Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms

Source abstract

We consider the Cauchy problem for ∂ t 2 u - ∂ x 2 u + u = - g ( ∂ t u ) 3 on the real line. It is shown that if g > 0 , the solution has an additional logarithmic time decay in comparison with the free evolution in the sense of L p , 2 ≤ p ≤ ∞ . Moreover, the asymptotic profile of u ( t , x ) as t → + ∞ is obtained. We also discuss a generalization. Consequently we see that the "null condition" in the sense of J.-M. Delort (Ann. Sci. École Norm. Sup., 34 (2001), 1--61) is not optimal for small data global existence for nonlinear Klein-Gordon equations.