Wave Equations Of Kirchhoff Type Research Articles

Existence and asymptotic behavior of solutions for non-linear wave equations of Kirchhoff type with viscoelasticity

"In this paper we discuss the global existence and the asymptotic behavior of solutions of an initial boundary value problem of a non-linear wave equation of Kirchhoff type. "

High-order iterative scheme for Kirchhoff-type wave equation with the source containing three unknown values

In this paper, a high-order iterative scheme is established in order to get a convergent sequence at a rate of order N, (N ≥ 2) to a local unique weak solution of a nonlinear Kirchhoff-type wave equation associated with Robin conditions.

Blow Up Of Solutions For Viscoelastic Wave Equations Of Kirchhoff Type With Variable Exponents

In this work, we consider the blow up of solutions for the viscoelastic wave equation of Kirchhoff type with variable exponents. The present result in this work improve the previous literature.

Well-posedness and stability for Kirchhoff equation with non-porous acoustic boundary conditions

In this paper we prove the well-posedness to the wave equation of Kirchhoff type. Under a portion of the boundary, we consider the acoustic boundary conditions. We also prove the exponential stability of the energy associated to the problem. Our result generalize the previous literature where only the Carrier model was considered (when the nonlinearity involves the L2(Ω) norm) or the Kirchhoff model with porous acoustic boundary conditions. The main tool is the Faedo-Galerkin method. Due to the presence of acoustic boundary conditions, we can not use special basis and new estimates are necessary. To prove the stability we use integral estimates.

Global existence and decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with logarithmic nonlinearity

This work deals with a system of Kirchhoff‐type equations with viscoelastic term and logarithmic nonlinearity. First, we proved the global existence by combining the potential well method with Faedo–Galerkin's procedure. Later, we established decay rate estimates by using multiplier method.

Well-posedness and energy decay of a transmission problem of Kirchhoff type wave equations with damping and delay terms

We consider a transmission problem of Kirchhoff type wave equations with delay and damping terms, subject to a memory condition on one part of the boundary. Under appropriate hypotheses on the relaxation function and the relationship between weights of damping and delay terms, we establish well-posedness of the problem. Using the Faedo-Galerkin approximation technique, and introducing suitable energyand Lyapunov functionals, we obtain estimates for exponential, polynomial, and logarithmic decay. For more information see https://ejde.math.txstate.edu/Volumes/2021/95/abstr.html

Numerical results for a fourth‐order nonlinear wave equation of Kirchhoff type with a viscoelastic term

AbstractIn this paper, numerical results for a fourth‐order nonlinear wave equation of Kirchhoff type with a viscoelastic term are established. First, the existence and uniqueness of a solution is proved by applying the Faedo–Galerkin method and the weak compact method. Next, an algorithm to the initial boundary value problem for a nonlinear evolution equation of fourth order is studied where the spatial‐variable discretization and time‐variable discretization are constructed to approximate derivatives by the difference schemes. Finally, the numerical results are shown to present the errors of proposed algorithm. Simulations of the approximate solution and the exact solution are also visualized.

A mixed finite element method for solving coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term

This paper is concerned with the numerical approximation of the solution of the coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term using a mixed finite element method. The Raviart‐Thomas mixed finite element method is one of the most prominent techniques to discretize the second‐order wave equations; therefore, we apply this scheme for space discretization. Furthermore, an L2‐in‐space error estimate is presented for this mixed finite element approximation. Finally, the efficiency of the method is verified by a numerical example.

Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity

<p style='text-indent:20px;'>In this paper, we deal with the initial boundary value problem of the following fractional wave equation of Kirchhoff type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} u_{tt}+M([u]_{\alpha, 2}^2)(-\Delta)^{\alpha}u+(-\Delta)^{s}u_{t} = \int_{0}^{t}g(t-\tau)(-\Delta)^{\alpha}u(\tau)d\tau+\lambda|u|^{q -2}u, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ M:[0, \infty)\rightarrow (0, \infty) $\end{document}</tex-math></inline-formula> is a nondecreasing and continuous function, <inline-formula><tex-math id="M2">\begin{document}$ [u]_{\alpha, 2} $\end{document}</tex-math></inline-formula> is the Gagliardo-seminorm of <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^\alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> are the fractional Laplace operators, <inline-formula><tex-math id="M6">\begin{document}$ g:\mathbb{R}^+\rightarrow \mathbb{R}^+ $\end{document}</tex-math></inline-formula> is a positive nonincreasing function and <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is a parameter. First, the local and global existence of solutions are obtained by using the Galerkin method. Then the global nonexistence of solutions is discussed via blow-up analysis. Our results generalize and improve the existing results in the literature.</p>

Exponential decay of the viscoelastic wave equation of Kirchhoff type with a nonlocal dissipation

The viscoelastic wave equation of Kirchhoff type with nonlinear and nonlocal damping is considered in a bounded domain of R^N. The existence of global solutions and decay rates of the energy are proved.

