Nonlinear Viscoelastic Equation Research Articles

Global Existence of Solutions for the Viscoelastic Kirchhoff Equation with Logarithmic Source Terms

In this paper, a nonlinear viscoelastic Kirchhoff equation in a bounded domain with a time-varying delay term and logarithmic nonlinearity in the weakly nonlinear internal feedback is considered, where the global and local existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo-Galerkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given. These results extend and improve many results in the literature.

Polynomial decay rate for a new class of viscoelastic Kirchhoff equation related with Balakrishnan-Taylor dissipation and logarithmic source terms

In this paper, a polynomial decay rate of Kirchhoff’s nonlinear viscoelastic viscoelastic equation solution related with Balakrishnan-Taylor dissipation solution and logarithmic source terms is obtained, where we obtain the result of energy decay of solutions without impose the usual relationship between certain relaxation functions and their derivatives. This result generalizes the previous results to an arbitrary decay rate, that is not necessarily of the polynomial or exponential decay.

Polynomial-exponential stability and blow-up solutions to a nonlinear damped viscoelastic Petrovsky equation

This work is concerned with the initial boundary value problem for a nonlinear viscoelastic Petrovsky equation $$\begin{aligned} u_{tt}+\Delta ^{2}u-\int _{0}^{t}g(t-\tau )\Delta ^{2}u(\tau )d\tau -\Delta u_{t}-\Delta u_{tt}+u_{t}|u_{t}|^{m-1}=u|u|^{p-1}. \end{aligned}$$We prove that the solution energy has polynomial rate of decay, even if the kernel g decays exponentially provided $$m>1$$ while decay rates is exponentially in the case of weak damping. The unbounded properties of solutions in two cases $$m=1$$ and $$p>m\ge 1$$ have been also investigated. For the first case, we prove the blow-up of solutions with different ranges of initial energy. For the second case, we prove blow-up of solutions under some restrictions on g when the initial energy is negative or non negative at less than potential well depth.

General decay for a coupled Lamé system of nonlinear viscoelastic equations

In this paper, we consider a coupled Lamé system of nonlinear viscoelastic equations with general source terms. Under some suitable conditions on the initial data and the relaxation functions, we prove an asymptotic stability result of global solution taking into account that the kernel is not necessarily decreasing. This work generalizes and improves earlier results in the literature.

Dynamic Properties of a Nonlinear Viscoelastic Kirchhoff-Type Equation with Acoustic Control Boundary Conditions. I

In this paper, we consider the nonlinear viscoelastic Kirchhoff-type equation $${u_{tt}} - M(||\nabla u||_2^2)\Delta u + \int_0^t {h(t - s)\Delta u(s)ds + a|{u_t}{|^{m - 2}}{u_t} = |u{|^{p - 2}}u} $$ with initial conditions and acoustic boundary conditions. We show that, depending on the properties of convolution kernel h at infinity, the energy of the solution decays exponentially or polynomially as t → + ∞. Our approach is based on integral inequalities and multiplier techniques. Instead of using a Lyapunov-type technique for some perturbed energy, we concentrate on the original energy, showing that it satisfies a nonlinear integral inequality which, in turn, yields the final decay estimate.

Dynamics of a Non-Linear Stochastic Viscoelastic Equation with Multiplicative Noise

Dynamics of a Non-Linear Stochastic Viscoelastic Equation with Multiplicative Noise

General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan‐Taylor damping and logarithmic nonlinearity

This work deals with the study of a new class of nonlinear viscoelastic Kirchhoff equation with Balakrishnan‐Taylor damping and logarithmic nonlinearity. A decay result of the energy of solutions for the problem without imposing the usual relation between a certain relaxation function and its derivative is established. This result generalizes earlier ones to an arbitrary rate of decay, which is not necessarily of exponential or polynomial decay.

Decay rates for a coupled quasilinear system of nonlinear viscoelastic equations

Abstract In this paper, we consider a nonlinear quasilinear system of two coupled viscoelastic equations and investigate the asymptotic behavior of this system. We establish an explicit and general formula for the energy decay rates. Our result allows a wider class of relaxation functions, which improves earlier results existing in the literature.

Global Nonexistence for a Nonlinear Viscoelastic Equation with Nonlinear Damping and Velocity-Dependent Material Density

We consider the initial boundary value problem of a nonlinear viscoelastic equation of Kirchhoff-type with nonlinear damping and velocity-dependent material density. We establish a nonexistence result of global solutions with positive initial energy and negative initial energy, respectively.

Dynamic Properties of a Nonlinear Viscoelastic Kirchhoff-Type Equation with Acoustic Control Boundary Conditions. I

В данной работе рассматривается нелинейное уравнение вязкоупругости типа Кирхгофа $$ u_{tt}-M(\|\nabla u\|^2_2)\Delta u +\int_0^t h(t-s)\Delta u(s) ds+a|u_t|^{m-2}u_t=|u|^{p-2}u $$ с начальными условиями и акустическими граничными условиями. В зависимости от свойств ядер свертки $h$ на бесконечности будет показано, что энергия решения убывает экспоненциально или полиномиально при $t\to +\infty$. Наш подход основан на использовании техники интегральных неравенств и множителей. Вместо того чтобы применять технику типа Ляпунова к некоторой возмущенной энергии, мы рассмотрим исходную энергию и покажем, что она удовлетворяет нелинейному интегральному неравенству, которое, в свою очередь, дает окончательную оценку убывания. Библиография: 33 названия.

