Damped Klein-Gordon Equation Research Articles

Exponential decay for damped Klein–Gordon equations on asymptotically cylindrical and conic manifolds

Exponential decay for damped Klein–Gordon equations on asymptotically cylindrical and conic manifolds

Equivalence between the energy decay of fractional damped Klein–Gordon equations and geometric conditions for damping coefficients

We consider damped s s -fractional Klein–Gordon equations on R d \mathbb {R}^d , where s s denotes the order of the fractional Laplacian. In the one-dimensional case d = 1 d = 1 , Green (2020) established that the exponential decay for s ≥ 2 s \geq 2 and the polynomial decay of order s / ( 4 − 2 s ) s/(4-2s) hold if and only if the damping coefficient function satisfies the so-called geometric control condition. In this note, we show that the o ( 1 ) o(1) energy decay is also equivalent to these conditions in the case d = 1 d=1 . Furthermore, we extend this result to the higher-dimensional case: the logarithmic decay, the o ( 1 ) o(1) decay, and the thickness of the damping coefficient are equivalent for s ≥ 2 s \geq 2 . In addition, we also prove that the exponential decay holds for 0 > s > 2 0 > s > 2 if and only if the damping coefficient function has a positive lower bound, so in particular, we cannot expect the exponential decay under the geometric control condition.

Lie Symmetry Classification, Optimal System, and Conservation Laws of Damped Klein–Gordon Equation with Power Law Non-Linearity

We used the classical Lie symmetry method to study the damped Klein–Gordon equation (Kge) with power law non-linearity utt+α(u)ut=(uβux)x+f(u). We carried out a complete Lie symmetry classification by finding forms for α(u) and f(u). This led to various cases. Corresponding to each case, we obtained one-dimensional optimal systems of subalgebras. Using the subalgebras, we reduced the Kge to ordinary differential equations and determined some invariant solutions. Furthermore, we obtained conservation laws using the partial Lagrangian approach.

Blow-up solutions of damped Klein–Gordon equation on the Heisenberg group

We prove the blow-up of solutions of the semilinear damped Klein–Gordon equation in a finite time for arbitrary positive initial energy on the Heisenberg group. This work complements the paper Ruzhansky and Tokmagambetov (J Differ Equ 265(10):5212–5236, 2018), where the global in time well-posedness was proved for the small energy solutions.

A heuristic approach to the prediction of a periodic solution for a damping nonlinear oscillator with the non-perturbative technique

The present work attracts attention to obtaining a new result of the periodic solution of a damped nonlinear Duffing oscillator and a damped Klein–Gordon equation. It is known that the frequency response equation in the Duffing equation can be derived from the homotopy analysis method only in the absence of the damping force. We suggest a suitable new scheme successfully to produce a periodic solution without losing the damping coefficient. The novel strategy is centered on establishing an alternate equation apart from any difficulty in handling the influence of the linear damped term. This alternative equation was obtained with the rank upgrading technique. The periodic solution of the problem is presented using the non-perturbative method and validated by the modified homotopy perturbation technique. This technique is successful in obtaining new results toward a periodic solution, frequency equation, and the corresponding stability conditions. This methodology yields a more effective outcome of the damped nonlinear oscillators. With the help of this procedure, one can analyze many problems in the domain of physical engineering that involve oscillators and a linear damping influence. Moreover, this method can help all interested plasma authors for modeling different nonlinear acoustic oscillations in plasma.

Global Stability Dynamics of the Quasilinear Damped Klein–Gordon Equation with Variable Coefficients

Global Stability Dynamics of the Quasilinear Damped Klein–Gordon Equation with Variable Coefficients

Energy asymptotics for the strongly damped Klein–Gordon equation

We consider the strongly damped Klein–Gordon equation for defocusing nonlinearity and we study the asymptotic behaviour of the energy for periodic solutions. We prove first the exponential decay to zero for zero mean solutions. Then, we characterize the limit of the energy, when the time tends to infinity, for solutions with small enough initial data and we finally prove that such limit is not necessary zero.

A blow-up result for a Nakao-type weakly coupled system with nonlinearities of derivative-type

In this paper, we consider a weakly coupled system of a wave and damped Klein–Gordon equation with nonlinearities of derivative type. We prove a blow-up result for the Cauchy problem associated with this system for nonnegative and compactly supported data by means of an iteration argument.

On a new generalized local fractal derivative operator

In this study, a new generalized local fractal derivative operator is introduced and we discuss its implications in classical systems through the Lagrangian and Hamiltonian formalisms. The variational approach has been proved to be practical to describe dissipative dynamical systems. Besides, the Hamiltonian formalism is characterized by the emergence of auxiliary constraints free from Dirac auxiliary functions. In field theory, it was found that both damped Klein-Gordon and Dirac equations are generalized, and for specific parameters, a field equation comparable to the Barut equation describing the electromagnetic interactions between N spin-1/2 particles in lepton physics is obtained. A Hamiltonian formulation of higher-order Lagrangian has been constructed and discussed as well. The reformulation of the problem based on fractal calculus has been also addressed in details and compared with the basic approach.

Non relativistic and ultra relativistic limits in 2D stochastic nonlinear damped Klein–Gordon equation

We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein–Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter dependence in the estimates of solution. In this paper we present two methods to confirm this parameter dependence. One is the classical, simple energy method. Another is the method via Strichartz estimates.

