Exponential Decay Of Energy Research Articles

Fractional pseudo-parabolic equation with memory term and logarithmic nonlinearity: Well-posedness, blow up and asymptotic stability

Considered herein is the initial–boundary value problem for a fractional pseudo-parabolic equation with memory term and logarithmic nonlinearity given by ut+(−Δ)su+(−Δ)sut=∫0tg(t−τ)(−Δ)su(τ)dτ+uln|u| under different initial energy levels. The local well-posedness of weak solution is firstly established by using Galerkin approximation and contraction mapping principle at arbitrary initial energy level. Secondly, the global well-posedness, polynomial and exponential energy decay estimates, finite time blow up are investigated at low initial energy level (u0∈Wδ,1) by utilizing modified potential well theory, Galerkin approximation, perturbed energy method, differential–integral inequality technique etc. Subsequently, based on the above conclusions of low initial energy, the global existence, polynomial and exponential energy decay estimates and finite time blow up are also derived at critical initial energy level (u0∈Wδ,2) by introducing some new approximation methods and techniques. Here, the sets Wδ,i(i=1,2) defined in Section 2.2 denote potential well families involving the parameter δ>0. Finally, we give a series of numerical examples used to illuminate above theoretical results.

A Moore-Gibson-Thompson heat conduction problem with second gradient

In this work, we study, from both analytical and numerical points of view, a heat conduction model which is based on the Moore-Gibson-Thompson equation. The second gradient effects are also included. First, the existence of a unique solution is proved by using the theory of linear semigroups, and the exponential energy decay is also shown when the constitutive tensors are homogeneous. The analyticity of the semigroup is also discussed in the isotropic case, and its spatial behavior is studied. The spatial exponential decay is also proved. Then, we provide the numerical analysis of a fully discrete approximation obtained by using the finite element method and an implicit Euler scheme. A discrete stability property is shown, and some a priori error estimates are derived, from which the linear convergence is concluded under suitable regularity conditions. Finally, some one-dimensional numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy.

Qualitative properties for a Moore–Gibson–Thompson thermoelastic problem with heat radiation

In this work, we study some qualitative properties arising in the solution of a thermoelastic problem with heat radiation. The so-called Moore–Gibson–Thompson equation is used to model the heat conduction. By using the logarithmic convexity argument, the uniqueness and instability of solutions are proved without imposing any condition on the elasticity tensor. Then, the existence of solutions is obtained assuming that the elastic tensor is positive definite applying the theory of linear semigroups, and the exponential energy decay is shown in the one-dimensional case. Finally, we consider the one-dimensional quasi-static version and we assume that the elastic coefficient is negative. The existence and decay of solutions are proved, and a justification of the quasi-static approach is also provided.

Exponencial stability of the energy in Naghdi shell model with localized internal dissipation and internal exact controllability

In this work we study the exponential stability of the energy associated with a Naghdi's shell model with localized internal dissipation. Using several tools from Riemannian Geometry we show the well possedness of the model, via semigroup theory, and obtain observability inequalities which allows us to prove the exponential decay of the total energy. As a consequence then we use Russell Principle to obtain exact controllability.

Exponential decay for the cubic Schrödinger equation on a dissipative waveguide

We derive an exponential decay rate of energy for the cubic Schrödinger equation on a dissipative waveguide with a damping term that is effective locally on a neighborhood at the boundary and at infinity. In this paper, we give a new example for which the geometric control condition is sufficient to obtain an exponential decay of energy Schrödinger equation. The proof is based on Strichartz's estimates and the smoothing effects argument associated with the linear Schrödinger equation(LS).

Exponential stability of a type III thermo-porous-elastic system from a new approach in its coupling

<p>In this article, we prove the well-posedness and stability of a one-dimensional thermo-porous-elastic system, considering a thermal coupling of the Green and Naghdi types III. In this scenario, we propose a new model where the temperature of the material directly affects the stress tensor of displacements and the equilibrated stress tensor on the volume fractions in the porous medium, thereby generalizing some existing results in the literature on the subject. Additionally, the exponential decay of energy is proven when the evolution equation corresponding to the dynamics of the elastic skeletal structure exhibits frictional-type damping.</p>

Well-posedness and Exponential Decay of Energy for the Solution of a Wave Equation with Nonlinear Source and Localized Damping Termes

We consider the wave equation with a locally damping and a nonlinear source term in a bounded domain. ytt − Δy + a(x)g(yt) = |y|p−2y, where p > 2. The damping is nonlinear and is effective only in a neighborhood of a suitable subset of the boundary. We show, for certain initial data and suitable conditions on g,a and p that this solution is global we use the Faedo-Galerkin method. Also we established the exponential decay of the energy when the nonlinear damping grows linearly by introducing a suitable Lyapunov functional.

