Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture

We study in this paper the small data Cauchy problem for semilinear generalized Tricomi equations with a nonlinear term of derivative type $$\begin{aligned} u_{tt}-t^{2m}\Delta u=|u_t|^p \end{aligned}$$ $$m\ge 0$$ . Blow-up result and lifespan estimate from above are established $$1<p\le 1+\frac{2}{(m+1)(n-1)-m}$$ If $$m=0$$ , our results coincide those wave equation. The novelty consists construction new test function, by combining cut-off functions, modified Bessel function second kind (generalized) eigenfunction Laplacian. Interestingly, if $$n=2$$ blow-up power is independent m. also furnish local existence result, which implies optimality at least 1-dimensional case.