An energy-preserving computational approach for the semilinear space fractional damped Klein–Gordon equation with a generalized scalar potential

Naturally preserving dissipative or conservative structures of a given continuous system in discrete analogs is of high demand when designing numerical schemes for dissipative or Hamiltonian partial differential equations. Armed with this fact, the proposed computational study focuses on the issue of considering energy-preserving discrete numerical schemes for the Riesz space-fractional Klein-Gordon equation with a generalized scalar potential. For the sake of more clearness, a combined numerical scheme that owes energy-preserving properties is a target to be achieved here. This is done by adapting Galerkin spectral approximation based on Legendre polynomials for Riesz space Laplacian operator side by side to invoke a Cranck–Nickolson scheme in time direction after making order reduction of the original system and a special second-order approximation of the scalar potential derivative. The numerical properties (stability, uniqueness, and convergence) of the scheme are well investigated. Numerical simulations also show that the proposed scheme can inherit the physical properties as the original problem and the numerical solution is stable and convergent to the exact solution.

Global Existence And Growth Of Solutions To Coupled Degeneratly Damped Klein-Gordon Equations

We study a system of nonlinear Klein-Gordon equations with viscoelastic and degenerate damping in a bounded domain. We prove the global existence and exponential growth of solutions with negative initial energy.

Asymptotics of solutions with a compactness property for the nonlinear damped Klein–Gordon equation

We consider the nonlinear damped Klein–Gordon equation ∂ttu+2α∂tu−Δu+u−|u|p−1u=0on[0,∞)×RNwith α>0, 2⩽N⩽5 and energy subcritical exponents p>2. We study the behavior of solutions for which it is supposed that only one nonlinear object appears asymptotically for large times, at least for a sequence of times.We first prove that the nonlinear object is necessarily a bound state. Next, we show that when the nonlinear object is a non-degenerate state or a degenerate excited state satisfying a simplicity condition, the convergence holds for all positive times, with an exponential or algebraic rate respectively. Last, we provide an example where the solution converges exactly at the rate t−1 to the excited state.

Description and Classification of 2-Solitary Waves for Nonlinear Damped Klein–Gordon Equations

We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein–Gordon equation $$\begin{aligned} \partial _{tt}u+2\alpha \partial _{t}u-\Delta u+u-|u|^{p-1}u=0 \end{aligned}$$on \(\mathbb {R}^N\), for \(1\leqslant N\leqslant 5\) and energy subcritical exponents \(p>2\). The description is twofold. First, we prove that 2-solitary waves with same sign do not exist. Second, we construct and classify the full family of 2-solitary waves in the case of opposite signs. Close to the sum of two remote solitary waves, it turns out that only the components of the initial data in the unstable direction of each ground state are relevant in the large time asymptotic behavior of the solution. In particular, we show that 2-solitary waves have a universal behavior: the distance between the solitary waves is asymptotic to \(\log t\) as \(t\rightarrow \infty \). This behavior is due to damping of the initial data combined with strong interactions between the solitary waves.

Numerical solution of two and three dimensional time fractional damped nonlinear Klein–Gordon equation using ADI spectral element method

At the present work, we propose a new numerical scheme for the solution of two and three dimensional time fractional nonlinear damped Klein–Gordon equation (DKGE). To this end, we use the Legendre spectral element method to discretize the equation in the spatial directions and for the time stepping, an alternating direction implicit (ADI) method based on a scheme of order O(τ2) is considered. We prove that the presented method is unconditionally stable and obtain an error estimate for the fully discrete scheme. The presented numerical experiments demonstrate the good accuracy and efficiency of the proposed ADI spectral element method.

Damped Klein-Gordon equation with variable diffusion coefficient

<p style='text-indent:20px;'>We consider a damped Klein-Gordon equation with a variable diffusion coefficient. This problem is challenging because of the equation's unbounded nonlinearity. First, we study the nonlinearity's continuity properties. Then the existence and the uniqueness of the solutions is established. The main result is the continuity of the solution map on the set of admissible parameters. Its application to the parameter identification problem is considered.</p>

The reducing rank method to solve third‐order Duffing equation with the homotopy perturbation

AbstractIn the current work, we apply a nonstandard scheme to solve the third‐order Duffing equation. This equation is produced from the strong damped Klein–Gordon equation under the traveling wave transformation. The solution and the stability conditions for the third‐order Duffing equation have been discussed for the first time. The present analysis is new and used the reducing rank method with the homotopy perturbation method. A nonoscillator solution with the oscillating solutions is derived individually and frequency formula is obtained. Besides, the stability analysis is discussed.

A note on finite time blowup for dissipative Klein–Gordon equation

This is a continuation of the study on an arbitrarily positive initial energy finite time blowup result of the solution to the initial value problem of strongly damped Klein-Gordon equation. Based on an adapted concavity method, both the conclusions and arguments of the present paper are not bounded by the coefficients of dissipative damping terms.

Asymptotic Profiles of Solutions for Structural Damped Wave Equations

We consider the Cauchy problem in the whole space for strongly damped Klein-Gordon equations. We derive asymptotic profles of solutions with weighted initial data by a simple method introduced by R. Ikehata. The obtained results show that the wave effect will be weak because of the mass term, especially in the low dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases.

Exponentially fitted methods for solving two-dimensional time fractional damped Klein–Gordon equation with nonlinear source term

In this paper, we introduce a numerical study for the two-dimensional damped Klein–Gordon equation with time fractional and nonlinear source term. The proposed scheme is based on the exponential fitting technique and gives rise to a class of fourth order or sixth order in both spatial dimensions and with the second order in time. This technique depends on a free parameter, which will be used to annihilate the error and increase the order of accuracy. The proposed method's stability and convergence are analyzed by Fourier analysis. Also, the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical tests.