Long time dynamics for damped Klein-Gordon equations

2015/05/22 by Burq, N., Raugel, G., Schlag, W. · 1 citation
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1505.05981

Abstract

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in H¹× L². In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities |u|^p-1 u, 1\textlessp\textless(d+2)/(d-2) as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems).

Cited by

• Blow-up, decay, and convergence to equilibrium for focusing damped cubic Klein-Gordon and Duffing equations

Related

• Long time dynamics for weakly damped nonlinear Klein-Gordon equations
• Asymptotics of solutions with a compactness property for the nonlinear damped Klein-Gordon equation
• A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy
• Global dynamics above the ground state energy for the one-dimensional NLKG equation
• Energy asymptotics for the strongly damped Klein-Gordon equation