We describe the large-time asymptotics of solutions to heat equation for fractional Laplacian with added subcritical or even critical Hardy-type potential. The is governed by a self-similar solution equation, obtained as normalized limit at origin kernel corresponding Feynman-Kac semigroup.

We consider viscous rotating accretion flows in which most of the viscously dissipated energy is stored as entropy rather than being radiated. Such advection-dominated flows may occur when the optical depth is either very small or very large. We obtain a family of self-similar solutions where the temperature of the accreting gas is nearly virial and the flow is quasi-spherical. The gas rotates ...

natural convection heat transfer from a vertical heated plate in a saturated porous medium is analyzed under thermal non-equilibrium assumptions. the plate is maintained at a non uniform temperature and the fluid is considered to be with internal heat generation. the coupled momentum and energy equations for fluid and solid phases are transformed to a similarity format, and solved numerically. ...