optimal and uniform decay rates of the energy are proved for global solutions.

Boundary Stabilization of Hyperbolic Hemivariational Inequalities

In this paper, we prove the existence of weak solutions and investigate uniform decay rates of global weak solutions for a hyperbolic hemivariational inequalitiy of dynamic elasticity.

Existence and asymptotic stability for the semilinear wave equation with boundary damping and source term

In this paper, we consider the semilinear wave equation with boundary conditions. This work is devoted to prove the existence of solutions and uniform decay rates of the wave equation with boundary damping and source term.

Uniform decay rates for second-order hyperbolic equations with variable coefficients

In this paper, we consider a second-order hyperbolic equations with variable coefficients and linear lower-order term. We use a new main multiplier and a suitable nonlinear version of a compactness uniqueness argument (Acta. Math. Sin. - Engl. Ser. 20(6) (2004), 1057-1072) to obtain the exponential stabilization of this system where no condition of smallness is needed.

Biodegradation rate of diesel range n-alkanes by bacterial cultures Exiguobacterium aurantiacum and Burkholderia cepacia

The biodegradation rate of hydrocarbons is an important consideration determining the time scale for bioremediation in oil-contaminated environments. Two naturally occurring bacterial cultures, Exiguobacterium aurantiacum and Burkholderia cepacia, were capable of utilizing diesel oil as the sole source of carbon and energy by induction of hydrophobic cell surfaces with water contact angle greater than 70°. The cultures demonstrated good degradation characteristics for diesel range n-alkanes (C9–C26) and were also capable of degrading pristane. A significant correlation was observed between maximum decay rate (MTR) of individual n-alkane peak area and initial abundance of n-alkanes in diesel ( r 2=0.79 and 0.97 for E. aurantiacum and B. cepacia, respectively). Thus, MTR (day −1) was essentially constant, in the range of 0.07–0.20, for n-alkanes with a wide range of carbon numbers from C12 to C26. Biodegradation altered the relative abundance of n-alkanes in diesel and resulted in a loss of symmetry in n-alkane distribution. C9, C17–C19, and C26 were completely degraded by both the cultures. In B. cepacia, the residual diesel was enriched in the higher carbon number n-alkanes C20–C25. The MTR of n-C16 present as a component of diesel was comparable to that for n-C16 when present as sole substrate for B. cepacia but not for E. aurantiacum. Scientific relevance Two cultures E. aurantiacum and B. cepacia were capable of utilizing diesel oil as sole substrate and exhibited uniform decay rates for a wide range of n-alkanes from C12 to C26. In contrast, most researchers have reported decreasing degradation rate with increasing carbon number.

Decay of Weak Solutions to the 2D Dissipative Quasi-Geostrophic Equation

We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data θ0 is in L 2 only, we prove that the L 2 norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For θ0 in L p ∩ L 2, with 1 ≤ p < 2, we are able to obtain a uniform decay rate in L 2. We also prove that when the \(L^{\frac{2}{2\alpha-1}}\) norm of θ0 is small enough, the L q norms, for \(q > {\frac{2}{2\alpha-1}}\) , have uniform decay rates. This result allows us to prove decay for the L q norms, for \(q \geq {\frac{2}{2\alpha-1}}\) , when θ0 is in \(L^2 \cap L^{\frac{2}{2\alpha-1}}\) .

Decoherence under many-body system-environment interactions: A stroboscopic representation based on a fictitiously homogenized interaction rate

An environment interacting with portions of a system leads to multiexponential interaction rates. Within the Keldysh formalism, we fictitiously homogenize the system-environment interaction yielding a uniform decay rate facilitating the evaluation of the propagators. Through an injection procedure we neutralize the fictitious interactions. This technique justifies a stroboscopic representation of the system-environment interaction which is useful for numerical implementation and converges to the natural continuous process. We apply this procedure to a fermionic two-level system and use the Jordan-Wigner transformation to solve a two-spin swapping gate in the presence of a spin environment.

Semi-discrétisation en espace du problème de la stabilisation interne de l'équation des poutres

We consider the finite-difference space semi-discretization of the beam equation with an internal stabilization. By using a frequency domain approach, we prove that this scheme yields a uniform exponential decay rate (with respect to the mesh size). This positive result is in contrast with previous negative ones for the case of boundary stabilization or for other evolution equations.

Uniform stabilization of an electromagnetic-elasticity problem in exterior domains

A coupled system of dynamic hyperbolic equations in electroma- gnetic-elasticity theory in the exterior of an open bounded obstacle $\mathcal{O}$ in 3-D is considered. In the presence of dissipative effects we obtain uniform decay rates of the solution as $t \rightarrow +\infty$. We do not require geometric assumptions on the obstacle or extra assumptions on the initial data. We apply our results to study the above system with a nonlinear perturbation, showing that the solutions hold the same rate of decay provided the initial data is small in a suitable sense. Previous results of this type have recently been obtained for the scalar wave equation by M. Nakao [18, 19] and R. Ikehata [8].

General decay rate estimates for viscoelastic dissipative systems

The linear viscoelastic equation is considered. We prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary, without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function. Our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity. The proofs are based on the multiplier method and on a general lemma about convergent and divergent series for obtaining the uniform decay rates.

Well-posedness and sharp uniform decay rates at the L2(Ω)-Level of the Schrödinger equation with nonlinear boundary dissipation

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.