Class Of Damped Wave Equations Research Articles

Rapid Exponential Stabilization of Nonlinear Wave Equation Derived from Brain Activity via Event-Triggered Impulsive Control

This paper investigates the problem of rapid exponential stabilization for damped wave equations. Based on a new event-triggered impulsive control (ETIC) method, impulsive control was designed to solve the rapid exponential stabilization of a class of damped wave equations derived from brain activity. The effectiveness of our control was verified through a numerical example.

On the solution of hyperbolic two-dimensional fractional systems via discrete variational schemes of high order of accuracy

In this work, we consider a general class of damped wave equations in two spatial dimensions. The model considers the presence of Weyl space-fractional derivatives as well as a generic nonlinear potential. The system has an associated positive energy functional when damping is not present, in which case, the model is capable of preserving the energy throughout time. Meanwhile, the energy of the system is dissipated in the damped scenario. In this work, the Weyl space-fractional derivatives are approximated through second-order accurate fractional centered differences. A high-order compact difference scheme with fourth order accuracy in space and second order in time is proposed. Some associated discrete quantities are introduced to estimate the energy functional. We prove that the numerical method is capable of conserving the discrete variational structure under the same conditions for which the continuous model is conservative. The positivity of the discrete energy of the system is also discussed. The properties of consistency, solvability, stability and convergence of the proposed method are rigorously proved. We provide some numerical simulations that illustrate the agreement between the physical properties of the continuous and the discrete models.

Global solutions and finite time blow up for damped semilinear wave equations

A class of damped wave equations with superlinear source term is considered. It is shown that every global solution is uniformly bounded in the natural phase space. Global existence of solutions with initial data in the potential well is obtained. Finally, not only finite time blow up for solutions starting in the unstable set is proved, but also high energy initial data for which the solution blows up are constructed.

Homoclinic bifurcations and dimension of attractors for damped nonlinear hyperbolic equations

A new method of obtaining lower bounds for the attractor's dimension is suggested which involves analysis of homoclinic bifurcations. The method is applied for obtaining sharp estimates of the attractor's dimension for a class of abstract damped wave equations which are beyond the reach of the classical methods.