Energy Method In Fourier Space Research Articles

Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay

Abstract In this study, we address a Cauchy problem within the context of the one-dimensional Timoshenko system, incorporating a distributed delay term. The heat conduction aspect is described by the Lord-Shulman theory. Our demonstration establishes that the dissipation resulting from the coupling of the Timoshenko system with Lord-Shulman’s heat conduction is sufficiently robust to stabilize the system, albeit with a gradual decay rate. To support our findings, we convert the system into a first-order form and, utilizing the energy method in Fourier space, and derive point wise estimates for the Fourier transform of the solution. These estimates, in turn, provide evidence for the slow decay of the solution.

New decay rates for a Cauchy thermoelastic laminated Timoshenko problem with interfacial slip under Fourier or Cattaneo laws

The objective of the present paper is to investigate the decay of solutions for a laminated Timoshenko beam with interfacial slip in the whole space subject to a thermal effect acting only on one component modeled by either Fourier or Cattaneo law. When the thermal effect is acting via the second or third component of the laminated Timoshenko beam (rotation angle displacement or dynamic of the slip), we obtain that both systems, Timoshenko–Fourier and Timoshenko–Cattaneo systems, satisfy the same polynomial stability estimates in the L2‐norm of the solution and its higher order derivatives with respect to the space variable. The decay rate depends on the regularity of the initial data. In addition, the presence and absence of the regularity‐loss type property are determined by some relations between the parameters of systems. However, when the thermal effect is acting via the first component of the system (transversal displacement), a new stability condition is introduced for both Timoshenko–Fourier and Timoshenko–Cattaneo systems. This stability condition is in the form of threshold between polynomial stability and convergence to zero. To prove our results, we use the energy method in Fourier space combined with judicious choices of weight functions to build appropriate Lyapunov functionals.

Decay rates for the Moore-Gibson-Thompson equation with memory

The main goal of this paper is to investigate the existence and stability of the solutions for the Moore–Gibson–Thompson equation (MGT) with a memory term in the whole spaces \begin{document}$ \mathbb{R}^{N} $\end{document} . The MGT equation arises from modeling high-frequency ultrasound waves as an alternative model to the well-known Kuznetsov's equation. First, following [ 8 ] and [ 26 ], we show that the problem is well-posed under an appropriate assumption on the coefficients of the system. Then, we built some Lyapunov functionals by using the energy method in Fourier space. These functionals allows us to get control estimates on the Fourier image of the solution. These estimates of the Fourier image together with some integral inequalities lead to the decay rate of the \begin{document}$ L^{2} $\end{document} -norm of the solution. We use two types of memory term here: type Ⅰ memory term and type Ⅲ memory term. Decay rates are obtained in both types. More precisely, decay rates of the solution are obtained depending on the exponential or polynomial decay of the memory kernel. More importantly, we show stability of the solution in both cases: a subcritical range of the parameters and a critical range. However for the type Ⅰ memory we show in the critical case that the solution has the regularity-loss property.

New decay rates for Cauchy problem of Timoshenko thermoelastic systems with past history: Cattaneo and Fourier law

In this paper, we investigate the decay properties of the thermoelastic Timoshenko system with past history in the whole space where the thermal effects are given by Cattaneo and Fourier laws. We obtain that both systems, Timoshenko–Fourier and Timoshenko–Cattaneo, have the same rate of decay and the regularity‐loss‐type property is not present in some cases. Moreover, new stability number χ is introduced, such new number controls the decay rate of the solution with respect to the regularity of the initial data. To prove our results, we use the energy method in Fourier space to build an appropriate Lyapunov functionals that give the desired results.

Decay estimate and asymptotic profile for a plate equation with memory

In this paper, the initial value problem for a plate equation with a memory is considered. Under weaker assumptions on the memory function, we obtain the decay estimates of solutions which is of the regularity-loss type by applying the energy method in Fourier space. Moreover, if the memory kernel is an exponential function, we derive the large-time behavior and asymptotic profile of solutions in the one-dimensional space by virtue of the technique of spectral analysis.

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.

Energy method in the partial Fourier space and application to stability problems in the half space

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space R n . In this paper, we study half space problems in R + n = R + × R n − 1 and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x ′ ∈ R n − 1 . For the variable x 1 ∈ R + in the normal direction, we use L 2 space or weighted L 2 space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t → ∞ . The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].