Weighted Fractional Commutator Estimate and Application to the Fractional Wave Equations with Time-dependent Damping

In this paper, we first establish a weighted bilinear fractional commutator estimate involving the spatial weight x ²ᵐ. This estimate provides a powerful tool to resolve the difficulties arising from the nonlocality of the fractional Laplacian (-Δ)σ/2 and the interaction between the unbounded weights and the nonlocal operator at infinity. We then apply this result to investigate the Cauchy problem for the fractional wave equation with time-dependent damping and source nonlinearity: $$ {cases} ∂_t^2 u(t,x) + (-Δ)σ/2 u(t,x) + b(t) ∂_t u(t,x) = |u(t,x)|^p, & (t,x) ∈ (0,T) × R^N, \\ u(0,x) = ε u_0(x), ∂_t u(0,x) = ε u_1(x), & x ∈ R^N, {cases} $$ where $σ∈ (0,2)$, the initial data scale with a small parameter $ε > 0$, and the time-dependent damping coefficient $b(t)$ satisfies $b(t) ≈ (1+t)-β$ for $β∈ [-1, 1)$. Utilizing scaling variables and constructing a weighted energy framework based on Theorem 1.1, we derive a priori differential inequalities for the total energy. For the supercritical case $p > p_c := 1 + σ{N}$, we prove the small-data global existence and uniqueness of strong solutions in weighted Sobolev spaces $Hσ/2,m(R^N) × H0,m(R^N)$ by closing a bootstrap argument. For the subcritical ($1 < p < p_c$) and critical ($p = p_c$) cases, we establish precise lower bounds for the lifespan $T_ε$ of blow-up solutions via delicate ODE analysis. Together with the corresponding upper estimates, our results fully determine the sharp lifespan estimates for all $β∈ [-1, 1)$, confirming that $p_c = 1 + σ{N}$ is indeed the critical exponent for the fractional wave equation with time-dependent damping.