Self-similar solutions satisfying or not the equation of the interface ✩

We study the existence of self-similar solutions for the porous medium equation with reaction and convection ut = (um−1ux )x + un−1ux +kup in R × [0,∞), where m,n > 1, 00.We are in particular interested in compactly supported self-similar solutions satisfying some good equation at the interface, the same equation appearing in the pure diffusion case. We prove that there exist such self-similar solutions only if k > 1/4n. An infinity of solutions with bad behaviour at least at one interface also exist. There exist no self-similar solutions with support arbitrarily small. We complete the study by considering the case 0