General Decay Research Articles

Asymptotic Behavior for a Coupled Petrovsky–Petrovsky System with Infinite Memories

The main goal of this article is to obtain the existence of solutions for a nonlinear system of a coupled Petrovsky–Petrovsky system in the presence of infinite memories under minimal assumptions on the functions g1,g2 and φ1,φ2. Here, g1,g2 are relaxation functions and φ1,φ2 represent the sources. Also, a general decay rate for the associated energy is established. Our work is partly motivated by recent results, with a necessary modification imposed by the nature of our problem. In this work, we limit our results to studying the system in a bounded domain. The case of the entire domain Rn requires separate consideration. Of course, obtaining such a result will require not only serious technical work but also the use of new techniques and methods. In particular, one of the most significant points in achieving this goal is the use of the perturbed Lyapunov functionals combined with the multiplier method. To the best of our knowledge, there is no result addressing the linked Petrovsky–Petrovsky system in the presence of infinite memory, and we have overcome this lacune.

Orientation Mixing in Active Suspensions

We study a popular kinetic model introduced by Saintillan and Shelley for the dynamics of suspensions of active elongated particles where the particles are described by a distribution in space and orientation. The uniform distribution of particles is the stationary state of incoherence which is known to exhibit a phase transition. We perform an extensive study of the linearised evolution around the incoherent state. We show (i) in the non-diffusive regime corresponding to spectral (neutral) stability that the suspensions experience a mixing phenomenon similar to Landau damping and we provide optimal pointwise in time decay rates in weak topology. Further, we show (ii) in the case of small rotational diffusion ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} that the mixing estimates persist up to time scale ν-1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u ^{-1/2}$$\\end{document} until the exponential decay at enhanced dissipation rate ν1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u ^{1/2}$$\\end{document} takes over. The interesting feature is that the usual velocity variable of kinetic models is replaced by an orientation variable on the sphere. The associated orientation mixing leads to limited algebraic decay for macroscopic quantities. For the proof, we start with a general pointwise decay result for Volterra equations that may be of independent interest. While, in the non-diffusive case, explicit formulas on the sphere allow to conclude the desired decay, much more work is required in the diffusive case: here we prove mixing estimates for the advection-diffusion equation on the sphere by combining an optimized hypocoercive approach with the vector field method. One main point in this context is to identify good commuting vector fields for the advection-diffusion operator on the sphere. Our results in this direction may be useful to other models in collective dynamics, where an orientation variable is involved.

Existence and General Decay of Solutions for a Weakly Coupled System of Viscoelastic Kirchhoff Plate and Wave Equations

In this paper, a weakly coupled system (by the displacement of symmetric type) consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain is considered. Under the assumptions on a more general type of relaxation functions, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions.

Decay of a Thermoelastic Laminated Beam with Microtemperature Effects, Nonlinear Delay, and Nonlinear Structural Damping

This article deals with a non-classical model, namely a thermoelastic laminated beam along with microtemperature effects, nonlinear delay, and nonlinear structural damping, where the last two terms both affect the equation which depicts the dynamics of slip. With the help of convenient conditions in both weight delay and wave speeds, we demonstrate explicit and general energy decay rates of the solution. To attain our interests, we highlight useful properties regarding convex functions and apply a specific approach known as the multiplier technique, which enables us to prove the stability results. Our results here aim to show the impact of different types of damping by taking into account the interaction between them, which extends recent publications in the literature.

General decay and blow up of solutions for a plate viscoelastic p(x)-Kirchhoff type equation with variable exponent nonlinearities and boundary feedback

In this paper, we consider a plate viscoelastic p(x)–Kirchhoff type equation with variable-exponent nonlinearities of the form associated with initial and boundary feedback. Under appropriate conditions on p(·), m(·) and q(·), general decay result along the solution energy is proved. By introducing a suitable auxiliary function, it is also shown that regarding negative initial energy and a suitable range of variable exponents, solutions blow up in a finite time.

Global existence and general decay for a weak viscoelastic equation with acoustic boundary conditions and a logarithmic source term

Global existence and general decay for a weak viscoelastic equation with acoustic boundary conditions and a logarithmic source term

Initial boundary value problem for a viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term: decay estimates and blow-up result

In this paper, we study the initial boundary value problem for the following viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term where the relaxation function satisfies g'(t)leq -xi (t)g^{r}(t), tgeq 0, 1leq r< frac{3}{2}. The main goal of this work is to study the global existence, general decay, and blow-up result. The global existence has been obtained by potential-well theory, the decay of solutions of energy has been established by introducing suitable energy and Lyapunov functionals, and a blow-up result has been obtained with negative initial energy.