Asymptotic behavior of a Balakrishnan-Taylor suspension bridge

<abstract><p>In this manuscript, we examine a nonlinear Cauchy problem aimed at describing the deformation of the deck of either a footbridge or a suspension bridge in a rectangular domain $ \Omega = (0, \pi)\times (-d, d) $, with $ d < < \pi $, incorporating hinged boundary conditions along its short edges, as well as free boundary conditions along its remaining free edges. We establish the existence of solutions and the exponential decay of energy.</p></abstract>

Analytical and numerical analyses of a viscous strain gradient problem involving type Ⅱ thermoelasticity

<abstract><p>In this paper, a thermoelastic problem involving a viscous strain gradient beam is considered from the analytical and numerical points of view. The so-called type Ⅱ thermal law is used to model the heat conduction and two possible dissipation mechanisms are introduced in the mechanical part, which is considered for the first time within strain gradient theory. An existence and uniqueness result is proved by using semigroup arguments, and the exponential energy decay is obtained. The lack of differentiability for the semigroup of contractions is also shown. Then, fully discrete approximations are introduced by using the finite element method and the backward time scheme, for which a discrete stability property and a priori error estimates are proved. The linear convergence is derived under suitable additional regularity conditions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy decay.</p></abstract>

Stability analysis for a Rao-Nakra sandwich beam equation with time-varying weights and frictional dampings

<abstract><p>In this paper, we studied the asymptotic behavior of solutions for a Rao-Nakra sandwich beam equation with time-varying weights and frictional damping terms acting complementarily in the domain. We studied the effect of the three damping on the asymptotic behavior of the energy function. Under nonrestrictive on the growth assumption on the frictional damping terms, we established exponential and general energy decay rates for this system by using the multiplier approach. The results generalized some earlier decay results on the Rao-Nakra sandwich beam equation.</p></abstract>

Analysis of a thermoelastic problem with the Moore–Gibson–Thompson microtemperatures

In this paper, we study, from both an analytical and a numerical point of view, a poro-thermoelastic problem with microtemperatures. The so-called Moore–Gibson–Thompson equation is used to model the contribution for the temperature and microtemperatures. An existence and uniqueness result is proved by using the theory of linear semigroups of contractions and, for the one-dimensional case, the exponential energy decay is found under some conditions on the constitutive coefficients. Then, a fully discrete approximation is introduced by using the finite element method and the implicit Euler scheme. We show that the discrete energy decays and we obtain some a priori error estimates from which, under some adequate additional regularity conditions on the continuous solution, we derive the linear convergence of the approximations. Finally, we perform some numerical simulations to demonstrate the accuracy of the approximations and the behavior of the discrete energy and the solution.

Simplified estimation of wave attenuation in marginal ice zone using homogenized scattering model

A theoretical model to explain the scattering process of wave attenuation in a marginal ice zone is proposed. Although many numerical methods have been developed to accurately estimate wave attenuation, it is not easy to incorporate this knowledge and results into practical use. Therefore, a simplified estimation method is developed here to explicitly and simply describe its fundamental mechanisms. We consider a periodic array of ice floes, where the floe is modeled by a vertical rigid cylinder. Using a homogenization technique, a homogenized free surface equivalent to the array is obtained. Then, we show that a dispersion relation of the homogenized free surface waves makes all wave numbers complex. As a result, the exponential energy decay in the scattering process is demonstrated. Under the deep water assumption, the wave attenuation coefficient is proportional to the open water’s wave number, ice concentration ratio, and imaginary part of the floe’s heave motion. To validate the proposed theory, a tank experiment was also conducted using cylindrical synthetic ice plates. Although our model is obtained under many simplifications, the theoretical results show the same tendency and order as the experimental results.

