2025/11/09 by Chao Liu, Liu, Chao, Yiqing Shi +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2511.06289
This work extends the previous work by the first author [arXiv:2409.02516] and [Math. Ann. 393 (2025), 317-363], analyzing the long-term behavior of solutions to a broader class of quasilinear wave equations with parameter 1<a≤30 and (1)/(3)\leqb≤(2)/(3): ∂2t \varrho- \biggl( \frac m2 (∂t\varrho )2(1+\varrho )2 + 4(k-m2)(1+\varrho )\biggr) Δ\varrho = F(t,\varrho,∂μ \varrho) where F is given by F(t,\varrho,∂μ \varrho):= b \varrho (1+ \varrho ) -(a-1) ∂t\varrho + (4)/(3) \frac(∂t\varrho )21+\varrho + \biggl(m2 \frac (∂t\varrho )2(1+\varrho )2 + 4(k-m2) (1+\varrho ) \biggr) qi ∂i\varrho - \mathttKij ∂i\varrho∂j\varrho . The results demonstrate that for this extensive family of quasilinear wave equations satisfying 1<a≤30 and (1)/(3)\leqb≤(2)/(3), self-increasing blowup solutions also exist, and self-increasing singularities emerge at certain future endpoints of null geodesics provided the inhomogeneous perturbations of data are sufficiently small.