New general decay results for a fully dynamic piezoelectric beam system with fractional delays under memory boundary controls

In this paper, we investigate the general stability results of a fully dynamic piezoelectric beam system with internal fractional delays under memory boundary controls. Firstly, by introducing two new equations and using two Volterra's inverse operators to deal with fractional delay terms and boundary memory terms, respectively, a new equivalent system is obtained. Secondly, by combining Faedo–Galerkin approximations with the compactness argument, we prove the local existence and uniqueness of weak solution. Finally, under a broader class of kernel functions, by constructing Lyapunov functionals and new equations and applying some technical lemmas, we establish the optimal and explicit energy decay results for two cases, namely, initial values and without imposing initial values equal to 0, that is, . It is worth pointing out that we focus more on the analysis of , because the stability results of are special cases of . This is the first time to analyze the stability of strong coupling problem by combining internal fractional delay and boundary viscoelastic damping. The obtained general decay results are not necessarily exponential or polynomial, which generalize the relevant results in the existing literature.

General decay and blow-up of solution for a viscoelastic wave equation with nonlinear degenerate damping and source terms

We are concerned with the initial-boundary value problem for a nonlinear viscoelastic problem with the degenerate damping and the nonlinear source term in this paper. Under appropriate assumptions on the initial data, memory kernel and parameters, we establish several results regarding the global solution, blow-up properties and general decay in the case of positive initial energy. We overcome the difficulties brought by positive initial energy by using the potential well theory.

The asymptotic behavior of solutions for stochastic evolution equations with pantograph delay

The polynomial stability problem of stochastic delay differential equations has been studied in recent years. In contrast, there are relatively few works on stochastic partial differential equations with pantograph delay. The present paper is devoted to investigating large-time asymptotic properties of solutions for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. We first show that the mild solutions of stochastic pantograph delay evolution equations with nonlinear multiplicative noise tend to zero with general decay rate (including both polynomial and logarithmic rates) in the [Formula: see text]th moment and almost sure senses. The analysis is based on the Banach fixed point theorem and various estimates involving the gamma function. Moreover, by using a generalized version of the factorization formula and exploiting an approximation technique and a convergence analysis, we construct the nontrivial equilibrium solution, defined for [Formula: see text], for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. In particular, the uniqueness, Hölder regularity in time and general stability, in the [Formula: see text]th moment and almost sure senses, of the nontrivial equilibrium solution are established.

Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms

Viscoelastic damping phenomena are ubiquitous in diverse kinds of wave motions of nonlinear media. This arouses extensive interest in studying the existence, the finite time blow-up phenomenon and various large time behaviors of solutions to viscoelastic wave equations. In this paper, we are concerned with a class of variable coefficient coupled quasi-linear wave equations damped by viscoelasticity with a long-term memory fading at very general rates and possibly damped by friction but provoked by nonlinear interactions. We prove a local existence result for solutions to our concerned coupled model equations by applying the celebrated Faedo-Galerkin scheme. Based on the newly obtained local existence result, we prove that solutions would exist globally in time whenever their initial data satisfy certain conditions. In the end, we provide a criterion to guarantee that some of the global-in-time-existing solutions achieve energy decay at general rates uniquely determined by the fading rates of the memory. Compared with the existing results in the literature, our concerned model coupled wave equations are more general, and therefore our theoretical results have wider applicability. Modified energy functionals (can also be viewed as certain Lyapunov functionals) play key roles in proving our claimed general energy decay result in this paper.

General decay anti-synchronization and [formula omitted] anti-synchronization of coupled reaction-diffusion complex-valued memristive neural networks

This paper studies the general decay anti-synchronization (GDAS) and general decay H∞ anti-synchronization (GDHAS) problems for coupled reaction–diffusion complex-valued memristive neural networks (CRDCVMNNs) without and with time delays. First of all, the concept of GDHAS under the complex-valued case is firstly presented based on the definitions of ψ-type function and ψ-type stability. According to this concept, we further prove a novel lemma which will be used throughout this paper under a new assumption. This lemma is very crucial which gives a new sufficient condition for achieving GDAS of CNNs in complex field. Then, by using the derived lemma and designing a suitable nonlinear feedback controller, some criteria for CRDCVMNNs without and with time delays to realize GDAS and GDHAS are established. At last, two numerical examples are used to verify the effectiveness of the anti-synchronization results.

