Asymptotic decay towards steady states of solutions to very fast and singular diffusion equations

We analyze long-time behavior of solutions to a class of problems related to very fast and singular diffusion porous medium equations having non-homogeneous in space and time source terms with zero mean. In dimensions two and three, we determine critical values of porous medium exponent for the asymptotic H-1-convergence of the solutions to a unique non-homogeneous positive steady state generally to hold.