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

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