An attractor for a nonlinear dissipative wave equation of Kirchhoff type

Global Existence and Energy Decay Rates for a Kirchhoff-Type Wave Equation with Nonlinear Dissipation

The first objective of this paper is to prove the existence and uniqueness of global solutions for a Kirchhoff-type wave equation with nonlinear dissipation of the form Ku'' + M(|A (1/2) u|(2))Au + g(u') = 0 under suitable assumptions on K, A, M(·), and g(·). Next, we derive decay estimates of the energy under some growth conditions on the nonlinear dissipation g. Lastly, numerical simulations ...

Representation of Dissipative Solutions to a Nonlinear Variational Wave Equation

The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuous and the discontinuities are not always crossed transversally. Solutions to the semilinear system...

A High Order Iterative Scheme for a Nonlinear Kirchhoff Wave Equation in the Unit Membrane

Global Attractor for the Generalized Dissipative KDV Equation with Nonlinearity

In order to study the longtime behavior of a dissipative evolutionary equation, we generally aim to show that the dynamics of the equation is finite dimensional for long time. In fact, one possible way to express this fact is to prove that dynamical systems describing the evolutional equation comprise the existence of the global attractor 1 . The KDV equation without dissipative and forcing was...

Nonlinear Perturbations of the Kirchhoff Equation

In this article we study the existence and uniqueness of local solutions for the initial-boundary value problem for the Kirchhoff equation u′′ −M(t, ‖u(t)‖)∆u+ |u| = f in Ω× (0, T0), u = 0 on Γ0×]0, T0[, ∂u ∂ν + δh(u′) = 0 on Γ1×]0, T0[, where Ω is a bounded domain of Rn with its boundary constiting of two disjoint parts Γ0 and Γ1; ρ > 1 is a real number; ν(x) is the exterior unit normal vector...