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

Abstract

In this paper, we first establish a weighted bilinear fractional commutator estimate involving the spatial weight $\langle x \rangle^{2m}$. 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: $$ \begin{cases} \partial_t^2 u(t,x) + (-Δ)^{σ/2} u(t,x) + b(t) \partial_t u(t,x) = |u(t,x)|^p, & (t,x) \in (0,T) \times \mathbb{R}^N, \\ u(0,x) = \varepsilon u_0(x), \quad \partial_t u(0,x) = \varepsilon u_1(x), & x \in \mathbb{R}^N, \end{cases} $$ where $σ\in (0,2)$, the initial data scale with a small parameter $\varepsilon > 0$, and the time-dependent damping coefficient $b(t)$ satisfies $b(t) \approx (1+t)^{-β}$ for $β\in [-1, 1)$. 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 $p > p_c := 1 + \fracσ{N}$, we prove the small-data global existence and uniqueness of strong solutions in weighted Sobolev spaces $H^{σ/2,m}(\mathbb{R}^N) \times H^{0,m}(\mathbb{R}^N)$ 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 $T_\varepsilon$ of blow-up solutions via delicate ODE analysis. Together with the corresponding upper estimates, our results fully determine the sharp lifespan estimates for all $β\in [-1, 1)$, confirming that $p_c = 1 + \fracσ{N}$ is indeed the critical exponent for the fractional wave equation with time-dependent damping.