Sharp boundedness and continuity results for the singular porous medium equation

Abstract

We consider non-homogeneous, singular (0 < m < 1) parabolic equations of porous medium type of the form $$ut - div{\kern 1pt} A\left( {x,t,u,Du} \right) = \mu {\kern 1pt} in{\kern 1pt} {E_T}$$ , where E T is a space time cylinder, and µ is a Radon-measure having finite total mass µ(E T ). In the range $$\frac{{\left( {N - 2} \right) + }}{N}$$ < m < 1 we establish sufficient conditions for the boundedness and the continuity of u in terms of a natural Riesz potential of the right-hand side measure µ.