Nonlinear three-dimensional diffusion models of porous medium in the presence of non-stationary source or absorption and some exact solutions

We study a general three-dimensional nonlinear diffusion model of porous medium with non-stationary source or absorption. We found nine basic models of the original model of the porous medium with non-stationary source or absorption, having different symmetry properties. For the model, admitting the widest group Lie of the transformations we found all invariant submodels. We found explicitly all essentially distinct invariant solutions describing invariant submodels of rank 0 of this model. In particular, we obtained the solutions, which we called “a layered circular pie”, “a layered spiral pie”, “ layered plane pie” and “a layered spherical pie”. The solution “a layered circular pie” describes a motion of the liquid or gas in a porous medium, for which at each fixed moment of a time at all points of each circle from the family of concentric circles a pressure is the same. The solution “a layered spiral pie” describes a motion of the liquid or gas in a porous medium, for which at each fixed moment of a time at all points of each logarithmic spiral, from the obtained family of logarithmic spirals a pressure is the same . The solution “a layered spherical pie” describes a motion of the liquid or gas in a porous medium, for which at each fixed moment of a time at all points of each sphere , from the family of concentric spheres a pressure is the same. A set of the solutions “a layered circular pie”, “a layered spiral pie” and “a layered spherical pie” contains the solutions describing a distribution of the pressure in a porous medium after a point blast or a point hydraulic shock. Also this set contains the solutions describing a stratified with respect to the pressure a motion of liquid or gas in a porous medium, with a very high pressure at infinity in a presence of a very strong absorption at a point. The solution “a layered plane pie” describes a motion of the liquid or gas in a porous medium, for which at each fixed moment of a time at all points of each plane, from the family of parallel planes a pressure is the same. A set of the solutions “a layered plane pie” contains the solutions describing a motion of the liquid or gas in a porous medium with a very high pressure near a fixed plane in a presence of a very strong absorption at infinity. Also this set contains the solutions describing a motion of the liquid or gas in a porous medium with a very high pressure at infinity in a presence of a very strong absorption on a fixed plane. The obtained results can be used to study to describe the processes associated with a underground fluid or gas flow, with water filtration, with the engineering surveys in the construction of the buildings, and also with shale oil and gas production.