In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation \(\mathcal {T}_\ell u=|\partial _t u|^p\), where \(\mathcal {T_\ell }=\partial _t^2-t^{2\ell }\Delta \). Smooth solutions blow up in finite time for positive Cauchy data when the exponent p of the nonlinear term is below \(\frac{\mathcal {Q}}{\mathcal {Q} -2}\), where \(\mathcal {Q}=(\ell +1)n+1\) is the quasi-homogeneous dimension of the generalized Tricomi operator \(\mathcal {T}_\ell \). Furthermore, we get also an upper bound estimate for the lifespan.

在此注释中，我们证明了带有导数类型为非线性项的半线性广义Tricomi方程的​​爆破结果，即方程\（\ mathcal {T} _ \ ell u = | \ partial _t u | ^ p \ ），其中\（\ mathcal {T_ \ ell} = \ partial _t ^ 2-t ^ {2 \ ell} \ Delta \）。当非线性项的指数p低于\（\ frac {\ mathcal {Q}} {\ mathcal {Q} -2} \）时，其中正值Cauchy数据的光滑解会在有限的时间内爆炸，其中\（\ mathcal { Q} =（\ ell +1）n + 1 \）是广义Tricomi算子\（\ mathcal {T} _ \ ell \）的准齐次维。此外，我们还获得了寿命的上限估计。