Long time asymptotics for 2 by 2 hyperbolic systems

Abstract

The goal of this paper is to study the behavior of the energy for 2 by 2 strictly hyperbolic systems. On the one hand we are interested in generalized energy conservation which excludes blow-up and decay of the energy for t→∞. On the other hand we present scattering results which take account of terms of order zero. © 2010 Elsevier Inc

A class of dissipative wave equations with time-dependent speed and damping

“…By density arguments, estimate (22) holds for any (u 0 , u 1 ) ∈ H 1 × L 2 . We address the interested reader to [22], where scattering results are proved for two by two systems with a similar technique.…”

“…We address the interested reader to [22], where scattering results are proved for two by two systems with a similar technique.…”

The threshold of effective damping for semilinear wave equations

“…The assumption λ ∈ L 1 which guarantees this latter property was first used in [5,6] to derive energy estimates in the setting of linear systems, eventually with the presence of a dissipative lower order term. On the other hand, if we consider the equation…”

Linear wave equations with time-dependent propagation speed and strong damping

“…for suitable real constants µ 1 , µ 2 , H, and α ∈ (0, 1). Many papers have been devoted to equations of this type (see for example the classical references [4,7] or the more recent ones [1,2,3,5,6,8,9,10,12,13,15,16]). Let us briefly discuss the previous results that are more relevant to our presentation.…”