Existence and uniqueness of radial solutions for fresh/salt water interface equation in a confined aquifer

We prove existence and uniqueness of radial weak solutions of the doubly nonlinear parabolic equationut=div[D(u)ϕ(∇u)],ut=div[D(u)ϕ(∇u)],whereD   and ϕϕ satisfy some hypothesis. This cover the particular case where D(s)=s(1-s)D(s)=s(1-s) and ϕ(p)≔p/(1+|p|2)ϕ(p)≔p/(1+|p|2) which is the hydrological case. Numerical simulations are provided also.