DYNAMIC PROPERTIES FOR NONLINEAR VISCOELASTIC KIRCHHOFF-TYPE EQUATION WITH ACOUSTIC CONTROL BOUNDARY CONDITIONS Ⅱ

In this paper, we consider the nonlinear viscoelastic Kirchhoff-type equation with initial conditions and acoustic boundary conditions. Under suitable conditions on the initial data, the relaxation function h(·) and M(·), we prove that the solution blows up in finite time and give the upper bound of the blow-up time T*.

General Decay and Blow-Up Results for Nonlinear Fourth-Order Integro-Differential Equation

This study aims at considering an initial-boundary value problem for nonlinear fourth-order viscoelastic equation in a bounded domain. Under suitable conditions of the initial data and of the relaxation function, it is proved that the solution energy is generally decayed. It is also shown that regarding arbitrary positive initial energy, certain solutions blow-up in a finite time.

Global Existence and Decay of Solutions for a Class of Viscoelastic Kirchhoff Equation

In this paper, a class of nonlinear viscoelastic Kirchhoff equation in a bounded domain with a time-varying delay in the weakly nonlinear internal feedback is considered, where the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo–Galerkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given.

Global existence of solutions to a viscoelastic non-degenerate Kirchhoff equation

ABSTRACT In this paper, a nonlinear viscoelastic kirchhoff equation in a bounded domain with a time varying delay in the weakly nonlinear internal feedback is considered, where the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo–Galarkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given.

On behaviour of solutions for a nonlinear viscoelastic equation with variable-exponent nonlinearities

This paper deals with a viscoelastic wave equation with variable-exponent nonlinearities, subject to nonlinear boundary feedback. Under appropriate conditions, a general decay result associated to solution energy is proved. It is also shown that regarding arbitrary positive initial energy, solutions blow-up in a finite time.

Asymptotics of viscoelastic materials with nonlinear density and memory effects

This paper is concerned with the nonlinear viscoelastic equation|∂tu|ρ∂ttu−Δ∂ttu−Δu+∫0∞μ(s)Δu(t−s)ds+f(u)=h, suitable to modeling extensional vibrations of thin rods with nonlinear material density ϱ(∂tu)=|∂tu|ρ, and presence of memory effects. This class of equations was studied by many authors, but well-posedness in the whole admissible range ρ∈[0,4] and for f growing up to the critical exponent were established only recently. The existence of global attractors was proved in presence of an additional viscous or frictional damping. In the present work we show that the sole weak dissipation given by the memory term is enough to ensure existence and optimal regularity of the global attractor Aρ for ρ<4 and critical nonlinearity f.

Maximum principle for optimal control of vibrations of a dynamic Gao beam in contact with a rigid foundation

ABSTRACTIn our preceding paper, we studied an optimal control problem of vibrations of a dynamic Gao beam in contact with a reactive foundation and derived the Pontryagin maximum principle for the controlled system in fixed final horizon case. As a follow-up, in this paper, we focus on the investigation of the Gao beam that may come in contact with a rigid foundation underneath it. In this case, the nonlinear viscoelastic beam equation is equipped with the Signorini condition. By the Dubovitskii and Milyutin functional analytical approach, we investigate the new optimal control problem with multiple inequality constraints and present further original results of current interests.

Stabilized finite volume element method for the 2D nonlinear incompressible viscoelastic flow equation

In this article, we devote ourselves to building a stabilized finite volume element (SFVE) method with a non-dimensional real together with two Gaussian quadratures of the nonlinear incompressible viscoelastic flow equation in a two-dimensional (2D) domain, analyzing the existence, stability, and error estimates of the SFVE solutions and verifying the validity of the preceding theoretical conclusions by numerical simulations.

General decay result for nonlinear viscoelastic equations

In this paper we consider a nonlinear viscoelastic equation with minimal conditions on the L1(0,∞) relaxation function g namely g′(t)≤−ξ(t)H(g(t)), where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of g at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H(s)=sp and p covers the full admissible range [1,2). We get the best decay rates expected under this level of generality and our new results substantially improve several earlier related results in the literature.

Nonlinear viscoelastic dynamic responses of bi-graphene/piezoelectric laminated films under moving particles

This paper reports the result of an investigation into the nonlinear viscoelastic dynamic characteristics of bi-graphene sheets/piezoelectric (BGP) laminated films subjected to moving particles (gas molecules, lithium ion, catalyst molecules, etc.). Based on the nonlocal elasticity theory and the von Kármán nonlinear geometric relation, and considering a rigorous van der Waals force between any two layers and the viscoelastic behavior of piezoelectric, the nonlinear viscoelastic dynamic equations of BGP laminated films is found by utilizing Hamilton's principles and Galerkin method. A reformulated differential quadrature method (DQM) is used to solve the nonlinear viscoelastic dynamic equation. Results show that the applied voltage and the mass of moving particles appear in larger effect on the nonlinear dynamic characteristics of BGP laminated films; the viscoelastic behavior of piezoelectric layer decreases the nonlinear dynamic amplitude of BGP laminated films only in free vibration part; the difference between linear and nonlinear solutions depends on the speed of moving particles. The obtained new features and interesting results about the nonlinear viscoelastic dynamic responses of bi-graphene/piezoelectric laminated films under moving particles in this paper are helpful for the design of chemical, physical and biomechanical sensors for the possible applications in sensing mass, force and charge, in which graphene/piezoelectric films act as basic elements, and will trigger a wave of research on the graphene/piezoelectric films based electrical elements or devises in the near future.