Global stability of travelling fronts for a damped wave equation

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

We consider the damped wave equation αutt +ut = uxx−V ′(u) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u(x, t) = h(x − st) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V . We show that, if the initial data are sufficiently close to the profile of...

Stability of Travelling Waves for a Damped Hyperbolic Equation

We consider a nonlinear damped hyperbolic equation in R, 1 ≤ n ≤ 4, depending on a positive parameter ǫ. If we set ǫ = 0, this equation reduces to the well-known Kolmogorov-Petrovski-Piskunov equation. We remark that, after a change of variables, this hyperbolic equation has the same family of one-dimensional travelling waves as the KPP equation. Using various energy functionals, we show that, ...

Global Nonexistence in a Nonlinearly Damped Wave Equation

Global existence‎, ‎stability results and compact invariant sets‎ ‎for a quasilinear nonlocal wave equation on $mathbb{R}^{N}$

We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type [ u_{tt}-phi (x)||nabla u(t)||^{2}Delta u+delta u_{t}=|u|^{a}u,, x in mathbb{R}^{N} ,,tgeq 0;,]with initial conditions $u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N geq 3, ; delta geq 0$ and $(phi (x))^{-1} =g (x)$  is a positive function lying in $L^{N/2}(mathb...

Finite Energy Travelling Waves for Nonlinear Damped Wave Equations

E(u,ut) = f |Vu|2 + m\u\2 + |ut|2 dx — [ |u|"+1 dx, (1.3) 2 JRn a+ 1 JR„ which represents a Lyapunov function of the problem, i.e., it is decreasing along any nonstationary trajectory of (1.1). Employing the potential-well arguments of PAYNE-SATTINGER [16] one observes that any solution of (1.1) emanating from sufficiently small initial data exists globally for all t e R+ and tends to zero with...