The forbidden physical mass interval conjecture for small-data global solvability

For the scalar semilinear equation, let $n$ denote the spatial dimension, $m$ the physical mass, and let the nonlinearity exponent satisfy $\alpha\in(0,\infty)$. Forbidden physical mass interval conjecture. The open interval

$$\left(\frac{\sqrt{n^2-1}}{2},\frac n2\right)$$

is a forbidden physical mass interval for the small-data global solvability of the Cauchy problem for all $\alpha\in(0,\infty)$. This claim concerns the range of physical masses in which small-data global solvability is expected to fail uniformly over all positive nonlinear exponents; the supplied text does not establish its resolution.