Strong Damping Terms Research Articles

Global nonexistence of solution for a nonlinear Klein–Gordon equation with strong damping, distributed delay and source terms

Global nonexistence of solution for a nonlinear Klein–Gordon equation with strong damping, distributed delay and source terms

Global solutions and blow-up for Klein–Gordon equation with damping and logarithmic terms

In this paper, the initial boundary value problem for Klein–Gordon equation with weak and strong damping terms and nonlinear logarithmic term is investigated, which is known as one of the nonlinear wave equations in relativistic quantum mechanics and quantum field theory. Firstly, we prove the local existence and uniqueness of weak solution by using the Galerkin method and Contraction mapping principle. The global existence, energy decay and finite time blow-up of the solution with subcritical initial energy are established. Then these conclusions are extended to the critical initial energy. Besides, the finite time blow-up result with supercritical initial energy is shown.

Global well-posedness of wave equation with weak and strong damping terms p-Laplacian and logarithmic nonlinearity source term

We consider the following damped p-laplacian type wave equation with logarithmic nonlinearity utt−Δut−Δpu+ut=|u|q−2uln|u|,x∈Ω,t>0 in a bounded domain with homogeneous Dirichlet boundary condition. Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and when the initial energy is subcritical, a sufficient condition for the solutions to blow up in finite time is derived, by combining the Nehari manifold with concavity argument. Then we parallelly extend the conclusions of global existence and energy decay for the subcritical case to the critical case by scaling technique. Besides, When the initial energy is supercritical, some new skills are invented to establish another finite time blow-up criterion for this problem.

Exponential growth of solutions with L_p-norm of a nonlinear viscoelastic wave equation with strong damping and source and delay terms

" In this work, we are concerned with a problem for a viscoelastic wave equation with strong damping, nonlinear source and delay terms. We show the exponential growth of solutions with $L_{p}$-norm. i.e. $\displaystyle\lim_{t\rightarrow\infty}\Vert u\Vert_{p}^{p}\rightarrow\infty.$"

Exponential Growth of Solutions with L p -Norm of a Klein-Gordon Wave Equation with Strong Damping, Source and Delay Terms

Exponential Growth of Solutions with L p -Norm of a Klein-Gordon Wave Equation with Strong Damping, Source and Delay Terms

Qualitative properties of solution to a viscoelastic Kirchhoff-like plate equation

This paper is concerned with the initial boundary value problem for viscoelastic Kirchhoff-like plate equations with rotational inertia, memory, p-Laplacian restoring force, weak damping, strong damping, and nonlinear source terms. We establish the local existence and uniqueness of the solution by linearization and the contraction mapping principle. Then, we obtain the global existence of solutions with subcritical and critical initial energy by applying potential well theory. Then, we prove the asymptotic behavior of the global solution with positive initial energy strictly below the depth of the potential well. Finally, we conduct a comprehensive study on the finite time blow-up of solutions with negative initial energy, null initial energy, and positive initial energy strictly below the depth of the potential well and arbitrary positive initial energy, respectively.

Blow up for a nonlinear viscoelastic wave equation with strong damping, source and delay terms

Blow up for a nonlinear viscoelastic wave equation with strong damping, source and delay terms

Magnetic Helicity Fluxes from Triple Correlators

Fluxes of the magnetic helicity density play an important role in large-scale turbulent dynamos, allowing the growth of large-scale magnetic fields while overcoming catastrophic quenching. We show here, analytically, how several important types of magnetic helicity fluxes can arise from terms involving triple correlators of fluctuating fields in the helicity density evolution equation. For this, we assume incompressibility and weak inhomogeneity, and use a quasi-normal closure approximation: fourth-order correlators are replaced by products of second-order ones, and the effect of the fourth-order cumulants on the evolution of the third moments is modeled by a strong damping term. First, we show how a diffusive helicity flux, until now only measured in simulations, arises from the triple correlation term. This is accompanied by what we refer to as a random advective flux, which predominantly transports magnetic helicity along the gradients of the random fields. We also find that a new helicity flux contribution, in some aspects similar to that first proposed by Vishniac, can arise from the triple correlator. This contribution depends on the gradients of the random magnetic and kinetic energies along the large-scale vorticity, and thus arises in any rotating, stratified system, even if the turbulence is predominantly non-helical. It can source a large-scale dynamo by itself while spatially transporting magnetic helicity within the system.

