In this paper, for the 1-D semilinear wave equation $\partial_{t}^{2} u-\partial_{x}^{2} u+\frac{\mu}{t} \partial_{t} u=|u|^{p}$ with scaling invariant damping, where $t\ge 1$, $p>1$ and $\mu \in(0,1) \cup\left(1, \frac{4}{3}\right)$, we establish the global weighted space-time estimates as well as the global existence of small data weak solution $u$ when the nonlinearity power $p$ is larger than some critical power $p_{\rm crit}(\mu)$. Our proof is based on a class of new weighted Strichartz estimates with the weight $t^{\theta}\left|(1-\mu)^{2} t^{\frac{2}{|1-\mu|}}-x^{2}\right|^{\gamma}$ ($\theta>0$ and $\gamma>0$ are appropriate constants) for the solution of linear generalized Tricomi equation $\partial_{t}^{2} \phi-t^{m} \partial_{x}^{2} \phi=0$ being any fixed positive number.
Qianqian LI
,
Huicheng YIN
. GLOBAL WEIGHTED SPACE-TIME ESTIMATES OF SMALL DATA WEAK SOLUTIONS TO 1-D SEMILINEAR WAVE EQUATIONS WITH SCALING INVARIANT DAMPINGS[J]. Acta mathematica scientia, Series B, 2025
, 45(6)
: 2330
-2353
.
DOI: 10.1007/s10473-025-0602-z
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