Global existence for the semilinear regular Euler-Poisson-Darboux-Tricomi equation

Abstract

For the semilinear Euler-Poisson-Darboux-Tricomi equation with a power nonlinearity, we derive that if the damping term dominates over the mass term, the small initial data solution exists globally if $p>p_F\left(\right(m+1)n+\frac{\mu -1-\sqrt{\delta }}{2})$. Combining with the blow-up results established by Palmieri [26] recently, we claim that $p_F\left(\right(m+1)n+\frac{\mu -1-\sqrt{\delta }}{2})$ is indeed the critical exponent for the equation in this case. Here, $p_F\left(l\right)=1+\frac{2}{l}$ is the well-known Fujita index.