A new strict decay rate for systems of longitudinal $ m $-nonlinear viscoelastic wave equations

<abstract><p>Recent years have been marked by a significant increase in interest in solving nonlinear equations that arise in various fields of natural science. This trend is associated with the creation of a new method of mathematical physics. The present study is devoted to the analysis of the propagation of $ m $-nonlinear viscoelastic waves equations in an unbounded domain. The physical properties are determined by the equations of the linear theory of viscoelasticity. This article shows the main effect and interaction between the different weak and strong damping terms on the behavior of solutions. We found, under a novel condition on the kernel functions, an energy decay rate by using an appropriate energy estimates.</p></abstract>

Qualitative results for a relativistic wave equation with multiplicative noise and damping terms

<abstract><p>Wave equations describing a wide variety of wave phenomena are commonly seen in mathematical physics. The inclusion of a noise term in a deterministic wave equation allows neglected degrees of freedom or fluctuations of external fields describing the environment to be considered in the equation. Moreover, adding a noise term to the deterministic equation reveals remarkable new features in the qualitative behavior of the solution. For example, noise can lead to singularities in some equations and prevent singularities in others. Taking into account the effects of the fluctuations along with a space-time white noise, we consider a relativistic wave equation with weak and strong damping terms and investigate the effect of multiplicative noise on the behavior of solutions. The existence of local and global solutions is provided, and some qualitative properties of solutions, such as continuous dependence of solutions on initial data, and blow up of solutions, are given. Moreover, an upper bound is provided for the blow up time.</p></abstract>

Global Well-posedness of Solutions for the $p$-Laplacian Hyperbolic Type Equation with Weak and Strong Damping Terms and Logarithmic Nonlinearity

In this paper, we consider the $p$-Laplacian hyperbolic type equation with weak and strong damping terms and logarithmic nonlinearity. By using the potential well method and a logarithmic Sobolev inequality, we prove global existence, infinite time blow up and asymptotic behavior of solutions in two cases $E(0) \lt d$ and $E(0) = d$. Furthermore, the infinite time blow up of solutions for the problem with $E(0) \gt 0$ ($\omega = 0$) is studied.

On the Global Behaviour of Solutions for a Delayed Viscoelastic-Type Petrovesky Wave Equation with p-Laplacian Operator and Logarithmic Source

This research is concerned with a nonlinear p-Laplacian-type wave equation with a strong damping and logarithmic source term under the null Dirichlet boundary condition. We establish the global existence of the solutions by using the potential well method. Moreover, we prove the stability of the solutions by the Nakao technique. An example with illustrative figures is provided as an application.

Global well-posedness of fourth-order Petrovsky equation with weak and strong damping terms

In this paper, a semilinear fourth-order Petrovsky equation with nonlinear weak damping term and linear strong damping term is considered in the frame of the potential well. We prove the existence and uniqueness of the global strong solution with sub-critical initial energy. In addition, we obtain the global existence, asymptotic behaviour and finite time blowup for the weak (non-steady state) solution with a critical initial energy level.

Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models

Abstract For studying the evolution of the transverse deflection of an extensible beam derived from the connection mechanics, we investigate the initial boundary value problem of nonlinear extensible beam equation with linear strong damping term, nonlinear weak damping term, and nonlinear source term. The key idea of our analysis is to describe the invariant manifold via Nehari manifold. To establish the results of global well-posedness of solution, we consider the problem at three different initial energy levels, i.e., subcritical initial energy level, critical initial energy level, and arbitrarily high initial energy level. We first obtain the local existence of the solution by using the contraction mapping principle. Then, in the framework of potential well, we obtain global existence, nonexistence, and asymptotic behavior of solution for both subcritical initial energy level and critical initial energy level. In the end, we establish the global nonexistence of solution for the problem with linear weak damping and strong damping at the arbitrarily high initial energy level.

Upper semicontinuity of optimal attractors for viscoelastic equations lacking strong damping

ABSTRACT This paper devotes to investigating the asymptotic behavior of viscoelastic equation with fading memory lacking strong damping term where . Its key feature is the lack of strong mechanical damping term replaced by weaker dissipative term (i.e. memory term). At present, the existing results only considered the existence and regularity of global attractors, but do not obtain the existence of the strong attractors, because the compactness is difficult to get in highly regular space. Accordingly, in current work, we will prove the existence of strongly global attractors with only weak dissipation when the nonlinearity f satisfies critical growth by using the dominated convergence theorem and the asymptotically contractive function method. Moreover, the upper semicontinuity of the strongly global attractors is also obtained when the perturbed parameter , which deepens some conclusions that exist in Conti et al. [Asymptotics of viscoelastic materials with nonlinear density and memory effects. J Differ Equ. 2018;264:4235–4259] and Qin et al. [Uniform attractors for a nonautonomous viscoelastic equation with a past history. Nonlinear Theory Methods Appl. 2014;101:1–15].

Blow up for a nonlinear viscoelastic wave equation with strong damping, source and delay terms

Blow up for a nonlinear viscoelastic wave equation with strong damping, source and delay terms

Initial boundary value problem for a damped wave equation with logarithmic nonlinearity

In this paper, a damped semilinear wave equation with logarithmic non-linearity is considered. Finite time blow-up criteria are established for solutions with both lower and higher initial energy, and an upper bound for the blow-up time is derived for each case. Moreover, by making full use of the strong damping term, a lower bound for the blow-up time is also obtained, for both subcritical and supercritical exponent. This partially extends some recent results obtained in [Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Analysis, Real World Applications, 51(2020), 102968] by Di et al.

Global existence and blow-up phenomenon for a quasilinear viscoelastic equation with strong damping and source terms

Considered herein is the global existence and non-global existence of the initial-boundary value problem for a quasilinear viscoelastic equation with strong damping and source terms. Firstly, we introduce a family of potential wells and give the invariance of some sets, which are essential to derive the main results. Secondly, we establish the existence of global weak solutions under the low initial energy and critical initial energy by the combination of the Galerkin approximation and improved potential well method involving with \(t\). Thirdly, we obtain the finite time blow-up result for certain solutions with the non-positive initial energy and positive initial energy, and then give the upper bound for the blow-up time \(T^\ast\). Especially, the threshold result between global existence and non-global existence is given under some certain conditions. Finally, a lower bound for the life span \(T^\ast\) is derived by the means of integro-differential inequality techniques.

The Family of Exponential Attractors and Inertial Manifolds for a Generalized Nonlinear Kirchhoff Equations

In this paper, we study the long-time behavior of a class of generalized nonlinear Kichhoff equation under the condition of n dimension. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup related to the initial-boundary value problem are proved, and then the existence of its exponential attractor is obtained. By extending the space E0 to Ek, a family of the exponential attractors of the initial-boundary value problem is obtained. In the second part, we consider the long-time behavior for a system of generalized Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectrum interval condition.

On a high-order iteration technique for a wave equation with nonlinear viscoelastic term

In this paper, a high-order iterative technique for solving a wave equation with strong damping and nonlinear viscoelastic term is constructed. For this purpose, we adapt the high-order iterative method used in the earlier works and establish the existence theorem of a recurrent sequence associated with the proposed problem. Thereafter, we prove a N-order convergence of the obtained sequence to a unique solution of the problem.