# Classic results for porous medium equation

### Selfsimilar solutions

 Let us seek for solutions of  satisfying the scaling hypothesis

where  and  are reparametrization of time and . The prefactor  ensures that  conserves mass, i.e.

The time derivative has to satisfy

Further, let us calculate the gradient of 

and the Laplacian evaluates to

Hence, the function  solves the equation

We want the coefficients to be time-independent and a comparison results in the condition

where  are constants, which can be specified later. From the first equality, we obtain , which is solved by . The second equality, leads to

and integrates to , where  is a further constant. Hence, we find the scaling relation

We are still free to choose the constants  and . A particular nice choice is given bys , ,  and , then we obtain the result: If  is a solution of the PME, then  solves

### Equilibrium solutions

From the self similar rescaled solution , we can derive the equilibrium solution. Stationary solutions are given by function  satisfying

Hence, by setting the flux inside of the divergence equal to zero

Hence,  is a trivial solution and in the case  it is easy to check that  is a solution (Compare this with the Ornstein-Uhlenbeck process, which is a special case of the Fokker-Planck equation}. Therefore, let us assume, that . Then, we have

which can be rewritten as

which determines  up to a constant 

We can only take the power  if the right hand side is non-zero, hence we set

hereby  denotes the positive part of . The constant  is chosen such that