Weighted decay estimate for the wave equation

Abstract

Abstract.The work is devoted to the proof of a new L°°-L weighted estimate for the solution to the nonhomogeneous wave equation in (3 + l)-dimensional space-time. The weighted Sobolev spaces are associated with the generators of the Poincaré group. The estimate obtained is applied to prove the global existence of a solution to a nonlinear system of wave and Klein-Gordon equations with small initial data.

A Note on $$L^{p}-L^{q}$$ Estimates for Semilinear Critical Dissipative Klein–Gordon Equations

“…Theorem 1 extends the results obtained in [42] at low frequencies to the full region 1 ≤ p ≤ q ≤ ∞, since d(1, ∞) = (n + 1)/2. Our results are consistent with (21) and (22), while we improve the rate in (23) to (1 + t) n+1 4…”

“…We stress that, however, some problems can be more successfully studied with estimates different from L p − L q estimates. For instance, see [23,24,25,29,32,41,55] for the critical exponent [45] of the nonlinear wave equation without damping, and [49] for the critical exponent of the classical damped wave equation.…”