The Strauss conjecture for the wave equation with critical damping

The Strauss conjecture concerns wave equations with power-type nonlinearities and seeks the optimal range of powers that admit global solutions for all sufficiently nice and sufficiently small initial data. In three spatial dimensions, it is well known that small data global existence is available provided that the power $p>p_c(3)=1+\sqrt{2}$, where $p_c(n)$ denotes the critical power in $n$-spatial dimensions. In the current article, the analogous problem is considered for damped wave equations where the damping coefficient is at the critical scaling. When $a$ denotes the constant multiple applied to the damping term, it is shown that the corresponding critical power is $p_c(3+a)$ provided that $a$ is sufficiently small.