A general decay criterion for a viscoelastic equation with nonlinear distributed delay and damping effects

In this paper, a viscoelastic equation with nonlinear distributed delay and nonlinear damping effects is considered. Under very general conditions that do not involve differential inequalities for the kernel function, a general decay criterion is established through the perturbed energy method and properties of convex functions. We extend the stability result of wave equation with linear delay and linear damping to the nonlinear case.

General Decay Synchronization of Fuzzy Inertial Memristive Neural Networks with Discontinuous Activation Function

General Decay Synchronization of Fuzzy Inertial Memristive Neural Networks with Discontinuous Activation Function

Some qualitative properties of weak solution for pseudo‐parabolic equation with viscoelastic term and Robin boundary conditions

AbstractIn this paper, we consider the initial boundary value problem of the generalized pseudo‐parabolic equation containing viscoelastic terms and associated with Robin conditions. We establish first the local existence of solutions by the standard Galerkin method. Then, we prove blow‐up results for solutions when the initial energy is negative or nonnegative but small enough or positive arbitrary high initial energy, respectively. We also establish the lifespan and the blow‐up rate for the weak solution by finding the upper bound and the lower bound for the blow‐up times and the upper bound and the lower bound for the blow‐up rate. For negative energy, we introduce a new method to prove blow‐up results with a sharper estimate for the upper bound for the blow‐up times. Finally, we prove both the global existence of the solution and the general decay of the energy functions under some restrictions on the initial data.

Theoretical and numerical decay results of a viscoelastic suspension bridge with variable exponents nonlinearity

AbstractStrong vibrations can cause considerable damage to structures and break materials apart. The main reason for the Tacoma Narrows Bridge collapse was the sudden transition from longitudinal to torsional oscillations caused by a resonance phenomenon. There exists evidence that several other bridges collapsed for the same reason. To overcome unwanted vibrations and prevent structures from resonating during earthquakes and winds, features and modifications such as dampers are used to stabilize these bridges. In this work, we study a nonlinear viscoelastic plate equation with variable exponents, which models the deformation of a suspension bridge. First, we establish explicit and general decay results of the system depending on the decay rate of the relaxation function and the nature of the variable‐exponents nonlinearity. Then, we perform several numerical tests to illustrate our theoretical decay results. Our results extend and generalize many earlier works in the literature.

General decay and lifespan of solutions for a system of nonlinear pseudoparabolic equations with viscoelastic term

In this paper, we consider the Robin-Dirichlet problem for a system of nonlinear pseudoparabolic equations with viscoelastic term. By the Faedo-Galerkin method, we first prove the existence and uniqueness of solution for the problem. Next, we give a sufficient condition to get the global existence and decay of the weak solution. Finally, by the concavity method, we prove the blow-up result of the solution when the initial energy is negative. Furthermore, we establish here the lifespan of the solution by finding the upper bound and the lower bound for the blow-up time.

Existence and general decay estimates of the Dirichlet problem for a nonlinear Kirchhoff wave equation in an annular with viscoelastic term

This paper is devoted to the study of a Kirchhoff wave equation with a viscoelastic term in an annular associated with homogeneous Dirichlet conditions. At first, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we establish a sufficient condition such that any global weak solution is general decay as t → +∞.

Decay results for a viscoelastic wave equation with distributed delay in boundary feedback

"In this work, a nonlinear viscoelastic wave equation is studied. By supposing distributed delay feedback acting on the boundary, we establish the general decay rate under suitable hypothesis."

On the Neumann-Dirichlet problem for a system of nonlinear viscoelastic equations of Kirchhoff type with Balakrishnan- Taylor term

In this paper, the Neumann-Dirichlet boundary problem for a system of nonlinear viscoelastic equations of Kirchhoff type with Balakrishnan-Taylor term is considered. At first, a local existence is established by the linear approximation together with the Faedo- Galerkin method. Then, by establishing several reasonable conditions and suitable energy inequalities, the solution of the problem admits a general decay in time.