Energy decay for small solutions to semilinear wave equations with weakly dissipative structure

Abstract

We consider the Cauchy problem for cubic nonlinear Klein-Gordon equations in one space dimension. We give the L p -decay estimate for the small data solution and show that it decays faster than the free solution if the cubic nonlinearity has the suitable dissipative structure.

Global existence for systems of nonlinear wave and Klein-Gordon equations in two space dimensions under a kind of the weak null condition

“…Even if the solution is asymptotically free, the asymptotic data can be away from the original data, and decay of the energy may also occur for some nonlinearity. For these results, see [16,17], Katayama-Murotani-Sunagawa [18], Nishii-Sunagawa [31], Nishii-Sunagawa-Terashita [32]. In these works, the main tool to obtain the asymptotic behavior is the profile system (4.11).…”