Asymptotic decay for defocusing semilinear wave equations in $\mathbb{R}^{1+1}$

TL;DR

This paper investigates the long-term decay of solutions to the one-dimensional defocusing semilinear wave equation, showing pointwise decay and polynomial decay rates for localized data, based on new weighted vector field methods.

Contribution

It improves understanding of decay rates for solutions in 1+1 dimensions and confirms a conjecture using novel weighted vector field techniques.

Findings

Finite energy solutions tend to zero pointwise.

Solutions with localized data decay polynomially over time.

New weighted vector fields are effective multipliers for decay analysis.

Abstract

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.