PDEs in inhomogeneous and anisotropic spaces with general data

I would like to present an introduction to equations describing diffusion-type processes in media of complicated structure. This type of problems require to employ unconventional functions spaces. I will try to give some intuition by showing their nice properties and difficulties one faces while working there. The object of my study is existence and regularity of certain type of solutions to related equations involving data too poorly integrable for the weak solutions to be defined. I will try to draw a general landscape of what is known, what has been recently done by me (partially in various collaborations) and what is about to be examined.