  This paper is devoted to the study of asymptotic behaviors of solutions to
the one-dimensional defocusing semilinear wave equation. We prove that finite
energy solution tends to zero in the pointwise sense, hence improving the
averaged decay of Lindblad and Tao. Moreover, for sufficiently localized data
belonging to some weighted energy space, the solution decays in time with an
inverse polynomial rate. This confirms a conjecture raised in the mentioned
work.
  The results are based on new weighted vector fields as multipliers applied to
regions enclosed by light rays. The key observation for the first result is an
integrated local energy decay for the potential energy, while the second result
relies on a type of weighted Gagliardo-Nirenberg inequality.

Analysis of PDEs