Existence and stability results of nonlinear higher-order wave equation with a nonlinear source term and a delay term

summary:We consider the initial-boundary value problem for a nonlinear higher-order nonlinear hyperbolic equation in a bounded domain. The existence of global weak solutions for this problem is established by using the potential well theory combined with Faedo-Galarkin method. We also established the asymptotic behavior of global solutions as $t\rightarrow \infty$ by applying the Lyapunov method

GENERAL DECAY OF SOLUTION FOR COUPLED SYSTEM OF VISCOELASTIC WAVE EQUATIONS OF KIRCHHOFF TYPE WITH DENSITY IN Rn

A system of viscoelastic wave equations of Kirchhoff type is considered. For a wider class of relaxation functions, we use spaces weighted by the density function to establish a very general decay rate of the solution

Existence and stability results of nonlinear higher-order wave equation with a nonlinear source term and a delay term

We consider the initial-boundary value problem for a nonlinear higher-order nonlinear hyperbolic equation in a bounded domain. The existence of global weak solutions for this problem is established by using the potential well theory combined with Faedo-Galarkin method. We also established the asymptotic behavior of global solutions as $t\rightarrow\infty$ by applying the Lyapunov method

Well-posedness and asymptotic stability for the Lamé system with internal distributed delay

In this work, we consider the Lamé system in 3-dimension bounded domain with distributed delay term. We prove, under some appropriate assumptions, that this system is well-posed and stable. Furthermore, the asymptotic stability is given by using an appropriate Lyapunov functional

Advanced neural network approaches for coupled equations with fractional derivatives

Abstract We investigate numerical solutions and compare them with Fractional Physics-Informed Neural Network (FPINN) solutions for a coupled wave equation involving fractional partial derivatives. The problem explores the evolution of functions u and v over time t and space x. We employ two numerical approximation schemes based on the finite element method to discretize the system of equations. The effectiveness of these schemes is validated by comparing numerical results with exact solutions. Additionally, we introduce the FPINN method to tackle the coupled equation with fractional derivative orders and compare its performance against traditional numerical methods. Key findings reveal that both numerical approaches provide accurate solutions, with the FPINN method demonstrating competitive performance in terms of accuracy and computational efficiency. Our study highlights the significance of employing FPINNs in solving fractional differential equations and underscores their potential as alternatives to conventional numerical methods. The novelty of this work lies in its comparative analysis of traditional numerical techniques and FPINNs for solving coupled wave equations with fractional derivatives, offering insights into advancing computational methods for complex physical systems

Stability for Weakly Coupled Wave Equations with a General Internal Control of Diffusive Type

The present paper deals with well-posedness and asymptotic stability for weakly coupled wave equations with a more general internal control of diffusive type. Owing to the semigroup theory of linear operator, the well-posedness of system is proved. Furthermore, we show a general decay rate result. The method is based on the frequency domain approach combined with multiplier technique

Decay Properties for Transmission System with Infinite Memory and Distributed Delay

We consider a damped transmission problem in a bounded domain where the damping is effective in a neighborhood of a suitable subset of the boundary. Using the semigroup approach together with Hille–Yosida theorem, we prove the existence and uniqueness of global solution. Under suitable assumption on the geometrical conditions on the damping, we establish the exponential stability of the solution by introducing a suitable Lyapunov functional

Проблема плотности некоторых функциональных пространств для изучения динамических уравнений на временных масштабах

In this paper we study some topological density properties of certain functional spaces on the time scales and its relationships to Lebesgue spaces in the sense of ∇-integrals on time scales. Our results are provided with applicationsВ этой статье мы изучаем некоторые свойства топологической плотности некоторых функциональных пространств на временных масштабах и их отношения с пространствами Лебега в смысле ∇-интегралов на временных масштабах. Наши результаты снабжены приложениям