v1.0: * Fixed: Missing prefactor in the expression of f-f*f * Fixed: 1/rho is convex, not concave. * Fixed: Miscellaneous typos and small changes to the formatting. v0.1: * Added: Theorem on positivity of solutions. * Added: Reference to [CJLL20] * Added: Theorem on decay rate is now more general. * Fixed: Clarified the discussion in point 2-2 of the proof of the theorem on decay. * Removed: open problem about positivity of solutions. * Fixed: Format of named theorems. * Fixed: Minor formatting fixes. * Fixed: In proof of decay: indenting error.