Blow-up of solutions to the Euler-Poisson-Darbox equation with critical power nonlinearity

In our recent precious work, we established the finite time blow up result and upper bound of lifespan estimate to the singular Cauchy problem of semilinear Euler-Poisson-Darboux equation in R^n with subcritical power type nonlinearity. By introducing an improved test function, we obtain an enhanced lower bound for the functional including the spacetime integral of the nonlinear term with an additional logarithmic growth, which finally yields the blow up result and upper bound of lifespan estimate for the corresponding Cauchy problem with "critical" nonlinear power. And this gives some partial answer to the open problem 1 posed by D'Abbicco (J. Differential Equations 286 (2021), 531-556).