The Dipole Solution for the Porous Medium Equation

We establish the existence and uniqueness of the Dipole Solution of the porous medium equation (PME), i.e. the solution of u t = (juj m?1 u) ; in Q = f(x; t) : x = (x i) 2 R N ; t > 0g ; with m > 1 and initial value u(x; 0) = M @ @x 1 (x) ; where is Dirac delta function in R N and M 6 = 0. It has the self-similar form u(x; t) = t ? U(xt ?) ; with exponents and determined from dimensional considerations. The proole U has compact support. The proof uses the so-called dual porous medium equation, z t = jzj m?1 z, and depends on the existence of a certain type of radially symmetric and self-similar solution in dimension N ? 1. We also prove that any solution of the PME in a half-space H = fx : x 1 > 0g with zero boundary data on @H converges as t ! 1 to the Dipole solution with the same moment.