Two partial differential equations are studied from the view-point of critical exponents. They are equations for a scalar unknown of one spatial variable, and produce self-similar solutions of the Navier-Stokes equations. Global existence and blow-up are examined for them, and the critical exponent separating them is determined.
"Blow-up problems in the strained vorticity dynamics and critical exponents." J. Math. Soc. Japan 65 (4) 1079 - 1099, October, 2013. https://doi.org/10.2969/jmsj/06541079