WELL-POSEDNESS AND ASYMPTOTIC BEHAVIOR FOR THE DISSIPATIVE P-BIHARMONIC WAVE EQUATION WITH LOGARITHMIC NONLINEARITY AND DAMPING TERMS

This paper concerns with the initial and boundary value problem for a p-biharmonic wave equation with logarithmic nonlinearity and damping terms. We establish the well-posedness of the global solution by combining Faedo–Galerkin approximation and the potential well method, and derive both the polynomial and exponential energy decay by introducing an appropriate Lyapunov functional. Moreover, we use the technique of differential inequality to obtain the blow-up conditions and deduce the life-span of the blow-up solution.

Analysis of two thermoelastic problems with the Green–Lindsay model

In this paper, we analyze, from the numerical point of view, two thermo-elastic problems involving the Green–Lindsay theory. The coupling term is different for each case, involving second order or first order spatial derivatives, respectively. The variational formulation leads to a linear coupled system which is written in terms of the velocity and temperature speed. An existence and uniqueness results and the exponential energy decay for the problem with the stronger coupling are recalled. The polynomial energy decay for the weaker coupling is then proved but using the theory of linear semigroups. Then, a fully discrete approximation is introduced using the finite element method and an implicit scheme. A discrete stability property and a main a priori error estimates result are shown, from which we can derive the linear convergence of the approximations. Finally, some numerical simulations are presented to demonstrate the accuracy of the algorithm, the discrete energy decay and the dependence on the relaxation parameter.

Anisotropy can imply exponential decay in micropolar elasticity

In this note, two problems arisen in micropolar elasticity are considered from the analytical point of view. Following the Kelvin–Voigt theory of micropolar viscoelasticity, two dissipative mechanisms are imposed: in the first problem, it is defined on the microscopic structure and, for the second problem, on the macroscopic structure. Then, an existence and uniqueness result, as well as an exponential energy decay, are proved for the first problem. Since similar arguments can be used for the second problem, only the main key points are commented.

Numerical Analysis of a Swelling Poro-Thermoelastic Problem with Second Sound

In this paper, we analyze, from the numerical point of view, a swelling porous thermo-elastic problem. The so-called second-sound effect is introduced and modeled by using the simplest Maxwell–Cattaneo law. This problem leads to a coupled system which is written by using the displacements of the fluid and the solid, the temperature and the heat flux. The numerical analysis of this problem is performed applying the classical finite element method with linear elements for the spatial approximation and the backward Euler scheme for the discretization of the time derivatives. Then, we prove the stability of the discrete solutions and we provide an a priori error analysis. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximations, the exponential decay of the discrete energy and the dependence on a coupling parameter.

Well‐posedness and exponential decay for the Moore–Gibson–Thompson equation with time‐dependent memory kernel

We are interested in the well‐posedness and exponential decay of Moore–Gibson–Thompson equation with memory Here, the main feature is that the memory kernel depends on time. By using the linear semigroup theory, we prove that the above system is global well‐posedness. In addition, the exponential decay of the related energy is shown to occur, provided that the memory kernel is controlled by a negative exponential.

Velocity Stabilization of a Wave Equation With a Nonlinear Dynamic Boundary Condition

This paper deals with a one-dimensional wave equation with a nonlinear dynamic boundary condition and a Neumann-type boundary control acting on the other extremity. We consider a class of nonlinear stabilizing feedbacks that only depend on the velocity at the controlled extremity. The uncontrolled boundary is subject to a nonlinear first-order term, which may represent nonlinear boundary anti-damping. Initial data is taken in the optimal energy space associated with the problem. Exponential decay of the mechanical energy is investigated in different cases. Stability and attractivity of suitable invariant sets are established.

Polynomial and exponential decay rates of a laminated beam system with thermodiffusion effects

In this paper we consider a laminated beam system with thermodiffusion effects with two kinds of boundary conditions, in which the mass diffusion introduces a new critical thickness in addition to the conventional critical thickness of thermoelastic damping. By using the method of semigroup, we prove the system is global well posed. The polynomial stability is established by using frequency domain method if the wave speeds are nonequal. We also establish exponential decay of energy under the assumption of equal wave speeds. At last, we present some numerical experiments to illustrate our theoretical results. The result is new and is the first time when the stabilization of the laminated beam system with thermodiffusion effects is obtained.

On the existence and decay in a new thermoelastic theory with two temperatures

In this work, we study a new two-temperatures thermoelastic model. Both thermodynamic and conductive temperatures are included, being related by means of an elliptic or parabolic equation. Then two problems are considered assuming the dependence or not on the rate of conductivity temperature. Existence of solutions for the three-dimensional setting and the exponential energy decay in the one-dimensional case are shown.