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 ⟨ 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: ∂t2 u(t,x) + (−Δ)σ/2 u(t,x) + b(t) ∂t u(t,x) = |u(t,x)|p, & (t,x) ∈ (0,T) × ℝN, u(0,x) = ε u0(x), ∂t u(0,x) = ε u1(x), & x ∈ ℝN, 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 > pc := 1 + σ/N, we prove the small-data global existence and uniqueness of strong solutions in weighted Sobolev spaces Hσ/2,m(ℝN) × H0,m(ℝN) by closing a bootstrap argument. For the subcritical (1 < p < pc) and critical (p = pc) cases, we establish precise lower bounds for the lifespan Tε 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 pc = 1 + σ/N is indeed the critical exponent for the fractional wave equation with time-dependent damping.
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BibTeX
@misc{lin2026weighted,
title = {{Weighted Fractional Commutator Estimate and Application to the Fractional Wave Equations with Time-dependent Damping}},
author = {Jiayun Lin and Masahiro Ikeda},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04247},
eprint = {2611.04247},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04247},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Weighted Fractional Commutator Estimate and Application to the Fractional Wave Equations with Time-dependent Damping
AU - Jiayun Lin
AU - Masahiro Ikeda
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04247
UR - https://arxiv.org/abs/2611.04247
AB - 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: ∂_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, 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 H^σ/2,m(ℝ^N) × H^0,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.
ER -