Weighted $L^2-L^2$ estimate for wave equation and its applications

TL;DR

This paper develops a new weighted $L^2-L^2$ estimate for 3-D inhomogeneous wave equations, enabling simplified proofs of global existence for certain semilinear wave equations with supercritical powers and damping.

Contribution

It introduces a novel weighted estimate using a Morawetz multiplier, providing new proofs and extending results on global existence for wave equations.

Findings

Established a weighted $L^2-L^2$ estimate for 3-D wave equations.

Proved global existence for small data semilinear wave equations with supercritical powers.

Extended global existence results to wave equations with scale-invariant damping.

Abstract

In this work we establish a weighted $L^2-L^2$ estimate for inhomogeneous wave equation in 3-D, by introducing a Morawetz multiplier with weight of power $s(1<s<2)$, and then integrating on the light cones and $t$ slice. With this weighted $L^2-L^2$ estimate in hand, we may give a new proof of global existence for small data Cauchy problem of semilinear wave equation with supercritical power in 3-D. What is more, by combining the Huygens' principle for wave equations in 3-D, the global existence for semilinear wave equation with scale invariant damping in 3-D is established.