Long-time dynamics of Kirchhoff equations with exponential nonlinearities

Our aim in this paper is to study the initial boundary problem for the two-dimensional Kirchhoff type wave equation with an exponentially growing source term. We first prove that the Kirchhoff wave model is globally well-posed in (H01(Ω)∩L∞(Ω))×L2(Ω), which covers the case of degenerate stiffness coefficient, and then obtain that the semigroup generated by the problem has a global attractor in the corresponding phase space. We also point out that the above results are still true in the natural energy space H01(Ω)×L2(Ω).

General Energy Decay Rates for a Nonlinear Wave Equation of Kirchhoff Type with Memory and Acoustic Boundary Conditions

General Energy Decay Rates for a Nonlinear Wave Equation of Kirchhoff Type with Memory and Acoustic Boundary Conditions

Attractors for the degenerate Kirchhoff wave model with strong damping: Existence and the fractal dimension

In this paper, we study the long time behavior of the Kirchhoff type wave equation in the space H01(Ω)×L2(Ω). We prove the existence of the global attractor for the equation which covers the case of possible generation of the stiffness coefficient. We also consider the geometrical property of the global attractor. By means of the Z2 index, we provide, under suitable assumptions, the fractal dimension of the global attractor is infinite.

Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term

A transmission problem for Kirchhoff-type wave equations with nonlinear damping and delay term in the internal feedback is considered under a memory condition on one part of the boundary. By virtue of multiplier method, Faedo-Galerkin approximation and energy perturbation technique, we establish the appropriate conditions to guarantee the existence of global solution, and derive a general decay estimate of the energy, which includes exponential, algebraic and logarithmic decay etc.

General decay of energy for a viscoelastic wave equation with a distributed delay term in the nonlinear internal dambing

In this article, we consider a viscoelastic wave equation of Kirchhoff type in a bounded domain with a distributed delay term present in the nonlinear internal dambing. By introducing a functional energy and suitable Lyapunov functional, under suitable assumptions, we establish a general decay result from which the exponential and polynomial decay are only special cases.

Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

The focus of the current paper is to investigate the initial boundary value problem for a system of viscoelastic wave equations of Kirchho type with a delay term in a bounded domain. At first,the energy decay rate is proved by Nakao's technique and expressed polynomially and exponentially depending on the parameter m. However and in the unstable set, for certain initial data, the blow-upof solutions is obtained.

Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

Energy decay rates for solutions of the Kirchhoff type wave equation with boundary damping and source terms

In this work, we are concerned with uniform stabilization for an initial-boundary value problem associated with the Kirchhoff type wave equation with feedback terms and memory condition at the boundary. We prove that the energy decays exponentially when the boundary damping term has a linear growth near zero and polynomially when the boundary damping term has a polynomial growth near zero. Furthermore, we study the decay rate of the energy without imposing any restrictive growth assumption on the damping term near zero.

Upper and Lower Bounds of Blowup Time to a Strongly Damped Wave Equation of Kirchhoff Type with Memory Term and Nonlinear Dissipations

In this paper, a wave equation of Kirchhoff type of the form $$\begin{aligned}&u_{tt}-M(\Vert \nabla u\Vert _{2}^{2})\Delta u + \int _{0}^{t}g(t-s)\Delta u(s)\mathrm{d}s+h(u_{t})-\mu \Delta u_{t}\\&\quad =|u|^{\alpha }u,\quad \mathrm {in}\quad \Omega \times (0,+\infty ) \end{aligned}$$ is considered. Under suitable assumptions on the viscoelastic term and initial data, we prove that the solutions blow up at a finite time. Our result improves the previous work by Chen and Liu (IMA J Appl Math 1–29, 2015) in which the authors obtained an exponential growth of solutions where the memory kernel depends on initial energy and we show that it is not necessary to have such a restriction. This also extends the work by Hu et al. (Bound Value Probl 2017:112, 2017) where the relaxation function has not been considered. Some estimates for lower bounds of blowup time are also given.

Decay of Solution for Degenerate Wave Equation of Kirchhoff Type in Viscoelasticity

In this paper we consider a class of degenerate viscoelastic wave equation with density in Kirchhoff-type. Under appropriate conditions on the initial datums, we prove a very general decay rate of solution using the Lyapunov functions. In order to compensate the lack of Poincare’s inequality in $$\mathbb {R}^{n}$$ , we use spaces weighted by density function.